Principles and Models of Biological Transport

Principles and Models of Biological Transport

Morton H. Friedman

Principles and Models of Biological Transport

123

Morton H. Friedman Duke University Department of Biomedical Engineering Durham NC 27708-0292 USA [email protected]

ISBN: 978-0-387-79239-2 DOI: 10.1007/978-0-387-79240-8

e-ISBN: 978-0-387-79240-8

Library of Congress Control Number: 2008931308 c 2008 Springer Science+Business Media, LLC  All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC., 233 Spring Street, New York, NY10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights.

Printed on acid-free paper 9 8 7 6 5 4 3 2 1 springer.com

For Ann, still …

v

PREFACE TO THE SECOND EDITION

Focus, Organization, and Content This book, like the first edition, deals with the mass transport processes that take place in living systems, with a focus on the normal behavior of eukaryotic cells and the organisms they constitute, in their normal physiological environment. As a consequence of this focus, the structure and content of the book differ from those of traditional transport texts. We do not start with the engineering principles of mass transport (which are well presented elsewhere) and then seek biological applications of these principles; rather, we begin with the biological processes themselves, and then develop the models and analytical tools that are needed to describe them. This approach has several consequences. First of all, it drives the content of the text in a direction distinctively different from conventional transport texts. This is because the tools and models needed to describe complex biological processes are often different from those employed to describe more well-characterized inanimate systems. Many biological processes must still be described phenomenologically, using methodologies like nonequilibrium thermodynamics. Simple electrical analogs employing a paucity of parameters can be more useful for characterization and prediction than complex theories based on the behavior of more well-defined systems on a laboratory bench. By allowing the biology to drive the choice of analysis tools and models, the latter are consistently presented in the context of real biological systems, and analysis and biology are interwoven throughout. Owing to its more biological focus, the book includes more biology and physiology than most texts on engineering in the life sciences, and some parts will be easier to follow for readers with some background in biology. To keep the text selfcontained in this respect, an early chapter is devoted to those aspects of cell biology most relevant to biological transport systems. A few words on the use of the term “models” in the title. The explosive growth of molecular biology in the past several decades demands that space be devoted to the molecular-level events that underlie the observables of biological transport. This means that “models” must now be understood to include physical and structural models of transport systems and processes at the molecular level, as well as the mathematical models of transport that continue to be developed to describe biological transport mechanisms at all levels. The text includes chapters that deal primarily with fundamental transport principles, including thermodynamics (Chapters 1, 2, 6); the cell, including intracellular transport (Chapter 3); the biological transport mechanisms, such as channels and carriers, employed by living organisms (Chapters 4, 5, 7), and applications of these mechanisms in control and in the function of tissues and organs such as kidney and vii

viii

PREFACE

lung (Chapters 8–11). A more thorough overview of the text can be gained by reviewing the detailed Table of Contents that follows. The models in latter chapters are derived using the principles presented in earlier ones, so the student appreciates the assumptions that underlie them, and their consequent limitations. Those aspects of transport that are closely associated with specific tissues — for instance, the HodgkinHuxley theory of axonal conduction — are presented in these latter chapters. Certain topics are dealt with less thoroughly in this text, or are outside its scope. Experimental techniques used in transport research are well described in the extensive methods literature and are not discussed here in great detail. Experimental data are presented primarily to illustrate specific transport mechanisms, so the selection of data is representative rather than complete. Transport phenomena that exist only in experimental settings — for instance, isotope interaction effects — are not discussed. On the other hand, the text will provide guidance to the experimenter regarding the appropriate tools to interpret experimental data, and the limits on their applicability. The emphasis of the text is on mass transfer. Both convection and diffusion are included, but purely convective transport, such as the transport of oxygen in large blood vessels or gas flow in the early generations of the respiratory tree, is not. The transfer of momentum, i.e., fluid mechanics, is not dealt with in any depth. Fortunately, there are several recent texts on biological fluid mechanics, particularly in the cardiovascular area. Heat transfer is also outside the scope. Some final comments on style: — the system of units used in the text is centimeter-gram-second (cgs) rather than Système International (SI), since the former units still dominate the literature. Conversions between the two systems are presented in Chapter 1. — the valences of ions are not shown in the text; sodium ion is desig+ nated by Na, not Na . The reader is assumed to know that metals and halogens are ionized in the biological milieu.

Use in the Classroom The text is designed for a first course in biological mass transport and is based on courses I have taught to students primarily, but not exclusively, in biomedical engineering at Johns Hopkins, Ohio State, and Duke. As is appropriate for an engineering course, the student is assumed to have a certain facility with modeling and mathematics. The material is presented at a level easily accessible to upper-class undergraduates or early graduate students. The combination of biological content and engineering approach should be attractive not only to biomedical engineering students, but also to students in chemical, mechanical, and environmental engineering who are interested in the life sciences. The text can also be used to provide a broader perspective for students in physiology and biophysics, or with students in the biological sciences who are interested in a more quantitative approach to transport. Though mathematical modeling is an essential component of the text, the equations are not exceptionally complex, for several reasons. Most biological transport

PREFACE

ix

processes can be described by steady-state equations, and even transients are generally quasisteady with respect to mass transfer. Most membrane transport models assume no variation in the variables of interest in the plane of the barrier. As a consequence, with rare exceptions (most notably Chapter 11 and to a lesser extent Chapter 9), the mathematics is limited to ordinary differential equations. On the life science side, an introductory knowledge of biology and chemistry is assumed, a requisite which my current students meet with a first college course in biology. A third discipline that supports the models developed in the text is thermodynamics, a subject not always included in the crowded curriculum. Accordingly, separate chapters are devoted to equilibrium thermodynamics in solution and nonequilibrium thermodynamics, prior to the treatment of diffusional and coupled transport, respectively. In addition to supporting a core or elective biological transport course, this text could also provide a framework for teaching biology and physiology per se, as well as biological modeling, to students in biomedical, chemical, and mechanical engineering. Transport is relevant to most biological systems and, in contrast to many other subdisciplines within biology, lends itself well to mathematical modeling and engineering analysis. The book includes examples that relate to the cardiovascular system, nerve transmission, kidney function, control systems, secretory activity, and other biological systems and functions. It should be possible to design an undergraduate course in engineering physiology that covers much of the subject in the context of biological transport, perhaps supplemented with a module on cardiovascular fluid mechanics that could be based on any of several recent texts in this area. To support the use of the text in class, exercises are included at the end of each chapter. Some of the exercises are thought problems; some require the student to apply the presented material quantitatively, and others encourage the student to explore beyond the boundaries of the book itself. More exciting and, if it meets its objectives, more useful, a website for the book has been set up on the server at the Pratt School of Engineering at Duke. The URL of the site is http://biotrans.pratt.duke.edu/ The site will be accessible to students, instructors, and other users and is intended to create a “community of the book” that will enhance both teaching and learning. Using the site, community members will be able to post additional exercises; suggest new material for inclusion in courses based on the text, either in general terms or with a link to a specific publication; and share course syllabi. It can also be used to post corrections or amendments to the text itself. Users will be encouraged to register on a listserv so that they can be alerted when something new has been posted on the site. If there is interest, the site can support forums on issues of common concern among the users. The website is intended to complement the text and be a continuing resource in this important field.

x

PREFACE

In Appreciation In a sense, this book has many authors besides myself. It reflects the research efforts of numerous investigators in biology and transport science, and insights from the authors of many former texts. Many of the figures are gifts from colleagues and their publishers, who graciously permitted me to reproduce or adapt their illustrations. Most of the exercises have been vetted by my students at Ohio State and Duke. I would like to single out a few people who either suffered through the writing or without whom it would never have been completed: — the students in my laboratory, to whom I was not always as available as I would have liked during the past few years, particularly Yun Liang, who found and copied all the references that had escaped digitization, and Ji Zhang, who computed and produced all the new numerical plots in this edition; — Tim Oliver, who managed the project and patiently translated my computer files into the pages that follow; the folks at HFS Imaging, who transformed my often ambiguous sketches into the precise figures that appear throughout; and my wife, Ann, who obtained the permissions that allowed us to reproduce the excellent artwork of earlier authors. Durham 2008

Morton H. Friedman

CONTENTS

Preface .....................................................................................................................

vii

1: Equilibrium Thermodynamics Introduction .......................................................................................................................... 1.1. Chemical Potentials and Activities .......................................................................... 1.1.1. Thermodynamic Preliminaries. The Electrochemical Potential ...................................................................................................... 1.1.2. The Interphase Equilibrium Condition ....................................................... 1.1.3. Electrochemical Potentials in Terms of Measurable Solution Variables: Introduction, and the Effect of Electrostatic Potential............... 1.1.4. The Gibbs–Duhem Equation ...................................................................... 1.1.5. Dependence of the Chemical Potential on Pressure.................................... 1.1.6. Dependence of the Chemical Potential on Composition............................. 1.1.7. Units ........................................................................................................... 1.1.8. Activity and Activity Coefficient ............................................................... 1.2. Ion Equilibrium across Membranes......................................................................... 1.2.1. The Nernst Equilibrium .............................................................................. 1.2.2. Origin of the Nernst Potential..................................................................... 1.2.3. Specific Ion Electrodes ............................................................................... 1.2.4. Activity Coefficient Considerations ........................................................... 1.2.5. The Donnan Equilibrium ............................................................................ 1.3. Chemical Equilibrium.............................................................................................. Problems...............................................................................................................................

1 1 2 4 5 5 6 7 11 12 14 14 15 17 17 19 21 24

2: Free Diffusion Introduction .......................................................................................................................... 2.1. Free Diffusion of Nonelectrolytes ........................................................................... 2.1.1. The Teorell Equation .................................................................................. 2.1.2. Integration of the Teorell Equation; Fick's First Law; Solute Permeability..................................................................................... 2.1.3. Unstirred Layers ......................................................................................... 2.1.4. Applications of Solution Theory................................................................. 2.1.5. Fick's Second Law and Convective Diffusion ............................................ 2.1.6. Justification of the Steady-State Assumption: Time Scales in Biological Transport........................................................... 2.2. Free Diffusion of Electrolytes ................................................................................. 2.2.1. Differences between Electrolyte and Nonelectrolyte Diffusion ................. 2.2.2. The Electrodiffusion Equation.................................................................... 2.2.3. Integration of the Electrodiffusion Equation ..............................................

29 30 30 31 36 38 40 41 43 43 44 47 xi

xii

CONTENTS

2.2.4. Some Special Cases .................................................................................... Equilibrium ......................................................................................... Uniform Composition ......................................................................... Diffusion Potential of a Bi-Ionic System............................................ Active and Passive Exchange with a Closed Compartment................ Equal Total Concentrations on the Two Sides of the Membrane: The Constant-Field Equation....................................... 2.2.5. Ionic Permeability and the Resting Potential of the Cell ............................ 2.2.6. Charged Membranes................................................................................... 2.2.7. Limitations of the Electrodiffusion Equation and Its Solutions.................. Problems...............................................................................................................................

49 49 50 51 52 53 57 59 61 62

3: The Cell Introduction .......................................................................................................................... 3.1. Overview ................................................................................................................. 3.2. The Structure of an Animal Cell.............................................................................. 3.2.1. Composition and Structure of the Plasma Membrane................................. Heterogeneity of Cell Membranes ...................................................... The Mosaic Model of the Plasma Membrane ..................................... The Glycocalyx and Surface Charge .................................................. 3.2.2. The Internal Structure of the Cell ............................................................... 3.3. Metabolism: The Production of ATP....................................................................... 3.4. Intracellular Transport ............................................................................................. 3.4.1. Intracellular Diffusion ................................................................................ 3.4.2. Protein Transport ........................................................................................ Transport into the Nucleus.................................................................. Transport into the Mitochondrion....................................................... 3.4.3. Vesicular Transport .................................................................................... Endocytosis......................................................................................... Exocytosis, Secretion and Transcytosis .............................................. 3.5. Cellular Motility and Locomotion ........................................................................... 3.5.1. Actin-Based Movement and Chemotaxis ................................................... 3.5.2. Cilia and Flagella........................................................................................ Problems...............................................................................................................................

67 67 68 68 68 72 74 75 78 84 84 87 87 88 90 91 95 100 100 105 107

4: Facilitated Diffusion: Channels and Carriers Introduction .......................................................................................................................... 4.1. Mechanisms of Channels and Carriers .................................................................... 4.1.1. Hallmarks of Mediated Transport............................................................... 4.1.2. Ion Selectivity of Channels......................................................................... 4.1.3. Energetics of Ion Selectivity, and Steric Effects......................................... 4.1.4. Ion Selectivity of Channels: Summary ....................................................... 4.1.5. The Structure of Ion Channels: Selectivity Filters, Gates, and Energy Profiles..........................................................................

111 112 112 114 116 119 120

CONTENTS

4.1.6. Regulation of the Gating Process................................................................ 4.1.7. Classification of Ion Channels. Aquaporins and Gap Junctions ................. 4.1.8. Carrier Models ............................................................................................ 4.1.9. Carriers and Channels: Convergences and Differences .............................. 4.2. Kinetics of Facilitated Transport ............................................................................. 4.2.1. Models of Ion Channel Transport: Overview ............................................. 4.2.2. Energy Barrier and Binding Models of Channel Transport ........................ Ionic Independence: Absolute Rate Theory........................................ Saturable Channels: Kinetic Analysis................................................. 4.2.3. The Patch Clamp and Two Applications .................................................... Acetylcholine Receptor Channel Kinetics .......................................... Voltage and Current Sensitivity of a Voltage-Activated Sodium Channel ............................................... 4.2.4. Stochastic Properties of Channels: Membrane Noise Analysis .................. 4.2.5. The Simplest Model of Carrier Transport: Assumptions............................ 4.2.6. The Simplest Model of Carrier Transport: Equations................................. 4.2.7. Monosaccharide Transport in the Erythrocyte............................................ 4.2.8. More Complex Carrier Models................................................................... 4.2.9. Exchangers and Cotransporters................................................................... 4.3. Inhibition of Facilitated Transport........................................................................... 4.3.1. Inhibition of Channel Transport: Channel Block........................................ 4.3.2. Inhibition of Carrier Transport ................................................................... Problems...............................................................................................................................

XIII

122 131 137 139 141 141 143 143 146 151 152 154 158 161 162 165 166 170 174 174 175 179

5: Active Transport Introduction .......................................................................................................................... 5.1. Active Transport: General Considerations .............................................................. 5.1.1. Metabolic Coupling and Affinity................................................................ 5.1.2. Classification of Active Transport Processes.............................................. 5.1.3. Identification of Active Transport Processes .............................................. 5.2. Mechanisms of Active Transport............................................................................. 5.2.1. Scalar Active Transport: Overview ............................................................ 5.2.2. Primary Scalar Transport ............................................................................ 5.2.3. Secondary Scalar Transport ........................................................................ Cotransport ......................................................................................... Countertransport ................................................................................. 5.2.4. Vectorial Active Transport, the Curie Theorem and Substrate Activation.................................................................................... 5.2.5. Sodium–Potassium Exchange..................................................................... 5.2.6. Pump Selectivity and Other Properties Shared with Passive Carriers ........ 5.3. Kinetics of Active Transport ................................................................................... 5.3.1. A Simple Secondary Scalar Transport Model: Assumptions...................... 5.3.2. A Simple Secondary Scalar Transport Model: Equations........................... 5.3.3. More Complex Symport Models; the Sodium/Glucose Transporter........... 5.3.4. Primary Scalar Transport ............................................................................ 5.3.5. Flux Equations for Primary Scalar Transport .............................................

185 186 186 188 190 190 191 192 193 195 197 199 202 204 205 206 208 209 213 214

xiv

CONTENTS

5.3.6. Relation between the Coupling Parameter * and the Affinity of the Metabolic Reaction ....................................................... 5.3.7. Vectorial Active Transport and the Nature of Na–K Exchange.................. 5.3.8. Pumps and Leaks ........................................................................................ Problems...............................................................................................................................

215 216 216 220

6: Nonequilibrium Thermodynamics Introduction .......................................................................................................................... 6.1. The Basic Phenomenological Equations.................................................................. 6.1.1. Conjugate Forces and Fluxes...................................................................... 6.1.2. Phenomenological Coefficients and Linear Thermodynamics ................... 6.1.3. Frictional Interpretation of the Phenomenological Equations..................... 6.1.4. A Cautionary Note before Proceeding........................................................ 6.2. Nonequilibrium Thermodynamic Description of Passive Transport ....................... 6.2.1. Setting the Stage ......................................................................................... 6.2.2. The Chemical Potential of the Solvent ....................................................... 6.2.3. A New Set of Forces and Fluxes; Osmotic Pressure................................... 6.2.4. The Kedem–Katchalsky Equations............................................................. 6.2.5. Physical Significance of the Reflection Coefficient: Semipermeable Membranes and the Osmometer, Steric Effects and Sieving........................................................................... 6.2.6. Osmotic Pressure of Solutions; Donnan Osmotic Pressure; Osmotic Effects on Cells ............................................................................ 6.2.7. Passive Transport of Multiple Nonelectrolytes........................................... 6.2.8. Passive Transport of Electrolytes: Electrokinetic Phenomena.................... 6.3. Nonequilibrium Thermodynamic Description of Active Transport......................... 6.3.1. Definition of Active Transport ................................................................... 6.3.2. Coupling between Nonconjugate Forces and Fluxes .................................. 6.3.3. Nonequilibrium Thermodynamics of Motor Proteins and ATP Synthase....................................................................................... 6.4. Limitations of Nonequilibrium Thermodynamics ................................................... 6.4.1. Closeness to Equilibrium: A Limitation Intrinsic to Linearized Nonequilibrium Thermodynamics ............................................ 6.4.2. The Concentration Dependence of the Phenomenological Coefficients................................................................................................. 6.4.3. Closeness to Equilibrium in Biological Systems ........................................ 6.4.4. The Information Content of Nonequilibrium Thermodynamics ................. 6.4.5. Approximations in the Derivation of the Kedem–Katchalsky Equations .................................................................................................... Problems...............................................................................................................................

225 225 225 227 229 230 231 231 232 234 235

237 242 246 247 248 248 250 252 254 254 257 258 259 260 261

7: Models of Transport across Cell Membranes Introduction .......................................................................................................................... 7.1. Transport across the Lipid Bilayer of Cell Membranes........................................... 7.1.1. Evidence for Nonelectrolyte Diffusion across the Lipid Bilayer................ 7.1.2. A Simple Model of Transbilayer Diffusion ................................................

265 265 266 267

CONTENTS

7.1.3. Potential Barriers in the Bilayer.................................................................. Models of Transport through Pores ......................................................................... 7.2.1. Classification of Pore Transport Models .................................................... 7.2.2. Hydraulic Conductivity of a Pore ............................................................... 7.2.3. Hindered Diffusion; Solute Permeability as a Probe of Pore Radius.......... 7.2.4. Other Factors Affecting Estimated Pore Size; The Equivalent Pore........... 7.2.5. Hindered Convection (Sieving); the Reflection Coefficient as a Probe of Pore Radius .................................................................................. 7.2.6. Combined Diffusion and Convection through Pores .................................. 7.2.7. Single-File Transport through Pores........................................................... 7.2.8. The Permeability Ratio of Larger Pores ..................................................... 7.3. Electrical Analogs.................................................................................................... 7.3.1. Equivalent Circuit for the Passive Flux of a Single Ion.............................. 7.3.2. Equivalent Circuit for the Passive Transport of Multiple Ions ................... 7.3.3. The Electrical Analog of a Rheogenic Pump.............................................. 7.3.4. Some Final Remarks................................................................................... Problems............................................................................................................................... 7.2.

XV

269 270 271 272 273 276 279 282 283 287 289 289 290 293 294 295

8: Regulation and Feedback Introduction .......................................................................................................................... 8.1. Regulation of Transport ........................................................................................... 8.1.1. Receptor-Mediated Second Messenger Systems: Cyclic AMP and Antidiuretic Hormone................................................................. 8.1.2. Direct Hormonal Regulation without an Extracellular Receptor: Aldosterone ................................................................................ 8.1.3. Calcium-Based Regulation ......................................................................... Maintenance of a Low Cytosolic Calcium Concentration .................. Calcium Signaling .............................................................................. Calcium Handling in Cells and Compartmental Analysis .................. 8.2. Feedback in Transport Systems: Insulin.................................................................. 8.3. Regulation by Transport .......................................................................................... 8.3.1. Regulation of Cell Volume ......................................................................... 8.3.2. Regulation of Cell pH................................................................................. Problems............................................................................................................................... Appendix 8.1. Nonequilibrium Binding in Compartmental Analysis ................................

299 301 301 306 307 307 307 309 315 317 317 322 325 327

9: Excitable Cells Introduction .......................................................................................................................... 9.1. Nerve ....................................................................................................................... 9.1.1. The Resting Neuron.................................................................................... 9.1.2. The Action Potential: Electrical Aspects .................................................... The Membrane Action Potential: Resting State.................................. Hodgkin and Huxley's Equations for the Dependence of Conductance on Membrane Potential ......................................... Excitation of the Membrane Action Potential.....................................

329 330 331 333 335 337 340

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CONTENTS

The Propagating Action Potential: Cable Theory ............................... The Role of Myelin............................................................................. 9.1.3. The Action Potential: Molecular Aspects ................................................... Hodgkin and Huxley's Channel Model ............................................... More Recent Models of the Potassium and Sodium Channels of Nerve ............................................................. 9.1.4. Synaptic Transmission................................................................................ 9.2. Muscle ..................................................................................................................... 9.2.1. The Resting Muscle Fiber........................................................................... 9.2.2. Excitation and Excitation–Contraction Coupling ....................................... Neuromuscular Transmission and Excitation: The Acetylcholine Receptor ........................................................... Excitation–Contraction Coupling ....................................................... 9.2.3. Molecular Mechanisms of Muscle Contraction .......................................... Problems...............................................................................................................................

343 347 349 349 353 359 367 368 370 370 378 382 387

10: Epithelial Transport Introduction .......................................................................................................................... 10.1. Organization of Epithelial and Endothelial Cell Layers and Some Consequences ................................................................................................ 10.1.1. The Organization of Epithelial and Endothelial Cell Layers ...................... 10.1.2. The Pathways across Leaky and Tight Epithelia ........................................ 10.1.3. Transport in a Parallel Path System ............................................................ 10.1.4. Coupling of Transepithelial Water Flow to Active Ion Transport .............. The Curran Model............................................................................... The Standing Gradient Model............................................................. The Sodium Recirculation Model....................................................... The Cotransporter Hypothesis ............................................................ 10.1.5. The Effect of Unstirred Layers on Transepithelial Diffusion and Osmosis: Concentration Polarization ................................................... 10.1.6. Electrical Analogs of Cell Layers ............................................................... 10.2. Examples of Epithelial and Endothelial Function ................................................... 10.2.1. Absorption .................................................................................................. Absorption of Sugars in the Small Intestine ........................................ Transport of Water and Sodium in the Proximal Tubule of the Kidney ...................................................................... Transport of Water and Sodium in the Distal Tubule of the Kidney .................................................................................. 10.2.2. Secretion ..................................................................................................... 10.2.3. Filtration ..................................................................................................... Problems............................................................................................................................... Appendix 10.1. Convection, Diffusion and Mass Addition in Channel Geometries ...........

391 392 392 395 400 405 405 407 410 411 412 414 418 419 419 422 427 430 434 439 443

11: Gas Transport Introduction .......................................................................................................................... 447 11.1. Partial Pressure and the Equations for Gas Flux...................................................... 448

CONTENTS

XVII

11.2. 11.3. 11.4. 11.5.

449 453 455 456 459 461 462 468 480

Overview of the Gas Transport Process .................................................................. Gas Exchange in the Lung....................................................................................... Oxygen Transport in the Blood ............................................................................... Transport from Red Blood Cells to Tissue .............................................................. 11.5.1. Reaction–Diffusion Processes .................................................................... 11.5.2. The Krogh Tissue Cylinder......................................................................... 11.5.3. Modifications to the Basic Krogh Model.................................................... 11.5.4. Beyond the Krogh Cylinder........................................................................ Problems...............................................................................................................................

References ............................................................................................................... 485 Index ........................................................................................................................ 497

1 EQUILIBRIUM THERMODYNAMICS

INTRODUCTION It is appropriate to begin this text with an examination of the equilibrium state, since departures from this state drive the transport processes that we will discuss in the chapters to follow. The equilibrium state is defined in the first section of this chapter, using the laws of thermodynamics to develop expressions for the activities and chemical potentials of species in the biological milieu. Because of the particular importance of the chemical potential (and the related electrochemical potential) to descriptions of transport, special emphasis is placed on its relation to solute activities and concentrations, and to the other measurable properties of biological solutions. Because of the close relation between equilibrium and transport, it is not surprising that a number of equilibrium thermodynamic concepts have become an integral part of the description of certain transport processes. Two of these concepts are developed in this chapter for later use: ionic equilibria across membranes and at interfaces, and chemical equilibrium and equilibrium constants.

1.1. CHEMICAL POTENTIALS AND ACTIVITIES Among the several equivalent thermodynamic definitions of equilibrium, the one that is based on chemical potential is particularly well suited to the purposes of this text. This is because gradients or differences in chemical potential are the natural driving forces for the fluxes of solute and solvent; they also drive the biochemical processes to which these fluxes may be coupled. By using the chemical potential to define thermodynamic equilibrium, we achieve a certain ease of movement between equilibrium considerations and transport events. In the first part of this section, the condition for thermodynamic equilibrium between two phases is developed in terms of the chemical and electrochemical potentials of the species in each phase. In biological transport, these phases are almost always solutions, and the bulk of the section is devoted to the development of expressions for the chemical and electrochemical potentials in terms of solution composition, presM.H. Friedman, Principles and Models of Biological Transport, DOI: 10.1007/978-0-387-79240-8_1, © Springer Science+Business Media, LLC 2008

1

2

CH. 1: EQUILIBRIUM THERMODYNAMICS

sure, and temperature. The section concludes with a brief discussion of solute activity and concentration as they relate to the chemical potential. A limited familiarity with thermodynamics is assumed. Accordingly, a few common thermodynamic terms are used here without definition, and the thermodynamic development includes a small number of equations that are presented without proof and whose derivation can be found in any thermodynamics text. These equations are indicated with an asterisk.

1.1.1. Thermodynamic Preliminaries: The Electrochemical Potential We begin with the First Law of Thermodynamics, in differential form: *dU = dQ – dW.

(1.1)

Here, dU is the change in the internal energy of a system of interest, dQ is a differential quantity of heat transferred into the system, and dW is a differential amount of work performed by the system. The internal energy of the system is increased when heat is added, and decreases when the system performs work. Different systems can perform different kinds of work, including: 1. the movement of volume, or expansion, against an external pressure; 2. the movement of charge, for instance that on an ion, against an electrostatic potential; and 3. the transfer of material. Corresponding to these,

*dW  PdV  \dq  œ Pi dni ,

(1.2)

i

where the sum includes all species in the system, P is hydrostatic pressure, V is the volume of the system, \ is electrostatic potential, q is electric charge, Pi is the chemical potential of the ith species, and ni is the number of moles of the ith species in the system. Equation (1.2) gives the differential work required to change, by differential amounts, the volume, charge, and composition of the system. We should note that there are other forms of work that systems can do, which we will not include here. Among those having biological relevance, an important work term appropriate to muscle is the work of shortening against a tensile force, F. If l is the length of the muscle segment, the differential work of contraction is –Fdl. The reversible addition or withdrawal of heat changes the entropy of the system, according to the Second Law of Thermodynamics: *dS 

dQ T

(1.3)

where S is entropy and T is absolute temperature. By substituting Eqs. (1.2) and (1.3) into (1.1), the Gibbs equation is obtained:

PRINCIPLES AND MODELS OF BIOLOGICAL TRANSPORT

3

dU  TdS  PdV \dq œ Pi dni .

(1.4)

i

The Gibbs free energy of the system is defined by: G = U + PV – TS.

(1.5)

dG = dU + PdV + VdP – TdS – SdT.

(1.6)

The total differential of G is:

Substituting Eq. (1.4) into (1.6), dG  \dq œ Pi dni VdP  SdT .

(1.7)

i

Equation (1.7) specifies the change in the Gibbs free energy of a system resulting from reversible changes in its charge, content, pressure and temperature. We shall see shortly that this expression is crucial for defining the conditions under which a system is in a state of equilibrium. In this text, the most common source of charges in solution is ionic solutes. The total charge qi carried by the ith ionic species is proportional to ni, and the first two terms on the right-hand side of Eq. (1.7) can be combined. The coulombs of charge carried by a mol or gram atom of the ith ion is: qi = zi  ni,

(1.8)

where zi is the valence of the ith ion and  is the Faraday, a conversion factor equal to about 96,500 coul/mol. The total charge in the system, q, is obtained by summing qi over i; its differential is dq  œ zi  dni.

(1.9)

dG  \ œ z i dni œ Pi dni VdP  SdT .

(1.10)

i

Substituting Eq. (1.9) into (1.7), i

i

Note that, although both sums are over all species, only charged species (zi z 0) contribute to the first sum. Define the electrochemical potential of the ith species by P i  Pi zi\ .

(1.11)

The electrochemical potential becomes identical to the chemical potential if the species is uncharged. Equation (1.10) can now be written as dG  œ P i dni VdP  SdT . i

(1.12)

4

CH. 1: EQUILIBRIUM THERMODYNAMICS

From Eq. (1.12), it can be seen that P i measures the contribution of the ith species to the Gibbs free energy of the system:  sG ¬­ ­­ P i  žžž Ÿž sni ®­

(1.13)

P, T , nj

where j includes all species except the ith; the electrochemical potential is thus the partial molal free energy of the ith species.

Figure 1.1. A system at equilibrium: P iI  P iII .

1.1.2. The Interphase Equilibrium Condition We are now ready to define interphase equilibrium in terms of electrochemical potentials. Consider the system in Figure 1.1, consisting of Phases I and II separated by a barrier that is permeable to species i. The temperature and pressure of the system is uniform. What is the meaning of the statement that the ith species is in equilibrium across the membrane? The thermodynamic condition for equilibrium is defined in terms of the Gibbs free energy: when a system is in a state of equilibrium, the change in Gibbs free energy accompanying an infinitesimal departure from that state is zero. Correspondingly, a species is said to be in equilibrium across a barrier like that in Figure 1.1 if the free energy of the system does not change when a differential amount of the species is transferred from one phase to the other. Consider the transfer of dni moles of species i from Phase I to Phase II. During this process, there is no change in pressure, temperature, or the number of moles of

PRINCIPLES AND MODELS OF BIOLOGICAL TRANSPORT

5

other species in either phase; furthermore, the amount transferred is too small to cause a change in the composition or partial molal properties of either phase. Phase I loses mass and its free energy is reduced proportionately; from Eq. (1.13),

 iI dni . dG I  P

(1.14)

The mass is received by Phase II and assumes the partial molal properties of that phase; thus, dG II  P iII dni .

(1.15) I

II

If species i is in equilibrium, then the total change in system free energy, dG + dG , is zero; thus, the desired equilibrium condition is: P iI  P iII .

(1.16)

An unstated key ingredient of this definition of equilibrium is the ability of the species to transfer from one phase to the other. If two phases containing a species in common are on the two sides of a barrier, that species can come to equilibrium (i.e., achieve identical electrochemical potentials in each phase) only if the interface between the phases is permeable to it. As a (rather obvious) corollary, when two phases are separated by a barrier that is impermeable to a particular species, that species cannot equilibrate between the phases. Finally, when phases or systems are described as being in equilibrium with one another, Eq. (1.16) is necessarily satisfied for only those species that can move between them.

1.1.3. Electrochemical Potentials in Terms of Measurable Solution Variables: Introduction, and the Effect of Electrostatic Potential Most biological solutions are characterized not by the chemical or electrochemical potentials of their constituents, but rather by more measurable quantities such as temperature, pressure, electrical potential, and composition. In the next few sections, we derive expressions for the chemical and electrochemical potential in terms of these measurable quantities. This allows us to rewrite Eq. (1.16) in terms of the measurables of the system. The lone contribution of the electrostatic potential to the electrochemical potential is the zi\ term in Eq. (1.11). It remains to express the chemical potential Pi in terms of temperature, pressure, and composition.

1.1.4. The Gibbs–Duhem Equation Before pursuing this task, it is necessary to derive one more key thermodynamic equation: the Gibbs–Duhem equation. We begin with the integrated form of the Gibbs equation: *U  TS  PV œ P i ni . i

(1.17)

6

CH. 1: EQUILIBRIUM THERMODYNAMICS

The total differential of U is obtained from Eq. (1.17): dU  TdS SdT  PdV  VdP œ P i dni œ ni dP i . i

(1.18)

i

The quantity dU is also given by the Gibbs equation [Eq. (1.4)], which can readily be written in terms of the electrochemical potential: dU  TdS  PdV œ P i dni .

(1.19)

i

The Gibbs–Duhem equation is obtained by subtracting Eq. (1.19) from (1.18): SdT  VdP œ ni dP i  0 .

(1.20)

i

The sum in Eq. (1.20) can be broken into two sums: œ ni dP i  œ ni dPi d\ œ ni zi . i

i

i

For electrically neutral solutions, œ ni zi  0 , i

so œ ni dP i  œ ni dPi , i

i

and the electrochemical potential in the Gibbs–Duhem equation can be replaced by the chemical potential: SdT  VdP œ ni dPi  0 .

(1.21)

i

1.1.5. Dependence of the Chemical Potential on Pressure The dependence of chemical potential on pressure will be developed first. Consider a solution at constant temperature and composition. Solving Eq. (1.11) for the chemical potential, and differentiating with respect to pressure,

 sPi ¬­ žž ­ žŸ sP ®­

T , ni

 sP ¬ s\ .  žž i ­­  zi  žŸ sP ®­ sP T , ni

(1.22)

The second term on the right-hand side of Eq. (1.22) is zero because the electrostatic potential and the pressure are independent variables. Equation (1.13) is substituted into the right-hand side of Eq. (1.22), which becomes

 sPi ¬­ žž ­ žŸ sP ®­

T , ni

  ¯ s ¡ž sG ¬­ ° ­­  ¡žž °. sP ¡Ÿž sni ®­P , T , n ° j ¢ ±

(1.23)

PRINCIPLES AND MODELS OF BIOLOGICAL TRANSPORT

7

The free energy is a continuous function of the independent variables, so the order of differentiation may be interchanged:

 sPi ¬­ žž ­ Ÿž sP ®­



T , ni

s sni

  sG ¬ ¯ ¡ž ­ ° ¡žžŸ sP ®­­ ° . T , ni °± ¡¢

(1.24)

The partial derivative of free energy with respect to pressure, at constant temperature and composition, is seen from Eq. (1.12) to be V, the volume of the system; thus,

 sV ¬­  sPi ¬­ žž ­ . ­­  žžž žŸ sP ®T , n žŸ sni ­­® P, T , nj i

(1.25)

The right-hand side of Eq. (1.25) is the partial molar volume of the ith species, Vi . Thus, at constant temperature and composition, dPi  Vi dP . Since the partial molar volume is essentially independent of pressure, we can write the indefinite integral:

Pi  Vi P Pic ,

(1.26)

c

where the “constant” of integration Pi is a function of only temperature and composition. In summary, we derived the dependence of Pi on pressure by fixing the remaining independent variables and obtaining a differential relation between chemical potential and pressure. Integrating this relation, we obtained an indefinite integral whose “constant” of integration (constant only with respect to changes in pressure) depends on only the variables that had been fixed, i.e., temperature and composition. The same c technique will now be used to derive the dependence of Pi on composition. Regrettably, the analysis does not proceed as simply — or as directly — as the one we have just performed.

1.1.6. Dependence of the Chemical Potential on Composition We begin by differentiating Eq. (1.26) and substituting the result into Eq. (1.21). c Since the composition dependence of Pi is sought, the remaining independent variable — temperature — is fixed. Consequently, the SdT term in the Gibbs–Duhem equation vanishes, giving

VdP œ ni (Vi dP dPic )  0 .

(1.27)

i

The coefficient of dP in Eq. (1.27) is œ niVi  V . This coefficient is zero, since the i

partial molar volumes are defined such that the volume of the system is essentially equal to the sum of the contributions niVi of each species in the system. Thus, the preceding equation simplifies to

œ ni dPic  0 . i

(1.28)

8

CH. 1: EQUILIBRIUM THERMODYNAMICS

Figure 1.2. A system demonstrating the effect of composition on chemical potential. The water level in Phase I is constant but that in the narrow standpipe can change when solvent crosses the membrane.

With Eq. (1.28) in hand, now consider the system depicted in Figure 1.2. The two phases, one (I) consisting of pure water, and the other (II) consisting of an aqueous solution containing k solutes, are in equilibrium across a membrane that is permeable to only the solvent. We now examine the consequence of a small change in the composition of Phase II. Before the composition change, the solvent, which can transfer between the phases, is in equilibrium across the membrane; that is, its chemical potential is the same in Phase I and Phase II. The chemical potential of the water in Phase I is not affected by the composition change in Phase II. However, the chemical potential of the water in Phase II is altered when the composition of the phase is changed. As a result, water crosses the membrane to restore its chemical potential in Phase II to that in Phase I, that is, to re-establish equilibrium. The transmembrane difference in solvent chemical potential, caused by the change in the composition of Phase II, is the driving force for the flow of water across the membrane. Water does not cross the membrane indefinitely. The chamber containing Phase II has a standpipe that translates any change in the volume of the phase into a change in its hydrostatic pressure. This, in turn, changes the chemical potential of the water in Phase II, according to Eq. (1.26), and equilibrium is eventually reestablished. If we consider a differential process, starting before the composition change and terminating after equilibrium has been reestablished, dP IIw  0 ,

(1.29)

PRINCIPLES AND MODELS OF BIOLOGICAL TRANSPORT

9

since the chemical potential of the water (denoted by the subscript “w”) in Phase II is equal to its chemical potential in Phase I before the process begins, and after it is complete. Equation (1.26) is used to rewrite Eq. (1.29) as Vw dP II dPcw, II  0 .

(1.30)

The second term in Eq. (1.30) is the change in chemical potential caused by the initial composition change, and the first term reflects the compensating pressure change. For this differential process, the effect of the water flow on the composition of Phase II is negligible. The remainder of this analysis will be concerned with the thermodynamics of only Phase II, so the superscript “II” will be omitted in what follows. Our next step is to use the Gibbs–Duhem equation to relate the pressure change in Phase II to the composition change that caused it. The sum in Eq. (1.28) can be split into a sum over the k solutes plus a term for the solvent: k

nw dP cw œ ni dPic  0 .

(1.31)

i 1

Substituting Eq. (1.30) into (1.31) and rearranging: k

œ ni dPic  nwVw dP

(1.32)

i 1

At this point we must use an empirical relation, the van 't Hoff Law. Refer once again to Figure 1.2. The van 't Hoff Law states that, when Phase II is dilute, the pressure change needed to stop water flow across the membrane is proportional to the initiating change in the composition of Phase II, according to the following equation: k

GP  RT œ Gci .

(1.33)

i 1

Here, R is the gas constant and T is absolute temperature; Gci is the small initial change in the concentration of the ith species in Phase II, and GP is the small compensating pressure change. Though this equation is based on measurements of finite changes in pressure caused by finite changes in concentration, it is reasonable to expect that it holds for differential changes as well: k

dP  RT œ dci .

(1.34)

i 1

The use of the van ‘t Hoff Law in this development means that the final result will be directly applicable to only dilute solutions; thus, additional dilute-solution approximations can now be made without further limiting the usefulness of the result. In particular, since such solutions are almost entirely water, the product nwVw in Eq. (1.32) can be replaced by V, the volume of the solution. The validity of this substitution can be demonstrated with a simple example. For instance, consider a liter of 150 mM NaCl solution, whose density is 1.0046 g/cc. The liter weighs 1004.6 g and con-

10

CH. 1: EQUILIBRIUM THERMODYNAMICS

tains 0.15 mols, or 8.8 g, of NaCl. By difference, it contains 995.8 g water. The molar volume of water is 18 cc/mol, so nwVw = (995.8/18) q 18 = 995.8 cc, close to the actual solution volume, V =1000 cc. Then, with Eq. (1.34), Eq. (1.32) becomes: k

k

i 1

i 1

œ ni dPic  VRT œ dci .

(1.35)

Furthermore, in dilute solutions, individual solute ions or molecules (“particles,” for short) are far apart, separated from one another by numerous solvent molecules. Thus, each solute particle is oblivious to all other solute particles; solute–solute interactions are absent. In particular, all of the particles of a given solute are oblivious to the presence of other solutes. Under such circumstances, the free energy and chemical potential of a particular solute are not expected to depend on the concentrations of other solutes. In the absence of interactions between different solutes, Eq. (1.35) can hold only if it applies to each species separately:

ni dPic  VRTdci .

(1.36)

We now have the desired differential relation between the compositional part of c the chemical potential of the ith solute (Pi ) in dilute solution and a variable (ci) that reflects the composition of the solution. To write the indefinite integral as we did in the previous subsection, we first divide both sides of Eq. (1.36) by ni, and make use of the definition ci = ni /V; then, dPic  RT

dci . ci

(1.37) c

We want to integrate Eq. (1.37) to obtain a relationship between Pi and ci. To perform c a definite integration, we need to know the value of Pi that corresponds to a particular c value of ci. We have no means for directly measuring Pi , so we make an arbitrary 0 definition: We define a standard concentration ci (usually 1 M = 1 mol/l for solutes) 0 and define the chemical potential term in this standard state as Pi . We can then integrate Eq. (1.37): Pic

ci

¨

dPic  RT ¨

Pi0

ci0

dci . ci

(1.38)

ci . ci0

(1.39)

Integrating and rearranging: Pic  Pi0 RT ln 0

Choosing molarity as the unit of concentration, ci = 1, and Eq. (1.39) becomes

Pic  RT ln ci Pi0 .

(1.40)

PRINCIPLES AND MODELS OF BIOLOGICAL TRANSPORT

11

Since Pi0 is evaluated at a specified concentration, ci0 , it is a function of only temperature. Substituting Eq. (1.40) into (1.26), we obtain the desired expression for the chemical potential of a solute in a dilute solution:

Pi  Vi P RT ln ci Pi0 .

(1.41)

Similarly, from Eq. (1.11), the electrochemical potential is

P i  Vi P RT ln ci zi  \ Pi0 .

(1.42)

An equation analogous to Eq. (1.41) can be written for the chemical potential of the solvent. It will be derived in Chapter 6, where it is first needed. The requirement that the argument of logarithms be dimensionless appears to be violated in Eq. (1.40) and the equations that follow. Strictly speaking, the argument is 0 ci /ci , which is dimensionless. If the standard state is 1 M, ci must be measured in molarity; i.e., mols/l. When the chemical potential is used to describe the driving force for transport, differences or gradients in this quantity are much more important. It is 0 easy to show that ci is absent from the expression for chemical potential difference, and the gradient is derived directly from Eq. (1.37), which does not contain ci0. As we will see below, more care is needed when using the chemical potential to describe chemical equilibrium.

1.1.7. Units All of the terms that comprise the electrochemical potential have the units of energy per mass. To express all of them in the same units, e.g., cal/mol, some conversions must usually be made: \ has the units of (coul/mol)-volt. 1 coul-volt = 1 joule = 0.2389 cal; thus 1 coul-volt/mol = 0.2389 cal/mol. Biological pressures are often measured in torr (1 torr = 1 mm Hg); thus, VP 2 commonly has the units of (cc/mol)-torr. 1 torr = 1333 dynes/cm and 1 dyne-cm = 1 –7 –5 erg = 0.2389 q 10 cal; thus 1 cc-torr/mol = 1333 ergs/mol = 3.18 x 10 cal/mol. Using R = 1.987 cal/mol-K, RT has the units of cal/mol. SI (Système International, or International System) units are based on the kilogram (kg), meter (m), second (s), ampere (A), and degree kelvin (K). The unit of substance remains the gram-mole, defined as that amount of material that contains as many elementary entities (atoms or molecules, depending on the substance) as there 12 are atoms in 0.012 kg of C. Force and energy are derived quantities; force is meas2 ured in newtons (1 N = 1 kg-m/s ), and energy is measured in joules (1 J = 1 N-m). 2 The SI unit of pressure is the pascal (1 Pa = 1 N/m ); 1 torr = 133.3 Pa. To convert quantities expressed in cal/mol to J/mol, multiply by the mechanical equivalent of heat, 4.187 J/cal. The gas constant in SI units is R = 8.32 J/mol-K.

12

CH. 1: EQUILIBRIUM THERMODYNAMICS

1.1.8. Activity and Activity Coefficient Equation (1.41) is limited in its applicability, since it is based on the van ‘t Hoff Law and a number of approximations that are valid only for solutions more dilute than the physiological solutions with which we will be concerned. This limitation is dealt with by introducing a quantity called the solute activity. The activity is defined by the requirement that:

Pi  Vi P RT ln ai Pi0 ,

(1.43)

where ai is the activity of the ith species. The activity of the solute is the value of ai that gives the correct chemical potential when substituted into Eq. (1.43). Thus, by definition, Eq. (1.43) is applicable to concentrated solutions as well as to dilute ones. Theoretical methods are available for predicting solute activity in simple and well-defined systems; however, the biological milieu is not simple and is rarely welldefined. Consequently, activities are generally determined experimentally when they are needed. More often, as will be illustrated in Chapter 2, the distinction between activity and concentration is neglected, Eq. (1.41) is used directly, and the error so introduced is “buried” in the analysis of the experimental data. The difference between activity and concentration is measured by the activity coefficient, Ji: Ji 

ai . ci

(1.44)

Comparing Eqs. (1.41) and (1.43), it is seen that the activity coefficient approaches unity (i.e., ai = ci) in dilute solutions. Its deviation from unity is a measure of the extent to which the chemical potential of the solute is affected by interactions among dissolved particles in the solution. The interactions of importance in biological solutions are almost exclusively electrostatic, so the distinction between activity and concentration is generally made only for charged species; the activity coefficients of nonelectrolytes at physiological concentrations are usually assumed to be unity. Two kinds of electrostatic interaction are usually considered. The first of these is the interaction between small ions that are close enough to “see” one other's electric field or the effects of the field on the intervening solvent. The best-known treatment of such interactions is the Debye–Hückel theory and its extensions, which give a theoretical basis for the concentration dependence of the activity coefficient in concentrated electrolyte solutions. These theories predict that the activity coefficient of an electrolyte in solution de2 pends on the ionic strength of the solution, Z = (6izi ci )/2. The activity coefficient initially decreases from unity as the electrolyte concentration is raised (Fig. 1.3). At higher concentrations, the slope of the activity coefficient-concentration curve becomes positive, but this occurs at concentrations beyond the physiological range; thus, in biological systems, ion–ion interactions consistently cause the activity of the solute to be less than its concentration.

PRINCIPLES AND MODELS OF BIOLOGICAL TRANSPORT

13

Figure 1.3. Solid line: concentration dependence of the mean activity coefficient of sodium chloride in aqueous solution at 25ºC. The molality of the solution is the number of mols of solute per 1000 g solvent, and is close to the molarity for dilute solutions. The mean activity coefficient of NaCl is related to the ionic activity coefficients by JNaCl = (JNaJCl)1/2. Dashed line: Mean activity coefficient predicted by the original Debye–Hückel theory. The original theory fails at NaCl concentrations below that corresponding to physiologic ionic strengths. The mean activity coefficient predicted by the extended Debye–Hückel theory, which includes an adjustable parameter that takes ion size into account, is not shown in the figure because it is virtually indistinguishable from the experimental curve.

A second kind of interaction that is more peculiar to biological systems is the binding of a charged solute to a soluble or insoluble particle in the solution. The binding step may or may not be reversible. Often, solute that is bound cannot participate in transport; thus the activity of unbound solute can be underestimated when it is derived from transport-related measurements. When the effect of Debye–Hückel interactions is small relative to the effect of binding, the apparent activity coefficient of the solute is close to the fraction of the solute that is unbound. Indeed, these two quantities — activity coefficient and unbound fraction — are often used interchangeably. A numerical illustration of the effect of binding on the apparent activity coefficient is given in the next section. There is a certain fuzziness in the distinction between the two kinds of interactions described above. The fuzzy middle ground is populated by a large number of biological solutes, such as proteins, charged polysaccharides, and other large, multiply charged molecules. The interactions of these species with small ions are describable by appropriate extensions of the Debye–Hückel theory, yet the presence of multiple charges along the polyion's length can give rise to cooperative effects strong enough to be described as binding, and they usually are. The description of these interactions is one of the more difficult and challenging problems in solution thermodynamics. Irrespective of the nature of the interactions, the activity coefficient in biological solutions is inevitably an experimental quantity. The solutions are too complex to

14

CH. 1: EQUILIBRIUM THERMODYNAMICS

be described by theories appropriate to simple electrolytes, and binding fractions must be measured. Fortunately, our inability to predict the activity coefficient is not as great a liability as one might think. First of all, as noted above, its deviation from unity is small in some cases and buried in the analysis in others. Additionally, as we shall see, the gradient of the chemical potential, and chemical potential differences, are more important determinants of the transport rate than is the chemical potential per se. To replace the activity by concentration in the expression for the chemical potential gradient, it is not necessary that the activity coefficient be unity, but only that it be uniform, a less strict condition that is more closely met in biological systems. This can be seen by taking the differential of the activity term in Eq. (1.43); if the activity coefficient is uniform, the second term in the rightmost side of the equation that follows is zero. d ( RT ln ai )  RT

 dc dai dJ ¬  RT žžž i i ­­­ . žŸ ci ai J i ®­

(1.45a)

Similarly, the contribution of the activity term in Eq. (1.43) to the difference between the chemical potentials in Phases I and II is RT ln aiI  RT ln aiII  RT ln(ciI J iI )  RT ln(ciII J iII )

 RT ln ciI  RT ln ciII RT ln

J iI . J iII

(1.45b)

Here, activity can be replaced by concentration if the activity coefficients are the same in both phases.

1.2. ION EQUILIBRIUM ACROSS MEMBRANES An important application of the concepts developed in the previous section is the description of ionic equilibria. The two such equilibria of greatest biological importance are presented below. The first of these is the Nernst equilibrium, which defines the potential difference across a membrane such that a permeable ion, present on both sides, is at equilibrium. This is followed by a description of the Donnan equilibrium, which arises when two phases are in equilibrium and one or more of the charged species in one phase cannot enter the other phase.

1.2.1. The Nernst Equilibrium The Nernst equilibrium describes the conditions under which an ion is in equilibrium between two phases. For discussion purposes, we will imagine that the two phases are separated by a membrane, and that the membrane is permeable to the ion. The equilibrium condition is given by Eq. (1.16): P iI  P iII

(1.16)

PRINCIPLES AND MODELS OF BIOLOGICAL TRANSPORT

15

In most considerations of ionic equilibria in biological systems, the effect of pressure on the equilibrium state can be shown to be very small; in addition, the system is generally isothermal. Under these circumstances, Eq. (1.16) becomes: RT ln aiI zi\ I  RT ln aiII zi\ II .

(1.46)

For a nonelectrolyte, zi = 0, and the equilibrium condition given above is satisfied only when the activity of the solute is the same in both phases. This is not the case for an ionic species, since interphase differences in activity can be compensated for by a potential difference across the membrane. The required potential difference for ionic equilibrium is found by rearranging Eq. (1.46): '\ 

RT aiII , ln zi  aiI

(1.47)

where the ' operator denotes the difference between the values of the operand in the I II two phases; e.g., '\ = \ – \ . Equation (1.47) is the Nernst equation, and the potential difference it defines is the Nernst potential of the ith ion, Ei. Each ion in solution has its own Nernst potential, which is given by Eq. (1.47) and depends on the activity of the ion in each phase. When a membrane separates two ionic solutions whose compositions differ, a potential ordinarily develops across the barrier. The factors that determine this potential difference will be discussed in later chapters; for now, it is sufficient to note that the Nernst potential of any given ion in such a system will not in general be equal to the potential across the membrane. Only those ions whose Nernst potential equals the membrane potential are at equilibrium.

1.2.2. Origin of the Nernst Potential To illustrate the use of the Nernst equation, consider two electrolyte solutions separated by a membrane that is permeable to only a single ion, P. The concentrations of the permeant in the two solutions are different. The potential difference between the two solutions is measured by a high impedance device, so no electrical current flows in the external circuit (Fig. 1.4). What potential difference will be measured when the permeant is in equilibrium across the membrane? Since no electrical current passes from one solution to the other through the external circuit, the current through the membrane must also be zero; otherwise, the solutions would accumulate charge, one becoming increasingly positive, and the other becoming increasingly negative. While the current through the external circuit would be carried by electrons, the charges that move across membranes are borne by ions. A general expression for the membrane current is: I   œ zi J i , i

(1.48)

16

CH. 1: EQUILIBRIUM THERMODYNAMICS

Figure 1.4. The Nernst potential. An electrolyte, PA, is dissolved in Phases I and II. There is no current through the external circuit. With the proper electrodes, the voltmeter measures the memI II brane potential \ – \ , which is equal to the Nernst potential of P. The accumulation of charge that gives rise to the Nernst potential when aPII > aPI is shown in the figure.

2

where Ji is the flux of the ith species across the membrane, in mols/cm -s, and I, the 2 current density, is in amps/cm . In the present application, only P can cross the membrane, so I = 0 implies JP = 0. For JP to be zero, the membrane potential '\ must be equal to the Nernst potential of P; then P will be at equilibrium and have no cause to cross the membrane. Thus, 

'\ = EP 

RT aPII ln . zP  aPI

(1.49)

How does the membrane potential arise? For illustrative purposes, assume that P is a cation and that its activity is higher in Phase II than in Phase I. The membrane is sitting in an empty chamber, and suddenly the solutions are added to each side. At that instant, there is no potential difference across the membrane, and P flows from Phase II to Phase I because its activity is higher in Phase II. Since there is no current in the external circuit, Phase II, which has lost some cation, develops a small negative charge, and Phase I correspondingly becomes positively charged. The membrane has become a charge separator — a capacitor. As such, it supports a potential that de-

PRINCIPLES AND MODELS OF BIOLOGICAL TRANSPORT

17

pends on the amount of charge transferred and the membrane capacitance. This potential difference, with Phase I positive with respect to Phase II, opposes the further transport of positively charged P from Phase II to Phase I. The flow of P across the membrane ceases when the membrane potential reaches EP. The amount of ion necessary to establish typical Nernst potentials (up to ca. 100 mV) across biological 2 membranes, whose capacitances are typically of the order of 1-2 Pfarads/cm , is very –7 2 –12 2 small, ca. 10 coul/cm or 10 mols/cm .

1.2.3. Specific Ion Electrodes I

II

To continue with the previous example, observe that, if aP is known, then aP can be found by measuring '\. This is the principle on which the use of specific ion electrodes is based. A solution, which is part of the electrode and contains the ion of interest, is separated from the solution to be analyzed by a barrier whose selectivity for the ion is extremely high. No current is permitted to flow between the solutions, so the electrode potential follows the Nernst potential of the selected ion. The pH electrode, which uses a glass barrier permeable to only hydrogen ions, is the best-known example of this kind of sensor. In most cases, the barrier is not perfectly selective, and ions other than the one of interest can also cross, though with more difficulty. As a consequence, the electrode potential no longer follows Eq. (1.49) exactly. Specific ion electrodes are best calibrated before use against solutions whose known composition is similar to that of the solutions to be analyzed. Equation (1.47) is often written in terms of logarithms to the base ten: Ei 

a II 2.303 RT log10 iI . zi  ai

(1.50)

At 37ºC, the coefficient RT/ = (l.987 cal/mol-K • 3l0 K / 96500 coul/mol) / 0.2389 cal/coul-V = 0.0267 V = 26.7 mV; multiplying by 2.303, the coefficient of the logarithm in Eq. (1.50) is 61.5/zi mV. Thus, if the activities of a monovalent ion on the two sides of a membrane differ by a factor of ten, then the Nernst potential of the ion is 61.5 mV at 37ºC. The Nernst potential is plotted against the activity ratio in Figure 1.5, for univalent and divalent ions.

1.2.4. Activity Coefficient Considerations For the activity ratio in the preceding equations to be well approximated by the conII I I II centration ratio ci /ci , it is necessary only that Ji | Ji ; the solutions need not be ideal. One characteristic of biological systems favors a uniform activity coefficient, and a second opposes it. First, the good news. As shown in the previous section, theories have been developed that describe quite well the activity coefficient in solutions of physiological ionic strength. Since the vast majority of solute particles in biological solutions are ions, and since the vast majority of dissolved ions are monovalent, the ionic strength closely approximates the

18

CH. 1: EQUILIBRIUM THERMODYNAMICS

total number of moles of dissolved material per unit volume. We shall see in Chapter 6 that powerful and potentially destructive osmotic forces arise when the total concentrations of dissolved particles on the two sides of a membrane are too different. Gradients in ionic strength sufficient to cause substantial activity coefficient gradients would also generate substantial osmotic gradients. Indeed, the ionic strength is quite uniform among the fluid phases in the body (though the composition is not!), suggesting that the activity coefficients should be too.

Figure 1.5. Dependence of the Nernst potential on the activity ratio of monovalent and divalent cations and anions at 37ºC.

The bad news was also alluded to in the previous section. Ion binding by charged macromolecules can significantly affect the apparent activity coefficient of the bound solute. Concentration is usually measured by total assay techniques, such as flame photometry or chromatography, which do not distinguish between free and bound ion. The degree of binding — or the reduction in activity coefficient due to cooperative effects — is not always known and can differ markedly from phase to phase, depending primarily on the number and nature of the macromolecules that are present. A common simplification when binding or cooperative effects are present (as, for instance, within a cell) is to regard the ion in question as residing in two distinct “compartments”: a bound compartment in which it is inactive (Jbound = 0) and a free compartment containing a near-ideal solution (Jfree | 1). Such compartments can be real — for instance, an intracellular organelle that preferentially sequesters an ion — or more diffuse, such as when a compartment represents the electrostatic interactions of an ion with multiple macromolecules distributed throughout the interior of the cell. When the distribution of the ion is represented in this way, the intracellular activity can be replaced by the concentration of free ion. Specific ion electrodes can be very useful for partitioning the total ionic content of a cell between the free and bound compartments, since they measure activity directly. The use of specific ion electrodes to determine the bound fraction in a cell is illustrated in one of the problems at the end of this chapter.

PRINCIPLES AND MODELS OF BIOLOGICAL TRANSPORT

19

As indicated earlier, the Nernst equation can be used to determine whether an ion is at equilibrium across a membrane. This assessment is usually made under the assumption that the activity coefficient of the ion is the same in the two phases that bathe the barrier; it will be flawed if the ion is preferentially bound in one of the phases and a total assay is used to measure concentration. The increasing use of intracellular ion-specific microelectrodes has generated a modest increase in the use of activity (rather than concentration) to characterize biological solutions. This is to be welcomed, since activity is the more correct variable for analyzing transport and interphase equilibrium. Of course, concentration is still used to measure the amount of each species in solution; it is particularly useful when the electroneutrality condition enters the analysis, as in the following subsection.

1.2.5. The Donnan Equilibrium The Donnan equilibrium describes a situation in which electrolyte solutions are in equilibrium across a barrier, and one or both of the phases contain charged species that cannot cross it (Fig. 1.6). For instance, two solutions, one of which contains a charged protein, might be separated by a membrane through which only small ions can pass. A charged (polyelectrolyte) gel in saline solution generates a Donnan equilibrium; the ions from outside can enter the gel and equilibrate between the gel and the solution, while the long-chain charged molecules that make up the gel are covalently linked to one another and cannot enter the saline outside. Another example is an electrolyte solution in which is suspended a biological cell whose cell membrane passes ions but not the charged proteins and other macromolecules in the cytoplasm. Consider the system represented in Figure 1.6a. A membrane separates two phases as in earlier figures. Phase II contains only a 1–1 electrolyte, CA, at a concentration cII = cAII = cCII. Both the cation (C) and anion (A) can cross the membrane, and consequently are present in Phase I as well. Phase I contains charged solutes (e.g., proteins) that cannot cross the membrane; the concentration of charge on these imI permeant solutes is zIcI . The Gibbs–Donnan equilibrium condition defines the equilibrium state of this system. I Before proceeding, it is important to emphasize that cI is the concentration of impermeant charge, not the concentration of impermeant molecules. If the concentration of impermeant molecules in Figure 1.6a is 1 mM and each molecule contains ten I I charged groups, then cI = 10 mM. With this definition of cI , zI is always r1, depending on the sign of the impermeant charge. This treatment of the impermeant charge is essential when Phase I is a gel, and the charge is part of the structure itself. In most biological systems, the impermeant molecules present both positive and negative I charges, and cI is the concentration of net charge. As hinted above, the description of the Donnan equilibrium proceeds more simply when concentrations rather than activities are used. Since both C and A can cross the membrane, both ions must satisfy the equilibrium condition (1.16) for the system as a whole to be at equilibrium. Put another way, their Nernst potentials must each equal the membrane potential, '\; from Eq. (1.47),

20

CH. 1: EQUILIBRIUM THERMODYNAMICS

Figure 1.6. The Donnan equilibrium. In (a), I represents a large, multiply charged macromolecule that cannot cross the membrane. In (b), many charges in the gel are part of its structure and cannot I leave; their total concentration is cI . In (c), a cell is immersed in an external bath or extracellular fluid. In all cases, C and A can freely exchange between the phases.

'\ 

RT c II RT c II ln I   ln I .  cC | cA

(1.51)

The two expressions on the right yield c II c AI  II . cCI c

(1.52)

PRINCIPLES AND MODELS OF BIOLOGICAL TRANSPORT

21

This ratio, or its reciprocal, is called the Donnan ratio. It is easy to show that when Phase II contains multiple monovalent ions, the Donnan ratio is the same for all ions of like charge. Phases I and II are both electrically neutral; that is, indexing the phases by j, œ zi cij  0 (j = I, II).

(1.53)

i

Note that the electroneutrality condition is properly written in terms of ionic concentrations, not activities, since the number of charges is sought. The electroneutrality condition applied to Phase II allowed us to equate the concentrations of C and A in that phase; its application to Phase I, which contains the charged impermeants, is cCI  c AI zI cII  0 .

(1.54)

Given the concentration of the solution in Phase II and that of the impermeant charge in Phase I, Eqs. (1.52) and (1.54) are solved simultaneously for the concentrations of C and A in Phase I. At this point, the simplicity afforded by using concentrations in Eq. (1.52), rather than activities, becomes apparent. The solutions are: cCI 

zI cII (cII )2 4(c II )2

c AI 

,

2 zI cII (cII )2 4(c II )2 2

.

(1.55)

The Donnan potential, '\D, is the membrane potential at equilibrium; substituting Eq. (1.55) into (1.51), '\ D 

I I 2 II 2 RT zI cI (cI ) 4(c ) , ln  2c II

(1.56)

2

where zI = 1 has been used. Figure 1.7 shows the variation of the Donnan ratio and the Donnan potential with ambient concentration divided by the concentration of imII I permeant charge. As c becomes much larger than cI , the Donnan potential approaches zero and the Donnan ratio approaches unity. This is referred to as suppression of the Donnan effect. The Donnan effect is suppressed in certain analytical and I experimental protocols to minimize uncertainties in the result when cI is unknown or varies widely among preparations.

1.3. CHEMICAL EQUILIBRIUM The concepts that we have just developed to describe the equilibrium of a solute between phases can be used in a very similar fashion to define the conditions under which a chemical reaction is at equilibrium. This is done below.

22

CH. 1: EQUILIBRIUM THERMODYNAMICS

Figure 1.7. Donnan ratio cAI / cII = cII / cCI and Donnan potential vs. cII / cII for zI = +1. When zI = –1, the ordinate gives cII / cAI and –'\D .

Any chemical reaction can be written in the following form: œ vi Ri l œ vi Pi ,

reactants

(1.57)

products

where Ri is the ith reactant, Pi is the ith product of reaction, and the {Qi} are the stoichiometric coefficients of the reaction, which specify the relative number of molecules of each species participating in the process. For instance, for the simple oxidation of glucose, where one molecule of the sugar yields six molecules of carbon dioxide and six of water, the stoichiometric coefficient of glucose is one and that of water is six. A chemical reaction is at equilibrium when the change in free energy accompanying the conversion of a differential amount of reactant(s) into product(s) is zero. For an isothermal, isobaric system, this change in free energy can be written as  sG ¬­ dG  œ žžž ­­ dni , i ž Ÿ sni ®­

(1.58)

where the sum includes the reactants (for which dni is negative) and the products (for which dni is positive). From Eq. (1.13), the partial derivative in the preceding equation is simply the electrochemical potential of the ith species: dG  œ P i dni .

(1.59)

i

Before setting dG = 0, we use the stoichiometric coefficients to relate the dni's of the several participants in the reaction to the dn of an arbitrarily selected product, P*, whose stoichiometric coefficient is Q*. Suppose dn* mols of P* are produced; then the corresponding values of dni for the other participants in the reaction are

PRINCIPLES AND MODELS OF BIOLOGICAL TRANSPORT

23

— for products: v ¬ dni  žž i ­­­ dn * , žŸ v * ®

(1.60a)

v ¬ dni  žž i ­­­ dn * . žŸ v * ®

(1.60b)

— and for reactants:

Substituting Eqs. (1.60) into (1.59),  dn * ¬­   ¯ dG  žž œ P i (vi ) œ P i vi ° , ¡ ­ ­ p °± Ÿž v * ® ¡¢ r

(1.61)

where œ r and œ p denote sums over reactants and products, respectively. At equilibrium, dG = 0. Since the coefficient dn*/v* of the second term on the right-hand side of Eq. (1.61) is nonzero: œ P i vi  œ P i vi  0 . p

(1.62)

r

Using the equations derived earlier in this chapter [Eqs. (1.11) and (1.43)], the electrochemical potentials of the species involved in the reaction can be written in terms of solution variables: electrostatic potential, pressure, activity and temperature. This procedure splits the left-hand side of Eq. (1.62) into four pairs of sums, which must add up to zero:  ¬  ¬ ž vi zi \  œ vi zi \­­ žœ viVi P  œ viVi P ­­ ­ žŸœ ž ® Ÿp ®­ p r r  ¬ žžœ vi RT ln ai  œ vi RT ln ai ­­­ Ÿp ® r  ¬ žžœ vi Pi0  œ vi Pi0 ­­­  0 . Ÿp ® r

(1.63)

Let us consider each pair separately. The first pair can be rewritten as  ¬ \ žžœ vi zi  œ vi zi ­­­ . Ÿp ® r Since charge is conserved during the reaction, the parenthesized term above (and hence the first pair of sums) is zero. The second pair of sums can be rewritten as:  ¬ P žžœ viVi  œ viVi ­­­ . Ÿp ® r

24

CH. 1: EQUILIBRIUM THERMODYNAMICS

Here, the parenthesized term measures the volume change accompanying the reaction. At atmospheric pressure and in solution, this contribution to the free energy change can generally be neglected relative to the remaining pairs of sums. The third and fourth pairs remain. At equilibrium, they must in combination equal zero. The fourth pair depends on only temperature and is termed the standard free energy change of the reaction, 'G0. Equating the third pair of sums to the negative of this quantity, and dividing through by RT, œ vi ln ai  œ vi ln ai   p

r

'G 0 . RT

(1.64)

We now use the properties of logarithms to rewrite the left-hand side of Eq. (1.64) in a somewhat different form: 

'G 0  œ ln aivi  œ ln aivi p r RT  ln 3 aivi  ln 3 aivi p

 ln

3 aivi p

3 aivi

r

w ln K a .

(1.65)

r

The quantity Ka is called the equilibrium constant of the reaction. It is an exponential function of the standard free energy change, so it too depends on only temperature. This constant defines the condition that is satisfied by the activities of the reactants and products when the reaction is at equilibrium: 3 aivi p

3 aivi

 Ka ,

(1.66)

r

0

where Ka = exp(–'G /RT). Equation (1.66) is the starting point for incorporating chemical equilibria into biological transport models. The equilibrium constant is dimensionless only if the sum of the stoichiometric coefficients of the products equals that of the reactants. This is generally not the case. The inconsistency of Eq. (1.65) with the requirement that the argument of logarithms be dimensionless has the same explanation as that presented earlier when deriving the 0 0 expression for the chemical potential. If the standard state for the {Pi } (and thus 'G ) is 1 M, then the activities (or concentrations, when ideality is assumed) in Eq. (1.66) must be expressed in moles/l.

PROBLEMS: CHAPTER 1 1. According to the van ‘t Hoff Law, written as 'P = RT'c, what is 'P in mm Hg when 'c = 1 mM and T = 37ºC?

PRINCIPLES AND MODELS OF BIOLOGICAL TRANSPORT

25

2. Confirm that the coefficient 2.303RT/ in the Nernst equation is 61.5 mV at 37ºC. 3. At constant temperature, the difference between the electrochemical potential of an ion in two phases is 'P i  V i 'P RT ' ln ai zi'\ . In cal/mol, what is the contribution to the electrochemical potential difference from the three terms if Vi I II = 60 cc/mol, 'P = 100 mm Hg, ci = 100 mM, ci = 10 mM, Ji = 0.75 in both phases, T = 310 K, zi = +1, and '\ = 100 mV?

4. A membrane separates two solutions, and an ion is in equilibrium across it. In Phase I, the ion concentration is 10 mM, and in Phase II it is 100 mM. The solutions are ideal and the pressure and temperature are the same on both sides. What + – ++ is the potential in Phase II, relative to that in Phase I, if the ion is Na ? Cl ? Ca ? T = 310 K. 5. The electrolyte concentrations inside a nerve axon (the axoplasm) are 14 mM Na, 140 mM K, and 3.5 mM Cl; the external concentrations are 142, 4, and 103 mM, respectively. At rest, the axoplasm is 90 mV negative with respect to the surroundings at 37ºC. Are any of the ions at equilibrium? Assume ideal solutions. 6. I measure the activities of several ions in the solutions on each side of a membrane, and the potential difference across the membrane. I calculate the Nernst potential of each ion and measure several of the ion fluxes across the membrane. For one of the ions, its Nernst potential is not equal to the membrane potential, but its flux across the membrane is zero. How can that be? 2

7. The capacitance of the axonal membrane is 2 Pfarads/cm . How many mols of 2 charge are needed to establish the resting potential across 1 cm of membrane? 8. A membrane permeable to only anions separates two solutions of NaCl at 37ºC. There is no external electrical contact between the solutions, so the current across I the membrane is zero. The concentrations of the solutions are c = 1 mM, dilute II I enough to be considered ideal, and c = 150 mM. The membrane potential \ – II \ is measured and found to be –125 mV. What is the activity coefficient of the chloride ion on Side II? 9. Two solutions of NaCl at 37ºC are separated by a membrane. In Phase I, cNaCl = 10 mM. In Phase II, cNaCl = 100 mM. The system is at open circuit. The membrane I II potential \ – \ = 61.5 mV. Assume all activity coefficients equal unity. (a) Is either ion at equilibrium? (b) The concentration of NaCl in Phase I is doubled. Does the membrane potential change? - yes (to what?), no (why?), I don't have enough information (what do you need to know?) (c) An amount of KCl is added to the original Phase I, sufficient to cause the potassium concentration to be 5 mM. Does the membrane potential

26

CH. 1: EQUILIBRIUM THERMODYNAMICS

change? — yes (to what?), no (why?), I don't have enough information (what do you need to know?) 10. Using literature values, calculate the ionic strength of extracellular fluid and blood plasma. Using Figure 1.3, compare the activity coefficients of NaCl solutions having these values of ionic strength. 11. I stick an electrode in an epithelial cell and find that its potential is –27 mV at 34ºC, relative to the bathing solution which is ideal and contains 5 mM K. I assay for intracellular potassium and find cK = 109 mM. Potassium is in equilibrium across the cell membrane. (a) What is the apparent intracellular activity coefficient of the ion? (b) Assume the intracellular potassium is in two compartments: in one it is bound and unavailable for exchange, and in the other it is free to exchange and has an activity coefficient of 0.9. How much of the 109 mM is in each compartment? 12. A dialysis bag permeable to sodium and chloride contains a charged impermeant. When the bag is allowed to equilibrate with a solution of 100 mM NaCl at 25ºC, the potential in the bag is 15 mV negative with respect to the saline outside. What are the sign and concentration of impermeant charge in the bag? Assume all solutions are ideal. 13. A dialysis bag permeable to sodium and chloride contains a negatively charged impermeant protein capable of binding cations. It is immersed in a solution of 100 mM NaCl at 25ºC, in which the activity coefficient of each ion is 0.78. Sodium and chloride diffuse into the bag and some of the sodium is bound by the protein, neutralizing those negative sites. All of the chloride in the bag is unbound. The solution in the bag can be regarded as ideal. At equilibrium, the total concentration of Cl in the bag is 50 mM and the total concentration of Na is 150 mM. (a) What is the potential in the bag relative to that outside? (b) What fraction of the sodium in the bag is bound? (c) What fraction of the protein sites are bound to sodium? 14. The connective tissue of the normal cornea (the corneal stroma) contains about 15 mM of nondiffusible negative charge. What are the Donnan potential and Donnan ratio between the corneal stroma and its surroundings when the concenII tration of electrolyte in the surroundings, c = 150 mM? 1.5 M? 15 mM? 1.5 mM? Assume T = 37ºC. 15. I drop a rigid dialysis bag containing an unknown concentration of impermeant into a large volume of NaCl solution of unknown concentration. After the bag has equilibrated with the solution, it is found to contain 194 mM Na and 27 mM Cl. T = 37ºC and all solutions are ideal.

PRINCIPLES AND MODELS OF BIOLOGICAL TRANSPORT

27

(a) What are the concentration and the sign of the charge of the impermeant in the bag? (b) What is the concentration of the sodium chloride solution into which the bag was dropped? (c) What was the potential difference between the solution in the bag and that outside? 16. A negatively charged gel (cI = 50 mM) is in equilibrium with a 150 mM NaCl solution at 37ºC. The NaCl solution is separated from a 30 mM KCl solution by a membrane that is permeable to only chloride. The entire system is open circuited. What is the potential difference between the gel and the 30 mM KCl solution? 17. Suppose I drop a negatively charged gel (cI = 50 mM) into a large amount of solution that contains 150 mM Na, 110 mM Cl, and a large macromolecule that produces a net negative charge of 40 mM and cannot enter the gel. What are the equilibrium concentrations of Na and Cl in the gel? 18. Suppose I have three phases in equilibrium: Phase I contains a negatively I charged impermeant, concentration cI ; Phase II contains a 1–1 electrolyte, conII centration c ; and Phase III contains a positively charged impermeant, concentraIII tion cI . Write the fully simplified expression for the potential difference between Phases I and III. All solutions are ideal. 19. Rederive Equations (1.55) and (1.56) for the case in which the cation is divalent. Go as far as you can — at least far enough to set up the equations to be solved — if the math gets hopeless. 20. Consider a chemical reaction at a cell membrane, where a solute A binds to a carrier X in the membrane. The reaction is A + X U
AX. I prepare a mixture of solute and carrier that contains 50 mM A and 60 mM X; when the binding reaction has reached equilibrium, cA = 10 mM, cX = 20 mM, and cAX = 40 mM. That is, 80% of the solute is bound. (a) What is the equilibrium constant for the reaction when concentrations are expressed in M? (b) Using Eq. (1.65), show that the same result will be predicted when a standard state of 1 mM is used.

2 FREE DIFFUSION

INTRODUCTION In this chapter, we consider the simplest of transport processes: the passive diffusion of a solute that occurs when its electrochemical potentials on the two sides of a permeable barrier are different. Indeed, this process is so simple that it fails to represent many aspects of transport in living systems. Nonetheless, it does describe some aspects of biological transport quite well, and it also provides a “base case” whose behavior can be compared against that of more complex transport mechanisms. This chapter is divided into two sections; the first deals with free diffusion of nonelectrolytes, and the second with that of ionic species. The principal property variable determining the flux of a nonelectrolyte is its permeability, a quantity that can in principle be related to the diffusion coefficient of the solute. Electrolyte diffusion in free solution is most rigorously described by classical electrodiffusion theory. The flux equations provided by this theory are very complex, and they have not seen nearly as much use as have approximations to them. Accordingly, emphasis will be placed here on the principles of electrodiffusion, and on the approximate solutions and special cases that are most commonly used. The equations of free diffusion can describe a wide variety of transport phenomena, including steady and unsteady transport processes; processes that can be described in one, two or three dimensions and in a variety of geometries; and processes in which chemical reactions and fluid flow take place simultaneously with diffusion. In this chapter, we will discuss a small subset of these, focusing on the tools that are applied to living systems. Comprehensive discussions of diffusional processes can be found in other texts, such as Crank's (1975) classic text, published thirty years ago and still being reprinted! A more limited set of solutions, but with more consistent biological applicability, can be found in Truskey et al. (2004).

M.H. Friedman, Principles and Models of Biological Transport, DOI: 10.1007/978-0-387-79240-8_2, © Springer Science+Business Media, LLC 2008

29

30

CH. 2: FREE DIFFUSION

2.1. FREE DIFFUSION OF NONELECTROLYTES The first transport process we will consider is the diffusion of a dissolved nonelectrolyte across a membrane or a similar barrier. The diffusive process is driven by the solute's concentration gradient. For now, the nature of the solvent is not particularly important, and it will usually be understood to be water, which is the most common biological solvent. As will be seen in Chapter 7, much of the material developed below is equally applicable to diffusion through a lipid film, such as the hydrophobic region of a cell membrane. In free diffusion through a membrane, the solute particles move about by random Brownian motion, like that in free solution. The solute flux, which is a measurable and reproducible quantity, is essentially the resultant of these separate motions. Even though the path of a single solute particle cannot be predicted, the consequence of an enormous number of these paths is quite reproducible.

2.1.1.

The Teorell Equation

The flux in free diffusion can be written very simply, in a form proposed by Teorell (1953): Flux = Mobility q Concentration q Driving force.

(2.1)

In the most commonly used units, the flux is the number of mols of solute crossing one square centimeter of membrane per second; it is proportional to the product of the solute mobility, which measures its ease of transport and depends jointly on the barrier/solvent and the solute, as well as the temperature; the solute concentration, which measures the amount of material available to participate in the process; and the driving force for the diffusion of the solute. The choice of a proper driving force is dictated by thermodynamic considerations that we will not examine until Chapter 6; for now, we will rationalize that choice by analogy with electrical phenomena. First, we recall that, when the chemical potential of the solute is the same in the two phases bounding the membrane, the solute is in equilibrium, and its flux across the membrane is zero. An analogous situation occurs in electrical circuits; when there is no electrical potential difference, there is no current flow. When the electrical potentials at two points are different, the potential gradient defines a field, and charged particles move in response to it. The force acting on the charges is the negative of the electrical potential gradient. The analogous driving force for solute flux is the negative of the chemical potential gradient: Driving force = –’Ps.

(2.2)

Almost every transport process with which we will be concerned can be described in terms of a single spatial coordinate perpendicular to the plane of the barrier. Calling that the x-direction, the driving force becomes:

PRINCIPLES AND MODELS OF BIOLOGICAL TRANSPORT

31

Driving force = 

dP s . dx

(2.3)

The implicit assumption in this one-dimensional treatment of gradients and fluxes is that these vectors are oriented perpendicular to the membrane plane and have negligible components parallel to that plane. This is reasonable if the extent of the membrane is much larger than its thickness, as is usually the case. The Teorell equation can now be written:

 dP ¬ J s  U s cs žž s ­­­ , žŸ dx ®

(2.4)

where Us and cs are the solute mobility and concentration, respectively. The flux Js is positive in the direction of increasing x. An integral driving force can also be defined, by integrating Eq. (2.3) across the membrane: a

Integral driving force  ¨  0

dP s dx  P Is  P IIs . dx

(2.5)

In Eq. (2.5), a is the thickness of the membrane. Phase I bathes the face of the membrane at x = 0, and Phase II the face at x = a. From Chapter 1, the integral driving force is zero at equilibrium. The integral driving force would appear to be far more convenient than the differential driving force [given by Eq. (2.3)] for describing transport, because it is based on the chemical potentials in the two phases external to the membrane. Chemical potentials inside the membrane, which must be known to find the local differential driving force, are generally unmeasurable. Fortunately, with a few reasonable assumptions, Eq. (2.4) can be integrated to give an expression that relates the transmembrane flux to the conditions in the ambient solutions. This we now do.

2.1.2. Integration of the Teorell Equation; Fick's First Law; Solute Permeability In integrating Eq. (2.4), the temperature is assumed to be uniform and the effect of ressure on the chemical potential of the solute is neglected; these are quite reasonable assumptions for the systems with which we will be dealing. If, in addition, the solutions are assumed to be ideal, then the chemical potential can be written very simply as μs = constant + RT ln cs.

(2.6)

Differentiating with respect to x,

 1 dcs ¬­ dP s d ln cs ­.  RT žžž  RT žŸ cs dx ®­­ dx dx

(2.7)

32

CH. 2: FREE DIFFUSION

Substituting Eq. (2.7) into (2.4), J s  U s RT

dcs . dx

(2.8)

The solute diffusion coefficient Ds is related to the solute mobility through the Nernst– Einstein relation, Ds = UsRT. Ds is often referred to as the binary diffusion coefficient (denoted Dij), as a reminder that its value depends on the identities of both solute and solvent. Introducing the diffusion coefficient into Eq. (2.8), we obtain: J s  Ds

dcs . dx

(2.9)

Equation (2.9) is known as Fick's first law of diffusion. Note that the flux is positive if the concentration gradient is negative. In the preceding derivation, it was assumed that Eq. (2.6) holds within the membrane, as though transport proceeded through aqueous pores in which the dependence of chemical potential on solution properties was identical to that in the aqueous solutions at the membrane faces. This is the first of several derivations in which the expressions for chemical or electrochemical potential in free solution will be used to describe the thermodynamic state of solute or solvent inside a transport barrier. The state of solutes and solvent inside a complex, heterogeneous biological barrier is not so neatly defined. Accordingly, it is convenient to think of cs(x) [and \(x) when describing electrolyte transport] as the concentration (and potential) of a free aqueous solution in equilibrium with a thin membrane slice at x. The concentration and potential of this equilibrium solution can be quite different from that of the true solution phase at that point in the membrane; however, since the two phases are defined to be in equilibrium, the chemical potentials of the solute and solvent are the same in each. A notable difference between the concentration of such an equilibrium aqueous solution and the true intramembrane solute concentration arises when the solubility of the solute in the membrane is different from that in the ambient aqueous phases. Such is the case for diffusion through the lipid bilayer of the cell membrane. The relationship between the solute concentration in the lipid and in an equilibrium aqueous solution is expressed in terms of the partition coefficient of the solute between the two phases. Diffusion through lipid layers will be described in Chapter 7.

Fick's first law assumes a somewhat more complicated form when the solutions are nonideal. In that case, the solute chemical potential must be written in terms of activity. The activity, in turn, is the product of the concentration and the activity coefficient. Thus, Eq. (2.7) is replaced by:  d ln cs d ln J s ¬­ dP s d ln as  RT  RT žž ­. žŸ dx dx dx dx ®­

(2.10)

For nonelectrolytes, the activity coefficient of the solute can be assumed to depend on only cs, which in turn is a function of x. Thus, the following substitution can be made: d ln J s d ln J s d ln cs  ¸ . dx d ln cs dx

(2.11)

PRINCIPLES AND MODELS OF BIOLOGICAL TRANSPORT

33

Substituting Eq. (2.11) into (2.10):  1 dcs ¬ dP s ­­žž1 d ln J s ¬­­ .  RT žžž ­­žž dx d ln cs ®­­ Ÿž cs dx ®Ÿ The Teorell equation then becomes  d ln J s ¬­ dcs ­ J s  U s RT žžž1 . žŸ d ln cs ®­­ dx

(2.13)

Define an augmented diffusion coefficient Ds* by  d ln J s ¬­ ­. D*s  Ds žžž1 žŸ d ln cs ®­­

(2.14)

For an ideal solution, Ds* = Ds. By substituting Eq. (2.14) into (2.13), a flux equation is obtained that looks almost identical to Eq. (2.9), and can be regarded as a generalization of Fick's first law to nonideal solutions: dc J s  Ds* s . dx

(2.15)

Fick's first law, as generalized above, is now integrated across the membrane to yield an expression for flux in terms of the transmembrane concentration difference. To set up the integration, Eq. (2.15) is rewritten as J s dx  Ds* dcs .

(2.16)

In the steady state, the solute flux is independent of x. Assume that the same is true of Ds*; Eq. (2.16) can then be integrated across the membrane and solved for Js:

Js  

II I D* s (cs  cs ) . a

(2.17)

The flux in Eq. (2.17) is based on a unit area of membrane, so it can be continuous at the interfaces x = 0 and x = a only if the entire cross-section of the barrier is available for transport. Furthermore, the assumption that the expressions for chemical potential as a function of concentration are the same in both barrier and bath implies that the solute diffuses through the same solvent as that in the ambient phases. The only barrier for which these assumptions hold would be a thin stagnant water film somehow maintained between two well-stirred aqueous baths. The solute permeability of such a thin film is defined as the solute flux per unit concentration difference: ks0 

J s0 J s0 Ds*   , a csI  csII 'cs

(2.18)

34

CH. 2: FREE DIFFUSION

where we have used the superscript “0” to indicate that diffusion takes place through a thin aqueous film. The form of Eq. (2.18) has been adopted to describe solute transport in biological systems. For such systems, the barrier is not a thin aqueous film, and the permeability is not given by Ds*/a. The solute permeability of a biological barrier is in general an experimental property, obtained by dividing the measured flux of a solute by its transmembrane concentration difference: ks 

J sa , 'cs

(2.19)

where J sa is the measured flux. Radiolabeled tracers are often used to measure permeability; the numerator and denominator of the right-hand side of Eq. (2.19) are replaced by the tracer flux and the transmembrane difference in tracer activity. Even when the solute does not cross the membrane by free diffusion, the experimental permeability is descriptive of the transport behavior of the system. Such empirical permeabilities, though not always easy to interpret in physical terms, are nonetheless useful for comparing solute transport rates and for predicting fluxes under similar conditions. There are some cases in which permeability can be estimated from a diffusion coefficient and membrane thickness. If the barrier is a stabilized thin film of a solvent immiscible with water, the permeability of the solute is determined by the partition coefficient, the film thickness and the binary diffusion coefficient of the solute in the solvent that comprises the membrane [see the second paragraph of the note following Eq. (2.9), and Chap. 7]. If the membrane possesses large interstices or pores such that diffusion through them is the same as that in free solution, the permeability is given by MDs*/a, where M is the void fraction in the membrane. Our inability to predict membrane permeability a priori reflects our ignorance of many factors that influence the transport of a given solute through a given membrane. Some of these factors, particularly applicable to transport through water-filled passages, are itemized below: — The void fraction mentioned above, or the fraction of the presented area of a membrane that is occupied by pores, are often unknown. — If the pores are not highly connected, their resistance to diffusion will depend on their tortuosity; if the solute must diffuse down a tortuous path, it will cross more slowly. — Transport depends critically on the diameter of the passage along the length of the diffusion path. The walls of pores give rise to a viscous drag that retards the diffusional process, and the degree of retardation remains significant for pores as large as ten times the solute diameter. This effect, which will be discussed in detail in Chapter 7, becomes greater when the diameter of the pore is closer to that of the solute. When the pore is only slightly larger than the solute, the latter must cross the membrane by single-file diffusion, and the augmented diffusion coefficient is no longer the

PRINCIPLES AND MODELS OF BIOLOGICAL TRANSPORT

35

appropriate measure of solute mobility. Of course, if the pore size is less than the solute size, then the solute does not cross the membrane at all, irrespective its aqueous diffusion coefficient. — In small pores, there is considerable opportunity for chemical and physical interactions between the solute and the walls. Such interactions include adsorption and binding. Furthermore, if the pore walls are charged, Donnan effects can cause the concentration of charged solutes inside the pore to be markedly different from the concentrations in the adjacent solutions. There can also be interactions between the solvent and the walls of the pore, which can cause the effective solvent viscosity to differ from that in free solution, thereby (see §2.1.4), affecting the solute diffusion coefficient. Once the permeability is specified, the transmembrane flux is predicted by Js = ks'cs.

(2.20)

Equation (2.20) is the expression most commonly used to describe the passive free diffusion of a nonelectrolyte across a barrier. It can also describe the transport of an ion in the absence of an electrical potential gradient. Some illustrative values of solute permeability are given in Table 2.1.

Table 2.1. Nonelectrolyte Permeabilities of Three Cell Membranes a

Ox erythrocyte Urea Glycol Diethylene glycol Glycerol

cm/s 7.8 q 10–5 2.1 q 10–6 7.5 q 10–7 1.7 q 10–8 b

Ehrlich ascites tumor cell (mouse) cm/s Galactose 2.1 q 10–6 3-methyl glucose 1.8 q 10–6 Sorbose 5 q 10–7 Glucosamine 2.5 q 10–7 Human erythrocyte Ethanol Glycerol Thiourea Urea a

c

cm/s 2.1 q 10–3 1.6 q 10–7 1.1 q 10–6 7.7 q 10–7

Davson and Danielli (1952). Crane et al. (1957). c From summary in Lieb and Stein (1986). b

36

CH. 2: FREE DIFFUSION

2.1.3. Unstirred Layers I

II

Assume that cs > cs ; by our conventions, 'cs > 0, Js > 0, and solute moves from Phase I to Phase II. Before a solute molecule can cross the membrane, it must first find its way from the bulk of Phase I to the membrane interface at x = 0. Two mechanisms are available to accomplish this: diffusion, which is described by equations like those given above, and convection, in which fluid motion carries solute from the main body of the phase to the proximity of the membrane. If one of the phases is the inside of a cell, convection is limited, and exchange between the bulk of the cytoplasm and the membrane interface is largely by diffusion. Similarly, diffusion is usually the dominant transport mechanism in the extracellular space on the other side of the cell membrane. In many experimental situations, however, convection can be introduced by stirring. The influence of stirring does not extend unattenuated to the membrane–solution interface; a thin, effectively unstirred layer adjacent to the membrane remains. Solute crosses this layer only by diffusion, and its flux is properly defined by Eq. (2.18); at x = 0,

Js 

DsI I (csb  csI ) , I G

(2.21) i

i

where Ds is the solute diffusion coefficient in the ith phase, G is the thickness of the corresponding unstirred layer, and the subscript “sb” denotes the bulk phase conceni tration of the solute; cs is the solute concentration at the interface between the membrane and the ith phase. For nonideal solutions, the augmented diffusion coefficient would be used in place of Ds. Similar considerations apply to the transport of solute from x = a to the bulk of Phase II. The solute concentration profile is shown in Figure 2.1. Three resistances in series separate the two bulk phases. The solute flux is equal to the overall concentration difference divided by the sum of these resistances, each of which is inversely proportional to a permeability: Js 

csbI  csbII . GI / DsI 1/ ks GII / DsII

(2.22)

Here, ks is the true permeability of the membrane. The apparent permeability of the membrane, Js /'csb, is the reciprocal of the denominator in the preceding equation. The characterization of the unstirred layer (or diffusion layer, as it is also known) as a sharply defined boundary layer containing all of the diffusional resistance outside the membrane is clearly an approximation. When the bounding phase is unstirred, there are concentration gradients throughout. In the presence of stirring, convective effects are absent at the membrane–solution interface and increase with distance from the membrane surface. Notwithstanding the limitations of the unstirred layer concept, it does provide a convenient means for including diffusional resistances outside the membrane in the equation for solute flux, and for characterizing the magnitude of such resistances. The quantity Gi can be regarded as the thickness of a layer of the external phase whose resistance to diffusion is the same as that actually present outside the membrane.

PRINCIPLES AND MODELS OF BIOLOGICAL TRANSPORT

37

Figure 2.1. Concentration profile in the presence of unstirred layers, and in the absence of solvent I II flow. The actual transmembrane concentration difference, cs – cs , is less than the overall concenI II tration difference, csb – csb .

It can be seen from Eq. (2.21) that cs approaches csb as the thickness of the unstirred layer approaches zero; otherwise, the solute flux would become infinite. When these two concentrations are assumed to be identical (an assumption that is often made in practice, and will be made liberally in the chapters to follow), the phase is said to be well-stirred. Although vigorous stirring can reduce the effective thickness of the unstirred layer, it cannot be reduced to zero; the well-stirred assumption is always an approximation. The effects of a variety of stirring motions on solute flux are analyzed in Pedley (1983). In the presence of unstirred layers, the concentration difference driving the transmembrane flux is less than the difference between the bulk phase concentrations (see Fig. 2.1). Solute permeabilities calculated using the latter driving force can be seriously underestimated if the resistance of the unstirred layers is an important fraction of the total interphase resistance. This is more likely to be the case if the membrane permeability is high. The neglect of unstirred layer effects can also lead to errors in the calculated parameters of carrier-based transport systems (Chap. 4). The diffusion coefficients of small solutes in the cytoplasm are not known very well, so it is difficult to make good estimates of the errors in cell membrane permeability caused by intracellular diffusional resistance. In such cases, it is common to

38

CH. 2: FREE DIFFUSION

assume that there is no diffusional resistance on the cytoplasmic side. The true cell membrane permeability is underestimated when this approach is used; however, permeabilities that are derived in this way can be compared with the permeabilities of other solutes derived similarly, or used to predict flux, as long as the bulk cytoplasmic concentration of the solute is used in the flux equation. We will see in Chapter 6 that, for many solute/membrane combinations, a transmembrane concentration difference induces a solvent flow, termed osmosis, in the direction of the more concentrated solution. The solute concentration profile in the unstirred layer is curved when osmosis (or any transmembrane solvent flow) is present. The effect of unstirred layers on transport in the presence of osmosis is discussed in Chapter 10. Equation (2.22) describes the steady-state transport of solute across a series of resistances, for the case in which two of the resistances are unstirred layers and the third is the membrane itself. The G/Ds ratios in the equation are simply the reciprocals of the permeabilities of the individual unstirred layers. Equation (2.22) can be regarded as a transport equivalent of Ohm's Law for the voltage-driven current through a number of resistors in series; here, the voltage is replaced by the bulk concentration difference, the current by solute flux, and the ohmic resistances by the reciprocals of the permeabilities of each barrier. This description of the flux through series barriers can be applied to many biological transport processes, such as transport though a single layer of cells, where solute enters across one face of the cell, crosses the cytosol, and then exits across the other face; transport through a cell supported by a permeable layer of extracellular matrix; or transport through a series of cell layers, as in epithelia (Chap. 10). In such cases, the general equation for solute flux is Js 

'cs , 1 œ i 1 k si m

(2.23)

where 'cs is the overall concentration difference and ksi is the solute permeability of the ith of m barriers. As above, the reciprocal of the denominator of Eq. (2.23) is the apparent permeability of the composite barrier.

2.1.4. Applications of Solution Theory A considerable body of theory has been developed to describe free diffusion in solution. Most of this theory cannot be directly applied to biological systems, for reasons that have already been presented. One applicable product of solution theory is the Stokes–Einstein equation, which identifies the variables that have the greatest influence on the diffusion coefficient. In general, the diffusion coefficient depends on the solute (naturally), the solvent, the concentration of the solute (or composition, for a multicomponent solution), and temperature. Einstein (1908) used Stokes' Law to derive the following approximate expression for the diffusion coefficient of a spherical solute:

PRINCIPLES AND MODELS OF BIOLOGICAL TRANSPORT

Dx

39

RT , 6 SKsN

(2.24)

where K is solvent viscosity, s is solute radius, and N is Avogadro's number. Stokes' Law describes the drag on a sphere moving through a homogeneous fluid of infinite extent. Implicit in this application of Stokes' Law are the assumptions that solute molecules are much larger than those of the solvent, and that the influence of the solution boundaries (e.g., the walls of a pore) is negligible. Equation (2.24) shows that the most important solute property affecting the diffusion coefficient is its size (and shape; the equation is more complex for nonspherical solutes), and the most important solvent property is its viscosity. Equation (2.24) predicts that the diffusion coefficient is inversely proportional to the solute radius; that is, the sD product is constant. This condition is met by the data in Table 2.2, even though the solute molecules are not much larger than those of the solvent. In biological systems, this simple inverse relation applies only to diffusion through large passages. When the size of the pore is not much greater than that of the solute, the permeability depends on pore radius as well as solute radius. The effect of pore size on solute permeability will be discussed in Chapter 7.

Table 2.2. Test of the Stokes–Einstein Equation

a

Solute

Solute a radius, s nm

Diffusion coefficient in aqueous solution a at 25ºC, D 2 cm /s

sD q 105 nm-cm2/s

Methanol Urea Glucose Glycerol Sucrose Raffinose

0.20 0.24 0.39 0.31 0.45 0.58

1.3 q 10–5 1.16 q 10–5 –6 6.8 q 10 8.3 q 10–6 5.5 q 10–6 4.2 q 10–6

0.26 0.28 0.26 0.27 0.25 0.24

Data from Schafer and Barfuss (1980).

The predicted effect of solvent viscosity on the diffusion coefficient has often been used to interpret and extrapolate experimental permeability data. From the Stokes–Einstein equation, the diffusion coefficient is expected to vary inversely with solvent viscosity. If the temperature dependence of permeability parallels that of the reciprocal of the viscosity of water, this is taken as evidence that the solute crosses the membrane via water-filled pores. The permeabilities of other diffusional transport routes (e.g., across the lipid phase of the cell membrane) are considerably more sensitive to temperature than is the permeability of an aqueous pore. Similarly, if it is known that a solute uses an aqueous pore to cross a membrane, then the temperature

40

CH. 2: FREE DIFFUSION

dependence of the viscosity of water can be used to predict the solute permeability at one temperature from the measured permeability at a different temperature.

2.1.5. Fick's Second Law and Convective Diffusion Fick's first law is one of the equations most commonly used to describe biological transport by free diffusion. It can readily be generalized to any coordinate system: Js = –Ds ’cs.

(2.25)

where Js is the flux vector in three-space. In the steady state, the law of mass conservation applied to the species s is 

’• Js = 0.

(2.26)

’• (Ds ’cs) = 0.

(2.27)

Substituting Eq. (2.25) into (2.26), 

Equation (2.27) is the steady-state form of Fick's second law of diffusion, also known as the diffusion equation. When the diffusion coefficient is uniform, the equation simplifies further to 2



’ cs = 0.

(2.28)

The diffusion equation has been solved in numerous geometries, for a wide variety of boundary conditions. Table 2.3 summarizes some useful forms of the steady-state diffusion equation.

Table 2.3. Some Forms of the Steady-State Diffusion Equation 1. Cartesian coordinates (x,y,z) d 2c a) 1-dimensional: Ds 2s  0 dx  s 2 cs s 2 cs s 2 cs ¬­ b) 3-dimensional: Ds žžž 2 2 2 ­­  0 žŸ sx sy sz ®­ 2.

Cylindrical coordinates (r =radial coordinate, z =longitudinal coordinate, no azimuthal variation) Ds d ž dcs ¬­ a) r-variation only: ­ 0 žr r dr žŸ dr ®­ b) r- and z-variation, different diffusion coefficients in r- and z-directions: Dsr s ž scs ¬­ s2c ­ Dsz 2s  0 žžr ­ r sr Ÿ sr ® sz

3. Spherical coordinates, r-variation only:

Ds d ž 2 dcs ¬­ ­ 0 žr r 2 dr žŸ dr ®­

PRINCIPLES AND MODELS OF BIOLOGICAL TRANSPORT

41

An additional contribution to the solute flux arises if the solution itself is moving. Let us return to the one-dimensional case described by Eq. (2.9). If the solution is flowing in the x-direction at a velocity v, the solute flux is augmented by a convective term equal to the product of the solution velocity and the local concentration: J s  Ds

dcs vcs . dx

(2.29)

The three-dimensional equivalent of Eq. (2.29) is Js = –Ds ’cs + csv,

(2.30a)

where v is now a vector. Substituting Eq. (2.30a) into Eq. (2.26), the equation for steady-state convective diffusion with a uniform diffusion coefficient becomes Ds ‹2 cs  v<‹cs  cs ‹v  0 . Biological solutions can be regarded as incompressible, and it can be shown that incompressibility implies ‹v = 0, so the steady-state convective diffusion equation becomes Ds ‹2 cs  v<‹cs  0 .

(2.30b)

When v is uniform (as it would be, in the one-dimensional case), Eq. (2.29) can be integrated to give an expression relating flux, the concentration boundary conditions, and velocity. It is easy to add a solute convection term to the more general forms of the diffusion equation given in Table 2.3, but it is not easy to solve the equations that result. Numerical simulation is usually required. The use of Eq. (2.26) to describe mass conservation in the steady state implies that the diffusing species is neither produced nor consumed in the region of interest. This assumption will generally apply throughout this text. In the Appendix to Chapter 10, the one-dimensional convective diffusion equation [Eq. (2.29)] will be extended to include changes in species concentration resulting from fluxes across the region boundaries. In Chapter 11, Fick's second law, with [Eq. (2.30b)] and without [Eq. (2.27)] convection, will be generalized to include chemical reactions within the region and time-dependent behavior.

2.1.6. Justification of the Steady-State Assumption: Time Scales in Biological Transport Virtually all of the transport processes described in this text are steady-state processes; that is, the concentrations in the system — both the external boundary conditions and the conditions inside the barrier — are assumed to be independent of time. When there is neither production nor consumption of the species of interest inside the membrane, mass conservation implies that the steady state flux satisfies Eq. (2.26). In the common one-dimensional description of membrane transport, Eq. (2.26) becomes

42

CH. 2: FREE DIFFUSION

simply dJs /dx = 0; that is, the flux based on a unit membrane surface area is the same at every cross-section in the membrane. We used this fact to integrate Fick's first law, and we will use it again. Another implication of the steady-state assumption is that the flux is constant in time. But this creates an apparent contradiction: how can the boundary conditions remain constant in the face of a perpetual flux? Clearly, they cannot, and the easy fix is to postulate, at least for the purposes of analysis, that the bounding solutions are infinite in extent, so their compositions do not change even when solute is lost or gained. Infinite systems are convenient to postulate but rare in the real world. Happily, most biological membranes do experience a relatively stable milieu because of homeostasis — the tendency of living systems to maintain a constant “internal environment,” which includes, for instance, the composition of the extracellular fluid that defines the boundary conditions for solute transport into and out of cells. The maintenance of reasonably constant boundary conditions for biological transport is thus accomplished by other agencies (such as the kidneys) outside the system under study. Of course, living systems do experience changes in their environment that cause changes in transport rates; these may reflect a failure of homeostatic mechanisms or a sudden challenge to the system occasioned externally or by the behavior of the organism. Some biological processes, such as regulatory events, are inherently dynamic. In such cases, the boundary conditions for transport cannot be regarded as constant, and steady-state solutions would no longer seem to apply. The proper approach to describing transport under these circumstances depends on the rate at which the boundary conditions change. If they change slowly compared to the rate at which the transport process can adapt to that change, transport can be regarded as quasisteady; that is, the transport rate at any time is equal to the steadystate flux corresponding to the boundary conditions at that time. If the boundary conditions change more rapidly, a full unsteady-state solution of the diffusion equation within the membrane is necessary. The time for the transport rate to adapt to changing boundary conditions is the time needed for the concentration profile in the membrane to change to the profile appropriate to the new boundary condition. Using a Fourier series solution in slab geometry, Weiss (1996a) obtained a time constant, td = a2/(S2Ds), for the approach to the steady state inside a homogeneous membrane with an arbitrary initial concentration profile, exposed at t = 0 to new concentration boundary conditions at each face. As might be expected, td is shorter for thinner membranes and for more rapidly diffusing solutes. One transport process for which the boundary conditions depend on time is the diffusion of a solute into or out of a closed compartment, such as a cell. The rate at which the concentration in the cell changes is related to the solute flux and the surface area and volume of the cell. Assume the interior of the cell is well-mixed and is Phase II, so flux into the cell is positive. The rate of change of the number of moles of solute in the cell, nsII , is given by dnsII  JsS , dt

(2.31)

PRINCIPLES AND MODELS OF BIOLOGICAL TRANSPORT

43

where S is the surface area of the cell. The concentration of solute in the cell is equal II II to the number of mols of solute per unit volume: cs = ns /V, where V is the cell volume (we assume for simplicity that all portions of the cell are accessible to the solute). Combining these two equations and Eq. (2.17), with Ds instead of Ds*, we obtain V

dcsII Ds (csI  csII )S  . dt a

(2.32) I

Eq. (2.32) is readily integrated for a constant ambient concentration, cs . The intracellular concentration follows a decaying exponential in time, with a time constant tc = aV/(DsS). The time constant is shorter when the solute passes through the membrane more readily (low thickness, high diffusion coefficient) and when the surface-tovolume ratio of the cell is large. 2 The quasisteady approximation is appropriate if td << tc , or a << S V/S. Interestingly, the diffusion coefficient does not appear in the criterion, because it affects the time constants of both processes similarly: even as a high membrane diffusion coefficient allows the concentration profile inside the membrane to adapt more quickly, it also causes the intracellular concentration to change more rapidly. Generally, the quasisteady approach has proven adequate for describing biological transport in the presence of changing boundary conditions. It is an important assumption underlying time-dependent applications of compartmental analysis, a modeling technique for complex systems that will be discussed in Chapter 8. Of course, changes in the boundary conditions are not the only tool by which living systems elicit changes in flux. As we shall see, such changes — particularly, rapid changes — are in most cases obtained by altering the transport properties of the membranes themselves.

2.2. FREE DIFFUSION OF ELECTROLYTES The free diffusion of electrolytes is considerably more complex than that of nonelectrolytes. The basic flux equation for electrolytes is the electrodiffusion equation. This nonlinear equation is soluble, but the general solutions are so complex that they have rarely been applied to biological systems. A general solution of the electrodiffusion equation, and a number of special cases, are given below.

2.2.1. Differences between Electrolyte and Nonelectrolyte Diffusion There are two principal differences between the diffusion of electrolytes and nonelectrolytes: 1. Charged solutes are subject to electrical forces when electrostatic potential gradients are present. Accordingly, the driving force for electrolyte transport is the gradient of the electrochemical potential rather than that of the chemical potential. 2. Since any electrolyte solution must contain at least one anion and one cation, there are always at least two solute species. The existence of mul-

44

CH. 2: FREE DIFFUSION

tiple species — and, correspondingly, multiple fluxes — leads to two concepts that arise only when electrolyte transport is considered. The first of these concepts is electroneutrality: the concentration of positive charges in a small sample volume equals the concentration of negative charges. This condition can be written as follows: œ zi ci  0 .

(2.33)

i

The second concept is ionic current. Ions moving in solution carry current just as electrons do in metal conductors. The contribution of each species to the current density is equal to the product of the species' flux and its charge. The current density is obtained by summing these contributions: I  œ I i  œ zi J i . i

(2.34)

i

where Ii = zi Ji is the contribution of the ith ion to the current. Note that the units of I as given above are mols of charge per square centimeter of transport area per second. If the right-hand side of Eq. (2.34) is multiplied by the Faraday, the units become coulombs per square centimeter per second; that is, amperes per square centimeter.

2.2.2. The Electrodiffusion Equation The flux of the ith ion in free solution, like that of the nonelectrolytes in the preceding section, is equal to the product of the mobility of the ion, its concentration, and the appropriate driving force, which in this case is – dP i / dx . The driving force can be written in terms of the chemical and electrostatic potential gradients: 

dP i dP d\   i  zi  . dx dx dx

(2.35)

The chemical potential gradient is treated as in the previous section, and the flux equation becomes

 RT dci dc d\ d \ ¬­ ­­  Ui RT i  Ui ci zi   zi J i  U i ci žžž . žŸ ci dx dx dx dx ®­

(2.36)

Equation (2.36) is the electrodiffusion equation, which is the most common starting point for describing free diffusion in electrolyte solutions. It is also known as the Nernst–Planck equation. As in the analysis of the Donnan equilibrium, concentrations are used rather than activities, to facilitate the use of the electroneutrality condition in solving the equation. As a consequence, the solutions of this equation neglect direct ion–ion interactions during the transport process, and the analysis to follow strictly holds only for solutions more dilute than those found in living systems. The electrodiffusion equation defines the dependence of ionic flux on the gradients in concentration and electrostatic potential in the membrane or barrier across which transport takes place. These gradients are not generally measurable. It is therefore desirable to integrate the equation, so that the fluxes can be related to the conditions at the membrane surfaces.

PRINCIPLES AND MODELS OF BIOLOGICAL TRANSPORT

45

The results of this integration will reveal a much more complex dependence of flux on boundary conditions than was the case when nonelectrolytes were considered. In dilute solutions, the flux of an uncharged species depends on only the concentration of that solute in the bathing solutions; this is so even when other solutes are present. In electrolyte solutions, however, the flux of each ion depends on the concentrations of all ions in the bounding solutions, and not in a simple fashion. In addition, for any particular pair of bounding compositions, the fluxes, and hence the transmembrane current density, depend on the transmembrane potential difference. This is illustrated in Figure 2.2a. Shown in Figure 2.2b are two common experimental situations. 1. Short Circuit. Electrodes in the two bathing solutions are connected by an external circuit, “shorting out” the membrane and bringing the potential difference across it to zero. The current density measured under shortcircuit conditions is called the short-circuit current. Passive ion fluxes at short circuit are driven by only concentration gradients and can therefore be described by the same equations as are used to describe the flux of nonelectrolytes. Accordingly, ionic fluxes at short-circuit can be expressed in terms of the membrane permeabilities of the ions, following Eq. (2.20). It is easy to calculate the short-circuit current from the membrane properties and bounding compositions, since at short-circuit the fluxes of the ions are independent of one another and the flux of each can be calculated from Eq. (2.20). 2. Open Circuit. At open circuit, there is no net transport of charge across the membrane; that is, I = 0. The zero-current condition is more typical of unmanipulated biological systems. The external path between the two sides of the barrier in Figure 2.2b does not ordinarily exist, and electroneutrality demands that equal amounts of positive and negative charge cross the barrier. At open circuit, the quantity of experimental interest is the potential difference across the membrane. The term “diffusion potential” is also used to describe the potential difference that develops across a membrane when the current is zero and all flux is passive. Under certain assumptions, the electrodiffusion equation can be integrated to compute the open-circuit potential and fluxes from the compositions of the solutions on the two sides of the membrane and the ionic mobilities within it; the more general solutions are presented first. Before proceeding, it should be remarked that many cellular and intracellular membranes contain active transport systems that generate a net ionic flux and corresponding active ionic current, I a . The zero-current condition in this case is I a I  0 , where I is the passive current described by the electrodiffusion equation.

46

CH. 2: FREE DIFFUSION

Figure 2.2. (a) A solution of the electrodiffusion equation, for an uncharged membrane at 25ºC. The composition of the solution in Phase I is 68 mM NaCl, 15 mM KHCO3, and 68 mM RCl, where R is a large cation whose mobility is one-tenth that of Na; in Phase II, the RCl has been replaced by NaCl. The mobilites of the ions are taken from Table 2.4, and the membrane is modeled as an aqueous film 1 cm thick. Dashed lines denote the short-circuited ('\ = 0) and open circuited (I = 0) conditions. After Friedman (1970). (b) Short-circuit and open-circuit conditions. In the former, the ammeter measures the short-circuit current; in the latter, the voltmeter measures the opencircuit potential difference.

PRINCIPLES AND MODELS OF BIOLOGICAL TRANSPORT

47

2.2.3. Integration of the Electrodiffusion Equation The integration of the electrodiffusion equation is complicated by the nonlinearity of the equation. The nonlinearity arises from the second term on the right-hand side, because the ionic concentration and potential gradient are functions of location in the membrane. The second term was absent when nonelectrolytes were considered. In the late nineteenth century, a number of investigators, including Planck (1890) and Behn (1897), reported solutions of the electrodiffusion equation. We will not reproduce these derivations here, limiting ourselves instead to the assumptions and final result. Both investigators made the same assumptions: 1. There are n ions in the system, and Eq. (2.36) holds for each of them; the mobility of each ion is independent of position (x) or local composition. The phases bounding the two faces of the membrane, whose thickness is a, are denoted Phase I and Phase II. The concentration of the ith ion in the j jth phase is denoted ci . 2. All ions are univalent. This restriction can be omitted, but the solution is more complicated when the valences of the ions are not all the same. 3. At every point in the membrane, the local composition is electroneutral. This assumption is strictly false whenever the electric field E = –d\/dx is nonuniform, but the deviation from electroneutrality is almost always trivial. The relation between the nonuniformity of the electric field and the departure of the solution from electroneutrality arises as follows: a volume of solution that is not electrically neutral contains a net charge called the space charge, U, whose local concentration is equal to 6i zici. Thus, when the electroneutrality condition [Eq. (2.33)] is satisfied, U = 0. The space charge concentration is related to the gradient of the electric field through the Poisson equation: dE/dx = ( / H )U , where H is the permittivity of the barrier. Thus, the electroneutrality assumption is strictly correct (i.e., U = 0) only when the field is uniform (i.e., dE/dx = 0). When electroneutrality is assumed and the electrodiffusion equation is solved accordingly, the calculated electrostatic potential \ is not generally a linear function of x; hence the field is not uniform and the electrolyte solution cannot be electrically neutral. For biologically relevant boundary conditions, this inconsistency is unimportant. Using the electric field gradient obtained by solving the electrodiffusion equation with the electroneutrality assumption, the space charge density can be computed from the Poisson equation. This value of U is inevitably orders of magnitude less than the concentration of the electrolyte solution itself.

4. The system is in the steady state, so all ionic fluxes are independent of x. The Planck solution gives a transcendental expression for the membrane potential I II '\ = \ – \ as a function of the bounding compositions. The open-circuit potential is obtained by solving

48

CH. 2: FREE DIFFUSION

NI  ln[ [U  U [N I  N II N II  ¸ V I  [V II NI N I  [N II ln II ln[ N I

II

ln

(2.37)

for [, where [ = exp('\/RT),

U j  œ U i cij , V j  œ Ui cij , cations

anions

j

and N is the total concentration of the jth phase, defined as n

N j  œ cij . i 1

The Behn solution provides the fluxes as well as the membrane potential, and consists of a set of n + 1 equations that are solved simultaneously: II   ¬­ žž ­ '\  g a ln N , žŸ RT ®­ NI

J i  ( g azi  1) ¸

U i RT ( N II  N I )(ciII e - zi'\ / RT  ciI ) (i = 1,…,n). ¸ a N II e - zi'\ / RT  N I

(2.38a)

(2.38b)

As before, flux from Phase I to Phase II is positive. I II If the compositions {ci } and {ci } are specified, and the mobilities are known, then Eqs. (2.38) constitute n + 1 equations in n + 2 unknowns: an unspecified constant, g; the membrane potential, '\; and n fluxes, {Ji}. Since there is one more unknown than there are equations, one of the unknowns, or a function of them, must be specified. Generally, this is either the membrane potential or the transmembrane current density. If the potential is specified, Eq. (2.38a) can be solved for gc, and Eq. (2.38b) gives the fluxes directly; a more difficult iterative procedure is required if the current is given and the membrane potential and fluxes are sought. Strictly speaking, the superscripts “I” and “II” denote the potentials and concentrations just inside the membrane faces. However, in applying the preceding solutions, the concentrations that are generally used are those in the bounding phases, and the membrane potential is measured in the external solutions as well. Often, no harm is done when this approximation is made, but errors can arise if a bounding phase contains charged species that cannot enter the membrane, or if the membrane contains fixed charges that cannot leave (Fig. 2.3). In such cases, the correct (i.e., intramembrane) boundary conditions for Eqs. (2.37) and (2.38) are related to the composition and potentials in the bathing solutions by the Donnan equilibrium expressions of the previous chapter. The solutions given above must be further modified when the membrane structure is charged, because the concentration of the charges on the membrane must be included in the electroneutrality condition (see §2.2.6).

PRINCIPLES AND MODELS OF BIOLOGICAL TRANSPORT

49

Figure 2.3. Boundary conditions for solutions of the electrodiffusion equation. In the example shown here, there is one mobile cation (C) and one mobile anion (A) in the system. Both bounding phases are well stirred. Phase I contains a negatively charged species that cannot enter the membrane, so there is a Donnan equilibrium at x = 0. Inside the uncharged membrane and impermeantfree Phase II, cC = cA, by electroneutrality. The correct boundary conditions for the electrodiffusion equation are those at the filled circles. The potential difference between Phases I and II equals the sum of the calculated membrane potential and the Donnan potential at the Phase I interface.

Before proceeding to the conditions under which simpler solutions of the electrodiffusion equation can be obtained, we should observe that the equation is linear in ci; therefore a partial integration can be carried out using an integrating factor. This procedure demonstrates that the ion flux is the product of three terms: UiRT, ciII exp(zi'\ / RT )  ciI , and the reciprocal of the integral of exp[zi\(x)] across the membrane. Since \(x) is generally not known a priori, this equation cannot give flux directly; however, the first term shows that the flux is proportional to the mobility of the ion, and the second demonstrates that the flux of an ion at equilibrium is zero (see also §2.2.4).

2.2.4. Some Special Cases Equilibrium. By setting Ji = 0 in Eq. (2.38b), we obtain the conditions under which the ith ion is in equilibrium across the membrane. The flux is zero when the last factor in the numerator is zero:

ciII e - zi'\ / RT  ciI  0 .

(2.39)

50

CH. 2: FREE DIFFUSION

Figure 2.4. Potential-driven current, uniform composition. The electron flow in the external circuit and the current across the membrane are both directed from Phase I to Phase II because the transmembrane current is defined as the flow of positive charge [Eq. (2.34)].

It is easy to show that Eq. (2.39) prescribes that the ith ion is in equilibrium when its Nernst potential is equal to the membrane potential. This conclusion was also reached in Chapter 1. Uniform Composition. The solution of the electrodiffusion equation proceeds more directly when the compositions of the solutions on the two sides of the membrane are the same, and the composition inside the membrane is consequently uniform. When a potential is applied across the membrane, the ions migrate across, driven by the electric field, and generate an ionic current; this situation is illustrated in Figure 2.4. The concentration gradient of each ion is zero, so Eq. (2.36) becomes:

 d\ ¬ J i  Ui ci zi žž ­­­ . žŸ dx ®

(2.40)

Since ci is independent of x, Eq. (2.40) can easily be integrated to give the ionic flux as a function of the imposed potential difference:

Ji 

U i ci zi '\ . a

(2.41)

Each flux is proportional to the potential difference across the membrane. It follows trivially from Eq. (2.34) that the membrane current density is similarly proportional to the membrane potential; hence, when the composition is uniform, the membrane obeys Ohm's Law. If the fraction of membrane area available for transport is M, and the transport paths are sufficiently large that the effects of the pore walls on the transport rate can be neglected, the conductance of the membrane is 2

g (mols/cm -sec-V) w

I M  œ Ui ci zi2 . a i '\

PRINCIPLES AND MODELS OF BIOLOGICAL TRANSPORT

51 –1

In electrical units, conductance is measured in siemens; 1 S = 1 ohm = 1 am2 pere/volt. The membrane conductance in S/cm is obtained by multiplying the previous value of g by . If, in Eq. (2.40), the electrostatic potential gradient is regarded as the driving force for transport, then, according to the Teorell equation, the product Ui zi assumes the role of a mobility. Indeed, the absolute value of this product is termed the electrical (or electrophoretic) mobility of the ion, ui = Ui |zi|. The electrical mobilities of several biologically important ions are presented in Table 2.4.

Table 2.4. Electrical Mobilities of Several Biologically Important Ions at 25ºC (Robinson and Stokes, 1965; Davies, 1968) Electrical mobility, cm2/sec-V, ui x 104

Ion H Li Na K NH4 Mg Ca Cl NO3 HCO3

36.25 4.01 5.19 7.62 7.62 5.50 6.17 7.92 7.41 4.62

Diffusion Potential of a Bi-Ionic System. An explicit solution for the diffusion potential can be obtained from Eqs. (2.38) if the system contains only one anion and one cation, of equal charge. In this case, the anion and cation fluxes are equal, since the current is zero, and the membrane potential is

'\ 

RT UC  U A c II ln , ¸  UC U A c I

(2.42)

where c is electrolyte concentration and the subscripts on the mobilities denote the cation and anion. The diffusion potential is independent of membrane thickness, and depends on only the mobility ratio UC /UA (divide numerator and denominator II I by UA to see this) and the concentration ratio c /c . The origin of the diffusion potential is easiest to explain for this case in which only two ions are present. If the membrane is permeable to both the anion and cation of a salt whose concentration is different on each side of the membrane, both ions will cross. The ionic fluxes must be equal when the current is zero, even though the mobilities of the two ions are not generally the same. The diffusion potential develops to compensate for this difference in mobility by increasing the electrochemical potential driving force for the ion having the lower mobility, and decreasing that for the more mobile ion. The potential “pulls” the less mobile ion across the membrane, while re-

52

CH. 2: FREE DIFFUSION II

I

tarding the flux of the more mobile species. Suppose c > c and UC >UA. Then the salt diffuses from Phase II to Phase I under its concentration gradient, and the potential of Phase I becomes positive relative to that of Phase II (i.e., '\ > 0), so as to retard C and increase the driving force for A. When UC = UA, there is no mobility difference to compensate for, and the diffusion potential is zero. Diffusion potentials can cause artifacts in certain electrophysiological experiments, and it is desirable to avoid them. Much use is made of concentrated KCl solutions (salt bridges) in such setups, because the mobilities of potassium and chloride are almost identical. Consider the other extreme, in which the mobility of one ion is much larger than that of the other, say UC >> UA. In this case, the membrane potential given by Eq. II I (2.42) approaches (RT/) ln c /c . Since there is only a single electrolyte in the system, II I the argument of the logarithm is also cC /cC ; thus the membrane potential approaches the Nernst potential of the cation. We shall see that the tendency of the membrane potential to approach the Nernst potential of the more (or most) permeable ion is evident for more complex electrolyte solutions as well. This tendency has been exploited experimentally to “clamp” the membrane potential at a selected value by bathing the two sides of the membrane with solutions containing different concentrations of an ion (often potassium) to which the membrane is particularly permeable. Active and Passive Exchange with a Closed Compartment. In the steady state, the net rate of entry of any species into a closed compartment equals the rate at which it is consumed; otherwise, its concentration in the compartment would change with time. Similarly, when a compound is synthesized in a closed compartment, the synthesis rate (less any consumption of the material inside the compartment) equals the rate at which the substance leaves the compartment. When the solute is neither consumed nor produced within the compartment, its net entry rate must be zero in the steady state. This is the case for most ions. As noted earlier, cell membranes are capable of actively transporting (“pumping”) ions between the interior of a cell and the extracellular fluid. Suppose that two cationic species with the same valence, z, are exchanged across the cell membrane, such that, for each ion of Species 1 that is pumped from Phase II to Phase I, r ions of Species 2 are pumped from Phase I to Phase II. In the steady state, the net rate at which each ion crosses the cell membrane — the active flux plus the passive flux — must be zero. Hence the passive flux of each ion is the negative of its active flux. Thus, for each ion of Species I moving passively from Phase I to Phase II, r ions of Species 2 move passively from Phase II to Phase 1:

J2 = –rJ1,

(2.43).

where r is the coupling ratio or coupling coefficient of the pump. Substituting Eq. (2.43) into Eq. (2.36), and rearranging,

U 2 RT Solving for d\/dx,

dc2 dc d\ d\  U 2 c2 z  rU1 RT 1 rU1c1 z . dx dx dx dx

(2.44)

PRINCIPLES AND MODELS OF BIOLOGICAL TRANSPORT

53

 ¬ ž rU dc1 U 2 dc2 ­­ d\ RT žžž 1 dx dx ­­­ .  žž dx z ž rU1c1 U 2 c2 ­­­ žžŸ ®­­

(2.45)

The function in the parentheses on the right-hand side is equal to

d (rU1c1 U 2 c2 ) d ln(rU1c1 U 2 c2 ) dx ;  rU1c1 U 2 c2 dx hence,

d\  

RT d < ln(rU1c1 U 2 c2 ) > . z

(2.46)

Integrating across the membrane, and letting z = 1,

'\ 

RT U 2 c2II rU1c1II . ln  U 2 c2I rU1c1I

(2.47)

This equation, which is obtained directly from the electrodiffusion equation, was used by Mullins and Noda (1963) to relate the membrane potential of skeletal muscle to the stoichiometry of active Na–K exchange across the muscle fiber membrane. Equal Total Concentrations on the Two Sides of the Membrane: The Constant-Field Equation. Even though the concentrations of individual ions vary widely in the body, the total ionic concentration, N, is quite uniform throughout (Table 2.5). The intracellular ionic content is only 10% less than the extracellular value, so the I II solution of the electrodiffusion equation for N = N is of some interest. This solution is also simpler — and much more frequently used — than the more general solutions given earlier.

Table 2.5. Typical Ionic Content of Intracellular and Interstitial Fluids, and Blood Plasmaa

a b

Ion

Plasma

Na K Ca Mg Cl HCO3 Phosphates SO4 Lactate Total

142 4 1 1 108 24 2 <1 1 283

Interstitial fluid 139 4 1 1 108 28 2 <1 1 284

Concentrations are in mM. Adapted from Guyton and Hall (2000). Includes larger molecules to which phosphate groups are attached.

Intracellular fluid 14 140 <<1 20 4 10 65b 1 2 256

54

CH. 2: FREE DIFFUSION

The solution for this special case cannot be obtained directly from Eqs. (2.38), which become indeterminate. We begin the derivation by using Eq. (2.36) to construct two sums: n

œ i 1 n

œ i 1

n  ¬ Ji dci d\ n ci zi ­­­ ,  žž RT œ  œ ž Ui dx dx i1 Ÿ ®­ i 1

(2.48)

n  ¬ J i zi dc d\ n ci ­­­ .  žž RT œ zi i  œ žŸ Ui dx dx i1 ®­ i 1

(2.49)

In Eq. (2.49), use has been made of the assumption, as was made in the Planck and 2 Behn solutions, that the ions are monovalent, so zi = 1. The predominance of monovalent ions is evident from Table 2.5. Consider each of the four sums on the right-hand sides of the two equations just written. The first sum on the right-hand side of Eq. (2.48) is simplified by interchanging the order of summation and differentiation:

dci

d

œ dx  dx œ c

i



dN . dx

(2.50)

The second sum on the right-hand side of Eq. (2.48) is zero, by the electroneutrality condition, Eq. (2.33). The first sum on the right-hand side of Eq. (2.49) is also zero, because it is the derivative of a quantity that is uniformly zero:

œz

i

dci d d0  œ zi ci  0. dx dx dx

(2.51)

The second sum on the right-hand side of Eq. (2.49) is N, by definition. Thus, Eqs. (2.48) and (2.49) can be rewritten:

œ

Ji dN ,  RT Ui dx

(2.52)

œ

J i zi d\ .  N Ui dx

(2.53)

These equations are not easy to solve in the general case, because the latter is nonlinear. However, the solution proceeds easily when N is the same on both sides of the membrane. First we recall that the steady-state flux is independent of x; if we assume that the ionic mobilities are also uniform, then the left-hand sides of Eqs. (2.52) and (2.53) are constants. Thus the right-hand sides must also be constants, independent of position in the membrane. From Eq. (2.52), dN/dx is constant, so N is a linear function of x. For the special case of interest here, N is the same at both sides of the membrane; therefore, it must be the same throughout. Since N is uniform and the right-hand side of Eq. (2.53) is constant, the electric field, –d\/dx, is also uniform. If the potential gradient d\/dx is the same everywhere in the membrane, it must be equal to

PRINCIPLES AND MODELS OF BIOLOGICAL TRANSPORT

55

(\ II  \ I ) / a  –'\/a. The electrodiffusion equation can then be written in the following form: J i  Ui RT

dci ž '\ ¬­ žU i zi  ¸ ­ ci . dx žŸ a ®­

(2.54)

The coefficient of ci in Eq. (2.54) is independent of x, so the equation can readily be solved for the flux as a function of the conditions on the two sides of the membrane. The result is one form of the Goldman–Hodgkin–Katz constant-field equation:

'\ ciI  ciII e -zi '\ / RT ¸ J i  ziUi ¸ . a 1  e -zi'\ / RT

(2.55)

The constant-field equation is the equation most commonly used to predict the ion fluxes across a membrane, and hence the membrane current, when the membrane potential and bounding compositions are specified. It clearly satisfies the equilibrium condition: the flux of an ion is zero if its Nernst potential equals the membrane potential. The equation can also be used to find the membrane potential when the current and bounding compositions are specified, but this generally requires a numerical solution. An exception is when the current is zero; in this case, the constant-field equation can be manipulated to predict the diffusion potential: œ Ui ciII œ Ui ciI RT cations anions . '\  ln | œ Ui ciI œ U i ciII cations

(2.56)

anions

As indicated, the sums in Eq. (2.56) include either all cations or all anions. Equation (2.55) can be written in nondimensional form, in which the nondimenJi a sional flux I  is a function of two nondimensional groups: a nondimenUi RTciI sional potential E  zi'\ / RT whose sign depends on the ionic charge, and the II I transmembrane concentration ratio C = ci /ci : I  E¸

1  CeE . 1  eE

(2.57)

Plots of I vs. E, parameterized by C, are shown in Figure 2.5. The ion flux is zero when the membrane potential equals the Nernst potential (in nondimensional units, E = ln C). As Eq. (2.55) indicates and Figure 2.5 demonstrates, plots of flux vs. potential are curved, so the membrane behaves as a rectifier: equal and opposite deviations from the Nernst potential do not in general induce equal and opposite ion fluxes. Although the electric field given by the electrodiffusion equation is independent I II of x only when N = N , the explicit solutions for flux and potential given above have seen considerably more use than the more unwieldy Behn or Planck solutions. As noted earlier, the use of the constant-field equation in many biological applications

56

CH. 2: FREE DIFFUSION

can be justified by the fortunate fact that N does not vary very much in living systems. What makes the constant-field equation more remarkable is its utility in spite of important differences between ion transport in real biological membranes and the continuum model implied by the electrodiffusion equation. These differences, which will become apparent in the chapters that follow, are more dramatic than a modest nonuniformity in N. An illustration of the use of the constant-field equation to interpret physiologic data is presented in the next subsection.

Figure 2.5. Nondimensional representation of transmembrane flux given by the Goldman– Hodgkin–Katz constant-field equation [Eq. (2.55)]. The variables are defined immediately preceding Eq. (2.57).

As was the case for nonelectrolyte transport, the ease with which an ion crosses a biological barrier is generally expressed in terms of its permeability, ki. The flux equations derived in §§2.2.3 and 2.2.4 — Eqs. (2.38), (2.41), and (2.55) — are based on free solution thermodynamics and are strictly applicable only to transport across a stagnant water film. For such transport, and neglecting nonideal effects, the permeability of the ith ion is related to its diffusion coefficient and mobility in free solution 0 by ki = Di /a = UiRT/a, analogous to the relationship for nonelectrolytes. And, as was the case for uncharged solutes, ion permeabilities in biological systems are experimental quantities, obtained by measuring the ion flux under known conditions and applying the flux equations presented above, with Ui replaced by kia/RT. With this substitution, the first two terms in Eq. (2.55) become ( zi'\ / RT )ki . An important application of the electrodiffusion equation is prediction of the relationship between ionic permeabilities and the membrane potential. In all such equations derived above — Eqs. (2.37), (2.42), (2.47), and (2.56) — the potential depends on the ratio of linear combinations of mobilities. In free solution, the mobility and permeability are proportional, with the proportionality constant RT/a. If there is a similar proportionality in biological membranes, it is easy to show that the mobilities

PRINCIPLES AND MODELS OF BIOLOGICAL TRANSPORT

57

in these equations can be replaced by permeabilities. This will be demonstrated in the following section.

2.2.5. Ionic Permeability and the Resting Potential of the Cell An electrostatic potential difference exists between the interior of biological cells and the extracellular fluid. Generally, the absolute value of this potential is below 100 mV, with the cell interior negative. Comparisons of the Nernst potentials of the primary biological ions — K, Na, and Cl — with the membrane potential show that Cl is generally close to equilibrium across the cell membrane, but the cations are not. The nonequilibrium state of the cations is maintained by active transport systems in the cell membrane that pump potassium ions into the cell in exchange for sodium ions, which are pumped out. The pump stoichiometry is such that the number of sodium ions pumped out exceeds the number of potassium ions pumped in; as a consequence, the pump generates an ionic current across the membrane. Equation (2.56) has been used to estimate the relative cation permeabilities of the cell membrane. As noted earlier, the total current across the cell membrane equals the active current due to the Na–K pump plus the passive current, which in this case is described by the constant-field equation. The total current across the membrane must be zero, or else charge accumulates in the cell. Thus, if there is an active current, the passive current cannot be zero. Equation (2.56) was derived under the assumption that the passive current is zero. Even though this is not generally the case for biological cells, the equation has been used to describe the dependence of the cell potential on ionic permeabilities, under the implicit assumption that the passive current is close enough to zero that its effect on the cell potential can be neglected. Considering only the three primary ions given above, and letting Phase I be the inside of the cell and Phase II the outside, Equation (2.56) becomes '\ r 

II U K cKII U Cl cClI RT U Na cNa ln , I  U Na cNa U K cKI U Cl cClII

(2.58)

where '\r is the cell potential. Since Cl is in equilibrium across the cell membrane, '\r = ECl: '\ r  

RT cClII RT cClI ln I  ln II .  cCl  cCl

(2.59)

Equating the arguments of the logarithms in the preceding two equations, and rearranging, II cClI U Na cNa U K cKII  . I cClII U Na cNa U K cKI

Substituting Eq. (2.60) into (2.59),

(2.60)

58

CH. 2: FREE DIFFUSION

'\ r 

II U K cKII RT U Na cNa ln . I  U Na cNa U K cKI

(2.61)

If the permeability and mobility of each cation are related by Ui = Dki, the argument of the logarithm becomes II DkNa cNa DkK cKII . I DkNa cNa DkK cKI

The proportionality constant cancels out, effectively replacing mobility by permeability. Dividing the numerator and denominator of the argument of the logarithm by kK,

'\ r 

 kNa ¬­ II žž ­ c c II K žŸ k ®­­ Na

RT . ln K  ž kNa ¬­ I I ž ­­ cNa cK žŸ kK ®­

(2.62)

Equation (2.62) has been used to estimate the Na/K permeability ratio from measurements of the resting potential and intracellular concentrations in solutions of known composition. Values for nerve and muscle at rest range from 0.01 to 0.2; the permeabilities of the two ions in the red cell membrane are closer to one another. The variation of resting potential with the permeability ratio is illustrated in Figure 2.6 for concentrations typical of a nerve fiber. Equation (2.62) is derived here as a special case of Eq. (2.56), which relies on the constant-field assumption. However, it can also be derived by setting r = 1 in Eq. (2.47), which does not assume a constant field. When the coupling coefficient is unity, there is no active current, so the passive current at open circuit is zero. Indeed, for this particular case, the constant-field assumption is unnecessary.

In this application of equations based on solution theory to transport in real biological systems, an important caveat must be stated. As we have already implied, ions and most other solutes do not cross biological membranes by diffusing down fluidfilled paths that can be regarded as simple extensions of the bounding solution into and through the membrane. Although the constant-field equation and others derived here from solution thermodynamics can describe with some success the effects of boundary conditions and membrane properties on fluxes and potentials, the parameters (such as permeability) that we derive to summarize the experimental results may have a very different physical origin than the same parameters when used to describe transport in simple solutions. The permeability ratio of a cell membrane tells us something about how readily various ions cross, but it tells us very little about the physical processes that accompany permeation.

PRINCIPLES AND MODELS OF BIOLOGICAL TRANSPORT

59

Figure 2.6. Effect of the sodium/potassium permeability ratio N on cell potential. The internal composition of the cell is 14 mM Na and 140 mM K, and the ambient solution is 142 mM Na and 4 mM K. T = 37ºC.

2.2.6. Charged Membranes The membrane matrix can contain dissociated polar groups and consequently possess a net charge. Other membranes, such as the membranes of biological cells, exhibit a surface charge that is due to exposed ionizable groups. We will discuss the first instance here, and the influence of surface charge in the following chapter. When extending electrodiffusion theory to membranes that contain free charges, the ionizable groups are assumed to be distributed uniformly within the barrier. This assumption is a good one for some systems, such as artificial membranes made from ion exchange resin, or gel-like extracellular structures, including certain connective tissues. It is less applicable to the charged cell membrane pores that we will be discussing in Chapter 4. The transport process in uniformly charged membranes is described by the model developed by Teorell (1935) and Meyer and Sievers (1936), and which is illustrated in Figure 2.7. When the membrane charge is known, the Donnan equilibrium condition can be used to compute the composition of the solutions just inside each membrane face; these compositions are the boundary conditions for the integration of the electrodiffusion equation across the membrane. The potential difference between the solutions at the two sides of the membrane is equal to the algebraic sum of the Donnan potentials at each face and the transmembrane potential difference obtained from the electrodiffusion equation.

60

CH. 2: FREE DIFFUSION

Figure 2.7. The Teorell–Meyer–Sievers (TMS) model. In the example shown here, a negatively charged membrane is bounded by two well-stirred solutions of the same electrolyte, CA. The concentrations cAI, cCI, cAII, and cCII—the boundary conditions for the electrodiffusion equation—are in Donnan equilibrium with the ambient phases. A typical potential profile is also shown, assuming C is more mobile than A, and I = 0. The difference \I – \II is obtained from the electrodiffusion equation.

The solution of the electrodiffusion equation is more complicated than before because the electroneutrality condition includes the fixed charge:

œ zi ci z X X  0 ,

(2.63)

i

where X is charge concentration and zX = ±1. The electrodiffusion equation was integrated by Behn (1897) for a uniformly charged membrane bathed by solutions all of whose ions have the same valence [Eqs. (2.38) were in fact obtained from Behn's original solution by setting X = 0]. The solution is given in Harris (1972); its complexity, coupled with uncertainties regarding both the fixed charge concentration in bio-

PRINCIPLES AND MODELS OF BIOLOGICAL TRANSPORT

61

logical membranes and the appropriateness of the electrodiffusion equation, have strongly inhibited its application to living systems. From a biological transport perspective, membrane charge is most relevant to the movement of ions through membrane channels. As already noted, the charge distribution in the membrane cannot be regarded as homogeneous for this transport mechanism, which will be discussed in Chapter 4.

2.2.7. Limitations of the Electrodiffusion Equation and Its Solutions The electrodiffusion equation and the equations derived from it have seen much use in the description of ion transport across biological membranes. It is important, when using these equations, to recognize their limitations and the limitations of conclusions and parameters obtained by applying them to biological data. These limitations depend on the particulars of the application of the electrodiffusion equation, and can be summarized as follows. The electrodiffusion equation as initially presented in Eq. (2.36) is already a simplification, since it is written in terms of concentration instead of activity; however, the fact that the activity coefficients of the major ions are far from unity has not limited its use as much as has its complexity. Used without further simplification, it can describe free diffusion in solution or through barriers in which the transport process can plausibly be described as free diffusion (i.e., a continuum process through large aqueous pores). In the case of transport through pores, those limitations noted in §2.1.2 that prohibit a priori predictions of nonelectrolyte permeability apply to electrolytes as well. For the special case of uniform total concentration, the electric field in uncharged barriers is uniform, and the constant-field equation applies. Small deviations in total concentration uniformity, of the order of those seen in biological systems, do not disqualify the constant-field equation. Since the constant-field equation derives from the electrodiffusion equation, the dilute-solution and large-pore restrictions continue to apply. Notwithstanding the former restriction, the constant-field equation is generally used to describe the transport of ions whose activity coefficients can be far from unity. Many biological transport pathways, such as the channels that ions traverse, are not crossed by free diffusion. The total concentration is neither uniform nor continuous at the membrane boundaries. The continuum assumption can fail too, so even the notion of concentration becomes a statistical concept. There are often multiple pathways in parallel, each selective for a different ion or class of ions. The kinetics of transport through such pathways will be discussed in the chapters to come. Notwithstanding their limited applicability to many biological transport processes, the equations of free diffusion have seen considerable use in describing such processes. They replicate many empirical features of biological transport, such as rectification and the dependence of membrane potential on permeabilities and ambient concentrations. Thus, they still provide a useful tool for correlating experimental data and predicting behavior under conditions not too dissimilar from those under which the experiments were carried out.

62

CH. 2: FREE DIFFUSION

PROBLEMS: CHAPTER 2 –5

1.

The permeability of urea in the ox erythrocyte membrane is 7.5 q 10 cm/s at –5 2 37ºC. The diffusion coefficient of urea in water is 1 q 10 cm /s at 20ºC. How thin would an unstirred layer have to be, at the temperature at which the ox erythrocyte data were obtained, for its resistance to be one-tenth that of the erythrocyte membrane?

2.

The erythrocyte membrane in the previous problem is isolated and put in a chamber to measure its urea permeability, as in Figure 2.1. The solutions on both sides of the membrane are stirred but a layer of thickness G remains at each face. Sketch the variation of the apparent membrane permeability as the unstirred layer thickness varies from 0 to 500 Pm. What are the values of apparent permeability at the two extremes of this range of thicknesses?

3.

The passive pathways for ions across cell membranes can sometimes be modeled as water-filled channels. The permeability of chloride across the red cell mem–4 brane has been estimated to be 2.5 q 10 cm/sec at 24ºC. If the effect of temperature on permeability were strictly a viscosity effect, what would the chloride flux be across short-circuited ('\ = 0) red cell membrane at 37ºC when 'cCl = 10 mM?

4.

A red blood cell can be modeled as a disc 8 Pm in diameter and 2 Pm thick. The lipid bilayer (Chap. 3) of the red cell membrane is about 5 nm thick. For diffusion through the bilayer, what is the ratio of the time constants td and tc? Can the quasisteady approximation be used to describe the fluxes across the bilayer that arise when the composition outside the cell is changed?

5.

Assuming a constant ambient concentration cs , integrate Eq. (2.32) from an iniII II II tial value of cs (0) = (cs )0 to obtain cs (t), and confirm that the time constant for the change in intracellular concentration is that given in the text.

6.

Assume that the permeability of a particular solute in a membrane, ks , is high enough that the quasisteady approximation applies. The membrane area is A, and I the membrane is bounded by two finite-sized compartments, one of volume V in II I which the initial solute concentration is (cs )0, and one of volume V in which the II initial solute concentration is (cs )0. Derive an expression for the time constant of the decay of the concentration difference between the two compartments.

7.

Urea is diffusing out of a spherical cell that is 10 μm in diameter, through large aqueous pores that occupy 5% of the cell surface. T = 37ºC. The diffusion coeffi–5 2 cient of urea in water is about 1.4 q 10 cm /s at 25ºC. The cell membrane is 5 nm thick. (a) Estimate the diffusion constant of urea at 37ºC.

I

PRINCIPLES AND MODELS OF BIOLOGICAL TRANSPORT

63

(b) Compute the time constant for the decay of the intracellular urea concentration at 37ºC. (c) How thick can the cell membrane be before the quasisteady assumption for urea diffusion fails (set td = 0.1tc)? 8.

Consider a water-filled membrane 100 μm thick. The electrostatic potential in the membrane rises quadratically from zero at one side of the membrane to 90 mV at the other side. Show that the space charge in the membrane is uniform throughout its thickness, and calculate its magnitude. The permittivity of water is about 7 –10 2 2 q 10 coul /N-m .

9.

Show using the Behn and constant-field solutions that when an ion is in equilibrium across a membrane, the flux of that ion across the membrane is zero. –4

2

10. The electrical mobility of the sodium ion in water is 5.19 q 10 cm /sec-V at 2 25ºC; what is its diffusion coefficient in cm /sec? 11. Explain in physical terms the dependence of membrane conductance on each variable (except the Faraday, which is just a conversion constant) in the expression for g that follows Eq. (2.41). Why does conductance depend on the square of each ion's charge? 12. A membrane 1 mm thick separates two 155 mM solutions of NaCl at 25ºC. It contains a square array of 100-μm diameter pores on 400-μm centers. What is the 2 conductance of 1 cm of this membrane? 13. (a) Derive Eq. (2.42) from the Planck and Behn solutions. (b) Use Eq. (2.42) to compute the diffusion potential across a membrane at 25ºC if Side I of the membrane is bathed by a 100-mM solution of NaX and Side II is bathed by 10 mM NaX. Perform this calculation for two values of the mobility of X: –4 2 (i) uX = 7.9 q 10 cm /s-V (chloride) –4 2 (ii) uX = 5.2 q 10 cm /s-V (same as sodium) (c) Compare the preceding results with the diffusion potential given by the constant-field equation [Eq. (2.56)]. (d) Repeat (b) and (c), assuming that all ionic mobilities are half of the values given above. Explain the result in physical terms. (e) Assuming ideal solutions, show that the diffusion potential becomes equal to the Nernst potential of the cation when the mobility of the anion is zero. Why is this so? 14. Derive Eq. (2.56) from Eq. (2.55). 15. Side I of a membrane is bathed by a 10-mM solution of AX and Side II is bathed by 100 mM AX; T = 37ºC, the solutions are ideal, and the ions are monovalent.

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Using either the Behn or constant-field solution, sketch the variation of diffusion potential as UA/UX varies from zero to infinity. What are the asymptotic values of the diffusion potential, and why? –4

2

16. The electrical mobility of the sodium ion in water is 5.19 q 10 cm /sec-V at –4 –4 25ºC; the corresponding values for K and Cl are 7.62 q 10 and 7.92 q 10 . (a) Using the Goldman equation, find the open-circuit potential across a water film separating a 100-mM KCl solution from a 100-mM NaCl solution. (b) Use the Goldman and Planck equations to predict the open-circuit potential across the water film when the concentration of the KCl solution is 90 mM. 17. A membrane is bounded by two solutions of NaCl, 100 mM on Side I and 10 mM on Side II. At open circuit, the absolute potential difference across the mem–5 2 brane is 13 mV and the sodium flux is 1.6 q 10 mols/cm -sec. (a) Is the potential at Side I greater or less than that at Side II? Why? Explain in physical terms. (b) Using the constant-field equation, calculate the sodium permeability of the membrane. 18. The intracellular concentrations of the major ions in human erythrocytes are: 135 mM K, 17 mM Na, and 77 mM Cl; the concentrations in plasma are: 4 mM K, 138 mM Na, and 116 mM Cl. Chloride is in equilibrium across the cell membrane. (a) What is the cell membrane potential? (b) What is the Na/K permeability ratio? (c) The composition of the plasma in the reference on which this problem is based differs from that in Table 2.5. Repeat (a) and (b), using the Table 2.5 values, to see how sensitive your results are to plasma composition. 19. Confirm that Eq. (2.47) reduces to Eq. (2.61) when the coupling ratio is unity. Why is this the case? 20. Consider the system described in the caption to Figure 2.6. Assume UNa/UK = 0.05. Now assume that increasing amounts of potassium replace the sodium in the external solution, while maintaining the total concentration of the two ions. (a) Plot the membrane potential as a function of external potassium conII II centration, from cK = 4 mM (cNa = 142 mM, the condition in Fig. 2.6) II II to cK = 140 mM (cNa = 6 mM). (b) Can further manipulation of the bathing solution, subject to the same constraint on the total external cation concentration, cause the cell potential to become positive?

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21. Some membranes have the ability to exclude virtually all ions bearing a particular charge, so the membrane becomes selective for only anions or only cations. Consider a cation-selective membrane of thickness a bounded on Side I by a soI lution of the 1–1 salt AX at a concentration, c , and on Side II by a solution of the II 1-1 salt BX at a concentration, c . (a) Write an expression for the short-circuit current in terms of mobilities, membrane thickness and bounding concentrations. (b) Using the constant-field equation, write an expression for the opencircuit potential of the membrane, in terms of the same variables.

3 THE CELL

INTRODUCTION Whereas the previous chapters in this text were as relevant to inanimate systems as to living ones (if not more so), many of the chapters to come, including the two after this one, will deal with transport mechanisms that are unique to living systems — that are, indeed, biological. Within the hierarchy of biological length scales discussed in the Preface, the cell is the best entry point into the field of biological transport. In the hierarchy of length scales, the cell is the smallest unit that exhibits all of the molecular activity that underlies transport processes. That is, with the inevitable few exceptions, cells contain all the engines of molecular biology, and no smaller structure can make this claim. Similarly, the cell is the largest unit that is common to all the tissues and organs that perform the transport processes necessary to life. In the hierarchy of scale, cells are the bridge between molecular biology and physiology. This chapter has two objectives, of which the first is to present the basic notions of cell biology needed to appreciate the mechanisms and models of transport to be discussed subsequently. Because of this focus, its content is selective and lacks the scope of a dedicated text in cell biology. Those wishing a broader perspective on the subject, or to pursue some of the material presented here in greater depth, are referred to more comprehensive texts such as Alberts et al. (2002) or Lodish et al. (2003). The second objective of this chapter is to deal with those aspects of transport that operate primarily within the cell, including the mechanisms that drive the motion of cells themselves. This material remains largely descriptive, but a limited number of relevant models will be discussed. The intracellular milieu is also important to exchanges between the cell and its environment, which will be treated in later chapters.

3.1. OVERVIEW We begin with a description of cellular structure, starting with the plasma membrane that separates the cell from its environment, and then moving inside to explore the cytoplasm and the organelles within it. As our scale of examination becomes smaller, M.H. Friedman, Principles and Models of Biological Transport, DOI: 10.1007/978-0-387-79240-8_3, © Springer Science+Business Media, LLC 2008

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we review in a highly selective way the chemistry that the cell uses to perform its functions, emphasizing the pathways by which the energy derived from food and oxygen is packaged to fuel a host of transport processes. We then move on to a review of intracellular transport, both diffusive within the cytoplasm, and directed by other cellular molecules. Exchanges with the environment via vesicular transport are included here because the mechanics of these processes relies heavily on intracellular events. Molecular motors within the cell are briefly discussed. Finally, we include cellular locomotion within the scope of biological transport, and describe how cells move and how some motile cells decide where to go.

3.2. THE STRUCTURE OF AN ANIMAL CELL Though functionally different animal cells can have different shape and structure (and several of these will be described in subsequent chapters), they generally share enough features that one can consider a generic animal cell. Such cells are of the eukaryotic type, a class which includes the cells of all animals, as well as other multicellular organisms, such as plants and fungi. Such a generic cell is shown in Figure 3.1.

3.2.1. Composition and Structure of the Plasma Membrane The plasma membrane of the cell is the primary barrier between the cytoplasm, which houses the internal machinery of the cell, and the chemically very different extracellular environment. All exchanges between the cell and its environment must cross the cell membrane, and most of the transport functions of cells and tissues are traceable to plasma membrane transport processes. The differences in ionic composition across the membrane give rise to the electrical behavior of cells that underlies the function of nerve and muscle. Furthermore, since the membrane is the cell's interface with the outside world, it contains the receptor proteins that sense the chemical and physical environment of the cell and initiate the cell's response to it. The structure of cell membranes has been a subject of intense study for many decades. Underlying this research has been the assumption that all cell membranes in higher organisms share a common organization, specializations of which meet the requirements of specific cell types. Any unifying model of cell membrane structure must explain, among other things, the variety of chemical compositions and transport functions exhibited by this versatile component of the cell. Heterogeneity of Cell Membranes. The major constituents of the plasma membrane are protein and lipid. As can be seen from Table 3.1, the protein/lipid ratio varies considerably, even within a single species. Similarly, the distribution of the lipid fraction among the major lipid classes (e.g., the cholesterol/phospholipid ratio) also varies. And even within the class of phospholipids, which is usually the primary lipid component of the membrane, there is considerable variability in the relative amounts of the various members of the class.

PRINCIPLES AND MODELS OF BIOLOGICAL TRANSPORT

Figure 3.1. (a) Generic animal cell and some of its contents. (b) Nuclear pore complex. Reprinted with permission from Alberts et al. (2002). Copyright © 2002, Garland Science.

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Table 3.1. Composition of Some Human Cell Membranes (Veerkamp, 1972)

Cell type Erythrocyte Platelet Schwann cell (myelin)

Percent protein (P)

Percent lipid (L)

P/L

Cholesterol/ phospholipid ratio

55 32 20

35 56 80

1.57 0.57 0.25

0.67 0.49 1.14

The number of different proteins in cell membranes far exceeds the number of different lipids. The list of membrane proteins includes many that are not related to transport, but also many that are, such as ion and water channels, numerous carriers and receptors, gap junctions that allow cells to exchange material with one another, and enzymes such as the transport ATPases, all of which will be discussed in the next two chapters. Most of the proteins associated with cell membranes are amphipathic; that is, the molecules have both polar (hydrophilic) and nonpolar (hydrophobic) parts. This is also the case for many lipids, including the phospholipids, each molecule of which (see Fig. 3.2, bottom, for an example) has a polar “head” resulting from the phosphate group, and two long nonpolar hydrocarbon “tails.” Cholesterol has a polar hydroxyl group at one end and a single hydrocarbon tail at the other. Thus, the cell membrane must be structured to allow for a variety of lipid and protein constituents, arranged in such a way that almost all of them extend between a hydrophilic (i.e., aqueous) environment and a hydrophobic (i.e., nonaqueous) one. For the reader unfamiliar with hydrophobic and hydrophilic interactions, here is a quick summary. The electrons that form the O–H bonds of solvent water are attracted to the oxygen nucleus, such that the oxygen develops a small negative charge and the hydrogens (protons) develop small positive charges; this separation of charge is called a dipole and the O–H bond is accordingly referred to as polar. Because of these dipoles, weak hydrogen bonds form between the (positively charged) proton of one water molecule and the (negatively charged) oxygen of an adjacent one. These transient bonds cause bulk water to organize into a dynamic lattice-like structure. Molecules or portions of molecules that contain polar bonds (e.g., O–H, N–H, C=O) develop dipoles in solution, and can interact with the water dipoles, essentially integrating themselves into the water lattice. These interactions are energetically favorable, and the molecule or portion of a molecule containing these bonds is referred to as hydrophilic; hydrophilic molecules are water soluble. Ions possess a permanent charge and also dissolve readily in water, surrounding themselves with a cloudlike “hydration shell” of oriented water molecules. Hydrophobic molecules or hydrophobic portions of larger molecules lack, or have relatively few, polar bonds, and are less soluble in water because they interfere with the formation of hydrogen bonds, which is energetically unfavorable. Hydrocarbons, with large numbers of nonpolar C–H bonds, are an example. Simply put, it is energetically favorable for hydrophilic molecules or hydrophilic portions of molecules to be in an aqueous milieu, while the opposite is true for hydrophobic moieties, which prefer to be in contact with other hydrophobic groups. As noted above, many molecules found in biological membranes have both polar (hydrophilic) and nonpolar (hydrophobic) regions.

PRINCIPLES AND MODELS OF BIOLOGICAL TRANSPORT

Figure 3.2. The mosaic model of the cell membrane. As suggested by the cartoon (and as will be seen in Chapter 4), the proteins embedded in the bilayer can extend a considerable distance into the aqueous phases adjacent to the membrane. The glycocalyx is not shown. The structures of two important constituents of the bilayer are shown below the model. The steroidal portion of the cholesterol molecule reduces its flexibility in the cell membrane and contributes to the low permeability of the bilayer.

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Furthermore, the cell membrane is crossed by solutes whose polarities also range from strongly hydrophobic to strongly hydrophilic. These solutes also vary substantially in size. An acceptable cell membrane model must therefore provide for a variety of transport pathways, capable of accommodating the different kinds of solute that are known to cross the membrane. The Mosaic Model of the Plasma Membrane. The mosaic model of the plasma membrane illustrated in Figure 3.2 is the currently accepted model of cell membrane organization. It consists of a continuous lipid bilayer about 5 nm thick, in which are embedded the proteins appropriate to the particular cell. The model puts no upper limit on the concentration of protein, as long as the bilayer remains continuous. The percentage of lipid in the membrane depends on the fraction of the membrane area that is occupied by the bilayer. The lipid bilayer is stabilized by hydrophilic and hydrophobic interactions. The hydrophobic interior of the bilayer is populated primarily by long hydrocarbon tails, providing a stable hydrophobic environment for them, while the polar heads of the lipids are exposed to the aqueous phases adjacent to the membrane, and can interact with the water dipoles. The lipid bilayer is the pathway used by hydrophobic solutes to cross the membrane. As will be discussed in Chapter 7, the transport of these species is usually modeled as free diffusion of dissolved solute. The lipid in the bilayer is fluid, so that the proteins that define the functions of the specific cell can change their conformation with ease. The proteins can rotate around an axis perpendicular to the plane of the membrane, or diffuse laterally in the membrane, carrying an annulus of lipid with them as they move. Some of the proteins embedded in the bilayer are channel formers (Chap. 4), providing polar paths across the membrane for the passage of water and small polar solutes, primarily ions. The nonpolar portion of these proteins is in contact with the interior of the bilayer, while the polar portions protrude into the aqueous surroundings of the membrane and line the transmembrane pore. The preference of hydrophobic residues on the outside of these proteins for the hydrophobic interior of the bilayer, and hydrophilic interactions of the polar portions of the protein with the phospholipid heads and polar species outside the membrane, stabilize the protein and inhibit its movement perpendicular to the membrane plane. The permeability to ions of channel-forming proteins is many orders of magnitude greater than that of the lipid bilayer, so a small number of channels can have a substantial effect on the electrical properties of the membrane. For instance, in the membrane of nerve cells, whose conduction properties are a cornerstone of nervous system function, there are 10–5 times as many channels as there are lipid molecules (Schauf and Bullock, 1979). Ion-selective channels in cell membranes are quite narrow: of the order of 0.5 nm wide at their narrowest point. This poses a problem for large polar molecules — mostly nonelectrolytes — that need to cross the membrane but are not particularly soluble in lipid and are too large to fit comfortably through the pores. Carrier proteins that also span the cell membrane play an important role in providing the cell with adequate amounts of such substances, whose rates of free diffusion across the bilayer are too small. These carriers may simply provide an easier route for them to cross the cell

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membrane passively (Chap. 4), or, for ions as well as nonelectrolytes, they may employ metabolic energy sources or gradients in the electrochemical potential of other species to actively transport solutes across (Chap. 5). As with channels in nerve, not many carriers are needed to do the job; for instance, the transport of glucose into epithelial cells is accomplished by a carrier that constitutes about 0.4% of the cell membrane protein (Klip et at., 1979; Kinne et al., 1980). The proteins that are stabilized in the membrane by hydrophobic and hydrophilic interactions are referred to as integral membrane proteins. Other proteins, called peripheral membrane proteins, adhere to one of the membrane faces through electrostatic interactions with the polar head groups of the bilayer lipids, or with other proteins. Peripheral proteins are much more easily displaced from the membrane than are integral proteins. As can be seen in Figure 3.2, not all of the integral proteins span the membrane. Accordingly, the mix of proteins facing the cytoplasm is not the same as those exposed to the extracellular fluid. The lipid compositions of the two leaflets (i.e., layers) of the bilayer are generally quite different; for instance, there is four times as much phosphatidylethanolamine in the outer leaflet of human erythrocytes as in the inner leaflet, and four times as much sphingomyelin in the inner leaflet vis-à-vis the outer one (Verkleij et al., 1973; see Haest, 2003). Furthermore, each leaflet is itself heterogeneous, containing isolated regions called lipid rafts, which differ from the surrounding lipid in composition and organization. The measured permeability of such a heterogeneous membrane is a weighted average of the permeabilities of its various domains. The motion of lipids in the plane of the cell membrane has been described by a lateral diffusion coefficient, evaluated by modeling this motion as a two-dimensional random walk (Lee et al., 1993). However, it appears that, for proteins at least, transport in the membrane plane cannot be approximated in this way (Jacobson et al., 1995). Owing to the heterogeneity of the bilayer, the effective diffusion constant of a protein in the membrane varies with location. In addition, the anchorages of the cytoskeleton (see below) in the cytoplasmic face of the membrane can transiently retain certain membrane proteins locally, diffusive forces notwithstanding. Some integral proteins may become tethered to the cytoskeleton. The cytoskeleton also seems to be capable of conveying these proteins along specific paths, at rates far faster than random diffusion would predict. Such motion may play a part in the locomotion of the cell, as adhesion proteins in the cell membrane are recycled to the front of the cell from more distal regions. Another impediment to the motion of proteins in cell membranes is cell junctions. When a layer of cells is stitched together by tight intercellular junctions, the protein composition of the membranes that form one side of the layer can be maintained very different from that of the other side. As we shall see in Chapter 10, this polarization of the cell layer permits it to exhibit directional properties such as absorption and the ability to “pump” water from one side to the other. The asymmetry of membrane protein composition is also reflected in the lipid composition of the outer leaflets of the two sides.

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The Glycocalyx and Surface Charge. A third class of compounds (in addition to protein and lipid) associated with the plasma membrane is carbohydrate. Much of this carbohydrate, in the form of polysaccharide chains, is bound to integral proteins protruding outward from the cell membrane, and to membrane lipids, forming a negatively charged coat of glycoprotein and glycolipid that is complemented by similar molecules adsorbed on the cell surface. This saccharide-rich coat on the outside surface of the cell membrane is called the glycocalyx. The glycocalyx may act as a transport barrier in some cases — it is negatively charged and can exclude anions through the Donnan effect — though it has other functions as well. Because of its charge, the glycocalyx is easily visualized by cationic stains. Figure 3.3 is an image of the glycocalyx of a myocardial capillary lined with endothelial cells, stained with Alcian blue.

Figure 3.3. Electron micrograph of the glycocalyx in a rat myocardial capillary. The lower image is a detail of the upper image. Reprinted with permission from van den Berg et al. (2003). Copyright © 2003, Lippincott Williams & Wilkins.

The negative charge on the glycocalyx, as well as acidic phospholipids in the bilayer and acidic residues on membrane proteins at the extracellular face, give rise to a small negative potential at the cell membrane surface. There is a similar potential at the intracellular face of the membrane. In principle, these zeta potentials can introduce errors when attempting to measure the true potential difference across the membrane

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This is because electrode measurements of membrane potential are made outside the range of the surface charges (Fig. 3.4). The zeta potential is typically only a few mV and decays rapidly with distance from the membrane surfaces; furthermore, it can be seen from Figure 3.4 that the error introduced by the zeta potential is the difference between these potentials at the two membrane faces, which is less than the potentials themselves. Accordingly, the effect of zeta potentials at the cell membrane surface is not usually considered in transport studies. However, the neglect of surface charge effects can lead to confusion when the transport properties of the membrane are strongly dependent on the true potential difference across it, as is the case for excitable tissue.

Figure 3.4. Error in membrane potential measurement caused by surface zeta potentials. Zeta poI II tentials cause both interfacial membrane potentials \ and \ to be less than the corresponding bulk potentials {\b} that are measured with microelectrodes. Since both ]potentials cause deviations in the same direction, the difference between the true potential difference '\ and the measured '\b depends on the difference between the two ]-potentials.

3.2.2. The Internal Structure of the Cell The interior of the cell consists of a continuous phase — the cytosol — in which is embedded the organelles that are necessary for the cell to survive and multiply and perform its particular functions. Alternatively, the interior of the cell can be partitioned into the nucleus and the cytoplasm; the latter includes the cytosol and all of the organelles except the nucleus. We will use the terms “cytoplasm” and “cytosol” almost interchangeably, but will distinguish between the continuous cytosol phase and the extranuclear organelles whenever necessary to avoid confusion.

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The distribution of cell volume between the cytosol and the organelles is summarized in Table 3.2. In describing these components of the cell, we will focus our attention on those aspects of cell structure that are particularly relevant to transport. Intracellular transport per se will be discussed in Section 3.4.

Table 3.2. Distribution of the Volume of a Liver Cell among Its Components Compartment Cytosol, including cytoskeleton Nucleus Mitochondria Endoplasmic reticulum and Golgi apparatus Vesicles, "somes"

Volume fraction 0.54 0.06 0.22 0.15 0.03

Adapted from Alberts et al. (2002).

Cytosol and Cytoskeleton. The cytosol is a solution composed primarily of electrolytes, as well as lipids and proteins, and a small amount of glucose. The composition of the intracellular fluid differs markedly from that outside the cell, as could be seen from Table 2.5. The difference in ionic composition is largely responsible for the electrical properties of biological cells, following the principles described in the preceding chapters. As noted in Chapter 1, constituents of the cytosol have the ability to bind small solutes, either electrostatically or through other mechanisms. When solutes are bound firmly within the cell, it is difficult to estimate the amount of material available for transport. When they are bound to a slowly exchanging macromolecule, the rate constants for binding and release become parameters of the transport process. These effects can make it difficult to extract basic transport data from experiments, and to identify the mechanism(s) of solute exchange between the cell and its surroundings. The binding elements within the cell may be localized, or distributed throughout the cytoplasm, and are difficult to isolate experimentally. A useful way to study transport in such systems is to measure solute fluxes under a variety of experimental conditions, and then see how well the data fit a hypothesized model of the process. Often, compartmental analysis (Chap. 8) is used for this purpose. Laced through the cytosol is the cytoskeleton, a three-dimensional network of actin filaments, microtubules, and intermediate filaments that is anchored to the plasma membrane and the organelles, and provides shape and structural integrity to the cell. The cytoskeleton contributes to a number of directed motions within and of the cell, such as secretion and motility, which will be discussed subsequently. Some properties of the major fibrillar components of the cytoskeleton are summarized in Table 3.3.

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Table 3.3. Properties of the Major Fibrillar Components of the Cytoskeleton Microfilaments

Microtubules

Intermediate filaments Vimentin, keratin, others Polymerize to bundles 8–11 Localizes organelles, participates in cell and junctional adhesion

Monomeric protein

Actin

Tubulin

Structure

Filaments, crosslinked bundles and networks 5–9 (filament) Determines cell shape; retards diffusion (cortical); participates in locomotion, muscle contraction, cell division, junctional adhesion, signal transduction

Cylindrical tubes of protofilaments 25 (outer diameter) Responsible for ciliary/flagellar motion; participates in cell division, intracellular transport

Diameter, nm Functions (in addition to load transmission)

We have already observed that the cytoskeletal network at the inner face of the plasma membrane (the cortical cytoskeleton) is capable of influencing the otherwise diffusive motion of integral membrane proteins. In addition, there is evidence [reviewed in Terzic and Kurachi (1998)] that the actin filaments of the cortical cytoskeleton can also affect the transport activity of certain of these proteins. Cellular Organelles. The cytosol surrounds a variety of membrane-bounded bodies called organelles, each of which has specialized functions. The major organelles are: The nucleus, which contains most of the genetic material of the cell and is responsible for the replication of DNA and the synthesis of RNA for protein manufacture. It is surrounded by a nuclear envelope, which, like the other membranes within the cell, is made of lipid bilayer. In contrast to the plasma membrane, the nuclear envelope is a continuous doubled membrane (see Fig. 3.1b), and is penetrated by pores so that materials can exchange between the nucleus and cytoplasm. Because of the design of the envelope, the pores do not traverse the strongly hydrophobic core of the bilayers; rather, they are bordered throughout their length by a nuclear pore complex, comprised of numerous proteins, that controls solute exchange between the nucleus and cytoplasm. The complex is capable of dilating to accommodate selected solutes. Mitochondria are the organelles in which the products of digestion are finally oxidized in a controlled way to produce adenosine triphosphate (ATP). Each molecule of ATP is a packet of chemical energy that the cell can use as an energy source for transport and many other purposes. The synthesized ATP is released into the cytosol, where it diffuses throughout the cell. The mitochondria tend to concentrate in regions of the cell where ATP consumption is greatest. The synthesis of ATP will be discussed in greater detail in the next section. Endoplasmic reticulum (ER) is a convoluted membrane-bounded space on a microtubule scaffold in the cytosol that is a primary site for the synthesis of proteins. Some of the synthesized proteins are used within the cell, some become integral membrane proteins in the periphery, and others are exported via secretory mechanisms. The ER also produces almost all the lipid for the cell, serves as a calcium ion

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storage site, and carries out many other enzymatic processes. The ER space is continuous with the space between the two bilayers of the nuclear envelope. Because of its many folds and tubes, the membrane area of the ER can be forty times that of the cell membrane. The ER is of two kinds. The rough ER supports a large number of ribosomes that are involved in the synthesis of a major portion of the cell's proteins, and give it its granular appearance. The smooth ER produces lipids and lipid-based molecules, and participates in intracellular calcium dynamics (Chap. 8). The Golgi apparatus serves as an assembly and dispatching center for proteins and lipids synthesized in the endoplasmic reticulum. Thus, the Golgi apparatus is generally the first stop for newly synthesized molecules on their way to their final destinations. Modifications to these molecules take place in the Golgi. Lysosomes, secretory vesicles, and other cytoplasmic components are formed here, as are molecules that contribute to the extracellular matrix. Not surprisingly, the Golgi apparatus is particularly large in secretory cells, and is located closer to the region of the cell membrane toward which the secretory vesicles are to be directed. A variety of vesicles and “somes.” Vesicles are small fluid-containing sacs, such as those that contain materials brought into the cell from outside through endocytosis (see §3.4.3). Vesicular transport of “cargo” molecules is also the most common mechanism for transport of synthesized molecules to the extracellular space. Clusters of vesicles carry newly synthesized molecules from the ER to the Golgi apparatus. The suffix “-some” derives from the Greek term for “body.” The bodies within the cytoplasm include: lysosomes, which contain digestive enzymes for various degradative purposes; endosomes, which are part of the endocytic pathway (see §3.4.3); peroxisomes, which contain enzymes that catalyze oxidation reactions in the cell; and the centrosome, a central organizing structure of the microtubule network that is found in the neighborhood of the nucleus.

3.3. METABOLISM: THE PRODUCTION OF ATP The metabolic activity of organisms is a subject to which entire books are devoted. The purpose of this section is to select from this enormous body of knowledge the basic information needed to understand how the cell obtains the energy it needs to power many of its transport processes. Thus, our scope will be narrow, focusing on the production of the high energy compound, adenosine triphosphate (ATP), which is the primary energy source for the cell. The production process begins with the intake of food. The complex molecules that comprise the food are converted into simpler compounds (catabolized) by the digestive process: carbohydrates are degraded into simple sugars, such as glucose; proteins into their constituent amino acids; and fats into glycerol and fatty acids. These molecules are then absorbed into the blood and carried to all the cells of the body. In the cell, the products of digestion, which are still quite varied, are further converted to a limited number of compounds (“substrates”) that serve as grist for the cell's own metabolic mill, the mitochondrion (Fig. 3.5). The primary substrates used by mitochondria are pyruvate and fatty acids. In the mitochondrial matrix, these substrates

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Figure 3.5. The mitochondrion and the transport processes in its inner membrane.

undergo a lengthy series of enzymatically catalyzed reactions by which they are oxidized to carbon dioxide and water; in the presence of insufficient oxygen, a less efficient process known as anaerobic glycolysis yields lactic acid as an end product. A small amount of the energy extracted from glucose is obtained during its conversion to pyruvic acid.

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The mitochondrial substrates are initially converted to a common product, acetyl coenzyme A (acetyl CoA), which enters the enzymatically controlled citric acid (or Krebs) cycle. In the citric acid cycle, carbon dioxide is removed from the acetyl group of acetyl CoA, while the energy content of the molecule is stored in the electron carriers NADH and FADH2. These compounds release their energetic electrons to a chain of enzymes in the inner membrane of the mitochondrion. The electrons are passed from one of these respiratory enzyme complexes to the next, incrementally releasing energy that is used by the enzymes to transport protons from the matrix into the intermembrane space. The last respiratory enzyme in the chain, cytochrome oxidase, completes the process by combining four electrons with molecular oxygen and four matrix protons to produce water. More than 90% of the consumption of oxygen by the cell proceeds through this mechanism. The cytochrome oxidase-catalyzed reaction, including the translocation of additional protons, is +

–

+

8H matrix + O2 + 4e o 2H2O + 4H intermembrane space. The respiratory enzymes use the energy extracted from the mitochondrial substrates to create a difference in the electrochemical potential of protons across the inner membrane of the organelle, higher in the intermembrane space than in the matrix. This potential energy is used to drive an endothermic reaction, the phosphorylation of (i.e., the addition of phosphate to) adenosine diphosphate (ADP), whose product is the high-energy compound, adenosine triphosphate (ATP) (see Fig. 3.6). Because the substrates are oxidized during the energy extraction process, this is referred to as oxidative phosphorylation. The phosphorylation of ADP is performed by an enzyme in the inner membrane, the F0F1–ATPase, or ATP synthase. Most ATPases involved in transport use ATP as an energy source to actively transport ions against their electrochemical potential gradient. In contrast, the F0F1–ATPase is driven backwards by the strong proton gradient generated by the respiratory chain. The inward flux of hydrogen ions through the ATPase causes it to synthesize the energetic compound rather than consume it. The linkage between ion transport and chemical reaction in ATP synthase is worth examining; more illustrations of this sort of linkage will arise in Chapter 5. Refer to the schematic of the synthase molecule in Figure 3.7. The F0 domain of the synthase is embedded in the inner mitochondrial membrane and consists of a rotary portion (“rotor”) closely apposed to a stationary “stator.” Protons move down their electrochemical potential gradient through the space between the rotor and stator (analogous to the rotor and stator of a turbine) in such a way that the rotor turns. The rotating rotor is attached to a stalk that extends into the head of the enzyme. The stalk and head extend into the mitochondrial matrix and constitute the F1 domain of the synthase. The rotating stalk proteins interact with the head proteins, causing rapid conformational changes of the latter. The E subunits of the head bind ADP and phosphate, and the induced conformational changes drive the phosphorylation reaction. A single molecule of synthase can produce more that 100 ATP molecules/sec, using the energy from 3 or 4 protons to synthesize each one.

PRINCIPLES AND MODELS OF BIOLOGICAL TRANSPORT

Figure 3.6. Phosphorylation of ADP to ATP.

Figure 3.7. Schematic model of ATP synthase. The head is comprised of six subunits: three D subunits and three E subunits. Reprinted with permission from Alberts et al. (2002). Copyright © 2002, Garland Science.

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The passive flow of hydrogen ions is thought to cause the rotor to turn by a “Brownian ratchet” mechanism (Junge and Nelson, 2005), illustrated in Figure 3.8. Let us set the scene first. The rotor contains negatively charged sites along its rim (from ionized glutamic acid) that are capable of being neutralized by hydrogen ions. Rotor sites adjacent to the hydrophobic lipid bilayer of the inner mitochondrial membrane are more stable when they are neutralized, rendering them less hydrophilic. The surface of the stator facing the rotor contains positive charges, so it is energetically favorable that the glutamate sites facing the stator be ionized. Accordingly, the rotor in Panel (a) of the figure is in a stable state, stabilized by hydrophobic interactions with the bilayer and electrostatic interactions with the stator. Finally, we add two proton access channels at the two ends of the stator, an inlet channel open to the protonrich intermembrane space, and an outlet channel that communicates with the matrix. This configuration is shown in Panel (a) of the figure The rotor is in a stable state, but it still undergoes Brownian rotary motion, fluctuating in both clockwise and counterclockwise directions without preference. Sometimes the fluctuation can get so large (say, counterclockwise) that an ionized glutamate finds itself under the inlet channel and close to the hydrophobic inner membrane; at the same time, a neutralized glutamate reaches the outlet channel and is close to the positively charged stator [Panel (b) in the figure]. At this point it is energetically favorable for the former glutamate to become neutralized and the latter one to become ionized, accompanied by the net transfer of one proton from a phase in which its electrochemical potential is high (the intermembrane space) to a phase in which its electrochemical potential is low (the mitochondrial matrix). This exchange leads to the configuration in Panel (c). The rotor cycles on to the Panel (a) configuration, having accomplished a counterclockwise step. The rotor also fluctuates in the clockwise direction, of course. However, to accomplish a step in that direction, the rotor must accept a proton from the outlet tube, where the proton concentration is less, and deliver a proton to the inlet tube, where the proton concentration is greater. Such a step is less favorable energetically and is less likely to happen. The result is that counterclockwise steps are unlikely to be reversed, and the rotor exhibits net counterclockwise motion (like a ratchet), thus rotating the stalk of the F1 domain. ATP is the principal carrier of metabolic energy in the cell, and most of it is produced in the mitochondria. When acted upon by an ATPase, it undergoes hydrolysis back to ADP and inorganic phosphate. This dephosphorylation reaction is accompanied by the release of the energy stored in the terminal phosphate bond (see Fig. 3.6); 0 the standard free energy change of this reaction is 'G = –7.3 kcal/mol. The equations in Section 1.3 show that the free energy change accompanying a chemical reaction 0 depends on the activities of the reactants and products as well as 'G . When typical values are used for cellular ATP, ADP, and phosphate activities, it is found that the energy available from ATP hydrolysis is in the range of 11–14 kcal/mol. When all of the steps of metabolism are considered, each molecule of glucose yields 38 molecules of ATP (Guyton and Hall, 2000); the mitochondrion recovers about 2/3 of the energy that would be released were the glucose oxidized completely to CO2 and water.

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Figure 3.8. Development of torque by rotary Brownian motion. View is a “top” view of the F0 domain as seen from the intermembrane space. One binding site is marked with an asterisk so that rotor rotation can be followed.

The specific effect of the release of energy from ATP is determined by the enzyme that causes the hydrolysis reaction to take place. For instance, Na,K–ATPase is responsible for an active transport system that shuttles sodium ions out of the cell and potassium ions in (Chap. 5).

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Another product of the citric acid cycle is GTP, guanosine triphosphate. This energetic compound activates GTP-binding proteins (G proteins, for short) by binding to them. Dephosphorylation of GTP to GDP deactivates the G protein. The larger, trimeric, G proteins play an important role in the transmission of receptor signals to regulate a variety of cellular processes that mediate membrane permeability, and we will discuss their mechanism in Chapter 8; smaller, monomeric G proteins are involved in the regulation of vesicle trafficking and locomotion, both of which are discussed in the next section. Both classes of G proteins are involved in the regulation of many other cellular processes unrelated to transport.

3.4. INTRACELLULAR TRANSPORT Three aspects of intracellular transport will be discussed. First, passive diffusion in the cytosol, which can be understood in the context of the previous chapter, will be described. Second, we discuss the mechanisms of directed transport within the cell, exemplified by the transport of newly synthesized protein to target organelles. Third, we describe how cells use small vesicles to exchange material directly with their surroundings.

3.4.1. Intracellular Diffusion As we saw in §3.2.2, the interior of the cell is a complex of organelles and macromolecules, laced with cytoskeleton. Passive diffusion through such a complex structure proceeds more slowly than in free solution. The description of intracellular diffusion in detail requires a correspondingly detailed description of the internal cellular structure. Owing to the difficulties of performing such an analysis, passive intracellular transport is usually described in terms of the continuum concepts used earlier to describe free diffusion. An example is Haugh and Lauffenburger's (1998) modeling of the dynamics of intracellular signaling molecules that have been phosphorylated at receptor sites in the cell membrane. The starting point for the description of the intracellular transport of the phosphorylated and dephosphorylated forms of these molecules was the unsteady diffusion equation, with an additional term to include chemical reaction. When the diffusion equation is used to describe passive transport through the cytoplasm, the ease of transport is usually characterized by an “effective” diffusion coefficient, which is evaluated by (1) carrying out a diffusion experiment that involves transport in the cell, and (2) analyzing the result as though the cytoplasm were a homogenous solution. The latter, of course, is not the case, and the calculated coefficient is in fact a composite property, contributed to by all the diverse components of the cell. On the other hand, if transport through the cytoplasm is to be modeled as though it were a homogeneous continuum, it is best that its transport properties be calculated under the same assumption. Typical values for the aqueous and intracellular diffusion coefficients of a variety of solutes are presented in Table 3.4a. The effective intracellular diffusion coefficient

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of a given solute is usually much less than the corresponding value in solvent water. There are several reasons for this. To start with, the cytosol is more viscous than water, owing to its macromolecular content. Estimates of the increase in viscosity have been as high as six-fold (Nitsche, 1999), which is more than enough to explain the observed retardation by cytoplasm of most of the solutes in the table.

Table 3.4a. Comparison of Free Solution and Intracellular Diffusion Coefficients

Compound Sorbitol Sucrose Inulin 24K Dextran Actin Bovine serum albumin

Molecular weight, kD 0.17 0.324 5.5 24 43 68

Equivalent molecular radius, nm 0.25 0.44 1.3 3.55 2.32 3.6

Aqueous diffusivity (Ds) 7 2 q 10 , cm /s 94 52 15 6.3 5.3 6.9

Intracellular c diffusivity (Ds ) 7 2 q 10 , cm /s

Retardation c = Ds /Ds

50 20 3 1.5 0.03 0.10

0.53 0.38 0.20 0.24 0.0057 0.0145

Adapted from Lightfoot and Duca (2000).

According to the Stokes–Einstein equation (2.24), if the solvent viscosity were the only cause of solute retardation, the retardation should be the same for all solutes, since the other variables in the equation are independent of the properties of the solvent. However, the data in Table 3.4a indicate that the retardation is greater for larger molecules. This occurs because of intervening structures in the cytosol, which can cause diffusion paths to be tortuous and narrow. Thus, larger particles will find certain paths too narrow to use, and will be retarded by structures that border the paths they can traverse. For the last of these effects to be small, the minimum dimension of the transport path must be many times larger than the solute size. The dependence of solute transport coefficients on the effective diameter of the transport path will be discussed in more detail in Chapter 7, in the context of diffusion through membrane pores. Luby-Phelps et al. (1987) measured the diffusion coefficient of a series of fluorescein-labeled Ficoll copolymers of different sizes in mouse 3T3 fibroblasts by following the recovery of fluorescence in a small region of the cell after bleaching by a strong argon-ion laser pulse. In this procedure, called Fluorescence Recovery After Photobleaching (FRAP), the time constant of the recovery process measures the rate of diffusion of unbleached molecules into the bleached area. Their results, summarized in Table 3.4b, clearly demonstrate that larger molecules experience greater retardation. Assuming that small solutes are retarded primarily by the greater viscosity of the cytosol, the largest Ficoll molecules experience a retardation from other causes that is close to ten times larger.

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Table 3.4b. Cytoplasmic Retardation of Ficoll Diffusion in Mouse Fibroblasts, Showing the Effect of Molecular Size (Luby-Phelps et al., 1987) Ficoll radius, nm

c

Retardation = Ds /Ds

3.2 6.2 10.6 14 18 22.7

0.277 0.223 0.167 0.116 0.098 0.037

The passage of many intracellular solutes is further retarded by transient binding to intracellular macromolecules. Naturally, those solutes that are bound most of the time exhibit a diffusion coefficient approaching that of the macromolecule–solute complex, which can be orders of magnitude less than that of the free solute (or zero if the macromolecule is bound to or part of the cell structure). When a fraction of the solute is bound and effectively immobile, the apparent diffusion coefficient will be correspondingly reduced from the true intracellular coefficient. Kao et al. (1993) estimated the diffusion coefficient of a small substituted fluorescein probe, BCECF, in mouse fibroblast cytoplasm using FRAP. The authors found that the diffusion coefficient of the test molecule in cytoplasm was 27% of that in water. They then hypothesized that the ratio of the cytoplasmic diffusion coefficient, Dsc, to that in water, Ds (i.e., the reciprocal of the retardation) was given by the product of three terms, each of which contributed independently to the reduction in the cytoplasmic coefficient. With slight modification, their expression was of the following form: c ¬ Dsc (1  fu )Dms K  E ­  c ¸ žžž fu ­­ exp(Dvob ) . Ds K žŸ Duc ®­

(3.1)

The first term is the ratio of water viscosity to the true viscosity of the cytosol, reflecting the viscosity dependence given by the Stokes–Einstein equation. The second term measures the effect of macromolecule binding on diffusion; the fraction of the solute that is unbound is fu, while the diffusion of the bound fraction, 1 – fu, is retarded according to the ratio of the cytosolic diffusion coefficients of the macromolecule-solute complex and the unbound solute. The third term measures the retardation due to the presence of obstacles in the cytoplasm; ob is the volume fraction of obstacles, and D and E are constants. Note that the second term goes to unity in the absence of binding (fu = 1), and the third term would be unity if cytoplasmic structures were absent (Qob = 0). Among the three factors, the authors estimated that the last accounted for a 60% reduction in diffusion coefficient. The remainder of the reduction was attributed in equal measure to binding and increased cytosolic viscosity.

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Finally, it should be remarked that the intracellular diffusion coefficient does not depend only on the identity of the solute. For a given solute, it varies among cell types, and it also depends on location within the cell.

3.4.2. Protein Transport A typical eukaryotic cell contains many thousands of different proteins, almost all of which are synthesized in the cytoplasm. These proteins include molecules intended to function in one of the many organelles of the cell, or in the plasma membrane; other proteins are intended for secretion into the extracellular space. There are several mechanisms by which specific proteins are directed to their final destinations within, or external to, the cell, and they generally involve processes beyond simple diffusion in the cell matrix. The provision of specific proteins to specific sites in the cell can be accomplished in two ways. Synthesized proteins can be exported into the cytosol and allowed to diffuse throughout the cell; each structure within the cell would then take up the appropriate molecules through some kind of recognition mechanism. Alternatively, proteins can be packaged and transported through direct means to their intended sites. Both of these mechanisms are used by the cell. Here we will examine how the former approach allows the protein requirements of two important organelles — the nucleus and the mitochondrion — to be met. The second mechanism is used to supply some proteins to the plasma membrane and to secrete others into the extracellular space; these will be described in the next subsection. Certain cells, such as those that line the kidney and intestine, form layers the faces of which have different properties. Methods for differentially distributing proteins to the membranes of such polarized cells will be discussed in Chapter 10. Transport into the Nucleus. Solutes, such as ions and macromolecules, that exchange between the nucleus and cytosol do so through bidirectional aqueous pores in the nuclear membrane (see Fig. 3.1). There are several thousand such nuclear pore complexes in the nuclear membrane of a typical vertebrate cell. From permeability experiments using proteins of various sizes, the equivalent radius of the pores is found to be 4–5 nm, so molecules of this size and smaller can exchange freely between the nuclear lumen and the cytosol. The permeability of molecules or molecular assemblies that do not exhibit specific interactions with the components of the nuclear pore complex is an inverse function of solute size. However, most proteins do not cross the nuclear membrane by simple diffusion. Larger cytosolic molecules (>50 kDa) destined for the nucleus contain a short amino acid sequence, called the nuclear localization signal, that signifies that they are to be imported by that organelle. While the protein is diffusing in the cytosol, the nuclear localization sequence is recognized by a corresponding nuclear import receptor, which binds to the protein and facilitates its entry into the nucleus by interacting with the pore complex. The protein/import receptor complex binds to the nuclear pore complex, docking first at the cytosolic fibrils, and then crosses the pore. After crossing the pore, the imported molecule is released. The protein complex evidently interacts with several components of the pore complex (the fibrils, the transporter, the bas-

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ket) en route to the nucleoplasm, suggesting that a sequence of association– dissociation steps may be involved. The pore complex can dilate and accommodate particles as large as 26 nm in diameter. The maximum pore size is species dependent, and also depends on the metabolic state of the cell. The pore complexes are equally capable of selectively exporting molecules from the nucleus into the cytosol, using similar mechanisms. The nuclear localization signal is replaced by a nuclear export signal. Among the molecules transferred out of the nucleus are the building blocks for the ribosomes that are used to synthesize proteins in the cytosol and endoplasmic reticulum, and the mRNA that carries the coded protein sequences. The basket on the nuclear side of the complex is believed to play a role in the exporting process, channeling large molecules through the central transporter region. The need for ATP in the transport process is unclear, although G proteins do modulate the interaction between the transported molecules and their import receptors. G proteins facilitate the dissociation, in the nucleus, of imported molecules from their receptors and the nuclear pore complex; they also stabilize exported protein complexes until they reach the cytosol. In this way, they create an appropriately directed gradient in the concentration of the protein complex, very similar to some of the active transport processes that are discussed in Chapter 5. We note parenthetically that other hallmarks of active transport are exhibited by transport across the nuclear membrane: certain solutes are accumulated in the nucleus against their concentration gradient, and many of the transport processes are saturable. The role of G proteins in the movement of protein/import receptor complexes across the nuclear membrane is described more fully in Talcott and Moore (1999). A relatively recent review of transport across the nuclear membrane, with citations to many other reviews, was published by Wente (2000). Transport into the Mitochondrion. The pathways taken by cytosolic proteins into and within the mitochondrion depend on their final destination. A large number enter the matrix space, requiring passage through both the outer and inner membranes of the organelle (Fig. 3.5). Similar to transport into the nucleus, the proteins are synthesized with a signaling sequence at one end. The terminal sequence allows the protein to be recognized by receptors in the outer membrane of the mitochondrion. In the cytosol, the protein molecules want to assume a folded structure in which their polar groups are on the outside, where they can engage in hydrophilic interactions with the solvent and thereby enhance the solubility of the protein. But proteins can traverse the channels in the outer mitochondrial membrane only if they are unfolded during transit. Chaperone proteins in the cytosol bind to the newly synthesized proteins to prevent them from folding. Upon reaching the outer membrane of the mitochondrion (Fig. 3.9), the protein binds to an import receptor, chaperone molecules are stripped off, and the receptor complex sends the protein, unfolded, through an associated channel into the intermembrane space. At locations along the mitochondrial surface where the inner and outer membranes are close to one another, there are small contact sites where the outer membrane receptor is in close proximity to a similar receptor in the inner membrane. Pro-

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teins destined for the matrix are handed off to the inner membrane translocator for transport into the central mitochondrion.

Figure 3.9. Protein transport into the mitochondrial matrix.

This process requires the presence of an electrostatic potential difference across the inner membrane (generated by the proton gradient that the respiratory enzymes create), and consumes ATP, whose energy is used to strip off the cytosolic chaperone molecules and the mitochondrial chaperones that help it cross the inner membrane. Upon entering the matrix, the signal sequence is enzymatically removed by a signal peptidase. We have seen that the inner membrane of the mitochondrion houses many enzymes needed for ATP synthesis. These proteins can follow the route described above to enter the matrix, and then use a second signaling sequence to attach to an inner membrane receptor. The second sequence lies between the terminal sequence that binds the protein to the outer membrane receptor, and the transported protein iteslf; it becomes the terminal sequence when the initial signaling sequence is split off in the matrix. The translocator associated with the inner membrane receptor conveys the protein into the inner membrane or the intermembrane space. The second sequence can be strongly hydrophobic to facilitate the insertion of the protein into the lipid-rich inner membrane.

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Figure 3.10. Vesicular transport flowchart.

An alternative mechanism for protein transport to the inner membrane has been proposed. In this model, the hydrophobic second signaling sequence interacts strongly with the inner membrane translocator as the protein is passing though on its way to the matrix. This inhibits the passage of the rest of the molecule, which remains fastened to the inner membrane while the primary signaling sequence extends into the matrix and is nipped off by the peptidase. The remainder of the protein is passed through the outer membrane and remains in the intermembrane space.

3.4.3. Vesicular Transport Many molecules move within the cell inside small sacs, called vesicles, made of bilayer membrane. Vesicular transport is used by cells to internalize nutrients, signaling

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molecules, growth factors and other solutes from the extracellular fluid; to release the products of cellular biosynthesis into the surroundings; and to move material between organelles. Figure 3.10 is a flowchart showing the fluxes and compartments that will be discussed in this brief overview of vesicular transport. Endocytosis. The internalization process, termed endocytosis, is illustrated in Figure 3.11. As can be seen from the figure, the cell membrane envelops a portion of the outside medium, eventually forming an intracellular sac, or vesicle, containing extracellular material. This process allows the cell to take up large molecules, such as entire proteins and other macromolecules, that find it difficult or impossible to cross the cell membrane by other means. Foreign species, such as toxins, antigens, and pathogens, are also taken up in this way by cells of the immune system.

Figure 3.11. Three kinds of endocytosis. Reprinted with permission from Dautry-Varsat and Lodish (1984). Copyright © 1984, Scientific American.

The most common kind of endocytosis is pinocytosis (from the Greek pinein, to drink). Pinocytic vesicles, found in capillary wall, epithelia, and other cell types, are small and contain only solution. Pinocytic vesicles are typically 100 nm in diameter. Pinocytic activity is constitutive; that is, the cell continues to form pinocytic vesicles, without requiring an external stimulus. Pinocytic vesicles stabilized by the protein caveolin begin as invaginations in the cell membrane known as caveolae. These vesicles participate not only in endocytosis, but also in transcytosis, in which the vesicular material is transported across the cell and deposited on the other side. Receptormediated endocytosis, described below, is a more specific pinocytic mechanism based on cell membrane receptors that bind to particular ligands.

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Certain cells, such as white blood cells, can form much larger vesicles (>250 nm in diamteter), which contain solid matter as well; the internalization process in this case is called phagocytosis (from the Greek phagos, one that eats). Receptor-mediated endocytosis involves a class of integral membrane proteins called cargo receptor proteins. The extracellular end of these proteins binds molecules needed by the cell and the other end extends into the cytosol. These receptors diffuse by thermal motion in the plane of the cell membrane, and different classes of receptors bind different ligand molecules. Prior to endocytosis, the receptors form clusters at specialized regions of the membrane, called coated pits. The pits are coated with the cytosolic protein clathrin, which binds to the cytoplasmic end of the cargo receptor through adaptin molecules. A cell might have a thousand of these protein-coated regions on its surface. Clathrin plays an important role in the initial formation of the coated pit, and in the deformation of the cell membrane to form the vesicle. The protein can selforganize at the cytoplasmic face of the plasma membrane to form a roughly spherical cage. Since the clathrin is attached to the cell membrane through the adaptin molecules, their self-assembly causes the membrane to invaginate, forming a coated vesicular “bud.” Breaking the neck (“scission”) of the bud to form a free intracellular vesicle requires energy. This is provided by a GTPase, dynamin, that employs the energy stored in the terminal phosphate bond of GTP. The dynamin finds its way to the bud with the help of another cytosolic protein that is capable of binding to both dynamin and adaptin. Powered by GTP hydrolysis, the dynamin pinches the neck of the bud, and breaks it. Once this happens, the vesicle seals the break and is free to enter the cytosol, with the cargo receptors and their ligands inside. The exact mechanism by which dynamin works is uncertain. The molecule may constrict the neck to the point that the free vesicle becomes the more energetically favorable configuration of the system, and the remaining material in the neck is extruded by the converging cell membrane. It has also been suggested that dynamin causes scission by undergoing a conformational change that stretches the neck. The clathrin cage, which encloses the neck, may also play a role in this process. The numerous proteins and processes that interact to determine the course of receptormediated endocytosis are reviewed in Marsh and McMahon (1999). Once the process beigns, it takes about a minute for the cell to form a vesicle from a coated pit. A series of electron micrographs illustrating the receptor-mediated endocytosis of lipoproteins by hen oocytes is shown in Figure 3.12. The newly formed free intracellular vesicles lose their clathrin coat within one or two minutes after internalization, and fuse into early endosomes. The endosomal pH is acidic, and many ligands dissociate from their receptors at this point and dissolve in the vesicular fluid; others remain bound to their receptors. The fate of the internalized solutes varies, but most of them are transferred either directly or via late endosomes to lysosomes, where they are enzymatically digested to produce simple molecules that the cell can use. Phagocytosed material is destroyed in a similar fashion. The entire process, from ligand binding to lysosomal degradation, takes about 15 minutes. The vesicular membrane and most of the receptors are then recycled to the cell surface.

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Figure 3.12. Stages in the endocytosis of lipoproteins. The bristle-like shapes that coat the outside of the vesicle are molecules of clathrin. Reprinted with permission from Perry and Gilbert (1979). Copyright © 1979, The Company of Biologists.

Recycling is an important feature of vesicular transport. By recycling receptors and membranes to their original site, the biosynthetic burden on the cell is significantly reduced. Another example of recycling, in secretion, is the return of membrane from the Golgi apparatus to the endoplasmic reticulum. The invagination of the plasma membrane and the formation of vesicles require an increase in membrane area. This is accomplished by local thinning of the membrane and is accompanied by the vigorous synthesis of membrane components by the cell. As one indication of the rate of the endocytic process, an activated fibroblast can interiorize each hour an amount of membrane equal to half of its surface area, while ingesting a quantity of extracellular solution equal to 5–10% of its volume. The process of reversible ligand (L) binding by a single class of receptors (R), followed by internalization and degradation, has been modeled (Truskey et al., 1985) by the following kinetic scheme (slightly modified here): k

k3 k2 1 ZZZ X l ( LR )int ¶¶ l Lint Rint , Lex Rex YZZ Z (LR )coated pit ¶¶ k

(3.2a)

k4 Lint ¶¶ l Ldegraded ,

(3.2b)

1

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where k1 and k–1 are the forward and backward rate constants for receptor–ligand binding, k2 is the first-order rate constant for internalization of the coated pits, k3 is the rate constant for the dissociation of the complex in the endosomes, and k4 is the rate constant for degradation in the lysosomes; Lex and Rex denote the ambient ligand and unbound receptors in the cell membrane, respectively. Except for the initial binding step, all of the reactions are first order. If it is assumed that the dissociation of the internalized complex is rapid (k3 >> k4), the conservation equations for the receptor-bound, internalized and degraded ligand are:

d[ LRc.p. ] dt

 k1[ Lex ][ Rex ]  k1[ LRc.p. ]  k2 [ LRc.p. ] ,

(3.3a)

d[ LRint ]  k2 [ LRc.p. ]  k4 [ LRint ] , dt

(3.3b)

d[ L d ]  k4 [ LRint ] , dt

(3.3c)

where the brackets refer to the concentration of unbound ligand or the number of receptors in each state. These equations were integrated in Truskey et al. (2004) under the assumption that the total number of receptors (unbound + bound) at the cell surface, and the external ligand concentration, [Lex], are constant; the first assumption can be written [Rex] + [LRc.p.] = [RTOT], and implies that internalized receptor is recycled to the cell membrane. Here our interest is in the effect of the variables of the system on its steady-state behavior, when the rate of passage of ligand from coated pit to lysosome is constant. At steady state, the numbers of receptors bound to ligand in coated pits, and the number bound to ligand in internalized vesicles, are constant, so the right-hand sides of Eqs. (3.3a) and (3.3b) are zero. Solving for the number of occupied receptor sites in the coated pits at steady state, [ LRc.p. ]steady state 

k1[ Lex ][ RTOT ] . k1 k2 k1[ Lex ]

(3.4)

Equation (3.4) shows that the amount of bound ligand at the membrane is proportional to the total number of receptors and decreases as the conversion processes (dissociation, measured by k–1, and internalization, measured by k2) become more rapid. The concentration is a saturable function of external ligand concentration; as [Lex] increases, the amount of receptor-bound ligand increases, but cannot exceed the total number of receptors. The number of internalized receptors in the steady state is obtained from the righthand side of Eq. (3.3b); Eq. (3.3c) gives the steady degradation rate, whose dependence on RTOT and [Lex] is similar to that of the ligand-bound receptor in the coated pits:  d[ Ld ] ¬­ žž ­ žŸ dt ®­

steady state



k2 k1[ Lex ][ RTOT ] . k1 k2 k1[ Lex ]

(3.5)

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It is worth noting that this model of transport is framed in the terminology of chemical kinetics. This is a departure from the approach taken in the previous chapter. No explicit account is taken of the motion, diffusive or otherwise, of the vesicle from the cell membrane to the endosome, nor of the ligand from the endosome to the lysosome. Kinetic models are often used to describe the progress of a biological process through a series of distinguishable states, such as might accompany the conformational changes of proteins or protein assemblies. We will see more of these in the next two chapters. For some steps, such as ligand binding or degradation, chemical kinetics is the appropriate vehicle for describing the process, but for other steps, such as transport into or out of endosomes, the transport processes are “buried” in the rate constants. Exocytosis, Secretion and Transcytosis. Secretory cells, such as those in the endocrine and neuromuscular systems, are capable of exocytosis, by which materials synthesized by the cell are discharged into the extracellular space. Synthesized molecules are transported in small vesicles from the endoplasmic reticulum to the Golgi apparatus, where they are sorted and stored in cytoplasmic vesicles until they are needed. The vesicles are guided toward the cell surface by microtubules. When signaled by an increase in cytosolic calcium level, the vesicles dock to, and then fuse with, the cell membrane, forming a fusion pore; the delay between the signal and the formation of the fusion pore can be as short as 0.2 msec in a nerve synapse or as long as one minute in mast cells. The fusion pore is a channel that traverses the membranes of both the vesicle and the cell; it allows the interior of the vesicle to communicate with the external environment. The pore widens, releasing the contents of the vesicle. The vesicle membrane is under tension, which is released during the fusion process. The vesicle tension generates a force that favors enlargement of the pore. Figure 3.13 illustrates the exocytosis of insulin by a pancreatic E-cell.

Figure 3.13. Electron microscopic image of two exocytic vesicles, prior to (left) and after (right) the formation of a fusion pore. Because of the electron density of vesicles such as these, reflecting a highly concentrated secretion, some secretory vesicles are often termed “granules.” Reprinted with permission from Orci et al. (1988). Copyright © 1988, Scientific American.

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The calcium signal can result either from an increase in calcium flux into the cell or the release of intracellular calcium stores, or both, and is usually prompted by activation of a cell surface receptor. G proteins have been shown to play a major role in secretion by a wide variety of cells; they are involved in the formation of secretory vesicles, as well as their transport to and fusion with the cell membrane. The translation of receptor binding into a calcium signal and vesicle release follows pathways similar to those that will be described in Chapter 8. The mechanism by which the increase in cytosolic calcium causes membrane fusion is not certain, but may involve calcium-binding proteins in the vesicle membrane. Calcium-dependent secretion is a form of regulated secretion. This mode of operation is very advantageous to the cell. It allows the cell to make and store proteins and other products during inactive periods when they are not needed. Also, the cell can release a sizable amount of the product in a short time from vesicles that protect it, while in storage, from degrading enzymes in the cytosol. Some cell products are secreted regularly without requiring an intracellular signal; this is known as constitutive secretion. The cell uses this mechanism to replenish its plasma membrane, for instance. Vesicles approaching the cell membrane are retarded by the cortical cytoskeleton described earlier. Calcium activation causes disruption of the cytoskeletal actin filament network and filament shortening, thereby facilitating movement of the vesicles to the cell border. Secretory vesicles are typically several hundred nanometers in diameter, and it is clear from Table 3.4 that they cannot rely on diffusion to reach the plasma membrane. Indeed, they are physically transported along cytoskeletal microtubules, which serve as physical roadways for the directed motion of secretory vesicles and other organelles within the cell. This motion is driven by a class of ATP-powered motor proteins termed kinesins. Different members of this class of ATPases appear to operate differently. Research on one of the more extensively studied kinesins, conventional kinesin (Rice et al., 1999), has yielded an intriguing mechanical model, one version of which is shown in Figure 3.14. Conventional kinesin transports membrane-bounded organelles, such as secretory vesicles, along microtubules. As can be seen in the figure, the protein binds its cargo and literally walks along the tubule. The kinesin in the figure is a dimer of two connected protein chains, each chain contributing one of the molecule's “heads” (which progress more like feet). The heads bind firmly but reversibly to the tubule. At the start of the cycle, the leading head is bound to the microtubule, and an activation site on the trailing head is occupied by ADP from the previous cycle. Walking occurs when ATP binds to the leading head, prompting a large conformational change in a portion of the connection region between the two chains, called the neck linker. The motion of the neck linker throws the trailing head ahead of the leading head, and it binds to the tubule. The ADP on the former trailing head (which is now the leading head for the next cycle) is released. The former leading head loosens from the tubule and is now the trailing head for the next cycle. The phosphate product of ATP hydrolysis is also released, while the ADP remains.

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Figure 3.14. Motion of conventional kinesin along a microtubule, as modeled by Howard (2001). The cycle starts with the shaded “head” adhering to the microtubule at the location marked with an asterisk. The states defined by Howard are indicated on the figure. A similar model was presented by Schnitzer et al (2000). It has been suggested that the trailing head may be bound to the microtubule at the time the leading head binds ATP (Yildiz et al., 2004).

Conventional kinesin has a “stride” of about 8 nm, and can take several hundred steps before detaching from the microtubule. Each step of each kinesin molecule requires the hydrolysis of a single molecule of ATP. The “pace” of kinesin molecules can be as fast as 2–3 μm/sec. A mathematical model of kinesin activity developed by Howard (2001) illustrates the fashion in which motor protein kinetics are described. The process is distilled below to three steps (refer to Fig. 3.14): (1) Reversible binding of ATP to the leading head, which is initially bound to the microtubule. Denote the initial state of the protein by Kbu, and let KbATP signify the state in which ATP is bound to the protein. ADP from the previous cycle is bound to the trailing head. (2) Hydrolysis of the bound ATP, accompanied by binding of the trailing head to the tubule ahead of the former leading head and release of ADP from the new leading head; call the new state KbADP/P. (3) Release of phosphate (Pi) from the former leading head after its release from the microtubule. The protein has returned to its original state Kbu, but the roles of the two heads have been exchanged, and the molecule is 8 nm + further down the tubule; call this state Kbu .

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The kinetic model for this process is as follows: k

k

1 2 ZZZ X K bu ATP YZZ l K bADP/P ADP , Z K bATP ¶¶ k

(3.6a)

1

k

3 K bADP/P ¶¶ l K +bu Pi .

(3.6b)

As in the model of endocytosis, we will now write the rate equations describing the kinetic scheme above. There are some differences from the endocytosis model, however. The first difference is the cyclic nature of the process, which is common for transport proteins. The protein continues to cycle through the same three states as it moves along the tubule. Assuming that the two heads of the protein are indistinguishable, and that the binding and conformational energies of the two bound states are the + same, state Kbu is kinetically indistinguishable from Kbu. This cyclic process can be diagrammed as shown in Figure 3.15.

Figure 3.15. Kinetic diagram of the kinesin cycle.

If we consider a large number of protein molecules independently passing through this cycle, some fraction of the molecules will be found in each state. We will denote this fraction by bracketing; if the transitions are rapid (so the protein spends negligible time between states), the sum of the bracketed values over all the states equals unity. In the steady state, the fractions will be constant, meaning that the number of protein molecules transitioning into a given state in a unit of time equals the number transitioning out. But the rate of transition into a given state is precisely the rate of transition out of the preceding state, and the transition rate out of a given state is the same as the transition rate into the subsequent state; since the process is cyclic, this implies that the rate of transition between any pair of successive states in the cycle is the same. Two caveats should be applied to this description. When the transition between two states is reversible, such as that between Kbu and KbATP, the “transition rate” from one to the next is properly a net transition rate, being the difference between the rates of the forward and reverse reactions. Furthermore, the foregoing description applies only to simple cycles in which there are no alternative, branching paths. In the earlier discussion, we regarded Eqs. (3.6) and Figure 3.15 as describing the aggregate behavior of a large number of molecules. We can also think of Fig-

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ure 3.15 as describing the behavior of a single molecule transitioning through a cyclic sequence of states. The rate constants of chemical kinetics are replaced by transition probabilities between states, and the brackets in the rate equations that follow can be thought of as the fraction of the cycle time during which the protein is in the indicated state. Equating the transition rates of each step in the cycle, k1[Kbu][ATP] – k–1[KbATP] = k2[KbATP] = k3[KbADP/P].

(3.7)

As noted above, the sum of the bracketed states equals unity: [Kbu] + [KbATP] + [KbADP/P] = 1.

(3.8)

Equations (3.7) and (3.8) can be solved for the three state occupancy fractions [note that Eq. (3.7) is actually two equations]. Then, any one of the terms in Eq. (3.7) gives the cycle rate: Cycle rate =

k1k2 k3 [ATP] . k3 (k2 k1 ) k1[ATP](k3 k2 )

(3.9)

Equation (3.9) shows that the rate is a saturable function of the available ATP that cannot exceed k2k3/(k2 + k3). At low concentrations of ATP, the cycle rate is proportional to [ATP], with a proportionality constant equal to k1k2/(k2 + k–1). The speed at which the kinesin molecule moves along the microtubule is the product of the cycle rate and the stride of the protein. Insights have been gained into the molecular mechanisms of such molecular motors by examining the effect of an external load on the velocity of the protein along a microtubule or actin filament (Meyhofer and Howard, 1995; Crevel et al., 1999; Visscher et al., 1999). Cantilever devices, such as atomic force microscopes, and optical traps are used to produce the piconewton forces that are needed in these studies. The effect of external forces are introduced into the rate equations by making the rate constants force dependent in an exponential fashion similar to the energy barrier models discussed in the next chapter [see Howard (2001) for more details]. It has also been proposed (Hua et al., 2002) that kinesin proceeds along the microtubule like an inchworm: starting with the heads separated along the microtubule, the trailing head is proposed to move forward until it is just behind the leading head; then the leading head moves 8 nm forward, and the two heads are separated once again. The majority opinion favors the first model. Kinesin is a member of a class of proteins that transduce chemical energy into mechanical work. Other members include dynamin, whose role in endocytosis was described earlier; the F0 unit of ATP synthase, discussed in the previous section; and myosin II, which plays a major role in muscle contraction. This aspect of myosin activity will be described in detail in Chapter 9. A different form of myosin (myosin V) is involved in vesicle transport along actin filaments, through a mechanism thought (Walker et al., 2000; Yildiz et al., 2003) to be similar to that of conventional kinesin. The roles of two additional motor proteins, actin and dynein, will be discussed in the section that follows. A second form of dynein also plays a role in vesicular transport

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within the cell. More details regarding molecular motors can be found in Howard (2001). Vesicular processes are also used to carry large extracellular solutes (such as lipoproteins and antibodies) across endothelial or epithelial cell layers, from one side to the other. This form of transcellular transport, which has both endocytic and exocytic components, is called transcytosis.

3.5. CELLULAR MOTILITY AND LOCOMOTION The directed motion of cells is a special kind of biological transport, essential to wound repair, the inflammatory response, and reproduction, among many other processes. Some cells crawl along surfaces, while others swim freely in fluids. Here, we will review cell motility primarily from a structural point of view, to understand how the cell uses internal mechanics to effect motion. The regulation of cell motility involves complex biochemical pathways, which will not be discussed here. Recent reviews are Horwitz and Parsons (1999), and Dekker and Segal (2000). The sequence of reactions used by membrane receptors to initiate cell motion fall into the category of second messenger systems, which are discussed in Chapter 8.

3.5.1. Actin-Based Movement and Chemotaxis Cell Migration. Many cells can pull themselves along the surface, or substrate, on which they reside. In vivo, the substrate is often extracellular matrix. Examples of cell migration include the migration of inflammatory cells to sites of infection, the renewal of skin, or the migration of metastasizing cancer cells to blood vessels. To move in this way, motile cells must have a way of fastening themselves to the substrate, which they do through structures known as focal adhesions and focal complexes. These complexes connect bundles of actin filaments from the cytoskeleton to the substrate through clusters of transmembrane receptor proteins called integrins, and several associated proteins. Motion is produced by variations in the force exerted by the cell on the substrate through the adhesion sites. A variety of mechanisms can prompt a cell to migrate along a surface, but the process is generally mediated by chemically or mechanically sensitive receptors at the surface of the cell. When the cell is motivated to move, often by the binding of signaling molecules to the receptors, the receptors send a signal to G proteins at the inner surface of the plasma membrane; the G proteins initiate a sequence of reactions that direct the growth and disassembly of actin polymer filaments in the cortical cytoskeleton in a fashion that extends the cell in the direction it wishes to go. These protrusions, known as lamellipodia, form adhesion sites with the substrate. The development of directionality within the cell is called polarization. Polarization can also enhance the sensitivity of cell surface receptors at the leading edge, which helps keep the cell moving in the same direction. In most cells that adhere to a substrate — even cells that do not move — interactions within the cytoskeleton create a contractile force within the cell. The contractile

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force exerts a traction against the adhesion sites. In the plane of the substrate, the traction is directed toward the interior of the cell from the perimeter of the cell's footprint. In the case of motile cells, the sites at the intended rear of the cell separate from the substrate, while the remaining complexes hold tight. The loosening of the adhesions at the rear of the cell seems to be coupled to the formation of lamellipodia at the front. As the tension at the rear is released, the rear of the cell retracts toward the front. The adhesion proteins that had fastened the rear of the cell to the substrate, and depolymerized actin from the region behind the leading edge, recycle through the cytoplasm toward the front of the cell. The selective dynamics of the turnover of the adhesion sites is regulated by microtubules that also insert into the adhesion complexes. The processes that underlie cell migration — polarization, protrusion and adhesion, and retraction — rely on the coordinated efforts of numerous structural and signaling proteins within the cell, including components of the cytoskeleton and the vesicular system; recent summaries of the molecular mechanisms of motility can be found in Pollard and Borisy (2003) and Ridley et al. (2003). Chemotaxis. Eukaryotic cells can employ transmembrane receptors, which extend into both the extracellular and intracellular environments, to respond to external signals that direct the cell to move in a preferred direction. For instance, suppose an inflammatory cell has found some dead tissue or bacteria and wants to recruit more inflammatory cells to help it in phagocytosis. To do so, it secretes a chemical known as a chemoattractant or chemokine, for which the target inflammatory cells possess membrane receptors. The attractant diffuses away from the recruiting cell, setting up a concentration gradient. Owing to the gradient, the fraction of the membrane receptors on the target cells that are bound to the attractant (the occupancy fraction) is greater on the side of the cell that is closer to the recruiter. Thus more receptors on that side transmit their signal to the regulatory molecules in the cell. This results in nonuniform actin polymerization, a redistribution of cytoskeletal elements toward the “upwind” edge of the cell, and motion in the direction of the chemoattractant source. The movement of a cell up a chemoattractant gradient is known as chemotaxis. The gradient need not be produced by the same cell type as that being recruited. In fact, some of the metabolic products of bacteria have chemoattractant properties, and inflammatory cells migrate up the gradient to its source and destroy it. Chemotaxis is also important to wound healing and embryonic development. Chemotaxis can be modeled as a biased random walk, in which the motion has a preferred direction. The analysis that follows is based on a one-dimensional model by Keener and Sneyd (1998), which was derived from a more comprehensive one by Tranquillo and Lauffenburger (1987). Assume that the cells can move either toward or away from the source along a coordinate, x (Fig. 3.16a). In general, their velocity, u, will depend on the local concentration of chemoattractant and will vary with x, since the chemokine level depends on the distance from the source. The number density of cells, n cells/cc, will also depend + on x. At any x, let n be the number of cells/cc moving in the direction of increasing x, – + toward the source, and n = n – n be the number density moving away from the source. We further assume that all cells at a given location have the same velocity,

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irrespective of direction, but that cells moving away from the source are more likely to turn toward the source than vice versa. It is this biased turning of the cells that is responsible for the chemotactic effect. We seek an expression for the net flux of cells when a steady-state distribution has been reached. To do this, we first solve the unsteady problem, and then set the time derivatives equal to zero.

Figure 3.16. Simple model of chemotaxis: (a) geometry and notation; (b) mass balance.

At any location x and time t, the net flux of cells toward the source is J c ( x, t )  J c  J c  u( x, t )[ n ( x, t )  n ( x, t )] , +

–

(3.10)

where Jc (x,t) and Jc (x,t) are the unidirectional fluxes toward and away from the + source. Let p dt be the probability that a cell moving away from the source turns to– ward it in a time interval dt, and let p dt be the probability that a cell moving toward

PRINCIPLES AND MODELS OF BIOLOGICAL TRANSPORT +

103

–

the source turns away from it; p > p , by hypothesis. Thus, at a location x, a dynamic process is going on, in which cells moving toward the source are changing direction to + – move away from it at a rate equal to n (x,t)p , and cells moving away from the source – + are changing direction to move toward it at a rate n (x,t)p . Consider a differential element of x, and carry out a mass balance for cells moving in each direction, over a time interval dt (Fig. 3.16b). For the cells moving away from the source,

 sJ  ¬ d (n dx )  žžž c ­­­ dxdt n p dxdt  n p dxdt . Ÿž sx ®­

(3.11a)

We use partial derivatives here because we are initially solving the unsteady problem, and the fluxes vary with distance and time. For cells moving toward the source,  sJ ¬ d (n dx )  žžž c ­­­ dxdt  n p dxdt n p dxdt . žŸ sx ®­ +

+

–

(3.11b)

–

Dividing by dx, replacing Jc by u(x)n (x,t) and Jc by u(x)n (x,t), and differentiating with respect to time,

sn s(un )  n p  n p , st sx

(3.12a)

sn s(un )   n p n p . st sx

(3.12b)

and

Differentiating Eq. (3.10) with respect to time,  sn sn ¬­ sJ c su ­ ( n  n ) .  u žžž  ­ ­ st st ® st Ÿ st

(3.13)

Substituting Eqs. (3.12) into (3.13),   s(un un ) ¯ sJ c su .  u ¡  2 n p  2 n  p ° ( n  n ) ¡ ° st sx st ¢ ±

(3.14)

Our interest is not so much in the number density of cells moving in each direction, + – but rather in the net flux and the total number density of cells. Since (n – n ) = Jc/u, J ¬ 1 + – by Eq. (3.10), and (n + n ) = n, by definition, simple algebra yields n  žžn c ­­­ ž 2Ÿ u® J ¬ 1 and n  žžn  c ­­­ . Substituting this into Eq. (3.14), 2 žŸ u®   s(un) ¯ J su sJ c .  u ¡ n ( p  p  ) °  J c ( p p ) c ¡¢ sx °± st u st In the steady state, the time derivatives are zero. Solving for Jc,

(3.15)

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CH. 3: THE CELL

  s(un) ¯ ° un( p  p ) u ¡ ¡ sx ±° ¢ Jc  (steady state) . p p

(3.16)

The second term in the numerator is the chemotactic term, since it reflects the contri+ – bution of the difference in turning rates p and p to Jc. Since, in bulk flow, the flux is equal to velocity times concentration, we can define the chemotactic velocity as uc 

u( p  p  ) . p p

(3.17)

For a complete description of chemotaxis, it is useful to have a model that describes how the imbalance of unidirectional fluxes arises from a gradient in chemoattractant concentration, c(x). We assume that the driving force for chemotaxis is proportional to the difference between the receptor occupancy fractions at the upwind and downwind ends of the cell. To first order, we can assume that the chemoattractant concentration gradient is small enough (typically, a few percent over the length of the cell) that the variation in the occupancy fraction, f, with concentration can be linearized. Then, the driving force for chemotaxis, Xc, is given by  df ¬ dc ¬ X c r L žž ­­­žž ­­­ , Ÿž dc ®Ÿž dx ®

(3.18)

where L is the length of the cell and df/dc is evaluated at the mean chemoattractant concentration seen by the cell. To relate this driving force to the expression for chemotactic velocity derived earlier, we suppose that uc is a saturable function of the driving force. That is, there is a maximum chemotactic velocity that the cell can achieve irrespective of the driving force. Clearly, that maximum is the migration velocity, u, achieved when all of the – cells consistently move toward the source (i.e., p = 0). A simple relation that exhibits saturation is the Michaelis–Menten equation, more commonly used to describe enzyme kinetics: uc 

uX c , Xc Kc

(3.19)

where Kc is a constant. This function is proportional to Xc when Xc is small (i.e., Xc << Kc), and approaches u as Xc becomes large. Using Eqs. (3.17) and (3.19), we can now relate the turning probabilities to the parameters of the chemoattractant–receptor interaction. Equating the values of uc/u given by each equation, and rearranging, Kc p  . 2 Xc Kc p

(3.20)

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105

Thus, when the driving force is zero, as in the absence of a chemoattractant gradient, the right-hand side of Eq. (3.20) is unity, the two turning probabilities are equal, and there is no chemotaxis. As Xc increases, the right-hand side of the equation falls below – + unity, so p becomes less than p , and chemotaxis proceeds. When Xc becomes much – + larger than Kc, the right-hand side approaches zero, as does p /p , and [by Eq. (3.17)], uc approaches u. In this model, the driving force for chemotaxis is the difference in occupancy fraction between the two ends of the cell. Intuitively, this difference would seem quite small. Recent studies suggest that biochemical processes within the cell produce a larger gradient in the crucial intracellular molecules that control the movement of the cell, than would be expected on the basis of occupancy fraction alone. Most of this intracellular gradient is near the leading edge of the cell. Furthermore, certain of the molecules involved in signaling evidently migrate toward the leading edge, or are preferentially synthesized there. The larger number of such molecules can amplify the signal generated by the receptors in that region. The centrosome is also thought to play a role in amplifying the signal induced by a chemokine gradient.

3.5.2. Cilia and Flagella While some cells crawl along a substrate, other cells swim using flagella or cilia. Flagella are specialized whiplike (hence the name) intracellular organelles that extend into the surrounding medium; a well-known example is the flagellum of sperm cells. Cilia are extensions of the cell that can be used for propulsion, but are also used to cause motion in the fluid adjacent to the cell, rather than move the cell itself. For instance, they are used to transport mucus in the lung, or eggs from the ovary to the uterus. To move cells or adjacent fluid, cilia and flagella must beat in a coordinated way. Although cilia and flagella differ from one another in length, number density on the cell surface, and function, they do share a similar structure and mode of operation. We will use the term “cilia” in the discussion to follow for both entities. Whereas actin plays a major role in locomotion on a substrate, the action of cilia depends largely on the microtubules. A generic cilium consists of a pair of microtubules along its axis, surrounded by and linked to a ring of microtubule pairs (doublets) that run parallel to the axis but closer to the membrane of the organelle (Fig. 3.17a). The membrane of the cilium is continuous with the cell membrane. Cilia are about 0.2 μm in diameter and can vary in length from 5 μm (short cilium) to 200 μm (long flagellum). Motion relies on the energy from cellular ATP and a motor protein, dynein, which has ATPase activity and is normally attached to one member of each doublet. The energy from the ATP is used by the dynein to cause one microtubule doublet to slide past the adjacent one. Specifically, the energized dynein bound to the A microtubule of one doublet (e.g., 9A in the inset to Fig. 3.17b) tries to walk along the nearby B microtubule in an adjacent doublet (1B in the inset). This creates a shearing force between the two doublets. Resistance to this shearing motion is not uniform; when the relative motion between the two doublets is prevented or retarded at some location, the pair will bend (Fig. 3.18). Repeated cyclic bending causes the cilium to beat. The

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local resistance to shear between a pair of doublets may originate from the radial spokes that extend from the doublets toward the central pair of tubules, or from the nexin links that connect adjacent doublets. Appropriate coordination of the sliding and resistive forces can lead to the normal ciliary beat.

Figure 3.17. A generic cilium. (a) Axial section. (b) Transverse section at the axial location indicated in (a). The plane in which the cilium moves is indicated by a dashed line that passes between the 5–6 pair. (c) Detail of microtubule doublet. Microtubule doublets can slide past one another, except for the 5–6 pair. Reprinted with permission from Kaneshiro et al. (2001), in which more detail can be found. Copyright © 2001, Elsevier.

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Figure 3.18. Interaction of shear and resistive forces to cause ciliary bending. (a) Doublet pair at rest. The dynein molecule is inactive. A region to the left resists shear between the two doublets. (b) The dynein is activated and tries to move Doublet N to the left relative to Doublet N + 1. The two doublets cannot move relative to one another in the shear resistant regions, causing them to bend as shown.

PROBLEMS: CHAPTER 3 4

1.

The cell membranes in the medulla of the kidney contain about 10 Na,K–ATPase 2 molecules per μm . One enzyme molecule can hydrolyze 100 ATP molecules/sec. What is the current density produced across the renal cell membrane by the pump? Express the current density in electrical units.

2.

The pH in the intermembrane space of the mitochondrion is 7, while that in the matrix is 8; also, the membrane potential across the inner mitochondrial membrane is 140 mV, with the intermembrane space positive with respect to the matrix. (a) What is the electrochemical potential difference of hydrogen ions across the inner membrane? (b) Assume ATP synthase uses an average of 3.5 protons to synthesize one molecule of ATP, and that ATP hydrolysis yields 12 kcal/mol. What is the energy conversion efficiency of ATP synthase, measured as the ratio of the energy released by hydrolysis to the energy given up by the protons as they pass through the enzyme?

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3.

Assume that the caloric value of your daily diet is 3000 kcal, and that 2/3 of this energy is stored in ATP and recovered during hydrolysis. (a) How many protons pass through all of the ATP synthase molecules in your body in a single day? (b) If the maximum production rate of a single synthase molecule is 100 ATP molecules/sec, what is the minimum number of synthase molecules in your body? Assume that ATP synthase uses an average of 3.5 protons to synthesize one molecule of ATP, that all ATP is produced by ATP synthase (which in fact is not the case) and that ATP hydrolysis yields 12 kcal/mol.

4.

Given that the standard free energy of ATP hydrolysis is –7.3 kcal/mol, compute the free energy available from ATP hydrolysis in the cell, using concentrations of intracellular ATP, ADP, and Pi obtained from the literature or the web.

5.

In a pure extract of cytosol, from which the organelles and other obstacles to diffusion have been removed, the diffusion coefficient of a solute at 25ºC is onequarter of its aqueous diffusion coefficient at the same temperature. Independent measurements of cytosolic viscosity at 25ºC show that the viscosity of the cytosol is three times that of water at that temperature. Since the reduction in diffusion coefficient cannot be explained as a viscosity effect alone, it is concluded that some fraction of the solute is bound to cytosolic macromolecules that diffuse at a negligible rate. What fraction of the solute is presumably bound in this way?

6.

Referring to Eq. (3.5), why doesn't the steady degradation rate depend on k4? Sketch the variation of the degradation rate with k1, k–1 and k2 at constant values of RTOT and [Lex], and explain the dependences you find.

7.

When diffusion is modeled as a random walk in three dimensions, the mean 2 square distance of a diffusing particle from its starting point, , grows with 2 time, t, according to = 6Dst. The insulin-loaded vesicle in Figure 3.13 is about 0.5 μm in diameter. How long would it take, on average, for it to diffuse 2 the 10 μm between the nucleus and cell border of a pancreatic E-cell ( = 100 2 μm ). Use the Stokes–Einstein relation to estimate the vesicle diffusion coefficient, assuming the temperature is 37ºC and that the effective viscosity of the cytoplasm is six times that of water.

8.

Draw the kinetic diagram and write the equations for the steady-state transition rates for a kinesin cycle in which the two heads are different; i.e., in the notation + of the text, Kbu is a different state than Kbu. Hint: This will enlarge the cycle by three additional steps.

9.

Show that the bracketed terms in Eqs. (3.7) and (3.8) can represent the fraction of the cycle time that the motor protein remains in each state, assuming the transi-

PRINCIPLES AND MODELS OF BIOLOGICAL TRANSPORT

109

tions are rapid. Also show that, in this representation, the terms are independent of the duration of the cycle. 10. (a) From Eq. (3.9), the maximum cycle rate for kinesin function is given by k2k3/(k2 + k3). Sketch the variation of the maximum rate with k2 at constant k3, and vice versa. Explain the shapes of the curves. (b) Similarly, at low concentrations of ATP, the cycle rate is proportional to [ATP], with a proportionality constant equal to k1k2/(k2 + k–1). Explain the dependence of the cycle rate on each of the rate constants in this expression. Hint: it may be helpful to first find the state occupancy fractions under the conditions of high and low ATP concentration. 11. How much ATP is needed for a kinesin molecule to carry the insulin vesicle in Problem 7 the 10 μm from the nucleus to the border of a pancreatic E-cell? 12. Assume that the maximum kinesin speed of 3 μm/sec corresponds to the maximum cycle rate given by Eq. (3.9) in the limit of large [ATP]. When the intracellular ATP level is 3 mM, the kinesin speed is only 2 μm/sec. When the intracellular ATP level is 2 mM, the kinesin speed is 1.5 μm/sec. What is the kinesin speed when the intracellular ATP level is 1 mM? 13. Inflammatory cells can detect bacteria at a distance of up to about 100 μm. Identify and explain the term in the equations for chemotactic velocity that limits the detection range of any cell, irrespective of its sensitivity to the chemokine or the statistics of its changes in direction.

4 FACILITATED DIFFUSION: CHANNELS AND CARRIERS

INTRODUCTION The descriptions of interphase equilibrium and diffusion in the preceding chapters are equally applicable to inanimate and living systems. In the latter, the interphase barrier is often a cell membrane or a layer of cells. These barriers retard transport by free diffusion and are absolutely essential to the viability and function of cells and tissues. Though the presence of barriers such as cell membranes is literally of vital importance, it is almost as important that they not be too impermeable. Nutrients must be allowed to enter the cell, wastes must leave, and the ionic composition of the cytosol and the interior of organelles must be capable of accommodating changes in the cellular environment. The tradeoff between barrier function and permeability poses a difficult design problem, particularly in view of the large variety of species, ranging from hydrophobic nonelectrolytes to highly polar ions, that must have access to the interior of the cell. The cell deals with this problem through a process known as facilitated diffusion. In facilitated diffusion, specific integral transmembrane proteins interact with particular solutes or classes of solutes, markedly increasing the rates at which they can cross the membrane. The membrane components responsible for facilitated diffusion are called transport mediators, so this process is also referred to as mediated diffusion. The two major classes of transport mediators are carriers and channels. In the first section of this chapter, the distinguishing differences between mediated transport and free diffusion are presented and interpreted in terms of the structure of the mediators and the mechanisms by which they work. The emphasis is on channels. The next section deals with mathematical models of channel and carrier transport. The notion that carriers can transport multiple substrates allows us to introduce exchangers and cotransporters in anticipation of the next chapter. Some models of channel transport are reserved for later chapters (pore models in Chap. 7, models of transport in nerve and muscle in Chap. 9, and transport across epithelia in Chap. 10). Inhibition of facilitated transport is the subject of the third section of this chapter. M.H. Friedman, Principles and Models of Biological Transport, DOI: 10.1007/978-0-387-79240-8_4, © Springer Science+Business Media, LLC 2008

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4.1. MECHANISMS OF CHANNELS AND CARRIERS Carriers and channels are the two major classes of transport mediator. A loose distinction between them is easily made: carriers bind solute molecules and carry them across the membrane; channels perforate the membrane, creating paths for diffusion or flow whose resistance is less than that of the bilayer in which they are embedded. In the former instance, the solute and carrier move jointly during at least part of the substrate's traverse of the membrane; in the latter, the solute moves relative to the pore that is formed by the channel protein. As more has been learned about the detailed mechanisms of these mediators, the distinction between them has become increasingly difficult to make. For instance, the original “ferryboat” image of the mobile carrier, shuttling back and forth across the membrane with solute on deck, has evolved into a conformational model in which the solute binding sites, relatively fixed with respect to the membrane, are alternately exposed to the phases on each side. The conformational change needed to accomplish this may involve only a small portion of the carrier molecule. Pores, on the other hand, are no longer seen as static pathways through the membrane, but rather as dynamic elements in which solute binding can play as great a role as it does in carriermediated transport.

4.1.1. Hallmarks of Mediated Transport Before discussing the mechanisms of channel- and carrier-mediated processes in detail, it is useful to review some of the characteristics of this mode of transport. First of all, facilitated diffusion is a passive process, driven by gradients in the chemical or electrochemical potential of the solute, just like free diffusion. The dependence of solute flux on the driving force is often different from that in free diffusion, but in some regimes, the effect of carriers or channels on the transport rate can be represented simply as an increase in the permeability of the membrane. Some of the features of facilitated transport that distinguish it from free diffusion are: 1. Specificity and Selectivity. Binding sites on carriers can be very selective. For instance, sugar carriers distinguish between d- and l-sugars. Presumably, the configuration of the binding site, or the charge distribution at the site, matches that of a distinctive portion of the desired substrate. At the same time, the binding forces are relatively weak (usually weaker than covalent), so that the solute can leave the carrier once it has access to the other side of the membrane. This frees the carrier to transport more substrate. Similarly, channels may pass cations readily but not anions, or potassium ions but not sodium ions. Within classes of transport mediators, there is in general a reciprocal relationship between the transport rate and the selectivity of the transport process, particularly when selectivity is accomplished by binding sites that discriminate among the available solutes. The interactions between the preferred solutes and the binding sites are generally stronger when the

PRINCIPLES AND MODELS OF BIOLOGICAL TRANSPORT

2.

3.

4.

5.

pathway is more selective; interactions with multiple sites may be necessary for positive identification of the substrate of a highly selective system. These strong interactions tend to retard transport. At the other extreme, certain channels pass ions at rates several orders of magnitude higher than carriers, but with little selectivity. Saturability. In contrast to the linearity of the integrated form of Fick's Law [Eq. (2.20)], the rate of facilitated transport approaches an asymptotic value as the substrate concentration difference across the membrane increases. For carriers, this occurs because there is a maximum rate at which carrier translocation can proceed. Channels can also saturate if they are narrow or possess a sequence of binding sites, along which ions pass in single file, or if there is a choke point along their length that limits the maximum transport rate. Inhibition, Activation, and Cooperativity. The rate at which a particular substrate is carried across the membrane can be influenced by other solutes in the system that either compete with the substrate for the transport site or affect the function of the mediator. Channel transport can be activated or inhibited by electric fields and mechanical forces. The most common inhibitory mechanisms will be described in the last section of this chapter. A related property is cooperativity, which can affect the transport rate either favorably (positive cooperativity) or unfavorably (negative cooperativity). Positive cooperativity and activation in channel transport will be described later in this section. Trans Effects. The trans side of the membrane is the side toward which the passive flux of substrate is directed; the cis side is the side from which the flux originates. For some forms of facilitated transport, the flux of a substrate can be altered by changing the concentration of other solutes on the trans side of the membrane. Temperature Coefficient. The energy barrier associated with the conformational change of a carrier is generally substantially higher than the activation energy of solvent viscosity, which determines the temperature dependence of free diffusion rates. Consequently, carrier transport rates increase more rapidly with temperature (i.e., have higher temperature coefficients) than free diffusion permeabilities. On the other hand, the temperature coefficient of the conductance of an open aqueous channel is not so different from that of a film of water. The activation energy of a temperature-dependent variable, v, is obtained by fitting the variable to an equation of the form v = A exp(–Ea/RT), where Ea is the activation energy. According to this equation, v increases with temperature. Since viscosity decreases with temperature, what is termed its activation energy is in fact that of its reciprocal, termed fluidity. Between 20 and 37ºC, the activation energy of the fluidity of water is 3.9 kcal/mol.

113

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4.1.2. Ion Selectivity of Channels Although carriers may transport electrolytes or nonelectrolytes, the species that pass through channels are almost always ions. Many channels are very selective, passing some ions readily, while being essentially impermeable to others. Based on structural models, channels are thought to narrow to no more than several tenths of a nanometer, similar to the diameters of the common biological ions and not much larger than the distance between the oxygen atoms in neighboring water molecules. This suggests that the selectivity properties of channels arise from interactions between the ions and the mouth or walls of the pore. These interactions do not lead to a single ranking of channel selectivities; as shown in Table 4.1, different channels exhibit different selectivities. The alkali ions whose selectivities are shown in the table have been used widely to probe channel selectivity. Their crystal and hydrated radii are listed in Table 4.2, along with another illustrative set of permeability data.

Table 4.1. Selectivity Sequences of Several Cation-Selective Channels Channel

Selectivity sequence Cs > Rb | K > Na > Li K > Rb > Cs > Na Cs > K > Na > Li Na | Li > K K > Rb >> Na | Li

Gramicidin A Alamethicin Acetylcholine receptor, rat Sodium channel, squid axon Potassium channel, squid axon

The inequalities indicate the relative magnitudes of the ion permeabilities in the channel.

Table 4.2. Properties of the Alkali Ions Ion Lithium Sodium Potassium Rubidium Cesium a

Crystal radius, Åa

Hydrated radius, Å

ki/kKb

0.60 0.95 1.33 1.48 1.69

2.31 1.78 1.22 1.18 1.16

0.09 0.07 1 0.74 0.18

1 Å = 1 Ångstrom = 10–10 m = 0.1 nm.

b

Potassium channel in snail neuron (Reuter and Stevens, 1980). Other potassiumselective channels exhibit similar sequences. The sequence here is: K > Rb > Cs > Li > Na.

A review by Diamond and Wright (1969) summarized 15 different permeability sequences that had been observed for the five alkali ions in biological systems. It is hard to imagine, at first, how so many sequences could exist. If the channel is regarded as simply a narrow water-filled pore, one would expect that the smallest ion

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would have the highest permeability, and that the permeability of larger ions would be less. The permeability of ions too large to fit through the pore would naturally be zero. However, we see from Table 4.2 that the relative size of the ions depends on whether one considers the ion in its crystalline form or in aqueous solution. Alkali ions of small atomic number (e.g., Li) have very small crystal radii, but when present in water, they interact more strongly with the water dipoles than do larger members of the family. The range and influence of these dynamic interactions increases as the ionic size decreases, and this is seen in the inverse relationship between atomic number and hydrated radius in the table. The Stokes–Einstein relation [Eq. (2.24)] predicts that the ionic mobility in solution is inversely related to the hydrated radius, and indeed the mobility of the alkali ions increases with atomic number. The interaction represented by the hydrated radius influence the ability of ions to pass through pores, since the cloud of water molecules that surrounds them may need to deform or be stripped away for the ion to get through. In such an event, the permeability sequence would depend on the energy cost of deforming or removing these hydration shells, as the polarized water cloud is known. Furthermore, most channels are lined by charged groups or dipoles (e.g., carboxyl, carbonyl) on the amino acid residues of the channel protein. These give rise to electrostatic forces, which affect, to varying degrees, the permeability of a given ion in the channel. The charges are opposite in sign to that of the ions for which the channel is selective. For instance, cation-selective channels have negatively charged walls, which can electrostatically and reversibly bind cations, or otherwise influence their rate of diffusion through the pore. In addition, the negative charges in the channel wall can, via the Donnan effect, increase the concentration of cations in the channel while excluding anions. The likelihood of an ion shedding its hydration shell, and the influence of charges or dipoles in the walls of the channel, are greatest when the channel is narrow. When it is wide, the ions need not alter their interactions with solvent water, and they pass through in hydrated form, which is the energetically favored state in aqueous solution. In addition, when the pore diameter is large, most of the ions traversing the channel do so at some distance from the wall, shielded from (and accordingly unaffected by) the charges thereon. There are such channels, which function essentially as perforations in the membrane. In these cases, the permeability sequence follows that of the hydrated radii (but in the opposite order, of course): Li < Na < K < }. This description also applies to some instances of transport through the gaps between cells. In narrow channels, ions are more likely to interact with sites at the wall, in some cases hopping from one site to the next as they work their way down the pore. This association–dissociation process is random in the same sense as free diffusion: not every hop is toward the trans side of the membrane, even though the net flux is so directed. Hopping from site to site can increase the mobility of an ion beyond its value in larger pores or in free solution.

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4.1.3. Energetics of Ion Selectivity, and Steric Effects The strength of the interaction between binding sites and ions is greater when the interacting charges can approach more closely; that is, when the ions are unhydrated, and when they are small. When the pore is so narrow that ions cannot enter without shedding or significantly attenuating their hydration shells, binding forces become predominant. Two energy differences are involved in this process: — The energy of hydration, 'Eh = Ehydrated ion – Eunhydrated ion. Since the hydrated state is energetically favored, 'Eh is negative. — The energy of binding the unhydrated ion to a site in the channel, 'Eb = Ebound ion – Eunhydrated ion. For an attractive binding force, 'Eb is also negative. Here, we have described the bound state as one in which the ion is “unhydrated,” implying that there are no dipole interactions between the bound ion and the solvent. In fact, ions can experience such interactions at some level throughout their passage through the channel. Thus, Eunhydrated ion is better regarded as the energy of the ion when it has entered the pore and its interaction with solvent is consequently reduced. We will continue to use the term “unhydrated” for simplicity.

Neglect the difference between the energy of the unhydrated ion in the ambient phase and in the pore; then, the energy change associated with the movement of a hydrated ion to the mouth of a pore, its dehydration, and its subsequent binding to a site on the wall, is 'Et = Ebound ion – Ehydrated ion = 'Eb – 'Eh.

(4.1)

This energy difference determines the ease with which an ion can take advantage of the binding site to enter and pass through the channel. The variation of this quantity from ion to ion is closely related to the permeability sequence in narrow channels. Now we can begin to see how the selectivity of a charged narrow channel may not be so predictable. Both ''s on the right-hand side of Eq. (4.1) have the same sign. Since electrostatic forces are responsible for both hydration and binding, both ''s are more negative for the smaller ions. It is not immediately clear how their difference might vary with ion size. The energy of hydration is a property of only the ion, but the energy of binding depends in addition on the field strength of the binding site. Different field strengths can lead to different selectivity sequences, as can be seen from the data in Table 4.3. In Case I, the (magnitude of the) hydration energy of the light ion is three units greater than that of the heavy ion, but the binding energy is also three units larger, so the transition energies ('Et) of the two ions are the same. When the field is stronger (Case II), the binding energies increase proportionately, while the hydration energies remain the same, and the transition energy now favors the binding and permeation of the lighter ion. Case III shows the opposite result of a reduction in field strength.

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Table 4.3. How Field Strength Can Affect Selectivity Case Field strength 'Eb (arbitrary units) Light ion Heavy ion 'Eh Light ion Heavy ion 'E t Light ion Heavy ion Selectivity:

I

II

III

Medium

High

Low

–12 –9

–16 –12

–8 –6

–5 –2

–5 –2

–5 –2

–7 –7 Comparable

–11 –10 Light ion favored

–3 –4 Heavy ion favored

The transition from a hydrated ion in the ambient solution to an unhydrated ion entering the pore to a bound unhydrated ion can be represented by an energy diagram. Figure 4.1 is a graphical representation of the data in Table 4.3. This explanation of the variability in the selectivity sequences of alkali ions was pioneered by Eisenman (1961), who proposed on theoretical grounds the existence of the eleven sequences in Table 4.4. Note that each of the cations can be the most permeable, if the field strength is in the appropriate range. Eisenman's theory has been shown to be capable of explaining most of the 15 alkali ion permeability sequences referred to earlier.

Table 4.4. Eisenman Selectivity Sequences Li> Na > K > Rb > Cs Na> Li > K > Rb > Cs Na > K > Li > Rb > Cs Na > K > Rb > Li > Cs Na > K > Rb > Cs > Li K > Na > Rb > Cs > Li K > Rb > Na > Cs > Li K > Rb > Cs > Na > Li Rb > K > Cs > Na > Li Rb > Cs > K > Na > Li Cs > Rb > K > Na > Li

Sequence Sequence Sequence Sequence Sequence Sequence Sequence Sequence Sequence Sequence Sequence

XI (at highest field strength) X IX VIII VII VI V IV III II I (at lowest field strength)

Other factors affecting ion selectivity are related to the size of the hydrated or unhydrated ions, and are called steric effects. Two of these are described below: 1. The selectivity of channels between anions and cations is often conferred by a collection of charged sites near the mouth of the channel; negatively

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charged sites promote cation passage, and positively charged sites favor anion transport. A small ion whose charge is of the "preferred" sign may have such a large hydration shell that it cannot get close to the sites at the mouth of the channel. The associated water molecules get in the way and screen the charge on the site. Such an ion is less likely to enter the pore. 2. If the ion cannot fit through the pore in its unhydrated form, its permeability will be zero. In terms of the energetics described above, it is not correct in this case to assume that the unhydrated ion has the same energy inside and outside the channel.

Figure 4.1. Energy diagram showing how field strength can affect selectivity. For each ion, the energy is relative to that of the hydrated ion in solution. The energy required to dehydrate Ion #1 (–'Eh1) is greater than that required to dehydrate Ion #2 (–'Eh2). The energies of binding for the two ions (–'Ebi, i = 1, 2) depend on field strength; when the field strength is low and –'Ebi = %Ebil , binding favors Ion #2, and when the field strength is high and –'Ebi = %Ebih , binding favors Ion #1.

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Figure 4.2. Factors affecting ionic selectivity. The curves on the figure represent the variation of ionic diameters with atomic number Z. A point on the Z–d plane corresponds to the transport of an ion whose atomic number is Z through a pore whose diameter is d. The plane has four regions. In Region A, the pore size is less than the diameter of the unhydrated ion, which is therefore too large to pass through the pore. In Region B, the pore size is less than twice the diameter of the unhydrated ion; the ions cannot pass each other in the pore, and cross in single file; this is a saturable transport mode. In Region C, the pore size is greater than that of the hydrated ion and the ions pass through the pore in hydrated form. In Region D, the ion must shed its hydration shell to pass, the permeability depends on field strength, and there can be a steric problem when the ion is small (hydrated diameter is large) and the pore is relatively narrow. The numbers on the horizontal cuts correspond to the numbers in the text.

4.1.4. Ion Selectivity of Channels: Summary The factors affecting the selectivity sequence of a channel for a class of ions (e.g., the alkali ions) are illustrated in Figure 4.2. The ordinate is the size of the pore at its narrowest point (its selectivity filter, which will be discussed subsequently), at which polar groups afford additional selectivity with respect to the sign of the ionic charge. The three curves in the figure represent the variation with atomic number of the ions' unhydrated diameters, twice their unhydrated diameters, and their hydrated diameters. Figure 4.2 shows how the selectivity sequence of a pore can depend on its diameter. Consider, for instance, the four horizontal cuts shown in the figure: 1. When the pore is very narrow, all but the smallest unhydrated ions are too large to cross the pore, which is highly selective for light ions. 2. For a larger pore, the lightest ion is sterically prevented by its hydration shell from nearing the closest binding sites at the pore mouth; the heaviest ions are still too large to fit; and the pore is selective for midsized ions, which are too large to pass each other inside the pore, and therefore cross single-file. 3. The selectivity sequence of the next largest pore depends on the field strength of the binding sites. 4. The permeability sequence of the largest pore is determined by the hydrated diameters of the ions, as described earlier.

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4.1.5. The Structure of Ion Channels: Selectivity Filters, Gates, and Energy Profiles. Figure 4.3 shows several ion channels at different levels of detail. The simplest such channel is formed by a molecule that has a pore down its center, oriented in the membrane thickness direction. The most well-studied channel of this kind, Gramicidin A, forms a pore lined with polar amino acid residues that cause the aqueous channel to be selective for cations. A sketch of the channel, which is comprised of two gramicidin monomers end to end, is shown in Figure 4.3a. The molecule is stabilized in the membrane by hydrophobic interactions of the nonpolar protein residues with the lipid bilayer; hydrogen bonds hold the two monomers in apposition within the membrane. Gramicidin is a member of a class of transport facilitators known as ionophores. Ionophores may be synthetic or natural peptides, and include several antibiotics; some, like gramicidin, function as channels, and others as carriers. Most transmembrane channels are oligomers of several protein molecules, all of which cross the membrane. These can be thought of most simply as assemblies of “rods” rather than tubes, arranged with their axes more-or-less parallel and more-orless perpendicular to the plane of the membrane. The pore lies along the axis of symmetry of the channel complex, occupying the space between and surrounded by the rods. The amino acid sequences of these proteins are such that they are stabilized in the membrane by hydrophobic interactions with the bilayer. The individual protein molecules that form the channel are known as the subunits of the channel. Each subunit consists of several helical domains that cross the membrane. Some sodium and calcium channels are made up of a single protein molecule that folds in such a way that it assumes a subunit architecture, although each “subunit” is actually a distinct domain of the same molecule. A cartoon of an oligomeric channel, voltage activated and selective for potassium, is shown in Figure 4.3b; in this channel, a portion of the pore is lined by a polar segment, called the P-loop, which is not part of a transmembrane domain. The P-loop forms the selectivity filter that preferentially passes potassium ions. A sketch of the acetylcholine receptor (AChR) channel, a cation channel crucial to neuromuscular transmission, is shown in Figure 4.3c. It has five subunits, and each subunit has four transmembrane domains. The receptor is sited in the endplate of the muscle fiber; when acetylcholine is released by the adjacent motor neuron and binds to the receptor sites on the two D subunits of the channel, it opens, allowing an enormous influx of depolarizing sodium ions that ultimately leads to muscle contraction. The other three subunits [the E, G (or H in some species), and J subunits] are each distinct. The AChR channel has been studied extensively and we will return to it later in this chapter and again in Chapter 9. The diameter and charge density of the pore vary along its length. A more detailed model of a potassium-selective channel in a bacterium, based on x-ray crystallography and since shown to be a good model of similar portions of eukaryotic potassium channels, is shown in Figure 4.3d. In this particular channel, the potassium ion relinquishes its hold on solvent water and interacts electrostatically with carbonyl oxygen dipoles that band the wall of the narrow selectivity filter; cation selectivity is provided by the

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acidic residues nearby and at the inner end of the pore. Both the carbonyl groups and the acidic residues are provided by the P-loop. Ions pass through the filter in single file, and more than one ion can be in the filter region at the same time. Repulsive forces between the ions promote swift passage through the filter. The filter is short [only 1.2 nm in length (Doyle et al., 1998)], to minimize its transport resistance. The selectivity filter does not pass Li or Na readily because they cannot pass through in their hydrated forms (see Table 4.2), while the unhydrated forms of the ions are too small to interact fully with the polar groups in the filter. As a consequence, the energy of binding of the lighter ions is less negative, and they are less apt to shed their hydration shells and pass through the pore. The rest of the pore is relatively nonselective. The potassium ion is hydrated in the wider central cavity, which is lined by hydrophobic residues to minimize retarding interactions with the wall. Each pore helix (see Fig. 4.3d) generates a dipole along its length, with the negative end of the dipole at the central cavity; these dipoles, and those of the polarized water in the cavity, help stabilize the ion within the heart of the bilayer. The development of this model earned Roderick MacKinnon, in whose laboratory the work was done, the 2003 Nobel Prize in chemistry. The selectivity filter in ion channels can be charged or lined with polar groups. In general, the wider the selectivity filter, the less selective the channel. The selectivity filter in the highly selective potassium channel is only 0.33 nm in diameter (Dwyer et al., 1980). The AChR channel rejects anions because of a collection of negatively charged regions near its selectivity filter; however, the characteristic dimension of the filter is 0.65 nm (Dwyer et al., 1980), and the channel exhibits only limited selectivity among cations. Though some channels are permanently open, most undergo transitions among conducting and nonconducting states. Transport through these channels is controlled by gates, so called because of their regulatory function; the rest of the pore is permanently open. Most models of gates treat them as physical barriers, blocking or severely narrowing the transport path. The gating process is often controlled by a receptor site, which can be part of the channel or a portion of an associated receptor protein. The gate opens via a conformational change when the site is activated; this is referred to as an allosteric response. For instance, the channel depicted in Figure 4.3b is gated by the S6 domain, which swings closed or open in response to the rotation or tilt of the voltage sensor, S4, whose motion is prompted by changes in transmembrane potential (Cha et al., 1999; Hille, 2001). Owing to the narrowness of the pore, only a small conformational change of the protein is needed, and the process takes only microseconds. The duration of the open state is typically of the order of milliseconds. The AChR channel is thought (Taly et al., 2005) to gate by a twisting motion of the entire pentamer, which narrows the pore. Another gating model, the ball-and-chain model, has been invoked to explain the inactivation of sodium and potassium channels in nerve, and will be described in Chapter 9. Figure 3.2e shows four images of a gated calcium release channel from skeletal muscle, which will be discussed in Chapter 9. The images show that the relative rota-

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tion of the two portions of the channel protein causes the pore to open so that calcium can pass through to initiate contraction. The gates that control the flux through the pore can be located anywhere along the length of the channel, and a channel can contain more than one gate. When ions pass through the pore by hopping from site to site, the energetics of the transport process can be represented by an energy profile such as that in Figure 4.4. The energetically favorable binding sites are represented by local minima in the profile; the depth of these potential wells differs from ion to ion, depending on the selectivity characteristics of the site. As shown in the figure, a gate can be modeled as a barrier whose height depends on whether the gate is open or closed. Rate equations for channel transport by site hopping have been developed on the basis of such energy profiles and will be described later in this chapter.

4.1.6. Regulation of the Gating Process A large number of agents can open or close channel gates. These include ions to which the channel may or may not be permeable, hormones, neurotransmitters (see Chap. 9), intracellular enzymes, changes in membrane potential, and membrane stretch. Some channels respond to these agents for as long as they are present, while others react only transiently. Channels whose gates are opened in this way are said to be activated. Chemical agents activate channels by binding to receptors that are associated with or part of the mediator; as a class, these channels are referred to as ligandactivated or ligand-gated. The ACh receptor channel described above exemplifies this class of mediators. The ligand may bind at the intracellular or extracellular face of the cell membrane. An important class of activators are the intracellular protein kinases that activate proteins by catalyzing their phosphorylation by ATP. Phosphorylation of a channel protein or an associated regulatory protein can increase the probability of the channel opening and the duration of its open state, and it has been proposed (Sperelakis and Schneider, 1976) that phosphorylation is a necessary condition for the opening of certain voltage-dependent channels. Other kinases can depress channel activity. The protein kinases are part of the cell's regulatory system and will be discussed further in Chapter 8. The intracellular mechanisms that regulate transport activity rely on other intracellular ligands as well, including cyclic adenosine and guanosine monophosphates (cAMP and cGMP), ATP itself, G proteins, and calcium ion. An example of a channel that requires phosphorylation to open is the cystic fibrosis transmembrane conductance regulator (CFTR) channel; a mutation in the gene for this protein is responsible for cystic fibrosis. This chloride-selective channel is activated when one of its domains, the R-domain, is phosphorylated by a protein kinase. It has generally been believed that the R-domain ordinarily plugs the channel at its cytoplasmic end, and that phosphorylation unplugs the channel to expose an ATP binding site. Intracellular ATP is then harnessed to provide the energy needed to open the channel. A more recent study (Chappe et al., 2005) suggests that the phosphorylated R-domain activates the channel by interacting with other domains in the

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Figure 4.3. Ion channel models. (a) Gramicidin A, a tubular ionophore. (b) A 1994 model of a voltage-gated potassium channel. [Reprinted with permission from Alberts et al. (1994). Copyright © 1994, Garland Science.] This channel has four subunits, two of which are shown. Each subunit has six membrane-spanning domains, which are numbered. The pore is surrounded by the subunits and partially lined by a loop (the P-loop) of amino acids between Domains 5 (S5) and 6 (S6). One of the six domains (S4) is charged and acts as the principal voltage sensor. (c) Acetylcholine (ACh) receptor channel: (1) an earlier cartoon (J NIH Res 1:152, 1989) and (2) a contemporary reconstruction based on cryoelectron microscopy [Reprinted with permission from Unwin (2000). Copyright © 2000, Royal Society.] It is now known that the ACh receptors are limited to the D subunits, as shown on the more recent image; asterisks mark the ACh binding sites. Each of the five subunits contributes the same transmembrane domain to line the pore. The parenthesized terms are specific to neuromuscular transmission. The recent image shows that sodium ions enter the muscle cell (sarcoplasm) laterally through gaps between the adjacent subunits. (d) Potassium-selective channel (KcsA) in Streptomyces bacteria (Doyle et al., 1998). The ions normally pass from the cytoplasm into the exterior medium. This channel does not possess the voltage activation “hardware” of eukaryotic potassium channels; each of the four subunits has only two transmembrane domains, M1 and M2. The gating region of the pore is near the cell interior. (e) Calcium release channel in skeletal muscle, top and side views [Reprinted with permission from Orlova et al. (1996). Copyright © 1996, Macmillan Publishers.] Rotation within the large cytoplasmic region (CY) of the protein gates the membrane pore that passes through the transmembrane region (TM).

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Figure 4.4. Energy profile for an ion passing through a channel.

protein rather than by dissociating from a blocking site. The need for ATP rather than a simple ligand to open the channel suggests that an unusual amount of energy is needed to do so. Channels can also be activated by the ion to which they are selective; this is known as substrate activation. The solute facilitates its own transport. Inhibition occurs when other solutes compete for the receptor sites and do not cause the gate to open when they are bound to them. The effectiveness of an inhibitor depends on the binding affinity of the receptor site for it, relative to the site's affinity for solutes that open the gate. Inhibitors can also act by binding to a site near the receptor and blocking activator access to it. Often a gate is controlled by multiple binding sites on different subunits, and the dependence of channel permeability on activator concentration (cA) is sigmoidal in shape. At low concentrations, most of the binding sites are unoccupied, and the probability that a given site is occupied is proportional to cA. Assume that n binding sites are associated with each gate and that the gate is much more likely to open if all of them are occupied. If the sites are independent, then the probability that they are all occupied is proportional to c An . When n > 1, this probability, which is also the fraction of channels that are open at any instant, increases rapidly with cA (Fig. 4.5). The fraction of channels that are open cannot exceed unity, and this asymptote is approached as cA becomes large and the fraction of binding sites that are unoccupied becomes negligibly small. We shall see later that the AChR channel exhibits this behavior, which has been termed positive cooperativity. The most widely used techniques for studying channel dynamics all rely on measurements of the electrical properties (e.g., current, voltage) of membrane preparations. Consequently, as illustrated by Figure 4.5, channels are more often characterized by

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their conductance than by their permeability. These properties are different, although the descriptive material that follows makes little distinction between them. The relationships among permeability, conductance, and mobility are presented in Chapter 7, where the distinction is important.

Figure 4.5. Variation of channel-mediated conductance, G, with activating ligand concentration, cA; n = number of gate-controlling sites. The conductance is Gmax when all channels are open.

The response of membrane conductance to membrane potential reflects the action of the transmembrane electric field on a charged segment of the channel or an associated receptor, known as the voltage sensor. The electric fields across biological membranes can be substantial: a cell membrane potential of 100 mV acting across a 57 nm bilayer creates a field of 2 q 10 V/m. A change in the field presumably causes translation or reorientation of the voltage sensor, accompanied by changes in the conformation of the protein and the conductance of the channel. The voltage sensor transduces the potential energy in the electric field into the mechanical work needed to open the gate. The transient motion of the charged sensor in response to a change in the local electric field is thought to be responsible for the brief gating currents that are observed under these conditions (e.g., Armstrong and Bezanilla, 1973). The sensor generally responds to changes in membrane voltage by generating a series of small gating currents, each accompanying a distinct conformational change of a subunit or domain, until sufficient charge is translated to open the channel. Voltage-gated channels are crucial to the function of excitable cells such as nerve and muscle, and we will discuss them further in Chapter 9. Using x-ray diffraction data, Long et al. (2005a,b) computed the structure of a voltage-dependent K channel from rat brain, including the voltage sensor, at 0.29-nm resolution. The structure is shown in Figure 4.6a.

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Figure 4.6. Please see insert for color version. (a) Stereo side view of the Kv1.2 potassium channel. A possibly regulatory E subunit is docked to the cytoplasmic end of the channel. Each subunit is a different color. The TM region spans the cell membrane and is the integral protein component of the channel complex. The T1 domains of the subunits form a tetrameric complex within the cytoplasm that lies on the axis of the pore, so the ions that exit the cell through the pore enter the channel through side portals. Negative charges at the rims of the portals contribute to the cation selectivity of the channel. [Reprinted with permission from Long et al. (2005a). Copyright © 2005, AAAS.] (b) Proposed voltage sensor. The S4–S5 linkers of three of the four subunits are shown in red; the tipping of the linker toward the cytoplasm upon closure can be seen clearly. The S6 segments are shown in blue. Left panel: open state. Right panel: closed state. [Reprinted with permission from Long et al. (2005b). Copyright © 2005, AAAS.]

The pore portion of this channel is quite similar to that of the bacterial channel in Figure 4.3d, including the selectivity filter and the organization of the helical domains. Voltage sensitivity in the eukaryotic potassium channel is provided by several positively charged amino acid residues, called gating charges, on the S4 transmembrane

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segment (Fig. 4.6b). The S4 and S5 segments are connected by a linker region that runs parallel to the bilayer just inside the cell. Long et al. propose (2005b) that electric field-induced motion of the S4 segment causes the nearer end of the linker region between S4 and S5 to move further into the cell, which in turn causes S6 to move in a way that closes the gate. The precise motion of the sensing segment, and the extent to which it moves when gating occurs, are still uncertain. Since gating reflects a conformational change of the protein, it is reasonable to describe the gating process using chemical kinetics. The simplest model would be: E

ZZX C YZZ O, D

(4.2)

where C designates the closed state, O the open state, and D and E are the first-order rate constants for closing and opening, respectively; thus, if Ni is the number of channels in the ith state, dN C  DN O  E N C . dt

(4.3)

A similar equation can be written for dNO /dt. Let NT equal the total number of channels under consideration, equal to NO + NC. The steady-state value of NO/NT is easily shown to be a function of the two rate constants:

 N O ¬­ žž ­  E . žžŸ N T ®­­ D E s.s.

(4.4)

These rate constants are influenced by membrane potential. Anderson and Stevens (1973) analyzed the current noise at the frog neuromuscular junction and found that the rate constant for channel closing, D, was an exponential function of membrane potential: A'\

D = Be

,

(4.5)

where A and B are (temperature-dependent) constants. This function also describes the voltage sensitivity of other preparations. Through relationships such as Eq. (4.5), the membrane potential can affect the frequency with which channels open and the mean duration of the open state. Naturally, an increase in the fraction of channels that are open at any instant (or, equivalently, the fraction of time that each channel is open) is seen experimentally as an increase in membrane conductance. We shall see in a later section that the membrane potential can also affect the conductance of the open channel by altering the energy profile within the pore. These effects allow membranes containing voltage-activated channels to exhibit a graded response, in which their macroscopic conductance is a smooth function of membrane potential. Gated channels can respond to more than one agent. For instance, potassiumselective channels in rat muscle are activated by calcium and by membrane depolarization. Calcium increases both the frequency of channel opening and the duration of the open state. At a given intracellular calcium level, depolarization further increases

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the frequency of transition to the open state (Fig. 4.7); similarly, the influence of depolarization is increased by raising the calcium level.

Figure 4.7. Effect of intracellular calcium level and membrane potential on the open fraction of K(Ca) (calcium-activated potassium) channels in rat muscle cell membrane. Reprinted with permission from Barrett et al. (1982). Copyright © 1982, Blackwell Publishing.

Mechanically activated channels are a relatively recent discovery. They are found in sensory organs and in nonexcitable cells unrelated to the senses, such as kidney, brain, and blood vessels (Oleson et al., 1988). Some pass only anions, and some only cations, with varying degrees of selectivity. These channels are of considerable interest as potential mediators of the biological and physiological response to mechanical forces in the body, such as hearing, the remodeling of bone under stress, or the response of arteries to shear stress caused by blood flow or tensile stress in hypertension. It appears that mechanosensitive channels use a variety of mechanisms to sense stretch, including a direct response to membrane tension. The cytoskeleton plays an important role as well, modulating the tension experienced by the channel in response to an external force, and transmitting the forces that accompany membrane distortion to the neighborhood of the responsive channel, or directly to the channel itself. A recent review is Hamill and Martinac (2001).

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Figure 4.8. Mechanical activation of a hair cell channel as proposed by Hudspeth and colleagues. (a) Stereo scanning electron micrographs of a sensory hair bundle of a bullfrog. The bundle is ca. 5 Pm in diameter. [Reprinted with permission from Hudspeth (1989). Copyright © 1989, Macmillan Publishers.] (b) Transmission electron micrograph of adjacent cilia and (arrows) attached tip links, which may serve as gating springs. The tip links, thought to be comprised of the protein cadherin (Siemens et al., 2004), connect the top of one cilium to the side of the higher adjacent one. [Reprinted with permission from Jacobs and Hudspeth (1990). Copyright © 1990, Cold Springer Harbor Laboratory Press.] (c) Proposed gating mechanism (Hudspeth, 1989). The tip of the cilium on the left is linked to the gate of a channel in the cilium on the right. An external force causes the two cilia to tip toward the right, so the cilium on the left slides down the cilium on the right, pulling on the gating spring and opening the gate.

Sensory mechanically gated channels, such as those in the hair cells of the inner ear, appear to rely on direct mechanical interaction with the gate. The channels in hair cells are on a bundle of cilia that extend from the cell, and respond directly to soundinduced ciliary motion. A model in which the relative motion of two adjacent cilia

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connected by an elastic gating spring opens a gate in one of them is illustrated in Figure 4.8. A more comprehensive review of the role of mechanically gated channels can be found in Gillespie and Walker (2001). Kinetic analyses of fluctuations in channel conductance suggest that the gating of channels is not a simple open-and-shut case. Channels can have a number of kinetically distinguishable closed states, and multiple open states whose durations or conductances differ. The existence of multiple open states suggests an alternative mechanism for positive cooperativity: when more of the sites controlling a given channel are occupied, the equilibrium among its open states favors those whose durations or conductances are larger. We will see some examples of this behavior later this in this chapter, and in Chapter 9. The kinetics of the transition of channels among closed, open, and inactivated states is a subject of continuing interest. More complex models include inactivated states that can be reached only from the open state, inactivation of channels only some of whose gate-controlling sites are occupied, and transitions among inactivated states.

4.1.7. Classification of Ion Channels. Aquaporins and Gap Junctions To this point, we have focused exclusively on ion channels. The selectivity of these channels varies, depending on the size and charge of the selectivity filter. The least selective channels are water filled and pass all small ions and even small nonelectrolytes. These nonselective channels appear to be very large, 2–3 nm in diameter, and have conductances comparable to those of aqueous solutions. Diffusion through these channels is not very different from free diffusion, and their conductances, in contrast to those of the narrower channels, do not saturate in concentrated solutions. These 8 9 channels can pass 10 –10 ions per second at 100 mV. As noted earlier, the selectivity filter is generally charged or lined with polar groups. As we examine channels with increasingly narrow filters, the filter first excludes ions on the basis of valence (although valence selectivity can also be provided by charges elsewhere in the pore or near its mouth), and then on the basis of size. Valence-selective channels pass only those ions having a particular valence, but are otherwise not very selective. The larger pore size implied by the limited selectivity of these channels is reflected in their conductances, which can be an order of magnitude larger than those of ion-selective channels. Ion-selective channels pass selected ions having a particular valence, and have been subjected to more study than the larger channels. Ion selectivity is governed by the selectivity filter of the channel and the steric factors described earlier. These pathways exhibit a wide range of unitary, or single channel, conductances, mostly in –12 –1 7 the 5–50 pS range (1 pS = 1 picosiemen = 10 ohm ), and pass about 10 ions per second when the driving force is 100 mV. These are discussed in detail in Hille (2001), which is the authoritative text on the subject. These channels can be further classified according to the ion to which they are most permeable, and whether they are ligand or voltage gated. Within the group of ion-selective channels, there is no clear relationship between selectivity and conductance.

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While Hille (2001) provides a comprehensive summary of the various families of ion channels and their members in excitable membranes, we present here a less detailed classification that includes non-excitable cells. The channels are sorted by the ion they chiefly transport. Several of these have been described already and more will be discussed in context of their function later in this text. — voltage-gated sodium channels. These channels are critical to action potential propagation in nerve. — epithelial sodium channels (ENaCs), responsible for absorption and secretion of the ion. — voltage-gated delayed rectifier potassium channels (e.g., the Kv1.2 channel, Fig. 4.6). These channels are responsible for most of the potassium current seen during action potential propagation in nerve. — voltage-gated fast transient potassium channels. These channels, also called A-type channels, open transiently and facilitate the translation by the nervous system of stimulus level into action potential frequency. — voltage-gated slow M-type potassium channels, that close during synaptic transmission to increase the responsiveness of the postsynaptic neuron (Chap. 9). — inward rectifier potassium channels. These channels exhibit their greatest conductance when the cell potential is more negative than the resting potential (the cell is hyperpolarized), and the potassium current is directed into the cell. — ATP-sensitive potassium channels. These inward rectifiers are found in smooth muscle and cardiac tissue, and play a role in the regulation of insulin secretion; they are gated according to the intracellular ADP/ATP ratio. — calcium-activated potassium channels. These ubiquitous channels are important mediators of the regulatory role of calcium in all cells, and are responsible for spontaneous pacemaker activity by individual excitable cells. — voltage-gated calcium channels (e.g., the calcium release channel in skeletal muscle, Fig. 4.3e). Extending beyond their role in muscle function, these channels mediate the entry of this important regulatory ion into all cells. — ligand-gated calcium channels, such as the IP3-activated calcium release channel in endoplasmic reticulum, which plays an important role in calcium signaling, as do calcium-dependent calcium channels in the plasma membrane of nonexcitable cells. — mechanically gated cation and chloride channels, discussed in the previous subsection. These channels are involved in the regulation of blood vessel diameter and cell volume. — various neurotransmitter-gated channels that pass sodium, potassium and chloride (e.g., the cation-selective AChR channel, Fig. 4.3c). Neuro-

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transmitter-gated chloride channels play a role in inhibitory chemical synapses (Chap. 9). — other ligand-gated cation channels involved in sensory transduction, such as photoreception and olfaction. — voltage-gated chloride channels. Some of these channels appear to play a role in kidney function; others contribute to cell volume regulation. — ligand-gated chloride channels. Some of these channels, which are important in secretion, are gated by calcium; others are regulated by cyclicAMP through protein kinases (Chap. 8). Kinase activation of the CFTR chloride channel was described earlier. A sampling of monovalent cation channels, including their unitary conductances, ionic selectivities, and number density on the cell surface, is presented in Table 4.5. The channels classified above facilitate the passive flux of solutes, primarily ions, between the cytosol and the extracellular domain. Not all channels share these characteristics. We have already met one such exception, in the previous chapter: the nuclear pore that allows solutes of all sizes to pass between the cytosol and the nucleoplasm. We now briefly describe two more classes of channels: aquaporins, which facilitate the flux of solvent, and gap junctions, which bridge the cytosols of adjacent cells. Aquaporins. Water crosses the cell membrane via several kinds of facilitators. These include the large nonselective channels mentioned above that also pass ions and small electrolytes; certain regulatory channels such as the cAMP-regulated Cl channel; and possibly some carrier systems. However, the primary transmembrane route for water is via specialized water channels called aquaporins, which are specific for H2O. Transport through large pores has been studied for many years, and solvent flux through them can be described using the formalism of nonequilibrium thermodynamics, to be discussed in Chapter 6. Aquaporins, which we describe below, are a more recent discovery. Before describing aquaporin channels, we should note the evidence that such channels, through which water flows rather than diffuses, exist. First, the notion that water crosses the membrane through an aqueous path rather than across the lipid bilayer is strongly supported by the observation that the temperature dependence of water transport follows the reciprocal of aqueous viscosity; the temperature dependence of transport through a lipid film is much stronger than this. Although this observation excludes transport through the bilayer, an expected result given the low solubility of water in lipid, it does not tell us whether solvent traverses the alternative route, an aqueous channel, via flow or by diffusion. This is because an inverse relation between solvent flux and viscosity is consistent with transport by either diffusion (recall the Stokes–Einstein equation in Chap. 2) or flow (see the expression for hydraulic conductivity in Chap. 7).

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Table 4.5. A Selection of Channels that Pass Monovalent Cations

Channel Na channel, squid axon Na channel, frog node of Ranvier Delayed rectifier K channel, squid axon Gramicidin A cationselective channel Ca-activated K channel, human erythrocyte Na channel, mouse skeletal muscle Inward rectifier K channel, frog skeletal muscle Acetylcholine receptor cation channel, rat muscle Ca-activated cation channel, rat heart Ca-activated high conductance (BK or maxi-K) K channel, rat muscle Maxi-K channel, rabbit sarcoplasmic reticulum

Unit conductance, a pS

Number density, channels/Pm2

Selectivity

Reference

4 8

330 1900

Na ~ Li >> K > Rb > Cs Li ~ Na >> K > Rb,Cs

(a, b) (c, d)

12

30

K > Rb >> Cs, Na, Li

(a)

15

Cs > Rb > K > Na > Li

(e)

18, 17

K >> Na

(e, f)

23.5

65

(g)

26

1.3

(h)

35, 25b

Cs > K > Na > Li

(e)

35

K ~ Na

(e)

187, 220

K > Rb >> Cs,Li,Na

(f, i)

130

K > Rb > Na > Li

(e)

Number density is obtained by dividing the maximum conductance per unit area of membrane by the unitary conductance of the channel. The number density is underestimated if not all of the channels are open when the maximum conductance is measured. (a) Conti et al. (1975); (b) Chandler and Meves (1965); (c) Conti et al. (1976); (d) Hille (1972); (e) taken from Latorre and Miller (1983); (f) taken from Petersen and Maruyama (1984); (g) Patlak (1988); (h) Matsuda and Stanfield (1989); (i) Blatz and Magleby (1984). a 1 pS = 1 picosiemen = 10–12 ohm–1.

The evidence for flow being the dominant mode of water transport across the cell membrane comes from experiments in which a water chemical potential difference was imposed across the membrane by adding a solute to one side, thereby generating a flow of solvent into the phase containing the added solute. This flow, termed osmosis, will be explained in much more detail in Chapter 6. The osmotic flow was five times greater than the diffusive rate measured using tritiated water as a tracer. It is estimated that up to 90% of the water flux across the erythrocyte membrane 9 crosses through aqueous channels; transport rates can exceed 10 water molecules per second (Agre and Kozono, 2003). The first identified aquaporin was isolated from erythrocyte membranes in 1988 (Denker et al., 1988). For this discovery, Peter Agre, in whose laboratory it was made, won the Nobel Prize in chemistry in 2003. Since 1988, aquaporins have been found in many other tissues, including the kidney, eye, brain, and gut. The original aquaporin protein, aquaporin-1, has a molecular weight of 28 kDa and consists of four subunits, each comprised of six membrane-spanning domains that surround a pore. Loops between two pairs of domains in each subunit extend part of the way into the bilayer —

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one loop from the intracellular side and one from the extracellular side — to line the pore (Fig. 4.9a). The entire oligomer consists of a cluster of four identical water channels, one from each subunit. This model of the aquaporin-1 channel is known as the hourglass model (Jung et al., 1994) because the loops that form the pore converge within the membrane. Most aquaporins are selective for water, but some do pass small nonelectrolytes. They are essentially impermeable to ions.

Figure 4.9. Aquaporin-1 (AQP-1). (a) Oblique view of a single subunit (Jung et al., 1994). The six domains are numbered in sequence. The loops between Domains 2 and 3 and between Domains 5 and 6 (denoted by a finer line for clarity) line the pore. The link between Domains 4 and 5 is partially obscured. (b) Stylized bipolar orientation of water molecules within the pore (Jensen et al., 2005). The water “wires” are aligned by hydrogen bonds (dashed lines), and the orientation of the molecules is altered by a strong positive dipole caused by amine groups lining the pore.

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Molecular dynamics simulations (de Groot and Grubmuller, 2001; Jensen et al., 2005) indicate that the water molecules pass through the channel pores in a single file, interacting through dipole forces with one another and with polar carbonyl and amine groups that line the pore. The water dipoles in the file are oriented in what has been termed a bipolar structure, with oxygen toward the center of the pore and the hydrogen atoms toward the ends of the pore (Fig. 4.9b). Selectivity to water relies on a complex electric field in the pore that results from the particular structure of the protein; only a small molecule with a large dipole moment, like water, can follow the + field and pass through. Protons, the most likely ion to pass the channel (as H3O ), are excluded by a positively charged region at the center of the pore. The dipole moment of a polar molecule measures the energy of its interaction with an electric field. When a molecule encounters a complex electric field, the force on it that tends to align it with the field is larger if its dipole moment is large. If, in addition, the molecule is small, less force is needed to align it. Consequently, a small molecule with a relatively large dipole moment is particularly nimble when trying to follow a complex electric field.

Gap Junctions. Gap junctions provide an aqueous path connecting the cytoplasms of adjacent cells, and are found in most tissues. The junctional complex in each cell consists of six subunits of a protein called connexin, which are arranged like the staves of a barrel to surround a pore that is about 1.5 nm in diameter (Fig. 4.10); each subunit consists of four transmembrane domains. The pore extends from the cell interior through the cell membrane and into the gap of about 2–4 nm between the outer leaflets of the cell membranes of the adjacent cells. The hemichannels in each cell membrane line up to produce a continuous aqueous path, consisting of 12 connexin subunits, between the interiors of the two cells.

Figure 4.10. Organization of gap junctions. 1 Å = 0.1 nm. Reprinted with permission from Purves et al. (2001). Copyright © 2001, Sinauer Associates.

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Gap junctions pass ions and small molecules up to 1 kD in size; since it is a conducting path (typical unitary conductances are in the 30–300 pS range), it also dissipates differences between the intracellular voltages of neighboring cells. Like ion channels that communicate with the extracellular space, gap junctions are gated and can fluctuate between open and closed states. Some are voltage gated, with conductances that depend on the difference between the intracellular potentials of the communicating cells. Their permeability can also be affected by hormones like dopamine, and by intracellular composition, decreasing when the cytosolic pH becomes too acidic or when intracellular calcium levels rise. Decreases in cytosolic pH and increases in intracellular calcium are signs of cell injury. It appears that the closure of the junctions acts as a quarantine at the cellular level, initiated in this case by the affected cell itself. Gap junctions allow cells to “communicate” with one another, and provide a pathway for electrical potential to spread, and solutes to diffuse, between cells. This serves several functions: — transmission of signals between excitable cells via gap junctions is more rapid than the chemical synapse that will be described in Chapter 9. This is particularly valuable when cellular activity must be synchronized, as in cardiac contraction and peristalsis. — many organs, like stomach and liver, are designed to respond to neurotransmitter signals from the autonomic nervous system, but not every cell in the organ is innervated; gap junctions transmit the signal from cells at which nerves terminate to cells where they do not. — there are natural cell-to-cell variations in metabolism that lead to differences in intracellular composition; these differences are dissipated when there is a path between cells for solute transport. — in embryonic development, cells destined to form specific organs communicate through gap junctions so that they can develop in a coordinated fashion.

4.1.8. Carrier Models A stylized model of the most common form of transmembrane carrier is depicted in Figure 4.11a. Like channels, these transport mediators span the membrane and consist of multiple subunits and/or domains. The operation of the dimeric, alternating access, carrier in the figure is straightforward. Suppose the concentration of the carrier substrate is higher in Phase I than Phase II. The carrier flips randomly between the two states in the figure. When the carrier is open to Phase I, it can bind a substrate molecule from that phase. Binding to one of the two sites on the carrier sterically inhibits binding to the other one, so carriers usually transport one molecule of substrate at a time. When the carrier flips to face Phase II (a process that substrate binding can accelerate; see §4.2.9), the substrate dissociates from the carrier and enters that phase. More than 1000 different carriers are believed to operate in this way.

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Figure 4.11. Stylized and molecular models of transmembrane carriers. (a) Alternating access model. An aqueous path extends from an external phase (I or II) to the receptor sites (solid circles) when the carrier is open in that direction. The “gate” that is formed at the abutment of the two protein molecules flips from one end of the pore to the other when the carrier undergoes a conformational change. Generally, only one substrate molecule fits in the channel between the two oligomers. The notation in the figure will be described in the following section of this chapter. (b) Two-gate model of the Cl–HCO3 exchanger (Knauf and Pal, 2003). See description in text. (c) Molecular model of the lactose permease of E. coli. The aqueous internal cavity of the molecule is open to the cytoplasmic side. A bound substrate molecule can be seen as a black shape between the domains within the membrane. The kissing arrows indicate the motion of the two domains that flips the gate from the external face to the internal one. Reprinted with permission from Abrahamson et al. (2003). Copyright © 2003, Federation of the European Biochemical Societies.

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An alternative carrier model (Knauf and Pal, 2003), which has been proposed for the Cl–HCO3 exchanger in red blood cells, in shown in Figure 4.11b. This exchanger will be described later in the chapter and is essential to CO2 transport in blood. In this model, the carrier is a channel with gates at both ends; the gates open and close randomly, but in a coordinated fashion such that they are not both open at the same time. When the internal binding site is unoccupied, an anion can enter the channel through an open gate, and bind. If the gate through which the anion entered closes, the anion is trapped in an occluded state, and it can cross the membrane if the first gate that opens subsequently is not the one through which it entered. Naturally, solute can cross either of these carriers in either direction. When only one substrate species participates in the transport process, the net flux of the substrate will be from the phase in which its concentration is higher to that in which it is lower, just as in free diffusion. The kinetics of single-substrate carrier transport will be modeled mathematically in the next major section of this chapter. The molecular structure of an alternating access carrier in the E. coli bacterium has recently been elucidated by Abramson et al. (2003) at 0.35-nm resolution, using xray diffraction techniques, and is shown in Figure 4.11c. This particular carrier actually functions as a countertransport system that accumulates lactose in the cell through secondary active transport, which will be discussed briefly in this chapter and in more detail in the next. The protein contains a single sugar binding site and is made up of two domains, each of which consists of six membrane-spanning helices connected by a flexible loop. The authors suggest that the translocating flip of the carrier can consist of a 30º clockwise rotation of the domain on the left, and a complementary 30º counterclockwise rotation of the one on the right, around an axis perpendicular to the plane of the figure and passing near the binding site. We mentioned earlier that carriers used to be thought of as ferryboats, binding substrate and crossing the bilayer from one side of the membrane to the other. Although this model is not the best description of native carriers, it is the mechanism used by carrier ionophores such as valinomycin, which contains a polar pocket for its ionic cargo and presents a hydrophobic exterior that allows it to pass through the bilayer with relative ease. Carrier ionophores are also ion selective; valinomycin, for instance, is specific for potassium.

4.1.9. Carriers and Channels: Convergences and Differences The convergence of carrier and channel models referred to at the beginning of this section is illustrated by the carrier models of Figure 4.11. Consider the alternating access model. Some of the characteristics of this model are unique to carrier transport: the solute is bound to a specific site and remains bound while transiting from one side of the membrane to the other (i.e., while the carrier changes conformation), and at no time is there an unobstructed path across the entire membrane. On the other hand, there are real similarities between the two: when the protein domains in the alternating access model block access to the binding sites from one phase or the other, the effect is much like that of a gate at the end of a channel; transport is enabled by a small con-

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formational change that opens a “gate”; and the binding sites on a carrier protein barely move, like the sites on the wall of a pore. Indeed, the two-gate model (Fig. 4.11b) stipulates that the carrier is effectively a channel with a gate on each side of the binding site. Some of the differences between the behavior of carrier and channel transport systems are summarized below. These are generalizations, and there are exceptions to almost all of them. 1. Carrier transport rates are generally more sensitive to temperature (i.e., have a higher activation energy) than channel-mediated fluxes. Solute passing through an aqueous channel experiences a viscous drag as it moves relative to the solvent in the pore. This is only one of the resistances the solute encounters but, as in free diffusion, its activation energy is much less than that of the conformational change of a carrier protein. The latter figure is typically 15–20 kcal/mol. 2. Carrier transport rates are less than channel-mediated rates. A single car2 5 rier might move 10 –10 substrate molecules across the membrane each second; we have seen that ions can pass through a single channel at rates 6 9 of 10 –10 per second. 3. When the carrier binding sites are not highly selective for a single solute, the flux of the primary substrate can be influenced more strongly by other solutes and their transmembrane concentration differences. Competition for binding sites on such a carrier leads to competitive inhibition, which reduces the transport rate of the primary substrate. Inhibition will be discussed in a later section in this chapter. The effect of one solute on the transport rate of another also underlies several mechanisms of active transport, which will be presented briefly in this chapter and discussed in greater detail in the following one. Two of these mechanisms are cotransport and countertransport. In cotransport, two solutes, the primary substrate and the cosolute, are both bound to the carrier and cross from Phase I (the cis side) to Phase II (the trans side) together. The flux of the primary substrate from Phase I to Phase II is accelerated by increasing the concentration of cosolute in Phase I. In countertransport, the carriermediated flux of the primary substrate is stimulated by adding cosolute to the trans side of the membrane. Carriers that perform countertransport are known as exchangers. Although solute interactions have a greater effect on carrier transport, channels are not immune to them. No channel is absolutely selective for a single species. For instance, the permeability of potassium in sodium channels is perhaps 5% of that of sodium. When more than one ionic species can compete for the transport sites in a pore, inhibitory effects are seen that are not very different from the inhibition of carrier transport. As a consequence, the selectivity of a channel can depend on the relative concentrations of the competing ions. Channels can also be blocked by ions that can enter the pore but cannot pass through the selectivity filter.

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Furthermore, as mentioned earlier, the rates of transport through gated channels are influenced by a variety of solutes that affect the dynamics of the gate. 4. Carriers transport ions and nonelectrolytes across membranes, but most channels pass only ions or water. Nonelectrolytes can traverse large and nonselective channels, but not the narrower, more selective ones; thus, if a facilitated transport system is found to be selective for a particular nonelectrolyte, especially a large nonelectrolyte, then the mediator is almost certainly a carrier. As noted earlier, carriers, which rely on specific binding interactions, are generally more selective regarding their substrate than are channels.

4.2. KINETICS OF FACILITATED TRANSPORT The transport of solutes via carriers and channels has been the subject of extensive kinetic analysis. Almost all of the carrier models that have been proposed obey very similar kinetics. Mathematical models of channel transport are more diverse, reflecting a broader range of assumptions regarding the biophysics of the transport process. Channel models tend to be specific to particular mediators, which can differ with respect to gating kinetics, the role of charges in the pore and electrostatic effects in general, and the relative sizes of the pore and solute. Several models of channel gating and transport will be presented below, as well as two less specific aspects of channel kinetics: energy barrier models and current noise analysis. Related topics reserved for later chapters include: (1) the use of models of solute diffusion and convection through narrow channels to estimate the size of pores in cell membranes (Chap. 7); (2) the Hodgkin–Huxley model of the kinetics of gated channels in nerve (Chap. 9); and (3) a continuum model of flow through large channels, developed to describe glomerular filtration (Chap. 10). With respect to carrier transport, we begin with the simplest model of an alternating access carrier and demonstrate its application to sugar transport across the red cell membrane. After that, we discuss more complex models, ending with models in which the carrier can bind multiple substrates. This will lead to a discussion of inhibition in the final section of this chapter, but in this section it is used to introduce the cotransporters and exchangers whose role in active transport will be an important part of the chapter to follow.

4.2.1. Models of Ion Channel Transport: Overview Whereas all models of carrier transport are essentially straightforward extensions of the alternating access model, models of channel transport fall into two distinct classes, which can be thought of as macroscopic and microscopic. The macroscopic approach uses the continuum description of ion transport embodied in the electrodiffusion equation and its solutions. The details and statistical aspects of channel function are ig-

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nored and the channel-containing membrane is regarded as a simple barrier with an experimentally determined permeability or conductance. Since these latter properties are obtained experimentally, the continuum approach can work well as long as the conditions to which it is applied do not differ too much from the conditions under which the membrane properties were measured. Permeability and conductance obtained in this way can of course be interpreted in terms of more microscopic channel properties, but these properties do not enter explicitly into the macroscopic description of the behavior of the membrane in which the channels are embedded. The most commonly used solutions of the electrodiffusion equation are the constant-field equation [Eq. (2.55)], and equivalent circuits or electrical analogs, which will be developed in Chapter 7. This was the approach taken by Hodgkin and Huxley (1952) in their groundbreaking study of conduction in nerve, which we will cover in Chapter 9. The microscopic approach, which will be the focus of the subsections to follow, builds the transport behavior of a channel-carrying membrane from the statistical properties of a large number of individual channels, each operating independently. All of these methods take gating into account and the consequent statistical aspect of transport through a single channel or a collection of channels. These models employ rate theory, noise analysis, and formal kinetics. An important tool for testing these models is the voltage clamp: measuring transmembrane current and its fluctuations while the potential difference across a portion of a channel-containing membrane is fixed. These data can be used to parameterize kinetic models of the transitions that accompany ion binding within channels, channel opening and closing, and the conformational changes that accompany activation. Although all of these models recognize the importance of gating and are developed in the context of solute movement through a single channel, they do fall into two distinct classes. The energy barrier and kinetic models of channel transport that we discuss first regard the pore as a sequence of energy barriers and wells that represent gates, filters, and binding sites. This profile is static, and the problem is either (1) to find the fluxes across an array of such pores when the boundary conditions are specified or, alternatively, (2) to determine the shape of the energy profile or the rate constants for barrier transit from experimental flux data. The effect of opening or closing a gate is represented by a change in the height of the energy barrier it presents to the approaching ion. A second class of models focuses on the statistics of the gating process itself. The channel is generally assumed to have a fixed conductance (which can be treated by the models just described) when in any given open state, and the challenge is to develop or parameterize a model of gating kinetics that describes experimental results. The seminal advance in this area was the voltage clamp, which allowed time-dependent measurements of the conductance of single channels in a small patch of membrane, with submillisecond resolution. When a larger patch of membrane containing voltageactivated channels is clamped at a fixed potential, the transmembrane current is characterized by a mean value on which are superimposed fluctuations as the individual channels in the membrane open and close. The use of both the mean current and current fluctuations to illuminate the gating process will also be described below.

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Finally, it should be noted that, beyond these latter models, high-performance computations can be made of molecular transport through discrete water-filled channels whose shape, charge distribution, and dynamics are specified in detail. These molecular model calculations may or may not retain a continuum description of the solution phase. With continuing increases in computing power, they are an exciting area of contemporary biophysical research, but they are outside the scope of this text.

4.2.2. Energy Barrier and Binding Models of Channel Transport We saw earlier that the binding sites along the wall of a narrow channel can be modeled as a series of energy barriers and wells. This is illustrated in Figure 4.12a. The barriers at the membrane–solution interfaces must be crossed when the ion enters or leaves the pore, while the barriers within the membrane are crossed when the ion hops from site to site. For analysis purposes, it is generally assumed that, apart from the gates, the energy profile in the membrane does not change with time and is unaffected by the binding of ions to sites in the pore. We begin this section by using absolute rate theory to develop expressions for the ion transport rate across a barrier system when the concentrations in the bounding phases are dilute enough that individual ions do not interfere with one another in the pore. Such a system exhibits independence: the transport of any given ion is unaffected by the presence or flux of any other ion in the system, and the flux equation assumes a form similar to that in free diffusion. Next, we develop a kinetic model that recognizes that there is a limited number of pores in any real system, and only a limited number of ions can be bound at any time. Since ions compete for binding sites, the independence principle no longer applies. This system is seen to demonstrate saturation, which has been observed in real channel systems. Another process in which ionic independence fails is single-file diffusion. In the simplest model of single-file diffusion, the ions all move down the pore by binding to the same sequence of sites. They cannot pass one another, and an ion cannot hop to an adjacent site if it is occupied. In dilute solutions, the rate equations for single-file diffusion approach the free diffusion form, because each ion is not likely to find another ion bound to an adjacent site. We will touch on this briefly in a description of occupancy states. Independence also fails if the energy barrier between two sites is altered by the binding of an ion elsewhere in the pore. There are certainly local effects of this kind. One model proposes a “repulsive” interaction between bound ions: the binding of one ion makes it more likely that a nearby ion will hop further away — or out of the channel.

Ionic Independence: Absolute Rate Theory. The kinetics of transport through a series of energy barriers and wells can be described by the theory of absolute reaction rates (e.g., Johnson et al., 1954). According to this theory, the rate at which ions of the ith species move from the jth well (or binding site) to the (j + 1)st well is

ki , j l j 1  ci , j ki , j l j 1 ,

(4.6)

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Figure 4.12. (a) Energy barrier model of an ion channel. There is no electrostatic potential gradient across the membrane, so PiI  PiII ( Pi is defined in the text). The energies of hydration and binding associated with ion selectivity at the entrance to the pore are indicated. (b) Effect of an applied potential on the energy profile. Curve 1: energy profile at the jth and j + 1st wells in the absence of an applied field. Curve 2: increment in local electrochemical potential, relative to that at x = a, caused by an applied transmembrane potential, '\. The potential profile is drawn as linear, though it need not be. Curve 3: energy profile at the jth and j + 1st wells in the presence of the applied field. Curve 3 is obtained by adding to Curve 1 the electrostatic contribution to Pi from Curve 2.

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where ci,j has been termed the internal concentration of the ith species in the jth well. The rate constant for the transition is

ki , j l j 1  Ae

( Pi , j / j 1 Pi , j ) / RT

,

(4.7)

where A is a constant and Pi w Pi0 ziᑠ\ is the concentration-independent part of the electrochemical potential (again neglecting pressure effects). As shown in Figure 4.12a, the first electrochemical potential in the exponent is that at the top of the barrier between the two wells, and the second is that in the well. The potential difference is the height of the barrier as seen from the jth well, and is the activation energy of the transition. The exponential term is the fraction of the ions in the jth well that have sufficient energy to cross the barrier. The preexponential term A can be regarded as the rate at which ions attempt the transition. The rate of transition across the barrier in the opposite direction is given by a similar expression, except that the activation energy is the height of the barrier as seen from the (j + 1)st well: J i , j 1l j  ci , j 1 Ae

( Pi , j / j 1 Pi , j 1 ) / RT

.

(4.8)

The internal concentrations {ci,j} are not the same as concentrations in free solution. For a membrane that contains a large number of identical pores, ci.j can be regarded as the number of these pores, per unit membrane area, whose jth sites are occupied by an ion of the ith species, divided by Avogadro's number. To relate the internal concentration to a free solution concentration such as ciI , consider the exchange of solute between Phase I and the first well inside the membrane. From Eq. (4.6), the net flux into the membrane at x = 0 is J i  J i ,Il1  J i ,1l I  ciI ki ,Il1  ci ,I ki ,1l I .

(4.9)

When Phase I and the first well are in equilibrium, Ji = 0. Substituting Ji = 0 into the preceding equation, ci ,1 

ciI ki ,Il1 ki ,1l I

(equilibrium) .

(4.10)

The rate constants in energy barrier models are sensitive to changes in membrane potential. Assume that a fraction, Dj, of the potential drop across the membrane occurs between the jth and (j + 1)st wells; Dj is referred to as the electrical distance between the wells. As can be seen from Figure 4.12b, the imposition of a potential difference across the membrane causes the activation energy on one side of the barrier to increase by ziᑠDj'\/2 (assuming the barrier is midway between the wells and the local field is constant); the activation energy on the other side falls by the same amount. For a cation, the transition from the jth well to the (j + 1)st well is favored by an increase in '\. The effect of membrane potential on ki , j l j 1 is obtained using Eq. (4.7):  zi ᑠD j '\ ¬­ ­. ki , j l j 1  (ki , j l j 1 )'\0 exp žžž žŸ 2 RT ®­­

(4.11)

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The rate constants in the opposite direction are given by an equation identical to Eq. (4.11), except that the sign of the exponent is changed. The energy barrier model has been used to interpret the current–voltage relationship of potassium channels in excitable cells. Like the constant-field equation, absolute rate theory predicts that the conductance of these channels increases when the extracellular potassium level is high and the intracellular potential is made more negative. This is known as inward rectification. Unless the membrane is assumed to contain very few barriers, an excessive number of rate constants (or, equivalently, barrier heights) must be evaluated to describe the transport process in quantitative terms. For instance, seven independent rate constants are needed to model a three-site channel that passes a single species. Because the number of experimental conditions that can be varied is small, the number of distinct rate constants that can be evaluated experimentally is also limited. Thus it has often been assumed that many of the barriers in a multibarrier membrane are identical; this reduces the number of distinct rate constants, and hence the number of transport experiments that are needed to characterize the energy profile. Another alternative is to model the pore as containing a limited number of internal binding sites, and correspondingly few barriers; we will see some examples of this in the next subsection. In analyzing the effect of potential differences across the membrane, assumptions must also be made regarding the partitioning of the potential drop across the several barriers; i.e., the {Dj}. The series of barriers that determines the permeability of a given channel is not the same for all ions. The factors affecting channel selectivity come into play here; indeed, the barrier heights that determine the rate constants at the membrane–solution interfaces can be related to the energies of hydration and binding discussed in the previous section. Saturable Channels: Kinetic Analysis. In the previous treatment of transport through a sequence of binding sites, no account was taken of the possibility that an ion might find an adjacent site occupied by another ion; all that was necessary for an ion to proceed over a barrier was that it have sufficient energy to do so. As pointed out earlier, such models implicitly assume that the solutions adjacent to the membrane are dilute enough — or the channels are numerous enough — that such interference is so rare that it can be ignored. In these systems, each ion proceeds independently of the others and is unaffected by the presence of other species. One characteristic exhibited by real biological channels is saturation; that is, for a given trans concentration of an ion, the ionic flux increases asymptotically to a limit as the cis concentration is raised. Systems in which ions cross independently are not expected to exhibit saturation; thus, when it is seen, it is taken as evidence that the transitions of individual ions across the membrane are no longer independent of one another. In the context of energy barrier models, independence fails when increasing numbers of ions occupy binding sites in the pores, thereby impeding the free passage of ions that characterizes independence. This is increasingly likely to happen as the number of ions seeking entry into the pores (i.e., the ambient ionic concentration) increases relative to the number of channel sites per unit area of membrane.

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To deal with occupancy issues, it is useful to model transport as a kinetic process, in which there is a finite concentration, CBT, of binding sites. Denoting the binding sites by B and the ion by S, and assuming that the ion binds to only a single site in transiting the membrane, the kinetic scheme is: k

k

1

2

II 1 2 ZZZ X ZZZ X S I B YZZ Z SB YZZ ZS B. k k

(4.12)

In the steady state, the concentration of occupied binding sites is constant; thus, the net rate at which S enters from Phase I and is bound equals the net rate at which S is released from the pore into Phase II. These rates, per unit area, are equal to the net flux across the pore array, JS.: J S  k1cSI cB  k1cSB  k2 cSB  k2 cSII cB .

(4.13)

Furthermore, the total concentration of sites is CBT: cB + cSB = CBT.

(4.14)

Equations (4.13) and (4.14) constitute three equations in cB, cSB, and JS. Solving for the flux, JS 

C BT (k1cSI  k1D) , 1 D

(4.15a)

where

cSB k1cSI k2 cSII D ,  cB k1 k2 or

J S  C BT

k1k2 cSI  k1k2 cSII . k1 k2 k1cSI k2 cSII

(4.15b)

I

For a given cis concentration cS , the flux is maximized when the trans concentration II cS = 0. Under these conditions, the flux simplifies to

J S  C BT I

II

k1k2 cSI . k1 k2 k1cSI

(4.16)

When cS is small and cS = 0, D is also small, indicating that most binding sites are unoccupied. Equation (4.16) shows that, under these conditions, the flux approaches I proportionality to cS , as in free diffusion (one of the exercises at the end of the chapter demonstrates that this is also the case for energy barrier models when the ions cross the membrane independently). Equation (4.16) also shows that, even when the ion is absent from Phase II, there is a maximum transit rate across the pore irrespective of

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the upstream concentration; the pore saturates. At very high values of cS , virtually all the pores are occupied (D is large; cB << cSB | CBT ), and the maximum flux is equal to k2CBT; every site is discharging ions into Phase II as fast as they can dissociate. This model can be generalized to include pores that contain a sequence of sites but which nonetheless can accommodate only one ion at a time. Such pores exhibit saturation properties similar to the one-site pore described above. We shall see that saturation is also a characteristic of carrier transport and that the II carrier-mediated flux follows a form identical to Eq. (4.16) when cS = 0, but the meaning of the coefficients is somewhat different. In contrast to the simplest model of carrier transport, this channel model does not predict symmetry; that is, the rates are different in magnitude if the concentrations on the two sides are reversed. Not surprisI ingly, the limiting flux when cS = 0 is determined by k–1 rather than k2, since the pore unloads into Phase I.

Figure 4.13. Modeling a saturable pore. (a) Dependence of the unitary conductance of glycine receptor channels in mouse spinal cord on ambient chloride ion activity, demonstrating channel saturation. Km is the chloride concentration such that the channel conductance is half of the maximum value Jmax; for a one-binding site model, it can be expressed in terms of rate constants using Eq. (4.16). (b) (opposite page) (1) Physical and (2) energy barrier models of a saturable neurotransmitter-gated anion channel with two binding sites in series. In (1), the carrier is in State SB1. Transport is single file. It is assumed in (2) that there is no electrical potential difference across the pore. The chloride free energy in the ordinate is referred to the ambient solutions, in which the chloride activity is the same. The analysis of the model depends on only the heights of the barriers and the depths of the wells, so the energy profile is conveniently represented by straight lines. Reprinted with permission from Bormann et al. (1987). Copyright © 1987, Blackwell Publishing.

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Saturation of biological channels has been regarded as evidence that internal binding sites are present. Hille (2001) points out that an internal binding site is sufficient for saturation, but it is not necessary, since continuum and dynamical models of pore transport predict that saturation can be caused by electrostatic repulsion within the channel. Figure 4.13a demonstrates the saturation of anion-selective glycine receptor channels in mouse spinal cord (Bormann et al., 1987). Bormann et al. modeled this channel by a symmetric energy profile containing one or two binding sites; the twosite model and the physical pore it is taken to represent are shown in Figure 4.13b.

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The kinetic analysis described above can be coupled to the original energy barrier model by defining the rate constants in terms of barrier heights, using Eq. (4.7). The one- and two-site anion channel models referred to above were treated in this way, using saturable kinetics. Similarly, a transmembrane potential gradient can be incorporated into the saturating pore model by modifying the rate constants according to Eq. (4.11). Both of these adaptations were used by Hille (1975) in a 3-well model of a saturable sodium channel in frog myelinated nerve (Chap. 9); such a model has seven independent barrier heights and three independent electrical distances. The energy profile and physical interpretation of its shape are shown in Figure 4.14.

Figure 4.14. (a) Energy profile for ion transport through a sodium channel in frog nerve at 5ºC. (b) Physical interpretation of (a). [Reprinted with permission from Hille (1975). Copyright © 1975, The Rockefeller University Press.] Passage through the pore is facilitated by electrostatic interactions with the negative carboxyl oxygens at binding sites at the “bottom” of the pore (B and D). The principal barrier (C) is at the narrow selectivity filter. Passage through the selectivity filter is rate limiting.

The behavior of saturable channels in which a traversing ion must bind to several sites in series depends on the number of ions that can be in the channel at one time. The sodium channel model just described is based on the assumption that only one ion can occupy the pore at any instant. If we describe the occupancy condition of each site in a three-site channel by O for occupied and E for empty, the occupancy state of such a channel passes through the following reversible transitions: EEE (all sites empty) l OEE (binding site on the left is occupied) l EOE l EEO l EEE (ion has been released into the solution on the right). When multiple ions can be present in the channel at a given time, additional states like OEO and EOO are possible, and the number of transitions can increase rapidly. The simplest such channel has two binding sites and can exist in four states: EE, OE, EO, and OO. The possible transitions for such a system are mapped in Figure 4.15. The kinetics of this system are described by first-order rate constants, as shown in the figure. The rate equations are straightforward and are presented in Keener and Sneyd (1998), along with references to earlier work on multi-ion models. As above, the rate

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constants and their voltage dependence can be expressed or interpreted in terms of the energy profile in the pore.

Figure 4.15. State diagram for a two-site channel with multiple occupancy. The first-order rate constants for each transition are subscripted according to the transitions they represent.

4.2.3. The Patch Clamp and Two Applications The patch-clamp technique played a crucial role in understanding and modeling the gating kinetics of ion-selective channels. In one application of this technique, a micropipette a few microns in diameter is used to isolate a small patch of membrane containing only one or a few channels. The micropipette is also used to maintain a fixed potential difference across the membrane (the voltage clamp). Under these conditions, the transmembrane current fluctuates from zero to an integral number of unitary currents, i, as the channel(s) open and close (as noted earlier, some channels have more than one open state, each with a corresponding unitary conductance, and this too is reflected in the patch clamp record). We will see in Chapter 7 that a reasonable driving force for ion transport through the channel is '\ – Ei, where '\ is the applied potential difference and Ei is the Nernst potential of the ion across the membrane. Dividing the unitary current by the driving force, we obtain the unitary (or unit) channel conductance: J

i . '\  Ei

(4.17)

The patch-clamp technique was demonstrated by Neher and Sakmann in 1976 (Neher and Sakmann, 1976). It was refined in 1981 (Hamill et al., 1981) to improve the electrical resistance and tenacity of the seal between the micropipette and the membrane, increasing the versatility of the technique. In 1991, Neher and Sakmann won the Nobel Prize in physiology or medicine for this important development and their research employing it. Figure 4.16a shows one of their original recordings, from

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an ACh receptor channel in frog. A more recent trace obtained in this way, showing the current through a patch of frog muscle membrane containing several channels, is presented in Figure 4.16b. The technique readily allows the influence of transmembrane voltage and ligand concentration on channel dynamics to be examined.

Figure 4.16. Patch clamp recordings of gating activity. (a) A frog ACh receptor channel. [Reprinted with permission from Neher and Sakmann (1976), as inverted by Aidley and Stanfield (1996). Copyright © 1976, Macmillan Publishers.] (b) A patch of frog skeletal muscle membrane containing several ATP-dependent potassium channels. [Reprinted with permission from Spruce et al. (1987). Copyright © 1987, Blackwell Publishing.] Gate opening is indicated by a downward pulse in (a) and an upward deflection in (b).

Two applications of the patch-clamp technique will be described below: analysis of the kinetics of the acetylcholine receptor channel, and the parameterization of a model of a voltage-gated sodium channel. Acetylcholine Receptor Channel Kinetics. The evolution of our understanding of the kinetics of the AChR channel is a good illustration of the influence of the patchclamp technique on the course of ion channel research. In this discussion, we borrow heavily from the more thorough description given by Hille (2001). We begin with a simple kinetic model in which a receptor on the channel binds a single acetylcholine molecule, and consequently opens: E ¶¶ l AR YZZ ZZX R A k¶ AR* , ¶ D

(4.18)

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where R is the receptor channel, A is the ligand or agonist acetylcholine, and the asterisk denotes an open channel. It is assumed that the acetylcholine binding and dissociation reactions are rapid, so that the left-hand reaction is at equilibrium. The first-order rate constants for channel opening and closing, respectively, are E and D. We shall see later that this model of gating kinetics — a single, rate-limiting transition from a single open state to a single closed state, with equilibrium binding of agonist — is consistent with the frequency spectrum of the measured channel current. The equilibrium binding condition allows us to condense the kinetic model to ZZX AR* . R A YZZ

(4.19)

However, this model is contradicted by other experiments on the channel. First of all, the endplate conductance at equilibrium was found to be a quadratic function of acetylcholine concentration, cA, when the concentration was sufficiently small that most of the receptors were unoccupied. We saw earlier that such positive cooperativity suggests that ACh must be bound to both receptor sites on the D subunits of the channel before the channel can open. The “improved” kinetic model looks like this: 2k

1 ZZZ X R A YZZ Z AR , k 1

E ZZZX Z ZZX AR A YZZZ A 2 R* , Z A 2 R YZZ D 2k k2

(4.20)

2

where the receptor sites are occupied sequentially, and the k's are rate constants. The rate constants for binding of the first agonist molecule to R, and for the dissociation of the first agonist molecule from A2R, are written as 2k1 and 2k–2, respectively, since both receptors are initially available prior to binding and either of the two ligands on A2R can dissociate first. Subsequent patch-clamp measurements at higher temporal resolutions (Colquhoun and Sakmann, 1981, 1985) showed that, while the receptor was in the open state, it flickered closed for brief submillisecond intervals. This process can be interpreted in terms of the preceding kinetic scheme. When a channel closes and becomes A2R, it can follow either of two paths: it can reopen, with rate constant E, or it can release one ligand molecule and become AR, with rate constant 2k–2. The most likely course of action depends on the relative magnitudes of the two rate constants. Reopening after a brief closure becomes more likely as E increases relative to 2k–2. For mouse skeletal –1 muscle, the two rate constants are comparable, in the 30–60 ms range. Figure 4.17 illustrates how the channel can vary among the several states, and how this is reflected in the conductance measurement. Although Eq. (4.20) describe many aspects of AChR channel kinetics, other kinetic pathways have also been identified. The open channel can lose one ligand without closing, becoming AR*, but the rate constant for this is much less than D; hence channel closure generally precedes dissociation. The channel can open without binding ACh (forming R*), or after binding a single ACh (forming AR*), but these transitions are relatively unlikely too.

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Figure 4.17. Hypothetical temporal variation of the state and conductance of a single acetylcholine receptor channel. The first time the open channel closes, it reopens promptly, but the second time, it loses a ligand before it can reopen. However, the empty receptor site is reoccupied before the channel loses its remaining ligand, and the channel ultimately opens again. Toward the end of the trace, a ligand binds to the naked channel but dissociates from the receptor before a second ligand can bind. The kinetics of AR binding and dissociation depend on the local ACh concentration.

In principle, the channel can occupy six different states: it can bind zero, one, or two agonist molecules, and it can be open or closed [it can also enter any of three closed desensitized states (with zero, one, or two binding sites occupied) after prolonged (>20 ms) exposure to acetylcholine, but this does not happen during normal physiological function]. However, it usually moves along a series of only four of the states in sequence: R, AR, A2R, and A2R*. The other states and additional reaction paths are largely excluded because the rate constants that describe the transitions among the six states do not favor these transitions; this is why open channels with unoccupied receptors are rare. This is a common characteristic of many complex biological systems: many reaction pathways are possible, but the rate constants define the fewer paths that the system almost always follows. The parameters of the reaction pathway given by Eq. (4.20) are given in the problem set at the end of this chapter. We will have more to say about the role of the AChR channel in neuromuscular transmission in Chapter 9. Voltage and Current Sensitivity of a Voltage-Activated Sodium Channel. We noted earlier that voltage-activated channels open when a voltage sensor in the protein moves in response to a change in transmembrane potential. This motion must therefore cause the open state to become energetically more favorable — that is, of lower free energy — than it was before. We can estimate this change in free energy and predict how the fraction of channels in the open state might change with membrane potential. To relate the fraction of channels that are open to the free energy of the opening process, consider the opening process as an equilibrium reaction: C U O . Let Ni be the number of channels in the ith state (i = C, O) and NT be the total number of

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channels, equal to NO + NC. From Chapter 1, the equilibrium constant for channel opening is K

 'G 0 ¬­ NO ­­ ,  exp žžž NC Ÿ RT ®­

(4.21)

0

where 'G is the standard free energy change of the transition to the open state. Substituting NC = NT – NO, and solving for the fraction of channels that are open, 1

 'G 0 ¬­¯ N O  ¡ ­­° .  ¡1 exp žžž N T ¢¡ Ÿ RT ®­°±°

(4.22)

Assume that, when the voltage sensor moves to open the channel, the accompanying movement of charge is equivalent to n electronic charges crossing the membrane, of thickness a, in which the channel is embedded. The electric field across the membrane is '\/a, where '\is the membrane potential. Thus, the potential energy given up by the sensor when it moves is nᑠ ¸ a ¸

'\  nᑠ'\ a

(using the Faraday instead of the electronic charge to be consistent with the use of 0 0 mols as the measure of quantity). Replacing 'G in Eq. (4.22) by 'G – nᑠ'\, 1

   'G 0 ¬­  nᑠ'\ ¬­¯ NO ­­ exp žž ('\)  ¡¡1 exp žžž ­­°° . ž ­ Ÿ ® NT RT RT Ÿ ® ¢¡ ±°

(4.23)

Taking the reciprocal of both sides, rearranging, and taking the logarithm, N ¬ 'G 0 nᑠ'\  ln žžž T  1­­­  žŸ N O RT ®­ RT

(4.24)

Let f be the fraction of channels that are open under a particular membrane potential 1 ¬ '\. Equation (4.24) indicates that ln žž  1­­­ is linear in '\, with a slope from which žŸ f ®­ the number of gating charges can be calculated. Hartshorne et al. (1985) used a voltage clamp to record the opening kinetics of sodium channels from rat brain, from which they obtained f as a function of '\(Fig. 4.18). Fitting this data to a form of Eq. (4.24) employed by French et al. (1984), they determined that the apparent gating charge, n, was approximately 4. From Eq. (4.17), the unitary current through an open sodium channel is iNa  J Na ('\  ENa ) .

(4.25)

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CH. 4: FACILITATED DIFFUSION: CHANNELS AND CARRIERS

Figure 4.18. Fraction of voltage-gated sodium channels from rat brain that are open, as a function of the potential across the bilayer into which they are incorporated (Hartshorne et al., 1985). The curve through the data is fitted by eye.

For a region of membrane containing NT sodium channels, the current INa is equal to the product of the unitary current and the number of channels that are open. Using Eq. (4.23), 1

   'G 0 ¬­  nᑠ'\ ¬­¯ ° ­ exp žž I Na ('\ )  J Na ('\  ENa ) N T ¡¡1 exp žžž ­ ­­° . ž ­ Ÿ ® RT RT Ÿ ® ¢¡ ±°

(4.26)

Equation (4.26) shows that the membrane potential affects the current carried by voltage-gated sodium channels through its influence on both the driving force for transport and the fraction of channels that are open. To predict this dependence, it is necessary to know the unitary conductance of the channel, the sodium activities on both sides of the membrane (needed to calculate ENa), the total number of channels, the standard free energy change of channel opening (or the equilibrium constant, K), and the apparent gating charge. For now, we just consider the shape of the current–voltage curve of a membrane patch containing these channels. The easiest way to get a sense of the shape of the current–voltage curve is to plot the driving force and open fraction separately, as is done in Figure 4.19. The driving force is a linear function of voltage; the number of open channels is a sigmoidal function of '\as it varies from very large negative values (NO/NT o 0) to very large positive values (NO/NT o 1). The slope of the driving force line is J and its '\-intercept is ENa; the asymptote of the number of open channels is NT, and K and n determine the steepness of the latter curve and its location along the '\axis. For very large negative values of '\, more negative than ENa, the current is zero because essentially all of the channels are closed. As the potential increases, channels start to open, but '\ < ENa, so the current is negatively directed, and becomes more so

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as more channels open with increasing membrane potential. However, as the membrane potential approaches the sodium Nernst potential from below, the driving force for ion flux falls sufficiently to cause the magnitude of the negative current to decrease, and it becomes zero when '\= ENa, since the ion is then at equilibrium. As '\continues to increase, both the driving force and the number of open channels increase; as the open fraction approaches unity, the slope of the current–voltage curve approaches the constant value of JNT.

Figure 4.19. The effect of membrane potential on membrane current through voltage-sensitive channels. (a) The separate effect of potential on the driving force for transport. (b) The separate effect of potential on the number of channels that are open. (c) The combined effect of potential on membrane current.

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In this analysis, the only transition undergone by the channel is from the closed state C to the open state O. We have noted several times that ion channels generally pass through several closed states before opening. In this simple model, C can be thought of as the closed state that immediately precedes channel opening. A more detailed model would apply equations analogous to those derived above to each transition of the channel, each with its own standard free energy change and equivalent gating charge. Also, the use of constant values of the {Ni} ignores the temporal variations in these quantities as individual channels open and close. Thus, an assumption of the analysis is that there are enough channels in the patch being modeled that such fluctuations perturb the mean only slightly. We will not make this assumption in the next subsection, where information regarding channel kinetics is derived from the current fluctuations under voltage clamp.

4.2.4. Stochastic Properties of Channels: Membrane Noise Analysis When the potential difference across a membrane containing a large number of channels is fixed, the membrane current fluctuates around a mean value as individual channels open and close. The statistics of these fluctuations contains information about the unitary conductance of the channel and the kinetics of the gating process. Assume that a patch of membrane contains N independent channels and that the unitary conductance of a single open channel is J. Let p be the probability of a given channel being open at any instant. On average, Np channels are open at any time, and the mean conductance is G  NpJ .

(4.27)

Since the channels operate independently, the number of channels, n, that are open at any instant follows the binomial distribution. Therefore, the variance of n is V2n  Np(1  p) .

(4.28)

Since the instantaneous conductance of the patch, G, is nJ, its variance is VG2  Np(1  p)J 2 .

(4.29)

Divide Eq. (4.27) into (4.29), to eliminate N:

VG2  (1  p)J . G

(4.30)

If the channels are closed most of the time, then p << 1 and the ratio of the variance of the conductance to the mean is a good estimate of the single-channel conductance, J. Note that this estimate does not rely on any assumptions regarding the reaction path that governs channel opening, as long as all of the open states have the same unitary conductance.

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When the potential difference across the membrane is held constant, the instantaneous current, I, is proportional to the instantaneous conductance; under these conditions, current fluctuations can also be used to find J: Jx

V2I ( p  1) , I (V  Veq )

(4.31)

where V is the potential at which the membrane is clamped and Veq is the membrane potential at which the mean current I  0 . Alternatively, Eqs. (4.27) and (4.29) can be combined to eliminate p. In terms of current fluctuations, V2I  iI 

I2 , N

(4.32)

where i is the contribution to the current from a single open channel. This equation was used (Sigworth, 1980) to derive the number of sodium channels in the frog node of Ranvier (Chap. 9), and their unit conductance, from a series of measurements of the mean and variance of the current. With this technique, a prior knowledge of p is not needed. The single-channel conductance can also be estimated from the frequency spectrum of the current (Anderson and Stevens, 1973). The current spectral density, SI(f), where f is frequency, is obtained from the current record by standard Fourier transform techniques (DeFelice, 1981) Assume that the channel has only two states — open (O) and closed (C) — and that the transition between the states follows firstorder kinetics. The gating reaction and the rate equation for the number of open channels are: E

ZZX O , C YZZ D

(4.2)

dN O  E N C  DN O , dt

(4.33)

where D and E are the first-order rate constants, NC and NO are the number of chan2 nels/cm membrane that are in the closed or open states, respectively; and t is time. For this simple reaction scheme, the spectral density of the current fluctuations is related to the unit channel conductance and the rate constant for channel closing by

SI ( f ) 

2 I J (V  Veq )

D[1 (2 Sf / D )2 ]

.

(4.34)

A frequency spectrum obtained by Anderson and Stevens for the endplate current at the frog neuromuscular junction is shown in Figure 4.20. The current fluctuations are caused by the gating of the AChR channels in the endplate (Chap. 9). Equation (4.34) fits their data well and has been used to extract the values of J and D from the experimental spectrum; note that D cannot be found from the mean and variance of the current.

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Figure 4.20. Spectral density of current fluctuations at the frog neuromuscular junction. The curve through the data is given by Eq. (4.34), with I = 80 nA, J = 32 pS, V = 60 mV, Veq = 0, and D = –1 0.132 ms . Based on the ratio of the variance of the current fluctuations to the mean [Eq. (4.31)], Anderson and Stevens obtained a value of 20.5 pS for the single-channel conductance of this pathway, not too different from that obtained from the frequency spectrum. Reprinted with permission from Anderson and Stevens (1973). Copyright © 1973, Cambridge University Press.

2

Equation (4.34) can be written as SI(f) = SI(0)/[1 + (2Sf/D) ], where SI(0) is the limiting spectral density as f o 0. SI equals half of the limiting value at a frequency f0.5 = DS. Thus, D can be obtained from f0.5, after which J can be found from SI(0), since all the other terms that comprise the limiting spectral density are known. The rate constants D and E determine the rate at which the gate closes and opens, and the average duration of the channel in the open and closed states. The average duration of either state is obtained from its probability distribution, pi(t) (i = C, O); pi(t)dt is the probability that a gate remains in the ith state for a length of time between t and t + dt. For the simple first-order processes of opening and closing given by Eq. (4.2), the form of the probability distribution is the same for both: pi(t) = ki exp (–kit), where kC = E and kO = D. The average time the channel spends in each state, , is the expectation value of t: d

 ti   ¨ tpi (t )dt  0

1 . ki

(4.35)

This is a reasonable result; the higher the rate of transition from one state to the other, the shorter the average duration of the initial state. Moreover, this equation shows that

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161

the transition rate constants can be determined from the distribution of the state durations measured in a patch clamp experiment. Membrane noise analysis is discussed comprehensively in the text by DeFelice (1981) cited earlier; a more compact summary can be found in Weiss (1996b).

4.2.5. The Simplest Model of Carrier Transport: Assumptions Almost all kinetic analyses of carrier transport rely on similar sets of assumptions. The assumptions underlying the simplest kinetic scheme are as follows: 1. The carrier, X, transports a single solute, A, and cannot leave the membrane. 2. Each molecule of X reversibly binds a single molecule of A. A carrier can transit from Side I to Side II, or from Side II to Side I, with or without A; A, on the other hand, can cross only as AX. 3. The binding/unbinding reaction A + X U AX is rapid, and is in equilibrium at both sides of the membrane. The equilibrium constant, K, for the dissociation of AX, is the same at both interfaces. It is usually assumed that all activities can be replaced by concentrations; thus, at Side i,

c iA cXi  K (i  I, II) . c iAX

(4.36)

In Eq. (4.36), the concentration cA is defined in the usual way, in moles per unit volume of ambient solution. The other concentrations, cX and cAX, are defined per unit area of membrane. This mix of units, which would ordinarily cause problems in the definition of K, is not a problem here bei i cause cX and cAX appear in ratio to one another; c X/c AX is the (dimensionless) ratio of the number of carriers on Side i that are not occupied by A to the number that are. When analyzing experimental data, it is common to assume that the solutions on both sides of the membrane are well stirred; this allows the interfacial concentrations of A, c iA , to be replaced in the equilibrium condition by the bulk concentrations. The assumption of equilibrium at the membrane interfaces implies that the rate-limiting steps in the transport process are the transitions of the loaded and unloaded carrier between the two sides of the membrane. 4. The phases on both sides of the membrane are of infinite extent, so their concentrations are constant in spite of the flux through the membrane, and a steady state can be reached. This condition is never rigorously met in transport experiments. It is approximated well enough in flat preparations, when there is a sufficient quantity of bathing solution on both sides. However, this may not be the case when one of the phases is the interior of a cell. When an ambient substrate level changes with time, the steady-state

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assumption is replaced by a quasisteadiness condition: the transport system is assumed to respond rapidly to changing boundary conditions, so the flux at any instant can be described by the steady-state equations, using the instantaneous concentrations in the two phases. The notion of quasisteadiness was introduced in §2.1.6. 5. The rate constant, Dc, for the movement of carriers from one side of the membrane to the other, is the same for transitions in either direction. For a carrier whose transition is accomplished by a conformational change, Dc is related to the transition frequency. Furthermore, it is assumed in this simplest model that Dc is the same for X and AX. It can be argued that the assumption that the transition rate of the carrier is unaffected by substrate binding is reasonable, since the mediator is large and its transition kinetics may be quite indifferent to whether a particular binding site is occupied. Of course, if one chooses to hypothesize the opposite — for instance, that the binding of substrate triggers the conformational change that carries it across the membrane — then one would not expect the transition frequencies of X and AX to be the same.

Under these conditions, the net flux of carrier j (j = X or AX) from Side I to Side II is

J j  D a(c Ij  c IIj ) .

(4.37)

The simplest carrier is shown schematically in Figure 4.21. The carrier model described here relies on a single conformational change to carry the substrate from one side of the membrane to the other. This model is most appropriate to the alternating access carrier, since a single transition causes the substrate binding site to face the other side of the membrane. Such a carrier is always in one of four states: facing Side I or Side II, with or without bound substrate. Thus four concentrations are sufficient to describe the condition of an alternating access carrier system. If the substrate transits from one side of the membrane to the other via a series of steps involving additional, intermediate states (e.g., the occluded state of the two-gate model), then more concentrations (and more rate constants) are needed to define the system.

4.2.6. The Simplest Model of Carrier Transport: Equations The following equations define the behavior of the simplest carrier in the steady state: 1. Conservation of Carrier. As noted above, the carrier is always in one of four possible states; if the total concentration of carrier is XT, then I II c AX c AX c XI cXII  XT .

(4.38a)

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163

Figure 4.21. Two representations of the simplest carrier. (a) Kinetic. (b) Cyclic.

2. Steady State. Since the carrier is confined to the membrane, the combined transmembrane flux of X and AX must be zero. Substituting Eq. (4.37) into JX + JAX = 0, I II D a(c XI  cXII c AX )0 .  c AX

(4.38b)

3. Definition of the Transport Rate. The unknown substrate flux, JA, is equal to the flux of AX, since A can cross in no other form: I II J A  D a(c AX ).  c AX

(4.38c)

Equations (4.36) and (4.38) constitute five equations in five unknowns: the concentrations of the four carrier states, and JA. Solving for the transport rate as a function of the boundary conditions ( c AI and c AII ) and the properties of the carrier system (XT, Dc, and K), JA 

D aXT 2

 c AI c AII ¬­ žž ­  žžŸ K c I K c II ®­­ . A A

(4.39) I

II

As expected for a passive process, flux is positive (from I to II) when cA > cA , and JA I II = 0 when cA = cA . The dependence of the transport rate on substrate concentrations in the ambient phases is sketched in Figure 4.22. In free diffusion, these graphs would be straight lines. The maximum flux attainable by the system is Vm = DcXT/2, achieved when the substrate concentration in Phase II is zero and the concentration in Phase I is raised to a value high enough to saturate the carrier system.

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Figure 4.22. Dependence of carrier transport rate on substrate concentrations. The cAI -intercept of the lower curve shows that the flux is zero when c AI and c AII are equal (to C). When c AII = 0 and cAI = K, JA is half of the maximum flux, Vm.

When the concentrations in both phases are low, such that c iA << K (i = I, II), Eq. (4.39) simplifies to JA 

D aXT I (c A  c AII ) . 2K

(4.40)

Under these conditions, most of the carrier binding sites are free [see Eq. (4.36)], and the flux equation has a form similar to that in free diffusion. This is known as the diffusion limit of the carrier process. When both ambient concentrations are much larger than K, most of the binding sites are occupied, and JA 

D aXT K ž 1 1¬ žž II  I ­­­ . 2 žŸ c A c A ®­

(4.41)

The simplest carrier is symmetric: when the substrate concentrations in Phases I and II are interchanged, JA changes sign (i.e., direction), but its magnitude remains the same. This is not surprising, because the model has no directionality; the equilibrium constants are the same at both sides, and the rate constants are the same in both directions.

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165

4.2.7. Monosaccharide Transport in the Erythrocyte The erythrocyte is an ideal model for studying cell membrane transport. It is easy to obtain from a living subject. Its function is not dependent on the presence of other cells (that is, each cell is an independently functioning unit, in contrast to being part of a structured tissue), and it exists as single cells in the body. The red cell lacks a nucleus or organelles, so its interior would appear to be a single compartment. Procedures exist for experimentally modifying the internal composition of the cell and for extracting the cell membranes for biochemical analysis. The erythrocyte is not only convenient to study; it also employs many of the transport mechanisms we deal with in this text. Here, we illustrate the equations derived above by applying them to the carrier-based transport of simple sugars (monosaccharides) across the red cell membrane. This discussion also shows how radioactive tracers are used to obtain the parameters of this transport process in intact cells. We will return to the erythrocyte after the next subsection, in the context of exchange and cotransport. The transport of monosaccharides, such as glucose, across the red cell membrane is a classic example of carrier-mediated diffusion, well described by Eq. (4.39). By adjusting the glucose concentration difference across the membrane, the carrier can be made to operate at its diffusion limit, or at saturation, or at any intermediate state. Each monosaccharide carrier is selective for a specific sugar. The constants of the transport model — Vm and K — are obtained from measurements of the initial rate of uptake of radiolabeled substrate by the cell from the surrounding medium. Initially, there is no labeled solute in the cell, and the tracer flux measures only the rate at which substrate is carried inward. This unidirectional rate is I D ac AX , where Side I is outside the cell. From the equations in the previous subsection, the unidirectional influx is J AIl II 

D aXT c AI . ¸ 2 K c AI

(4.42)

The flux of tracer, denoted by the subscript “A*,” is proportional to the unidirectional influx of A: J A*  DJ AIl II ,

(4.43)

where D  c AI * / c AI is the specific activity of the tracer in the external medium. The maximum transport rate of the carrier system, DcXT/2, is reported in a somewhat modified form when measured in this way. This is because tracer concentration (measured as radioactivity) in the cell, rather than flux through the cell membrane, is the primary experimental quantity. To express the maximum transport rate in terms of concentrations, we must relate the rate of change of the intracellular tracer level to the transmembrane flux. The number of moles of tracer entering the cell per unit time is equal to the product of the tracer influx and the membrane area:

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dnAII*  J A* Acell . dt

(4.44)

Assuming that the interior of the cell is well mixed, the concentration in the cell is uniform and equal to the number of moles per unit volume, so

dc AII* 1 dnAII* .  dt Vcell dt

(4.45)

Substituting Eqs. (4.42)–(4.44) into (4.45), the following equation for the initial rate of increase of intracellular tracer concentration is obtained:

A dc AII* D aXT c AI  D ¸ cell ¸ ¸ . 2 dt Vcell K c AI

(4.46)

The maximum transport rate that is commonly reported is the initial rate of increase in the intracellular tracer concentration when the carrier is saturated (i.e., c AI >> K), divided by the specific activity. It can be seen from Eq. (4.46) that this “rate” is equal to the true maximum flux of which the carrier is capable — DcXT/2 — multiplied by the area-to-volume ratio of the cell. More comprehensive measurements of sugar transport across the erythrocyte membrane, including measurements of efflux as well as influx, and their dependence on sugar concentration, indicate that the solute that enters the cell does not reside in a single intracellular compartment, but is partitioned among three compartments: a submembranous compartment, a binding complex associated with clusters of carrier protein, and the cytosol (Cloherty et al., 1995). The identification of these distinct compartments was made using compartmental analysis, a powerful tool for describing the interaction of fluxes and concentrations in complex biological systems. Compartmental analysis will be introduced in Chapter 8, and the compartmental parameters determined by Cloherty et al. will be used in one of the exercises. Note that the organization of the compartments in the red cell does not affect the determination of the maximum transport rate, which depends on only the amount of tracer entering the cell, irrespective of the compartment within which each tracer molecule subsequently resides.

4.2.8. More Complex Carrier Models The usual practice in characterizing a transport process thought to operate by a carrier mechanism is to see if its behavior fits the predictions of the simplest model (provided, of course, that the assumptions of the model make sense for the case of interest). The most common way to do this is to measure the unidirectional flux through the carrier as a function of the concentration of substrate on the side from which the flux originates. This can be done most easily using tracers and Eqs. (4.42) and (4.43). Substituting Eq. (4.43) into (4.42) and taking the reciprocal of both sides,

PRINCIPLES AND MODELS OF BIOLOGICAL TRANSPORT

1

J

I l II A



¬ D 1 K  žžž I 1­­­ . * J A Vm žŸ c A ®­

167

(4.47)

Thus, a plot of the reciprocal of the unidirectional flux against the reciprocal of the cis solute concentration will be a straight line if the carrier follows the simple model. The I slope of the line is K/Vm, and the 1/cA -intercept is –1/K. If the carrier data do not fit the simple model, some of the assumptions of the model can be relaxed. Relaxation of these assumptions is inevitably accompanied by an increase in the number of unknown parameters of the system, and one must be careful when fitting a limited amount of experimental data to models having many adjustable parameters. The modifications to the model can relate to the boundary conditions at the membrane surface or the mechanism of the carrier itself. Regarding the former category, Thomson and Dietschy (1977) showed that when unstirred layers are in series with a simple carrier, the dependence of the substrate flux on the bulk concentrations is very different from Eq. (4.39). Some of the complexities that have been added to the model of the carrier itself are described below. 1. The Rate Constants Are Substrate Dependent. As indicated earlier, the rate constants for X and AX may differ. In this event, the rate constant Dc is replaced by two constants — DcX and DcAX — which govern the flux of free and loaded carrier, respectively. The flux equations are given by Eq. (4.37), with Dc replaced by D aj . Equations (4.38b) and (4.38c) are modified accordingly. Solving for JA,

JA 

a DXa XT K (c AI  c AII ) DAX . a c AII c AI K (c AI c AII )( DXa DAX a ) 2 DXa K 2 2 DAX

(4.48)

The transport system remains symmetric. Equation (4.48) reduces to (4.39) when DcX = DcAX. See also #5 below. 2. The Equilibrium Constants Are Asymmetric. The equilibrium constants i at the two sides of the membrane may differ, in which case K is replaced by K in Eq. (4.36). 3. The Rate Constants Are Asymmetric. The rate constant for transitions from Side I to Side II may be different from that for transitions from Side II to Side I. Such asymmetry is plausible: these mediators have different conformations when facing each side of the membrane, and it is not unreasonable that different conformations might exhibit different transition rates, as well as different substrate affinities. The solution of the rate equations when the rate constants depend on the direction of the transition but are the same for X and AX is left as an exercise. When the rate constants for X and AX are also assumed to differ, the single rate constant of the simplest model is replaced by four rate constants, and Eq. (4.37) becomes

J j  D aj c Ij  Da j c IIj ( j  X , AX ) .

(4.49)

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CH. 4: FACILITATED DIFFUSION: CHANNELS AND CARRIERS

Equations (4.38b) and (4.38c) change accordingly. In Eq. (4.49), the unidirectional rate constants are distinguished by their subscripts: Dcj is the rate constant for the transition of the jth form of the carrier from Side I to Side II, and Dc–j is the rate constant in the opposite direction. Asymmetric carrier systems exhibit rectification: when the concentrations of the solutions bounding an asymmetric carrier are interchanged, the magnitude of the substrate flux changes. The “efficiency” of the rectifier can be expressed in terms of the ratio of the fluxes in the two directions (Krupka and Deves, 1983). The equilibrium constants and transition rate constants of asymmetric systems are not independent. Facilitated diffusion is a passive process, and the substrate flux must be zero when c AI  c AII . It can be shown that, for this requirement to be met, the following must hold: II

I

DcAXDc–XK = Dc–AXDcXK .

(4.49a)

This condition is met trivially for symmetric systems. Asymmetry can arise if the substrate A is charged and is being transported through an electrostatic potential difference (see #5 below), or if a significant standard free energy change accompanies the transition of the bound or free carrier from one side to the other. In the latter case, the partition of free carrier between the two sides at equilibrium is

 'G 0 ¬­  c XI ¬­ ­­  žž II ­­ K X  exp žžž Ÿ RT ®­ žŸž cX ®­

,

(4.50)

equilibrium

0

where 'G is the difference between the standard state free energy of the carrier when it is facing Side I, and that when it is facing Side II, and KX is the equilibrium constant II I for the transition between X and X . At equilibrium, the rates of transition in each direction are equal: I

II

DcXcX = Dc–XcX (equilibrium), whence Dc–X = KXDcX (equilibrium). 0

If 'G is small, KX | 1, and the two rate constants are approximately equal. Otherwise, the rate constants are asymmetric. 4. The Uptake and Release of Substrate Are Not at Equilibrium. In this case, Eq. (4.36) is replaced by rate equations for the binding and unbinding processes. Setting the substrate flux equal to the net rate of binding at Side I, and that of release at Side II, I II J A  kbI c AI c XI  krI c AX  krII c AX  kbII c AII c XII ,

where kbi (kri ) is the rate constant for binding (release) at Side i.

(4.51)

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169

Weiss (1996a) presents the most general case of single substrate binding, in which binding and release are regarded as rate processes as in Eq. (4.51), and the transition rate constants are asymmetric and substrate dependent. The requirement that the flux II I I II be zero when 'cA = 0 implies DcAXDc–X kr kb = Dc–AXDcXkr kb . An interesting result is that the relation between flux and the bounding concentrations can be expressed in terms of only four parameters that in turn are functions of the four interfacial rate constants, the four transitional rates, and XT. Thus, it is impossible to obtain all the rate constants of the model from experimental flux measurements. 5. The Carrier–Substrate Complex Bears a Charge. If, for instance, X is neutral and A is an ion, then AX is charged. In that event, the kinetics of the transition of the complex across the membrane are influenced by the membrane potential, since the potential difference adds to the carrier concentration difference, in a fashion similar to the energy barrier models of channel transport described earlier. If X is neutral and AX is charged, DcX z DcAX and Dc–X z D'–AX because the charge status of the two species differs, and D'AX z D'–AX because the electrostatic potential gradient favors one transition and opposes the other. The flux equations for ion transport via a charged complex, when the uptake and release reactions are not at equilibrium, are summarized in Lauger (1980). 6. The Carrier Mechanism Follows the Two-Gate Model. If the carrier follows the two-gate model (Fig. 4.11b), the kinetic model of transport includes two additional states: the occluded state, with or without a substrate molecule in the space between the two gates. Referring to Figure 4.21, the vacant occluded state is interI II I II posed between X and X , and the occupied occluded state lies between AX and AX . The kinetic model is straightforward and is included among the exercises at the end of this chapter. 7. The Carrier Can Bind Multiple Substrates. When a carrier can bind and transport more than one substrate (call the second one B), its repertoire of functions is greatly increased. The versatility afforded carriers by the ability to bind multiple substrates is even greater if the binding affinities (i.e., the equilibrium constants for binding) or the transition rates of the carrier depend on what is bound to it.

When a carrier can transport more than one solute, it may function in several ways: if it carries two substrates in the same direction, it is called a cotransporter; if it carries them in opposite directions, it is called a countertransporter or exchanger. The rates of transport through such mediators depend on the transmembrane differences in the concentrations or electrochemical potentials of both A and B. Under some conditions, the carrier may employ the electrochemical potential gradient of one substrate to move a second substrate against its electrochemical potential gradient; this process is called secondary active transport and will be discussed in the next chapter. On the other hand, if both A and B bind to the same site, the transport of one substrate can be inhibited by the other as the two compete for available sites on the carrier; inhibition will be discussed in the next section of this chapter.

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Exchangers and cotransporters are found in great variety and in every cell. They deserve a subsection of their own.

4.2.9. Exchangers and Cotransporters In discussing exchangers and cotransporters, we will focus again on the red blood cell. The major anions in the erythrocyte — chloride, sulfate, bicarbonate, phosphate — are in equilibrium across the red cell membrane. The anions enter and leave the cell by a process known as exchange diffusion. The exchanger rarely crosses the membrane without an accompanying substrate molecule. Two examples of the operation of an exchange diffusion carrier are represented in Figure 4.23. Figure 4.23a shows the normal function of the carrier. The mediator can transport two substrates, A and B. It can bind either substrate at either side of the membrane, but is very unlikely to undergo a conformational transition to the other side unless its single binding site is occupied. This restriction on the behavior of the carrier can arise if the energy barrier for the conformational change of the unbound exchanger is greater than those for the exchanger–substrate complexes. The rate constants for the transition are sensitive functions of the barrier height, so D'X would be much less than D'AX or D'BX.

Figure 4.23. Exchange diffusion processes. (a) Heteroexchange. In this example, cBI and c AII are negligibly small, so the carrier rarely transits from Side I to Side II as BX, or from Side II to Side I as AX. (b) Tracer measurement of the exchange rate. The tracer is initially absent from Side II, and the flux of tracer from Side II to Side I remains negligible. The tracer flux is proportional to the unidirectional flux of A from Side I to Side II. In all cases, the unidirectional fluxes of unloaded carrier are minor and are not represented in the cycles.

Consider the behavior of such a carrier when the concentration of B in Phase I, and the concentration of A in Phase II, are negligible. Then the carrier binds A at Side I, transits to Side II, releases A into Phase II and binds B, and then returns to Side I, where it releases B into Phase I and once again binds A. This process is termed heteroexchange and can involve more than two substrates. If a substrate is present on both sides of the membrane, the carrier can carry it in either direction, just as in the

PRINCIPLES AND MODELS OF BIOLOGICAL TRANSPORT

171

single substrate model described earlier. As noted above, inhibition can take place if either bounding solution contains both substrates. Returning to Figure 4.23a, the transition rate constants of the exchanger are generally asymmetric and substrate dependent. For an exchanger whose purpose is to transfer a molecule of A from Side I to Side II, in exchange for a molecule of B transported from Side II to Side I, the rate constants DcAX and Dc–BX will be greater than Dc–AX and DcBX. When these inequalities are very strong, X will carry only A from Side I to Side II, and only B from Side II to Side I. This is an example of obligatory, or pingpong exchange. A similar result is obtained if the relative binding affinities of the carrier for the two solutes depend on whether the binding sites are facing Side I or Side II; we will see examples of this in the next chapter.

Figure 4.24. Kinetic representation of a two-substrate ping-pong system. The substrates are A and B. Only the carrier states are shown.

The most well-studied exchanger in erythrocytes exchanges one chloride ion for one bicarbonate ion; bicarbonate leaves the cell and chloride enters it. This exchanger plays an important role in maintaining the volume and internal pH of most cells (Chap. 8), in anion transport across epithelial cells, and in carbon dioxide transport in the blood (Chap. 11). The transport is obligatory and can be represented by the kinetic I cycle in Figure 4.24, with AX representing carrier bound to extracellular chloride and II BX representing carrier bound to intracellular bicarbonate. Evidence suggests that chloride and bicarbonate bind to different sites on the protein; at any time, both sites are exposed to the same side of the erythrocyte membrane, and they alternate in tan-

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dem between the two sides as the carrier undergoes conformational change. The ex4 changer normally transports about 5 q 10 ions per second. A cotransporter in the erythrocyte membrane that we will discuss in Chapter 5 transports one potassium ion, one sodium ion, and two chloride ions into the cell. This carrier, like the anion exchanger, is electrically neutral, and no net charge is transferred across the cell membrane. Other molecules can “piggyback” on exchangers that are carrying ions across the membrane. An example of such a carrier is the serotonin transporter, which takes up this neurotransmitter after synaptic transmission has occurred. This exchanger actually accumulates serotonin in the cell, pumping it in against its concentration gradient via secondary active transport, a mechanism that will be discussed in detail in the next chapter.

Figure 4.25. Kinetic representation of the serotonin (S) transporter, modified from, and following the rules stipulated by, Rudnick (1996). Normal cyclic behavior is counterclockwise. The unidirectional arrows are written to reproduce the stipulated rules. In fact, these transitions are reversible, but with equilibrium constants that strongly favor the indicated directions. In Rudnick's model, the carrier binds chloride, transmitter and sodium in sequence from the extracellular synaptic cleft (Chapter 9) and discharges transmitter, sodium, and chloride in sequence on the cytosolic side. Potassium is then bound and the carrier undergoes a conformational change to expose its binding sites to the extracellular region once more.

PRINCIPLES AND MODELS OF BIOLOGICAL TRANSPORT

173

The serotonin transporter cycle is shown in Figure 4.25. The exchanger brings one sodium ion and one chloride ion into the cell and carries one potassium ion out. Serotonin enters the cell in its cationic form, so the transporter is electrically neutral. Each transporter molecule possesses four distinct binding sites, one for each substrate. The transition rate constants of the exchanger depend on which of its binding sites are occupied, and this dependence governs its function. One proposed set of “rules” for the serotonin transporter (Rudnick, 1996) leads to the behavior in the figure; these stipulate that: — the transporter can flip from the extracellular side to the intracellular side only when sodium, chloride, and serotonin are all bound — the transporter can flip from the intracellular side to the extracellular side only when its binding site for potassium is occupied, or when none of the sites are occupied As shown in the figure, Rudnick's model also stipulates a specific sequence of binding and unbinding steps on the two sides of the membrane. Figure 4.23b illustrates the usual technique for measuring the exchange rate of the carrier. The concentration of a selected substrate, A, is the same on both sides of the membrane. A small amount of radiolabeled substrate is added to the solution on one side of the membrane (Side I in Fig. 4.23b). The exchange rate is computed from the rate of accumulation of tracer in Phase II. When the concentration of A is the same on both sides of the membrane, and no other substrate concentration difference is imposed, the unidirectional flux of A from Phase I to Phase II, measured by the tracer, is the same as that from Phase II to Phase I, and is termed the exchange flux; the net flux of A, equal to the difference between the two unidirectional fluxes, is zero under these conditions. Exchange diffusion systems are subject to trans stimulation, sometimes called accelerative exchange diffusion. Consider the cycle depicted in Figure 4.23a. Here, the flux of A from Side I to Side II is enhanced by raising the concentration of B in Phase II. Increases in cBII promote the binding of B to the carrier, facilitating its transition to Side I. As a result, the concentration of binding sites available to the molecules of A in Phase I is raised. Viewed another way, raising the concentration of B in Phase II increases the overall carrier transition rate from Side II to Side I. In the steady state, this rate must be the same as that from Side I to Side II. Thus, the carrier cycles more frequently, increasing the flux of A as well. In a ping-pong system, every carrier transition carries a substrate molecule. The equation for the exchange rate of a single-substrate ping-pong system is derived in Dalmark (1975); its form is similar to that of the flux equation for the simplest carrier: J Aex 

Mc AI . K1/ 2 c AI

(4.52)

In ping-pong transport, and in contrast to the simple carrier model described earlier, both M and K1/2 depend on the transmembrane concentration ratio, c AI / c AII . Equation

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(4.52) predicts a relatively simple dependence of exchange rate on substrate concentration, provided that the concentration ratio is fixed. The obligatory nature of ping-pong transport was used in an interesting way by Jennings (1982) to estimate the number of chloride exchangers in the red cell membrane. Cells containing radiolabeled chloride were placed in a medium, none of whose anions were readily carried by the exchange system. Under these conditions, few carriers that are facing outside flip to the inside; carriers initially facing inside carry one ionic load into the medium, and stop. The number of transport sites per cell was estimated from measurements of the initial loss of labeled chloride from the cells.

4.3. INHIBITION OF FACILITATED TRANSPORT Both channels and carriers are subject to inhibition: the decrease, limit, or block of their action or function as transport mediators. Here we focus on the inhibition of facilitated transport by a solute other than an “intended” transport substrate. Transport may also be inhibited by physical forces, such as by voltage clamping of gated channels, as described earlier. The mechanism of inhibition depends on the identity of the mediator. Inhibitors can interfere with channels by blocking the path of the desired substrate, either by virtue of their size or by attaching to binding sites intended for the substrate; this is called channel block. Inhibitors can enter from either end of the channel. They can also affect gating kinetics. Inhibitors of carrier transport operate primarily by interfering with substrate binding; ligand-activated channels can also be inhibited in this way.

4.3.1. Inhibition of Channel Transport: Channel Block It should be clear by now that much of the research on channel behavior has been motivated by the important role of ion channels in nerve and muscle function. Correspondingly, a large number of studies of channel block have dealt with the effect of toxins, anesthetics, and other drugs on native channels. These studies and others involving substances that are not natural to the organism, though they have lent considerable insight into channel structure and function, are outside the scope of this text, and the reader is once more referred to Hille (2001) for a thorough discussion of the subject. An example of channel inhibition by competition between naturally occurring ions is the voltage-dependent block of inward-rectifying potassium channels by intracellular ions. As noted earlier, these channels are called inward rectifiers because they exhibit a greater potassium conductance when the electrochemical potential gradient for potassium is such that the ion flux is directed inward. Block is seen as a further reduction in outward current. We saw earlier that potassium ions pass through these channels in a queue. Other intracellular ions, such as magnesium and larger molecular ions, will also migrate into the channel, but will not easily pass through the selectivity filter. The interfering ion will be trapped at the filter, with potassium ions backed up behind, and the transport channel will remain blocked until the blocking ion either passes through the filter or, by Brownian motion, works its way out of the pore and back into the cytosol.

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Hille and Schwarz (1978) modeled this kind of inhibition using an energy profile model consisting of three wells representing a series of binding sites accessible to both potassium and the inhibitor, terminating in an energy barrier at the extracellular end of the pore that the inhibitor could not cross. The model hypothesized that more than one binding site could be occupied at a time, and that the presence of an ion in one well affected the rate constants for the binding or unbinding of a second ion at an adjacent well; the latter was included to reflect electrostatic repulsion between ions of like charge at closely spaced binding sites. A family of current–voltage curves predicted by the model is shown in Figure 4.26. It is notable that the outward potassium current exhibits a maximum as the intracellular potential becomes increasingly positive; at high membrane potentials, it is virtually impossible to extract a plugging cation from the pore once it has reached the impermeable energy barrier.

4.3.2. Inhibition of Carrier Transport In the models of carrier transport analyzed in the previous section, the transport rate depended on the ambient concentrations of only the designated substrate, A. Often, other solutes can compete with A for the carrier's binding sites, or block A's access to these sites. When this happens, JA decreases. Competitive Inhibition. In competitive inhibition, other solutes, in addition to A, can bind to X. To examine mathematically the implications of substrate competition, consider the simplest case, in which a carrier has a single binding site that reversibly binds B as well as A. The equations of the previous section are modified as follows: I II 1. Conservation of Carrier. Two additional concentrations, cBX and cBX , are added to the left-hand side of Eq. (4.38a). 2. Steady State. The governing equation becomes JX + JAX + JBX = 0, where I II J BX  D a (cBX  cBX ) . The transition rate of BX is assumed to be the same as that of X and AX. 3. Equilibrium. Two additional equations describe the BX U B + X equilibrium at the two sides of the membrane: cBi cXi  K B (i  I, II) . i cBX

(4.53)

These last two additional equations are needed, since there are two new unknowns, I II cBX and cBX . Solving for JA, JA 

D aXT 2

  ¯ c AI c AII ¡ °.  ¡ c I K (1 c I / K ) c II K (1 c II / K ) ° B B A B B ± ¢ A

(4.54)

Weiss (1996a) presents the flux equation for competitive inhibition with rate constant asymmetry, while retaining the assumption that the transition rates of the carrier remain unaffected by substrate binding. The dependence of the flux on the terms in the bracket is the same as in Eq. (4.54), but the maximum flux is equal to DcjDc–jXT /(Dcj+Dc–j), which reduces to DcXT/2 when the rate constants in each direction are equal.

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Figure 4.26. Energy barrier model of voltage-dependent block. (a) Energy barrier model. The outside phase contains potassium; the inside phase contains potassium and an cationic inhibitor, B. Potassium can cross all barriers, and B can cross all but the leftmost one. (b) Model predictions of potassium flux as a function of membrane potential (inside relative to outside). Curves are shown for fixed internal concentrations of B and K and an external concentration of K that varies over a factor of 100, with Curve 1 corresponding to the lowest value and Curve 5 corresponding to the highest value, equal to the fixed internal concentration. The units of the ordinate, which measures outwardly directed potassium flux, are arbitrary. Reprinted with permission from Hille and Schwartz (1978). Copyright © 1978, The Rockefeller University Press.

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Comparing Eq. (4.54) to (4.39), we see that the binding of B effectively increases II the value of K in both terms in the brackets. When cA = 0, the second term vanishes I and K in the first term is increased by KcB /KB. Referring to Figure 4.22, an increase in I K raises the value of cA needed to achieve a flux equal to Vm/2; thus, the flux curve is I shifted to the right, reducing the transport rate of A for any value of cA . The reduction I in the rate of transport of A increases as cB is raised. The effectiveness of B as an inhibitor depends on the ratio K/KB, which is a measure of the selectivity of the carrier. For instance, if the carrier's affinity for B is stronger than it is for A, then KB is small compared to K, and the influence of a given concentration of B on the flux of A is relatively large. When B binds irreversibly to the carrier, the inhibitor eventually occupies all the carrier binding sites and, in the only attainable steady state, JA = 0. This result is obtained from Eq. (4.54) by setting KB = 0. Carriers subject to competitive inhibition exhibit trans stimulation even when the transition rate constants are symmetric and independent of the occupancy of the carrier binding sites. The flux given by Eq. (4.54) is proportional to the difference between two ratios, each of which is reduced by the addition of B. By our convention, the first ratio — evaluated at Side I — is the larger, and adding B to Phase I lowers JA. However, adding B to Phase II reduces the second ratio, increasing the difference between the two, and raising JA. This instance of trans stimulation can be understood in physical terms as follows: II . This increases the driving force for B competes with A for X at Side II, reducing c AX the transition of AX from Side I to Side II, causing the transition rate JAX to increase. Equation (4.54) is the basis of a simple experimental test for the presence of competitive inhibition. The flux of radiolabeled A (call it A*) from Phase I to Phase II is measured at several inhibitor concentrations, cBI . The concentration of tracer in Phase II is negligible, and Eq. (4.54) simplifies to J A* 

ac AI * , c AI * b

(4.55)

where a = DcXT/2, a constant. The second term in the denominator, b = K(1 + cBI /KB), is a linearly increasing function of inhibitor concentration. Rearranging Eq. (4.55), J A*  a 

bJ A* . c AI *

(4.56)

Thus, in the presence of competitive inhibition, plots of JA* vs. JA* / c AI * are linear, with an intercept (a) that is independent of inhibitor level. The slopes of these plots (–b) are negative, and become more so as the concentration of inhibitor in Phase I is raised. Substrate B can competitively inhibit the transport of A without crossing the membrane. Suppose DcBX = 0. Then, the carrier is effectively immobilized when B is bound to it. Modifying the earlier equations accordingly (DcBX = 0, JBX = 0), and assuming that cBII = 0, the flux of A is

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JA 

D aKXT K B (c AI  c AII ) . ( K c AII )(2 K B c AI 2 KK B KcBI )

(4.57)

It can be seen from Eqs. (4.54) and (4.57) that the effect of the inhibitor can be overcome to some extent by raising the concentration of A in Phase I; indeed, when A is absent from Phase II, the maximum substrate transport rate (which is obtained by letting c AI become large) is equal to the maximum rate in the absence of inhibitor.

Figure 4.27. Kinetic representations of two inhibited carrier systems. For clarity, only the carrier states are shown. (a) Two substrates (A1, A2) and an immobilizing inhibitor, B. (b) Irreversible inhibition by an inhibitor that can bind to all forms of the carrier. An expression for the carrier inactivation rate in terms of the inhibition rate constants and the equilibrium constants for substrate binding is given in Krupka and Deves (1983).

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The cases shown above are among the simplest models of reversible competitive inhibition. Many more situations have been treated by Krupka and Deves (1983). These include: the transport of two competing substrates in the presence of an immobilizing inhibitor; the formation of a ternary complex of substrate, inhibitor, and carrier; and a combination of these, whose kinetic scheme is shown in Figure 4.27a. Krupka and Deves also treated the effects of asymmetry in the equilibrium constants for inhibitor binding, KB, and competition between immobilizing inhibitors on the same side of the membrane. As remarked earlier, when the inhibitor binds irreversibly to the carrier, there is no substrate flux in the steady state. The experimental measurement of interest in this case is the rate at which the carrier is inactivated. The kinetic representation of an irreversible inhibition process is shown in Figure 4.27b. Noncompetitive Inhibition. Carrier-mediated fluxes can also be inhibited by solutes that do not bind to transport sites. For instance, the flux of A can be inhibited by added B if the latter binds to a site close to A's transport site, thereby blocking A's access to the carrier. The monosaccharide carriers in the erythrocyte can be blocked by large molecules, such as glycosides and corticosteroids, and noncompetitive inhibition is thought to be the mechanism by which this takes place.

PROBLEMS: CHAPTER 4 1.

The selectivity of a membrane for three of the alkali metals is Na > Li > K. Recognizing that smaller ions have more negative energies of hydration and binding, provide a set of values of these energies, in arbitrary units like those in Table 4.3, that explains this selectivity sequence.

2.

Derive Eq. (4.4) from the preceding equations. Show that the fraction of channels that are open depends only on E/D, and sketch the dependence. Why does doubling both rate constants not affect NO/NT? What does doubling both rate constants do to the kinetics of channel opening?

3.

The membrane of the squid axon is called the axolemma. Suppose a piece of axolemma is clamped between two identical physiological solutions of sodium chloride and a 90-mV potential difference is applied across it. What is the current density through the sodium channels in the membrane? The needed channel properties are in Table 4.5.

4.

Represent transport through a pore that contains a single binding site by an energy profile in which the barriers on each side of the site are of different height. In general, transport through such a pore is defined by four rate constants. (a) Solve for the steady-state flux across the membrane in terms of the rate I II constants and the bounding concentrations ci and ci ; i.e., derive Eq. (4.15b).

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(b) Using the relationships between the rate constants and the energy levels of the barriers and wells with respect to the external solution, show that the flux across the membrane is proportional to 'ci. (c) Repeat (a) for the case in which there is a transmembrane potential difI II ference '\ = \ – \ and the binding site is electrically midway between the faces of the membrane. 5.

In Eq. (4.16), the ion flux depends on the total concentration of channels, the concentration of ion on the cis side, and three rate constants. Describe how the ion flux varies as each of the rate constants is separately increased, and explain why, in physical terms.

6.

Start with the data for the conductance dependence of the glycine receptor channel in Figure 4.13. (a) Relate Km and Jmax for a single channel to the rate constants in Eq. (4.16). Km is the Cl activity at which the flux through the channel is half of its maximum value. (b) Show how the conductance–activity curve changes as each of the rate constants is separately doubled, and explain the change.

7.

Construct the state diagram for a channel containing three sites in series, where the sites are so close that two adjacent sites cannot be occupied at the same time.

8.

The parameters of the mouse ACh receptor reaction pathway given by Eq. (4.20) have been derived by Auerbach and Akk (1998) from patch clamp measurements 8 on transfected kidney cells at 22–24ºC; they are k1 = k2 = 1.1 q 10 /M-s, k–1 = k–2 4 4 3 = 1.8 q 10 /s, E = 5 q 10 /s and D = 1.2 q 10 /s. Using these data, (a) Calculate the fraction of doubly bound receptors that are open in the steady state; i.e., N A2 R* /( N A2 R* N A2 R ) . (b) Calculate the mean lifetime of a single opening. (c) What fraction of the transient closures to the A2R state result in reopening? Why is this state seen as a brief flicker when the channel reopens?

9.

A useful parameter of voltage-dependent gating is '\0.5, the membrane potential at which half of the channels are open. Use Eq. (4.24) to relate '\0.5 to the standard free energy change of channel opening, and rewrite Eq. (4.23) in terms of 0 '\0.5 instead of 'G .

10. Pick the data points off Figure 4.18 and fit them to Eq. (4.24) to find the effective gating charge of the sodium channel in brain, and the equilibrium constant for channel opening. Then compute the predicted open fraction at each potential from the best fit and plot the results on the original figure to see how well the model fits the original data.

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11. Assume that the sigmoidal curve describing the open fraction of a voltageactivated potassium channel is very steep, so that the fraction of open channels increases discontinuously from zero to one at a threshold voltage Et. Sketch the current–voltage curve for NT of these channels, when (a) Et < EK (b) Et > EK EK is the Nernst potential of potassium across the membrane. 12. Sketch a current–voltage curve, like that in Figure 4.19c, for a membrane that contains voltage-gated cation channels that are increasingly likely to be open as the membrane potential becomes more negative. Explain any differences from the curve in the figure. 13. From the variance of sodium current fluctuations at the frog node, Sig4 worth (1980) determined that the node possessed N = 4.2 q 10 channels, each carrying a unitary current of i = 0.34 pA (picoamperes) under the voltage clamp employed. (a) Sketch the variation of the current variance with mean current, as given by Eq. (4.32). How would you find N and i from this curve? (b) The driving force for the current, including the Nernst potential of Na, was 59 mV; what is the conductance of a single channel? (c) The maximum current seen during these experiments was 8.2 nA; what fraction of the channels was open at that time? 14. From the spectral density curve in Figure 4.20, derive the rate constant for gate closing, and the unitary conductance, of the endplate channel studied by Anderson and Stevens in 1973. 15. Using the values of J and D obtained by Anderson and Stevens (1973) (Fig. 4.20), calculate the quantity of charge that passes through an AChR channel during a open state of average duration when the potential across the channel is 100 mV. 16. Derive the expression for average state duration given by Eq. (4.35). 17. According to Eq. (4.39), the maximum flux attainable by a simple carrier is D'XT/2. Provide a physical explanation of why this is so, for an alternating access carrier. 18. For the simplest carrier model, what fraction of the carriers are in each state (i.e., I I II what are cAX /XT, ...) when cA = cA ? 19. A membrane containing a simple carrier specific for A separates a solution of the substrate (Phase I) from a solution in which there is no A. When the concen-

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tration of A in Phase I is 5 mM, the flux across the membrane is 1 q 10 I 2 mols/cm -s; when cA = 20 mM, the flux is doubled. (a) What is the equilibrium constant at the interfaces? (b) What is the maximum attainable flux? I (c) What is the flux of A when cA = 0 and the concentration on the other II side of the membrane, cA , is 30 mM? –9

–10

2

20. A simple carrier system transports A at a rate of 6 q 10 mols/cm -s when the concentration of A on the cis side of the membrane is 6 mM, and there is no A on the trans side. When the trans concentration of A is raised to 3 mM, the flux falls to one-third of what it was before. What are K and Vm for this carrier system? 21. Transport across a monolayer of cells can often be treated as transport across two membranes in series: one membrane faces one bath, and the second membrane faces the other bath, with the cytoplasm between the two. Suppose that the cells in such a monolayer produce a carrier for a particular solute, A, and that the concentration of the carrier in both membranes is the same. The concentration of A I in one bath is cA , and the concentration in the other bath is zero. Write an expresI sion for the intracellular concentration of A in terms of cA , and the Vm and K of the carrier. Assume that Vm and K are the same in each membrane. 22. A carrier system whose parameters are Vm and K is placed between two solutions I II and c Ab . There are unstirred layers of thickwhose bulk concentrations are c Ab ness d at each membrane surface; the solute diffusion coefficient in the unstirred layers is DA. (a) Write an expression that can be solved for the solute flux JA as a function of the bulk concentrations in the bathing solutions, the unstirred layer thickness, the diffusion coefficient in the unstirred layer, and the transport parameters of the carrier. You do not need to solve for JA. (b) Show that JA approaches the solution given by Eq. (4.39) as d approaches zero. 23. Derive Equation (4.41). 24. You run an experiment on the uptake of sugars by red blood cells, a system you suspect is describable by simple carrier kinetics. You use radioactive tracers to make your flux measurements, which give you the flux Js as a function of the I II sugar concentration on Side I (cs ), when cs = 0. How would you plot these data to get a straight line, and how are K and DcXT related to its slope and intercept? 25. Demonstrate mathematically that Eq. (4.42) for unidirectional flux follows from the unidirectional rate of transition of the bound carrier from Phase I to Phase II, I Dcc AX.

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26. The values of K and Vm for glucose transport across the cell membrane of human 2 erythrocytes are 5.6 mM and 0.0403 μmol/cm -min. What is the initial rate of decrease of cytoplasmic glucose concentration when the plasma glucose level is reduced to half of the 4.3 mM concentration inside the cell? The cell can be modeled as a disk 7.2 microns in diameter and 2.2 microns thick. 27. The glucose concentration in the cytoplasm of the erythrocyte described above is 10 mg/100 cc and that in the blood is 85 mg/100 cc. In the steady state, the rate of entry of glucose into the cell equals the rate of glucose consumption by metabolism. What is the metabolic rate of the red cell, in kcal/hr-cell? One mol of glucose yields 456 kcal of usable energy. Both the bath and the cytoplasm may be regarded as well mixed. 28. (a) Derive Eq. (4.48) for the carrier flux when the rate constants are substrate dependent. (b) Show that JA is symmetric for this case. (c) Show that Eq. (4.48) reduces to Eq. (4.39) when DcX = DcAX = Dc. 29. Derive an equation for the flux through the simplest carrier when the equilibrium constants at the two sides of the membrane are different, and show that the carrier flux remains zero when the concentrations of substrate are the same on both sides of the membrane. 30. Derive an equation for carrier flux when the rate constants for the transition of the carrier from Side I to Side II differ from those for the transition from Side II to Side I, assuming that the rate constants for AX and X are the same. 31. (a) Construct the state diagram for the two-gate channel model. Denote each state by abc, where a is the state of the inside facing gate (Open or Closed), b is the state of the substrate binding site (occupied by A or Empty), and c is the state of the outside facing gate (O or C). Thus, the state of the channel when both gates are closed and the binding site is occupied is CAC. Recall that both gates cannot be open at the same time. (b) Assume that the transitions in the state diagram in which substrate binds to or dissociates from the channel binding site (E U A) are at equilibrium with dissociation equilibrium constant K, and that all gating transitions (O U C) have the same rate constant Dc. Write the equations from which the flux of A through the system, JA, can be found, usout in ing XT, cA and cA as for the simple carrier. (c) Solve the equations for JA. 32. Explain why the Cl–HCO3 exchanger in the normal red cell is expected to be asymmetric, and give one reason that the Na–K–2Cl transporter might be.

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33. Explain the shape of the curves in Figure 4.26, and their dependence on external potassium concentration. 34. Using Eq. (4.54) for carrier transport in the presence of competitive inhibition, derive an equation for the flux of A as a function of 'cB when the concentration of A is the same on both sides of the membrane. Sketch the variation of JA with 'cB for positive and negative values of the latter quantity. 35. Using Eq. (4.54) for carrier transport in the presence of competitive inhibition, II I derive an equation for the flux of A as a function of cB when cB = 0. Sketch the II variation of JA with cB . 36. A membrane contains two sets of carriers for the same solute, A. They are similar in all respects, including their equilibrium constant for binding A, except that one has a binding site that allows it to be irreversibly inhibited by B. In the absence of –9 2 B, the flux through the membrane is 6 q 10 mols/cm -s when the concentration of A at Side I is very large and the concentration of A at Side II is zero. The flux I –10 2 is halved when cA is reduced to 40 mM. The flux falls to 8 q 10 mols/cm -s II I when cA is raised to 20 mM, maintaining cA at 40 mM, and enough B is added to inactivate the inhibitable fraction. What fraction of the carrier is inhibitable?

5 ACTIVE TRANSPORT

INTRODUCTION In free diffusion, and in facilitated transport, the driving force for the flux of a particular solute depends on only the concentrations or thermodynamic potentials of that species at the two faces of the transport barrier. Such transport processes are passive, and can have either of two outcomes: 1. If at least one of the phases is not practically infinite in extent and the solute is neither consumed nor synthesized, then the process will “run down” to equilibrium as the passive flux acts to equalize the chemical potentials on the two sides of the barrier. 2. If the solute is consumed or synthesized on one side of the membrane, then a nonequilibrium steady state can be achieved, such that the rate of solute transport across the membrane equals the rate of consumption or synthesis. The most cursory inspection of biological systems quickly reveals situations very different from either of those described above. In some cases, it is found that species move “uphill” from a phase in which their chemical (or electrochemical) potential is low to one in which it is higher. In other cases, a nonmetabolized solute is present on both sides of a membrane across which it can pass, but no net flux of the species can be demonstrated even though its chemical (or electrochemical) potential on one side of the membrane is different from that on the other side. These situations, illustrated in Figure 5.1, are most commonly the result of an interaction between metabolism and transport called active transport. Active transport is the subject of this chapter and is unique to biological systems. This chapter is organized rather like the preceding one. The first section is a general overview of active transport and includes a few words about its driving force. The distinction between primary and secondary active transport is presented, and means for identifying the presence of active transport are briefly reviewed. The second section describes the transport mechanisms in some detail. Finally, in the last section, kinetic analyses are presented for some of the more tractable cases, and the effects of internal slippage and external leaks on the net flux produced by active systems are discussed. M.H. Friedman, Principles and Models of Biological Transport, DOI: 10.1007/978-0-387-79240-8_5, © Springer Science+Business Media, LLC 2008

185

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Figure 5.1. Two transport situations that cannot be explained by the passive transport concepts in the previous chapters. (a) Uphill transport. (b) A nonequilibrium state without net flux.

5.1. ACTIVE TRANSPORT: GENERAL CONSIDERATIONS The two situations in Figure 5.1 can easily be distinguished from passive transport effects by locating them on a plot of flux versus electrochemical potential difference. This is done in Figure 5.2. In all the passive processes we have considered so far, the flux of a solute is zero when it is in equilibrium across the membrane; that is, Ji = 0  i = 0. The flux increases as the electrochemical potential difference inwhen 'P creases, and is always directed toward the phase in which the electrochemical potential is less. For thermodynamic reasons to be described with greater rigor in the next  i is termed the conjugate driving force for the passive species flux, Jip. chapter, 'P

5.1.1. Metabolic Coupling and Affinity In Figure 5.2, the curve for passive transport lies in the first and third quadrants and passes through the origin. In contrast, uphill transport processes occupy the second (or  i -axis that appear to fourth) quadrant, and give rise to nonzero intercepts on the 'P violate the equilibrium condition. As shown in Figure 5.2, both situations in Figure 5.1 reflect a displacement of the flux curve from that for a purely passive process:

J i  J ip J ia .

(5.1)

In thermodynamic terms, the quantity J ia reflects a coupling of the flux of the ith species to a driving force that is not its conjugate driving force. When the nonconjugate driving force that gives rise to J ia derives from a metabolic reaction, we refer to this component of the total flux as the rate of active transport of the ith species.

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Active transport systems can be regarded as transducers that convert chemical energy, obtained from metabolic reactions, into transport work. This is generally accomplished by using chemical energy from ATP to induce a change in the conformation of a carrier protein, which is also the ATPase that catalyzes the reaction. The energy is transferred to the protein by phosphorylation, in which the high-energy terminal phosphate on the ATP is relocated to the carrier. When the energy from ATP drives active transport directly (as in primary active transport; see below), the conjugate driving force for the active flux is the affinity, A, of the phosphorylation reaction:

A = 'Preaction  œ vi Pi  œ vi Pi , reactants

(5.2)

products

where {vi} are stoichiometric coefficients, as in Chapter 1. Since these reactions necessarily provide energy, A > 0.

Figure 5.2. Solute flux vs. electrochemical potential difference. (a) Passive transport. (b) Flux in the presence of an active transport contribution. The static head condition corresponds to that in Figure 5.1b. In the present figure, the active transport rate J ia is much less sensitive to %P i than is the passive flux J ip .

Many active transport systems also convert chemical energy into electrostatic potential energy by contributing to the potential difference across the membrane. The membrane potential can influence a variety of cellular and transport events (e.g., the modulation of voltage-dependent channels) not directly related to the active process itself.

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5.1.2. Classification of Active Transport Processes When phosphorylation by ATP is directly coupled to the flux of the actively transported solute, the process is referred to as primary active transport. This mechanism is exemplified by a number of ion transport systems in cell and organelle membranes, and is shown schematically in Figure 5.3a. Generally, this process causes the chemical potential of the transported solute to be higher on the side of the membrane toward which active transport is directed; hence, such systems are commonly described as “pumps,” pumping “uphill.” Primary active transport creates a driving force for passive, downhill backflux of the substrate. This passive flux can in turn be coupled to the flux of a second species, driving it uphill against its chemical potential gradient. Such a coupled process is called secondary active transport and is illustrated in Figure 5.3b. The solute whose pump-generated gradient drives the secondary flux is called the cosolute of the secondary transport process. Secondary transport is used by cells to take up many solutes, such as amino acids and sugars, for which there are no specific ATPases. The most common cosolute used by secondary transport systems is sodium ion.

Figure 5.3. Two classes of active transport. (a) A primary transport system that moves sodium ions from the inside (i) of the cell to the outside (o), and potassium ions in the opposite direction. (b) A secondary transport system that uses the passive influx of sodium to accumulate a sugar, S, in the cell. Na is the cosolute.

Primary active transport carriers that enzymatically phosphorylate themselves are known as P-type ATPases. In animal cells, only three primary transport systems of this kind are known. These are: 1. The sodium–potassium exchange pump, which exchanges two potassium ions for three sodium ions per ATP molecule hydrolyzed. Potassium is transported into the cell and sodium is carried out. This ubiquitous pump is responsible for the nonequilibrium state of these ions across the cell

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189

membrane; the pump creates a large transmembrane difference in the electrochemical potential of sodium, which drives most of the secondary transport systems that bring nutrients and metabolites into the cell. We will review the mechanism of this pump later in the chapter. 2. The plasma membrane calcium pump, which transports calcium out of the cell, reducing its intracellular concentration to submicromolar levels, while the extracellular calcium level is ca. 1.2 mM. The transporter has a much higher affinity for calcium when it is facing the interior of the cell, allowing it to bind the ion from cytosolic levels of 0.1–0.2 μM [note that this usage of “affinity” refers to the strength with which the carrier binds substrate, rather than the chemical energy available for transport defined in Eq. (5.2)] The sarcoplasmic/endoplasmic reticulum calcium (SERCA) pump, which is found in endoplasmic reticulum and muscle and will be described in Chapter 9, differs structurally from the plasma membrane pump but probably operates by a similar mechanism. The plasma membrane calcium pump transports one calcium ion out of the cell per ATP molecule hydrolyzed. The pump can be activated by other phosphorylating sources, such as the kinases that are part of the cell's regulatory system (Chap. 8). 3. The hydrogen–potassium exchange pumps that have been found in stomach, colon and kidney. The gastric H,K–ATPase acidifies the stomach with hydrochloric acid. The steps in H–K exchange are similar to those of the Na–K exchanger described below, including the existence of two conformational states of the protein and the presence of occluded states (Sachs et al., 1995). The pump transfers one proton into the secretory canaliculus, which empties into the stomach, and one potassium ion into the cell, per molecule of ATP hydrolyzed. Since primary active transport systems are often referred to as pumps, it is understandable that two key features of the active process are commonly described using centrifugal pump nomenclature. These are: level flow, which describes the transport rate when the transmembrane difference in the electrochemical potential of the substrate is zero; and static head, which is the opposing electrochemical potential difference that reduces the flux to zero. These two operating points of the pump are shown in Figure 5.2. At level flow, the flux is generated solely by nonconjugate driving forces. In primary transport, the nonconjugate force is the affinity of the ATP reaction; in secondary transport, it is the electrochemical potential difference of the cosolute. At static head, the absence of net flux can be due to backflux (“leak”) through a passive path in parallel with the pump, “slippage” of the pump itself, or both. It has been argued that secondary active transport is not strictly active, since the nonconjugate force to which the substrate flux is coupled is the electrochemical potential difference of the cosolute and not the affinity of a metabolic reaction. We will examine this argument from a thermodynamic perspective in the next chapter.

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CH. 5: ACTIVE TRANSPORT

5.1.3. Identification of Active Transport Processes  i curve As noted earlier, active transport processes cause a deviation from the J i  'P for passive transport. Therefore, to demonstrate the presence of active transport, it is necessary to measure the flux and the electrochemical potential difference of the transported species. Since these pumps reside in cellular and intracellular membranes, at least one of the required chemical potentials is intracellular. The measurement of the appropriate intracellular activity poses problems that have not been fully solved, even with the use of intracellular microelectrodes. As discussed in Chapter 1, when some of the solute is bound or otherwise reduced in activity, and a total assay is used, the chemical potential of the solute can be overestimated. An added complication is the likelihood that the distribution of active solute is not uniform throughout the cell. For instance, suppose that only a portion of the cell is accessible to solute entering from (or leaving for) the extracellular phase, so that the solute is entirely or disproportionately contained in this “transport” compartment. The concentration of solute in this compartment cannot be assayed separately. The errors in measured activity due to compartmentalization cannot be eliminated by using ionselective electrodes, and may be compounded by compartment-to-compartment variations in the activity coefficient of the solute. An uncertain fraction of the water in the cytosol is associated with the large number of polar macromolecules that inhabit the cell. This structured water is a poorer solvent than the free fraction. The presence of structured water acts to lower the size of the transport compartment. In many cases, the uncertainties in the operative chemical potential difference are so large that one cannot be certain that apparent deviations from passive transport behavior are real. As a result, the identification of active transport is usually made by observing the effects of various agents on the transport rate. For instance, the supposed pump should be slowed or stopped by low temperatures, metabolic inhibitors such as cyanide, or transport inhibitors such as ouabain. However, even these approaches can be ambiguous, particularly when metabolic inhibitors, which reduce the availability of ATP and thereby affect a multitude of cellular processes, are used. The kinetic mechanisms of active transport proteins are usually examined in isolated preparations or as expressed in other cells, such as oocytes, in which flux and activity measurements can be more readily be made

5.2. MECHANISMS OF ACTIVE TRANSPORT The mechanisms by which solutes are actively transported across membranes can be regarded as extensions of the mechanisms of carrier-mediated diffusion discussed in the previous chapter. The main difference is that, in active transport, ATP hydrolysis or the transport of a cosolute is coupled into one or more of the steps in the carriermediated process, driving it to maintain a nonequilibrium state. Any of these steps may be involved: the binding of the substrate to the carrier at the cis side of the membrane, or its release at the trans side, or the conformational change of the carrier itself. This section will be devoted to these mechanisms.

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191

The transport models we will discuss in the first part of this section and in much of the following section will be developed as extensions of the simple carrier model of the previous chapter. These will demonstrate the basic mechanisms on which biological active transport relies. The real systems are often more complex than the models we present below, and will be described along the way. The term scalar active transport has been used to describe transport driven by alterations in the binding and unbinding rate constants at the membrane interfaces; when modifications in the translocation kinetics are involved, the term vectorial active transport has been used. Real active transport mechanisms can have both scalar and vectorial aspects. For clarity, we will begin by examining a model that is unambiguously scalar, in which ATP or a cosolute act on only the substrate affinity of the carrier, and on only one side of the membrane. Throughout, we will assume that the pump is oriented to drive substrate from Phase I (the cis side) to Phase II (the trans side).

5.2.1. Scalar Active Transport: Overview In the simplest model of passive carrier transport that was analyzed in Chapter 4, the flux of carrier-bound solute across the membrane was proportional to the concentraI II tion difference c AX  c AX . If c AI  c AII , the concentration difference is positive, and solute flows passively from Phase I to Phase II. In scalar active transport, ATP or the cosolute modify the interfacial carrier concentrations so that 'cAX is positive even when the concentration of A, the substrate, is less in Phase I than in Phase II. This cannot happen in facilitated transport, even if the equilibrium constant for binding is different at the two faces of the membrane. How does it happen in active transport? Suppose the carrier exists in two forms, X and Y, normally in equilibrium with one another. The forms can be chemically different, or differ only in conformation. Form X is the transport state, which has a high affinity for the substrate. Form Y has a much lower affinity for the substrate, and binds A weakly, if at all. The transport cycle for such a carrier is shown in Figure 5.4. It is not very different from those in Chapter 4. The transformation of the carrier from Y to X is termed activation, occurs at Side I, and promotes carrier binding of substrate. Substrate crosses the membrane passively as AX. After the substrate has been released at Side II, X deactivates to Y, which returns to Side I, again passively. In this model, X does not cross the membrane unaccompanied by A. We will assume for now that Y does not bind A. The principles discussed below continue to hold (though the analysis becomes more difficult) if X can cross the membrane, or if Y can also transport A. We will discuss these alternatives later in this chapter. I The cyclic process shown in the figure can be driven in either of two ways: c AX can be raised by facilitating the conversion of Y to X at Side I, thereby increasing the II availability of active carrier; or c AX can be lowered by accelerating the conversion of X to Y at Side II. Either of these effects will increase the driving force for the translocation of AX.

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CH. 5: ACTIVE TRANSPORT

Figure 5.4. Cyclic representation of a scalar active transport carrier. The activation step takes place at Side I, and deactivation takes place at Side II. Dashed lines indicate the processes by which the pump can be driven: (1) primary activation [Eq. (5.3a)] or secondary cotransport [Eq. (5.5)]; (2) primary deactivation [Eq. (5.3b)] or secondary countertransport [Eq. (5.6)]. Either the activation step or the deactivation step is driven, but not both.

In principle, both primary and secondary scalar transport mechanisms are possible. However, while secondary systems are well described by the models that follow and their natural extensions, the sequence of reactions and conformational changes that P-type ATPases undergo are far more complex than the one-step scalar activation process contemplated here. For the sake of completeness and to increase our understanding of metabolic coupling, we will describe how a primary transport process can follow this simple model, but will emphasize secondary transport as the dominant example of the scalar mechanism.

5.2.2. Primary Scalar Transport In this simple model of primary scalar transport, the cycle in Figure 5.4 is driven by phosphorylation of the carrier. Activation is described by the following reaction: I

I

Y + ATP U
X + ADP + Pi ,

(5.3a)

where X is the high-affinity form of the carrier. The superscript on the carrier symbol denotes the side of the membrane at which the reaction takes place. A corresponding equation for deactivation at Side II can also be written: II

II

X + ATP U
Y + ADP + Pi .

(5.3b)

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193

The notion that a protein can be “deactivated” by phosphorylation seems contradictory, but it's only a matter of terminology. The two forms of the carrier, X and Y, are distinguished only by their different transition rate constants (only Y crosses the membrane without A) and substrate affinities (only X binds A). Thus “deactivation” by ATP signifies only that phosphorylation alters one or both of these properties, which is quite plausible.

How can coupling promote (for instance) the activation of Y at Side I? To answer this question, we compare the reaction given by Eq. (5.3a) to the activation process in the absence of coupling, which is simply I

I

Y U
X .

(5.4)

The free energy change accompanying Eq. (5.3a) is the sum of the free energy changes of (1) the activation step in the absence of coupling [Eq. (5.4)] and (2) hydrolysis of the ATP. Since the hydrolysis of ATP liberates energy, the standard free energy change of the latter reaction is negative; thus, its effect is to make the free energy change of the activation step more negative. As was shown in Chapter 1, this causes the equilibrium constant of the activation reaction to be larger, raising the ratio of activated to inactivated carrier at Side I. More colloquially, the hydrolysis of ATP drives the activation reaction to the right.

5.2.3. Secondary Scalar Transport Secondary scalar transport is the principal means by which nonelectrolytes such as sugars and amino acids, and certain inorganic solutes, are actively taken up by cells. The immediate driving force for secondary scalar transport is a transmembrane gradient of the electrochemical potential of a cosolute, B. The cosolute is usually an ion. Often, it is sodium that is driven by the Na–K exchange pump to a nonequilibrium distribution across the cell membrane. The passive backflux of this ion carries the substrates of many secondary transport systems into the cell. In one form of secondary transport, the carrier is activated by binding the cosolute: I

I

I

Y + B U
X .

(5.5)

X subsequently binds the substrate, A, at a different binding site, and then transits to Side II. The complex that undergoes the transition consists of carrier, cosolute, and substrate. This process is called cotransport, or symport, or positive coupling, because B and A cross to Side II together. A kinetic representation of a cotransport system is shown in Figure 5.5a. As far as transport rates are concerned, there is no difference between the substrate A and the cosolute B in this simple model. Here, the two solutes are cosubstrates, both of which must be bound to Y at Side I before it can transit to Side II. A distinction remains in terms of mechanism if the association of one species with the inactive carrier must precede that of the other species; for instance, if binding the first species in-

194

CH. 5: ACTIVE TRANSPORT

creases the affinity of the carrier for the second. Finally, there is an energetic difference between A and B if B is giving up energy by traveling down its electrochemical potential gradient and A is gaining energy by virtue of its transport into a region where its (electro)chemical potential is higher.

Figure 5.5. Kinetic representation of secondary scalar transport systems. The rate processes in these figures are reversible; their net directions when the pump is transporting A from Phase I to Phase II are shown with heavier arrows. (a) Cotransport. In this representation, B is bound first at Side I, and A is released first at Side II; alternatively, the binding and dissociation sequences can be random. (b) Countertransport. B binds to X at Side II to form Y. The systems as shown here are obligatory: the cotransporter transits between Sides I and II only as Y, in which form neither A nor B are bound, or as AX, in which form both A and B are bound; the countertransport carrier transits between Sides I and II only when either A or B (but not both) are bound.

The fact that B passes through the pump from Phase I to Phase II is crucial to the coupling process. In secondary active transport, the energy that drives the pump is effectively stored in the electrochemical potential gradient of the cosolute. The cosolute releases this energy by crossing from the phase in which its electrochemical potential is high to the phase in which it is low. In countertransport, the cycle is similar to that in cotransport, except that the cosolute binds to X at Side II. II

II

II

X + B U
Y .

(5.6)

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195

Thus, while X = BY in cotransport, Y = BX in countertransport. The cosolute crosses the membrane from Side II to Side I and is released into Phase I. By increasing cYII , the electrochemical potential gradient of the cosolute drives the transition of Y to Side I, increasing the availability of the carrier on that side. When B is released at Side I, the carrier, once more in the active form X, can now bind A. For this pump to function, the electrochemical potential of B must be larger in Phase II than in Phase I. The flux of the cosolute is opposite to that of the substrate. Accordingly, this process is termed countertransport, antiport, or negative coupling. It is illustrated in Figure 5.5b. Cotransport. Cotransport exhibits a number of identifying features: 1. Raising the concentration of B in Phase I increases the active flux of A. This is called cis stimulation of the pump, and occurs because an increase in the concentration of B drives the reaction given by Eq. (5.5) to the right. Thus more activated carrier is available to bind A. (Similarly, countertransport is trans stimulated by raising cBII .) 2. Raising the concentration of A in Phase I, while holding that in Phase II constant, increases the rate at which B crosses the membrane via the pump. As with any carrier system, the effect of raising c AI is to increase the rate at which the substrate crosses the membrane from Phase I to Phase II. If Y cannot transport A, then substrate crosses only in the company of B, so the flux of cosolute by this pathway also increases. This is called reciprocal stimulation. 3. When the chemical potential of the substrate is the same on both sides of  B changes the sign of JA. This is so the membrane, changing the sign of 'P because the only driving force for the flux of A at level flow is the electrochemical potential gradient of the cosolute. 4. When A and B cross from Phase I to Phase II only by cotransport, the flux of A is stoichiometrically related to that of B; that is,

J B  vB J A .

(5.7)

where Ji is the flux of the ith solute through the pump. The coupling coefficient vB is the number of molecules of B that accompany each molecule of A through the pump. 5. The rate of secondary active transport does not respond as rapidly as that of primary transport when inhibitors are added to the solutions bathing the tissue. Rather, the transport rate decays slowly as cosolute flows through the pump while the driving gradient, no longer maintained by the inhibited primary transport process, dissipates. If the cosolute gradient is maintained artificially by addition of cosolute to the appropriate phase, the rate of active transport of A and its dependence on the concentrations of B are not affected by inhibitors of primary transport. Similarly, reciprocal stimulation can be demonstrated in the presence of inhibitors, provided that cosolute is available at Side I.

196

CH. 5: ACTIVE TRANSPORT

6. The overall free energy change associated with the secondary active transport process must be negative. For instance, in cotransport, more free energy must be lost by the cosolute, as it passively returns to Phase II, than is gained by the substrate. This is a necessary condition for any irreversible process. For cotransport of a nonelectrolyte by an ion, the following inequality must therefore be satisfied:  IB  P  IIB )  J A (P IIA  P IA ) . J B (P

(5.8)

This condition was once a source of controversy, because it appeared to fail for some systems that were thought to operate by cotransport. It turned out that this seeming inconsistency was due to an overestimate of the intracellular electrochemical poten IIB in Eq. (5.8). Early estimates of this quantity were based tial of cosolute sodium, P on total assay concentrations; however, it is now known from ion-selective microelectrode measurements that the intracellular sodium activity can be less than half of the total sodium concentration. When the intracellular electrochemical potential of sodium is computed correctly, the inequality is satisfied.

It is easy to show that the electrochemical potential difference of the cosolute can also be written in terms of the potential difference '\ across the  B = z Bᑠ('\  E B ) . membrane and the cosolute Nernst potential, EB: 'P

In cotransport, at least two molecules or ions bind to the carrier on one side of the membrane (this can also be the case in countertransport if the stoichiometry of the exchange is not 1:1). The sequence of binding can be random or ordered. If the affinity of the inactive carrier for A is lower than that for B, and is enhanced by the prior binding of B, then most carriers will bind B first and then A. This characteristic is exemplified by the Na–K–2Cl (NKCC) cotransporter, a secondary symporter that is found in nearly all cell types and pumps chloride into the cell against a modest electrochemical potential difference. The stoichiometry of the electrically neutral pump is given by its title; the carrier transits from one side to the other only when no ions, or all four substrates, are bound to it. The ions bind and unbind in a specific sequence, mediated by the effect of the binding of one ion on the affinity of the carrier for the next one in the sequence. Figure 5.6 illustrates a model of this sequential process proposed by Lytle et al. (1998). When the fluxes of A and B from Phase I to Phase II are coupled stoichiometrically, Eqs. (5.7) and (5.8) can be combined to provide an upper bound on the B : accumulation of A that can accomplished by a pump whose driving force is 'P II I B . PA – PA < vB 'P

(5.9)

 B , then either vB > 1, or If the chemical potential difference of A is found to exceed 'P A is actively transported across the membrane by other mechanisms. This test was used by Bichara et al. (1983) to infer the presence of multiple active transport systems for hydrogen ion in the proximal tubule of the rabbit kidney.

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Figure 5.6. Model of Na–K–2Cl cotransport. In normal function, the net sequence of states proceeds clockwise. The empty carrier E0 opens to the extracellular space, revealing an Na binding site near the intracellular gate. Sodium binding prompts a conformational change revealing a binding site for Cl. Chloride binding similarly prompts K binding, which in turn reveals a binding site for the second Cl. After the carrier passes through an occluded state E4 closed to both the extracellular and intracellular fluids, the ions leave the carrier for the cytosol in the order in which they were bound from the external phase. After the second chloride has entered the cell, the carrier is once more in the E0 state. Reprinted with permission from Lytle et al. (1998). Copyright © 1998, The American Physiological Society.

When the chemical potentials are expressed in terms of experimental quantities, pressure effects are neglected, and the cosolute is assumed to be a monovalent cation, Eq. (5.9) becomes ln

 aBI ¬­ a AII ᑠ žln ­. v  '\ ž B ­ žžŸ a II RT ­ a AI ® B

(5.10)

The maximum accumulation ratio a AII / a AI is never achieved in real systems, because of inefficiencies in the pump and passive leaks of substrate from Phase II, which in symport is usually the inside of a cell. The accumulation ratio is also reduced if the substrate is consumed in the cell. Therefore, the nonconjugate driving force needed to obtain a specified accumulation ratio is always greater than the theoretical minimum. The losses that degrade the performance of active transport systems will be discussed at greater length in the next section. Countertransport. Similar considerations apply to antiport systems, including those, like the Cl–HCO3 exchanger described in the previous chapter, that exchange

198

CH. 5: ACTIVE TRANSPORT

ions across the cell membrane. Here we consider the Na–Ca exchanger, in which the influx of three cosolute sodium ions (from Phase II to Phase I) drives the removal of one calcium ion from the cell (i.e., QNa = 3). Using the notation presented earlier, AX is Ca-bound carrier and Y = BX is Na-bound carrier. It is believed that, for this exchanger, the same set of binding sites carries each ion. Since unbound carrier can bind either ion on either side of the membrane, not all transitions carry the “intended” ion. Tracer studies have demonstrated the so-called Na–Na and Ca–Ca exchanges that are a consequence of this imperfect selectivity. The net loss in free energy accompanying a single cycle of the exchanger is

 a II ¬­  a II ¬ ᑠ 2ᑠ ­­  žžln Ca   Q Na žžžln Na '\ '\­­­ p 0 , I I ž žŸ aNa RT ®­ Ÿž a Ca RT ®­ I

(5.11)

II

where '\ \ – \ is the cell membrane potential as usually defined. When the left-hand side of Eq. (5.11) equals zero, no change in free energy accompanies Na–Ca exchange; accordingly, the system is at equilibrium, there is no net flux of either ion, and static head has been achieved. The membrane potential such that the exchanger is at equilibrium for a specified intracellular and extracellular composition is called the reversal potential of the exchanger. Solving Eq. (5.11) for the minimum intracellular calcium level sustainable by the exchanger, which occurs at static head, I II ln aCa,min  ln aCa  Q Na ln

II aNa ᑠ'\ (Q Na  2) . I RT aNa

(5.12)

Equation (5.12) shows the leverage afforded by the coupling coefficient and membrane potential in reducing the minimum intracellular calcium level. Exponentiating both sides of Eq. (5.12), and setting QNa = 3, 3

 I ¬  ᑠ'\ ¬­ I II ž aNa ­ aCa,min  aCa žž II ­­ exp žžž ­. žŸ aNa ®­ Ÿ RT ®­

(5.13)

Because of the larger number of cosolute molecules carried by each cycle of the pump, a tenfold transmembrane ratio of cosolute activities translates into a thousandfold ratio of calcium activities at static head. This effect derives from the activity dependence of the chemical potential, is based on only pump stoichiometry, and does not require that either solute be charged. The third factor in the right-hand side of Eq. (5.13) arises because each cycle of the pump brings one net positive charge into the cell, which is favored by a negative intracellular potential; if QNa were equal to two, the pump would be electrically neutral and this term would vanish. Examples of symporters include the carriers that allow absorptive epithelia such as intestine and kidney to accumulate amino acids and sugars; these systems, for which sodium ion is the cosolute, will be discussed in Chapter 10. Typical antiporters include the Cl–HCO3 and Na–Ca exchangers that have already been discussed. The effect of membrane potential on transporter performance seen for the Na–Ca ex-

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199

changer is also seen for sodium-driven cotransporters. Even when the substrate is negatively charged (e.g., certain amino acids, carboxylic acids, sulfate and phosphate ion), the coupling coefficient is such that each cycle of the pump carries net positive charge into the cell. It has been proposed that the concentration gradient of an amino acid brought into the cell via a secondary transport system can drive by countertransport the accumulation of a second amino acid by the cell. The uptake of the second amino acid by this mechanism has been termed tertiary active transport. Figure 5.7 shows the coupled cycles for this tertiary transport process and the primary and secondary processes that accompany it. Aronson (1983) has suggested that a similar mechanism may be partially responsible for hydrogen secretion by the kidney proximal tubule; the tertiary process in this case is driven by the efflux of organic anions that are accumulated in the tubule cell by a sodium gradient-driven cotransport system.

5.2.4. Vectorial Active Transport, the Curie Theorem and Substrate Activation In vectorial coupling, a nonequilibrium state is maintained by driving the translocation of the substrate from one side of a membrane to the other. Only primary active transport systems exhibit this kind of coupling: the energy acquired by the ATPase as a result of its phosphorylation is used to drive the conformational change that transports the substrate. Before describing the mechanisms by which vectorial coupling can take place, we must first consider the limitations on these mechanisms that follow from the Curie theorem. The Curie theorem states that forces of one tensorial order cannot couple with fluxes of a different order (tensors are outside the mathematical scope of this text, but the following reasoning does not depend on any specialized knowledge). The affinity of a chemical reaction is a scalar, having no physical direction. A scalar is a zerothorder tensor. Thus the energy released by ATP hydrolysis can couple into other chemical reactions, such as the activation or deactivation of a carrier, which are also scalars. The transition of a carrier from one side of a membrane to the other is clearly vectorial, having a direction, and is accordingly a first-order tensor. The Curie theorem says that the latter process cannot be driven by a chemical reaction. The Curie theorem notwithstanding, mechanisms of vectorial coupling have been proposed that are biochemically plausible. A universal feature of these mechanisms is anisotropy: the driving reaction is still a scalar, but it functions differently at each face of the membrane. Indeed, the phosphorylation reaction that drives primary active transport is generally believed to take place only when the ATPase is facing a preferred side of the membrane. Such an enzymatic reaction can easily be anisotropic, since the conformation of the enzyme when it faces Side I of the membrane differs from its conformation when it is facing Side II. This anisotropy gives a directional aspect to the chemical process, evidently sufficient to satisfy the Curie theorem.

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CH. 5: ACTIVE TRANSPORT

Figure 5.7. Tertiary amino acid transport. The potassium exchange process is omitted from the primary transport cycle. Carrier #1 is activated by Na; Carrier #2 is deactivated by AA1.

PRINCIPLES AND MODELS OF BIOLOGICAL TRANSPORT

201

In vectorial transport, the energy transferred to the ATPase by phosphorylation is used to drive loaded carrier from Side I to Side II (or, in principle, unloaded carrier in the opposite direction); there is no need to postulate the existence of active and inactive forms (though they can be added to the picture). In the first instance, I

II

AX + ATP U
AX + ADP + Pi .

(5.14)

Here, the ATPase, X, exists in only one state (there is no inactive state) and binds A via an association reaction identical to that of the simple carrier of Chapter 4. It catalyzes its phosphorylation by ATP only when it is bound to A and only when it is facing Side I; this provides sufficient anisotropy to satisfy the Curie theorem. The loaded and phosphorylated carrier then undergoes a conformational change, exposing the bound substrate to Phase II. Similarly to primary scalar transport [Eq. (5.3a)], the energy in the terminal phosphate bond of the ATP drives the reaction to the right. The phosphate is subsequently released. The transport cycle corresponding to Eq. (5.14) is shown in Figure 5.8. The carrier transition that is not driven by ATP proceeds as a passive process in the opposite direction; for the cycle shown here, cXII  c XI , and X returns to Side I passively.

Figure 5.8. Cyclic representation of vectorial active transport, as described by Eq. (5.14).

The enzymatic activity of the ATPase that carries A from Side I to Side II must be greater when A is bound to it; otherwise, it would be equally likely to transition to Side II when its binding site is unoccupied. Like other enzymes, its activity could increase as a result of a conformational change, in this case induced by binding A. Such an enzyme is said to be activated by the substrate.

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CH. 5: ACTIVE TRANSPORT

The kinetics of real primary transport systems are more complicated than either the scalar or vectorial examples given above, and indeed include features of both. This will be apparent from the description of the ubiquitous sodium–potassium exchange pump that follows.

5.2.5. Sodium–Potassium Exchange The sodium–potassium exchange pump is a primary active transport system driven by a cell membrane ATPase that carries sodium ions out of the cell and potassium ions in. The Na–K ATPase was discovered by Jens Skou in 1957, for which he won the Nobel Prize in Chemistry in 1997 (transport science has produced a large number of Nobel laureates, but sometimes the wait is lengthy). The stoichiometry of the pump is such that, for each molecule of ATP that is hydrolyzed, three sodium ions and two potassium ions are usually carried across the cell membrane. Although this is the most common stoichiometry, it can vary from 3:2, depending on the concentrations of the ions on the two sides of the cell membrane (Lauger, 1991). Both species are transported against their electrochemical potential gradients. The pump is present in almost all animal cells and is of immense physiologic importance, being responsible for functions as diverse as intestinal absorption and signal transmission in nerve. It produces the sodium electrochemical potential gradient that drives most secondary scalar transport processes, and plays an important role in cellular pH and volume regulation. As another indication of its importance, it consumes perhaps one-third of the ATP produced by the cell, and more in excitable tissue, where its maintenance of transmembrane differences in ion electrochemical potential is essential to nerve and muscle function. The commonly accepted mechanism of the pump, known as the Post–Albers scheme (Albers et al., 1963; Post et al., 1965), has both scalar and vectorial aspects. Na–K ATPase is the carrier and exists in two conformations. In Form E1, the enzyme's binding sites are at Side I (the cytoplasmic side) and have a higher affinity for sodium than potassium. In Form E2, the sites face Side II (the extracellular side) and prefer potassium. The sodium and potassium movements across the membrane take place consecutively in a sequence of elementary steps: I

E1•ATP + Na U Na • E1 • ATP,

(5.15a)

Na • E1 • ATP U
Na • E1 ~ P + ADP,

(5.15b)

Na • E1 ~ P U
Na • E2 – P,

(5.15c)

II

(5.15d)

E2 – P + K U
K • E2 – P,

(5.15e)

K • E2 – P U
K • E2 + Pi ,

(5.15f)

Na • E2 – P U
E2 – P + Na , II

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203

K • E2 + ATP U
K • E2 • ATP,

(5.15g)

K • E2 • ATP U K • E1 • ATP,

(5.15h)

I

K • E1 • ATP U K + E1 • ATP.

(5.15i)

The symbols “Na” and “K” in these equations represent three sodium ions and two potassium ions, respectively. Here, Eqs. (5.15a) and (5.15b) describe substrate binding to, and phosphorylation of, the ATPase. Magnesium ion is necessary for the phosphorylation step to take place. Equation (5.15c) represents the consequent conformational change of the protein from Form E1 to Form E2. During the course of this change, which requires that Na be bound to the enzyme, some energy is extracted from the phosphate bond that was formed when the enzyme was phosphorylated. Thus the E2–P bond is less energetic than the E1~P bond. Facing Side II and in Form E2, the enzyme no longer has a high affinity for sodium, and releases it (5.15d). Since it does have a high affinity for potassium when it is in Form E2, the ATPase binds this ion from Phase II (5.15e). Thereupon, the enzyme releases its phosphate (5.15f) and binds a fresh molecule of ATP (5.15g) prior to returning to its original conformation. When the conformational change is complete (5.15h), the carrier loses its affinity for potassium, which it releases into Phase I (5.15i). The ATPase is now ready to bind sodium once more. The overall stoichiometry of the normally functioning pump is: 3NaI + 2KII + ATP U 3NaII + 2KI + ADP + Pi .

(5.16)

The overall reaction given by Eq. (5.16) ordinarily proceeds to the right, hydrolyzing ATP, accumulating potassium in the cell, and pumping sodium out. However, by the law of mass action, the reaction can be driven to the left if the opposing ion electrochemical potentials get too high (e.g., large cNaII/cNaI or cKI/cKII ). In this event, the pump operates in reverse and phosphorylates ADP to produce ATP. The primary Ca pump does this too, and it is the normal mode of operation of the ATP synthase in the mitochondrion.

There are several variants and elaborations of the Post–Albers scheme described above, some of which involve additional intermediate states of the enzyme. Robinson (1997) presents a cyclic model of the pump that includes 21 steps, but generally follows the sequence in Eq. (5.15). In another version (Sachs, 2003), K•E2 can either follow the bind–transit–release sequence in Eqs. (5.15g–i), or transit to the E1 form first, and then release its potassium before binding ATP. Included in these more recent models are occluded states, similar to those seen in the two-gate carrier model (Fig. 4.11b) and the Na–K–2Cl cotransporter model (Fig. 5.6), in which bound ions can access neither ambient phase. Furthermore, the binding of the three sodium ions, and that of the two potassium ions, to the transport protein is clearly more likely to proceed one ion at a time, rather than all at once, and the same is true for their release.

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A modification of the Post–Albers scheme that includes occluded states and sequential binding and unbinding has been offered by Balshaw et al. (2001), based on recent work by others in the field. Referring to the sequence of steps in Eq. (5.15), — sodium binds sequentially, first two ions, and then, more slowly, the third. Some transitions take place when only the first two sodium ions have bound to the enzyme. — phosphorylation of E1 is accompanied by occlusion of the (2 or 3) sodium ions. — after the enzyme has assumed the E2 form, the extracellular gate opens and the sodium ions are released sequentially: last in–first out. — potassium also binds sequentially, with positive cooperativity. Some carrier transitions take place with only one potassium ion bound to the enzyme. — dephosphorylation of E2 is accompanied by occlusion of the potassium ion(s). — the intracellular gate opens when the enzyme assumes the E1 form, and the potassium ions are released sequentially into the cytosol. Models similar to Eqs. (5.15) have been used to describe the function of other Ptype ATPases. As noted earlier, the reaction cycle of the SERCA pump will be described in Chapter 9. Eisenberg and Hill (1985) have pointed out an important relationship between the kinetic parameters of the transport process and the efficiency of the pump. In particular, the rate constants of competing steps that would decrease this efficiency are small. This ensures that the binding and release of substrates occur at the proper point in the cycle. For example, the rate constant for the reaction E1 • ATP o E1 ~ P + ADP is much less than that for Eq. (5.15a); as a consequence, E1 has time to bind sodium before autophosphorylating. Similar relationships among rate constants ensure that ions are released from the enzyme before it transits back to the side at which they were bound. In the authors' words, the rate constants “channel the enzyme” along a reaction path in which the hydrolysis of ATP is tightly coupled to the transport process.

5.2.6. Pump Selectivity and Other Properties Shared with Passive Carriers Since the proteins responsible for active transport function similarly to carriers, they share many of the characteristic carrier properties described in the previous chapter. Selectivity. Similar to passive carriers, the ATPases that participate in active transport are selective for their substrates; accordingly, each primary transport pump relies on a different transport ATPase. ATPases that exchange different ions across the cell membrane possess two sets of binding sites, one for each ion. Each site would be expected to have a higher affinity for its substrate when it is facing the side of the

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membrane that is the cis side for that ion (for sodium, the inside of the cell; for potassium, the outside). Indeed, in the Na–K pump, there is considerable asymmetry in the binding constants for both ions. The overall selectivity of the Na–K ATPase reflects the binding constants of both sets of sites and is not perfect (although, for simplicity, the schemes described above are written as though it is); internal K competes for the Na sites in the E1 state, and external Na competes for the K sites in the E2 state. Indeed, 20% of the ions bound from the cytoplasm by the Na–K ATPase in nerve are potassium (Trachtenberg et al., 1981). Similarly, in red blood cells, the Na–K ATPase supports sizeable Na–Na and K–K exchange fluxes, which can occur only if these ions pass in both directions through the pump. Saturability and Inhibition. Other carrier properties discussed in the previous chapter, such as saturability and inhibition, are also exhibited by pump-associated carriers. As with passive carriers, the maximum pump rate is ultimately limited by the highest frequency with which the protein can undergo conformational change. Transport by the Na–K pump is inhibited by the cardiac glycoside ouabain at a concentration of 10–5 M. This is in the same range (10–6–10–5 M) as that required to inhibit the dephosphorylation of ATP by the membrane ATPase. Other agents also have corresponding effects on the transport rate and the activity of the ATPase. Other enzymes responsible for specific transport processes have been identified in this way. Ouabain binds to the ATPase at only the outside of the cell. Red blood cells can be reconstituted with their cell membranes inside out, i.e., with the cytoplasmic face of the membrane exposed to the bathing medium. The pump is also turned around, and these cells actively accumulate sodium. This uptake is not inhibited by ouabain in the bathing medium, presumably because the ouabain binding sites now face the interior of the cell. Structure. The ATPases also share some of the structural features of the carriers discussed in the previous chapter. ATPases are oligomers and, as we have already seen, have preferred orientations in the membranes in which they are embedded. Among the P-type ATPases, the structure of the SERCA pump has been characterized in greatest detail, and will be presented in Chapter 9. In addition to sharing structural features with passive carriers, the physical processes by which ATPases are thought to translocate species from one side of the membrane to the other are among those described earlier for carrier transport. The pump conformations include occluded states like those in the two-gate carrier model, and other models of active transport are similar to the alternating access model, modified to include distinct binding sites for multiple substrates.

5.3. KINETICS OF ACTIVE TRANSPORT As in the previous section, we begin with scalar active transport. Secondary scalar transport is dealt with first because it is the most natural extension of the carrier transport models discussed in Chapter 4. We then move on to primary scalar transport and

206

CH. 5: ACTIVE TRANSPORT

vectorial transport. This section closes with a discussion of the leakage processes internal and external to the pumps that limit the degree of substrate accumulation that they can achieve.

Figure 5.9. Kinetic model of a simple symport system.

5.3.1. A Simple Secondary Scalar Transport Model: Assumptions Figure 5.9 is a kinetic model of the simplest symport system, in which activation precedes substrate binding, and A binds to only X. The kinetic equations that describe this system are extensions of the equations that govern the simplest carrier, and are based on similarly extended assumptions: 1. The carrier, in either the active or inactive form, is restricted to the membrane; Y reversibly binds only B, and X (= BY) reversibly binds only A. For simplicity, it is assumed that only one molecule of A or B binds to the carrier in each cycle. 2. The only species that can cross the membrane are Y and AX; X and unbound A cannot cross the membrane. 3. The binding and unbinding reactions of cosolute and substrate are at equilibrium at both sides of the membrane. The equilibria at the two membrane interfaces are defined by two dissociation equilibrium constants, yielding four equations: Activation: cBi cYi  K a (i = I, II). c iX

(5.17a)

c iA cXi  K (i = I, II). c iAX

(5.17b)

Substrate binding:

PRINCIPLES AND MODELS OF BIOLOGICAL TRANSPORT

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As in Chapter 4, activities have been replaced by concentrations, and the assumption that the interface reactions are at equilibrium implies that the translocation of the carrier is rate limiting. 4. A steady or quasisteady state exists. 5. The rate at which each form of the carrier passes from one side of the membrane to the other is given by Eq. (4.37):

J j  D a(c Ij  c IIj ) (j = Y, AX).

(4.37)

A weakness of this assumption that is particularly relevant to active transport is its failure to include the effect of the transmembrane potential difference on the transition rate constants. The membrane potential is rarely small under physiological conditions, though it can be brought to zero in experimental preparations by short circuiting. Almost all active transport processes move charges across the membrane. Symport is no exception, since the cosolute is usually sodium. Consequently, the active transport rate is generally affected by the membrane potential. For example, the sodium-coupled solute flux into vesicles made from epithelial brush border membranes can be raised by lowering the potential inside the vesicle (Freel and Goldner, 1981). Parent et al. (1992a) measured the substratecoupled sodium flux through sodium/glucose cotransporter molecules expressed in frog oocytes, under voltage clamp, and fit the results to a saturating function analogous to Eq. (4.42); the maximum sodium current and the apparent dissociation equilibrium constant both exhibited a strong dependence on the membrane potential. Since the maximum sodium current is proportional to Dc, Parent et al. attributed the dependence of maximum current on membrane potential to a voltage-sensitive transition rate. The voltage dependence of the apparent equilibrium constant was taken at face value, i.e., as evidence of the effect of membrane potential on the activation equilibria at the membrane interfaces. They also found that negative membrane potentials enhanced sugar uptake, a result to be expected from the increased sodium driving force. The omission of membrane potential from the analysis limits the applicability of the flux equations that follow to strongly depolarized or short-circuited membranes (unless, of course, the pump is not voltagesensitive). Fortunately, many experimental investigations of active transport are carried out under just such conditions. One cannot uncritically assume that pump kinetics measured at short circuit apply when the normal transmembrane potential is present. To model rheogenic (i.e., currentgenerating) active transport across a membrane that is not short-circuited, it is necessary to relate the fluxes to driving forces that are expressed in terms of the electrochemical potential. The sensitivity of pump rates to membrane potential also arises from the conformational changes of the carrier that effect transport. These

208

CH. 5: ACTIVE TRANSPORT

changes generally require the movement of charged residues and dipoles that are part of the pump protein but are not directly involved in substrate binding or translocation. The movement of these groups, like that of the binding sites and substrate, is influenced by the electric field in which the transitions occur. As one consequence, the potential difference across the membrane can affect the transition rate constants and hence the equilibrium distribution of carrier between Side I and Side II. The influence of the electric field on the transition rate constants was examined mathematically by Lauger (1984), who represented the pump molecule as an assembly of point charges. This effect is of somewhat less concern for most contemporary carrier models (e.g., Fig. 4.11a), in which the binding sites move only a short distance, and in a direction that is more or less in the plane of the membrane. The analysis of antiport is not substantially different from that of symport and is included as an exercise. There is one important difference between the two: since in antiport A and B are never bound to X at the same time, the carrier need have only a single binding site for both substrate and cosolute. The assumptions of the symport model are modified as follows: 1. The activated carrier X binds either A or B, and neither solute binds to any other form of the carrier; when A binds to X, it forms the carrier–substrate complex, AX; when B binds to X, it forms the “inactivated” form of the carrier, BX = Y. 2. As in the symport model, the only forms of the carrier that can cross the membrane are Y and AX. 3. The activation equilibrium constant Ka is replaced by the deactivation i i i equilibrium constant Kd = cX cB /cY .

5.3.2. A Simple Secondary Scalar Transport Model: Equations 1. Conservation of Carrier. The carrier has six possible states: inactive (Y), activated (X = YB), and bound to substrate (AX = AYB), at Side I or Side II: I II cYI cYII c XI cXII c AX c AX  XT .

(5.18a)

2. Steady State. For the simplest case, we assume, as we did for the simplest passive carrier model, that all rate constants have the same value, Dc. This assumption can be relaxed as in Chapter 4; see also the next subsection: I II D a(cYI  cYII c AX  c AX )0 .

(5.18b)

Recall that the activated carrier does not cross the membrane without substrate. 3. The Transport Rate is equal to the flux of AX: I II J A  D a(c AX  c AX ).

(5.18c)

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Equations (5.17) and (5.18) constitute seven equations in seven unknowns: the concentrations of the six carrier states, and JA. The substrate flux depends on the concentrations of A and B in Phases I and II, and the properties of the carrier system (XT, Ka, and K). Solving for JA , JA 

D aX T K a K (cBI c AI  cBII c AII ) (cBI c AI K a K Kc BI )(cBII c AII K a K ) (cBII c AII K a K KcBII )(c BI c AI K a K )

.

(5.19)

The effect of the cosolute gradient on the flux of A is seen more clearly at level flow; when the substrate concentration in both phases is cA,

JA 

D aXT K a Kc A (cBI  cBII ) , den

(5.20)

where den is the denominator in Eq. (5.19). Since the cosolute concentrations appear in the denominator, the substrate flux at level flow is not as simple a function of these concentrations as it might first appear. The level flow solution is, however, symmetric with respect to cosolute concentration; when cBI and cBII are interchanged, the sign of JA is reversed, but the magnitude of the substrate flux remains the same.

5.3.3. More Complex Symport Models; the Sodium/Glucose Transporter Many of the assumptions on which the preceding equations were based can be relaxed, usually at the price of increased complexity. Since symport is a carrier-based process, the simple transport model developed above can be subjected to all of the complicating modifications applied in Chapter 4, such as the introduction of ligandsensitive transition rate constants, asymmetries, or nonequilibrium at the interfaces. Some modifications that are more specific to active transport are described below: 1. The Activated Carrier Can Cross the Membrane without Substrate. The I II quantity Dc(cX – cX ) is added to the left-hand side of Eq. (5.18b), and the expression for substrate flux becomes

JA 

c AI cBI ( K a cBII )  c AII cBII ( K a cBI ) D aXT K ¸ . 2 [ K ( K a cBII ) c AII cBII ][c AI cBI K ( K a cBI )]

(5.21)

In an analysis of the Na/glucose cotransporter, Centelles et al. (1991) point out that the ability of an activated carrier to cross the membrane without substrate constitutes a “short circuit” of the cyclic behavior of the system (see Fig. 5.10, below), which in the case of this particular symport system seriously compromises the coupling of substrate glucose transport to the energy contained in the transmembrane difference in the electrochemical potential of cosolute sodium. They conclude that this transition is prevented in the cotransporter either by minimizing the transition rate constant for X (as in the simple model given above), or through kinetic parameters that allow activation to occur only after the substrate has bound to the carrier.

210

CH. 5: ACTIVE TRANSPORT

Figure 5.10. Kinetic representation of a general cotransport system. X is BY. The dissociation constants for the three interfacial binding equilibria are shown. The kinetics of the simplest system are within the shaded region demarcated by heavy dashed lines.

2. The Inactive Form of the Carrier Can Also Bind and Transport A (as AY). I The presence of this side reaction introduces two more unknown concentrations, c AY II and c AY . Minor modifications must be made to Eqs. (5.18a–c) (one of the exercises at the end of the chapter asks the student to make them). Two more equations are needed and are provided by the equilibrium constant for the side reaction:

ciA cYi  K s (i = I, II). ciAY

(5.22)

Competitive inhibition does not ensue when both A and B can bind to Y, because the binding sites for the substrate and the cosolute are distinct. Activation of the carrier by the cosolute is reflected in a greater affinity of X for A compared to that of Y for A; i.e., K << Ks (recall that these equilibrium constants are for the dissociation of the ligand from the binding site on the carrier; thus large equilibrium constants correspond to weaker binding and lower affinities). Put another way, when B binds to Y (forming X), it increases the substrate affinity of the A-binding site on the carrier. AY can also bind B, forming AX. The binding rate constants may be such that this is the preferred sequence of events: Y binds A, and then B, rather than B, and then A. This is the sequence contemplated earlier by Centelles et al. for the sodium/glucose i i i transporter. The equilibrium constant for the binding of B by AY, Kb = cAY cB /cAX , is not independent of the other three; indeed, Kb /Ka = K/Ks. When the latter ratio is small, as

PRINCIPLES AND MODELS OF BIOLOGICAL TRANSPORT

211

in the previous paragraph, the former is too, implying that the affinity of AY for B is greater than the affinity of Y for B; thus, in this case, substrate binding to the inactive carrier increases the affinity of the B-binding site on the carrier for the cosolute. The increase in the affinity of the carrier for one solute, caused by the binding of another solute, is termed binding cooperativity. Figure 5.10 is a kinetic representation of a general symport system in which Y can bind either A or B first. For the simple model analyzed earlier and delineated in the figure, the inactivated carrier must bind the cosolute before it can bind the substrate; in the more general system, either solute can be bound first, and binding of the first solute increases the affinity of the carrier for the second solute. If neither X (= BY) nor AY can cross the membrane, transport is obligatory and the carrier crosses only when both solutes or neither solute are bound. Under such circumstances, continuous transport requires that both solutes be present on the cis side of the membrane. The distinction is often made between an “ordered” process in which only the substrate or cosolute must be bound first, and a “random” process in which the inactivated carrier can bind either solute first. Ordered processes can be regarded as limiting cases of the more general random process. For instance, the ordered sequence in which the inactivated carrier binds B first and then A is a limiting case of random binding in which Ks >> Ka.

For the cosolute gradient to be well coupled to the active transport of A, the activated carrier must have either a greater affinity for substrate than Y, or a faster transition rate. Heinz and coworkers (Heinz et al., 1972, Geck and Heinz, 1976) classified symport processes as affinity-type or velocity-type, depending on which effect of activation dominates. In a pure affinity-type process, the transition rates of all forms of the carrier are the same, but K is much less than Ks. The simplest model given above is of this kind; Ks is so large that AY is absent. In the simplest velocity-type process, K = Ks, but DcAX >> DcAY. Truskey et al. (2004) modeled a limiting case of a velocity-type scalar transport process in which a side reaction is present but only Y and AX can cross the membrane; nine independent parameters were needed to define this transport system. 3. When The Reactions at the Interfaces Are not at Equilibrium, the equilibrium constants are replaced by rate constants for binding and dissociation of the substrate and cosolute, as was done in the previous chapter for a passive carrier. The rate constants must be consistent with the equilibrium constant. At equilibrium, the rates of binding and dissociation are equal; for instance, for substrate binding at Side I, I kbI c AI cXI  krI c AX (equilibrium),

(5.23)

where kbi (kri ) is the rate constant for binding (release) at Side i. Referring to Eq. (5.17b), I

I

I

K = kr /kb .

(5.24)

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CH. 5: ACTIVE TRANSPORT

Similar equations apply at Side II, and for activation. 4. The Cosolute and/or Carrier are Charged. The implications of this added complexity have already been discussed. 5. The Coupling Coefficient QB is not Unity.

Figure 5.11. Kinetic model of the sodium/glucose cotransporter in rabbit intestine (Parent et al., 1992b). C is the unbound carrier, equivalent to Y; CNa2 is the carrier activated by two sodium ions, equivalent to X; and SCNa is the activated carrier bound to glucose, equivalent to AX. The superscripts denote the external and internal phases between which transport takes place. All transitions and binding reactions are defined by rate constants as given in the figure; interfacial equilibrium is –1 not assumed. The units of the transition rates are s ; activation is regarded as a third-order reaction –2 –1 and the units of the rate constants are mols s ; sugar binding to the activated carrier is a secondorder reaction and the units of the rate constants are mols–1s–1. E  ᑠ'\ / RT , where '\ is cell potential.

The sodium/glucose cotransporter, an important contributor to intestinal absorption, exemplifies most of the complexities described above. Two sodium ions accompany each transition of the activated carrier, which carries one glucose molecule (S) at a time; thus, the activated carrier X = CNa2, where C = Y is the inactive carrier, and the loaded carrier AX = SCNa2. The inactive carrier possesses two negative charges, so the activated carrier is electrically neutral. Parent et al. (1992b) have thoroughly characterized the kinetics and potential dependence of this transport system, using the intestinal transporter of rabbit, and their results are summarized in Figure 5.11. All three forms of the carrier — C, CNa2, and SCNa2 — can cross the membrane. The transition rate constants for CNa2 and SCNa2 are symmetric, but the rate constants for the transition of the charged form of the carrier, C, are asymmetric and potential dependent. The rate constants for the activation/deactivation and substrate binding/unbinding reactions in the two carrier conformations are all different, and the activa-

PRINCIPLES AND MODELS OF BIOLOGICAL TRANSPORT

213

tion/deactivation rate constants when the carrier sites are exposed to the external medium are potential dependent. Centelles et al. (1991) examined the fluxes through vesicles containing porcine renal sodium/glucose transporters and favor a different model. In contrast to the sixstate model of Parent et al., they found that all eight states in Figure 5.10 were important. In the model preferred by Centelles et al., the inactivated carrier C can bind either the cosolute or substrate first. If it binds the substrate first, it can cross to some extent as SC (this is absent from the Parent model); however, if it binds sodium first, it rarely crosses as CNa2 (recall that the transition rate for the sodium-bound carrier was thought to be small, to reduce short-circuiting of the glucose flux). Therefore, it must either lose the sodium, or bind glucose, before transitioning to the other side of the membrane. Binding cooperativity promotes the formation of the ternary complex SCNa2 from either CNa2 or SC.

5.3.4. Primary Scalar Transport The equations that describe primary transport can be derived from a model and assumptions that are very similar to those used to treat secondary transport. The only important difference between the equations for primary and secondary scalar transport is in the activation/deactivation equilibria at the two sides of the membrane. Whereas in secondary transport, these equilibria are given by Eq. (5.17a), the corresponding equations in primary transport naturally do not involve B, and they are different for each side of the membrane. Consider, for instance, the simple illustration given earlier in which the primary transport process is driven by activation at Side I [Eq. (5.3a)]. For simplicity, write the metabolic reaction to which the activation process is coupled as R (reactants) o P (products). Then, at Side I, the activation step is I

I

I

I

Y +R UX +P.

(5.25a)

In Eq. (5.3a), R is ATP and P is ADP + Pi . The carrier deactivates passively at Side II: II

II

X UY .

(5.25b)

The equilibrium constants for these reactions are designated KA and KD, respectively: KA 

cXI cPI , cYI cRI

(5.26a)

cYII . cXII

(5.26b)

KD 

We assume that the equilibrium constant between X and Y, in the absence of activation, is symmetric. Thus, if there were no coupling to ATP hydrolysis at Side I, then:

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CH. 5: ACTIVE TRANSPORT

 c I ¬­ žž X ­ žžŸ c I ®­­ Y

no coupling



1 . KD

(5.27)

The ratio cXI / cYI is a measure of the fraction of A-free carrier sites at Side I that are activated; call it the activation ratio. Its value in the presence of ATP is obtained by rearranging Eq. (5.26a), assuming Eq. (5.25a) is at equilibrium: c XI K A cRI  . cYI cPI

(5.28)

A parameter, *, that measures the increased availability of activated carrier when phosphorylation by ATP is coupled into the activation process, can be constructed from the activation ratios given by the two preceding equations: *w

cXI / cYI K A K D cRI  . (c XI / cYI )no coupling cPI

(5.29)

Since the coupling drives the activation reaction Y o X to the right, the activation ratio given by Eq. (5.28) is larger than that given by Eq. (5.27), and * > 1. When there is no coupling, * = 1, by definition.

5.3.5. Flux Equations for Primary Scalar Transport As noted earlier, the only difference between the governing equations for the preceding models of primary and secondary scalar transport lies in the description of the activation/deactivation equilibria at the membrane–solution interfaces. This difference can be dealt with by an algebraic trick that allows us to write the primary transport flux equations directly from those for secondary transport. Table 5.1 presents the activation ratios at each side of the membrane, for primary and secondary scalar transport, obtained from Eqs. (5.17a), (5.26b), (5.27), and (5.29). The expressions for the activation ratio in secondary transport can be converted into those for primary transport by replacing cBI by Ka*/KD and cBII by Ka /KD. Among the seven equations that describe the simple symport model, the only ones that contain the cosolute concentrations are the pair given by Eq. (5.17a). Thus we can make the substitutions indicated above without altering any of the other equations, and the flux equation for primary transport can be obtained directly from Eq. (5.19) by making the same substitutions:

JA 

D aXT KK D (*c AI  c AII ) . (*c AI KK D K *)(c AII KK D ) (c AII KK D K )(*c AI KK D )

(5.30)

At level flow, JA 

D aXT KK D c A (*  1) , den

(5.31)

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215

where den is the denominator in Eq. (5.30). When there is no coupling, * = 1, and JA = 0, as it should.

Table 5.1. Activation Ratios for Scalar Active Transport Driven by Carrier Activation Side I Secondary transport

Primary transport

cXI I Y

c c

I X I Y

c

 

Side II

cBI

cXII

Ka

II Y

c

II X II Y

*

c

KD

c

 

cBII Ka 1 KD

5.3.6. Relation between the Coupling Parameter * and the Affinity of the Metabolic Reaction The relation between * and the affinity, A, follows from the thermodynamic equations in Chapter 1. Since symmetry with respect to the X U
Y equilibrium is assumed, the superscripts that designate each side of the membrane are not needed here and will be 0 omitted. From Eq. (1.65) and the definition of 'G , RT ln K D  PY0  P0X ,

(5.32a)

RT ln K A  P0X P 0P  PY0  P0R .

(5.32b)

RT ln( K D K A )  P0P  P0R .

(5.33)

Adding these equations,

Next, Eq. (1.43) is used to express the right-hand side of Eq. (5.33) in terms of the chemical potential change accompanying the metabolic reaction. Equation (1.43) is first simplified by (1) neglecting the effect of pressure on the chemical potential and (2) assuming that the activity coefficients of P and R are unity; it is then rearranged to give Pi0  Pi  RT ln ci .

(5.34)

Substituting Eq. (5.34) into (5.33), and rearranging: P R  P P  RT ln( K D K A ) RT ln

cR , cP

(5.35)

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CH. 5: ACTIVE TRANSPORT

where cP replaces the product of the ADP and phosphate concentrations. From Eq. (5.2), the left-hand side of Eq. (5.35) is precisely the affinity, A; from Eq. (5.29), the right-hand side is RT ln *. Thus, * = eA/RT.

(5.36)

When the affinity is positive, * is greater than one.

5.3.7. Vectorial Active Transport and the Nature of Na–K Exchange When energy from ATP is coupled into the transition of the carrier from one side of the membrane to the other, the active transport process is regarded as vectorial. Such transport is necessarily primary, since ATP is directly involved in the transport process. The Na–K exchange pump described earlier is a good model for examining the vectorial aspect of active transport. The Post–Albers scheme shown in Eqs. (5.15) is typical of P-type ATPases: ATP phosphorylates the enzyme on one side of the membrane [Eq. (5.15b)], and then that energy is used to drive the conformational change of the carrier that exposes its binding sites to the other side [Eq. (5.15c)]. Such a process has both scalar and vectorial aspects. Equation (5.15b) clearly describes a scalar activation process, which is preceded by binding of one of the substrates (Na) to the pump. However, the transition of the product of the phosphorylation reaction to Side II is not entirely passive (as it would be in pure scalar transport), and relies on the energy in the terminal phosphate bond to effect the conformational change. The energy in the ATP first activates the carrier, and then is expended in driving it across the membrane. The two steps described above can be combined: Na • E1 • ATP U Na • E2 – P + ADP.

(5.37)

This equation fits the definition of vectorial transport and can readily be modeled as I an extension of passive carrier transport. Referring to Eq. (5.14), AX is the Na • E1 II complex, and AX is Na • E2 – P. In the simple model of passive carrier transport that we developed in Chapter 4, the rate constants for the transition of AX from either side of the membrane to the other are the same. However, the exothermic hydrolysis of ATP drives the reaction given by Eq. (5.37) to the right; that is, the rate constant for the transition of the ATPase from E1 to E2 is greater than that in the opposite direction: DcAX > Dc–AX. Thus this portion of the Post–Albers scheme can be described in the same fashion as facilitated diffusion in the presence of rate constant asymmetry [cf. Eq. (4.49)]. A crucial difference is that the passive transport condition given by Eq. (4.49a) does not apply here.

5.3.8. Pumps and Leaks In many experimental studies of active transport, a level flow condition is sought by short-circuiting the experimental preparation and bathing it on both surfaces by identical solutions at similar pressures. This is done to eliminate driving forces for addi-

PRINCIPLES AND MODELS OF BIOLOGICAL TRANSPORT

217

tional, passive fluxes that might confuse the interpretation of the results. In short, the data are easier to analyze and to interpret when they are taken at level flow. In vivo, the pump drives the system away from level flow: cAII becomes greater than c AI . Indeed, the purpose of most active transport systems is to accumulate selected solutes at the trans side of the membrane, which is often the interior of a cell. Under these conditions, the desirable features of level flow experiments are absent. The electrochemical potential gradient created by the pump drives two kinds of leakage fluxes, which are discussed below and illustrated in Figure 5.12.

Figure 5.12. (a) Slippage of a pump. Fluid leaks back through the pump, driven by the pressure head. (b) A shunt. Fluid leaks back through an external path in parallel with the pump. The pumps in this figure transport water against a hydrostatic pressure difference; the analogous pumps in cell membranes transport substrate against an electrochemical potential difference.

Slippage (Fig. 5.12a). For a fixed value of c AI , the substrate flux given by Eqs. (5.19) and (5.30) decreases monotonically as c AII is raised. This decrease in pump rate is not the result of transport paths in parallel with the pump. Rather, it reflects a slippage in the pump itself, analogous to the slippage in a centrifugal pump when it is operating against too high a pressure head. When c AII increases, the dissociation of AX at Side II is suppressed. As a result, the carrier more often returns to Side I with its binding site(s) still occupied by substrate. Similarly, as c AI is reduced, the rate at which activated carrier binds substrate becomes less, increasing the likelihood that the cosolute will dissociate from the carrier, allowing it to cross to Side II without substrate. At static head, the rate at which AX crosses from Side II to Side I becomes equal to the rate at which it crosses from Side I to Side II, and the net substrate flux is zero.

218

CH. 5: ACTIVE TRANSPORT

From Eqs. (5.19) and (5.30), it is seen that the static head condition (JA = 0) occurs when cAII/cAI = cBI/cBII (secondary transport) = *(primary transport).

(5.38)

The left-hand side of Eq. (5.38) is the maximum accumulation ratio of the pump. Note that a static head condition is reached even when the pump is the only path by which substrate crosses the membrane. Another kind of internal leak, seen in cotransport, occurs when the activated carrier can cross the membrane without the substrate, or when the side reaction is present. In both cases, the coupling between the cosolute gradient and the substrate flux is diminished. Shunts (Fig. 5.12b). Once the pump raises the electrochemical potential of its substrate in Phase II above that in Phase I, a driving force for passive transport in opposition to the pump has been created. If there are passive pathways in parallel with the pump, solute will leak through them from the trans side of the pump to the cis side. Such leakage paths are called shunts. When Phase II is a closed compartment, such as the inside of a cell, shunts reduce the accumulation ratio attainable by the pump. In the steady state, the net flux (active plus passive) of substrate into the cell must equal the rate at which the solute is consumed. Denoting the active flux of A by the superscript “a” and assuming that the leak rate follows the equations of free diffusion, [ J Aa k A (c AI  c AII )] Acell  mVcell ,

(5.39)

where Acell and Vcell are the surface area and volume of the cell, respectively, kA is the permeability of A in the cell membrane, and m is the rate at which the substrate is consumed, per unit volume. Since cAII > cAI, the passive flux is oppositely directed to the II pump flux. Solving for cA , c AII  c AI

J Aa  mVcell / Acell . kA

(5.39a)

For a given pump rate, the intracellular concentration is reduced by increasing either the permeability of the cell membrane or the rate of consumption of substrate within the cell. When the substrate is not consumed by the cell (as is the case for most ions) or the consumption rate is sufficiently small, the right-hand side of Eq. (5.39) can be set equal to zero. Then, in the steady state, the pump rate into the cell equals the leak rate out: J Aa  k A (c AII  c AI ) .

(5.40)

When the leak rate is defined by a different expression — for instance, if a carrier, not at its diffusion limit, is involved — the right-hand side of Eq. (5.40) is changed accordingly.

PRINCIPLES AND MODELS OF BIOLOGICAL TRANSPORT

219

Figure 5.13 illustrates the relationship between the fluxes through the pump and leak pathways. In the figure, the solid line describes the variation in the active flux II from Phase I to Phase II as a function of the trans concentration cA for a fixed value of I the cis concentration cA . The active transport rate decreases as the opposing concenII I II tration difference cA – cA increases, falling to zero when cA reaches the static head II value, c A,s.h. . The maximum accumulation ratio of the pump is c AII,s.h. / c AI .

Figure 5.13. Dependence of pump rate (JAa, solid line) and leak flux magnitude (dashed lines) on the trans concentration of substrate, cAII, for a fixed cis concentration, cAI. See text for description.

The flux through the leak is directed toward Phase I when cAII > cAI. The magnitude II of the leak flux is plotted against cA in Figure 5.13, using dashed lines, for two differII I ent leak permeabilities, kA. Naturally, the leakage flux is zero when cA = cA , and it inII creases in proportion to the difference between the two concentrations as cA becomes larger. The magnitudes of the (oppositely directed) pump and leak rates are equal where the dashed lines intersect the operating curve of the pump; the abscissas at the intersection points, cA* and cA**, are the steady-state values of cAII corresponding to I each shunt permeability. This steady-state value decreases from c AII,s.h. toward cA as the shunt permeability increases. Thus, shunts act to reduce the maximum accumulation ratio generated by active transport systems. A sketch of the dependence of this ratio on shunt permeability is shown in Figure 5.14, for secondary scalar transport. When kA = 0, there is no shunt, and the ratio is given by Eq. (5.38). The pump becomes less and less capable of concentrating A in Phase II (i.e., the accumulation ratio approaches unity) as the shunt permeability increases.

220

CH. 5: ACTIVE TRANSPORT

Figure 5.14. Dependence of the accumulation ratio of a secondary scalar transport system on shunt permeability, kA. At each point on the curve, the passive flux through the shunt is equal and opposite to the active flux through the pump.

PROBLEMS: CHAPTER 5 1.

The operating curve of an active transport system that pumps A from Phase I to a Phase II is approximated by a simple linear expression JA = a + b'cA. (a) In terms of a and b, what is the flux at level flow? (b) In terms of a and b, what is the concentration difference across the membrane at static head?

2.

The pump in the previous problem is sited in a cell membrane and pumps into the –11 2 –12 cell. The parameters of the pump are a = 4 q 10 mol/cm -s and b = 1 q 10 2 mol/cm -s-mM. The substrate A is not metabolized by the cell, but it does leak –11 2 out via a simple carrier system whose parameters are DcXT /2 = 9 q 10 mol/cm s and K = 40 mM. Write an equation that can be solved for the steady-state concentration of A in the cell when the external concentration is 10 mM.

3.

The layer of cells that line the intestine absorb glucose (S) and can be modeled as two membranes in series. The portion of the cell membrane that faces the inside of the intestine (the luminal membrane) contains sodium/glucose symporters that pump the sugar from the digestive tract into the cell. The rest of the membrane (the abluminal membrane) contains glucose carriers that transport intracellular sugars into the interstitium and thence to nearby capillaries. The sodium ion gradient that drives the pump in the luminal membrane is created by a primary transport mechanism. Assume that the actively transported flux of glucose across the luminal membrane can be written JS = LSS'PS + P, where LSS and P are constants, and that the

PRINCIPLES AND MODELS OF BIOLOGICAL TRANSPORT

221

flux of S across the abluminal membrane can be described by the simple carrier of Chapter 4 in the diffusion limit. These are the only pathways for glucose transport. Glucose is consumed negligibly by the cell. (a) Write an equation that can be solved for the intracellular concentration of glucose in the steady state, in terms of the glucose concentrations in the intestine (cSI) and interstitium (cSII), and the properties of the two transport systems, You need not solve the equation. Neglect the effect of pressure on the chemical potential and assume the system is isothermal. (b) What is the initial effect on the rate of the symport system (does the rate change continuously or discontinuously; does it rise or fall?) if: (i) The primary transport ATPase is inhibited? (ii) cSI is raised? II (iii) cS is raised? 4.

Can the coupling coefficient for symport be less than 1? If so, how? If not, why not?

5.

Show that the expression for the electrochemical potential difference of the coso B that follows Eq. (5.8) is correct. lute 'P

6.

If the potential in the interior of a cell at 37ºC is 60 mV negative relative to the surroundings, and the extracellular and intracellular sodium concentrations are 110 and 5 mM, respectively, and the extracellular glucose level is 90 mg%, how high could the intracellular glucose concentration be raised by sodium symport were it not consumed in the cell? Assume one Na ion crosses the cell membrane with each glucose molecule, that there is no glucose leak, and that all activity coefficients are unity.

7.

Derive an equation analogous to Eq. (5.10) that describes the maximum accumulation ratio achievable by an antiport system.

8.

(a) Using Eq. (5.11), derive an expression for the reversal potential of the Na– Ca exchanger in terms of the Nernst potentials of the two ions across the cell membrane. (b) Typical intracellular and extracellular concentrations of Na and Ca are: intracellular Na = 14 mM, intracellular Ca = 0.1 PM, extracellular Na = 139 mM, extracellular Ca = 1.2 mM. What is the reversal potential of the exchanger under these conditions? (c) The extracellular Na concentration in (b) is reduced gradually from 139 to 14 mM while the other three concentrations remain the same. Plot the reversal potential of the exchanger over this sodium concentration range and explain the shape of the curve.

222

9.

CH. 5: ACTIVE TRANSPORT

The active transport system in Figure 5.7 is in steady state, with sodium being pumped out of the cell and the two amino acids in. The Na–K ATPase is suddenly inhibited by the addition of ouabain, which binds to its active sites. Plot and explain the time course of the fluxes of Na and the two amino acids through the transport system, subsequent to inhibition. Assume for simplicity that there are no other pathways for any of the species to cross the cell membrane.

10. The cell membranes of a particular portion of the kidney (the thick ascending limbs of the loops of Henle) contain about 104 Na–K ATPase molecules per μm2. One enzyme molecule can hydrolyze 100 ATP molecules/sec. What is the current density produced across the renal cell membrane by the pump? Express the current density in electrical units. 11. The electrical potential inside the human red blood cell is –9 mV relative to plasma. Using literature values for the plasma and intracellular sodium and potassium levels, determine whether either ion is actively transported across the red cell membrane, and in which direction. What kind of transport system would explain your result? 12. Using the stoichiometry in Eq. (5.16), show that the free energy available from the hydrolysis of ATP is sufficient to drive the Na–K pump for normal values of intracellular and extracellular cation concentrations, and a typical cell potential of –60 mV. 13. Starting with Eqs. (5.17) and (5.18), determine whether reciprocal stimulation can take place when B can cross the membrane either as X or as AX. 14. Derive a flux equation for antiport analogous to Eq. (5.19). 15. Use Eq. (5.19) to derive an expression for substrate flux across a symporter when the cis concentration of substrate is zero. Show that the dependence of JA on cAI has the same form as the corresponding equation for a simple carrier. Explain in terms of the mechanism of the pump why the substrate flux is independent of cBII under these conditions. 16. Show that the substrate flux for secondary scalar transport at level flow is symmetric with respect to cosolute concentration. Show that this symmetry does not generally hold when the substrate concentrations on the two sides of the membrane differ, and explain why. 17. Derive Eq. (5.21). 18. Specify the changes that need to be made to Eqs. (5.18a–c) to include the side reaction in which Y binds A.

PRINCIPLES AND MODELS OF BIOLOGICAL TRANSPORT

223

19. The affinity of ATP hydrolysis under normal intracellular conditions is 11–14 kcal/mol. Using a value of 11 kcal/mol, calculate the coupling parameter * for a primary scalar system in which the carrier is activated by the hydrolysis of a single molecule of ATP. 20. In a pump–leak system in which the pump substrate is not metabolized, how does the pump rate change as the permeability of the leak is increased? Does it approach a limit? If so, what is the limit? If not, why not? 21. A primitive cell has only two pumps. Pump #1 is a primary pump that uses ATP to accumulate B in the cell from the outside medium. Pump #2 is a symport system that uses the gradient of B to extrude C from the cell. C enters the cell via a passive leak, but B crosses the cell membrane only via the pumps. (a) The cell is functioning normally. Then all ATPases are suddenly inhibited. Sketch the time course of the transport rates through Pumps #1 and #2, starting just prior to inhibition, and explain the difference between them. (b) The cell is functioning normally. Then the extracellular concentration of C is raised. Sketch and explain the time course of the intracellular concentration of B, starting just prior to the change in the extracellular concentration of C.

6 NONEQUILIBRIUM THERMODYNAMICS

INTRODUCTION Chapter 5 described a class of transport processes in which the substrate flux is driven by either the affinity of a metabolic reaction or the electrochemical potential difference of a cosolute. This coupling of a flux to a force that is not its conjugate driving force is often seen in biological transport. Nonequilibrium thermodynamics, also called irreversible thermodynamics, is the formalism that describes the relationships among the many forces and fluxes that are simultaneously present in living systems. Our interest is primarily in linear nonequilibrium thermodynamics, where the fluxes depend linearly on the several driving forces in the system. This linear relation is presented in the first section of this chapter, following some important definitions. In subsequent sections, the basic equations of linear nonequilibrium thermodynamics are applied to passive and active transport. In the section on passive transport, we deal for the first time with the thermodynamics of solvent flow, and introduce the process of osmosis. The chapter concludes with a discussion of the limitations of nonequilibrium thermodynamics.

6.1.

THE BASIC PHENOMENOLOGICAL EQUATIONS

6.1.1. Conjugate Forces and Fluxes As indicated above, nonequilibrium thermodynamics concerns itself with the transport phenomena that occur when multiple forces are acting simultaneously on the species in the system, causing a number of simultaneous fluxes. In such cases, the fluxes and the forces can be enumerated in a number of equivalent ways. For instance, when a voltage is applied across a membrane bounded by solutions of a single electrolyte, the resulting ionic fluxes can be described in terms of cation flux and anion flux, or in terms of cation (or anion) flux and current density. Similarly, when a membrane separates two nonelectrolyte solutions whose pressures and concentrations differ, the driving forces can be expressed in terms of the concentration and pressure differences, or equivalently in terms of the chemical potential differences of solvent and solute. M.H. Friedman, Principles and Models of Biological Transport, DOI: 10.1007/978-0-387-79240-8_6, © Springer Science+Business Media, LLC 2008

225

226

CH. 6: NONEQUILIBRIUM THERMODYNAMICS

Among these options are particular sets of conjugate forces and fluxes which, we are taught by nonequilibrium thermodynamics, possess certain useful properties. These properties will be described in more detail later in this section. The identification of conjugate forces and fluxes is based on a thermodynamic quantity called the dissipation function, I: IT

di S . dt

(6.1)

The quantity di S/dt is the rate at which entropy is locally generated in a unit volume of membrane as a result of irreversible transport processes taking place in it; di S/dt and the dissipation function are necessarily positive. The local variable I can be integrated across the membrane to give a

a

)  ¨ Idx  T ¨ 0

0

di S dx . dt

(6.2)

From the thermodynamic equations in Chapter 1, it can be shown that I and ) are the rates at which free energy is dissipated during transport. The units of I and ) are 3 2 2 cal/cc-sec (in SI units, J/m -s) and cal/cm -s (in SI units, J/m -s), respectively. Each flux Ji contributes to the dissipation function. Paired with each flux is a corresponding conjugate force Xi, such that

I  œ Ji Xi .

(6.3a)

i

The {Xi} can also be defined such that

)  œ J i Xi .

(6.3b)

i

The passage of the ith species across the membrane dissipates free energy at a rate  i / dx ) cal/cc-sec. Therefore, by Eq. (6.3a), the electrochemical potenequal to J i (dP tial gradient is conjugate to the species flux. The integrals across the membrane of the conjugate forces defined by Eq. (6.3a) are also conjugate forces, because their products with the {Ji} satisfy Eq. (6.3b). The integral of the electrochemical potential gradient is the electrochemical potential difference, which accordingly is also an acceptable conjugate. Equation (6.3b) also identifies the affinity as the force that is conjugate to the progress of a chemical reaction. As remarked earlier, the coupled transport processes taking place in a complicated system can often be described in terms of alternative sets of fluxes. Corresponding to each of these sets is a matching set of conjugate driving forces. Any set of force–flux pairs whose pairwise products add up to the dissipation function is equally amenable to nonequilibrium thermodynamic analysis. As we will see, some sets are better for some purposes than others, and sometimes different sets are used to examine different aspects of the same transport system. Because the dissipation function can be written in either the differential or integral form, both the gradient and the difference of the electrochemical potential are usable

PRINCIPLES AND MODELS OF BIOLOGICAL TRANSPORT

227

as conjugate driving forces. The gradient would seem to have a better claim to the title of “force” since a force is, after all, the gradient of a potential. However, for reasons of simplicity and convenience (and ignorance, because we know much more about the conditions at the membrane surfaces than we do about the gradients inside it), we will use the integrated forms almost exclusively. The differential forms find more use in free solution studies or experiments on inanimate systems, which can be more thoroughly instrumented.

6.1.2. Phenomenological Coefficients and Linear Thermodynamics The relationships among the forces and fluxes in a nonequilibrium system are described in nonequilibrium thermodynamics by a set of phenomenological coefficients. A convenient starting point for introducing these coefficients is the expression presented in Chapter 2 for the free diffusion of a single nonelectrolyte when the only driving force for solute flux is the chemical potential gradient of the solute itself:

 dP ¬ J s  U s cs žž s ­­­ . žŸ dx ®

(2.4)

In Chapter 2, we used the expression for the chemical potential in terms of solution variables [Eq. (1.41)] to replace cs(–dPs/dx) in Eq. (2.4) by –RTdcs /dx, from which we obtained Fick's First Law of Diffusion, which was subsequently integrated across the membrane. Here, for reasons that will become clear later, we set up the integral differently: P II

a

s J ¨ s dx  ¨ (dP s ) . 0 U c P Is s s

(6.4)

Since the flux is independent of position in the membrane, it can be brought out in front of the integral sign; in addition, the integration of the right-hand side is trivial, and Eq. (6.4) becomes a

Js ¨ 0

dx  P Is  P IIs . U s cs

(6.5)

By our earlier definition of the ' operator, the right-hand side of Eq. (6.5) is 'Ps. The integral on the left is called the integral or generalized resistance of the membrane, Rss. Equation (6.5) becomes

JsRss = 'Ps.

(6.6)

Corresponding to this version of the integrated flux equation is a similar version of the original differential form Eq. (2.4): J s rss  

dP s , dx

(6.7)

228

CH. 6: NONEQUILIBRIUM THERMODYNAMICS

where rss = 1/(Uscs). The two resistances, Rss and rss, are related by a

a

Rss  ¨ rss dx  ¨ 0

0

dx . U s cs

(6.8)

Continuing this seemingly trivial exercise, we note that the integrated form (6.6) can also be written: Js = Lss'Ps,

(6.9)

where Lss is the integral or generalized conductance of the system; for this particular case, the conductance is simply the reciprocal of the resistance. These generalized resistances and conductances, which define the dependence of a flux on its conjugate driving force, are referred to as straight phenomenological coefficients. It is not apparent that Eq. (6.6) or (6.9) should be preferable in any way to the proportionality between flux and concentration difference that was developed in Chapter 2. For one thing, the chemical potential difference is clearly more difficult to express in terms of measurable quantities than is the concentration difference. In addition, it is found experimentally that the solute permeability is generally a weak function of solute concentration. Since the permeability is proportional to the solute mobility, the latter quantity is also insensitive to concentration. Thus, referring to Eq. (6.8), it is clear that Rss and Lss are considerably more concentration dependent than is the permeability, since concentration appears in the defining integral. What, then, do the new equations have to offer? When there is only one force and one flux in the system, as in the example above, they have nothing to offer; to describe the flux of a single solute, the concentration difference is clearly the preferred driving force. But, when there are multiple forces and fluxes, Eqs. (6.6) and (6.9) can be generalized to a relatively simple set of equations that includes all the interactions among the forces and fluxes in the system. Suppose n pairs of conjugate fluxes and driving forces are present; then the flux of the ith species is given by n

J i  œ Lij X j (i = 1,….,n).

(6.10)

j 1

More complete theory allows for higher powers of {Xj} on the right-hand side, but the linear expression above is sufficient for our purposes. The {Lij} are the phenomenological coefficients of the system. The straight coefficients are those for which i = j; those coefficients for which i z j are termed the cross-phenomenological coefficients. As an example, consider the coupled fluxes of a nonelectrolyte and solvent (subscript “w”) across a membrane. By Eq. (6.10), the fluxes of the two species are Js = Lss'Ps + Lsw'Pw,

(6.11a)

Jw = Lws'Ps + Lww'Pw.

(6.11b)

PRINCIPLES AND MODELS OF BIOLOGICAL TRANSPORT

229

The number of phenomenological coefficients needed to describe all the fluxes in a system rises rapidly as the number of species increases. This rise is not as rapid as it might at first appear, because the phenomenological coefficients are not independent (we will see another reason in §6.2.7). In particular, the matrix of the phenomenological coefficients is symmetric; that is, Lij = Lji.

(6.12)

For example, in Eqs. (6.11), Lsw = Lws. Equation (6.12) is the Onsager reciprocal relation, one of the most powerful principles of nonequilibrium thermodynamics. Since each flux can be written as a linear sum of all the forces, each force can be written as a linear sum of all the fluxes: n

X j  œ R ji J i (j = 1,…,n).

(6.13)

i 1

Reciprocity also holds for the {Rji}:

Rji = Rij.

(6.14)

The R’s are related to the L’s by a general expression that is easily derived using linear algebra. The equation for a specific R in terms of the L-matrix is R ji 

L ji L

,

(6.15)

where |L| is the determinant of the L-matrix and |Lji| is the minor determinant of the matrix corresponding to Lji. In most practical applications of nonequilibrium thermodynamics to biological transport, n does not exceed three, and it is easy to move between the L-representation and the R-representation without using matrix algebra. As we have seen, when n = 1, L11 = 1/R11. The choice of representation to use depends on the problem being addressed. The two formulations are equivalent, but usually they are not equally convenient. In this chapter, we will use the L-representation to describe passive transport processes, and the R-representation to examine active transport.

6.1.3. Frictional Interpretation of the Phenomenological Equations In 1961, Kedem and Katchalsky (1961) rewrote the phenomenological equations of the previous subsection in yet another form, in which the phenomenological coefficients were replaced by a set of frictional coefficients. According to the frictional model, the flux of each species is determined by a balance between (1) its conjugate driving force and (2) the frictional drag on the species by all of the other components of the system, including the membrane. The frictional drag on the jth species by the ith component was given by Fji = fji(vi – vj),

(6.16)

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CH. 6: NONEQUILIBRIUM THERMODYNAMICS

where fji is a frictional coefficient and vi is the nominal velocity of the ith component, defined as Ji /ci. The membrane is chosen as the frame of reference, so its velocity is zero. The force balance that determines the fluxes is similar to Eq. (6.13): n

X j   œ Fji  Fjm (j = 1, ..., n).

(6.17)

i 1 iv j

For instance, when a single nonelectrolyte is present, Eqs. (6.16) and (6.17) give J J ¬ f J X s   fsw žžž w  s ­­­ sm s , žŸ cw cs ®­ cs

(6.18a)

J J ¬ f J X w   fws žžž s  w ­­­ wm w . žŸ cs cw ®­ cw

(6.18b)

The relationship between the frictional coefficients of Eqs. (6.18) and the generalized resistances of Eq. (6.13) is straightforward. The reciprocal relation Rsw = Rws becomes fsw f  ws . cw cs

(6.19)

Thus, the number of independent frictional coefficients required to describe the transport system is the same as the number of phenomenological coefficients. An advantage of the frictional representation is that the frictional coefficients are generally less sensitive to concentration than are the phenomenological coefficients; also, when the transport mechanism is known, the frictional coefficients are easier to interpret in terms of that mechanism and the properties of the constituents of the system. However, unless the transport mechanism is well understood — and this requires information beyond that provided by nonequilibrium thermodynamics — it is unwise to assign any more physical content to measured frictional coefficients than to the phenomenological coefficients to which they are related.

6.1.4. A Cautionary Note Before Proceeding In the last section of this chapter, we will discuss in some detail the limitations of the nonequilibrium thermodynamic description of coupled transport processes; one of these, alluded to above, is worth stressing now. No direct information regarding mechanism is provided by nonequilibrium thermodynamics. As a result, one cannot predict the values of the phenomenological coefficients (or the frictional coefficients to which they are related) using only this methodology. The coefficients are, indeed, phenomenological: even though the relationships among them must satisfy the principles of nonequilibrium thermodynamics, they are experimental quantities, whose values are ultimately dependent on mechanisms to which nonequilibrium thermodynamics is blind.

PRINCIPLES AND MODELS OF BIOLOGICAL TRANSPORT

231

6.2. NONEQUILIBRIUM THERMODYNAMIC DESCRIPTION OF PASSIVE TRANSPORT As in Chapter 2, we begin our discussion of passive transport by considering a system consisting of a single nonelectrolyte dissolved in a solvent, say water. Most of this section will deal with the application of nonequilibrium thermodynamics to passive transport in this attractively simple system. The description of the coupled transport of solvent and a single nonelectrolyte is followed by extensions to electrolytes and multiple solutes.

6.2.1. Setting the Stage Two solutions of a nonelectrolyte are separated by a membrane; the concentrations and/or the hydrostatic pressures of the solutions are unequal. The only necessary difference between this system and that in Chapter 2 is that the membrane in this case is permeable to water. We have already seen the equations that describe the resulting fluxes: Js = Lss'Ps + Lsw'Pw,

(6.11a)

Jw = Lsw'Ps + Lww'Pw,

(6.11b)

where the Onsager reciprocal relation has been used to replace Lws by Lsw. The chemical potentials whose differences appear in Eqs. (6.11) are those just inside each face of the membrane. Chemical potential is a continuous function of position across the membrane–solution interface; if it were discontinuous, then infinite gradients would exist, a condition inconsistent with the presence of a finite flux. Thus, an arbitrarily small error is introduced by basing the driving forces on the chemical potentials in the solutions immediately adjacent to the membrane (see Fig. 6.1). The driving forces can then be written in terms of the concentrations and pressures in the solutions, which are more accessible than the inside of the membrane. When the solutions are assumed to be well stirred, the chemical potentials can be written in terms of the bulk concentrations of the solutions. The membrane properties are reflected in only the phenomenological coefficients: Lss, Lsw, and Lww. From a practical point of view, Eqs. (6.11) can be made much easier to use by rewriting the chemical potential differences 'Ps and 'Pw in terms of the more readily measured concentrations and pressures in the two solutions. The first of these differences is easy to translate, using Eq. (1.41): 'P s  Vs 'P RT ' ln cs .

(6.20)

For now, we assume the solutions to be ideal (Js = 1) and dilute; the effects of nonideality and nondiluteness will be dealt with later. We also assume that there are no temperature gradients across the membrane; this is why no 'P 0s term appears in Eq. (6.20). Indeed, if there were such gradients, a third force/flux pair reflecting the dissipation due to the flow of heat would have to be included in the phenomenological equations.

232

CH. 6: NONEQUILIBRIUM THERMODYNAMICS

Figure 6.1. Chemical potential profile across a membrane. The chemical potential difference 'P is measured immediately inside the faces of the membrane. The difference 'Pc is based on solution properties at a distance H from the faces. Because P is continuous, lim 'Pa  'P . The chemical poHl 0 tentials in the bulk solutions are denoted by the subscript b.

Another assumption that will prove very useful is that the concentration difference across the membrane is small relative to the average concentration in the bathing solutions. When this is the case, 'ln cs can be approximated as follows: ' ln cs 

'cs , cs

(6.21)

where cs  (csI csII ) / 2 . The error of this approximation is a function of the concentration ratio r  csI / csII . Both sides of Eq. (6.21) can be written in terms of r, and are plotted in Figure 6.2. The error of the approximation does not reach 10% until r = 3.

6.2.2. The Chemical Potential of the Solvent No equation comparable to Eq. (1.41) was derived for the solvent in Chapter 1; now is the time to do so. To obtain a general expression for solvent chemical potential, we consider a system identical to that used in Chapter 1, consisting of k solutes, indexed by i, in solvent water. Begin by taking the differential of Eq. (1.26), which holds for solutes and solvent alike: dP w  Vw dP dP cw .

(6.22)

Equations (1.31) (the Gibbs–Duhem equation) and (1.35) (from the van 't Hoff Law in differential form) are used to express dP cw in terms of measurable solution properties. First, Eq. (1.31) is rearranged to give dPcw  

1 k œ ni dPic . nw i1

(1.31)

PRINCIPLES AND MODELS OF BIOLOGICAL TRANSPORT

233

Figure 6.2. % ln cs and %cs / cs as functions of the concentration ratio, r.

Equation (1.35) is then used to replace the right-hand side of Eq. (1.31). In this case, the result is simpler if we do not replace nwVw by V (as we did in Chap. 1): k

dPcw  Vw RT œ dci .

(6.23)

i 1

Equations (6.22) and (6.23) prescribe the chemical potential of the solvent in differential form. Since Vw is essentially constant, dPw can easily be integrated across the membrane to give the required chemical potential difference:

'P w  Vw 'P  Vw RT œ 'ci .

(6.24)

When there is only one solute,

'P w  Vw 'P  Vw RT 'cs .

(6.25)

As was the case when we derived the expression for the chemical potential of the solute, the use of the van 't Hoff relation limits the applicability of the preceding equations to solutions that can be treated as ideal and dilute.

234

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6.2.3. A New Set of Forces and Fluxes; Osmotic Pressure Now that the chemical potential differences can both be written in terms of pressures and concentrations, we can rewrite Eqs. (6.11) accordingly. If we linearize the concentration dependence of 'Ps according to Eq. (6.21), then Js and Jw become linear functions of 'P and 'cs. However, these functions are rather awkward; for instance,

L ¬ J s  ( LssVs LswVw )'P žžž ss  LswVw ­­­ RT 'cs . žŸ cs ®­

(6.26)

This complexity arises because the solute and solvent fluxes are not conjugate to 'P and 'cs. The dependence of the transport rates on 'P and 'cs can be expressed more conveniently by finding a new set of fluxes whose conjugate driving forces are at least proportional to the pressure and concentration differences. To do this, we return to the dissipation function for the system. Using Eqs. (6.3b), (6.20) with (6.21), and (6.24), and regroupingҏ, )  J s 'P s J w 'P w  RT 'cs ¬­ ­ J wVw ( 'P  RT 'cs )  J s žžžVs 'P žŸ cs ®­­

(6.27)

J ¬  ( J wVw J sVs )'P žžž s  J wVw ­­­ RT 'cs . žŸ cs ®­

Equation (6.27) shows that the coupled fluxes of solute and solvent, driven by 'Ps and 'Pw, can be represented by an alternative set of fluxes and forces. The first flux, J wVw J sVs , is simply the volume flux across the membrane, denoted Jv. The units of 2 Jv are cc/cm -sec. The volume flux is conjugate to 'P, the hydrostatic pressure difference across the membrane. The second flux, J s / cs  J wVw , is termed the exchange flux, JD; it is conjugate to RT'cs. This latter driving force also has the units of pressure, and is the osmotic pressure difference across the membrane. Osmotic pressure will be discussed in more detail later in this section. A physical meaning for JD can be derived, as follows. We saw earlier that the nominal velocity of solute or solvent in the membrane, vi, is defined as Ji /ci . Accordingly, J s / cs approximates the mean velocity of the solute. We also saw in Chapter 1 that, in dilute solutions, Vw x V / nw  1/ cw ; it follows that J wVw x J w / cw  vw . Hence the exchange flux is the difference between the velocities of the two species in the membrane; put another way, it is the nominal velocity of the solute relative to the solvent.

Corresponding to these new conjugate pairs is a new set of phenomenological equations, equivalent to Eqs. (6.11): Jv = Lvv'P + LvD'S

(6.28a)

JD = LvD'P + LDD'S,

(6.28b)

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where 'S = RT'cs. We should note that, in most texts, the three phenomenological coefficients in Eqs. (6.28) are denoted Lp (not Lvv), LpD (not LvD), and LD (not LDD). The subscripts used here are more consistent with the general notation used earlier in this chapter; in any case, these coefficients will shortly be replaced by a different set whose notation is standard.

6.2.4. The Kedem–Katchalsky Equations The next step in the derivation of the final working equations for transmembrane flux is to manipulate Eqs. (6.28) and obtain a pair of expressions that give Jv and Js in terms of the hydrostatic and osmotic pressure differences. Although the solute flux is not given explicitly by Eqs. (6.28), it can easily be found by adding the volume and exchange fluxes: J v J D  J sVs

Js J  s (1 Vs cs ) . cs cs

(6.29)

The product Vs cs is the volume of solute per unit volume of solution (cc solute/mol solute q mols solute/cc solution), and is much less than unity in reasonably dilute solutions. Neglect it in Eq. (6.29) and solve for Js: J s  cs ( J v J D ) .

(6.30)

The solute flux is seen to be a linear function of the volume and exchange fluxes; therefore, its dependence on the hydrostatic and osmotic pressure differences across the membrane should also be linear and expressible in terms of the three phenomenological coefficients in Eqs. (6.28). In practice, these coefficients are replaced by three related quantities: Hydraulic conductivity or flow conductivity: J ¬ L p w žž v ­­­  Lvv ; žŸ 'P ® 'S0

(6.31a)

Reflection coefficient: Vw

LvD ; Lvv

(6.31b)

Solute permeability: J ¬ Z w žž s ­­­ . žŸ 'S ® J v 0

(6.31c)

These three transport coefficients characterize the passive flux of a given solute and a given solvent through a given membrane as comprehensively as did the phenomenological coefficients in the earlier equations. The physical significance of two of the

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new coefficients is rather clear: the hydraulic conductivity measures the volume flow induced by a hydrostatic pressure difference, and Z measures the solute flux induced by a concentration difference (like the permeability in Chap. 2). The meaning of the reflection coefficient is less obvious and will be discussed later, once the final flux equations have been obtained. It remains to write Jv and Js in terms of these new coefficients. The easiest way to do this is to use Eqs. (6.31) to express the phenomenological coefficients of Eqs. (6.28) in terms of the new coefficients; then Eqs. (6.28) and (6.30) will provide the results we seek. Replacing Lvv by Lp and LvD by –VLp in Eqs. (6.28) and (6.30): J v  L p 'P  VL p 'S ,

(6.32a)

J s  cs ( J v  VL p 'P LDD 'S) .

(6.32b)

Next, Eq. (6.31c) is used to replace LDD in Eq. (6.32b) by the solute permeability. Equation (6.31c) defines Z in terms of the solute flux at zero volume flux. From Eq. (6.32a), when the volume flux is zero, 'P = V'S.

Substituting this expression into Eq. (6.32b), again at zero volume flux,

J s  cs (V2 L p LDD )'S . By substituting this expression for the solute flux into Eq. (6.31c), the following relation between Z and LDD is obtained: Z  cs (V2 L p LDD ) .

(6.33)

Equation (6.33) is solved for LDD: LDD 

Z V2 L p . cs

(6.34)

Substituting Eq. (6.34) into (6.32b),  ¬ Z'S J s  cs žžž J v  VL p 'P V2 L p 'S­­­ žŸ cs ®­  cs [ J v  V( L p 'P  VL p 'S)] Z'S

(6.35)

The parenthesized quantity multiplying V in Eq. (6.35) is simply Jv; thus, J s  cs J v (1  V) Z'S .

(6.36)

Equations (6.32a) and (6.36) are the equations most commonly used to characterize the coupled flux of solute and solvent through a membrane permeable to both species. They are often referred to as the Kedem–Katchalsky equations, named after the au-

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thors who introduced them to the biological community (Kedem and Katchalsky, 1958). A numerical example illustrating their use is given in Table 6.1. Table 6.1. Numerical Illustration of the Kedem–Katchalsky Equations Volume flux, J v  L p 'P  VL p 'S Solute flux, J s  cs J v (1  V) Z'S Consider the transport of sucrose across a Cuprophan dialysis membrane at 37ºC when: csI = 155 mM = 1.55 q 10–4 mols/cc csII = 125 mM = 1.25 q 10–4 mols/cc  'P = 100 torr = 100 q (1333 dynes/cm2-torr) = 1.333 q 105 dynes/cm2 The transport coefficients for this system are (Meyer and Friedman, 1977): Lp = 4 q 10–11 cc/dyne-s  V= 0.2 Z = 6 q 10–15 mols/dyne-s From csI and csII , cs = 1.4 q 10–4 mols/cc  'S= (1.987 cal/mol-K) q (310 K) q (3 q 10–5 mols/cc) q (4.186 q 107 dyne-cm/cal) = 7.74 q 105 dynes/cm2 Then, Jv = 4 q 10–11 q (1.333 q 105 – 0.2 q 7.74 q 105) cc/cm2-s = –8.60 q 10–7 cc/cm2-s Js = 1.4 q 10–4 q (–8.60 q 10–7) q (1 – 0.2) + 6 q 10–15 q 7.74 q 105 mols/cm2-s = 4.55 q 10–9 mols/cm2-s

Each term in the Kedem–Katchalsky equations can be associated with a specific physical process. The first term in the volume flow equation can be regarded as the hydraulic flow induced by the hydrostatic pressure difference, and the second term, called osmotic flow or osmosis, is the contribution to the volume flux resulting from the osmotic pressure difference across the membrane. Similarly, the first term in the solute flux equation can be regarded as the rate at which solute is carried across the membrane (i.e., convected) by the volume flux (this is often termed solvent drag), while the second term has the form of a diffusional component driven by the solute concentration difference. However intuitively satisfying this decomposition of the Kedem–Katchalsky equations may be, it must be remembered that each molecule of solvent and solute is acted upon by all driving forces, so the processes described above are not really separable; for instance, it is meaningless to ask whether a particular solvent molecule crossed the membrane hydraulically or by osmosis.

6.2.5. Physical Significance of the Reflection Coefficient: Semipermeable Membranes and the Osmometer, Steric Effects and Sieving The physical significance of the reflection coefficient might be expected to be somewhat less clear than that of the other two transport coefficients, if only because the

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reflection coefficient appears in both Kedem–Katchalsky equations. It measures the effectiveness with which osmosis is driven by a given osmotic pressure difference [Eq. (6.32a)], and also influences the rate of convection of solute by the volume flux [Eq. (6.36)]. Like solute permeability, the value of the reflection coefficient depends on the properties of both the solute and the transport barrier.

Figure 6.3. (a) Pure osmosis. Only solvent crosses the membrane. (b) (top to bottom) Hydrostatic pressure, concentration, and solvent chemical potential profiles between the two phases in (a). The dashed line shows the hydrostatic pressure profile at osmotic equilibrium, where there is no water flow. The entire pore is inaccessible to solute, and unstirred layers are ignored.

We begin by examining the dependence of volume flux on osmotic pressure. Consider the system shown in Figure 6.3a; the membrane passes water but is impermeable to solute (such a membrane is termed semipermeable), and the pressures in both phases are the same. Since the concentrations of solute are different in each phase, there is a corresponding interphase difference in the chemical potential of the solvent, which can cross the membrane. Since 'cs > 0 and 'P = 0, it follows from Eq. (6.25) that 'Pw < 0; thus, solvent flows from Phase II, where its chemical potential is higher, to Phase I, where it is less. This flow is pure osmosis, driven by only the osmotic pressure difference across the membrane, and is directed toward the phase having the higher osmotic pressure. Also note that, contrary to statements that still appear in the

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transport literature, the driving force in this case is not the concentration gradient of the solvent. The most plausible physical driving force for osmosis is a hydrostatic pressure gradient in the membrane pores. Since the membrane is semipermeable, there must be a portion of each pathway across it that is inaccessible to solute and is therefore occupied by water alone. Assume for clarity that this is the case for the entire pore. As noted in the context of Figure 6.1, the chemical potential of the solvent must be a continuous function; a discontinuity would correspond to an infinite gradient and thus an infinite driving force for water flow. This means (see Fig. 6.3b) that μw is continuous at the membrane faces x = 0 and x = a, where a is the length of the pore. Thus, from Eq. (6.25), the discontinuous drop in solute concentration at the left end of the pore (x I = 0), from cs to zero, must be compensated for by a corresponding drop in hydrostatic pressure. The situation is the same at the other end of the pore (x = a), but the decrease II I in pressure is less because cs < cs . Thus, a hydrostatic pressure gradient, driving hydraulic flow toward Phase I, develops in the pore. The profiles of hydrostatic pressure, solute concentration and solvent chemical potential, in and adjacent to the pore, are shown in Figure 6.3b. Returning to Figure 6.3a, the flow of solvent can be stopped by raising the pressure in Phase I until 'Pw becomes zero; then the solvent will be at equilibrium across the membrane and flow will cease. From Eq. (6.25) and the definition of osmotic pressure, it is apparent that the hydrostatic pressure difference necessary to bring the solvent flow to a halt is equal to the osmotic pressure difference across the membrane; when 'P = 'S'Pw = 0. This relationship between the hydrostatic and osmotic pressure differences at zero solvent flow (and zero volume flow, since the membrane does not pass solute) can be used to design an osmometer, as shown in Figure 6.4. Water crosses the semipermeable membrane, raising the hydrostatic pressure in Phase I. The difference between the hydrostatic pressures in Phase I and Phase II is proportional to the difference, h, between the heights of the two fluid surfaces: 'P = Ugh,

(6.37)

where U is the density of the solution and g is the gravitational constant. At equilibrium, this pressure head is equal to the osmotic pressure difference across the membrane. The hydrostatic pressure profile across the osmometer membrane at equilibrium is shown by the dashed lines in Figure 6.3b. Osmosis across a semipermeable membrane is described by the Kedem– Katchalsky equations. We have already seen that the volume flux across any membrane is zero when 'P = V'S. Thus, an alternative definition of the reflection coefficient is  'P ¬ . V  žž ­­­ žŸ 'S ® J v 0

From the preceding, it is evident that V = 1 for a semipermeable membrane.

(6.38)

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CH. 6: NONEQUILIBRIUM THERMODYNAMICS

Figure 6.4. An osmometer. The osmotic pressure is higher in Phase I than in Phase II. 'P = PI – PII is equal to 'S = SI – SII. If SI is known, then SII can be calculated from 'P.

When the reflection coefficient is unity, the osmotic pressure difference has as great an influence on volume flow as does the hydrostatic pressure difference [see Eq. (6.32a)]. When the substitution V = 1 is made in Eq. (6.32a), it takes the form of Starling's Law, originally developed to describe the volume flux across the capillary wall under combined differences in hydrostatic and osmotic pressure. In this context, it is worth noting that relatively small concentration differences generate considerable osmotic pressure differences: at 37ºC, when 'cs = 1 mM, RT'cs = 'S= 19 torr. Since the hydrostatic pressure in the capillaries is only about 40 torr more than that in the interstitial fluid outside, the osmotic pressure difference across the capillary wall necessary to reduce the volume flow to a modest level corresponds to a concentration difference of only a few millimoles per liter. If the membrane is leaky and passes solute, then the hydrostatic pressure difference needed to stop osmosis is less than 'S; that is, V < 1. This behavior can also be understood in terms of the profiles in Figure 6.3. When the membrane is semipermeable, the solute cannot enter the pore, and a hydrostatic pressure difference develops at each end of the pore to maintain continuity of the solvent chemical potential; that is, to compensate for the difference between the osmotic pressure of the ambient solution and that in the solute-free pore. When solute can enter the pore, a solute-free region remains, but it does not include the entire cross-section of the pore. Assume that the solute is a sphere of radius s, and let the center of the sphere define the radial position,

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U, of the solute molecule in the pore. Clearly, if the pore radius is r, the radial position of the solute cannot exceed r – s (see Fig. 7.3 in the next chapter). The fraction of the 2 2 2 pore cross-section accessible to the solute is )s = [(r – s) /r ] = (1 – s/r) . This exclusion of the the solute from a portion of the pore is known as the steric effect, and we will discuss it more in the next chapter. Anderson and Malone (1974) derived an expression for the reflection coefficient in terms of a radial potential energy profile that described the partial or complete exclusion of solute from the pore; when applied to a profile representative of steric exclusion, they obtained 2

   s ¬2 ¯ V = (1 – )s) = ¡¡1  žž1  ­­­ °° . ž ¡¢ Ÿ r ® °± 2

(6.39)

As the solute radius approaches that of the pore, the reflection coefficient approaches unity, in agreement with the result for a semipermeable membrane. Leakier membranes have smaller reflection coefficients. The reflection coefficient approaches zero for nonselective barriers, such as membranes whose transport pathways are much larger than the solute. This can also be seen from Eq. (6.39), as s/r o 0. A solute concentration difference across such a barrier will induce very little osmosis, even when the osmotic pressure difference is high. Sometimes the reflection coefficient is described as the ratio of the “apparent” or “effective” osmotic pressure difference to the true difference. An alternative interpretation of the reflection coefficient can be arrived at from the solute flux equation, Eq. (6.36). As the reflection coefficient rises from zero to unity, the flux of convected solute given by the first term in the equation falls from cs J v to zero. Noting that the convected solute flux when V = 0 is the flux that would be expected if solute were simply carried along in the solvent, the reflection coefficient has been interpreted as a measure of the retarding effect of the membrane on the solute relative to its resistance to solvent flow; that is, as a measure of the sieving effect of the barrier. The sieving effect of membranes is used in two important separation processes: ultrafiltration and reverse osmosis. In ultrafiltration, a solution of a relatively large membrane-impermeable species (often protein) is concentrated by raising its hydrostatic pressure to drive solvent across the membrane while the solute stays behind. The pore size in the membrane is typically 2 nm. A similar process takes place in reverse osmosis, which is used for desalination. Water is forced under hydrostatic pressure across a membrane whose pores, in contrast to those of ultrafiltration membranes, are so narrow (ca. 0.5 nm) that its reflection coefficient for sodium chloride and other small solutes is very close to unity. The osmotic pressure of the impure water is much higher than that of the desalted product, and the volume flow is driven against the osmotic pressure gradient. This is why the process is called reverse osmosis.

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6.2.6. Osmotic Pressure of Solutions; Donnan Osmotic Pressure; Osmotic Effects on Cells Osmotic Pressure of Solutions. In the preceding derivations, the concept of osmotic pressure arose from the expression for the chemical potential of the solvent. If the solution is sufficiently dilute, Equation (6.24) applies, and each solute contributes separately to the ideal osmotic pressure difference: 'Sideal = RT œ i 'ci . The summation includes all solutes, ions as well as nonelectrolytes. This expression can also be I II j written as 'Sideal = Sideal – Sideal , where Sideal = RT œ i cij , the ideal osmotic pressure of the jth phase. Since the osmotic pressure of an ideal solution depends on only its total concentration N = 6ci , irrespective of the identity of the solutes, it can be obtained simply by adding up the “particles” of solute per unit volume of solution. Thus, the total concentration of a solution containing 100 mM NaCl and 100 mM sucrose is 300 mM (100 mM each from Na, Cl, and sucrose). The osmolarity of a real solution is defined as the total concentration of an ideal solution that has the same osmotic pressure as the real solution in question. By this definition, if a solution is sufficiently dilute, its total concentration and osmolarity are essentially identical. Osmolarity is generally measured in milliosmols/liter, mOsm/l, to distinguish it from total concentration, which is measured in millimoles/liter. Two solutions having the same osmolarity are called isosmolar. For instance, in a world in which all solutions are ideal, the NaCl/sucrose solution described above, and a 100 mM solution of Na2SO4, would be isosmolar. Their osmolarity would be 300 mOsm/l. The molality of a solution is defined similarly to the molarity, except that the concentration is referred to 1 kg of solvent rather than 1 liter of solution. When molality is used instead of molarity, the associated terms described above are altered accordingly; for instance, the number of milliosmols/kg solvent is termed the “osmolality,” and two solutions having the same osmolality are isosmolal. For dilute solutions, molarity and molality are virtually identical. Most solutions encountered in living systems are not ideal, primarily because ion– ion and ion–solvent interactions reduce the effective number of independent particles in solution. For instance, the osmolarity of a 155-mM solution of NaCl at 25ºC is only 288 mOsm/l even though its total concentration is 310 mM; its osmotic pressure is 7% less than it would be were the solution ideal. The osmotic coefficient of a solution, M, is the ratio of the osmolarity of the solution to its total concentration (which is the same as the ratio of its osmotic pressure to the osmotic pressure it would exert if it were ideal); for the 155-mM NaCl solution, M 0.93. The osmotic coefficient of most physiological solutions (plasma, intracellular and interstitial fluid) is near this value. With this definition of the osmotic coefficient, the osmolarity, O, is equal to the product of the osmotic coefficient and the total concentration, so that the true osmotic pressure of a solution, Sreal = RTO. Other authors identify O with the total concentration, in which case Sreal = RTMO; the product MO is then referred to as the effective or corrected osmolarity of the solution.

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Figure 6.5. Dependence of the osmotic coefficient on molar concentration (M) for (1) D-glucose, (2) NaCl, and (3) KCl at 20ºC. From data in Lide (2004).

In general, the osmotic coefficient depends on temperature, concentration and the identity of the solute. Although it is less than unity for electrolyte solutions at physiological concentrations, it can be greater than unity for more concentrated ionic and nonelectrolyte solutions and for protein solutions. For instance, the osmotic coefficient of the highly concentrated hemoglobin inside red blood cells is near 4 (Baumgarten and Feher, 2001). The osmotic coefficient of plasma protein solutions is a nonlinear function of concentration that can be described by a second-degree polynomial with positive coefficients. The variation of osmotic coefficient with concentration is shown in Figure 6.5 for several common solutes. Another term used to describe osmotic pressure is tonicity. The tonicity of a solution measures the osmotic flow it can induce across a specific membrane; thus, in contrast to osmolarity, which depends on only the composition of the solution, the tonicity depends on the reflection coefficients of the solutes in the membrane under consideration. For ideal solutions, the tonicity, T  œ i Vi ci . Solutions having the same tonicity are said to be isotonic; when the solutions on the two sides of a membrane are isotonic, osmosis is absent. The terms hypotonic and hypertonic specify whether the solution in question has a lesser or greater tonicity than a reference solution, often the intracellular fluid. When the membrane in question is impermeable to all of the solutes in the solution, all of the reflection coefficients are unity, and the tonicity and osmolarity are identical.

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Figure 6.6. Normalized Donnan osmotic pressure vs. cII / c II .

Donnan Osmotic Pressure. As shown in Chapter 1, when charges are trapped in a phase that can still exchange some electrolyte with a second phase, a Donnan equilibrium develops across the interface between the two phases. It is easy to show from Eq. (1.55) that the total concentration of mobile ions in the first phase [Phase I in Eq. (1.55)] exceeds that in the electrolyte solution (Phase II) with which it has equilibrated:

'c  c AI cCI  c AII  cCII  (cII )2 4(c II )2  2c II  0 .

(6.40)

This concentration difference gives rise to an osmotic pressure difference SD = RT'c between the two phases; the osmotic pressure in the charged phase is always higher. The dependence of SD (the Donnan osmotic pressure) on cII is plotted in nondimensional form as Figure 6.6.

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Equation (6.40) does not include the contribution of the impermeant charged molecules in Phase I to the osmotic pressure of the phase. In most cases, the impermeant charges are carried on large macromolecules, which are relatively few in number, and whose contribution to the osmolarity of Phase I can safely be ignored (recall that cII is the concentration of impermeant charge in Phase I, not the concentration of charge-bearing molecules). The Donnan osmotic pressure promotes a volume flux into Phase I and is one of the reasons that charged gels tend to swell. Osmotic Effects on Cells. We noted in Chapter 2 that the total concentrations of the intracellular and extracellular fluids are similar, and they are indeed very close to isosmolar. This is a good thing: for the cell to maintain its normal volume, there cannot be any net flux of water into or out of it. Thus, any difference in osmolarity across the cell membrane must be balanced by a difference in hydrostatic pressure, the magnitude of which would depend on the reflection coefficients of the solutes in the cell and its environment. Because the osmotic pressure difference between the intracellular and extracellular fluids is very small, the hydrostatic pressure difference that arises to balance it is within the load bearing capabilities of the cytoskeleton and plasma membrane. When, in nonphysiological experiments, the extracellular osmotic pressure is suddenly increased or decreased from normal, the driving force for water flow rises markedly and the major initial flux is that of the solvent. The cell shrinks or swells, depending respectively on whether the external solution has been made hypertonic or hypotonic. The deformation of the cell alters the internal hydrostatic pressure, but this change is minor compared to the imposed change of external osmotic pressure; accordingly, the change in cell volume continues until the intracellular fluid becomes nearly isotonic with the external solution. During the initial rush of water flow, the solute fluxes into or out of the cell are not sufficient to alter substantially the intracellular solute content (as opposed to intracellular concentration). If, furthermore, the intracellular and extracellular fluids are essentially isotonic when water flow subsides — that is, if we neglect the small hydrostatic pressure difference across the cell membrane — it is easy to calculate the dependence of the final cell volume on the experimentally imposed extracellular osmotic pressure. The cell is considered to consist of two compartments: an aqueous compartment of variable volume V a that contains N moles of dissolved solute, and an inert phase of volume Vs that consists of fixed volume organelles, solid matter, and water that is bound to intracellular macromolecules and thus is osmotically inactive. Assume that the cell has achieved osmotic equilibrium with a solution whose osmotic pressure is Sc. Since we are neglecting hydrostatic pressure differences, the internal and external fluids are isotonic at equilibrium, so the intracellular osmotic pressure is also ScThe intracellular osmotic pressure is related to N and V a by Sa 

RT MN . Va

(6.41)

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Solving for V a , and adding in the volume of the inert phase, the cell volume, Vc, is given by Vc 

RT MN Vs . Sa

(6.42)

Thus, a plot of cell volume vs. the reciprocal of the extracellular osmotic pressure (or extracellular concentration) should be a straight line with a y-intercept equal to the volume of the inert phase. This behavior has been demonstrated for a variety of cells, including red blood cells (LeFevre, 1964). When the osmotic pressure ambient to a suspension of erythrocytes is reduced, the cells can swell considerably, since they are normally disc shaped and can increase their volume substantially without increasing their surface area; acute increases in surface area are accompanied by tensile stresses in the cell membrane that can lead to rupture. Naturally, the cell cannot enlarge indefinitely as the external osmotic pressure is reduced, and when the concentration of the external solution becomes less than a threshold value, the cell membrane ruptures, causing hemolysis.

6.2.7. Passive Transport of Multiple Nonelectrolytes The Kedem–Katchalsky equations are only slightly more complicated when more than one solute is present. Of course, there are more phenomenological coefficients; referring to Eq. (6.10), the rank of the matrix of coefficients is n, where n is the number of 2 species (the solvent and n – 1 solutes). Thus, there are n coefficients, of which n are straight coefficients. Thanks to the Onsager reciprocal relation, only half of the re2 maining n – n cross coefficients are independent, so the total number of independent 2 coefficients is n + (n – n)/2 = n(n + 1)/2. Thus, for a single solute, n = 2, and (as we have seen) three coefficients are sufficient to describe the transport behavior of the system. When there are two solutes (n = 3), six coefficients are needed. This number rises quadratically as the number of solutes increases. To reduce the number of transport coefficients needed to describe a system containing multiple solutes, it is usually assumed that there is no direct coupling between the chemical potential difference of any given solute and the fluxes of all the other solutes. Then, the only cross coefficients are of the form Lsi w , where the subscript “i” denotes the ith solute. There are n – 1 of these, because there are n – 1 solutes. Since there are n straight coefficients, a total of 2n – 1 phenomenological coefficients are sufficient to describe the system; these can be translated into the hydraulic conductivity, n – 1 reflection coefficients (one for each solute), and n – 1 solute permeabilities. The Kedem–Katchalsky equations become n1

J v  L p 'P  L p œ Vi 'Si ,

(6.43a)

J i  J v ci (1  Vi ) Zi 'Si (i  1,..., n  1) ,

(6.43b)

i 1

where 'Si  RT 'ci .

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6.2.8. Passive Transport of Electrolytes: Electrokinetic Phenomena When the solutes crossing a barrier are charged, the nonequilibrium thermodynamic description of the transport processes is more difficult, and the transport system exhibits some new features. Some of these new features can be identified simply by examining the dissipation function, without attempting the derivation of the transport equations, which are described fully in Kedem and Katchalsky (1963). Assume that the system contains only ions (indexed by i) and solvent; the extension to a combination of charged and uncharged solutes is straightforward. The driving force conjugate to ionic flux is the electrochemical potential difference, so the dissipation function is i . )  J w 'P w œ J i 'P

(6.44)

i

Proceeding in a fashion analogous to that used in Eq. (6.27), but using Eq. (6.24) instead of (6.25) because multiple solutes are involved,  ¬ RT 'ci )  J wVw ('P  RT œ 'ci ) œ J i žžžVi 'P ziᑠ'\­­­ žŸ ci ®­ J ¬  ( J wVw œ J iVi )'P œ žžž i  J wVw ­­­ RT 'ci (ᑠ œ J i zi )'\ žŸ ci ®­

(6.45)

The dissipation function given above is similar in some ways to that for a single nonelectrolyte [Eq. (6.27)], and different in others. As before, the first product shows that the volume flux (which now includes contributions from each solute) is conjugate to the hydrostatic pressure difference. This product is followed by the sum of the contributions of the exchange fluxes to the dissipation rate; the exchange flux of each solute is conjugate to RT'ci, which is that solute's contribution to the osmotic pressure difference across the membrane. The last product in Eq. (6.45) did not appear in Eq. (6.27), and arises specifically because charged solutes are present. The driving force is the electrostatic potential difference across the membrane; the flux to which that force is conjugate is the current density. Nonequilibrium thermodynamics tells us that the members of this last force–flux pair can be coupled to the other forces and fluxes in Eq. (6.45). Such interactions are known as electrokinetic phenomena. Among these are: (1) electroosmosis, in which a volume flow is driven by an imposed potential difference, and (2) streaming effects, which arise when a component of the transmembrane current is driven, or a component of the transmembrane potential is generated, by a hydrostatic pressure-driven flow. The phenomenological coefficients defining these phenomena are derived in the paper by Kedem and Katchalsky (1963) referred to earlier.

248

CH. 6: NONEQUILIBRIUM THERMODYNAMICS

6.3. NONEQUILIBRIUM THERMODYNAMIC DESCRIPTION OF ACTIVE TRANSPORT Active transport is incorporated into the nonequilibrium thermodynamic description of coupled transport by introducing the affinity, A, of the driving metabolic reaction and its conjugate flux, the rate of the reaction per unit membrane area, Jr. The reaction rate and the affinity of the reaction must be defined such that the reaction dissipates free energy at a rate equal to Jr A. Often, Jr is taken to be the mols of ATP hydrolyzed per unit area per unit time, and A is then the affinity of the hydrolysis reaction. The flux equations for active transport given by nonequilibrium thermodynamics are considerably simpler than those derived from kinetic analysis in Chapter 5, but they tell us nothing about the mechanism of the process. They do have their place, however, and we will use them to try (unsuccessfully) to construct an objective definition of active transport. Furthermore, since the forte of nonequilibrium thermodynamics is the description of coupled processes, we will also use the phenomenological equations to examine the several proposed measures of the “tightness” with which the metabolic reaction is coupled to the flux of substrate.

6.3.1. Definition of Active Transport For the simplest case of primary transport of a single uncharged solute, A, the phenomenological equations are JA = LAA 'PA + LAr A,

(6.46a)

Jr = LrA 'PA + Lrr A.

(6.46b)

Because of the notation adopted throughout the text, A in Eqs. (6.46) designates both the pump substrate and the affinity of the driving reaction. In context, there should be little confusion regarding the meaning of A at any point in the discussion to follow. The terms “substrate” and “affinity” will be used instead of the symbol wherever possible. For a system described by Eqs. (6.46), the requirement for active transport of substrate is that LAr be nonzero. This criterion is independent of the direction of either the flux of substrate or its chemical potential gradient; “downhill” active transport in the same direction as the passive component of the flux is fully within this definition. The definition of active transport is more difficult in systems containing multiple solutes and more coupling of forces and fluxes. In such systems, it is convenient to regard active transport as a coupling of the substrate flux to Jr (rather than to the affinity), and to write the phenomenological equation for JA in terms of resistances. To do this, we start with Eq. (6.13): X j  œ R ji J i . i

The sum includes all species fluxes, and Jr. Partition the sum as follows:

(6.13)

PRINCIPLES AND MODELS OF BIOLOGICAL TRANSPORT

X j  R jj J j R jr J r

249

œ

R ji J i .

(6.47)

other species

Solve Eq. (6.47) for Jj , and let the jth species be the substrate. Then, the substrate flux is given by

 ¬­ ­­ 1 žžž JA  žž X A  RAr J r  œ RAi J i ­­­ . RAA ž other žŸ ®­ species

(6.48)

In Eq. (6.48), the first term on the right-hand side is the contribution to the flux of substrate that is driven by the substrate's own chemical (or electrochemical) potential difference, XA; the straight resistance, RAA, is positive. The second term measures the contribution of the metabolic reaction, and the third term can be regarded as the drag or entrainment of the substrate by the fluxes of other species. For instance, the convection of substrate in moving solvent would be included in the third term, with i = w and RAw < 0 (since solvent drag carries the solute in the same direction as the solvent). The sum in Eq. (6.48) also includes inhibitory effects. If species B is a competitive inhibitor of the flux of A, then RAB is positive, since JA is diminished when more B uses the carrier to cross the membrane; i.e., when JB increases. Equation (6.48) takes on a particularly simple form when the substrate is in thermodynamic equilibrium across a membrane (i.e., XA = 'PA = 0) and the substrate flux is not coupled to that of any other species. In that case, JA = –RArJr /RAA. For an active  A , and thermodytransport system whose substrate is an ion, 'PA is replaced by 'P namic equilibrium is achieved by short-circuiting the membrane (so that '\ = 0) and maintaining identical concentrations of substrate on the two sides. Under these conditions, the substrate flux should be directly proportional to the rate of ATP hydrolysis. Primary active transport is the direct coupling of substrate flux to Jr; that is, RAr z 0. Similarly, secondary transport via the cosolute B would be characterized as follows: RAr = 0 (primary transport of A is absent), RBr z 0 (B is the substrate of a primary active transport system), and RAB z 0 (the flux of A is coupled to that of B). The preceding description suggests a means for deciding whether any particular flux (e.g., of substrate A) has an active component: if RAr is nonzero, A is actively transported; if RAr is zero, examine those fluxes (call them {JAc}) to which JA is coupled (i.e., RAAc z 0), and see if any of these fluxes are coupled to a Jr (i.e., RAcr z 0). If none are, look for metabolic coupling to any species fluxes to which the {JAc} are coupled, and so forth. This is an attractively straightforward and objective algorithm; the problem in using it is that every flux in a living organism can ultimately be traced to a metabolic reaction of some kind. The arbitrariness of such a definition of active transport has prompted some to suggest that only primary active transport should bear that title. At the very least, the claim that a particular flux has an active component should be accompanied by an indication of the process to which that component is thought to be coupled.

250

CH. 6: NONEQUILIBRIUM THERMODYNAMICS

Some investigators have proposed that active transport be defined as the flux of substrate against its electrochemical potential gradient. Although uphill transport is one indication of an active process, this definition does have some problems. One difficulty is that it makes no mention of metabolic coupling, which underlies all active transport. Furthermore, active transport systems that pump uphill are also capable of supporting a downhill flux when the electrochemical potential of the substrate is lower at the trans side of the membrane. In this case, the pump and leak fluxes proceed in the same direction. This mode of operation is more common when the coupling between ATP hydrolysis and transport is weak.

6.3.2. Coupling Between Nonconjugate Forces and Fluxes Nonequilibrium thermodynamics provides a formal means for characterizing, through the phenomenological coefficients, the degree to which a flux is coupled to a force that is not its conjugate. Thus, it is perfectly suited to describe coupling in active transport, in which the substrate flux is coupled to a nonconjugate driving force, the affinity of the metabolic reaction. The degree of coupling, q, between the ith flux and the jth force is given by q

Lij ( Lii L jj )1/ 2

.

(6.49)

Since Lij = Lji, the same value of q holds for the coupling of the jth flux to the ith force. The coupling can be positive or negative, depending on the sign of Lij. Thermodynamic considerations dictate that L2ij d
LiiLjj; thus, the absolute value of q lies between zero and one. When q = 0, there is no coupling; when coupling is complete, |q| = 1. In applying Eq. (6.49) to active transport, i is set equal to A, the substrate. For primary transport, j = r, the driving reaction; for secondary transport, j = B, the cosolute. In cotransport, A travels down B's (electro)chemical potential gradient, so JA and  B have the same sign. Therefore, LAB > 0, and q is positive. In countertransport, the 'P cross coefficient is negative, and so is q. When two force–flux pairs are completely coupled, the ratio of the fluxes is fixed — one cannot proceed without the other — and is independent of either driving force. 1/2 This can be seen by substituting Lij = (LiiLjj) into the flux equations: 1/2

Ji = LiiXi + (LiiLjj) Xj, 1/2

Jj = LjjXj + (LiiLjj) Xi. 1/2

Multiply Ji by (Ljj /Lii) : 1/ 2

 L jj ¬ J i žžž ­­­ žŸ L ®­ ii

 ( Lii L jj )1/ 2 X i L jj X j = Jj ( | q | = 1).

(6.50)

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251

The stoichiometry of the coupled fluxes (i.e., Ji /Jj) is determined by the ratio of the straight phenomenological coefficients. The fluxes may be similarly or oppositely directed, since the sign of the square root of Ljj /Lii can be positive or negative. We previously referred to such transport as obligatory. Values of |q| less than unity reflect incomplete coupling between the driving process — the dephosphorylation of ATP or the flux of cosolute — and the flux of the substrate. One class of processes that diminish the degree of coupling are the leakage fluxes described in Chapter 5. Recall that two kinds of leak can be identified: internal and external. An internal leak (called “slippage” in Chap. 5) occurs within the pump itself; examples include the transition of unloaded carrier from the cis side of the membrane to the trans side, or of loaded carrier in the opposite direction. External leaks include transport pathways in parallel with — but physically separate from — the pump. The degree of coupling can be measured in several ways. It can be written in terms of: 1. the dependence of the fluxes on one another:  sJ ¬­ q 2  žžž i ­­ < žŸ sJ j ®­­ Xi

 ¬ žž sJ j ­­ , žžŸ sJ i ®­­ Xj

(6.51a)

or 2. the driving forces at which each flux is zero: q2 

( X i ) Ji 0 ( X i ) J j 0

,

(6.51b)

or 3. the flux of one species when either the flux or driving force of the other is zero: q2  1

( J j ) J i 0 ( J j ) Xi 0

.

(6.51c)

Equation (6.51c) shows that, when coupling is incomplete, ( J j )Ji 0 z 0; that is, the jth flux is nonzero even when the ith flux is zero. Consider the case in which the subscript i denotes the substrate and j denotes the chemical reaction that drives the active substrate flux. With this designation of i and j, we see that, when the coupling is incomplete, metabolic activity is necessary ( J j v 0) to maintain the nonequilibrium state created by the pump, even in the absence of a substrate flux ( J i  0) . This is a result of the leaks referred to above. Two other thermodynamically based measures of flux coupling have been applied to active transport. The first of these is the efficacy, which is based on the driving forces at static head. The efficacy with which the jth force drives the ith flux is

252

CH. 6: NONEQUILIBRIUM THERMODYNAMICS

 X ¬­ H  žžž i ­­ Ÿž X j ®­­

.

(6.52)

J i 0

Of greater interest is the efficiency, which is the power associated with the active substrate flux, divided by the rate at which energy is dissipated by the driving process. Consider a secondary scalar transport process in which the substrate is an uncharged amino acid and the cosolute is sodium ion. The work of transporting nA mols of the amino acid A against a chemical potential difference P IIA  P IA is

nA (P IIA  P IA )  nA 'P A ; when the substrate flux is JA, the rate of transport work is equal to –JA'PA. Similarly, the rate at which energy is dissipated by the sodium ion B  B (for a chemical passing down its own electrochemical potential gradient is J B 'P reaction, the dissipation rate is Jr A). The efficiency with which the jth process drives the ith flux is K

JX J 'P power out  i i  A A . B power in JjXj J B 'P

(6.53)

The efficiency depends strongly on the experimental conditions. Consider a pump whose operating curve is given by Figure 6.7. The pump normally operates in the second quadrant, pumping substrate from Phase I to Phase II (JA > 0) against its chemical potential gradient ('PA < 0). From Eq. (6.53), its efficiency in doing so is generally greater than zero. But at the two intercepts that bound that segment of the operating curve, the efficiency is zero! At level flow, XA = 'PA is zero, and at static head, JA is zero. Thus, as the operating point moves from the static head in the direction of increasing flux, the efficiency rises from zero, passes through a maximum, and then falls as the level flow condition is approached. It can be shown that the maximum efficiency depends on the degree of coupling, and is unity when |q| = 1.

6.3.3. Nonequilibrium Thermodynamics of Motor Proteins and ATP Synthase Motor proteins, such as the kinesins of Chapter 3 and the actin and myosin of the myofibril in Chapter 9, employ the energy in ATP to create motion against a resisting force, such as the viscous drag on secretory vesicles being carried along a microtubule or a force opposing the contraction of muscle. These proteins convert chemical energy into mechanical work rather than solute flux, but the nonequilibrium thermodynamic treatment of both conversions is similar. The conjugate force–flux pair for motor proteins consists of the force, F, against which the motor acts, and the velocity, v, of the protein. These are coupled with the progress of the metabolic reaction and its affinity; using the linear formulation,

v = LvvF + LvrA,

(6.54a)

Jr = LvrF + LrrA,

(6.54b)

where the subscript “v” in the generalized conductances denotes the protein.

PRINCIPLES AND MODELS OF BIOLOGICAL TRANSPORT

253

Figure 6.7. Operating curve of a pump in the absence of external leaks. The variation of efficiency between static head and level flow is also shown.

Since the straight phenomenological coefficients are always positive, the linear formulation implies that the velocity vector of the motor is in the same direction as the force. And so it would be, if the protein responded passively to the force in the absence of ATP. The energy provided by ATP hydrolysis lifts the system to a decidedly nonequilibrium state, driving the motor in a direction opposite to the force. We will see below that when the system is far from equilibrium, additional, higher-order terms must be added to the linear formulation to describe properly the dependence of the fluxes on the forces. Similar considerations apply to active ion transport. We saw in Chapter 3 that the molecular motor ATP synthase is driven by the electrochemical potential difference of hydrogen ions across the inner mitochondrial membrane. The synthase, in turn (and in turning), drives the phosphorylation of ADP. Using the formalism of Eq. (6.48), 1. The rotation of the synthase rotor (JA) is conjugate to the torque it experiences (XA) and is also coupled to the flux of hydrogen ions across the membrane [Ji in Eq. (6.48)]. 2. The phosphorylation reaction (Jr ) is conjugate to its affinity and is also coupled to the motion of the rotor. The first of these processes could be close to equilibrium since the torque is derived from Brownian motion, but the second can be far from equilibrium, since the phosphorylation reaction is highly endothermic.

254

CH. 6: NONEQUILIBRIUM THERMODYNAMICS

6.4. LIMITATIONS OF NONEQUILIBRIUM THERMODYNAMICS Although nonequilibrium thermodynamics is a powerful tool for analyzing and characterizing coupled processes, it is not without its limitations. Some of these have already been noted. In this section, we will examine the impact of these limitations in a more organized way, with particular emphasis on the usefulness and range of applicability of the flux equations. The applicability of these equations is further limited by the approximations made in deriving them, and we will also deal with these restrictions.

6.4.1. Closeness to Equilibrium: A Limitation Intrinsic to Linearized Nonequilibrium Thermodynamics The basic equation underlying the applications of nonequilibrium thermodynamics described in this chapter is the linear relationship between the fluxes and the forces given by Eq. (6.10). We will first consider the limitations on the use of this equation. When Eq. (6.10) was first presented, it was noted that a more complete expression would include higher-order terms in the driving forces. Indeed, the linear relation can be regarded as the first terms in a Taylor series expansion for the ith flux: n

n

n

J i  œ Lij X j œ œ Lijk X j X k ... . J 1

(6.55)

j 1 k 1

The justification for truncating the series after the linear terms is that the expansion variables — in this case, the driving forces — are small. Since the driving forces are zero at equilibrium, the requirement that the forces be small is equivalent to a requirement that the system be close to equilibrium. Such a requirement is not very useful from a quantitative point of view. How close to equilibrium is close enough? How small a driving force is small enough? The answer depends on the problem at hand. If one is trying to measure a particular transport property, then “small enough” might mean: sufficiently small that the error in the final answer introduced by neglecting higher-order terms is less than the experimental error. If nonequilibrium thermodynamics is being used in a more qualitative way — say, to identify the existence of a particular interaction — “small enough” might mean: sufficiently small that the neglect of higher-order terms does not cause us to conclude that the interaction exists when it doesn't, or vice versa. Since the phenomenological coefficients are experimental quantities, it is in most cases impossible to estimate, a priori, the errors in flux prediction resulting from the departure of the transport system from equilibrium (ironically, such departure is necessary for transport to take place at all). Some insight into the factors that affect these errors can be gained by considering the free diffusion of a single nonelectrolyte; for this process, the flux equation derived in Chapter 2 can be used as a gold standard against which the nonequilibrium thermodynamic result can be compared. That gold standard is Eq. (2.20): Js = ks'cs  ks (csI  csII ) .

(2.20)

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255

In the nonequilibrium thermodynamic formulation, the flux equation for the same system is Eq. (6.9):

Js = Lss'Ps.

(6.9)

For an isothermal, isobaric system, Eq. (6.9) becomes

Js = LssRT 'ln cs  Lss RT ln

csI . csII

(6.56)

Equating the right-hand sides of Eqs. (2.20) and (6.56), and solving for Lss:

ks c I  c II (6.57) ¸ s I sII . RT ln(cs / cs ) If the solute flux is proportional to its conjugate driving force 'Ps, then Lss is constant. Assume that Lss approaches a limiting value, Leqss , when the driving force approaches Lss 

zero. Then the deviation of Lss from Leqss as 'Ps increases from zero is a measure of the departure of the nonequilibrium thermodynamic description of the system from linearity. Alternatively, the constancy of Lss measures the validity of the linear thermodynamic description of the diffusion process. Returning to Eq. (6.57), we see that Lss is the product of two terms, the first involving the permeability and the second the concentrations at the two sides of the membrane. The permeability depends on these concentrations, too, and we will approximate that dependence here by assuming that the permeability is a function of only the mean concentration, cs . Now consider an experiment in which the initial concentration of solute is the same at both sides of the membrane. Then the concentration at Side I is raised, and that at Side II is lowered, such that the mean concentration is unchanged. When the boundary conditions are varied in this way, the permeability remains constant by our previous assumption, and the variation of Lss with the chemical potential difference is given by the second term in Eq. (6.57). Let r  csI / csII ; Eq. (6.57) can then be written as

Lss 

ks csII r  1 ¸ . RT ln r

(6.58)

The driving force is a particularly simple function of r: 'Ps = RT ln r.

(6.59)

The quantities r and csII in Eq. (6.58) are related by the requirement that the mean concentration, given by Eq. (6.60), is fixed:

cs 

csI csII c II (r 1)  s . 2 2

(6.60)

256

CH. 6: NONEQUILIBRIUM THERMODYNAMICS

Equation (6.60) is rearranged to express csII in terms of cs ; substituting this expression into Eq. (6.58), Lss 

ks cs 2(r  1) kc w s s ᑬ(r ) . ¸ RT RT (r 1) ln r

(6.61)

Figure 6.8. The function ᑬ, defined in Eq. (6.61). Note that ᑬ is the ratio of the functions in Figure 6.2; thus, the appropriateness of the linearization of 'ln cs, and the closeness to equilibrium of the transport system, go hand in hand.

Since the mean concentration and permeability are constant, the variation of Lss as the driving force rises from zero is given by the function ᑬ(r). This function is unity at equilibrium (i.e., when r = 1), so Leqss is equal to ks cs / RT . ᑬ(r) is plotted against r in Figure 6.8. Clearly, this transport system is close enough to equilibrium even when the concentration at one side of the membrane is twice that at the other, since L55 is only 4% less than its equilibrium value when r = 2. This kind of analysis can also be applied to facilitated diffusion. For the simplest carrier, Eq. (4.39) is used as the gold standard to measure the variation of LAA with 'PA, much as Eq. (2.20) was used for free diffusion; the function corresponding to ᑬ(r) in Eq. (6.61) is ᑬ

I

II

where r = cA /cA .

ln r r 1  r ( K / c AII ) 1 ( K / c AII )

,

(6.62)

PRINCIPLES AND MODELS OF BIOLOGICAL TRANSPORT

257

In primary active transport, the affinity of the metabolic reaction that drives the pump must be small. “Small” in this case means: A << RT. When this inequality holds, the expression for * given by Eq. (5.36) can be approximated by * 1

A , RT

(6.63)

and the expressions for active flux in the previous chapter become linear in the affinity. For instance, the active flux at level flow, given by Eq. (5.31), becomes JA 

D aXT K cA K D A ¸ ¸ , 2 [ K ( K D 1) c A ][c A KK D ] RT

(6.64)

where A/RT << 1 has been used. At level flow, Eq. (6.46a) reduces to JA = LArA, so Eq. (6.64) gives LAr a physical meaning in terms of the parameters of a model of the transport mechanism. Phenomenological coefficients have physical meaning only in the context of such models. When the affinity is large and the strong inequality given above does not hold, higher-order terms in A/RT are needed to approximate the exponential in Eq. (5.36); this in turn leads to higher-order terms in the flux equation, which assumes the general form of Eq. (6.55).

6.4.2. The Concentration Dependence of the Phenomenological Coefficients Referring to Eq. (6.61), we see that LSS depends not only on the concentration ratio, but also on the mean concentration of the bathing solutions. This behavior is also exhibited by the phenomenological coefficients of more complex systems. Consequently, one must be careful when using phenomenological coefficients measured at one concentration to predict fluxes at a substantially different concentration. In practice, this caution is not that restrictive. There are two reasons for this. First, living systems are viable over only limited concentration ranges, so it is rare that measurements on a biological transport process at one concentration would be used to describe that process at a markedly different concentration. Second, some of the transport coefficients that appear in the flux equations are less sensitive to concentration than are the phenomenological coefficients from which they are derived. A good example is the solute permeability, Z, given by Eq. (6.33). This coefficient, which is the product of the mean concentration and a function of the phenomenological coefficients, can be as insensitive to concentration as the free diffusion permeability, ks. Presented in Figure 6.9 are experimental data (Meyer et al., 1981) showing the concentration dependence of the transport coefficients that describe the passage of sucrose and water across the dialysis membrane Cuprophan. For this system, the coefficients are rather constant, considering that the sucrose concentrations vary over a sixfold range.

258

CH. 6: NONEQUILIBRIUM THERMODYNAMICS

Figure 6.9. Transport coefficients for sucrose and water transport across Cuprophan 150 PM dialysis membrane at 37ºC (Meyer et al., 1981). Error bars are ±1 S.E.M. The largest concentration ratio in any of these experiments was 1.31.

6.4.3. Closeness to Equilibrium in Biological Systems Moving from the simple diffusion case described above to real biological systems, where the transport processes and their coupling are usually more complex and less well understood, there is both good news and bad news regarding the requirement that these processes be close to equilibrium. The bad news is that when the mechanism of the transport process is not well understood, there is no basis on which to predict the range of driving forces over which the system will be linear. Since considerable deviations from linearity can occur, data on such coupled flows must be handled with due regard for possible nonlinear effects. Ultimately, the range of linearity of the phenomenological equations must be found experimentally. The good news is that when these experiments are performed, linearity is often observed even when the driving forces assume values that might be expected to lead to considerable nonlinearity. This extended linearity has been shown by Essig and coworkers (Essig and Caplan, 1979) to be a feature of sodium transport across amphibian epithelia. They found that the active sodium flux and Jr varied linearly with membrane potential, even when transmembrane differences as high as 80 mV were used. An 80-mV potential change has the same effect on the electrochemical potential difference as a 20-fold change in the concentration ratio. The close-to-equilibrium limitation applies to only those flux equations that are based on the linear relation [Eq. (6.10)] between conjugate forces and fluxes. Other flux equations, particularly those derived in Chapters 2 and 4, describe processes in

PRINCIPLES AND MODELS OF BIOLOGICAL TRANSPORT

259

which each flux is coupled to only its conjugate driving force. These equations are not based on Eq. (6.10), and can be used to describe decidedly nonequilibrium situations. The flux that usually determines whether the close-to-equilibrium limitation can be avoided is the solvent flux; if there is none, or if it is not coupled to the solute fluxes, then the latter can usually be described by the equations derived in the earlier chapters. For instance, the distribution of most interesting ions across biological barriers is very far from equilibrium, and almost all treatments of their passive transport are traceable to the electrodiffusion equation, which requires that the passive flux of any ion be driven by only that ion's electrochemical potential gradient. Why do we put up with these limitations? The reason is that nonequilibrium thermodynamics is the only available means for dealing with coupled flows in the many systems whose structure is too complex, or whose transport mechanisms are too poorly understood, to admit a more detailed model. Ironically, this virtue is associated with yet another of the method's limitations, which we now discuss.

6.4.4. The Information Content of Nonequilibrium Thermodynamics Nonequilibrium thermodynamics prescribes the relationships among a collection of interacting forces and fluxes without regard to the mechanism of these interactions or the structure of the physical system in which the interactions take place. This is, as stated above, a virtue, but it is also a vice, since it means that nonequilibrium thermodynamics can tell us very little about those interactions or that structure. This limitation has several implications. Perhaps most important, one must be extremely careful when interpreting experimental phenomenological coefficients in physical terms. Such interpretations are only appropriate if the transport mechanism is known from separate experiments. Though nonequilibrium thermodynamics cannot provide a model of the physical process, it can provide guidance in model development by supporting some mechanisms and opposing others. Suppose, for instance, it is hypothesized that solvent and solute cross a membrane via aqueous pores. We will see in Chapter 7 that the size of these supposed pores can be estimated from the phenomenological coefficients. If the pore size is reasonable, the hypothesis is supported. If the pore size is unreasonably large or depends strongly on the identity of the solute, the hypothesis may be questioned. Another use of nonequilibrium thermodynamics in model building was illustrated in the previous section, where it was seen that certain active transport processes can be indicated or excluded on the basis of the sign or magnitude of the relevant cross coefficient. It is also dangerous to use nonequilibrium thermodynamics to predict the behavior of one transport system from the phenomenological coefficients of another. For instance, without additional justification (usually in the form of a reasonably certain transport mechanism), it is unwise to predict the effect on transport of substituting one solute for another, or to predict the flux of one solute on the basis of the phenomenological coefficients for the transport of another.

260

CH. 6: NONEQUILIBRIUM THERMODYNAMICS

6.4.5. Approximations in the Derivation of the Kedem–Katchalsky Equations The Kedem–Katchalsky equations are subject not only to the intrinsic limitations of linear nonequilibrium thermodynamics that have already been discussed, but also to limitations implied by the approximations that were made in their derivation. There were four such approximations: 1. The chemical potential differences were written in terms of concentration rather than activity. 2. The van 't Hoff Law, valid only for dilute solutions, was used in deriving Eq. (6.24) for the chemical potential difference of the solvent. 3. Equation (6.21) was used to express the chemical potential difference of the solute in terms of the concentration difference and the mean concentration. The effect of the concentration ratio on the validity of this approximation has already been discussed. 4. The volume fraction of solute, Vs cs , was assumed to be much less than unity in deriving Eq. (6.30). Approximation #3 must be satisfied if the flux equations are to be linear; the remaining approximations are valid only for ideal dilute solutions. A set of equations analogous to the Kedem–Katchalsky equations and applicable to more concentrated, nonideal solutions can be derived from the Kirkwood formulation of the Stefan–Maxwell equations (Lightfoot, 1974) by retaining activity coefficients and the solute volume terms (Friedman and Meyer, 1981). The result looks very much like the Kedem– Katchalsky equations: ) , J v  L p ('P  V'S

(6.65a)

 . J s  J v (1  V)cs Z'S

(6.65b)

The effects of nondiluteness and nonideality are included in the modified osmotic  , whose transmembrane difference is defined by pressure, S  1 * ¬­ ­   žž 'S žŸž1  v ®­­ 'S , s

(6.66)

where the quantity * w ( d ln J s / d ln cs )cs cs is a measure of the nonideality of the solutions and vs w Vs cs is a measure of their nondiluteness. A tilde is placed over the solute permeability in Eq. (6.65b) to indicate that this transport coefficient is defined differently from Z. In ideal dilute solutions, * and vs both approach zero, and Eqs. (6.65) reduce to the Kedem–Katchalsky equations.

PRINCIPLES AND MODELS OF BIOLOGICAL TRANSPORT

261

PROBLEMS: CHAPTER 6 1.

Repeat the calculation in Table 6.1 entirely in SI units, starting by specifying the bounding concentrations, the pressure difference, and the transport coefficients in these units. Then solve for the fluxes and convert them back to the units in the text as a check.

2.

The following transport-related quantities were obtained from measurements on an experimental preparation of a rat glomerulus (the glomerulus is a portion of the kidney that we shall discuss in more detail in Chap. 10; refer to Figs. 10.15 I II and 10.21. I = capillary blood; II = Bowman's space): P = 49 mm Hg, P = 14 mm Hg, 'S = 17 mm Hg, SGFR (single glomerulus filtration rate) = 44 nl/min. Assume that the entire osmotic pressure difference is due to albumin, which cannot cross the glomerular filtration barrier. The barrier area of a single glomerulus 2 is estimated to be 1 mm , based on histological measurements. What is the hydraulic conductivity of the barrier?

3.

Oocytes can be cryopreserved for women at risk of losing their ovarian function. Relatively concentrated solutions of cryoprotectants are used to prevent damage to the cells during the freezing process. The phenomenological coefficients of the mouse oocyte membrane for the cryoprotectant ethylene glycol are: Lp = 0.18 μm/min-atm, Z = 0.03 μm/sec, and V = 0.97 (Paynter et al., Cryobiology 39:169– 176, 1999). Suppose the cell is initially in osmotic equilibrium with its environment and enough cryoprotectant is added to the ambient solution to achieve a glycol concentration of 1.5 M, a typical value. There is no hydrostatic pressure difference across the membrane. T = 10ºC. (a) What is the initial rate of osmosis across the cell membrane? (b) A slow rate of cell shrinkage is desirable. This could clearly be accomplished by using a cryoprotectant with a lower reflection coefficient. In spite of this, most cryoprotectant reflection coefficients are near unity. Why do you think this is the case?

4.

A membrane is bounded by two sucrose solutions whose concentrations are 120 mM at Side I and 150 mM at Side II. When the hydrostatic pressure at Side II is raised to 400 mm Hg above that at Side I, the volume flow across the membrane is zero. What is the reflection coefficient of the membrane for sucrose? T = 37ºC. The sucrose solutions are then replaced with identical solutions of inulin, which cannot cross the membrane. The hydrostatic pressure difference is unchanged. The volume flow across the membrane is measured. (a) In which direction does the volume flux go? –7 2 (b) The magnitude of the volume flux is 8 q 10 cc/cm -s. What is the flow conductivity of the membrane?

5.

You wish to measure the hydraulic conductivity of a membrane and its reflection coefficient for a particular solute. To do this, you impose a concentration differ-

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ence of 10 mM across the membrane, and measure a volume flux of 1 q 10 2 cc/cm -s toward the more concentrated side; the pressure difference across the membrane is zero during this part of the experiment. The volume flux is reduced to zero by raising the pressure of the more concentrated phase by 50 mm Hg. T = 37ºC. (a) What are the hydraulic conductivity and reflection coefficient? (b) You then remove the pressure difference across the membrane, but the concentration difference remains. What is the direction and magnitude –15 of the solute flux if cs = 100 mM and Z = 6 q 10 mols/dyne-s? 6.

I have developed a new experimental technique that allows me to measure the volume flow across a membrane as a function of the transmembrane concentration difference of a single nonelectrolyte solute, while holding the hydrostatic pressure difference constant. Using this apparatus, I find 2

–5

–6

Jv (cc/cm -s) = 2 q 10 – 1 q 10 'cs (mM) when 'P = 100 mm Hg. T = 37ºC. (a) What are Lp and V? (b) What is the solute flux when there is no pressure difference across the I II –15 membrane, cs = 105 mM, cs = 95 mM, and Z = 6x10 mols/dyne-s? 7.

Write an interesting numerical problem based on the Kedem–Katchalsky equations, using a set of real phenomenological data from the biological literature. Do not use the transport data in Table 6.1.

8.

A U-shaped tube has a membrane at the base of the U that separates the two arms, both initially filled with distilled water to a height of 30 cm. The cross2 sectional area of each arm is 2 cm . Five millimoles of glucose, a solute to which the membrane is impermeable, is added to the solution in one of the arms. Water crosses the membrane to bring the system back to equilibrium. What is the height of fluid in each arm at equilibrium? In solving this problem, take into account the change in the concentration of the glucose solution as a consequence of water transport across the membrane; assume that the density of the fluid is 1 g/cc throughout. T = 20ºC.

9.

A leaky semipermeable membrane is bounded by two solutions of a single nonelectrolyte, S. On Side I, the concentration of S is 100 mM; on Side II, it is 60 mM. Side II is at atmospheric pressure. The system temperature is 37ºC. The solutions are ideal. (a) When the absolute pressure at Side I is 1.5 atm, there is no volume flow across the membrane. What is its reflection coefficient for S? (b) When the absolute pressure at Side I is 2 atm, the measured volume –8 2 –12 2 flow is 5 q 10 cc/cm -s, and the solute flux is 5 q 10 mols/cm -s.

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What are the hydraulic conductivity of the membrane and its solute permeability for S? (c) Suppose you made these measurements, and after doing so you decide to measure the osmotic coefficients of the solutions. You find that the osmotic coefficient of the 60-mM solution is 0.96, and that of the 100mM solution is 0.94. Assume you were to redo the calculations using the correct osmotic pressure; for each transport coefficient, do you expect an increase or a decrease from the earlier value, or no change? Answer this question without redoing all the calculations, and explain your reasoning. 10. Nonelectrolytes usually form more ideal solutions than electrolytes. To measure the osmotic coefficient of a 155-mM NaCl solution, you bathe one side of an ideal semipermeable membrane with the solution, and the other side with 311mM glucose; there is no pressure difference across the membrane. Water flows from the NaCl solution into the glucose solution, and you have to raise the pressure on the glucose side by 475 mm Hg to stop the flow. What is the osmotic coefficient of the NaCl solution? The osmotic coefficient of glucose at this concentration is close to unity. 11. The inner chamber (Phase I) of the osmometer in Figure 6.4 contains 135-mM KCl solution and Phase II contains a glucose solution of unknown concentration. The osmometer membrane passes neither solute. At 20ºC, fluid rises in the osmometer tube until it is 1 m above the level in Phase II. The osmotic coefficient of the KCl solution is 0.93. Assuming that the glucose solution is ideal, what is its concentration? 12. An osmometer can be used to measure the molecular weight of a solute. Suppose you dissolve 100 grams of a solute whose molecular weight is unknown in a liter of water and fill the outer container of the osmometer with the solution. Then fill the inner chamber with the 135-mM KCl solution that was used in the previous problem. Choose an osmometer membrane that passes neither solute. At 20ºC, fluid rises in the osmometer tube until it is 1 m above the level in Phase II. Assuming that the solution in Phase II is ideal, what is the molecular weight of the solute it contains. 13. A dialysis bag permeable to sodium and chloride contains a charged impermeant. When the bag is allowed to equilibrate with a solution of 100-mM NaCl at 25ºC, the potential in the bag is 15 mV negative with respect to the saline outside. What is the difference between the osmotic pressure in the bag and that outside? Assume all solutions are ideal. 14. The connective tissue of the normal cornea (the corneal stroma) contains about 15 mM of nondiffusible negative charge. What is the Donnan osmotic pressure of

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the stroma when the concentration of electrolyte in the surroundings, c = 150 mM? 1.5 M? 15 mM? 1.5 mM? Assume T = 37ºC. 15. Red blood cells are often lysed by immersing them in hypotonic medium. We know that red cells neither swell nor shrink in isotonic saline. Suppose a red cell is fitted with an internal pressure transducer that telemeters internal hydrostatic pressure, and is then dropped into distilled water. (a) What is the instantaneous pressure reading at t = 0+? (b) The pressure eventually decays. Name two possible causes of this pressure drop. 16. The human red cell in osmotic equilibrium with plasma can be modeled as a disc 8 μm in diameter and 2 μm thick. The osmolarity of plasma is 282 mOsm/l. Red blood cells are suspended in plasma, and distilled water is added to the plasma to reduce its osmotic pressure. What is the external osmolarity at which the cell membrane begins to go into tension? Assume that the inert volume of the cell is 25% of the cell volume when the cell is in equilibrium with plasma, and that the plasma osmotic coefficient is unaffected by dilution. 17. A spherical membrane vesicle of radius r0 is initially in osmotic equilibrium with an external solution whose osmotic pressure is S0. The solutions inside and outside the vesicle are made up of solutes to which the membrane is impermeable. There is no inert phase within the vesicle. The vesicle membrane can stretch, but its tension contributes negligibly to the hydrostatic pressure inside the vesicle, which remains equal to that outside. The external osmotic pressure is suddenly changed to S1. Write the differential equation that describes the variation of vesicle radius or volume with time, in terms of the initial conditions, the new osmotic pressure, and the hydraulic conductivity of the membrane. 2

18. Given the inequality Lij  LiiLjj, what inequality is implied among Lp, V and Z for a single nonelectrolyte? 19. Prove one of Eqs. (6.51) and use one of the others to design an experiment to measure the degree of coupling between the fluxes of i and j.

7 MODELS OF TRANSPORT ACROSS CELL MEMBRANES

INTRODUCTION This chapter is a transitional one. The chapters preceding this one described a variety of transport mechanisms that are present in biological systems, and the models and formalisms that are used to describe them. With rare exceptions, each of the processes was treated in isolation. In contrast, the chapters that follow integrate these individual processes to describe how they maintain the viability and support the function of cells and tissues in living systems. In most of these applications, the most important barrier to transport is the plasma membrane that bounds the cell. Nearly all of the specific transport mechanisms described earlier reside in the cell membrane. A number of models of transmembrane transport have been motivated by specific features of real cell membranes, or of tissues. Three of these will be discussed here: lipid layer models, which have been used to describe the passage of hydrophobic solutes through the lipid bilayer; pore transport models, which describe the passage of more polar species through aqueous pathways; and electrical analogs, which represent ion transport across the cell membrane by a set of equivalent electronic circuits. In the last of these, we use equivalent circuits to describe how the ion channels and pumps that are embedded in the cell membrane act in concert to determine its current–voltage relationship, anticipating the integrative models that will be described in later chapters.

7.1. TRANSPORT ACROSS THE LIPID BILAYER OF CELL MEMBRANES Sufficiently hydrophobic nonelectrolytes can traverse the cell membrane by dissolving in the lipid bilayer and crossing by free diffusion. This is not the route taken by most metabolically important molecules; sugars and amino acids, for instance, generally cross the cell membrane either by facilitated transport or secondary active transport. However, many drugs enter the cell by dissolving in the bilayer and diffusing across. M.H. Friedman, Principles and Models of Biological Transport, DOI: 10.1007/978-0-387-79240-8_7, © Springer Science+Business Media, LLC 2008

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An understanding of the mechanisms of nonelectrolyte diffusion across the lipid bilayer is also valuable for what it teaches us about the structure of the cell membrane.

7.1.1. Evidence for Nonelectrolyte Diffusion Across the Lipid Bilayer The reasons for believing that solutes can enter the cell by free diffusion across the lipid bilayer are: 1. There is a good correlation between the permeabilities of some nonpolar solutes, and their lipid solubilities or lipid/water partition coefficients. The partition coefficient, Ks, relates the concentrations of the solute in two immiscible phases, when it has equilibrated between them. Denoting the two phases by the subscripts “l” (for lipid) and “a” (for aqueous), the partition coefficient is defined as Ks 

csl csa

(7.1)

at equilibrium. 2. The permeability of cell membranes to certain classes of nonelectrolytes is much more sensitive to solute size or molecular weight (Ms) than are the diffusion coefficients of the same solutes in aqueous solution. When the relation between the aqueous diffusion coefficient, Dsa, and molecular –n weight is approximated by Dsa r Ms , n is found to be approximately 1/3– 1/2. However, the molecular weight dependence of the permeability of the bovine red cell membrane to a series of amides, and polyhydric alcohols and their esters, corresponds to n = 6 (Lieb and Stein, 1969). Values of n this large are characteristic of diffusion in polymers, where the solvent molecules are much larger than those of the solute. 3. When the transmembrane diffusion rates of nonelectrolytes having similar molecular weights are compared, the less spherical (i.e., more rodlike) molecules are generally found to cross more rapidly. This behavior is also characteristic of diffusion in polymers. In water, the smaller solvent molecules are easier to displace, and more spherical solutes have higher diffusion coefficients. 4. The permeability of cell membranes to large molecules increases more rapidly with increasing temperature than does the permeability to smaller molecules. This is explained in the following way. The considerable sensitivity of cell membrane permeability to solute molecular weight, described above, presumably reflects the much greater difficulty encountered by the larger solute molecules as they try to pass between the hydrocarbon chains that are undergoing random thermal motion in the bilayer. As the temperature of the bilayer is raised, the chains become more fluid, and this hindrance to large molecule diffusion lessens. Thus, larger molecules benefit more from increases in temperature than do smaller ones.

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If the solutes were diffusing in an aqueous solution, then the percent change in permeability caused by a specified temperature change would be nearly independent of molecular size and would parallel the temperature dependence of the reciprocal of solvent viscosity [recall the Stokes– Einstein Eq. (2.24)]. The permeabilities of cell membranes to species that diffuse across the bilayer are more sensitive to temperature than are the diffusion coefficients of these solutes in aqueous solution. 5. The addition of low-molecular-weight plasticizers to biological membranes raises their permeability. This is accomplished at plasticizer concentrations too low to have any effect in aqueous solutions. Presumably, the plasticizers cause the bilayer to become more fluid.

7.1.2. A Simple Model of Transbilayer Diffusion The diffusion of nonelectrolytes across the lipid bilayer can be described by the equations developed in Chapter 2; in particular, Eq. (2.20) gives Js = ksl'csl,

(7.2)

where ksl is the permeability of the solute in the lipid phase. The quantity 'csl is the difference between the solute concentrations in the lipid at the two surfaces of the membrane. These concentrations, in turn, are related through the partition coefficient to the concentrations of the solute in the adjacent aqueous phases. This is shown in Figure 7.1 and Eq. (7.3):

'csl  cslI  cslII  K s csaI  K s csaII = Ks'csa,

(7.3)

where Ks is assumed to be independent of solute concentration and the same at both sides of the membrane. Combining Eqs. (7.2) and (7.3),

Js = kslKs'csa.

(7.4)

The concentration of solute in the solutions adjacent to the membrane is much easier to measure than is the concentration in the lipid film. Accordingly, the measured solute permeability, ks, is usually based on 'csa, rather than 'csl. In computing the permeability, the solutions are assumed to be well stirred, so that the bulk concentrations of the solute can be used to calculate 'csa. The permeability is

ks 

Js = kslKs. 'csa

(7.5)

Equation (7.5) explains the correlation between the experimental solute permeability and the partition coefficient. Although the partition coefficients between water and the heterogeneous lipid bilayer of cell membranes are unknown, the factors that might affect them are not. Stein (1967) reasoned that, among solutes of a given size, the partition coefficient should be less for those that interact more strongly with the aqueous

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solvent, and greater for those that contain more hydrophobic groups. This prediction was supported by permeability data, which showed that, after correction for solute size, solutes forming more hydrogen bonds in aqueous solution had lower permeabilities and, among those with the same hydrogen-bonding capacity, the ones containing more nonpolar –CH2– groups were more permeable.

Figure 7.1. Chemical potential and concentration profiles for solute diffusion across a lipid film. The solutions adjacent to the film are well stirred. As illustrated here, the partition coefficient is usually much less than unity.

Lieb and Stein (1969) expressed the true solute permeability in the lipid, ksl, as

ksl  Po M sn ,

(7.6)

where Po is a membrane property and contains the membrane thickness. The second factor reflects the molecular weight dependence of the diffusion coefficient of solutes in the lipid. Combining Eqs. (7.5) and (7.6), –n

ks = P0Ms Ks.

(7.7)

Equation (7.7) predicts that, for a series of solutes crossing a lipid layer, a plot of log (ks/Ks) vs. log Ms should be a straight line with a slope equal to –n. Figure 7.2 shows such a plot, for a series of human red cell permeants selected by Lieb and Stein

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269

(1986). Since the partition coefficient of the solutes between the erythrocyte bilayer and water had not been measured, that between the nonpolar hydrocarbon hexadecane and water is used instead. The figure shows a good correlation, with n = 3.83 and a correlation coefficient of 0.80.

Figure 7.2. Logarithmic plot of the solute permeability (ks) – hexadecane/water partition coefficient (Ks) ratio vs. solute molecular weight (Ms) for the passage of six alcohols, urea, and water across the human red cell membrane. From the slope of the linear fit, n = 3.83 in Eq. (7.7). The correlation coefficient is 0.80.

7.1.3. Potential Barriers in the Bilayer Solute molecules cross the bilayer by working their way between the fluctuating hydrocarbon chains, moving into spaces between the chains as they become available. This process of “hopping” from hole to hole was modeled by Danielli (in Davson and Danielli, 1952) as a sequence of transitions between energy wells in the bilayer (Fig. 7.3). Energy barriers at the membrane/solution interfaces were added to reflect the interfacial resistance to transport of the solute. The solute flux given by this model is also proportional to 'csa, and the permeability is a function of the number of potential wells and the transition rates in the bilayer and at the interfaces. Danielli's model is reminiscent of the models of ion transport in channels that we saw in Chapter 4, in which the ions were assumed to hop from one charged site to the next as they moved through the pore. The analyses of energy barrier models of bilayer and channel transport are similar, except that, for ions, the transition rates between wells depend on the electrostatic potentials at the sites. A problem shared by all potential well models — whether of lipid layers or ion channels — is that, in their most general form, they contain a large number of adjust-

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able parameters related to well depths, barrier heights, and the associated rate constants for the transitions from each well to the next. There are two ways of dealing with this. One is to limit the number of wells, as was done in the ion channel models. The other is to allow for a larger number of wells, but with the assumption that many of the parameters are identical, so that their values are fewer and can be evaluated from transport experiments. Danielli's treatment, for instance, is based on the assumption that there are only three distinct rate constants: one for the transition from lipid to water, one for the transition from water to lipid, and one between the potential wells in the lipid phase.

Figure 7.3. Energy profile for a relatively polar molecule in a lipid film, according to Danielli (Davson and Danielli, 1952). The barrier 'Pa is encountered upon entering the membrane, 'Pb is encountered on leaving the membrane, and 'Pe is the barrier to hopping inside the membrane. For strongly hydrophobic molecules, 'Pb > 'Pa, and the profile looks like a corrugated well rather than a corrugated mesa.

7.2. MODELS OF TRANSPORT THROUGH PORES Toward the end of the previous chapter, we distinguished between two approaches to the description of passive transport: the nonequilibrium thermodynamic approach, which is appropriate when the solute flux is coupled to solvent flow, and which requires that the system be close to equilibrium; and the earlier analyses of free and facilitated transport, in which solutions were presented for the solute flux in systems

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that are farther from equilibrium, but in the absence of solvent drag. The models of pore or channel transport that have been proposed can be classified similarly.

7.2.1. Classification of Pore Transport Models One class of models describes transport through ion-specific channels in the cell membrane. The approach taken can be microscopic, as in the energy barrier models treated in Chapter 4, or phenomenological. An example of the latter is Hodgkin and Huxley's (1952) pioneering work on nerve, which will be discussed in Chapter 9. When the phenomenological approach is taken, the open channel is assumed to have a particular conductance (or conductances, if the channel has multiple open states), which is found experimentally, and the rate at which ionic current passes through the channel depends on the fraction of time that its gate is open. The mechanisms that underlie the unitary conductance are not of primary concern. Since any macroscopic membrane patch contains a large number of channels, the current density can also be expressed in terms of the fraction of gates that are open at any instant. In both of these approaches — the microscopic and the phenomenological — the sole force driving an ion through the channel is the gradient of its electrochemical potential. The second class of model, which will be discussed in this section, describes a less selective pore, through which nonelectrolytes and water also pass. These models attempt to interpret the phenomenological transport coefficients of nonequilibrium thermodynamics in terms of the mechanics and kinetics of flow and diffusion through a pore. A particular application of these models is the estimation of the effective pore radius from transport data. These estimates are not quantitative; still, if the calculated radius is unrealistically high or unrealistically low, this is certainly an indication that other transport pathways are likely to be present. Similarly, a reasonable radius is evidence for (but by no means proof of) transport through pores in the membrane. Pore sizes computed from such models have also been used to decide whether solutes or water cross cell layers by passing through the cells, or between them. A calculated pore size much larger than the expected size of cell membrane pores is taken as evidence for significant transport through the wider passages between the cells. Broadly speaking, one can define three regimes of pore transport, depending on the ratio of the solute radius, s, to the pore radius, r. Examples of all three will be presented in the subsections to follow. These models of pore transport will assume that spherical, uncharged solute molecules (including tracer solvent molecules) are passing through a cylindrical pore. A reference will be given subsequently to extensions of the theory that treat nonspherical and charged particles, and pores of different shape. The three regimes are as follows: 1. If s << r, flow through the pore is identical to pipe flow, and the conventional formulations of free diffusion and fluid mechanics apply. This is illustrated in the subsection that follows. 2. As s/r increases to about 0.05–0.1, transport of solute through the pore begins to be influenced by the pore walls, which obstruct entry into the pore and give rise to additional viscous drag because solvent must flow

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between the solute particle and the wall as the solute passes through. This regime is known as hindered diffusion, and is used to model pore transport for s/r values up to ca. 0.4. Models of hindered diffusion are based on continuum concepts, and their applicability can be questioned when the pore is only several solute or solvent molecules wide, but they have been used successfully under these circumstances. Correction terms reflecting the influence of the wall are expressed as polynomials in s/r. 3. When s/r > 0.5, solute particles can no longer pass one another, and transport is single file. Single-file transport is treated using approaches similar to those used earlier for ion channels, which are similarly restrictive. Many of the features of specific ion channels, such as selectivity filters and electrostatic forces, are generally absent from single-file transport models, though the notion of binding sites is retained to enumerate the number of solute molecules that a fully occupied pore might contain.

7.2.2. Hydraulic Conductivity of a Pore The easiest phenomenological coefficient to interpret in terms of pore transport is the hydraulic conductivity. The pore is simply regarded as a straight cylindrical tube of radius r and length equal to the membrane thickness, a; the pore axis is perpendicular to the membrane face. The pressure drop, 'P, required to drive a flow, q, through such a pore is given by the Hagen–Poiseuille equation: 'P 

8aKq , Sr 4

(7.8)

where K is the solvent viscosity. The volume flux is equal to the flow rate divided by the cross-sectional area of the pore: Jv 

q . Sr 2

(7.9)

The hydraulic conductivity is defined by Eq. (6.31a): Lp 

Jv . 'P

(6.31a)

Substituting Eqs. (7.8) and (7.9) into (6.31a), Lp 

r2 . 8 aK

(7.10)

The hydraulic conductivity of the pore is directly proportional to the square of the pore radius and inversely proportional to the membrane thickness and the viscosity of the solvent. A similar expression can also be derived for a membrane that contains a specified number of pores, of specified radius, per unit area; the derivation of this expression is left as an exercise.

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If the axis of the pore is not perpendicular to the membrane faces, its length, l, replaces the membrane thickness, a, in the denominator of Eq. (7.10). The ratio l/a p 1 is known as the tortuosity of the pore, W.

Figure 7.4. (a) Nonelectrolyte diffusion through a cylindrical pore. The solute toward the “bottom” of the pore (in the side view) cannot enter because the radial distance from its center to the pore wall is less than s. The sterically available area, Ast, is outlined by the dashed circle in the front view of the pore. (b) Solvent film with pore-sized patch. The area of the patch is A0 = Sr2.

7.2.3. Hindered Diffusion; Solute Permeability as a Probe of Pore Radius The radius of membrane pores can be estimated from the dependence of nonelectrolyte permeability on solute size. Ions are not as good for this purpose, because their effective radius depends strongly on whether they are hydrated when they are in the pore (recall Chap. 4). In addition, when ions are used, the analysis is complicated by the possibility of electrical effects arising from the membrane potential or charged sites in the pore. Nonelectrolyte diffusion through a pore is most simply modeled by considering a spherical solute, of radius s, diffusing through a straight cylindrical channel, as shown in Figure 7.4a. Our objective is to compare the rate of solute transport through the pore to that across a film of solvent whose thickness is the same as the length of the pore. Transport through the film proceeds by free diffusion and is given by Eq. (2.18):

J s0  ks0 'cs , where the superscript denotes the solvent film.

(2.18)

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The flux through the pore is less than that through the film, for two reasons. First, the walls of the pore at the membrane face impede the entry of the solute into the pore; this is termed the steric effect. Second, once the solute has entered the pore, it encounters a viscous resistance that, because of the nearby walls of the channel, exceeds the resistance in free diffusion. We will consider each of these effects in turn. The Steric Effect. As can be seen from Figure 7.4a, the center of any solute molecule that enters the pore must cross the x = 0 plane at least a distance s from the wall of the channel. Thus, the area of the pore accessible to solute is 2

 r  s ¬­ Ast  S(r  s)  Ao žž , žŸ r ®­­ 2

(7.11)

2

where Ao is the cross-sectional area of the pore, Sr . The Viscous Effect. The viscous effect causes a further reduction in the flux through the pore. Consider a small circular patch of solvent film whose area is the same as that of 0 the pore (Fig. 7.4b). Let n be the number of mols of solute crossing the film through the patch, per unit time:

n0  J s0 Ao  ks0 Ao 'cs .

(7.12)

The number of moles, n, crossing through the pore per unit time, can be described by an equation similar to Eq. (7.12):

n = ksAst'cs.

(7.13)

Equation (7.13) defines the pore permeability, ks; ks measures the ease of transport through the pore, once the solute has gotten inside. Since the patch and the pore present the same area to the solute, any differences 0 between n and n are of either steric or viscous origin. The ratio of the two transport rates is k A n  0s ¸ st . 0 n ks Ao

(7.14)

The second factor on the right-hand side of this equation is the steric contribution to 0 the difference between n and n , given by Eq. (7.11). It is the ratio of the rate at which molecules enter the pore to the rate at which they enter the patch. The viscous contribution is contained in the first factor, which is the ratio of the ease with which the particle diffuses through the pore once it has entered, to the ease with which it crosses the (wall-less) solvent film. An equation for this ratio was derived by Faxen (1922, cited in Renkin, 1954): 3

5

s¬ s¬ s¬ ks s  1  2.104 2.09 žž ­­­  0.95žž ­­­ w 1  P žž ­­­ . 0 žŸ r ® Ÿž r ® Ÿž r ® ks r

(7.15)

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In deriving this equation, it was assumed for simplicity that the spherical solute molecules are centered along the pore axis and that the solution in the pore is sufficiently dilute that each such molecule crosses the membrane independently. Equation (7.15) is an approximation to the numerical solution of the governing flow equations that applies when s < 0.4r. The viscous effect given by Eq. (7.15) is plotted against s/r in Figure 7.5. The figure shows how ks decreases relative to the permeability of the solvent film as the radius of the solute approaches that of the pore.

Figure 7.5. (a) Faxen's function [Eq. (7.15)] for the viscous effect on solute diffusion through a pore. (b) Combined steric and viscous effects [Eq. (7.16)] on solute diffusion. (c) Anderson and Quinn's (1974) function [see Eq. (7.23)] for the viscous effect on solute convection through a pore. Note that the steric and viscous terms, here and in Figure 7.8, approach unity as the pore diameter becomes much larger than the solute diameter (s/r o 0).

Substituting Eqs. (7.11) and (7.15) into (7.14), 2  s ¬­¯  s ¬­ n   ž ž ¡ °  1  P ž ­­ ž1  ­­ . žŸ r ®° Ÿž r ® n 0 ¢¡ ±

(7.16)

This ratio is also plotted against s/r in Figure 7.5. Equation (7.16) can be used to estimate pore size from the dependence on solute radius of the permeabilities of a series of nonelectrolytes. When Eq. (7.16) is used in this way, it is assumed that these solutes all use the same pores, and that the transport channels are all of similar size. The applicability of Eq. (7.16) is also limited by the assumptions and approximations used to obtain Eq. (7.15).

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Equations (7.12) and (7.16) can be used to predict the rate of diffusion of a solute of radius s through a pore of radius r, in terms of the free-diffusion permeability: 2

   s ¬¯  s ¬ n  ks0 Ao ¡1  P žž ­­­° žž1  ­­­ 'cs . žŸ r ®° Ÿž r ® ¢¡ ±

(7.17)

Note that, throughout this derivation, ks0 is the permeability of a solvent film whose thickness is the same as the length of the pore. If the tortuosity of the pore, W, is greater than one, ks0 will be proportionately less than that of a film whose thickness is the same as that of the membrane: if the membrane thickness is a, the pore length is l, and the free diffusion coefficient of the solute is Ds , ks0 

Ds ž Ds ¬­ 1  ž ­­ W žŸ a ® l

(7.18)

The flux of solute through the pore, Js, is equal to n/Ao ; using Eq. (2.18), 2 2      s ¬­¯  s ¬­  s ¬­¯  s ¬­ 0 ž ž ž ž ¡ ° ¡ ° J s  k 1  P ž ­­ ž1  ­­ 'cs  J s 1  P ž ­­ ž1  ­­ . žŸ r ®° žŸ r ® ¡¢ ¡¢ Ÿž r ®°± Ÿž r ® ± 0 s

(7.19)

2

Thus, the polynomial in s/r, [1 – P(s/r)](1 – s/r) , measures the ratio of the flux through the pore to that through a film of thickness, l; call this quantity )d . The ratio of the flux through the pore to that through a film of thickness, a, is )d /W. Equation (7.19) shows that the apparent permeability of a pore, ksapp , is given by ksapp w

Js  ks0 ) d (s / r ) . 'cs

(7.20)

Equation (7.20) was used in an interesting way by Wang and Veenstra (1997), who employed it to estimate the pore radius of the rat gap junction protein connexin43 from measurements of the conductance of the channel to a series of six cations. They used a slightly different expression for )d (s/r) and assumed that ksapp was proportional to )d , neglecting the variation among the free diffusion coefficients of the solutes (the effect of this omission is examined in an exercise). The hydrated radii of the six ions were used and the apparent permeability of each ion normalized by that of lithium. The value of r that best fit their data was 0.63 nm; this was interpreted as the radius of the “modest” selectivity filter of the channel, and was consistent with estimates based on the size of several sugars that were too large to pass through. The fit to their data is shown in Figure 7.6.

7.2.4. Other Factors Affecting Estimated Pore Size; The Equivalent Pore The pore size of larger channels, such as those between cells, can also be estimated from solute permeabilities, using solutes much larger than those used to size pores in cell membranes. Deen and coworkers (Deen et al., 1980, 1983) estimated the intercellular pore size in the glomerular capillary wall (in the kidney; see Chap. 10) from the

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277

fluxes of a series of charged and neutral macromolecules. Their model was based on the convective diffusion equation (2.29), modified to include the hindrance to diffusion and convection arising from the wall of the pore. Steric effects were included through a partition coefficient at the ends of the pore. This coefficient, like the hindrance factors, depended on s/r, but not on solute charge. When the solute was charged, a Donnan equilibrium condition was applied at the ends of the pore; the contribution of the transmural electrostatic potential gradient to the driving force for solute flux was expected to be negligible and was omitted.

Figure 7.6. Correlation of the normalized ionic permeability of rat connexin Cx43 vs. hydrated ion radius. The permeabilities are normalized by that of lithium, indicated by the open circle. The curve is the predicted dependence for a pore radius r = 0.63 nm. Reprinted with permission from Wang and Veenstra (1997). Copyright © 1997, The Rockefeller University Press.

Their results showed that pore theory can explain the effect of molecular size on glomerular permeability. The derived pore radius was found to depend on the shape of the polymers that were used. In rat, the pore radius that best described the size dependence of the fluxes of a series of dextrans was 4.7 nm, while that for a series of horseradish peroxidases was 3.7 nm. This difference reflects deviations between the assumptions of the pore transport model (e.g., simple, uniform geometry), and reality. Because of these deviations, a pore radius obtained by applying such models to transport data is often referred to as an equivalent pore radius; it is the radius of a cylindrical pore whose transport properties would match the experiments if the other assumptions of the model held. The equivalent radius that explains the permeability of one class of molecule cannot necessarily be used to predict the permeability of members of another class. When the solute bears a charge, its flux depends on the membrane potential, the charge density in the pore, ionic strength, and the variables that govern the flux of uncharged molecules. Figure 7.7, taken from Deen et al. (1983), is a prediction by

278

CH. 7: MODELS OF TRANSPORT ACROSS CELL MEMBRANES

their model of the dependence of the filtrate/plasma concentration ratio in the glomerulus (proportional to solute flux through the glomerular capillary wall) on solute radius, solute charge, and the concentration of fixed charge in the pore. The figure shows that the solute flux is strongly dependent on these variables when r and s are comparable. This dependence is separate from the effect of molecular shape, which can add another decade of variability to the solute flux (Deen et al., 1983).

Figure 7.7. Glomerular filtrate-to-plasma concentration ratio (T) as a function of solute charge (Z) and radius (s). Results are shown for two values of membrane fixed charge concentration (Cm), and a pore radius r = 5 nm. The ratio T is proportional to solute flux. In this model, solute charge affects flux only through Donnan equilibria at the ends of the pore; when Cm = 0, Donnan effects are absent, and T is independent of Z. 1 Å = 0.1 nm. Reprinted with permission from Deen et al. (1983). Copyright © 1983, Springer-Verlag.

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279

7.2.5. Hindered Convection (Sieving); the Reflection Coefficient as a Probe of Pore Radius It is possible to obtain an equivalent pore radius from the reflection coefficient of a single solute. Consider a sieving experiment, in which a solution is driven through a cylindrical pore by hydrostatic pressure, and the solute concentration is the same on both sides of the membrane. Except for a small entrance region near its mouth, the pressure-driven solvent velocity profile in the pore is parabolic:

 U2 ¬ v  vmax žžž1  2 ­­­ , Ÿ r ®­

(7.21)

where v(U) is fluid velocity in the axial direction and U is radial distance from the axis of the pore. The maximum velocity is on the centerline, at U = 0. Because of the velocity profile, more of the solute convected through the pore does so close to the pore axis, as compared to the diffusional case discussed earlier. For this experiment, the steric ratio is (Ferry, 1936)

Ast r s  ¨ 2SUvdU 0 Ao

r

2

4

 s¬  s¬ ¨ 2 SUvdU  2 žž1  ­­­  žž1  ­­­ . žŸ r ® Ÿž r ® o

(7.22)

The steric ratios given by Eqs. (7.11) and (7.22) are plotted against s/r in Figure 7.8. Owing to the velocity profile, as described above, the ratio is closer to unity — that is, the steric effect is less — for a sieving experiment than for a diffusion experiment.

Figure 7.8. Steric ratio as a function of solute radius/pore radius for (a) diffusion [Eq. (7.11)] and (b) convection (sieving) [Eq. (7.22)].

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CH. 7: MODELS OF TRANSPORT ACROSS CELL MEMBRANES

When transport through the pore is by convection rather than diffusion, a different polynomial (Anderson and Quinn, 1974) represents the viscous effect, subject to the same restrictions as Eq. (7.15); for the ith species: ni ¡  ž si ¬­ ž si ¬­  2 ž1  ­  ž1  ­ r ®­ Ÿž r ®­ ni0 ¡¡ žŸ ¢ 2

2

4

¯  ¯ ° ¡1  P 'ž si ¬­­° w Q* , žž ­° i °¡ Ÿ r ®± ±° ¢

(7.23)

3

where Pc(si /r) = 0.667(si /r) + 0.163(si /r) . This expression for the viscous effect in convection is also plotted against s/r in Figure 7.5. We see that viscous retardation is considerably less in convection than in diffusion. The difficulties encountered by the solute in entering and passing through the pore are shared by the solvent molecules as well. The solute is sieved by the pore if it is retarded more than the solvent. When the pore is very large, convection through it is the same as that through a solvent film, and there is no sieving: ns0  csVw nw0 ,

(7.24)

where cs is the uniform solute concentration. The sieving effect of the pore can be related to the solute reflection coefficient, using the Kedem–Katchalsky equation for solute flux [Eq. (6.36)] and approximating the volume flux through the pore by J wVw . In the absence of a concentration gradient ('S = 0, cs = cs ), Eq. (6.36) becomes J s  cs J wVw (1  V) .

(7.25)

Dividing the flux terms on each side of the equation by the area of the pore, ns  cs nwVw (1  V) ,

(7.26)

where ni = Ji /Ao is the rate at which the ith species crosses the pore, in mols/pore-s. Equations (7.23), (7.24), and (7.26) can be combined to express the reflection coefficient as a function of the radii of the solute, solvent, and pore: V  1

vs* . vw*

(7.27)

Equation (7.27), with Eq. (7.23), can be used to estimate the pore size from a single reflection coefficient, provided that the solute and solvent pass through the same pores. The reflection coefficient need not be measured by a sieving experiment. Goldstein and Solomon (1960) carried out osmotic equilibrium experiments to measure the reflection coefficients of the human red blood cell membrane for nine solutes whose molecular size could be estimated. They used P(si /r) given by Eq. (7.15) rather than Pc(si /r) in the definition of vi* . The equivalent pore radius that best fit their data was 0.42 nm, not an unreasonable value. The fit to their data is shown in Figure 7.9. The implications of their choice of polynomial is examined in an exercise. Equations (7.23) and (7.27) can also be used to estimate the reflection coefficient of a solute in a pore from the solute and pore radii.

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Figure 7.9. Reflection coefficients of the nine solutes used by Goldstein and Solomon (1960) vs. solute radius. The curve is a plot of vs* / vw* (see text) vs. solute radius for pore radius r = 0.42 nm and solvent radius sw = 0.15 nm. Reprinted with permission from Goldstein and Solomon (1960). Copyright © 1960, The Rockefeller University Press.

If the flow of solvent through the pore is not appreciably retarded, vw* | 1 and V = 1 – vs* . The concentration of the solution exiting the pore, cf, is equal to the mols of solute crossing, ns, divided by the accompanying volume of solvent, Vw nw ; using Eq. (7.26) and the relation between V and vs* ,

c f  cs vs* .

(7.28)

Thus, under these conditions, vs* is the ratio of the concentration of the filtrate exiting the pore to the upstream concentration of the solute. This ratio is referred to as the sieving coefficient of the solute in the membrane. Equation (7.26) is based on the assumption that the concentration in the pore is uniform and the same as that at both faces of the membrane. When this assumption holds, Eq. (7.26) describes the relative amounts of solute and solvent passing through the pore. If, however, the downstream compartment is comprised of the less concentrated filtrate, cs is no longer uniform in the pore, and these assumptions fail. The sieving coefficient can still be defined as cf /cs, as it often is, but it is no longer equal to vs* . Another treatment of sieving that is based on continuum concepts rather than nonequilibrium thermodynamics, and that includes transmembrane concentration differences, is presented in the following subsection and illuminates this problem further.

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CH. 7: MODELS OF TRANSPORT ACROSS CELL MEMBRANES

A variety of models of solute diffusion and convection through pores that are small enough to affect both access and transport, yet large enough that continuum modeling can still be applied, can be found in a review by Deen (1987). This review also includes alternative expressions for the viscous effect in cylindrical pores and similar equations for slit-shaped pores (which are used to model gaps between capillary endothelial cells), a model of simultaneous diffusion and convection like that described below, and modifications to the theory that apply to charged and nonspherical solutes and more concentrated solutions in which solute–solute interactions cannot be neglected.

7.2.6. Combined Diffusion and Convection through Pores We saw in Chapter 2 that, when free diffusion takes place in the presence of solvent motion, the solute flux equation contains two terms, the first for diffusion and the second for convection: J s  Ds

dcs vcs , dx

(2.29)

where v is the local fluid velocity. This equation can readily be modified to describe convective diffusion through cylindrical pores in the cell membrane or slit-like gaps between cells. We saw in the previous subsections that both components of the solute flux are reduced relative to transport in free solution as a consequence of steric and viscous effects resulting from the pore geometry. Let the ratios of each flux component in the pore to that in free solution be designated ), appropriately subscripted (d = diffusion, c = convection). Furthermore, replace v by vm, the average fluid velocity in 2 the pore. With these substitutions, Js becomes the flux through the pore, in mols/cm pore area-s. Equation (2.29) becomes J s  Ds ) d

dcs vm ) c cs . dx

(7.29) I

In the steady state, Js is the same throughout the length of the pore. Letting cs denote II the concentration of solute at the x = 0 face of the pore, and cs denote the concentration at the x = a face, integration of Eq. (7.29) yields csII exp(Pe p ) csI I J s  vm ) c cs ¸ , 1  exp(Pe p ) 1

(7.30)

 ) ¬ v a ¬ where Pe p  žžž c ­­­žžž m ­­­ is the pore Peclet number, which measures the relative con­ž Ds ®­ žŸ ) d ®Ÿ tributions of convection and diffusion to the solute flux through the pore. The second term in the expression for Pep is the Peclet number, Pe, for convective diffusion in free solution, and the first term modifies this quantity to reflect solute and solvent interac-

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283

tions with the pore. The characteristic length a is generally taken to be the distance over which diffusion takes place, usually the length (as it is here) or radius of a channel. In free solution, convection dominates when Pe >> 1, and diffusion dominates when Pe << 1. In terms of the variables used here, the filtrate concentration is given by cf = Js /vm. Equation (7.30) shows how this quantity depends on the concentration boundary conditions, the hindrance factors )c and )d, and the free-solution Peclet number. Note that the fraction in Eq. (7.30) becomes unity when either (1) the pore Peclet number gets very large or (2) the concentration difference across the membrane is zero. In I each case, cf = )ccs . In the former case, the large Peclet number signifies that the solute flux is dominated by convection, and the solute is readily carried across the membrane with the solvent, hindered by the pore so that its flux lags that of the solvent ()c < 1), but at a rate so high that it is unaffected by the concentration of the downstream solution. In the latter case, the assumptions of Eq. (7.26) apply, and Eq. (7.30) simplifies to cf = )ccs, equivalent to Eq. (7.28).

7.2.7. Single-File Transport through Pores Some of the assumptions on which the preceding models of pore transport are based become increasingly questionable as the pore radius becomes small. For instance, the solvent in the channel certainly cannot be regarded as a continuum when the width of the pore is only a few times the diameter of a water molecule. Under such conditions, the Hagen–Poiseuille equation does not apply, and the concept of a solvent velocity profile is meaningless. When solute molecules pass through the pore in single file, the factors that determine the permeation rate differ from those that govern diffusion in a larger pore, and solute–solute interactions necessarily play a dominant role when the solution is not highly dilute and the solute molecules cannot pass each other. The presence of these narrow pores in a membrane can be inferred from measurements of its osmotic and diffusive water permeabilities. The osmotic permeability (Pf ) is found from the solvent flux induced by an osmotic pressure difference. The diffusive water permeability (PDw ) is obtained from the diffusive flux of radiolabeled solvent (e.g., tritiated water, HTO) in the absence of bulk flow. Of particular interest is the permeability ratio, Pf /PDw, since the ratio is determined in large part by the number of water molecules, n, that line up within the pore. Several models predicting the permeability ratio in single-file transport have been proposed; before examining them, we use the Kedem–Katchalsky equations to make certain that Pf and PDw are expressed in consistent units. The units we select are those of J/'S. Osmotic Permeability. Generally an impermeant is used to generate the driving osmotic pressure difference, so JI = 0 (where I denotes the impermeant) and V = 1. The system is maintained at a uniform hydrostatic pressure. Then, Eq. (6.32a) becomes:

J v  J wVw  L p 'S , and

284

CH. 7: MODELS OF TRANSPORT ACROSS CELL MEMBRANES

Pf w 

Lp Jw  . 'S Vw

(7.31)

Diffusive Permeability. A tracer amount of tritiated water (denoted by the subscript “w*”) is the solute in a diffusion experiment at zero volume flow. The tritiated water is added to only Phase I. Equation (6.36) gives

Jw* = Z'S , where 'S*  RTcwI * . The diffusive permeability is

PDw w

J w* Z. 'S*

(7.32)

Thus,

Pf PDw



Lp ZVw

(7.33)

We will use the two forms of the permeability ratio in Eq. (7.33) interchangeably in this subsection and the one to follow.

Figure 7.10. Pore model of single-file transport. (a) Osmotic experiment. Small open circles are water molecules, n of which fill the pore. The water molecules are numbered according to the site they occupy. Large shaded circles are impermeant molecules, whose concentration is higher in Phase II. (b) Tracer experiment. Shaded circles are tritiated water, added to Phase I.

Transport Models. The pore model used by Lea (1963) is shown in Figure 7.10. Water molecules pass through the pore single file; n molecules fill the pore, and the pore is always full. It is convenient to think of the pore as containing n sites in series, each of which may or may not be occupied by a solute molecule. Here, the physical meaning of a “site” is less specific than it is in an ion channel; it is simply one of a number of locations along the pore at which a solute molecule may be found. In the osmotic experiment (Fig. 7.10a), the membrane faces are bathed by solutions of im-

PRINCIPLES AND MODELS OF BIOLOGICAL TRANSPORT

285

permeant, and the concentration of impermeant is higher in Phase II. Under these conditions, the driving force for water flux can be expressed in terms of the solvent concentration difference, with cwI  cwII . The rate at which water molecules from Phase I occupy Site 1 in the pore (thereby knocking a solvent molecule from Site n into Phase II) is proportional to cwI . Similarly, the rate at which solvent molecules from Phase II occupy Site n is proportional to cwII . The net osmotic flux from Phase I to Phase II is the difference between these rates, and the latter rate is less than the former. Thus, flow is toward the phase whose osmotic pressure is higher. Note that interphase transport accompanies each entry of a solvent molecule into the pore, because a second molecule must simultaneously leave the far end of the pore and enter the adjacent bathing solution. The diffusive transport of tracer (Fig. 7.10b) proceeds differently, because a tracer molecule must traverse the entire pore to contribute to Jw*. (the tracer concentration is low enough that the simultaneous occupancy of the pore by more than one tracer molecule is extremely unlikely). Indeed, a tracer molecule entering from Phase I can be knocked back into Phase I and never cross the membrane. Lea described the movement of tracer through the pore as a one-dimensional random walk. He found that the fraction of tracer molecules initially at Site 1 that reach Phase II is 1/(n + 1); that is, for every n + 1 tracer molecules that enter the pore, only one contributes to Jw*. In an osmotic experiment, every entry of a solvent molecule into the pore contributes to Jw; in this context, osmosis is n + 1 times more “efficient” than tracer diffusion. In terms of the measured permeabilities,

Pf PDw

 n 1.

(7.34)

For a series of mammalian erythrocytes, the values of n calculated from the permeability ratio range from 1 to 6 (Dick, 1966). Permeability ratios of 3 to 9 have been reported for the proximal tubule of the kidney (Weiss, 1996a). Levitt's (1974) model is more intuitive than Lea's, and is based on an extension of the Einstein relation for diffusion in free solution to transport through the pores in a membrane. Levitt's result is expressed in terms of phenomenological coefficients and is slightly different from Lea's:

Lp

ZVw

n.

(7.35)

Finkelstein (1987), in a fine monograph for the Society of General Physiologists, considered and critiqued the then current models of single-file transport, including a more critical analysis of Levitt's work that arrives at the same result. Single-file transport can also be analyzed in terms of the state diagrams introduced in Chapter 4. In §4.2.2, we illustrated this concept for a sodium channel that contained three binding sites in series, only one of which could be occupied at any instant. If we remove the occupancy limitation and replace it by the restriction that a solute molecule cannot move to an adjacent site that is occupied, we have defined single-file transport; the corresponding state diagram is shown in Figure 7.11a.

286

CH. 7: MODELS OF TRANSPORT ACROSS CELL MEMBRANES

Figure 7.11. (a) State diagram for single-file transport through a three-site pore. Sites are either Occupied or Empty. The heavily marked transitions define the one-vacancy model. (b) Passage of a tracer molecule (filled circle) through a three-site pore. The transitions are numbered sequentially. The tracer occupies Site #1 from the cis (left) side in Step 1 and leaves Site #3 for the trans (right) side in Step 10.

In contrast to the earlier models of single-file transport, in only one of the eight possible states is the pore filled with water; in most of the states, only a fraction of the sites are occupied. The cycle of four states connected by heavier lines in Figure 7.11a describes the passage of solute molecules through the pore when no more than one of the sites is unoccupied at a given time: the so-called one-vacancy model (Kohler and Heckmann, 1979; Hernandez and Fischbarg, 1992). For a particular solute molecule, such as a tracer water molecule, to cross the membrane, the cycle is repeated twice, as shown in Figure 7.11b. In each cycle, the tracer molecule, denoted by a filled circle, moves one more step toward the trans side of the membrane. After the tracer is released at the trans side, a third cycle is needed to bring the vacancy back to the cis side of the membrane. Thus, for a membrane containing n sites in series, n such cycles are needed: n – 1 cycles to bring the tracer from Site 1 to Site n, and one more cycle to return the vacancy to the cis side so that the next tracer molecule can enter the pore.

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287

Each cycle in Figure 7.11b is a sequence of n + 1 states; thus, the transport of a single tracer molecule across the membrane takes n(n + 1) steps. In contrast, the osmotic flow of a water molecule through a pore that contains n sites takes only n + 1 steps (an entry step, n – 1 internal transitions, and an exit step), since the water molecules move through the pore in a coordinated fashion, each molecule replacing the one in front of it as it passes through the membrane. Therefore, this model gives a permeability ratio of [n(n + 1)]/[n + 1] = n, like the earlier models. This approach was expanded upon by Hernandez and Fischbarg (1992), who considered cases in which vacancies are transient and in which the system is far from equilibrium. An advantage of state models of this kind is the ease with which they may be translated into rate equations, as was done in Chapter 4. From the point of view of mechanism, a permeability ratio significantly greater than unity is a good indicator of single-file transport; the exact number of molecules that fill the pore — whether the ratio equals n or n + 1 — is of less consequence. Following Finkelstein (1987), the precise relation between the permeability ratio and the number of sites in the pore depends on (1) the ratio of (a) the rate constant for water exchange, at the two membrane faces, between the ambient solution and the first site inside the pore, to (b) the rate constant for the site-to-site steps within the pore, and (2) the maximum number of sites that are stipulated to be vacant during the transport process. Since transport is by definition single file, pore radius is not a parameter of any of these models.

7.2.8. The Permeability Ratio of Larger Pores The possibility that water crosses a membrane through larger pores can also be tested by measuring the permeability ratio. Starting with the frictional representation of transport described in Chapter 6, Thau et al. (1966) expressed Lp and Z in terms of frictional coefficients, assuming only that the thermodynamic and kinetic properties of tritiated and unlabeled water are the same, and that the tritiated water (HTO) is present in tracer amounts. Their result simplifies to

Lp 

Z

( fTH

Vw , fwm a

(7.36a)

1 , fwm )a

(7.36b)

where fTH measures the frictional interaction between HTO and H2O. Constructing the permeability ratio,

Lp

ZVw

 1

fTH . fwm

(7.37)

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CH. 7: MODELS OF TRANSPORT ACROSS CELL MEMBRANES

The permeability ratio is seen to depend on the relative importance of interactions among water molecules (proportional to fTH), and interactions between water and the membrane (proportional to fwm). For large pores, fTH | RT/Dw, where Dw is the self-diffusion coefficient of water (i.e., the aqueous diffusion constant of a tracer molecule of water). The coefficient fwm can be related to pore size through Eqs. (7.36a) and (7.10). Equation (7.37) then becomes

Lp

ZVw

1

RT r 2 ¸ . Dw 8Vw K

(7.38)

This equation can be used to obtain the equivalent radius of a pore without knowing its length. From Eqs. (7.37) and (7.38), it can be seen that Pf /PDw is large when transport takes place through wide pores, where interactions among solvent molecules dominate the influence of the wall (fTH >> fwm), and the second term on the right-hand side of Eq. (7.38) is much greater than unity. In the presence of a pressure driving force, transport is predominantly by convection. In smaller pores, the effect of the wall (fwm) is greater, and the permeability ratio is closer to unity. When pores are absent, or so narrow that solvent molecules cannot pass through, water crosses the membrane only by dissolving and diffusing in it, and Pf and PDw should become identical. Equation (7.38) can be simplified in an approximate way by using the Stokes– Einstein relation [Eq. (2.24)] to express Dw in terms of other variables. For selfdiffusion, the denominator in the relation is four rather than six (Bird et al., 2002). Replacing L p / ZVw by the permeability ratio, Eq. (7.38) becomes

Pf PDw

1

Ssw N 2 r , 2Vw

(7.39)

where sw is the radius of a water molecule and N is Avogadro's number. The quantity Vw is the volume of a mole of water, approximately (4/3)Ssw3 N , whence Pf PDw

2

 3 ­¬ž r ¬­  1 žž ­­ž ž ­ . Ÿž 8 ®Ÿž sw ®­­

(7.40)

Recognizing the approximations in its derivation, Eq. (7.40) shows that, notwithstanding the additional variables in Eq. (7.38), the permeability ratio is primarily determined by the ratio of pore size to solute size. The equation predicts that the interactions between solvent molecules become nearly ten times the wall effect when the pore radius is five times that of the solvent. The more correct Eq. (7.38) predicts a radius ratio between five and seven. Finally, note that the models of both single-file transport and transport through larger pores predict a permeability ratio greater than unity. Thus, the observation that Pf /PDw > 1 indicates the presence of pores, but it cannot by itself identify the mechanism by which solute passes through them.

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289

7.3. ELECTRICAL ANALOGS Many of the material properties and transport variables associated with ion flow across biological barriers have analogs in the properties and variables that describe electrical circuits. Some examples are listed in Table 7.1. In view of these similarities, it is not surprising that the methods used to describe current flow and voltage effects in electrical circuits have also been used to model and analyze ion transport and its interaction with membrane potential in biological systems.

Table 7.1. Parallels between Ion Transport and Electrical Circuits Ion transport

Electrical circuit

Ion flux Membrane potential Ion mobility Rheogenica pump Membrane capacitance

Current density Voltage difference Conductance Current source Capacitance of a capacitor

a

A rheogenic pump transfers net charge across the membrane. Such pumps have often been referred to as electrogenic, since their activity contributes to the membrane potential. However, an electrically neutral (i.e., nonrheogenic) pump can also affect the membrane potential through the concentration differences it produces. The term “rheogenic” is more precise.

The principal differences between biological ion flows and electrical circuits are: (1) biological and electrical currents are both driven by electrical potential differences, but biological currents are also driven by differences in concentration; (2) the current in biological systems is generally carried by more than one species; and (3) the biological system is rarely as linear as its electrical analog. We will deal with the first two of these differences in this section, reserving the third for Chapter 10.

7.3.1. Equivalent Circuit for the Passive Flux of a Single Ion The construction of an equivalent circuit for a single ion crossing a membrane under concentration and electrostatic potential gradients begins with the electrodiffusion equation (2.36):  d ln ai d \ ¬­  ziᑠ J i  U i ci žžRT ­. žŸ dx dx ®­

(2.36)

For this application, it is unnecessary to approximate the activity of the ion by its concentration, as was done in Chapter 2. 2 Let Ii be the current carried by the ith ion, in mols of charge/cm -s; Ii = ziJi (to 2 convert Ii to amps/cm , the usual measure of electrical current density, multiply it by the Faraday, which has the units of coul/mol). From Eq. (2.36),

290

CH. 7: MODELS OF TRANSPORT ACROSS CELL MEMBRANES

 RT d ln ai d \ ¬­ ­. I i  zi2ᑠUi ci žžž žŸ zi ᑠ dx dx ®­­

(7.41)

Eq. (7.41) can be rearranged to give Ii dx RT d ln ai  d \ . ¸  2 zi ᑠ ᑠ Ui ci zi

(7.42)

Equation (7.42) is integrated across the membrane: II   1 a dx ¯ °   RT ln ai (\ I  \ II ) . Ii ¡ 2 ¨ ¡ ᑠz 0 U c ° ziᑠ aiI i i± ¢ i

(7.43)

The first term on the right-hand side of Eq. (7.43) is the negative of the Nernst potential of the ith ion, Ei [see Eq. (1.47)]. The second term is the membrane potential, '\. Since the left-hand side of the equation must also be a voltage, the coefficient of Ii must be a resistance; call it ᑬi. Define the membrane conductance of the ith ion by Gi = 1/ᑬi. Then, Eq. (7.43) becomes Ii = Gi('\– Ei).

(7.44)

Equation (7.44) summarizes the dependence of the current carried by the ith ion on the potential difference across the membrane and the ionic activities at each side. The activities determine the ion's Nernst potential, and the current carried by the ionic flux is proportional to the difference between the membrane potential and the Nernst potential. The proportionality constant is the ionic conductance of the membrane. When Ei = '\, the ion is in equilibrium across the membrane, and Ii (and Ji) is zero. An analogous equation [Eq. (4.17)] was presented in Chapter 4 to describe the relation between the unitary conductance and unitary current of a single open ion channel. Equation (4.27) related the unitary conductance of a single channel to the measured (mean) conductance of a population of such channels operating in parallel, in terms of the number density of channels and the fraction that are open at any given time. Equation (7.44) predicts a linear dependence of flux on membrane potential, if the conductance is constant. This is not a bad assumption if the potential is not varied over too large a range. Since the conductance depends on the concentration profile of the ion in the membrane, it is sensitive to the ionic concentration in the external solutions. The equivalent circuit described by Eq. (7.44) is shown in Figure 7.12.

7.3.2. Equivalent Circuit for the Passive Transport of Multiple Ions The circuit shown in Figure 7.12 describes the flux of only a single ion. The passage of multiple ions across the membrane is represented by a collection of single-ion equivalent circuits in parallel, as illustrated in Figure 7.13 for the most commonly studied trio of ions — sodium, potassium, and chloride. This approach to modeling multi-ion transport implies that the ion fluxes do not interact; such would be the case if each species crossed the membrane via its own set of ion selective channels.

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Figure 7.12. Equivalent circuit for a single ion. The orientation of the battery in the membrane depends on the sign of the Nernst potential.

Figure 7.13. Equivalent circuit for multiple ions.

The model of multi-ion transport shown in Figure 7.13 leads to a particularly useful relationship between the open-circuit potential and the ionic conductances of the membrane. The total passive current across the membrane in Figure 7.13 is p

I = GNa('\ – ENa) + GK('\ – EK) + GCl('\ – ECl),

(7.45)

where the superscript “p” indicates that only passive fluxes are considered. Solving p Eq. (7.45) for the membrane potential at open circuit (I = 0), '\ o.c. 

GNa ENa GK EK GCl ECl œ Gi Ei p  (I = 0). GNa GK GCl œ Gi

(7.46)

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Equation (7.46) shows that the membrane potential is a weighted average of the Nernst potentials of the permeant ions; the weighting factors are the fractions of the total conductance contributed by each ion. Therefore, as the membrane conductance of any ion increases relative to the other conductances, the open-circuit potential of the membrane approaches the Nernst potential of that ion. Increases in a single-ion conductance can occur because of an increase in the concentration of the ion or an increase in its effective mobility, as would occur if the channels normally traversed by that ion were open a larger fraction of the time. Recall that the solutions of the electrodiffusion equation in Chapter 2 showed the same dependence of membrane potential on the relative mobilities of the ions. The relationship given by Eq. (7.46) is the basis of an electrochemical (as opposed to electronic) method for voltage clamping a membrane. The membrane whose potential is to be fixed is loaded with valinomycin, a potassium-selective ionophore, thereby increasing the potassium conductance of the membrane by up to several orders of magnitude. Consequently, the membrane potential becomes nearly equal to EK. The Nernst potential of potassium, at which the membrane is clamped, can be adjusted by changing the ambient potassium concentration. This technique is particularly helpful when voltage clamping by means of an intracellular electrode is difficult. Even in the absence of added potassium-selective ionophore, the potassium permeability of the plasma membrane is considerably greater than that of sodium, owing to a large number of potassium channels (so-called potassium leak channels) in the membrane that are open most of the time. Accordingly, the resting potentials of most cells are within 10 mV of EK. The several parallel paths in Figure 7.13 can be reduced to a single path by simple algebraic manipulation of the preceding equations. Start with the more general form of Eq. (7.45): p

I = 6Gi('\– Ei) = 6Gi('\) – 6GiEi.

(7.47)

Equation (7.46) is used to replace the second term in Eq. (7.47) by '\o..c..6Gi , yielding an expression having the same form as Eq. (7.44): p

I = (6Gi)('\ – '\o.c.).

(7.48)

Thus, the current–voltage behavior of a circuit representing the transport of multiple ions can be replaced by a Thevenin equivalent circuit with a single conductance equal to the sum of the conductances in the original circuit, and a single battery potential equal to the open circuit potential. This identity allows complex parallel transport systems to be modeled — at least as far as electrical behavior is concerned — by a simpler system consisting of a single nominal path. The general rule for constructing a Thevenin equivalent circuit is given by Thevenin's theorem. In terms of conductances, it states that any combination of batteries and conductances with two terminals can be reduced to a single battery, whose voltage is equal to the open circuit potential of the system, and a single conductance equal to the short-circuit current divided by the battery voltage.

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If the compositions at both sides of the membrane are the same, then all Nernst p potentials are zero, and Eq. (7.45) simplifies to I = '\˜6Gi , which is Ohm's Law. The conductance of the membrane is the sum of the individual Gi's.

7.3.3. The Electrical Analog of a Rheogenic Pump As noted in Table 7.1, the most obvious electrical equivalent of a rheogenic pump is a current source. A current source, in turn, can always be modeled by a battery and series conductance. The latter representation is usually used when constructing equivalent circuits for actively transporting cells and tissues. The pump pathway is placed in parallel with the passive paths across the membrane in which the pump resides. When the substrate concentration is the same at both sides of the membrane, the pumpa generated current, I , is given by an equation very similar to Eq. (7.44): a

a

a

I = G ('\ – E ).

(7.49)

According to this expression, the pump rate is a linear function of the potential difference across the membrane; this is not unreasonable, since the pump moves net charge from one side of the membrane to the other, and the ease of doing so should depend on the membrane potential. a The “conductance” of the pump, G , measures the sensitivity of the pump rate to a '\. The quantity E is referred to as the electromotive force (EMF) of the pump; multiplied by the conductance, it gives the pump rate at short circuit. As '\is raised from a a a a value less than E to a value greater than E , the pump current changes sign. Since I a crosses zero when the membrane potential equals the EMF of the pump, E is also termed the reversal potential of the pump. a As in passive systems, the open-circuit potential can be driven toward E if the a experimental conditions (usually nonphysiological) are such that G is much greater than the passive conductance of the membrane; for instance, if the individual ion conductances were made very small. The EMF of the pump is the open-circuit potential of the membrane in the absence of passive flux. Since passive leaks are always present a under physiological conditions, '\ never reaches E . If the membrane potential is always much smaller than the EMF of the pump, then the fractional change in pump rate accompanying the normal variations in membrane potential will be small, and the pump may be regarded as a voltage-independent, constant-current source. Under these conditions, the equation for open-circuit potential is particularly simple. The total current across the membrane, equal to zero at open circuit, is a

p

I=I +I .

(7.50)

Substituting Eq. (7.47) into (7.50), setting the total current equal to zero, and solving for the membrane potential,

'\ o.c. 

œ Gi Ei  I a (I = 0). œ Gi

(7.51)

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Equation (7.51) can be used in the same way as Eq. (7.46) was in the previous subsection, to derive the Thevenin equivalent circuit for a multi-ion system in the presence of active transport. The total current–voltage relation is identical to Eq. (7.48), with '\o.c. given by Eq. (7.51); proof of this is left as an exercise. The first term in the numerator of Eq. (7.51) measures the contribution of the passive currents to the membrane potential, and usually dominates the second term, which reflects the rheogenic contribution of active transport. Note, however, that the passive currents themselves arise from the electrochemical potential differences created by the active transport systems in the cell membrane. The overall conductance of a number of conducting paths in parallel is the sum of the individual conductances, and the overall resistance is the reciprocal of the overall a a conductance, so the second term in Eq. (7.51), I /6Gi, can be equated to I Rm, where Rm is the membrane resistance. This term has sometimes been referred to as the voltage drop of the pump current across Rm. This is clearly wrong, since the pump current does not pass through the resistive channels that are traversed by the passive ion fluxes. In fact, the current that generates the additional voltage represented by the second term in Eq. (7.51) is the additional passive current that arises to balance the pump current and restore the total current to zero. This “neutralizing” passive current is oppositely directed to the pump current, so the term is negative.

7.3.4. Some Final Remarks All the equivalent circuits described above are purely resistive. Thus, they should apply equally well to transient and steady-state processes. However, when using equivalent circuits to model biological systems that are changing with time, care must be taken when the time scale is very short, or when the experiment is long. The capacitance of the membrane must be taken into account when the membrane potential can change rapidly. Whenever the potential difference across the membrane is altered, the capacitive charge on the membrane changes. The transient current associated with the change in membrane charge is proportional to the rate at which the membrane potential varies. Equivalent circuits including membrane capacitance are used to model excitable tissue, and will be illustrated in Chapter 9. The ionic conductances and the Nernst potentials both depend on the composition of the phases adjacent to the membrane. These compositions can change during the course of an experiment, particularly if nonphysiological conditions (e.g., short circuiting) are imposed or if any of the compartments are of limited size, like the inside of a cell. In this event, purely resistive models remain appropriate, but the battery voltages and conductances are not constant. When this problem is anticipated, every effort is made to conclude the experiment before significant changes in composition have taken place. Two applications of electrical analogs will be discussed in subsequent chapters. In Chapter 9, the relation between membrane potential and ion conductances will be used to describe the mechanism by which the action potential in nerve is generated. In Chapter 10, we will discuss the use of equivalent circuits to interpret the electrical properties of membranes and tissues in terms of the EMFs and conductances of the

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transport pathways. This discussion is reserved for the chapter on epithelia because electrical analogs have been used most frequently to interpret experiments on this class of tissues. As mentioned above, the implications of the nonlinearities that are seen in the current–voltage relations of many biological membranes and tissues are also deferred to Chapter 10.

PROBLEMS: CHAPTER 7 1.

Suppose we construct a bilayer membrane each layer of which has a different partition coefficient for a particular solute. Layer I faces Phase I and Layer II faces Phase II. The two layers meet in the middle of the membrane. Assume that I II Ks << Ks , independent of concentration. The bulk concentration of the solution in Phase II is greater than that in Phase I: csaI  csaII ; however, because of the inequality of the partition coefficients, cslI  cslII . (a) Which way does the flux go? (b) Sketch the solute concentration profile in the membrane.

2.

The fraction of a membrane's area that is occupied by pores is f. Starting with Eq. (7.10), which gives the hydraulic conductivity of a single pore, derive an expression for: (a) the Lp of the membrane as a function of pore radius and f; (b) the Lp of the membrane in terms of pore radius and the number of pores per unit area of membrane.

3.

You purchase a membrane that is prepared in such a way that all pores have the same known radius; call it r. You measure the thickness of the membrane with a micrometer; it is a. You want to compute the number of pores in the membrane per unit membrane area, n. (a) Describe a transport measurement or measurements you could carry out to obtain the information necessary to calculate n. (Note: looking at the membrane under a microscope is not allowed.) Assume you know the physical properties of any solute or solvent you use. (b) Derive an expression for n in terms of r, a, and the measurements and/or experimental conditions in (a).

4.

The viscous retardation function used by Wang and Veenstra (1997) was more complex than the relation given in Eq. (7.15), consisting of the ratio of two poly3 5 6 nomials in s/r: [1 – 2.105(s/r) + 2.0865(s/r) – 1.7068(s/r) + 0.726(s/r) ]/[1 – 5 0.75857(s/r) ]. (a) Repeat their analysis using the function 1 – P(s/r) as presented in Eq. (7.15), to assess the sensitivity of the diameter of the hypothesized selectivity filter to the form of the viscous term. Don't forget to include the steric term in )d. Note from Figure 7.6 that the dependent variable app app that Wang and Veenstra fit to their model was ki /kLi .

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0

(b) Equation (7.20) suggests that ki /ki is a more appropriate variable than app ki alone for correlation against s/r; after normalization by the correapp 0 app 0 sponding value for lithium, this function becomes (ki /ki )/(kLi /kLi ) = app app 0 0 (ki /kLi )/(ki /kLi ). The ratio of the ion permeabilities in the solvent equals the ratio of their aqueous mobilities, Ui, as discussed in Chapter 2. Since all ions have the same valence, the ratio of their electrical moapp app bilities, ui, serves as well. Repeat (a) using (ki /kLi )/(ui/uLi) as the dependent variable. app app The values of ki /kLi , hydrated radii si, and electrical mobilities for the six ions are tabulated below.

Cation

Permeability ratio, kiapp/kLiapp

Hydrated radius**, si, nm

Rb Cs K Na Li TMA* TEA*

1.63 1.53 1.35 1.05 {1 0.74 0.43

0.329 0.329 0.331 0.358 0.382 0.367 0.400

Electrical mobility, ui q 104, cm2/sec-V 8.09 8.01 7.62 5.19 4.01 4.66 3.39

* TMA = monovalent tetramethylammonium ion, TEA = monovalent tetraethylammonium ion. ** Nightingale (1959).

5.

Using a molecular radius of 0.15 nm for water, plot the variation of solute reflection coefficient with pore size for solutes whose radii are 0.25 and 0.3 nm. Use a range of pore sizes such that the reflection coefficients vary from 0.1 to 0.95.

6.

The reflection coefficient of the human red cell membrane for malonamide was measured by Goldstein and Solomon and found to be 0.83. The effective radius of this solute is 0.257 nm. If a radius of 0.15 nm is assumed for the water molecule, what is the pore radius of the red cell? Do the calculation using P(s/r) [Eq. (7.15)] to describe the viscous effect, as Goldstein and Solomon did, and using Anderson and Quinn's Pc(s/r) [Eq. (7.23)]. Compare the two results.

7.

Explain why, in the analysis of combined pore diffusion and convection leading to Eq. (7.30), the filtrate concentration is equal to Js/vm.

8.

A porous membrane is bounded by two sucrose solutions. Plot the variation of filtrate concentration with pore Peclet number for 100 p Pep p 0.01 when: (a) the upstream and downstream sucrose concentrations are 100 mM; (b) the upstream concentration is 100 mM and the downstream concentration is 10 mM; (c) the upstream concentration is 100 mM and the downstream concentration is 200 mM;

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To focus on the effects of boundary conditions and Peclet number, assume that the hindrance factors are unity. 9.

Consider a filtration process in which the downstream solution is the filtrate itself. (a) Derive an equation for filtrate concentration as a function of pore Peclet number and upstream concentration. Assume that the hindrance factors are unity. (b) Plot filtrate concentration vs. pore Peclet number over the same range of Pep as in the previous problem, for an upstream concentration of 100 mM. In this problem, the downstream concentration is not specified as it was in the previous problem, but is determined by the solute and volume fluxes.

10. Construct the state diagram for a one-vacancy model of a two-site pore. Show the sequence of transitions that accompany the passage of a tracer molecule through the pore. 11. Using Eq. (7.38) and literature values of the self-diffusion coefficient, molar volume, and viscosity of water, estimate the ratio of the osmotic and diffusive permeabilities of water in a pore whose diameter is 0.42 nm, at 37ºC. 12. Why doesn't the ionic charge, zi, appear in Eq. (7.44) for the contribution of the ith ion to the membrane current density? Explain in physical terms or in terms of the effect of charge on the quantities in the equation. 13. A membrane that passes only ions is short circuited. It can be shown that under these conditions the concentration profile for any ion crossing the membrane is linear in the membrane. Use the equivalent circuit formulation to derive an expression for ion flux at short circuit in terms of ion mobility, membrane thickness, and the bounding concentrations of the ion, assuming all activity coefficients are unity. 14. Three ions important in neuromuscular transmission at the muscle endplate (Chap. 9) are Na, K, and Ca. Under a certain set of experimental conditions, the extracellular and intracellular concentrations of these ions were as summarized in the table below:

Ion

Extracellular concentration, mM

Intracellular concentration, mM

Na K Ca

119 5 2.5

14 140 0.0001

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(a) Draw an equivalent circuit for the muscle membrane when all three ions can cross. Be certain that the batteries are oriented properly with respect to their positive and negative terminals. (b) The acetylcholine-gated cation channel in the membrane shows little selectivity, and passes Na, K, and Ca with relative ease. If the concentrations bounding the membrane do not change, what element(s) in the equivalent circuit change, and in what direction, when these channels open? (c) When the channels open, the flux through the membrane consists almost entirely of Na ions. Why not K? Why not Ca? Phrase your answers in the terms of the parameters of the electrical analog. 15. A membrane at 37ºC is ten times more permeable to potassium than to sodium, and passes no other ions. The solution on one side of the membrane (Side I) contains Na at an activity of 14 mM and K at an activity of 140 mM; the other side of the membrane (Side II) is exposed to 142 mM Na and 4 mM K (activities again). (a) What is the membrane potential at zero current? Use the equations from electrical analogs, assuming conductance is proportional to permeability. (b) Sketch the variation of membrane potential with GK/GNa as this ratio is reduced from ten to zero. 16. Use the Thevenin theorem to derive Eq. (7.48) directly. 17. Construct the general Thevenin equivalent circuit for the transport of multiple ions across a membrane, in the presence of active transport. Apply this circuit specifically to a membrane containing sodium, potassium, and chloride channels and the Na–K exchange pump.

8 REGULATION AND FEEDBACK

INTRODUCTION In this chapter we shall discuss biological transport as it relates to cellular regulation. Cellular regulation is a large and important area within cell biology, and our perspective on the topic, focusing narrowly on transport, will be more limited than that of more comprehensive texts on cell and molecular biology. This does not mean that transport does not play a significant role in cellular regulation; indeed, its role is of considerable importance. Two aspects of the relationship between transport and regulation will be considered: (1) the regulation of the activity of a transport system by agents within or outside the cell; and (2) the regulation by transport of the compositions (and concentrations) of neighboring solutions. By the activity of a transport system is meant the rate at which the transport molecules of a given kind carry substrate across the membrane in which they reside, for a given driving force. Generally, activity is altered either by changing the number of transporter molecules in the membrane, or by increasing the transport capability of such molecules; the latter can be accomplished, for instance, by changing the gating kinetics of an ion channel. Regulation of transport and regulation by transport can both be present in the same control system: for example, an external agent like a hormone can promote the insertion of transport mediator molecules into a target membrane (regulation of transport); the mediator can then alter the cis or trans levels of its substrate (regulation by transport). We have already referred to some mechanisms by which biological transporters are regulated. Direct activation of ion channels by ligand binding and membrane potential changes was discussed in Chapter 4 and will be revisited in the next chapter in the context of neurotransmission. Mechanical forces transmitted through the cell membrane and cytoskeleton can also alter transporter activity. In the present chapter, we will deal with mechanisms in which intermediary molecules translate a chemical signal into an increase in transporter activity. Often the process is initiated by the binding of an extracellular agent, such as a hormone, to a receptor in the cell membrane; then, through a series of intermediate processes and biomolecules, the transporter system is activated. One possible outcome of this cascade of reactions is a sigM.H. Friedman, Principles and Models of Biological Transport, DOI: 10.1007/978-0-387-79240-8_8, © Springer Science+Business Media, LLC 2008

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nal to the nucleus to synthesize more transporter molecules. Several techniques for transmitting the external “message” to its target transport system will be discussed in the first part of this chapter.

Figure 8.1. Relationships among the mediators and consequences of transport. Heavy arrows indicate the regulation of transport activity by mechanical forces, ambient compositions and chemical stimuli, and membrane potential; dashed arrows indicate that transport activity influences transmembrane fluxes and hence the composition of the cis and trans compartments. Additional relationships are shown with plain arrows.

The relationships among the variables that are involved in regulation of and by transport are shown in Figure 8.1. As we saw in Chapter 4, transporter activity can be influenced by mechanical forces, membrane potential, and chemical signals, as well as the concentration of the substrate itself. The fluxes across a membrane depend on transporter activity and the compositional and electrostatic contributions to the electrochemical potential driving force. These fluxes can affect the composition of the solutions bounding the membrane. Meanwhile, the membrane potential, which influences both the driving force for transport and the activity of voltage-activated transporters, depends (as we saw in Chap. 2) on the bounding compositions and the permeabilities of the permeable ions; the permeabilities, in turn, reflect the gating kinetics, i.e., the activity, of the transporters. It should be no surprise that feedback is a prominent feature of regulatory processes that involve transport. When the activity of a transport process is altered, either by increasing the capacity of existing transport molecules or by increasing their number, the transmembrane fluxes and potential, and the compositions of the cis and trans phases, will generally change, as the system seeks a new steady state. Transport models can predict the time course of these changes in response to alterations in transport activity. The analysis of this response, in both the unsteady and steady states, is facilitated by a methodology known as compartmental analysis, which will be described and illustrated in the context of intracellular calcium dynamics. We have already examined two of the assumptions made in compartmental analysis: the quasisteadiness assumption, in Chapter 2, and the well-mixed assumption, in Chapter 4. Compartmental analysis allows us to

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construct integrated transport models of isolated cells or cell layers that contain multiple barriers, transport routes, and solutes. An important feature of compartmental analysis that distinguishes it from most of the modeling we have done to this point is that some or all of the compartments are regarded as finite in size; accordingly, their compositions, which determine the boundary conditions for transport, are generally time dependent. As remarked above, the multiplicity of arrows in Figure 8.1 virtually guarantees feedback in regulatory systems that include biological transport processes. Indeed, the two arrows joining “Transporter activities” and “Membrane potential” define the feedback loop that is responsible for the propagation of the action potential in muscle and nerve, as will be demonstrated in the next chapter. In this chapter, we will examine the feedback between insulin secretion and response in insulin-secreting pancreatic cells. Since transport systems directly affect local compositions, it is not surprising that they are the primary mechanisms for maintaining the compositions of the extracellular and intracellular fluids in a viable range. The maintenance of the appropriate extracellular environment is the responsibility of the kidney, which will be discussed in Chapter 10 (though a few remarks are made here); accordingly, the focus here will be on the role of transport in regulating the intracellular environment — in particular, the regulation of cell volume and cytosolic pH.

8.1. REGULATION OF TRANSPORT Transport within cells or across the cell membrane can be regulated by biological agents originating inside the cell, or by external agents, such as hormones secreted elsewhere. Hormones are often thought of as messengers, carrying signals via the blood to the target cells that have the appropriate receptors, instructing them to perform in a particular way. The external agents can act directly by activating the transporters that are present or by signaling the nucleus to produce more transporter molecules; they can also act indirectly through intermediate compounds in the cytosol. These intermediate compounds, known as second messengers (the hormone is the “first messenger”), are generally formed as a consequence of the binding of the hormone, or agonist, to a cell surface receptor. We will describe this mechanism first.

8.1.1. Receptor-Mediated Second Messenger Systems: Cyclic AMP and Antidiuretic Hormone Second messenger systems are defined according to the identity of the second messenger. The cyclic AMP system is a well-studied one, and we will use it to illustrate the activation process. Many hormones, such as antidiuretic hormone (ADH, also known as vasopressin), which regulates water reabsorption in the kidney, operate by this mechanism. A schematic of the operation of the cyclic AMP system is shown in Figure 8.2a. Let us introduce the membrane players first:

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Figure 8.2 (opposite). The cyclic AMP system. (a) The cycle. R = receptor; DEJ = subunits of G protein, initially associated with R; E = adenylyl cyclase; H = hormone; C = ion channel. (1) R binds H and stimulates G to discharge guanosine diphosphate (GDP) and bind GTP, activating it. (2) The activated D subunit migrates to E, activating it; the activated cyclase transforms ATP to cAMP; the hormone dissociates from the receptor; the EJ complex of the G protein migrates toward C to modulate its activity. (3) Autohydrolysis (see text) of GTP causes the D subunit to stop activating E; cAMP synthesis stops and Pi is released. (4) the D subunit dissociates from the effector, and the G protein reassembles at the receptor. (b) The ATP o cAMP reaction. Only the part of the ATP molecule that changes is shown; the structure of the entire molecule appears in Figure 3.6.

— an integral transmembrane receptor protein (R) with an extracellular binding site for the hormone; — a GTP-binding protein, or G protein, typically consisting of three different subunits (heterotrimeric), and anchored by fatty acid chains to the cytoplasmic leaflet of the bilayer. The G protein is activated when its binding site, on the D subunit of the protein, is occupied by energetic GTP. In the cycle depicted in Figure 8.2a, the site initially is occupied by GDP, so the protein is inactive. In some cases, the G protein and receptor are associated at rest in a single complex; and — an effector molecule (E), adenylyl cyclase. The cyclic AMP-mediated response is initiated when a hormone molecule binds to its receptor in the membrane of a target cell. Upon binding the hormone, the receptor protein undergoes a conformational change and interacts with the G protein at the cytoplasmic face of the membrane, causing it to release its GDP ligand and bind GTP from the cytosol. Activated by GTP, the G protein dissociates from the receptor protein and its subunits cleave into the D and EJ fragments; the receptor is now free to promote the activation of additional G protein molecules, for as long as the hormone remains bound. Both of the G protein fragments can participate in transport regulation. For instance, EJ fragments can cause potassium channels in the heart to open, and neuronal calcium channels to close. However, the D subunit, to which the GTP is bound, is the principal player in the cyclic AMP system. It migrates along the inner face of the bilayer until it finds an effector molecule to bind to and activate. Different G proteins activate different effector molecules. The effector molecule for the cyclic AMP system is adenylyl cyclase. The activated adenylyl cyclase catalyzes the conversion of local cytosolic ATP to cyclic 3c,5c-AMP, or cAMP (Fig. 8.2b). At this point, the G protein has done its job. The GTP on the D subunit is hydrolyzed to GDP by a GTPase sequence within the subunit itself, inactivating the subunit, releasing phosphate into the cytosol, and terminating effector activation. The D subunit dissociates from the effector, and the two fragments of the G protein reassociate (sometimes forming a G protein–receptor complex) and await their next signal from an agonist-bound receptor protein.

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Cyclic AMP, the second messenger, can produce a variety of physiological responses, including alterations in membrane permeability, the activation of other enzymes, protein synthesis, or the secretion of other hormones. The effects of the second messenger are cell specific. Since the messenger molecule, and others involved in the regulatory process, diffuse within the cell, the response can be remote from the site of cyclic AMP synthesis. This is illustrated by the mode of action of ADH. As will be described in greater detail in Chapter 10, the target cells for ADH form a layer one surface of which (the mucosal, or apical, surface) lines the kidney tubules through which a blood filtrate passes, while the other surface (the serosal, or basolateral, surface) is adjacent to the renal capillaries. ADH increases the water permeability of the mucosal surface; by regulating water reabsorption into the circulation, the hormone maintains the normal extracellular osmolarity (the control loop is completed by osmoreceptors in the hypothalamus of the brain that signal increased secretion of the hormone when the extracellular osmolarity rises). Since the hormone is carried in the circulating blood, it engages target kidney cells at their serosal surface, and that is where most of the ADH receptors reside. Cyclic AMP is produced there, and diffuses into the cell to initiate further reactions that ultimately promote the release of aquaporincontaining vesicles at the mucosal surface. With respect to transport effects, cAMP can act directly on channels, such as those involved in olfaction, and cause them to open. More commonly, however, a cyclic AMP-dependent protein kinase is involved. Protein kinases are cytosolic proteins that catalyze the phosphorylation by ATP of particular target proteins, thereby activating them. Phosphorylation by kinases is a principal intracellular control mechanism. One such kinase is protein kinase A (PKA), which consists of four subunits, two of which can bind cAMP and are termed “regulatory,” and two of which are catalytic. When sufficient messenger (>2 molecules of cAMP per kinase molecule) has bound to the regulatory subunits, they undergo a conformational change and cleave from the catalytic subunits, which are activated. The latter subunits then diffuse in the cytosol until they find a target protein, such as a transport molecule or a regulatory protein in the nucleus, to activate. This process, far downstream from the initial binding of the hormone to its receptor, accomplishes the task that the hormone originally signaled the cell to perform. Recall that when multiple ligand molecules (cAMP in this case) are needed to activate a substrate molecule (such as PKA), the dependence of substrate activity on ligand concentration curves sharply upward when the concentration of ligand is far from saturation (Fig. 4.5). For this reason, the stoichiometry of cAMP–PKA activation (n > 2) increases the sensitivity of the cellular response to the cytosolic cAMP level. The kinase can act in several ways. Suppose the hormone signals an increase in the activity of a transport system. There are three principal ways in which this can be accomplished: by promoting the synthesis of additional transporter molecules; by activating channels and increasing the fraction of time each channel is open; and by promoting the migration of transporter-loaded storage vesicles to, and the insertion of their contents into, the intended membrane. As noted earlier, ADH uses the last of

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these mechanisms, promoting the release of aquaporin-2-loaded vesicles into the mucosal membrane to increase its flow conductivity. The response to the agonist depends on the cell type, and its strength depends on the increase in cyclic AMP level that is induced. The amount of second messenger produced within the cell depends, in turn, on the hormone level in the blood and the density of receptors on the cell surface. An increase in either of these causes the number of receptors whose binding sites are occupied to rise, accelerating the rate of cyclic AMP synthesis, and elevating the concentration of cAMP in the cell. Once the level of stimulating hormone in the blood falls, the cellular response diminishes, since the enzyme phosphodiesterase continuously degrades the cyclic AMP to an inactive form. The activity of this enzyme, in turn, is inhibited by the drug theophylline, which consequently prolongs and strengthens the hormone-induced response. The duration of the response is also limited by the autohydrolysis of the GTP on the D subunit of the G protein, which limits the duration of cyclase activation. Additionally, intracellular protein phosphatases can remove the phosphate group from activated transport molecules, rendering them inactive. Cyclic AMP is capable of inhibiting certain protein phosphatases, thereby increasing the duration and intensity of the response of the transporter system. We can construct a simple model to relate the regulatory response to the agonist stimulus. If agonist binding and unbinding are rapid, relative to changes in agonist level, the binding process can be regarded as being in equilibrium, and the number of cell receptors occupied by agonist, Nb , is given by the simple Michaelis–Menten expression for saturable binding employed in Chapters 3 and 4: Nb  NT ¸

cA , K cA

(8.1)

where NT is the total number of receptors on the cell surface, cA is the ambient agonist concentration, and K is the equilibrium constant for the dissociation of the agonistreceptor complex. If the rate of production of cyclic AMP is proportional to the number of occupied receptor sites, and the rate of degradation of the messenger is proportional to the amount that is present in the cell, then the dynamics of the intracellular messenger concentration are given by dccAMP  k f N b  kr ccAMP . dt

(8.2)

Here, kf reflects the kinetics of the G protein and cyclase, while kr reflects that of the phosphodiesterase. The steady-state level of cyclic AMP is obtained by setting the right-hand side of Eq. (8.2) to zero:

(ccAMP )s.s. 

kf kr

NT ¸

cA . K cA

(8.3)

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CH. 8: REGULATION AND FEEDBACK

G proteins are also involved in regulatory processes that use calcium as a second messenger, and operate directly on channels as well. Table 8.1 lists several G proteins involved in transport and the actions of their subunits.

Table 8.1. Selected G Proteins Involved Directly or Indirectly in the Regulation of Transport G protein

1

Functions

Gs

Activates adenylyl cyclase Activates Ca channels Downregulates Na channels

Gi

Inhibits adenylyl cyclase Inhibits Ca channels Activates K channels

G0

Inactivates Ca channels Activates K channels 1 Activates phospholipase C-E

Gq

Activates phospholipase C-E1

See §8.1.3.

8.1.2. Direct Hormonal Regulation Without an Extracellular Receptor: Aldosterone The hormone aldosterone controls sodium reabsorption in the kidney and acts on its target cells without the help of either an extracellular receptor or a second messenger. Lipid soluble, it diffuses through the cell membrane and into the cytoplasm, where it combines with specific receptor proteins. The hormone–receptor complex then diffuses into the nucleus of the cell, where it promotes the transcription of genes to mRNA that codes for proteins that increase the sodium throughput of the cell layer. In particular, newly synthesized Na–K ATPase molecules are inserted into the serosal surface of the cell to actively transport sodium toward the renal capillaries, while additional sodium channels are inserted into the mucosal surface to increase its sodium conductance, thereby maintaining a physiological intracellular sodium level while providing additional substrate for the serosal pump. When a hormone influences both the entry and exit of a species that is transported across the cell, as is the case here, throughput can be more efficiently regulated and the intracellular concentration of the transported species can be more stable. When a hormone or other agonist regulates transport activity by promoting biosynthesis rather then by increasing the open probability of channels already in the membrane or by inserting ready-made transporters from storage vesicles, there is a time delay between the stimulus and the response. In the case of aldosterone, new mRNA appears 20–30 min after the cell has been exposed to the hormone. Fif-

PRINCIPLES AND MODELS OF BIOLOGICAL TRANSPORT

307

teen to 90 min later, the effect of newly synthesized protein is seen in the sodium transport rate.

8.1.3. Calcium-Based Regulation Calcium ion is an important player in transport regulation. Increases in cytosolic calcium initiate changes in a host of transport-related processes. Thus, calcium is a regulator of transport, like cyclic AMP. Calcium ion concentrations and dynamics are also regulated by transport. In resting cells, the cytosolic calcium concentration is very low, as a result of ATPase and antiporter molecules that transport the ion into the extracellular fluid or into sequestering organelles; the increase in cytosolic calcium that constitutes the signal arises from passive transport back into the cytosol from external fluid or internal stores. Thus, calcium dynamics reflects both the regulation by transport that serves to maintain the resting low-calcium state, and the regulation of transport that follows an increase in the cytosolic calcium level. We will consider both aspects of the role of calcium in order. In the last part of this subsection, we will use the modeling of calcium transport inside the cell as a point of departure for a discussion and illustration of compartmental analysis. Maintenance of a Low Cytosolic Calcium Concentration. Calcium is pumped out of the cytosol by three active transport systems. Ca ATPases in the cell membrane and the membranes of the endoplasmic reticulum (ER) carry out primary active transport, extruding calcium from the cell or sequestering it in the ER; in muscle, where calcium dynamics plays a crucial role, the sarcoplasmic reticulum functions similarly to the endoplasmic reticulum in other cell types. As pointed out in Chapter 5, the Ca ATPase in both reticulum membranes is the sarcoplasmic/endoplasmic reticulum calcium (SERCA) pump. Secondary active transport also plays a role here: the Na–Ca antiporter in the plasma membrane uses the transmembrane free energy difference of three sodium ions to pump a single ion of calcium out of the cell. As a consequence of these active transport systems, the intracellular calcium concentration can be drawn down to as low as 0.1 μM, while that in the extracellular fluid is around 1 mM. Calcium is also accumulated in mitochondria and by calcium binding molecules in the cytosol that further reduce the internal activity of the ion. Accumulation by itself cannot maintain a low intracellular calcium level indefinitely, since the capacity of these sequestering sites is limited; as they become saturated, calcium entering the cell passively across the cell membrane would eventually cause the cytosolic level to rise. The active transport systems in the plasma membrane, which extrude calcium ions into the effectively limitless capacity of the extracellular space, are the only mechanisms that can oppose passive entry of calcium in the long term. Similarly, prolonged elevation of the intracellular calcium level — and thus a prolonged response to the original agonist — requires influx from the extracellular space as well as the release of ions sequestered in organelles. Calcium Signaling. Although the magnitudes of the calcium ion concentrations inside and outside the cell are small relative to the concentrations of the major ions, the calcium concentration ratio across the cell membrane is much larger, leading to a calcium Nernst potential that is even more positive that that of sodium. In addition,

308

CH. 8: REGULATION AND FEEDBACK

the electrostatic potential inside the cell is negative relative to that outside. As a consequence, the driving force for calcium entry into the cell is enormous. The primary route for calcium to reenter the cytosol — and initiate its signaling role — is through calcium channels in the endoplasmic reticulum (ER) or plasma membrane; mitochondrial calcium stores may also be released. When calcium reentry is signaled by agonist–receptor binding at the cell membrane, a G protein is activated, which subsequently activates the membrane-bound enzyme phospholipase C-E, which in turn acts on a membrane phospholipid to produce a messenger molecule, inositol 1,4,5-triphosphate (IP3). IP3 binds to calcium release channels in the ER, causing them to open and release stored calcium into the cytosol. The consequent increase in cytosolic calcium increases the calcium conductance of the plasma membrane, and additional calcium floods into the cytosol from the extracellular space through calcium channels and/or nonselective cation channels. Voltage-gated calcium channels in the cell membrane can also open, without requiring prior calcium release from the ER; this mechanism is particularly relevant to muscle function. Mechanical stretch can cause mechanically sensitive calcium-permeable channels in the plasma membrane to open if the cell volume increases sufficiently; we shall see below that this allows calcium activation to play a role in cell volume regulation. Once its concentration in the cytosol has risen, calcium ion can act directly on those channels for which it is an activating ligand; these include Ca-activated potassium and chloride channels. Many of the effects of calcium in the cell are mediated by the calcium-binding protein calmodulin. When calmodulin's binding sites are occupied, which occurs at a cytosolic calcium concentration of about 1 μM, it undergoes a conformational change and activates an associated kinase. The kinase can then phosphorylate target molecules, similarly to PKA. Among the transport or transport-related proteins influenced by calcium, either directly or through calmodulin, are adenylyl cyclase, cyclic AMP phosphodiesterase, cyclic GMP-gated sodium and calcium channels, the plasma membrane Ca ATPase, and at least one sodium channel. The interaction of the calcium and cyclic AMP systems indicated by this list adds an additional level of complexity to the modeling of these regulatory processes. Some of the intracellular calmodulin molecules are bound via tethers to the cytosolic tail of voltage-gated calcium channels. When these channels are opened (by membrane depolarization), calcium enters the cytosol through the pore, and some of these ions bind to the calmodulin. When its sites are occupied, the calcium-binding protein undergoes a conformational change and inactivates the channel. Since the binding sites are exposed to the higher calcium concentrations at the pore exit, they become occupied more quickly than sites on calmodulin molecules freely diffusing in the cytoplasm, thereby expediting this negative feedback process. Calcium also mediates the effect on transport of a second product of phopholipase C-E activity, diacylglycerol. Diacylglycerol activates a protein kinase, PKC, which can inhibit certain ion channels when signaled to do so by a rise in cytosolic calcium. Additional transport-related effects of alterations in intracellular calcium levels include changes in gene expression, and the stimulation of exocytosis and secretion, as described in Chapter 3. Another calcium-binding protein, troponin C, is important in muscle contraction and will be discussed in Chapter 9.

PRINCIPLES AND MODELS OF BIOLOGICAL TRANSPORT

309

Figure 8.3. (a) Schematic showing the principal compartments involved in cytosolic calcium regulation and the transport mechanisms responsible for calcium fluxes between them. Included is an artificial compartment that represents calcium-binding molecules that equilibrate with unbound (“free”) calcium in the cytosol. Other organelles not in the figure that participate in calcium handling include mitochondria and certain vesicles. Open circles denote primary active transport systems and closed circles denote exchangers/cotransporters. Channels are indicated by gaps in the membrane, and mechanically activated channels have springs. (b) Representation of calcium dynamics using the compartmental analysis formalism. Note that the extracellular fluid is not regarded as a compartment, since its composition is assumed to be unaffected by calcium exchange; note also that in some cases a single pair of arrows represents more than one transport process. The paired arrows reflect the bidirectionality of intercompartmental transport, which is the case here even though several unidirectional transporters are present. Note also that compartments need not be physically discrete; the calcium-binding molecule “compartment” is distributed throughout the cytosol. Two arrows describe transport across the plasma membrane; the inward-directed arrow represents passive channel influx, and the outward-directed arrow represents primary and secondary active transport.

Calcium Handling in Cells and Compartmental Analysis. The dynamics of cytosolic calcium depends on the fluxes between four principal regions, as shown in Figure 8.3a. Proceeding from the outside in, they are the extracellular fluid, the cytosol, various anionic molecules in the cytoplasm that can bind calcium, and the endoplasmic reticulum. The major calcium transport systems that move calcium between these compartments are also indicated. We see that the figure consists of (1)

310

CH. 8: REGULATION AND FEEDBACK

compartments containing solute and (2) interfaces between compartments. The interfaces contain transport systems that convey solute between the compartments on the two sides of the interface. Calcium transport between these compartments and across these interfaces can be modeled using a formalism known as compartmental analysis. We have already used some equations from compartmental analysis: when we examined, in Chapter 2, the conditions for quasisteadiness in the face of changing boundary conditions, and in Chapter 4, when we modeled tracer kinetics in the erythrocyte. Here we will be more formal and general in our presentation of the method. General formulation of compartmental analysis. Classical compartmental analysis relies on two fundamental assumptions: 1. The rate at which solute enters or leaves any given compartment is slow enough, relative to the rate of diffusion of the solute in the compartment, that the concentration gradients in the compartment are negligible; that is, the concentration in the compartment is always uniform. Then, that concentration, cij, is equal to nij /Vj, where nij is the number of moles of the ith species in the jth compartment, and Vj is the volume of the compartment. This is often referred to as the well-mixed assumption even though mechanical mixing is generally absent from living systems. 2. The rate of change of the concentrations in the two compartments at an interface is slow enough that transport across the interface can be regarded as quasisteady. The requirement on cell dimensions and membrane thickness for this condition to be met was discussed in Chapter 2. Since the equations of compartmental analysis are essentially mass balances, we add the condition, readily met, that the quantity of solute contained within the interface itself (i.e., material in the process of passing through) is negligible relative to that in the compartments. The condition that must be satisfied for the first requirement to be met is similar to that previously derived for the quasisteady state, except that tc must now be greater than the diffusional time constant for the compartment rather than for the membrane. We will derive this expression for a spherical cell (in contrast to a planar membrane) using a notation consistent with that in Chapter 2. For a sphere subject to a change in concentration at its surface, a reasonable time constant for the intracellular concentration to approach uniformity is (Truskey et al., 2004) t d ,cell 

r2 , 2 Dc ,cell

(8.4)

where r is cell radius. The time constant for changing concentration is as before: tc 

aV , Ds S

(8.5)

PRINCIPLES AND MODELS OF BIOLOGICAL TRANSPORT

311

where Ds is the solute diffusion coefficient in the cell membrane. The volume and surface area of the cell can be written in terms of its radius, and the ratio Ds /a can be replaced by the permeability, ks; requiring td,cell << tc, the cytosol will be well-mixed if ks 

2 Ds ,cell 3r

.

(8.6)

Thus, the concentration of a species in a compartment is more likely to be uniform if the compartment is small, if the solute diffuses rapidly, and if the interfaces that bound the compartment are not very permeable. Given the lack of mixing in most transport compartments and the factors discussed in Chapter 3 that retard diffusion, this assumption can fail when the solute is large or charged and its passage through the interface is facilitated. Compartmental analysis describes the change in the composition of interacting compartments as a function of time. It does so by relating the change in the composition of a compartment to the flow of matter into and out of it, while relating the flows across each interface to the composition of the compartments at each side. Our definition of the flow across an interface is different from that of the flux. The flow of the ith species, Ji, is equal to the total number of mols crossing the interface in unit time (i.e., 2 mols/sec), while the flux, Ji, is based on a unit area of interface (i.e., mols/cm -sec). The two variables are related by Ji = JiA, where A is the transport area upon which the flux is based. Indeed, the flux of a species is generally obtained by measuring its flow and dividing by a nominal transport area, such as the measured area of a membrane. The number of moles of the ith species in the jth compartment changes with time, t, as a consequence of three kinds of processes: — flows from or into neighboring compartments, — flows from or into regions external to the system, and — production or destruction of the species within the compartment. Generally, there are multiple pathways across an interface; indexing the pathways by r, and identifying the interface between the jth and kth compartments by the subscript j/k, Jij/kr is the flow of the ith species from the kth compartment to the jth compartment via the rth pathway. Similarly, we denote the flow of the ith species from the external region into the jth compartment via the rth pathway by Jij/0r. Lastly, we let ±rij be the rate at which the ith species is generated or consumed in the jth compartment. Thus, the total rate of change of nij is dnij dt

 œœ J ij / kr œ J ij / 0 r o rij . k( j)

r

(8.7)

r

The index k(j) on the first sum indicates that the sum is (of course) taken over only those compartments that communicate with the jth compartment. The second term is nonzero only if the external compartment communicates with the jth compartment. The sum over r indicates that all transport pathways across each interface are included in the sum.

312

CH. 8: REGULATION AND FEEDBACK

The last term in Eq. (8.7) is generally zero for most ions, which are neither consumed nor produced within the body; however, it can be important for metabolites, drugs, or products of biosynthesis. An important — and likely the most common — application of compartmental analysis is in pharmacokinetic modeling, which is used to track the distribution of drugs in the body; pharmacokinetic models almost always include loss terms to reflect the metabolism, degradation, and inactivation of the drug. When the fundamental assumptions of compartmental analysis are met, the flow of a species between two compartments is a function of only the state of the two compartments and the properties of the interface. Usually, electrostatic potential effects and solute–solute interactions are ignored, so the flow of the ith species is assumed to depend on only cij and cik and the activity of the transport system. This allows us to write Jij/kr = Gij/krFr(cij,cik),

(8.8a)

Jij/0r = Gij/0rFr(cij,ci0).

(8.8b)

and

where Gij/kr is the activity of the rth transport system carrying the ith species between the jth and kth compartments and Fr (cij,cik) is the driving force for the transport of the ith species through the rth pathway across the j/k-th interface; as before, the subscript “0” distinguishes exchange with the external phase from that with other compartments in the system. These equations, though heavily subscripted, can be quite simple. For instance, for the free diffusion of solute across a simple membrane, the transport system activity 3 is the product of the membrane permeability and membrane area (cm /s), and the driv3 ing force is the concentration difference across the membrane (mols/cm ). For carrier transport, Eq. (4.39) applies, with JA replaced by JA and XT replaced by XT, the total number of carriers in the interface. Primary active transport systems are often regarded as independent of the cis and trans concentrations of the substrate (if they don't change too much), or are described by a saturable function of cis substrate concentrav  c ¬­ cv ž tion, c. Two such functions are and ž , where Q is the number of ions žŸ K c ®­­ K cv transferred per pump cycle and K is a constant. For exchangers, a product of these terms is used, one for each substrate, using different K's and the appropriate cis concentration for each. Since the compartments are well mixed, the rate of change of the concentration of the ith species in the jth compartment is given by dcij dt



1 dnij . V j dt

(8.9)

Combining Equations (8.7) to (8.9), an ordinary differential equation in cij is obtained:

PRINCIPLES AND MODELS OF BIOLOGICAL TRANSPORT

dcij dt



1 Vj

313

  ¯ ¡ œœ Gij / kr Fr (cij , cik ) œ Gij / 0 r Fr (cij , ci 0 ) o rij ° . ¡ k( j) r ° r ¢ ±

(8.10)

Certain formalisms are employed when compartmental relationships are described graphically: compartments are designated by circles, intercompartmental fluxes are denoted by double arrows between the compartments, and external sources and sinks are generally denoted by single arrows directed into or out of the appropriate compartment. This formalism is illustrated for calcium regulation in Figure 8.3b. In this figure, two oppositely directed arrows cross the plasma membrane, since the direction of the net Ca flux across the membrane changes during calcium signaling. Useful overviews of compartmental analysis can be found in Lewis (1996) and Cobelli et al. (2006). The relation between total and free cytosolic calcium. When applying compartmental analysis to intracellular calcium dynamics, a distinction must be made between the total and free calcium (i.e., calcium not bound to intracellular molecules) in the cytosol. While the fluxes into or out of the intracellular fluid compartment affect its total calcium content, they depend on the level of free calcium, since bound ion does not contribute to the driving force for transport. So we begin by defining that relationship prior to applying the compartmental model. Two equations define the relationship between free and total cytosolic calcium. The first is the Michaelis–Menten relation for the binding sites:

cS ,Ca  cS ,T

cCa,F K cCa,F

.

(8.11)

where cS,Ca is the concentration of occupied sites (which is the same as the concentration of bound calcium), cS,T is the total concentration of sites on the calcium-binding molecules, cCa,F is the concentration of free calcium in the cytosol, and K is the dissociation equilibrium constant of the binding sites. The second equation states that the total calcium concentration, cCa, equals the sum of the bound and free fractions:

cCa = cS,Ca + cCa,F .

(8.12)

Equation (8.11) is substituted into Eq. (8.12), yielding the desired relationship between total and free calcium:

cCa 

cCa,F ( K cCa,F cS ,T ) K cCa, F

.

(8.13)

Compartmental modeling of calcium dynamics. Applying Eq. (8.10) to the system in Figure 8.3 and with a simpler notation, the rates of change of total calcium concentration, c, in the endoplasmic reticulum (subscript "1") and cytosolic (subscript "2") compartments are given by

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CH. 8: REGULATION AND FEEDBACK

dc1 1   ¢) ER  GER (c1  c2, F )¯± , dt V1

(8.14a)

dc2 1   ¢) ER GER (c1  c2, F )  ) PM  ) NC GPM (cEF  c2, F )¯± , dt V2

(8.14b)

where the plasma membrane and SERCA pump rates ĭPM and ĭER depend on the concentration of free calcium in the cytosol, c2,F ; all passive transport is described as the product of a membrane conductance (similarly subscripted) and a transmembrane concentration difference; the Na–Ca exchange rate, ĭNC, is regarded as constant; and cEF is the calcium concentration in the extracellular fluid. The calcium binding equilibrium in the ER has not been included. It might be useful to review how Eq. (8.10) translates into Eqs. (8.14). We will do this here for the ER (j = 1); the translation for the cytosol is left as an exercise. Ca is neither produced nor consumed in the ER, so rCa,1 = 0. The second sum in Eq. (8.10) is also zero, because the ER does not communicate with the extracellular fluid. The ER communicates with only one compartment, the cytosol (k = 2). There are two pathways that connect the ER and the cytosol: the SERCA pump and the Ca channels (r = 1,2). The rate of calcium entry through the pump, JCa,1/21 = )ER , is a function of the free substrate level in the cytosol. The rate of calcium influx through the channels, JCa,1/22,, is negative and equal to the product of the channel conductance, GER, and the transmembrane difference in free calcium concentration.

The total calcium concentration in the cytosol appears only on the left-hand side of Eq. (8.14b). The total concentration can be replaced by the free concentration by dividing both sides of the equation by dc2/dc2,F; the left-hand side becomes dc2,F/dt, and the divisor of the right-hand side is obtained by differentiating Eq. (8.13):

KcS ,T dc2 1 . ( K c2,F )2 dc2,F

(8.15)

where c2,F has replaced cCa,F in Eq. (8.13). Thus, the rate of change of free calcium concentration is given by dc2, F dt



2 1 <) ER GER (c1  c2, F )  ) PM  ) NC GPM (cEF  c2, F )>( K c2, F )

( K c2, F )2 KcS ,T

V2

.

(8.14c)

When the cell is at rest, the calcium levels in the cytosol and ER have reached a steady state, so the right-hand side of each mass balance equation is zero. This allows us to solve for the resting concentration in each compartment, denoted by the superscript “0”:

c10 

) ) NC ) ER cEF  PM , GER GPM

(8.16a)

PRINCIPLES AND MODELS OF BIOLOGICAL TRANSPORT

c2,0 F  cEF 

) PM ) NC . GPM

315

(8.16b)

The calcium dynamics initiated by IP3 can be described by integrating Eqs. (8.14a) and (8.14c), using Eqs. (8.16) for the initial conditions, and driving the pair of equations with step increases in the channel conductances represented by GER and GPM .

8.2. FEEDBACK IN TRANSPORT SYSTEMS: INSULIN As noted earlier, transport processes can regulate the level of agonists that in turn can affect the regulatory process, giving rise to feedback effects. Some biological feedback systems serve to maintain a reasonably constant cellular environment, while others are designed to amplify the effects of changing control variables. The regulation of insulin secretion is a good illustration of the role of feedback in biological control. This hormone is produced in specialized pancreatic E-cells and is crucial to carbohydrate metabolism. Increases in blood insulin levels stimulate cells to take up more glucose, maintaining a normal concentration of glucose in the blood. Most cells express insulin receptors on their external surface, with muscle, liver, and adipose tissue being particularly well endowed in this respect. Decreases in insulin responsiveness or production are responsible for the symptoms of diabetes. The cytoplasmic end of the insulin receptor protein has ATPase activity and autophosphorylates when an extracellular insulin molecule binds to the external receptor. In this form, the protein can phosphorylate and activate substrate molecules that bind to it from the cytosol. The substrate molecules initiate a sequence of intracellular signals that result in the insertion into the cell membrane of glucose carriers that had been stored in the membranes of intracellular vesicles for rapid release. Because of the ready availability of glucose carriers, the effect of insulin on cell metabolism is seen in minutes. Since the secreted insulin is responsible for regulating the glucose level in the blood, the E-cells must have a way to sense that level. They do this by metabolizing glucose that enters the cell; the E-cell membrane contains glucose carriers, and the higher the glucose level in the pancreatic capillaries, the higher the substrate influx and insulin release rate. The precise mechanism by which the glucose level in E-cells is translated into the insulin secretion rate is not absolutely clear, but many of the components of the process have been identified, and the summary that follows includes the most well-accepted steps. An increase in intracellular glucose metabolism leads to increased formation of ATP, which blocks ATP-sensitive potassium channels in the cell membrane. When the potassium channels are closed, the cell potential becomes less negative (i.e., depolarizes), as it moves away from the Nernst potential of potassium. Cell membrane depolarization causes voltage-gated calcium channels in the membrane to open, leading to an increase in intracellular calcium, which in turn causes insulin-loaded vesicles within the cell to discharge their contents via exocytosis (recall Fig. 3.13). When glucose levels are reduced, the ATP concentration in the cell decreases, ATP molecules dissociate from their binding sites on the potassium channels, and the channels

316

CH. 8: REGULATION AND FEEDBACK

reopen, repolarizing the cell. This shuts down the calcium channels in the cell membrane and allows the binding and pumping mechanisms in the cell to bring calcium levels back down to resting values, thus preventing excessive hormone secretion and hypoglycemia. More details can be found in Misler (2001).

Insulin has the same effect on E-cells that it does on other cells in the body: it alters the permeability of the E-cell membrane to glucose. The E-cell uses two glucose transporters; one of them (GLUT1) is activated by extracellular insulin, while the other (GLUT2) is downregulated. The activation of GLUT1 is a form of positive feedback, since the effect of the hormone on this carrier favors an increase in intracellular glucose, which promotes increased insulin secretion. Correspondingly, the downregulation of GLUT2 constitutes negative feedback. The combination of positive and negative feedback leads to oscillations in insulin secretion, which have been modeled by Maki and Keizer (1995) for an experimental system in which a steady flow of glucose solution was passed through a bed of cultured E-cells. The situation is more complex in vivo, because the response of the insulin-sensitive cells to changes in insulin levels affects the glucose concentration ambient to the E-cells. Figure 8.4 shows the feedback loops that are present in vivo.

Figure 8.4. Feedback between insulin-sensitive cells and pancreatic E-cells. Positive and negative feedback are indicated by “+” and “–”" on the arrows linking each control variable to the variable(s) it influences. The arrows containing a circle define the basic control system for the E-cell. Increases in glucose permeability act to reduce plasma glucose levels since more of the sugar is taken up by insulin-sensitive cells.

PRINCIPLES AND MODELS OF BIOLOGICAL TRANSPORT

317

The feedback system described by Figure 8.4 is designed to maintain a relatively stable level of plasma glucose in the face of external disturbances such as those occasioned by eating and exercise. In Chapter 10, we will see how feedback in epithelial cells allows them to maintain normal intracellular sodium levels while supporting a variable transcellular sodium flux. Another interesting feedback system, modeled in detail in Sherman et al. (2002), arises in hormone-activated endocrine cells because of the interaction between (1) the calcium-activated potassium conductance of the plasma membrane and (2) the nonlinear dependence of ER release channel conductance on cytosolic calcium; feedback in this case leads to oscillations in intracellular calcium and a corresponding pulsatile release of the endocrine secretion. In some cases, feedback systems are designed to be highly nonlinear, to increase the strength of a response or shorten its delay. Surely the most well-studied system of this kind is the feedback between membrane potential and channel-mediated membrane permeability that underlies the generation of action potentials in excitable tissue, which will be the subject of the next chapter.

8.3. REGULATION BY TRANSPORT In the first section of this chapter, we described several ways in which cells can regulate the activity of their resident transport systems. Through such regulation, the organism maintains the balance of water and electrolytes in the extracellular milieu, ensures the health of individual cells, and allows the organism and its component parts to carry out their functions and respond successfully to changes in the external environment and other stresses. The text that follows gives some examples of how biological transport processes interact and complement one another in achieving these objectives. There are numerous control loops in the body that employ a host of channels, carriers, pumps, and other pathways. The systemic regulation of blood glucose was described in the previous section; here we will focus on two regulatory processes at the cellular level: the maintenance of cell volume and osmolarity, and the regulation of intracellular pH.

8.3.1. Regulation of Cell Volume We saw in Chapter 6 that the cell is very sensitive to osmotic forces driven by differences between the osmolarity of the cytosol and that of the extracellular fluid. Such differences drive osmotic flows that can shrink or lyse the cell. For the cell volume to be stable, the cytosol must achieve the same tonicity as the extracellular fluid. It follows that the key variable determining cell volume is the number of osmols of solute in the cell: osmotic flows across the cell membrane will cease, and the cell volume will stabilize, when the internal osmols have been diluted to a tonicity essentially identical to that in the extracellular fluid. This relationship was summarized in Eq. (6.42).

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CH. 8: REGULATION AND FEEDBACK

Thus, the main goal of cell volume regulation is to maintain the solute content of the cell in a viable range. The maintenance of cell volume is critical to the cell's survival, and many transport systems play a role. The Na–K exchange pump is particularly useful, since it is found in all cells and it generates a net outward flux of solute by virtue of its 3Na:2K stoichiometry. Intracellular electroneutrality requires that the “excess” cation that leaves the cell during each pump cycle be balanced by the flux, via other pathways, of net positive charge into the cell. In the steady state, when the internal composition of the cell is unchanging, the net positive charge is carried by the equal and opposite flux of sodium into the cell, through passive or other active routes, less the corresponding potassium flux out. However, if the steady-state requirement is relaxed, as when the cell is responding to an external challenge to its volume control, some of the excess cation efflux is balanced by the flux of anion, generally chloride, in the same direction. This actually enhances the ability of the pump to reduce the solute content of the cell. Indeed, inhibiting the Na–K exchanger with ouabain can cause cells to swell and sometimes burst. Keener and Sneyd (1998) have presented an interesting model to describe the dependence of cell volume on the Na–K pump rate. The model assumes that ions cross the cell membrane passively and via the pump. The extracellular fluid (superscript “e”) contains a specified electrically neutral mix of sodium, potassium and chloride; i the intracellular fluid (superscript “i”) contains these ions and nI mols of negatively charged impermeant. Equivalent circuits are used to describe the ionic currents across the cell membrane: p

a

INa = INa + INa = GNa('\ – ENa) + 3p,

(8.17a)

IK = GK('\– EK) – 2p,

(8.17b)

ICl = GCl('\ – ECl).

(8.17c) 2

Here, the currents are directed outward and p is the number of pump cycles/cm -sec. The intracellular electroneutrality condition is i i cNa cKi  cCl

zI nIi , Vc

(8.18)

where zI is the average negative charge on an impermeant molecule and Vc is the volume of the cell. Thus, the last term in Eq. (8.18) is the intracellular concentration of impermeant charge. It is assumed that the cell has achieved a steady state (i.e., the intracellular composition is constant) and that osmotic balance has been achieved. As noted earlier, the steady-state assumption implies that the net current carried by each ion is zero; since chloride is not actively transported, it is in equilibrium across the cell membrane. Assuming ideal solutions, the osmotic balance condition is i i cNa cKi cCl

nIi e e .  cNa cKe cCl Vc

(8.19)

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In our derivation of the Donnan osmotic pressure in Chapter 6 [Eq. (6.40)], we neglected the contribution of the impermeant to the osmotic pressure of the phase in which it was contained; the assumption was that the impermeant charge resided on a relatively small number of macromolecules that contributed minimally to the osmolarity of the phase. Accordingly, the Donnan osmotic pressure was defined by only the concentrations of the mobile ions in the two phases. Here, it is assumed that each mol of impermeant in the cell generates the same contribution to the intracellular osmotic pressure as a mol of permeating ion. This additional term reflects the osmotic contribution of other anions in the cell, such as phosphates and bicarbonate. The pump rate, ionic conductances, extracellular composition, and intracellular impermeant content are specified. The preceding five equations can then be solved for the three intracellular concentrations, the cell membrane potential, and the cell volume [recall that the Nernst potential, needed for Eqs. (8.17), can be expressed in terms of the intracellular concentrations, since the extracellular concentrations are specified]. The solution for cell volume is particularly simple if we let zI = 1, that is, if we assume that the impermeant is a monovalent anion. Then the solution for cell volume is Vc 

nIi .      3ᑠp ¬­¯  2ᑠp ¬­¯ e ¡ e ž ž ­­° c K ¡1  exp ž ­° cNa ¡1  exp žž ¡ žŸ RTG ®­­° Ÿ RTGNa ®­°°± ¡¢ ¡¢ K °±

(8.20)

For this solution to be physically realistic, the cell volume must be positive. This requirement implies constraints on the ability of the Na–K pump to stabilize the cell volume in a specified extracellular milieu. To find out what these constraints are, we must examine the equation in more detail. i Since nI is specified positive, a physically realistic solution requires that the denominator of Eq. (8.20) be positive. The variation of the two terms in the denominator with pump rate is sketched in Figure 8.5. Because the sign of the exponent in each term is different, they behave differently: the exponential in the sodium term dee creases as p increases, and the sodium term asymptotically approaches cNa ; in contrast, the potassium term becomes exponentially more negative as the pump rate increases. Thus, the sign of the denominator depends on the sum of an asymptotic positive term and an exponential negative term. Inevitably, if the pump rate is large enough, the denominator will become negative. But, if the sodium term initially rises faster with p than the potassium term falls, there will be a range of p such that the denominator, and cell volume, will be positive. For this to be so, d(Na term)/dp|p=0 must exceed –d(K term)/dp|p=0; performing the differentiations and rearranging, this ree e quirement becomes: 3cNa /GNa > 2cK /GK. We will call this the necessary condition. Happily, this condition is ordinarily met, since the extracellular sodium concentration is substantially higher than that of potassium, and its conductance is generally lower. As shown in Figure 8.5, satisfaction of the necessary condition permits a positive cell volume over a limited range of pump rates, p < p*. As the pump rate increases, the sum of the sodium and potassium terms first increases from zero, but the denominator eventually decreases and crosses zero at p = p*. The cell will rupture

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when the pump rate is very small (for instance, when the pump is inhibited or the cell is dead) or approaches p*, and exhibits a minimum volume at an intermediate value of the pump rate.

Figure 8.5. Variation with pump rate, p, of (1) the two terms in the denominator of Eq. (8.20); (2) the denominator when the necessary condition (see text) is met; (3) cell volume. The units of Curves 1 and 2 are concentration; the unit of Curve 3 is volume.

As can be seen in Figure 8.6, many other transporters are involved in cell volume regulation. The potassium leak channels mentioned in Chapter 7 contribute to the high potassium conductance that allows the necessary condition to be satisfied. In physical terms, the most effective contribution of the Na–K pump to volume control is the removal of osmotically active sodium ions from the cell. Unfortunately (at least from a volume control point of view), sodium extrusion is accompanied by an active influx of potassium ion. The high potassium conductance of the cell membrane (accompanied by a low external potassium concentration) facilitates the potassium leak that prevents the intracellular level of this ion — and its contribution to the intracellular osmotic pressure — from rising any higher than it does. When cells swell under osmotic forces, two kinds of stretch-activated channels open to increase potassium conductance still further. These include mechanosensitive potassium channels and less selective channels that are permeable to a number of

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ions, including calcium. Calcium enters the cell and increases the open time of the Caactivated potassium channels in the cell membrane. Volume-regulated anion channels, which transport primarily chloride, are also activated by cell swelling. As ions cross the cell membrane, electroneutrality must be maintained, and the exit of cations from the cell is facilitated when the resistance to a neutralizing efflux of chloride is reduced.

Figure 8.6. A selection of transport systems involved in the regulation of cell volume. Systems counterclockwise from the Na–K pump act to reduce cell volume; systems clockwise from the pump act to increase cell volume.

The Na–K pump keeps the cells from swelling abnormally, but a delicate balance is needed to avoid harmful shrinkage as well. Secondary transport systems and exchangers play a role in this latter aspect of volume control by providing paths for the entry of osmotically active ions into the cell. Most symport systems bring sodium ions into the cell with each substrate molecule, and an increase in their transport rate raises the steady-state level of sodium in the cell; the chloride level increases concomitantly to maintain electroneutrality, further increasing intracellular osmotic pressure. We noted in Chapter 5 the presence of reciprocal stimulation in symport systems: when the cis concentration of the substrate is raised, the fluxes of the substrate and cosolute both increase. Indeed, increasing the amino acid concentrations ambient to hepato-

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cytes (the cells of the liver, which are important contributors to protein synthesis, and whose plasma membranes accordingly contain a variety of amino acid cotransporters) causes them to swell (Haussinger, 1998). Similar behavior has been seen in intestine and kidney (Baumgarten and Feher, 2001). The Na–H exchanger and the Cl–HCO3 exchanger are activated by cell shrinkage to carry sodium and chloride into the cell and hydrogen and bicarbonate out. Although they are electrically neutral, they are not osmotically neutral. The entering ions are osmotically active, but hydrogen and bicarbonate are buffered inside the cell to maintain the normal intracellular pH, and their loss does not diminish the internal osmotic pressure. The Na–K–2Cl cotransporter, which plays an important role in stabilizing cardiac cells and erythrocytes, is a potent opponent of cell shrinkage, delivering four osmotically active ions per cycle. It is activated by a volume-sensitive protein kinase and is turned off by a protein phosphatase that is inhibited by cyclic AMP (Lytle, 1998).

8.3.2. Regulation of Cell pH Like cell volume, maintenance of the proper cytosolic pH is essential to cell function and viability. The proteins responsible for cell function contain both acidic and basic groups, and their activity is strongly pH dependent; changes in the ionization status of charged portions of these molecules can lead to significant, and potentially undesirable, conformational changes. The conductance of ion channels is pH sensitive, and changes in intracellular pH induce dramatic changes in the excitability of nerve and muscle, and in the cell potential. The effectiveness of signaling molecules like cyclic AMP and calcium is also pH dependent; if the intracellular pH is too low, protons may displace calcium from its intracellular binding sites, inappropriately increasing the cytosolic levels of the latter ion. The normal cytosolic pH of mammalian cells is in the range of 7.0–7.2, and it is in this environment that the cellular proteins perform their intended functions. However, there are many mechanisms by which hydrogen ions can accumulate in the cytoplasm and lower the cell pH to harmful levels: — Hydrogen is produced when the carbon dioxide produced by aerobic metabolism dissolves in water, forming carbonic acid, which dissociates into protons and bicarbonate. — Hydrogen may leak in across the plasma membrane of the cell. The normal pH of the extracellular fluid is 7.4, but the cell potential of most cells is sufficiently negative that there is an electrochemical potential driving force for hydrogen entry across the cell membrane. — Hydrogen leaks into the cytosol from intracellular organelles, such as endosomes, lysosomes, and secretory vesicles, whose internal pH can be as low as 5. The high acidity of these organelles is maintained by primary proton pumps in their bounding membranes. Alkaline organelles, like mitochondria, actively extrude hydrogen into the cytosol.

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Excess protons are either pumped out of the cytosol or neutralized by bicarbonate to form carbon dioxide according to the reaction

ZZX CO 2 H 2 O . H HCO3 YZZ

(8.21)

Several secondary transport systems act to prevent the pH from becoming too low (see Fig. 8.7). An Na–H antiporter driven by the transmembrane electrochemical potential gradient of sodium carries one proton out of the cell for each sodium ion that enters. A more important secondary transport system, exemplifying both symport and antiport, uses the sodium potential gradient to pump bicarbonate into the cell, and chloride (and possibly hydrogen) out; the precise stoichiometry of this (Na + HCO3)–Cl exchanger is still uncertain. A simpler Na–HCO3 cotransporter has also been identified in renal and other cell membranes; the stoichiometry of this pump can vary, with one, two, or three bicarbonate ions entering the cell with each sodium ion. All of these transporters are regulated by feedback mechanisms that increase their activity when the cytosolic pH falls.

Figure 8.7. A selection of transport systems involved in the regulation of cell pH. Solid lines denote sources of acidification and dashed lines denote alkalinization; sodium, potassium, and chloride fluxes maintain electroneutrality. The influx of sodium and efflux of potassium is a consequence of the Na–K exchange pump in the cell membrane (not shown).

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Another important transport mechanism involved in pH regulation is the Cl–HCO3 exchanger. In normal operation, the exchanger carries bicarbonate out of the cell, driving Eq. (8.21) to the left, producing hydrogen ions, and decreasing intracellular pH. Thus, this exchanger acts in opposition to those described above, preventing the cytosol from becoming too alkaline. Another system that increases cytosolic acidity is an electrically neutral K–H antiporter, which is driven by the transmembrane difference in the electrochemical potential of potassium that is produced by the Na–K pump: cosolute potassium leaves the cell and substrate hydrogen enters. A K–Cl cotransporter found in the cell membrane of erythrocytes (Gibson et al., 2000) acts indirectly to acidify the cell: the symporter, driven by the potassium gradient, is directed outward; the accompanying decrease in the intracellular chloride concentration increases the driving force for the Cl–HCO3 exchanger and thus the rate of bicarbonate extrusion by the latter pathway. Like the acidity-reducing transport systems, the acidification systems are responsive to cellular pH. Decreases in pH increase the activity of the former, and decrease the activity of the latter. The sources of excess protons enumerated above — metabolism, passive transport across the cell membrane, and leakage from intracellular organelles — diminish as the cytosolic pH decreases.

Figure 8.8. Balance of intracellular proton sources and sinks. (a) Removal and neutralization of cytosolic protons by alkalinizing transport systems. (b) Influx and production of cytosolic protons by acidifying transport systems. (c) Metabolic, leakage, and organelle sources of cytosolic protons. (d) Combined sources of cytosolic protons. Curve d = Curve b + Curve c. The cytosolic pH is the pH at which the proton generation rate (Curve d) equals the proton loss rate (Curve a).

The interaction of the several components of pH regulation is shown in Figure 8.8. The sink term (Curve a) increases as the cytosol becomes more acidic, and the source terms (Curve d = Curve b + Curve c) increase as the cytosol becomes more alkaline. An equilibrium is reached at a desired pH where the source and sink rates are

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equal; i.e., protons leave and enter the cytosol at the same rate. Importantly, this equilibrium is stable: if, for instance, the cell becomes more alkaline than the set point, the source term then exceeds the sink term, and the pH returns to its normal value. Figure 8.8 can be used to illustrate how the steady cytosolic pH changes when a drug or other intervention is used to alter the activity of these regulatory transport systems; an example of this is given as an exercise. A number of transport systems, such as the Na–H and Cl–HCO3 exchangers, participate in the regulation of both cell volume and pH. Accordingly, the regulatory processes that control these critical cell properties are interrelated. An important difference between the two, noted in the previous subsection, is that the hydrogen and bicarbonate fluxes that are so important to pH control do not play a significant role in volume regulation.

PROBLEMS: CHAPTER 8 1.

Explain why, in Figure 8.1, “Composition” and “Fluxes” are connected by arrows in each direction. Explain how this can cause feedback between these variables.

2.

Use Eq. (8.2) to express the variation in intracellular cyclic AMP level with time following a step increase in hormone level from zero to cH. Assume that the binding reaction is rapid so that the number of receptors bound to agonist undergoes a step increase as well. Use your solution to calculate the time constant for the approach of the cAMP level to its steady state value. Explain why each of the terms appear in the expression for the time constant.

3.

(a) Using values of the extracellular and intracellular concentrations of sodium and calcium given in Problem 14 in the previous chapter, and an equivalent circuit representation of the driving force for ion flux, calculate the ratio of the driving forces for the passive transport of the two ions across the cell membrane when the cell potential is –60 mV. (b) At rest, the passive flux of calcium into the cell is much less than that of sodium. Explain why this is the case.

4.

Derive Eq. (8.6) from the preceding equations.

5.

Many cells are flat and can be modeled using the same plane-parallel geometry as that used to describe the cell membrane. Thus, the well-mixed condition for such a cell is formally the same as the quasisteadiness condition in a membrane, except that the diffusional time constant is that of the cell. Derive the well-mixed condition for a flat cell.

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6.

After an egg is fertilized, calcium is released into the cytoplasm from internal stores just below the cell membrane. How soon after Ca release is complete can the cytoplasm be regarded as well mixed? The diameter of the human ovum is about 150 Pm. Assume the temperature is 37ºC and that the cytoplasmic diffusion coefficient of calcium is half of that in water.

7.

Explain how Eq. (8.14b) follows from Eq. (8.10), using as a pattern the explanation of Eq. (8.14a) given in the indented paragraph following Eq. (8.14b).

8.

The compartmental analysis of intracellular calcium dynamics can be simplified by assuming that a constant fraction of the total cytosolic calcium is bound. Using this assumption, derive the appropriate differential equations to replace Eqs. (8.14).

9.

Cloherty et al. (1995) have developed and parameterized a compartmental model of sugar transport in erythrocytes. The model hypothesizes that sugar enters the cell through GLUT1 transporters, which deposit it in a submembrane compartment distinct from the cytosol. From there, it can exit directly into the cytosol or associate with a binding complex from which it can dissociate into the cytosol. (a) Construct a model of the sugar transport system analogous to Figure 8.3b. (b) Write the differential equations that describe the rate of change of the concentration of sugar in each compartment (submembrane, binding complex, cytosol), assuming that the GLUT1 transporters follow carrier kinetics. Assume that the fluxes between the binding complex and its neighboring compartments are proportional to the sugar concentration differences across the interfaces.

10. Solve the Keener–Sneyd model for cell volume control for an arbitrary value of zI, the average charge on an impermeant particle, introduced in Eq. (8.18). –17

11. Consider a spherical cell that contains 5 q 10 mols of negative charge, immersed in extracellular fluid containing 139 mM Na and 4 mM K. The sodium –14 –13 2 and potassium conductances of the cell are 1.4 q 10 and 7 q 10 mols/cm -smV. What is the Na–K pump rate that minimizes cell volume, and what is that minimum volume? 12. How would the constraints on the ability of the Na–K pump to stabilize the cell volume (i.e., the “necessary condition”) be altered if the cell contained net positive charge? Justify your answer. 13. In physical terms, why does the cell volume grow toward rupture as the pump rate approaches p*?

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14. Show how the curves in Figure 8.8 change if: (a) the mitochondria are inhibited so that the rate at which they extrude protons into the cytosol is reduced; (b) the Na–H antiporter is inhibited; (c) the K–H antiporter is inhibited. Assume that the Na and K fluxes that are inhibited in (b) and (c) pass instead through passive pathways that do not affect cell pH.

APPENDIX 8.1. NONEQUILIBRIUM BINDING IN COMPARTMENTAL ANALYSIS In this chapter, we illustrated compartmental analysis by using it to describe the transport of calcium across interfaces in the presence of intracellular equilibrium binding. In some cases, such as sugar transport in erythrocytes, binding cannot be assumed to be at equilibrium, and the algebraic relationships in Eqs. (8.11) and (8.13) must be replaced by rate equations. This we do here for Eq. (8.11); the replacement for Eq. (8.13) is found as it was in the equilibrium case. Assume a solution compartment containing substrate A at a concentration cA is in intimate contact with a compartment containing binding sites S whose total concentration is cS,T. There is no barrier resistance between the two compartments; indeed, the binding “compartment” can represent a collection of substrate-binding molecules dispersed throughout the solution. Let cS be the concentration of free binding sites in the “binding compartment,” and cAS be the concentration of binding sites that are occupied by substrate. Then, naturally, cS,T = cS + cAS .

(A8.1)

The rate of change of concentration in the solution compartment due only to its exchange with the binding compartment is  dc A ¬­ žž ­ Ÿž dt ®­

 k1c A cS k2 c AS ,

(A8.2)

binding exchange

where k1 and k2 are the second-order rate constant for binding and the first-order rate constant for dissociation, respectively. Substituting Eq. (A8.1) into Eq. (A8.2) to eliminate cS, and grouping terms,  dc A ¬­ žž ­ Ÿž dt ®­

 k1c A cS ,T c AS (k1c A k2 ) .

(A8.3)

binding exchange

It is easy to show that, when binding is at equilibrium and there is no further exchange, the Michaelis–Menten equation is recovered from Eq. (A8.3), with K = k2/k1.

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Equation (A8.3) provides a differential equation for cAS to replace the equivalent equilibrium equation, Eq. (8.11). If all concentrations are referred to the same volume, as would be the case if the binding “compartment” were dispersed throughout the solution, any change in cA due to binding causes an equal and opposite change in cAS: dcAS/dt = –(dcA/dt)binding exchange. Thus, dc AS  k1c A cS ,T  c AS ( k1c A k2 ) . dt

(A8.4)

9 EXCITABLE CELLS

INTRODUCTION The earlier chapters reviewed the many different transport processes that are found in living systems. Although every cell in the body possesses transport systems that allow it to take up nutrients and maintain its internal composition, these processes have been investigated more thoroughly in some cell types than in others. One class of cells that employs a large variety of transport mechanisms, and has been studied in great detail, is the class of excitable cells. Two prominent examples of excitable cells are the neuron and the muscle fiber, the constituent cells of nerve and muscle. If the potential across a portion of the nerve cell membrane is altered sufficiently from its resting value, this potential disturbance (the action potential) can propagate along the length of the cell, and is the basis of signal transmission in the nervous system. A similar disturbance propagating through a muscle fiber induces contraction. In this chapter, we take a serial approach to the function of these cells. Starting with the resting neuron, we review how the action potential originates and propagates. This description focuses on the electrical and electrodiffusive phenomena that accompany propagation. Then, we describe the molecular level events that are responsible for the unusual transport properties of the neuron that allow it to exhibit this highly nonlinear electrical behavior. The section on nerve concludes with a summary of the transport processes that take place during synaptic transmission, when one neuron communicates with a second neuron. Our discussion of the muscle fiber begins again at rest, and follows the chronology of muscle contraction: excitation by neurons at the neuromuscular junction, the transmission of an action potential through the muscle fiber, and the coupling of the action potential to muscle contraction. As with nerve, both the electrodiffusive and molecular aspects of these processes are described. Nerve and muscle have been the objects of considerable study, and this lone chapter is necessarily selective in content. The fourth edition of a classic volume by Aidley (1998) provides a fuller review of the field, with more experimental information and more detailed biology that is included here. The third edition of Keynes and Aidley's (2001) introductory text is a bit more recent, but less comprehensive. M.H. Friedman, Principles and Models of Biological Transport, DOI: 10.1007/978-0-387-79240-8_9, © Springer Science+Business Media, LLC 2008

329

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Figure 9.1. A myelinated neuron.

9.1. NERVE The nerve cell, or neuron, is responsible for the transmission and integration of signals 10 in the nervous system. In the human brain, there are 5 q 10 of them, interconnected in a complex network. The structure of a neuron is sketched in Figure 9.1. Inputs to the cell from other neurons or sensory receptors are received at the dendrites or on the cell body; in brain, a single neuron may collect the signals from as many as a thousand other nerve cells. These signals are integrated at the cell body and may prompt the neuron to send a signal of its own to the cells at which its nerve endings terminate. These receiving cells include other neurons, gland cells (which are stimulated to secrete), and muscle fibers (which are stimulated to contract). Thus, two kinds of signal transmission take place in nervous tissue: 1. Cellular, from the dendrites and cell body of a neuron to its nerve endings. Almost the entire path traveled by the nerve signal is along cells. The signal exists as a transient change in the potential difference across the cell membrane, called the action potential. The action potential is conducted from the cell body to the nerve endings through the slender axon (see Fig. 9.2). A segment of axon carrying an action potential is said to be excited. When the signal has to be carried a long distance, as from the end of a limb, the transmission path is almost entirely along axons. Nerve cells with axons up to 3 feet in length are found in man; the whale has axons that are more than 60 feet long!

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Figure 9.2. Potential profiles along a length of excited unmyelinated axon. A single action potential is shown at two times during its progress down the axon; t1 precedes t2. The cell potential at rest is '\r | –90 mV. The lower part of the figure shows the length of axon along which the action potential is propagating.

2. Transcellular, from the synaptic terminals at the nerve endings of one neuron, to the dendrites or cell body of the next. In contrast to the cellular mechanism, which is primarily electrical, the signal at the synapse is generally transmitted chemically, by means of compounds called neurotransmitters.

9.1.1. The Resting Neuron When it is not propagating an action potential, the neuron is not very different from other, nonexcitable cells. The cell body, or soma, is the primary biochemical factory for the entire cell; materials synthesized in the cell body must travel through the axoplasm (i.e., the cytoplasm in the axon) to reach the rest of the neuron. At rest, the interior of the neuron is ca. 90 mV negative with respect to its surroundings; this potential difference is higher than that across the plasma membranes of most nonexcitable cells. A sodium–potassium exchange pump in the nerve cell membrane creates a

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nonequilibrium distribution of these ions, as can be seen in Table 9.1. The intracellular composition is essentially unaltered by the transient fluxes that will be seen to accompany the propagation of the action potential. Other ions are also actively transported across the neuronal membrane, but their role in the propagation of the action potential is secondary to that of sodium and potassium.

Table 9.1. Ion Gradients across a Mammalian Axon

Species Na K Cl a

Axoplasmic concentration I ( ci ), mM

Extracellular concentration II ( ci ), mM

Nernst potential,a mV

142 4 103

62 –95 –87

14 140 4

Based on concentrations. The cell potential is ca. –90 mV. T = 37ºC.

The electrostatic potential inside the neuron is ca. 150 mV more negative than the Nernst potential for sodium, so the driving force for passive entry of this ion into the cell is enormous. For potassium, the interior of the cell is only 5 mV more positive than the Nernst potential, so there is a much smaller driving force for the passive flux of this ion, directed outward. The resting potential of the cell is much closer to the Nernst potential of potassium than it is to that of sodium because the potassium conductance of the resting cell membrane is much higher than that of sodium (recall Chap. 2). There also appears to be a small driving force for chloride, directed toward the extracellular phase. Considering the uncertainties in the intracellular activity coefficients of the ions, it is reasonable to consider chloride to be in equilibrium across the neuronal cell membrane. As discussed in Chapter 2, the membrane potential depends on: (1) the permeabilities (or mobilities) of the permeating ions; (2) their concentrations (more correctly, their activities, though concentrations will be used here) at the two sides of the cell membrane; and (3) the current carried by their passive fluxes across the membrane. In nerve, these fluxes pass through independent, ion-selective channels. When the neuron is at rest, it is in a steady state, and its intracellular composition is constant. Thus, the sum of the passive and active fluxes of any ion across the cell membrane must be zero: Jip + Jia = 0.

(9.1)

Multiplying each term in Eq. (9.1) by zi , and summing over all the ions, the following relation between the passive and active current densities at rest is obtained:

I p Ia  0 .

(9.2)

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We saw in Chapter 7 that Eq. (9.2) is also the open-circuit condition for a barrier containing passive and active pathways in parallel; thus, the open-circuit potential of the cell membrane and the resting potential of the cell are the same. The dependence of the cell potential on permeability, concentration, and current has historically been described in terms of either the constant-field equation (Chap. 2) or electrical analogs (Chap. 7). In both cases, the reasonable approximation is made that only the most abundant ions — sodium, potassium, and chloride — need be included in the equation for the potential. The resting potential given by the constant field equation [Eq. (2.56)] is

'\ r 

II U K cKII U Cl cClI RT U Na cNa , ln I ᑠ U Na cNa U K cKI U Cl cClII

(9.3)

where the subscript “r” denotes the resting state. As before, Phase I is inside the cell and Phase II is outside. As noted in Chapter 2, the permeability and mobility are related by ki = Ui RT/a, where a is membrane thickness. Since the proportionality constant relating permeability and mobility is the same for all ions, Eq. (9.3) can be written in terms of permeabilities, simply by replacing each U by the corresponding k. Equation (9.3) continues to be used to describe the potential difference across the nerve cell membrane, even though none of the conditions required for its validity are likely to be met: there is considerable evidence that the passive ion fluxes across the nerve cell membrane, which proceed through ion channels, do not satisfy the electrodiffusion equation; the total concentrations of diffusible ions at the two sides of the membrane are very different; and since the Na–K pump is rheogenic, the passive current is nonzero, by Eq. (9.2). The equation for resting potential obtained from electrical analogs was given in Chapter 7 [Eq. (7.46)]. It, too, is based on the electrodiffusion equation and the assumption that the passive current is zero. The reason that both of these equations have been so successful, their limited appropriateness to the true situation notwithstanding, is that they do correctly predict the qualitative response of the membrane potential to changes in ionic permeability. Much of the mechanism of membrane excitation and the action potential can be understood by applying the principle, presented in Chapter 7 and embodied in Eq. (7.46), that when the conductance of a particular ion increases, the membrane potential moves toward the Nernst potential of that ion. The constant-field equation predicts this behavior as well.

9.1.2. The Action Potential: Electrical Aspects The action potential is an extraordinarily important transport-related event. It is the language of the neuron, responsible for our actions and sensations, our thoughts and emotions. All action potentials propagated by a given neuron have the same amplitude; signals are coded by frequency modulation. The strength of a neuronal signal is the frequency with which action potential spikes pass down the axon to the nerve end-

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ings. Depending on the nerve fiber, signals can be coded at frequencies as low as one spike per second or as high as several hundred spikes per second. Hodgkin and Huxley's (1952, and references cited therein) pioneering work on the origin of the action potential was based largely on a series of elegant experiments using the giant axon of the squid. Their experiments and brilliant analysis earned them a Nobel Prize in 1963. The giant axon is ca. 0.5 mm in diameter, and it was selected because its size made it relatively convenient to study. The experimental results shown below are taken from Hodgkin and Huxley's work, and the details of the feedback process derived from these data are strictly applicable to only the squid axon. However, the basic mechanisms that underlie the action potential in squid axon also explain the excitability of vertebrate axons, nerve cell bodies, and muscle.

Figure 9.3. Time course of the action potential at a point.

Figure 9.2 presented a pair of snapshots of a propagating action potential; the abscissa was distance. A similar picture is obtained by plotting the potential as a function of time at a point on the membrane as the action potential passes by. This is done in Figure 9.3. Two aspects of the action potential, evident in these figures, require explanation: 1. For the action potential to propagate, there must be a mechanism by which resting axon immediately ahead of the spike in the potential is stimulated to undergo the potential changes shown in Figure 9.3. Since these changes are not spontaneous, the resting axonal membrane must somehow be excited by the approaching spike. How does this happen? 2. After the membrane potential begins to deviate from its resting value under the influence of the approaching action potential, what is responsible for the changes that follow?

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It is convenient to answer these questions superficially at first, and later in detail. As it approaches, the propagating potential spike drives a depolarizing current through the resting membrane. Excitation of the resting membrane occurs when the depolarization exceeds a threshold. Upon excitation, a feedback loop involving potential-dependent cation conductances is entered. This feedback loop causes the membrane potential to follow the course shown in Figure 9.3, thereby generating the current necessary to excite the next element of membrane.

Figure 9.4. The feedback process that drives the action potential. Step 1 is defined by the Hodgkin–Huxley equations. Step 2 is governed by the electrical analog of the axolemma, the cell membrane of the axon. Dashed lines show how the changing membrane potential can be computed from the constant-field equation (Step 2c).

The Membrane Action Potential: Resting State. We begin our analysis of the propagating action potential by considering the feedback process that answers the second of the two questions posed above. This process is easier to understand if we imagine it to take place in a patch of membrane that is isolated from the rest of the axon. Such a patch can be excited, and its potential will vary with time in a fashion similar to the profile in Figure 9.3. This dynamic potential is referred to as the membrane action potential. The feedback process that produces the membrane action potential is shown in Figure 9.4. Depolarization of the membrane patch causes the sodium and potassium conductances of the patch to change. These changes alter the membrane potential further, driving more changes in the membrane conductances. If the initial depolarization of the membrane exceeds a threshold value, then a potential spike results. As mentioned earlier and indicated in Figure 9.4, the dependence of the membrane potential on the transport properties of the ions has been analyzed in terms of mobilities, using the constant-field equation, and in terms of conductances and equivalent circuits. Hodgkin and Huxley's original work used the latter formulation, and we will too.

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Figure 9.5. Electrical analog of the axolemma. The sodium and potassium conductances are variable; the fixed leak conductance reflects the permeability of the membrane to chloride and other ions.

The electrical analog of the axolemma (i.e., the cell membrane of the axon) used by Hodgkin and Huxley is shown in Figure 9.5. Since voltage transients are of interest, the membrane capacitance, C, is included. A leak pathway, subscripted by “l”, is also included to represent the flux of ions other than sodium and potassium. The Na– K pump was absent from the Hodgkin–Huxley analog. According to this model, the current density across the axolemma is I C

d ('\ ) GNa ( '\  ENa ) GK ('\  EK ) Gl ( '\  El ) , dt

(9.4)

where '\(t) is the instantaneous membrane potential, the {Gi} are conductances, ENa and EK are the Nernst potentials of the cations, and El is the membrane potential at which the leak current is zero. Consider the membrane at rest in the steady state. Since the membrane potential is constant by definition in the steady state, the first term on the right-hand side of Eq. (9.4) is zero. Furthermore, since the membrane patch is isolated from the rest of the axon, there can be no current across it; if there were, the adjacent axoplasm would cease to be electrically neutral. So I = 0. Under these conditions, '\ becomes the resting potential, and Eq. (9.4) becomes 0 = GNa('\r – ENa) + GK('\r – EK) + Gl ('\r – El).

(9.5)

Solving for '\r, '\ r 

GNa ENa GK EK Gl El . GNa GK Gl

(9.6)

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As noted earlier, the ion fluxes across the membrane that accompany the action potential do not cause the composition of the axoplasm to change significantly. Thus, the Nernst potentials that appear in the preceding equations can be regarded as constant. Hodgkin and Huxley's Equations for the Dependence of Conductance on Membrane Potential. From their experimental data, Hodgkin and Huxley deduced that, during the course of the action potential, the variation of the potassium and sodium conductances of the axolemma could be described by a set of differential equations with potential-dependent coefficients: 4

3

GK = GK,max n , GNa = GNa,max m h,

(9.7a,b)

where dn  D n (1  n)  En n , dt

(9.7c)

dm  D m (1  m)  Em m , dt

(9.7d)

dh  D h (1  h )  Eh h , dt

(9.7e)

and the first-order rate constants {D} and {E} are functions of '\r – '\, as shown in Figure 9.6. Because of the form of the differential equations, the values of n, m, and h remain between zero and one. Thus, the ionic conductances cannot exceed the corresponding values of Gi,max. Hodgkin and Huxley (1952) also developed empirical expressions for the dependence of the rate constants on the difference between the resting potential and the instantaneous potential difference across the membrane, '\. Hodgkin and colleagues defined the resting potential as positive, with activation accompanied by a negative deviation from that potential (Hodgkin et al., 1952). Here we define the resting potential as negative, with activation generating a positive deviation. This distinction is important when using their empirical relations for the rate constants and in interpreting some of their figures.

An alternative formulation of Eqs. (9.7c-e) is dn nd  n ,  dt Wn

(9.7f)

dm md  m ,  dt Wm

(9.7g)

dh hd  h ,  Wh dt

(9.7h)

338

where nd 

CH. 9: EXCITABLE CELLS

Dn is the steady-state value of n at any fixed membrane potential, D n En

1 is a characteristic time constant, and similarly for m and h. Both of these D n En parameters are functions of the membrane potential, since the {D} and {E} are. The steady-state values and time constants of the activation parameters n, m, and h are plotted against membrane potential in Figure 9.7.

Wn 

Figure 9.6. Potential dependence of the rate constants in Eqs. (9.7c–e), for the giant axon of the squid. The solid lines are empirical correlations of the rate constant data. (a) Dn and En at 6ºC. Open symbols designate Dn data, filled symbols designate En data. The resting values of Dn and En are indicated on the figure. (b) Dm and Em. (c) Dh and Eh. Reprinted with permission from Hodgkin and Huxley (1952). Copyright © 1952, Cambridge University Press.

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Figure 9.7. (a) Steady-state values and (b) time constants of the activation parameters in Eqs. (9.7a,b). Note the abscissa is directed oppositely to Figure 9.6. Adapted from Keener and Sneyd (1998).

When the membrane is clamped at a fixed potential, the rate constants {D} and {E} that define the kinetics of the activation parameters are fixed, as are the steadystate values and time constants. In this case, Eqs. (9.7f–h) can be integrated from an initial value to give (e.g., for n)

n(t )  nd  (nd  no )et / Wn (voltage clamp),

(9.8)

where no is the initial value of n. If the initial condition is the resting state of the memD nr , where the rate constants are evaluated at the brane, then n0  nd ('\ r )  D nr Enr

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resting potential. Of course, the response of the activation parameters to a voltage clamp is very different from their variation during a normal action potential, where the membrane potential varies over a wide range, and Eq. (9.8) does not apply. However, voltage clamping was an extremely valuable technique used by the early investigators to understand the kinetics and voltage dependence of the ion conductances of the axon. Excitation of the Membrane Action Potential. The variation with time of the potential difference across the membrane patch is obtained by integrating Eqs. (9.4) (with I = 0) and (9.7c–e); these constitute four first-order ordinary differential equations in '\, n, m, and h. Equations (9.7a) and (9.7b) are used to express the sodium and potassium conductances in terms of n, m, and h. The membrane capacitance, Nernst potentials, leak conductance, and maximum conductances in Eqs. (9.7a) and (9.7b) are all constant.

Figure 9.8. Calculated response of the membrane potential of a patch of squid axon at 6ºC to a step depolarization, G, applied at t = 0. The values of G (in mV) are indicated on the curves. The threshold for initiating the membrane action potential lies between 6 and 7 mV. Reprinted with permission from Hodgkin and Huxley (1952). Copyright © 1952, Cambridge University Press.

The response of the membrane patch, initially at rest, to a depolarizing stimulus G is modeled by setting the initial value of the membrane potential equal to '\r + G, and integrating. The initial values of n, m, and h are the resting values. Either of two responses are obtained (Fig. 9.8): 1. If G is small enough, the membrane potential decays back to its resting value. 2. When G exceeds a threshold value, the depolarization rises rapidly and almost immediately to values in the neighborhood of 100 mV before the membrane potential returns to its resting value. This is the membrane action potential. The existence of a threshold for initiation of the membrane action potential is a consequence of the same nonlinear time-dependent conductances that underlie the action potential itself. The interested reader is referred to Weiss (1996b) for an illuminating discussion of how a system of nonlinear conductances can exhibit threshold behavior.

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Figure 9.9. Variation during the course of a propagating action potential of: (a) membrane potential and conductances; (b) ionic currents. Positive currents are directed inward; I is total current density across the axolemma. The axon temperature is 18.5ºC. Reprinted with permission from Hodgkin and Huxley (1952). Copyright © 1952, Cambridge University Press.

Once initiated, the membrane action potential exhibits a particular waveform that depends on the kinetics of the ion conductances and is independent of the size of the initiating stimulus. The transient changes in the sodium and potassium conductances of the membrane are shown in Figure 9.9a for a propagating action potential; the curves for the membrane action potential are similar. We will follow the time course of the action potential in terms of the changes in the activation parameters n, m and h (Fig. 9.7 will be useful in this discussion). Hodgkin and Huxley interpreted these parameters and their voltage dependence in terms of the gating kinetics of the voltagedependent cation channels that conduct the ions across the axolemma. We will discuss more recent kinetic models of these channels later in this chapter.

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The sharp initial depolarization of the membrane that follows suprathreshold excitation is caused by a rapid increase in GNa, which drives the membrane potential toward ENa.. In terms of the activation parameters, the sodium channels respond soonest to the excitation because of the short time constant of m (Fig. 9.7b). Indeed, these channels are often referred to as “fast sodium channels.” The exciting stimulus increases m (Fig. 9.7a), so more channels open and GNa rises. The effect of raising m is substantial because the resting value of m is very small. Furthermore, m becomes larger as the membrane depolarizes. This gives rise to a positive feedback loop, in which an increase in GNa causes depolarization, and depolarization causes m to rise, driving further increases in m and GNa. This feedback process is responsible for the rapid upstroke of the action potential. Eventually, the positive feedback becomes spent, as m approaches its limiting value of unity. At this point, h, which has been decreasing during the upstroke, becomes controlling. The reduction in h, driven by the potential-dependent drop in h and accelerated by the simultaneous decrease in the time constant Wh,, eventually causes GNa to diminish. This causes the depolarization to decrease as well. The voltage-gated potassium channels also open upon depolarization, as can be seen from the potential dependence of n in Figure 9.7a. However, they exhibit slower kinetics than the sodium channels, so the potassium conductance peaks well after the sodium conductance has begun its decline (Fig. 9.9a). These channels are often called delayed rectifiers because of the relative slowness of their response, and because their conductance changes with the membrane potential. The time constant for n is similar to that for h (Fig. 9.7b), so potassium conductance rises at a pace similar to that of the sodium conductance decay. The increase in GK accelerates the repolarization of the cell membrane, which tends to decrease n, opposing further increases in potassium conductance. Eventually GK reaches a peak, and then falls slowly. While the potassium conductance is still significant, GNa becomes so small that the membrane potential is dominated by GK. During this period, the membrane potential becomes more negative than the resting value (i.e., hyperpolarized), approaching EK. The steady-state value of n is low at these potentials and GK subsides further as the membrane approaches its resting state. Thus, depolarization causes both the sodium and potassium conductances to rise initially; however, the two conductances affect the membrane potential in different ways. Increases in sodium conductance are depolarizing, leading to the positive feedback noted earlier and a rapid increase in GNa. In contrast, increases in potassium conductance oppose depolarization, since EK is even more negative than the resting potential. This self-limiting negative feedback between depolarization and potassium conductance also retards the potassium conductance dynamics. The peak values of the ionic conductances can be read from Figure 9.9a. The values at rest {Gi,r} are harder to read because they are so small. The Hodgkin–Huxley 2 2 equations yield values of 0.25 mS/cm for GK,r and 0.003 mS/cm for GNa,r, which explains why the resting potential of the cell is so close to EK.. The numbers also illustrate the dynamic range of the excitation process: potassium conductance varies over a 4 50-fold range, and sodium over a factor of 10 !

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The transmembrane fluxes during the course of a propagating action potential are shown in Figure 9.9b. Their variation with time follows directly from the changes in conductances and potential described above. The Propagating Action Potential: Cable Theory. To understand how an action potential propagates down a nerve axon, we borrow a result from cable theory, so called because it was originally developed to describe the variation in time and space of currents and voltages in submarine cables. The analogy is not a bad one: insulated by the axolemma (and often a myelin sheath as well), the axoplasmic conductor is surrounded by interstitial fluid whose similarity to seawater recalls the environment in which life itself originated. Indeed, the analogy is even better for Hodgkin and Huxley, who used cold seawater to bathe the axons they harvested from coastal squid. Figure 9.10a shows a length of axon down which an action potential is propagating. The action potential has reached the neighborhood of z = 0, and the membrane potential elsewhere is near the resting value. In this case (and in contrast to the membrane action potential), the region experiencing the action potential is not isolated from the rest of the axon, and the transmembrane current is not zero. Cable theory teaches that the axial variation in membrane potential induces loop currents that pass through the active region and recross the axolemma through resting membrane. These loop currents depolarize adjacent axolemma, increasing its sodium conductance, and thereby cause the potential disturbance to propagate along the axon. The current loops in the axoplasm and external solution are driven by axial potential gradients inside and outside the cell. Inside the axon, the axial current is iI   I

1 s\ I , r I sz

(9.9a)

I

where i is a function of z, and r is the resistance of the axoplasm per unit length. Similarly, the axial loop current outside the axon is i II  

1 s\ II , r II sz

(9.9b)

II

where r is the resistance to current flow in the extracellular fluid. The axial gradient of membrane potential is obtained by rearrangement and subtraction: s\ I s\ II s( '\ ) s I   (\  \ II )  sz sz sz sz

r II i II  r I i I .

(9.10)

The current loops are closed; therefore, at any z, the current moving away from the site of the action potential in the axoplasm is equal to the current toward the site in the extracellular fluid: I II i = –i . (9.11) Substituting Eq. (9.11) into (9.10), s('\ )  (r I r II )i I . sz

(9.12)

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Figure 9.10. Propagation of the action potential along: (a) an unmyelinated neuron; (b) a myelinated neuron. The action potential is at z = 0 and is propagating to the right. Dashed lines in (a) illustrate Eq. (9.13). (c) Electrical analog of a unit length of axon, showing loop currents. In this simple model, the axolemma is described by a single resistance and capacitance in parallel. Note II that the axoplasmic potential becomes equal to '\(z) when the extracellular potential \ is set equal to zero.

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The membrane current density (im, in amperes per unit length of axon) at any value of z is proportional to the density of current loops crossing the membrane at that location. Outward-directed membrane current occurs at the expense of the axial axoplasmic current (Fig. 9.10a): im  

si I . sz

(9.13)

The classic cable equation is obtained by differentiating Eq. (9.12) with respect to z, and using Eq. (9.13) to express the result in terms of the membrane current:

s 2 ('\ )  (r I r II )im . 2 sz

(9.14)

Hodgkin and Huxley assumed that the extracellular resistance was negligible, relative II to the axoplasmic resistance. It is easy to show from Eq. (9.14) that, when r = 0, the membrane current density I, expressed in amperes per unit area of axolemma, is related to the axial variation in membrane potential by I

r s 2 ('\ ) , 2U sz 2

(9.15)

where r is the inner radius of the axon and U is the specific resistance (resistivity) of U the axoplasm. The needed substitutions are: r I  2 and im = 2SrI. Since the action Sr potential propagates at a constant velocity, the spatial derivative in Eq. (9.15) can be replaced by a time derivative: I

r 1 s 2 ('\) . ¸ 2U T2 st 2

(9.16)

In Eq. (9.16), T is the propagation, or conduction, velocity of the action potential. The variation with time of '\ at a point on the axon is found by integrating Eqs. (9.4) and (9.7c–e) as before, replacing I in Eq. (9.4) by the right-hand side of Eq. (9.16). In this application, the partial derivative in Eq. (9.16) can be replaced by the ordinary derivative d2('\)/dt2, since z is fixed. The empirical expressions for the dependence of the rate constants on membrane potential, referred to earlier, are used in the integration. There are a number of software packages available for simulating action potential propagation and other related phenomena: two recommendations are Neuron (http://neuron.duke.edu/) and HHsim (http://www.cs.cmu.edu/~dst/HHsim/). Note that cable theory does not stipulate that the action potential must propagate in a preferred direction. The potential spike proceeds toward the nerve endings (antegrade) only because it is initiated at the cell body. Retrograde propagation can be produced by electrically stimulating the axon in the neighborhood of the nerve terminals. Stimulation of the axon elsewhere along its length can trigger two simultaneous

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action potentials, one propagating antegrade to the nerve terminals, and the other retrograde toward the cell body. For any family of axons, characterized by a given set of conductance properties, there is only one value of the coefficient r / 2UT2 for which the solution for '\(t) converges. This implies that all action potentials in a given axon propagate at the same velocity. Furthermore, if we consider a collection of axons having the same channel conductance properties but differing in size, their propagation velocities should be directly proportional to the square root of the axon radius (and inversely proportional to the square root of the axoplasmic resistivity). This prediction is partially tested in Table 9.2, where the radii and conduction velocities of several nerve fibers, and the ratio T/r1/2, are presented. The ratios vary considerably among the fibers. This reflects differences in their potentialdependent conductances and associated rate constants, as well as differences in myelination (see below).

Table 9.2. Radii and Conduction Velocities of Some Neurons Description Motor fibers, skeletal muscle Typical myelinated axons Sensory fibers, pressure and touch Pressure and pain receptors Nonmyelinated pain fibers Typical nonmyelinated axons Squid giant axon

Radius, r, μm

Conduction velocity, T, m/s

T/r1/2, m1/2/s

7.5–10 10 6 2.5 2.5–5 1–2.5 0.25–0.5 0.75 0.5 250

50–120 120 70 30 30–70 5-25 0.5–2 2 1.3 25

28700 37900 28600 19000 25800 11300 2040 2310 1840 1580

Reference a b b b a a a b b b

a

Aidley (1998), based on midpoint of range of r and T. b Sperelakis (2001). The best-fit exponent, n, that minimizes the variation in T/rn among the myelinated fibers, is n = 1.07 (r2 = 0.96), indicating that, empirically, conduction velocity is more or less proportional to axon radius. Interestingly, when the nonmyelinated fibers (excluding the squid axon) are included in the correlation, a high 2 quality of fit can be still be obtained, with n = 1.50 and r = 0.97.

A nerve is a bundle of neurons, which vary to some extent in propagation velocity and excitation threshold. When a nerve receives a stimulus, each neuron that experiences suprathreshold depolarization delivers an action potential which propagates at the velocity that is characteristic for that neuron. The measured response of the nerve, called the compound action potential, is the sum of the responses of the individual neurons that comprise it. In contrast to the response of a single neuron, the response of a nerve is graded; that is, the amplitude of the response depends on the amplitude of the stimulus. This is a consequence of the variation of excitation thresholds among the neurons in the nerve: the stronger the stimulus, the larger the number of neurons that are excited, and the larger the number of action potentials that are summed. Similarly,

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the compound action potential is broader than a single action potential, because of the variation in propagation velocities among the excited neurons. Cable theory is a bridge between the physiological and electrical descriptions of action potential propagation, a bridge between electrophysiology and bioelectricity. The starting point for the electrical description of the neuron is shown in Figure 9.10c. Purely electrical modeling of conduction in nerve (and muscle) is outside the scope of this text; the interested reader is referred to an excellent volume by Plonsey and Barr (2000). The Role of Myelin. Many axons are sheathed by the lipid-rich membrane, myelin, as shown in Figure 9.1. The sheathing is laid down by specialized cells known as Schwann cells, which wrap around the axon and produce multiple layers of myelin. The myelin coat is interrupted periodically between successive Schwann cells, at the nodes of Ranvier. Myelin is an insulator whose presence around the axon increases the velocity at which the action potential propagates. As a result, while typical conduction velocities in unmyelinated nerves are between ½ and 2½ m/s, myelinated fibers can transmit signals at velocities up to 130 m/s (recall Table 9.2). Most large mammalian nerves are myelinated; the squid giant axon is not. The mechanism by which the myelin sheath works can be seen by referring to Figure 9.10b. The kinetics of the channels at the nodes of Ranvier are qualitatively similar to the kinetics of channels in the unmyelinated axons that were the subject of Hodgkin and Huxley's original work; as before, propagation results from the depolarization of resting membrane by current loops caused by the action potential. In myelinated nerve, the electrical resistance between the axoplasm and the extracellular fluid is much higher than in unmyelinated nerve, except at the nodes. Thus, the loop currents preferentially cross the membrane at the nodes, leapfrogging over the axolemma between the nodes; this is called saltatory conduction. Because of saltatory conduction, an action potential in a myelinated nerve can excite (nodal) axolemma farther ahead of it than can an action potential in an unmyelinated neuron of similar size. Indeed, the action potential at one node of a myelinated neuron can depolarize as many as five or six nodes in the direction of propagation. As a consequence, the propagation speed is much higher in the myelinated fiber. Since almost all of the action potential–generated current in myelinated nerve passes through the nodes, the current density at these sites is higher than that in unmyelinated nerve. This is reflected in a higher concentration of ion channels at the nodes: typically, there might be as many as 103–104 voltage-gated sodium channels in 2 2 1 μm of nodal membrane, compared to fewer than 100 channels/μm in the axolemma of an unmyelinated neuron. Table 9.3 gives representative sodium channel densities at various sites on typical mammalian neurons. Potassium channels in myelinated nerve exhibit a relatively higher internodal concentration, extending under the Schwann cell. Figure 9.11 demonstrates the distribution of Na and K channels at the node of Ranvier in rat optic nerve.

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Figure 9.11. Please see insert for color version Fluorescent micrograph of rat optic nerve labeled for potassium channels (blue), Caspr (red) and sodium channels (green). Caspr is a protein that marks the edge of the Schwann cell. The strong green signal highlights the highly concentrated sodium channels at the nodes of Ranvier, and the more diffuse blue signal demonstrates the presence of potassium channels in the internodal axolemma. Reprinted with permission from Rasband and Shrager (2000). Copyright © 2000, Blackwell Publishing.

Table 9.3. Sodium Channel Densities along the Length of a Typical Mammalian Neuron (Catterall, 1984) Site Cell body Axon, initial segment (hillock) Axon, unmyelinated Axon, myelinated, node of Ranvier Axon, myelinated, internodal Terminals

Channels/μm2 50–75 350–500 110 2,000–12,000 <25 20–75

There are some interesting differences between the conductance dynamics of the channels at the nodes, and those of the squid axon. For instance, in the node of Ranvier of the toad, there are two sodium conductances in parallel, presumably reflecting

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two different kinds of channels. The kinetics of each is described by an equation of the same form as Eqs. (9.7), but different in detail (Weiss, 1996b): 2

2

GNa,1 = GNa,1,max m h, GNa,2 = GNa,2,max p ,

(9.17)

where p follows the same kind of kinetics as n, m, and h. Another feature of the nodal channels is that the maximum conductances at the nodes can vary with membrane potential, which was not the case for the squid axon.

9.1.3. The Action Potential: Molecular Aspects Hodgkin and Huxley's Channel Model. Surely one of the most remarkable features of the Hodgkin–Huxley theory is the dependence of potassium and sodium conductance on membrane potential described by Eqs. (9.7) and Figure 9.6. It is this dependence that is ultimately responsible for the excitability of the neuron. Although the form of Eqs. (9.7) and the functions in Figure 9.6 were empirical, Hodgkin and Huxley did suggest a molecular mechanism that was consistent with them. Consider first the simpler equations that define potassium conductance. It was pointed out in Chapter 4 that most ion channels are gated and exhibit open and closed states. Assume that the number of potassium channels (NK) in a patch of membrane is such that, if all of them were open, the potassium conductance of the patch would be GK,max; that is,

GK,max = NK JK,

(9.18)

where JK is the conductance of a single potassium channel. The potassium channels (and the sodium channels too) are assumed to have but one open state, and it is further assumed that all potassium channels have the same unitary conductance when they are open. Then, the fraction of channels that are open at any instant is GK/GK,max, and the variation of membrane conductance with time actually measures the variation in this fraction. This ratio is also the probability that any particular potassium channel is open 4 at any instant. From Eq. (9.7a), GK/GK,max = n . What mechanism would give a probability of this form? Suppose the gate to each potassium channel is controlled by four independent activation sites, all facing one side of the membrane, Side A. The gate opens only when all four sites are occupied by four corresponding activator molecules associated with the channel. Assume that these molecules can migrate back and forth across the membrane, and are reversibly bound to their matching activation sites when they are at Side A. The probability of a particular binding site being occupied is equal to the probability that the activator molecule associated with that site is at Side A. If this probability is n, then the probability that the four sites controlling a single gate are occupied simultaneously is n4, the desired result. The rate equation for n can be related to the kinetics of the migration of the activator molecules. Call the other side of the membrane Side Not-A. Assuming the simplest kinetics for the movement of the activator molecules across the membrane, the (unidirectional) rate at which they cross from Side Not-A to Side A is proportional to

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the number of them at Side Not-A. Similarly, the rate of crossing from Side A to Side Not-A is proportional to the number of molecules at Side A. The rate of change of the number of activator molecules at Side A is the difference between the two unidirectional rates: dN A  AN N Not-A  BN N A , dt

(9.19)

where Ni is the number of activator molecules at Side i, and AN and BN are the unidirectional first-order rate constants. In terms of these variables, n, the fraction of activator at Side A, is NA/NT, where NT = NA + NNot–A is the total number of activator molecules in the patch. Clearly, NNot–A/NT = 1 – n; thus, when both sides of Eq. (9.19) are divided by NT, the result is Eq. (9.7c), with Dn = AN and En = BN. The dependence of Dn and En on the membrane potential (Fig. 9.6) is explained by supposing that the activator molecules are charged. Then the rates at which they cross the membrane depend on '\, and this is reflected in the rate constants.

Figure 9.12. Energy diagram for the movement of charged activator molecules across a potential barrier. The activator binding sites are at the outside of the barrier.

To get a sense of the effect of '\ on the movement of the activator molecules, imagine that the molecule must cross an energy barrier to move from one side of the membrane to the other, much as an ion did in Chapter 4 (Fig. 9.12). The molecule is at equilibrium across the membrane when dn/dt = 0, that is, when n = n. How does n depend on '\? The unidirectional rate constants that determine n are: Dn, which defines the activator flux toward Side A; and En, which defines the flux away from Side A. Suppose the activator molecules are positively charged. Since depolarization increases potassium conductance, and hence n, and also decreases the attraction of the axoplasm for

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the cationic activator molecules, we identify the outer surface of the axon with Side A. Therefore, from Eq. (4.11),  ᑠ'\ ¬­ , D n  (D n )'\0 exp žž žŸ 2 RT ®­­

(9.20a)

 ᑠ'\ ¬­ . En  (En )'\0 exp žž žŸ 2 RT ®­­

(9.20b)

and

This behavior is qualitatively similar to that in Fig. 9.6a. We saw earlier that Dn nd  ; using Eqs. (9.20), D n En

1

nd  1

(En )'\0 (D n )'\0

 ᑠ'\ ¬­ exp žž žŸ RT ®­­

.

(9.21)

Note that: (1) depolarization decreases the exponent, raising the steady-state value of n; and (2) n cannot exceed unity. This simple energy barrier model correctly predicts the qualitative dependence on membrane potential of both the rate constants and the steady-state values of the activation parameter that determines potassium conductance. Hodgkin and Huxley used empirical expressions of a similar form to correlate their experimental measurements of these parameters. The explanation of the sodium conductance in terms of activation molecules is somewhat more complex, since two variables, m and h, require explanation. In this case, two controlling species are hypothesized. The gate is controlled by three activation sites and one inactivation site, and opens only when the three activation sites are all occupied and the inactivation site is not. The activation and inactivation sites are presumed to be independent and to bind different species. In Eq. (9.7b), m is the fraction of activating molecules at the activation side of the membrane (like n in the potassium model), and h is the fraction of inactivating molecules not at the inactivation side. The dynamics of the potassium and sodium conductances are also consistent with a model in which the channels are controlled by multiple independent gates, whose kinetics are equivalent to those of the activator molecules postulated above. According to this interpretation, the potassium channel is controlled by four gates and passes ions only when all four are open (in contrast, the channel in the original model was controlled by one gate). Similarly, the sodium channel is open when the three activation gates are open and the channel is not inactivated by closure of the h-gate. A third kinetically equivalent model supposes that the channel protein consists of four subunits, all of which must be in the appropriate conformation for the channel to conduct. We will use the terms “gating” and “gates” in the discussion to follow to describe any of these alternative but kinetically indistinguishable processes.

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Gating-type models are readily described by state diagrams like those used in Chapter 4. The rate constants for the transitions between states are equivalent to the AN and BN of Eq. (9.19). In its most general form, a system of four distinguishable gates, 4 each of which can be open or closed, leads to 2 = 16 distinct states (only one of which — OOOO — is open). The construction of the state diagram for such a system is left as an exercise. State diagrams have been used to extend the Hodgkin–Huxley concept of cation channel gating in nerve to include the possibility that the rate constants of gates of a particular kind (i.e., the four n-gates of the potassium channel or the three m-gates of the sodium channel) are distinct. The rate constants may also depend on whether other gates are open or closed, leading to cooperativity effects. As one example, in some treatments of inactivation kinetics using the Hodgkin–Huxley model, it has been assumed that the rate constant for closure of the h-gate (i.e., inactivation) depends on the state of the m-gates. In particular, the value of Eh increases as more of the m-gates controlling a given channel open, and the channel becomes increasingly susceptible to inactivation. The rate constant for closure of the inactivation gate is highest when the channel is open. When Eh is small except when the channel is open, the channel exhibits a single inactivated state that, for all practical purposes, can be reached only from the open state. This is indicated in Figure 9.13.

Figure 9.13. A four-gate model of the fast sodium channel. The rate constants for the m-gates are Dm and Em; the rate constants for the h-gate are Dh and Eh. The rate constants are multiplied by the number of kinetically indistinguishable gates that can undergo the indicated transition; for instance, consider the second state on the top row, which represents all (kinetically indistinguishable) states in which two m-gates are closed and the h-gate is open. For the transition to the third state on the top row, either of the two closed m-gates may open, so the effective rate constant for a single opening is twice Dm. Depolarization drives the conformation of the gates from that at the upper left toward the open conformation at the upper right; the kinetically preferred sequence of states is indicated by heavy arrows. In contrast to the Hodgkin–Huxley model of sodium channel kinetics, the m-gates open serially as the channel progresses from rest toward the open state. This three-step process causes a lag between the application of a depolarizing voltage and the activation of sodium conductance. The open arrowheads describe inactivation from the open state, and recovery. Modified from French and Horn (1983).

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The original kinetic model employed by Hodgkin and Huxley is recovered when the n- and m-gates are kinetically indistinguishable and their kinetics are independent of the status of the other gates. The potassium channel state diagram reduces to five states in series (0, 1, 2, 3, and 4 gates open); the sodium channel diagram reduces to eight states, and is shown in Figure 9.13. In state diagrams, transitions between states generally represent only a single event, such as the opening or closing of a single gate. However, complex molecules like channel proteins can undergo conformational changes in which multiple transitions are tightly coupled. Such motions have been considered in more recent models, to which we now turn. More Recent Models of the Potassium and Sodium Channels of Nerve. Although the preceding models are consistent with the equations that Hodgkin and Huxley derived from their data, the authors never claimed that their original proposal was the only mechanism that could explain the observed conductance changes. We have 4 3 already noted that a variety of mechanisms can reproduce the n and m h dependence of conductance that they found. Similarly, the dependence of the activation parameters — whatever their physical correlates in real channels may be — on membrane potential can be explained in several ways. For instance, the potential dependence of the rates at which hypothetical charged activator molecules shuttle across the membrane can also reflect the dynamics of a charge-containing channel-forming protein, or gating subunits associated with the pore, whose conformation responds appropriately to the electric field generated by the membrane potential. This dependence can result from the charge distribution, or the presence of permanent or inducible dipoles, on the macromolecule. Thus, the Hodgkin–Huxley model does an excellent job of describing the Hodgkin–Huxley data, and the equations they used can readily be interpreted in terms of our current understanding of the molecular dynamics of channel-forming proteins. Most of the more recent models of cation channel kinetics in nerve were developed to explain new data obtained using later techniques such as voltage clamping and noise analysis, and from more refined measurements of (1) gating current and charge and (2) the time course of activation. The potassium channel. An attractive feature of the Hodgkin–Huxley model of the potassium channel is the notion that channel opening is controlled by four identical activator molecules (or their contemporary equivalents), which could presumably reflect the four identical subunits that comprise the channel. This symmetry is retained a later model proposed by Zagotta et al. (1994). They suggest that four gates must open for the channel to open (presumably one gate in each subunit), equivalent to the Hodgkin–Huxley model, but that the opening of each gate is a two-step process (R1 o R2 o A), with different rate constants for each step (but the same for each gate). The channel opens when all four gates are in the A-state. Once the channel is open, it can close in either of two ways: by transitioning to a closed state (Cf) different from those through which the channel passed en route to opening, or through the reversion of one of the gates to its R2 state. When the channel closes by the latter mechanism, the rate constant for that particular closure is less (by a factor T = 9.4) than that for any other A o R2 transition; this is the only cooperative effect in the model.

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Figure 9.14. State diagram for the Zagotta–Hoshi–Aldrich (1994) model of potassium channel opening. The authors modeled the behavior of a Shaker potassium channel from which the inactivation gate had been removed, so inactivation is absent from the state diagram. Each state is defined by two numbers: the number of gates in the R1 state and the number in the R2 state; the number in the A state is obtained by difference; N A  4  N R1  N R2 . Note that the numeral “0” is used here, not the letter “O”; the open state is “00” because none of the gates are in either of the R states. A few illustrative rate constants are given; the rest are left to the student. The Shaker potassium channel is one of the most commonly studied ion channels. Its proper function is dependent on the Shaker gene, so called because a mutation in this gene in the fruit fly, Drosophila, causes the fly's legs to shake uncontrollably as a result of defective neurotransmission.

Even when all gates are identical, the Zagotta–Hoshi–Aldrich (ZHA) model contains 14 closed states, plus the Cf state. The state diagram is shown in Figure 9.14. A later model by Zheng and Sigworth (1998) built on the ZHA model by including subconductance levels that pass ions but are partially gated; these states, whose conductances are less than that of the fully open state, had earlier been proposed by Chapman et al. (1997) on the basis of single channel recordings. Zheng and Sigworth suggested that these levels could be the 01 (three gates open) or 02 (two gates open) states of the ZHA model. They proposed that when the protein enters the 00, 01, or 02 states, it is briefly nonconducting before transitioning to its corresponding — and distinct — conducting state. Recent experiments by Chapman and vanDongen (2005) strongly support Chapman's earlier hypothesis; in their channel model, the subconductance levels were associated with the 02 and 03 states in the ZHA model, in which only one or two gates are open. They also interpret the R1 o R2 transition as the voltage activation of the subunit via its S4 voltage sensing domain, followed by the opening of the gate, R2 o A; the gate cannot open unless the subunit is activated. This is consistent with the observation of gating currents prior to an increase in channel conductance, indicating that charge reorientation precedes the creation of an open channel. The state of knowledge regarding the voltage gating of potassium channels was reviewed by Bezanilla (2005) in a commentary on the Chapman and vanDongen paper.

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Figure 9.15. State model of a sodium channel in nerve. The sequence of steps normally followed by a sodium channel during the passage of an action potential is indicated numerically. Note that all transitions are reversible. Adapted from Patlak (1991).

The sodium channel. More recent studies of the kinetics of the voltage-activated sodium channels of nerve and muscle provide alternative explanations of the conductance behavior observed by Hodgkin and Huxley. Like subsequent models of the potassium channel, they depart from the notion of activator molecules and focus instead on the sequence of closed and open states through which the channels pass. Many of these models have been reviewed by Patlak (1991) and by Keynes (1994), who included a sodium-channel model of his own, and similar models are found in a more recent review by Ulbricht (2005). The models share some common features, some of which are also shared by the potassium channel models. Because the sodium channel domains are not all identical, fourfold symmetry is not generally invoked. Characteristic features of more recent sodium channel models are illustrated in Figure 9.15 and are summarized below: 1. A sequence of nonconducting closed states (C0, ..., C3), in which at least one activation gate is closed, and through which the channel must pass before opening. The state in which all gates are closed is presumably the favored state at rest. 2. An open state (O), in which all gates are open and the pore is clear. This is the state that a closed sodium channel rapidly (in about 0.25 msec) converts to upon depolarization, during the upstroke of the action potential. The channel closes if the cell is repolarized promptly. 3. An inactivated state (I1) that can be entered from the open state, in which the pore no longer passes ions, but the gates remain open. The open state inactivates when depolarization is maintained; the time constant for this process can be as much as 3 ms. The I1 state is thermodynamically favored over the open state while the membrane is depolarized, which is why open channels eventually inactivate. Inactivation appears to take place when a segment of the channel protein, different from the gating sequence, blocks the pore; this segment is often termed the inactivation gate.

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4. Upon repolarization, the closed states are preferred to the open state and the thermodynamically preferred transition from State I1 is to one of the closed states (C3 or C2) that originally preceded channel opening. As shown in the figure, this transition can proceed through a second inactivated state (I2); in this illustration, I2 transitions to C3. The I2–C3 transition is reversible, so the channel can inactivate without opening. Some comments on this last step: (1) When the inactivated state must transition to a closed state, models such as C1 l ... l Cn l O l I, in which there is a single sequence of states, and which therefore require that inactivated sites reopen before closing, are excluded. The existence of an inactive state accessible from a closed state becomes more reasonable when it is recalled that, in State C3, for instance, three of the activation gates have already moved out of the way, potentially leaving enough room for the inactivation gate to block the pore. (2) Three of the activation gates are open in State I2, since it is derived from State C3; likewise, all gates are open in State I1. The existence of a transition between the two inactivated states implies that the blocking segment can prevent conduction while allowing a single gate to close and reopen.

The time course of the sodium conductance in Figure 9.9a can be explained by the preceding model. Most of the sodium channels are closed at rest. Excitation causes some of them to open, increasing macroscopic sodium conductance and flux, and causing more depolarization, which causes more channels to open. Two mechanisms drive the subsequent reduction in GNa.. After less than 1 ms, the number of channels that have become inactive is sufficient to cause the sodium conductance to fall. In addition, sodium channels that open late may close their gates in response to GK-induced repolarization, before they even have time to inactivate. The state behavior of the sodium channel is consistent with what is known of its structure and amino acid sequence. The protein consists of four subunit domains that span the bilayer and, as noted earlier, are not identical. Negatively charged portions of each domain extend into the channel to form a narrow selectivity filter. The gates are activated by a sequence of positively charged amino acids in the S4 segments that sense the local electric field. Although the sensors in each domain presumably respond independently to changes in potential, all must respond for the gate to open; the similarity of this process to the model in Figure 9.15 is evident. The inactivation gate is believed to be located near the cytosolic end of the protein, because it can be destroyed by injecting proteolytic enzymes into the cell. It is tempting to associate this region with the inactivation site in the Hodgkin–Huxley conductance model. A cartoon of the channel in three of its states is shown in Figure 9.16a. 3 Since the domains of the sodium channel differ, the m h dependence of the Hodgkin–Huxley model could reflect the existence of three domains that only activate and one that activates and also contains the inactivation gate (Fig. 9.16a). This situation was modeled by Chanda and Bezanilla (2002) for the sodium channel in rat skeletal muscle. In their model, there are two kinds of domains (the notation here is changed somewhat from that used by Chanda and Bezanilla):

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Figure 9.16. (a) Three states of the sodium channel: resting (C0), open (O), and inactivated (I1). Only two domains (the B and D domains) are shown, indicated by the labeling of the voltage sensors. The inactivation gate is placed on the fourth, or D, domain. [Reprinted with permission from Guy (1988). Copyright © 1988, Oxford University Press.] (b) The ball-and-chain model of inactivation. Here, the receptor site is formed when the channel opens. [Adapted from Levinson and Sather (2001).]

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— three activate-only domains, which are equivalent in the state diagram and will be referred to collectively as the E-domains; and — the remaining domain that contains the inactivation gate and will be referred to as the I-domain. A fundamental hypothesis of the model is that the I-domain can inactivate the channel only when the I-domain is activated, and it cannot activate until one of the E-domains has been activated.

Figure 9.17. State diagram of a sodium channel in skeletal muscle. Each state is defined by a sequence of E's and I's, indicating which domains are activated, followed by C/I/O, indicating whether the channel is closed, inactivated, or open. In the original model the kinetics of the activation of the individual E-domains differ, so there is a preferred route through the diagram to the open states, and thence to the inactivated state. Here, distinctions among the E-domains are neglected for simplicity. The number of equivalent states (again neglecting quantitative differences in kinetics) are shown in parentheses for each state in the figure. For instance, the EI–C state is a closed state in which one E-domain and the I-domain are activated; there are three equivalent states because any one of the three E-domains may be activated. The domain-dependent kinetics of the several activation steps favors the path shown with the heavier arrows. Adapted from Chanda and Bezanilla (2002).

At rest, none of the domains have yet been activated by depolarization, and the channel is closed. The channel has two open states: State O1, in which only the three E-domains are activated, and State O2, in which all four domains are activated. The model operates as follows; see Fig. 9.17: 1. The first response of the channel is activation of one of the three Edomains. At this point, the channel can be in any one of three closed states, labeled E-C 2. Activation of the first E-domain permits the I-domain to activate, so the next domain to activate can be any one of the three still at rest. When the second activation step is complete, there are six possible states, three closed states (EI-C) in which an E-domain and the I-domain are activated,

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and three closed states (EE-C) in which two E-domains are activated. Since the I-domain is activated in the former set of states, a channel that finds itself in one of these states may inactivate from that state, which is still closed; the three resulting inactivated states are labeled EI-I. 3. A third domain activates next. Channels in the EI-I state remain inactivated and enter one of three EEI-I states. When the third domain to activate in an EE-C channel is the I domain, or if another E-domain activates in an EI-C channel before it has time to inactivate, the channel enters one of three EEI-C states, which can inactivate to the corresponding EEI-I state. When the third domain to activate in an EE-C channel is the remaining E domain, the channel enters the EEE-O1 open state. 4. The last domain activates. If the channel is initially in the EEE-O1 state or still in the EEI-C state, it converts to the fully activated open EEEI-O2 state; if it has inactivated, it remains inactivated in the EEEI-I state. 5. The open EEEI-O2 channel eventually inactivates to the EEEI-I state. An intriguing physical explanation of inactivation, for both sodium and potassium channels, is the ball-and-chain model (Armstrong and Bezanilla, 1977). This model, illustrated in Figure 9.16b, proposes that the open channel is blocked when a globular protein sequence, tethered to the channel by a chain of amino acids, swings into the pore from the cytoplasmic side and binds to a receptor. The protein “ball” serves as the inactivation gate. The receptor site may form when the channel opens, or the site may be preexisting. We have seen that, in more recent models, partially activated closed channels, as well as fully open ones, can inactivate. It was recently shown (Oliver et al., 2004) that channel inactivation can be inhibited by negatively charged phospholipids in the cytosolic leaflet of the plasma membrane; evidently the lipids interact electrostatically with the positively charged ball, retarding its movement toward the receptor site. The response of the sodium channel to changes in potential, including inactivation and reactivation, can be modeled by the differential equations of chemical kinetics, as was the movement of activator molecules in Eq. (9.19). Because of the large number of channels in a patch of membrane, the fraction that are open at any instant can be expressed using the rules of probability, as in Hodgkin and Huxley, or using a kinetic description based on mass action principles, as will be illustrated in the next section for the acetylcholine receptor.

9.1.4. Synaptic Transmission The passage of a nerve signal from one neuron to the next is accomplished at the synapse. The large majority of synapses employ a chemical transport process, and this mechanism will be described first. An alternative mechanism, involving direct electrical connection between neurons, will be discussed briefly at the end of this subsection. A diagram of a chemical synapse is shown in Figure 9.18. Synapses can occur between pairs of dendrites, axons, and even the cell bodies of two neurons, but we will focus here on the more common transmission between the nerve endings at the far end

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of the axon and the dendrites or cell body of the neuron receiving the signal. When an action potential reaches the nerve endings of the presynaptic neuron, a chemical neurotransmitter synthesized in the terminal region is released by exocytosis (Fig. 9.18b) into the small (20–100 nm) cleft that separates the presynaptic neuron from the dendrites or cell body of the postsynaptic neuron. This is called excitation–secretion coupling. Each synapse employs only a single transmitter. The release process is calcium dependent (Chap. 3) and is initiated when the depolarizing action potential opens voltage-dependent calcium channels in the terminal membrane, leading to an influx of calcium into the terminal cytosol. The process is expedited by the close proximity of the clustered calcium channels and neurotransmitter-loaded vesicles, and the proximity of the vesicles to the presynaptic membrane.

Figure 9.18. A chemical synapse. Panel (b) is a magnification of the cross-hatched region in (a). Reprinted with permission from Eccles (1965). Copyright © 1965, Scientific American.

Since exocytosis is a vesicular transport process, neurotransmitter is released in packets; the release is said to be quantized. The number of quanta that are released at a synapse in response to a single propagation down the presynaptic neuron depends on (1) the free calcium concentration that develops in the cytosol at the release sites in the presynaptic membrane upon depolarization, and (2) the availability of the neuro-

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transmitter. The latter is a function of (a) the rates at which the transmitter is synthesized, stored in vesicles, and reabsorbed by the cell after release (see below), and (b) the proximity of the vesicles to the presynaptic membrane at the time the calcium channels open. Neurotransmitter release is also modulated by other chemicals acting on the presynaptic membrane, as described below.

Figure 9.19. Recycling of neurotransmitter vesicles in the synaptic terminal. Adapted from Suszkiw (2001).

The recycling of vesicle membrane is outlined in Figure 9.19. After the vesicle has emptied at the release site (Step 1 in the figure), it collapses (Step 2) into the presynaptic membrane. The patch of vesicular membrane retains its identity and diffuses laterally away from the active zone where the release sites are. Distant from the release sites, is reforms as an endocytic vesicle (Step 3) and reenters the terminal cytosol (Step 4), where it is refilled with newly synthesized transmitter (Step 5) and returns to the release site (Step 6). The half-life of these vesicles is more than a week; they are replaced by transport down the axon (axoplasmic transport) of new ones synthesized in the cell body. Excitation–secretion coupling is regulated by an interaction between the voltage-sensitive calcium channels and calcium-activated potassium channels in the nerve terminal membrane. When calcium enters the nerve endings to stimulate exocytosis, it also increases the conductance of the calcium-dependent potassium channels. The increased potassium conductance has a hyperpolarizing effect, decreasing the conductance of the calcium channels. We saw in Chapter 8 that a similar feedback mechanism based on ATP-sensitive potassium channels is used by the pancreas to regulate insulin secretion.

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The neurotransmitter diffuses across the synaptic cleft to the postsynaptic neuron. The surfaces (the postsynaptic membranes) of the dendrites and soma of the postsynaptic cell directly across from the active zone contain receptor proteins that are responsive to particular neurotransmitters. In some cases, binding of the neurotransmitter by the postsynaptic receptor is rapidly (<0.1–1 ms) followed by a transient change (usually an increase) in the local conductance of the postsynaptic membrane, the result of a conformational change of an ion channel closely associated with the receptor. This is known as fast synaptic transmission. In slow synaptic transmission, G proteins are used by the receptor to signal selected channels elsewhere in the membrane to open. The activated G protein can act directly on the channel or through intracellular second messengers such as cyclic AMP. The second messengers and their associated kinases can influence multiple transport molecules in the postsynaptic neuron, including active transport systems, ion channels, and receptors for other neurotransmitters. Synapses that use G protein-mediated mechanisms respond more slowly to the presynaptic stimulus, with time delays (latencies) in the tens of milliseconds and responses that can extend over seconds or minutes. The distinction between fast and slow transmission is also seen during excitation– secretion coupling. Where the synapse is fast, neurotransmitter-laden vesicles dock at the vesicular grid (Fig. 9.18) near the calcium channels in the presynaptic membrane. Thus, they sense an increase in local calcium concentration as soon as the channels open, and release their contents promptly. Even so, transmitter release is the slowest step in the fast synapse, contributing most of the 0.3–0.5 ms delay that precedes a response in the postsynaptic neuron. Release is complete in about 1 ms. In slow synapses, the vesicles are farther from the cleft and have to migrate through the cytoplasm to reach the presynaptic membrane; furthermore, they are not exposed to the high calcium levels adjacent to the channels, and must wait until the bulk cytoplasmic calcium concentration reaches a level sufficient to prompt exocytosis.

The receptors that are closely associated with channels and participate in fast transmission are termed ionotropic, while those that participate in slow transmission and use G proteins are called metabotropic. Many neurotransmitters, including acetylcholine, J-aminobutyric acid (GABA), and serotonin, bind to receptors of both kinds, while others, such as norepinephrine and dopamine, bind only to metabotropic receptors. Synapses — and their associated neurotransmitters — may be either excitatory or inhibitory. Excitatory synapses transiently raise the sodium conductance of the postsynaptic membrane, causing local depolarization (GK is raised as well, but the sodium effect dominates). The change in potential caused by a single synapse is called the excitatory postsynaptic potential (EPSP); in brain, it is up to 2 mV in amplitude and a few ms in duration. This local potential change spreads passively along the neuronal membrane, away from the postsynaptic region. Most inhibitory synapses raise the chloride conductance; this has a hyperpolarizing effect but the effect is small, since chloride is close to equilibrium across the resting cell membrane. The increase in chloride conductance does reduce the depolarizing effect of increases in sodium conductance caused by excitatory synapses elsewhere on the neuron, and in this way it

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opposes action potential initiation. Other inhibitory synapses increase potassium conductance and produce larger hyperpolarizing inhibitory postsynaptic potentials (IPSPs). The size of either postsynaptic potential depends on (among other things) the number of quanta that are released by the presynaptic neuron. Each synapse, in its own small way, affects the membrane potential of the postsynaptic neuron. If the interval between the action potentials traveling down a particular presynaptic neuron is briefer than the duration of the postsynaptic potential they individually elicit, then the effects of multiple action potentials at that synapse are to some extent additive. This summation converts the frequency-coded information in the presynaptic axon into amplitude-coded depolarization of the postsynaptic cell membrane. The postsynaptic neuron generally receives stimuli at multiple sites, from multiple neurons. Consequently, the input to the cell consists of a collection of transient postsynaptic potentials separated in space and time. Each of these potentials spreads out from its origin, decaying over time, and its peak amplitude at any location diminishes with distance. The spatial and temporal characteristics of this electrotonic potential spread depend on the capacitance and resistance of the cell membrane, and the resistivity of the cytosol of the postsynaptic neuron. As a consequence, the depolarization of the postsynaptic cell membrane varies with time, and with location. If, at any time, any excitable site on the cell is depolarized beyond its excitation threshold, then one or more action potentials will originate at that location. In practice, the action potentials almost always start at the axon hillock (Fig. 9.1), because it has the lowest threshold for excitation. Some of these considerations are illustrated in Figure 9.20. Neurotransmitters are biologically powerful compounds. The geometry of the synapse, where only a narrow cleft separates the two neurons, limits the unintended diffusion of these chemicals to nearby cells. Unintended neurotransmission is further limited by processes originating in the synaptic membranes. For many neurotransmitters, a high-affinity Na cotransport system in the presynaptic membrane reabsorbs the unbound chemical. GABA uses a 2Na–Cl cotransporter. As discussed in Chapter 4, serotonin is reabsorbed as a cation with one sodium ion and one chloride ion, in exchange for a potassium ion to maintain electroneutrality. The neurotransmitter glutamate has adverse effects on the brain when present in the extracellular fluid, and must be reabsorbed almost completely. Accordingly, its uptake stoichiometry involves three sodium ions, which gives the transporter enough free energy to work with that it can operate against a million-fold glutamate concentration ratio! Reabsorbed transmitter molecules are either repackaged into vesicles or enzymatically destroyed. Acetylcholine remaining in the synaptic cleft is destroyed within milliseconds by the enzyme, acetylcholinesterase, which is synthesized by the postsynaptic neuron. Thus, by reabsorption or digestion, the cleft is readily cleared of neurotransmitter, and the stage is set for the next action potential to arrive at the presynaptic nerve endings. This process of rapid removal permits the frequency of signal transmission to be higher than would be the case if the cleft were left to clear by passive diffusion alone. Since the input to the postsynaptic cell depends on the concentration and residence time of the neurotransmitter in the cleft, these removal systems also affect the size of the PSP produced by an incoming action potential.

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Figure 9.20. Postsynaptic activation. (a) Additivity of multiple EPSPs at a single synapse. Three examples are shown, for different relationships between (1) the time delay, ', between successive postsynaptic potentials, and (2) the time, td, required for a single EPSP to decay. Individual potentials are indicated by solid lines, and the total potential by dashed lines. As the time between successive EPSPs decreases, the likelihood that the postsynaptic neuron will be excited increases. (b) Interaction of electrotonic spread and excitation threshold. The vertical lines denote the initial (t = 0) location of two pairs of EPSPs, and the solid lines show how their contribution to membrane depolarization varies spatially, as a result of electrotonic spread, at a later time, tm. The dashed lines show the sum of the two EPSPs at tm. The two potentials at the left originated too far apart to excite the membrane by summation, while the pair at the right started closer together, and their sum at tm exceeds the excitation threshold. For clarity of presentation, the cell membrane is drawn as a onedimension line, rather than the two-dimensional surface it is.

Transmitter release by the presynaptic neuron declines as the neuron repolarizes. Repolarization causes the calcium channels in the terminal membrane to close, and the cytosolic calcium level is drawn down by active transport, as described in Chapter 8. The synaptic terminal is now ready for the next action potential. In addition to causing the excitatory and inhibitory effects described above, neurotransmitters can also interact with specific receptors on ion-selective channels to lower sodium conductance, or to change the potassium or calcium conductances of the post-

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synaptic membrane. They can also influence cell metabolism, thereby altering membrane conductance or potential. A given neurotransmitter can have different effects on different neurons, or multiple effects on a single neuron. Shown in Figure 9.21 are the chemical structures of some of the more than 20 known neurotransmitters.

Figure 9.21. Some common neurotransmitters.

The chemistry of synaptic transmission is further complicated by the process of neuromodulation, which relies on receptors in the synaptic membranes. In heterosynaptic modulation (Fig. 9.22a), the release of neurotransmitter by a presynaptic neuron (Neuron #2 in the figure, transmitting to Neuron #3) is facilitated or inhibited by a

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Figure 9.22. Examples of neuromodulation. (a) Heterosynaptic modulation. (b) Automodulation. R denotes neuromodulator receptors on the modulated Neuron #2.

modulator that is released by another neuron (Neuron #1) that forms a synapse between its presynaptic membrane and the terminal of Neuron #2. In automodulation (Fig. 9.22b), the terminal membrane of Neuron #2 responds to the neurotransmitter that is stored in that terminal, setting up the possibility of a feedback loop to control the level of transmitter release. In both cases, the neuromodulator binds to G proteincoupled receptors in the target terminal membrane; acting through mechanisms described in the previous chapter, the G protein alters the activity of either potassium or calcium channels in the membrane. Suppression of the voltage-gated calcium channels in the presynaptic membrane can suppress the synaptic signal, or even block transmission, by decreasing the amount of neurotransmitter that is released. Deactivation of hyperpolarizing potassium channels in the presynaptic membrane increases the strength of the action potential and the quantity of neurotransmitter that is released, which is seen at the system level as increased sensitivity to the initiating stimulus. Similarly mediated increases in the potassium conductance of the postsynaptic membrane can produce a hyperpolarizing IPSP; depression of postsynaptic potassium channels can enhance synaptic excitation. Many neurotransmitters — including serotonin, norepinephrine, and acetylcholine — are known to act as neuromodulators. Rather than using a chemical intermediary to propagate the action potential between cells, some synapses rely on direct electrical communication. In electrotonic transmission, the membrane potential of the postsynaptic cell is altered by the passage of current from the presynaptic cell through gap junctions (Chap. 4) that provide a

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low-resistance pathway between the interiors of the two cells. The general architecture of the synapse is the same as that of chemical synapses, except that the presynaptic and postsynaptic membranes are in close contact at the gap junctions; away from the junctions, the cleft remains. Because there is minimal resistance to current flow through the junctions, the loop currents generated by the action potential in the presynaptic neuron pass into the postsynaptic cytoplasm through the junctions, and exit through the postsynaptic cell membrane, depolarizing it and prompting an action potential in the postsynaptic neuron. In chemical transmission, the resistance of the postsynaptic membrane is too high and the loop currents that accompany the incoming action potential exit through the synaptic cleft. Other modulating molecules, such as ATP and second messengers, can also cross from one cell to the next through the gap junctions. The current paths in the two kinds of synaptic transmission are shown in Figure 9.23.

Figure 9.23. Current paths in synaptic transmission. (a) Chemical transmission. (b) Electrotonic transmission.

As noted earlier, this mechanism is less common than chemical transmission. However, the rapidity of electrotonic transmission does offer advantages when synchronization among neurons is desired, for instance, in the cardiac pacemaker cells of some species. Mixed synapses, in which both gap junctions and neurotransmitter vesicles are present, have also been observed, in mammalian spinal cord (Rash et al., 1996).

9.2. MUSCLE Muscle can be classified as either striated or smooth. Two important kinds of striated muscle are cardiac muscle and skeletal muscle. The latter, which connects to bones

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(via tendons) and is responsible for voluntary movement, has been the most popular preparation for studies of muscle transport and physiology. This section will deal primarily with skeletal muscle, although some aspects of the transport-related behavior of cardiac muscle that distinguish it from skeletal muscle will also be discussed. Smooth muscle, like cardiac muscle, is not under voluntary control, and is responsible for functions such as peristalsis in the intestine and the control of blood vessel caliber. The function of smooth muscle relies on the coordinated contraction of the many individual spindle-shaped smooth muscle cells that comprise the tissue. In contrast to skeletal muscle, the contraction of smooth muscle cells depends on a host of inputs, some of which are activating and some of which are inhibitory. These inputs may be electrical (i.e., changes in cell potential), biochemical (including a large variety of neurotransmitters, hormones and local secretions, and changes in the local microenvironment) or mechanical (e.g., stretch). Regulatory mechanisms like those in Chapter 8 have also been implicated in smooth muscle activation. The greater variety of smooth muscle stimuli offers many opportunities for force modulation, permitting the strength and periodicity of the contraction to be adjusted as needed without conscious intervention. Correspondingly, the electrical pulses that travel through these collections of cells lack the uniformity of velocity, amplitude and shape seen in nerve and striated muscle. The ability to coordinate contraction, and the availability of multiple signaling pathways for activation, allow smooth muscle to maintain strong contractions for long periods of time. Smooth muscle generally contracts and relaxes more slowly than does striated muscle. While the varieties of smooth muscle differ from striated muscle in many respects (and differ from one another as well), at their core they rely on similar transport processes and the same families of contractile proteins. Except for occasional remarks in the text that follows, the behavior of smooth muscle will not be discussed here. Skeletal muscle is comprised of numerous (104–106) muscle fibers in bundles, lying side by side. The fibers are multinucleate cells, 10–100 μm in diameter, that can be as long as the muscle of which they are a part. The structure of a striated muscle fiber is shown in Figure 9.24. The fiber consists of a bundle of cylindrical myofibrils, which are the contractile elements of the tissue, wrapped in sarcolemma, and interspersed with sarcoplasmic reticulum (SR). Myofibrils are typically 1–2 μm in diameter. The functions of the sarcolemma, the transverse tubules (“T-tubules”) with whose walls the sarcolemma is continuous, and the SR will be described below.

9.2.1. The Resting Muscle Fiber Transport in the resting fiber is not very different from that in the resting neuron. The intracellular concentrations and Nernst potentials of the ions in the two cell types are very similar. The resting potential in muscle is also similar to that in nerve, and arises from a rheogenic sodium–potassium exchange pump in the sarcolemma. The pump can cause the muscle fiber membrane potential to become even more negative than the Nernst potential of potassium. The reason for this is found in Eq. (7.51). The first term in this equation, 6GiEi/6Gi, is the membrane potential when the passive current is zero; compare Eq. (7.46). Rewriting Eq. (7.51) accordingly,

PRINCIPLES AND MODELS OF BIOLOGICAL TRANSPORT

'\  ('\)I p 0 

369

Ia . œ Gi

(9.22)

The first term on the right-hand side of the equation is the potential across the membrane when the passive current is zero; from Eq. (7.46), this potential is a weighted average of the Nernst potentials of the permeating ions, and therefore cannot fall outside their range. The pump current, I a , depends on the pump rate and the stoichiometry of the Na–K exchange. For the usual 3Na:2K ratio, I a  0 , and the pump has a hyperpolarizing effect, which can bring '\ to values more negative than the most negative Nernst potential.

Figure 9.24. Structure of a striated muscle fiber. Reprinted with permission from Peachey (1965). Copyright © 1965, The Rockefeller University Press.

Calcium is actively extruded from the sarcoplasm by two pumps in the sarcolemma: a Ca–ATPase that drives a primary calcium pump, and an antiport system that uses sodium as the cosolute. Neither of these is likely to have a significant direct effect on the membrane potential. A third calcium pump in muscle, located in the SR membranes, will be described below. The resting conductance of the muscle fiber membrane is due primarily to chloride and potassium. The chloride permeability at rest is particularly high in skeletal muscle, 3–7 times that of potassium (Sperelakis, 1998). When the pH surrounding the fiber is raised, the GCl/GK ratio decreases, suggesting that some of the permeation pathways in the membrane contain acidic groups that are dissociated at high pH. The

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exposed negative charges would enhance the permeation of cations and tend to exclude anions from the pathway, via the Donnan effect. Under physiological conditions, the distributions of potassium and chloride across the resting sarcolemma are close to equilibrium, owing to the high permeability of the membrane to these ions. Sodium is much less permeable than potassium and chloride. As in nerve, its distribution across the fiber membrane is far from equilibrium and is maintained by the sodium–potassium exchange pump. Indeed, the idea that sodium might be actively transported was first proposed as an explanation of the nonequilibrium distribution of the ion in skeletal muscle.

9.2.2. Excitation and Excitation–Contraction Coupling This section and the one that follows describe the transport and molecular processes that take place from the arrival of a neuronal action potential at a synapse between nerve and muscle, through muscle contraction at the molecular level. To proceed further is to enter the realm of muscle mechanics, which is outside the scope of this text. Neuromuscular Transmission and Excitation: The Acetylcholine Receptor. The signal that tells a muscle fiber to contract is delivered by a motor neuron whose terminals are enfolded by grooves (the synaptic trough) in the sarcolemma at the neuromuscular junction (Fig. 9.25). The neuromuscular junction is the site of synaptic signal transmission between these two excitable tissues. Most muscle fibers are innervated by a single nerve ending at a single junction; however, the axons of motor neurons have a branching structure, so one neuron can excite up to several hundred fibers. In many respects the neuromuscular junction is similar to the nerve–nerve synapse described earlier. Acetylcholine (ACh) is the neuromuscular transmitter. As in the nerve–nerve synapse, its release is quantized and initiated by an increase in calcium concentration in the nerve ending, which results from depolarization-induced transient opening of voltage-gated calcium channels in the terminal membrane. A cytoskeletal lattice maintains the ACh vesicles and calcium channels in close proximity 4 (Harlow et al., 2001). Typically, two or three vesicles, each containing about 10 ACh molecules, are released per arriving action potential. Acetylcholine binds to receptors in the muscle membrane, prompting their associated cation channels to open, and causing local depolarization. The cation conductance and membrane depolarization of the muscle fiber membrane are further enhanced by voltage-gated sodium channels located nearby (Fig. 9.25). The ACh receptor is one of the most well-studied ligand-gated channels, and its structure and gating kinetics were described in some detail in Chapter 4. There are 4 2 typically about 1.5–2 q 10 of these channels per μm of muscle membrane, perhaps 2 5 q 10 in a single junction. Each channel consists of five subunits (Fig. 4.3c), surrounding a water-filled transmembrane pore. As can be seen in the figure, most of the receptor extends into the synaptic cleft, and the channel narrows towards the cytosolic end. One of the four membrane-spanning domains in each subunit (the M2 domain) runs along the pore and has an inward kink that creates a throat that functions as a weak selectivity filter. When the channel closes, the throat narrows and exposes hydrophobic residues, blocking the passage of ions.

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Figure 9.25. The neuromuscular junction at increasing levels of detail. The term end-plate is often used to refer to the portion of the fiber that has been altered to accept the neural signal. The acetylcholine receptors at the necks of the postjunctional (also called junctional) folds are directly opposite the transmitter release sites in the active zone of the presynaptic membrane; voltage-gated sodium channels are also present deeper in the folds. The calcium channels in the active zone are not shown. Reprinted with permission from Bloom and Fawcett (1975), copyright © 1975, W.B. Saunders; and Aidley (1998), copyright © 1998, Cambridge University Press.

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The channel opens when the M2 domains undergo a concerted conformational change, prompted by ACh binding. This enlarges the throat to 0.8  0.2 nm in diameter [recall the estimate of 0.65 nm by Dwyer et al. (1980)] and causes it to become lined by polar residues (Unwin, 1995). It has been proposed that that the channel's selectivity for cations when open is due to several rings of negatively charged amino acid residues, arising from the five M2 segments that surround the pore (Fig. 9.26).

Figure 9.26. Rings of negatively charged groups that confer cation selectivity to the AChR channel. The extracellular, intermediate and cytoplasmic rings contain negatively charged glutamate (E) and aspartate (D) residues and confer cation selectivity; the central ring contains polar serine (S) and threonine (T) residues and presumably facilitates ion passage when a nearby hydrophobic ring that gates the pore moves out of the way upon ligand binding. The single letter notation for amino acid residues is used here. Reprinted with permission from Imoto (1993). Copyright © 1993, New York Academy of Sciences.

As described in Chapter 4, the unitary conductance and kinetics of the ACh receptor channel have been estimated (Anderson and Stevens, 1973) from the frequency spectrum of the current fluctuations that reflect the opening and closing of the channels, and from the ratio of the variance of the current fluctuations to the mean current. The conductance of a single ACh channel was subsequently measured by Neher and Sakmann (1976) in Nobel Prize-winning work. Using the patch clamp technique described in Chapter 4, they isolated a portion of denervated frog muscle fiber distant from the endplate (Fig. 9.25), clamped the voltage across it, and measured the current across the patch. The number density of ACh receptors on the fiber itself is much smaller than that at the endplate, so the current through a single channel could be measured. Each time the channel opened, a current of a few pA passed through it, for some tens of milliseconds; then the channel closed again. Their results are illustrated in Figure 4.16a. Figure 9.27 shows a more recent recording, in rat. Neher and Sakmann also measured the reversal potential of the channel, that is, the potential at

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which current ceased even when the channels were open. The single-channel conductance could then be calculated: J ACh 

i , ('\ )applied  ('\ )I 0

(9.23)

where i is the single-channel current under the applied potential. A value of 43 pS was found for the conductance of a single ACh channel in frog, similar to the values obtained earlier using noise analysis (Fig. 4.20). Note the similarity between Eqs. (9.23) and (4.17); in the present equation, the Nernst potential is replaced by the reversal potential. When Eq. (4.17) is used to obtain the unitary conductance from the unitary current at a given voltage clamp, it is necessary to calculate the Nernst potential of the permeating ion. This requires knowledge of the activity of the ion on both sides of the membrane. Furthermore, Eq. (4.17) assumes that the current is carried only by that ion, i.e., that the channel is perfectly selective. On the other hand, the reversal potential in Eq. (9.23) is an experimental quantity that is easily measured by varying the clamp voltage and does not rely on channel selectivity, because it is defined on the basis of total current. If channel selectivity is high enough, measurements of reversal potential can be used to estimate ionic activity coefficients, as illustrated in one of the problems at the end of this chapter.

Figure 9.27. Patch-clamp current through a single acetylcholine channel in rat. The voltage clamp is –70 mV. Reprinted with permission from Sakmann (1992). Copyright © 1991, The Nobel Foundation.

In Chapter 4, we reviewed the kinetics of the AChR channel as deduced from patch-clamp recordings. Most open channels have two molecules of transmitter bound to them; this state is reached when a closed channel with two ligands opens, rather than via the addition of a second ACh molecule to an already open channel. The equilibrium between the open and closed states of the (ACh)2–receptor complex strongly favors opening, the intended effect of the transmitter. This is consistent with evidence that the strength of the ACh–receptor bonds is greater when the channel is open.

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The channels remain open for about 1 msec. Acetylcholine is rapidly destroyed by acetylcholinesterase after being released into the synaptic cleft; the diminishing local concentration of ACh shifts the binding equilibrium on the receptors towards dissociation, and the channel closes.

Figure 9.28. (a) Electrical analog of the postsynaptic membrane for an excitatory synaptic potential. The EMF of the transmitter-gated channel pathway (left path), EEPP, depends on the openchannel Na and K conductances and the cation concentrations. The synaptic “switch” opens when the channels close. The current through the channels is carried primarily by sodium. The remaining pathways represent the unexcited fiber membrane adjacent to (and electrically in parallel with) the excited region, and include a capacitance (center path) and a generalized leak (right path); the latter is characterized by an equivalent circuit consisting of a battery, Em, and its accompanying resting conductance, Gm. Arrows indicate the positive direction of current through each path. The current generated by the Na–K exchange pump is ignored for simplicity. (b) Time course of the end-plate potential, identifying the four phases of excitation described in the text.

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The ACh receptor does not discriminate strictly between Na and K, and the fluxes of both ions across the postsynaptic membrane increase when the channels open. At rest, potassium is much closer to equilibrium across the postsynaptic membrane than is sodium, so the principal effect of an increase in conductance is a large influx of sodium, accompanied by membrane depolarization. As in nerve, depolarization can induce an action potential in the muscle fiber, by opening voltage-gated sodium channels in the endplate membrane. The time course of the endplate potential, '\EP, can be divided into four phases and understood through an equivalent circuit representation of the post-synaptic membrane (Fig. 9.28; see caption): 1. At rest, the synaptic “switch” is open and only the leakage path can carry current. Electroneutrality requires that there be no net current across the synaptic membrane, so Im = 0, and the (open-circuit) endplate potential is given by the Nernst potentials and resting conductances of the permeant ions, according to Eq. (7.46). In the notation of Figure 9.28, '\EP equals the resting potential, Em. 2. When transmitter is released, receptor binding causes the AChR channels to open, the synaptic switch closes, and EEPP drives an inward ionic current through the channel pathway (Is < 0). Some of the current discharges the capacitance of the postsynaptic membrane (Ic > 0), and the remainder (termed the leakage flux) exits through the leakage pathways: –Is = Ic + Im. Initially, most of the current discharges the capacitor and depolarizes the membrane. This is because the driving force for the leakage flux is equal to '\EP – Em , and the membrane potential is still close to the resting value. 3. As the membrane depolarizes further, the influx through the channels decreases because of the lower driving force, and the outward leakage flux increases as its driving force rises. Since the inward current through the channels is balanced by the outward capacitive and leakage currents, both of these trends reduce the rate of capacitor discharge and depolarization. Eventually the two fluxes become equal, and depolarization ceases to grow (Ic = 0). At this point, –Is = Im, and the membrane potential has achieved its most positive value, '\EP*. Equating the two currents, –Gs('\EP* – EEPP) = Gm('\EP* – Em) (at maximum depolarization). The maximum depolarization, '\EP* – Em, is readily obtained: '\*EP  Em 

Gs ( EEPP  Em ) . Gs Gm

(9.24)

The amount of ACh released when an action potential in the motor neuron reaches the junction is sufficient to raise the conductance of the channel pathway, Gs, to about ten times Gm. Consequently, the maximum depolari-

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zation given by Eq. (9.24) is close to EEPP – Em. The value of '\*EP  Em , for vertebrate muscle, is about 65 mV. 4. When the channels close and the synaptic switch opens, a slow outward current through the leakage pathways recharges the membrane capacitance as the cell reassumes its normal resting polarization. In contrast to neuron–neuron synapses, the result of the neuromuscular synapse is always excitation, which causes the muscle to contract. The action potential initiated at the endplate by ACh propagates through the fibril by a mechanism similar to axonal conduction. The threshold for initiation is about half the value of '\*EP  Em given above, so the neural signal rarely fails to get its message across; local depolarization at the endplate invariably prompts an action potential in the neighboring sarcolemma. The action potential rapidly spreads down the fiber and along the walls of the Ttubules. This fine network of tubules, whose surface area may be as much as four times that of the non-tubule sarcolemma, distributes the propagating potential throughout the fiber within milliseconds. The action potential initiates contraction of the muscle. The contraction process is slow relative to the speed with which the action potential sweeps through the fiber; thus, even though the fiber is initially excited only at the neuromuscular junction, its contraction is synchronized. Synchronization is crucially important for cardiac muscle, and is facilitated by structural features that are different from skeletal muscle. Cardiac muscle is a network of individual mononuclear cells joined to one another at intercalated discs at which gap junctions permit electrotonic coupling. Specialized high-frequency pacemaker and Purkinje fibers, and others, embedded in the muscle carry the excitation throughout the heart more rapidly than in skeletal muscle. This increased level of coupling allows the action potential (and subsequent cardiac contraction) to sweep through cardiac tissue rather than proceed linearly as it does in nerve. A multidimensional version of the electrical analog of Figure 9.10c, the bidomain model (Henriquez, 1993), has been used to describe conduction in cardiac tissue. The shapes of the action potential in skeletal and cardiac muscle differ. In skeletal muscle, it is similar to that in nerve, lasting only a few milliseconds (Fig. 9.29a). In cardiac muscle, the duration of the depolarization phase is much longer, and the action potential can last several hundred milliseconds (Fig. 9.29b). The long action potential in cardiac muscle can be explained within the Hodgkin–Huxley framework by postulating a different set of parameters in the rate equations for GK and GNa, but the difference is primarily due to voltage-sensitive Ca channels in cardiac muscle. These slow channels are depolarizing when open because calcium, like sodium, is actively extruded from the sarcoplasm, and ECa is large and positive. The calcium channels inactivate much more slowly than the sodium channels, protracting the depolarization phase of the action potential. The kinetics of slow calcium channels in muscle are incorporated into the Hodgkin–Huxley formalism in a fashion similar to Eqs. (9.7): 2

GCa = GCa,max d f,

(9.25)

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where d is the activation parameter and f is the inactivation parameter. The physical interpretations of d and f, and of the inactivation that these channels also exhibit, are similar to those described earlier for the fast sodium channel, but with two activation sites instead of three. The rates of activation, inactivation, and recovery are slower for the calcium channels.

Figure 9.29. Intracellular recordings of action potentials in muscle. (a) A skeletal muscle fiber. [Reprinted with permission from Sperelakis et al. (1973). Copyright © 1973, The American Physiological Society.] (b) A cardiac ventricular cell. [Adapted with permission from Wahler (2001). Copyright © 2001, Academic Press.]

The duration of the depolarization plateau in smooth muscle can be as short as that in skeletal muscle, or longer than that in cardiac muscle. Plateaus over one second in length are developed in smooth muscles responsible for prolonged contractions, as in the ureter and uterus. Even though their durations are similar, there are differences between the shapes of the action potentials in nerve and skeletal muscle. For one, the delayed rectifier in muscle passes sodium ions more readily than that in nerve. This decrease in selectivity opposes the dominance of potassium conductance that causes hyperpolarization in nerve. As can be seen in Figure 9.29, the action potential in muscle decays monotonically to the resting potential as the channels close, without hyperpolarizing. Skeletal muscle exhibits a very sharp repolarization phase compared to nerve or other kinds of muscle. This is a consequence of the high permeability of skeletal muscle to chloride. In skeletal muscle, chloride equilibrates easily across the cell membrane, reaching an intracellular concentration such that its Nernst potential equals the resting potential of the cell. When the action potential sweeps through the fiber, depo-

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larization creates a driving force for passive chloride influx across the sarcolemma. This influx rapidly quenches the rise in cell potential, sharpening the repolarizing downstroke. Excitation–Contraction Coupling. The process by which the propagating action potential causes the muscle to contract is referred to as excitation–contraction coupling. It can be described in three steps: 1. Maintenance of Low Sarcoplasmic Calcium Levels: The SERCA Pump. During the relaxation period between contractions, the membranes of the SR actively transport calcium ions from the sarcoplasm into the reticulum, reducing the cytosolic calcium level to ca. 0.1 μM. Magnesium ions must be present for this to happen. The calcium pump is a primary active transport system, in which the calcium carrier is a phosphorylated ATPase complexed with magnesium. The sarcoplasmic/endoplasmic reticulum calcium (SERCA) pump is probably better characterized than any of the other P-type ATPases described in Chapter 5. In skeletal muscle, each molecule of ATP drives two calcium ions into the SR. In cardiac muscle, the stoichiometry is more variable, ranging from 0.4–1.0 Ca ions/ATP. The calcium is bound within the reticulum to a protein, calsequestrin; binding reduces the “head” against which the pump must work by decreasing the activity of the ion in the reticulum. One mol of calsequestrin can bind 40–50 mols of calcium. The reaction cycle of the pump is shown in Figure 9.30a. It is similar in some respects to the Na–K pump cycle described in Chapter 5. After the enzyme in the E2 state has discharged its calcium burden into the reticulum, it loses the terminal phosphate from the previous cycle, and undergoes a passive transition to the E1 form. The passive step has been proposed to be rate limiting (Edes and Kranias, 2001). When the enzyme is in the E1 form, its binding sites are exposed to the sarcoplasm and have a higher affinity for Ca. The enzyme binds ATP, followed by two Ca ions in sequence. After all substrates are bound, the carrier enters an occluded state (the E1(Z) state in the figure), and cannot release its bound ions to either phase (see the last panel in Fig. 9.30b). It is then phosphorylated (at an aspartate residue, termed the phosphorylation site) by the bound ATP, and uses the released energy to revert to the E2 state, in which the binding sites face the reticulum. Accompanying the transition to the E2 state, ADP is released and the calcium gate facing the reticulum opens (Toyoshima et al., 2004). In the less energetic E2 state, the pump has a much smaller affinity for Ca, and the ions are discharged into the SR. The gate closes upon the dissociation of the remaining phosphate from the E2 form of the ATPase. Tanford et al. (1987) proposed a cooperative model of sequential calcium binding by the Ca-ATPase. In this model, binding of the first calcium ion induces a conformational change of the carrier that exposes a second, higher affinity binding site. This model is shown in Figure 9.30b. They also proposed that the occluded EZ conformation is the only one in which the ATP-binding domain of the carrier is close enough to the phosphorylation site for activation to take place. Thus, the enzyme can bind ATP before it binds calcium, but the ATP will not drive the pump until it has its full complement of ions.

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Figure 9.30. (a) The reaction cycle of the calcium pump in sarcoplasmic reticulum. The asterisk ( ) denotes the start of the cycle as described in the text. In an alternative model (Edes and Kranias, 2001), ATP interacts with the carrier only after the two calcium ions have been bound from the cytosol. In any event, the full complement of substrates must be bound before phosphorylation can take place. The pump also countertransports 2–3 protons per cycle (Yu et al., 1993), which are taken up from the reticulum by the E2 form of the enzyme. This makes the pump less rheogenic and less voltage sensitive than it would be otherwise. The countertransported protons are not shown. (b) Tanford's model of cooperative calcium binding (Tanford et al., 1987).

The structure of the SERCA pump in the Ca2E1 state has been determined by x-ray crystallography at 0.26-nm resolution (Toyoshima et al., 2000), and is shown in Figure 9.31. The protein has four domains: the A, N, and P domains extending into the cytoplasm, and the transmembrane M domain. ATP binds to the N domain and the phosphorylation site is in the P domain. Calcium binding induces the ATP-containing

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N domain to rotate toward the phosphorylation site, driving phosphorylation and the conformational change of the protein. Gating of the pump is governed by the conformation of the A domain (Toyoshima et al., 2004).

Figure 9.31. Please see insert for color version. Architecture of the SERCA ATPase (Toyoshima et al., 2000). The ATP binding and phosphorylation sites are marked. The structure was determined while two calcium ions were bound within the protein, which is in the Ca2E1 state. Toyoshima's group subsequently published (Toyoshima and Nomura, 2002) an analogous study of the pump in the calcium-free E2 state, detailing the substantial conformational changes that accompany the transition to that state from the E1 state depicted here. These changes also explain the reduction in the calcium affinity of the ATPase that accompanies the transition. See also recent structural studies by Olesen et al. (2004). Reprinted with permission from Toyoshima et al. (2000). Copyright © 2000, Macmillan Publishers.

2. Excitation and Calcium Release. The passage of the action potential along the walls of the T-tubules causes the SR to release accumulated calcium ions into the sarcoplasm, causing its calcium level to increase by a factor of 10–100. The exact mechanism by which the action potential induces calcium release is unclear, but there is some useful machinery in the triad region indicated in Figure 9.24. A more detailed view of the triad region is shown in Figure 9.32. At localized sites called feet, voltagesensitive calcium channels in the tubule membrane are close to calcium release chan-

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nels in the reticulum membrane. The voltage-sensitive channels in the tubule are known as the dihydropyridine receptors (DHPRs) because they respond to this family of calcium blockers, and the calcium release channels in the SR are called the ryanodine receptors (RyRs) because they have a high affinity for ryanodine, a plant alkaloid (the structure of the tetrameric ryanodine receptor was presented in Fig. 4.3e). As can be seen in Figure 9.32, the large cytosolic domain of the release channels almost touches the tubule (
Figure 9.32. Stylized view of the triad region in skeletal muscle. Only a fraction of the release channels are closely associated with the voltage-sensitive channels, which are arranged in foursomes, called tetrads.

As shown in Figure 9.32, not all of the release channels are adjacent to the DHPRs in the T-tubule. There are two theories of how these release channels are induced to open. One holds that release channels not associated with tetrads are activated by the increase in sarcoplasmic calcium that results when the allosterically coupled channels open, in a positive feedback loop. The other theory maintains that the release channels are in coordinated, connected clusters that respond as a group even though not every one has a partner tetrad in the T-tubule. During calcium release (and calcium accumulation as well), there must be additional ion fluxes across the SR membrane to maintain the electroneutrality of the reticulum. Passive potassium and chloride channels have been found in the SR mem-

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brane of striated muscle and may facilitate the maintenance of charge balance after excitation. The calcium release channel itself also passes monovalent cations. 3. Contraction and Relaxation. The high concentration of calcium in the sarcoplasm activates contraction by the myofibrils, discussed in the next section. Subsequently, the release channels in the SR inactivate, and within about 30 msec the calcium pumps in the SR (and, to a lesser extent, in the sarcolemma) bring the sarcoplasmic calcium concentration back to resting levels, so that the myofibril can relax. Heiny (2001) has estimated the duration of each of the steps in the contraction process, from the initiation of the action potential at the neuromuscular junction to peak muscular tension, for frog skeletal muscle at 18ºC. Her results are summarized in Table 9.4. Some of the items in the table will be described in the following subsection. Table 9.4. Duration of Key Steps in the Activation of a Hypothetical Fast-Twitch Skeletal Muscle in Frog at 18ºC Step

Duration, ms

Action potential propagation along sarcolemma Action potential propagation along T-tubules Activation of calcium release channels by local depolarization Calcium release and binding to troponin C Crossbridge binding and achievement of maximum tension

5–10 ca. 0.7 ca. 0.5 2–3 15-25

Paraphrased from Heiny (2001).

9.2.3. Molecular Mechanisms of Muscle Contraction The functional unit of the muscle fiber that accomplishes contraction is the myofibril. As can be seen in Figure 9.24, the myofibril appears as an organized collection of parallel lines that are visible under electron microscopy. The lines are aligned in such a way that the fiber appears to be transversely striped, or striated, under lower magnification. This structure is presented more clearly in Figure 9.33, so we can study it in more detail. The composition of a vertebrate myofibril is given in Table 9.5. Table 9.5. The Major Protein Constituents of a Typical Vertebrate Skeletal Muscle [Maruyama (1986), as cited in Aidley (1998)] Protein Contractile proteins Myosin Actin Regulatory proteins Troponin Tropomyosin Structural proteins Titin Nebulin

Molecular weight, kDa

Weight fraction

520 42

0.43 0.22

70 66

0.05 0.05

2800 750

0.10 0.05

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Figure 9.33. (a) Low-power electron micrograph of vertebrate striated muscle, showing filament alignment and striations. [Reprinted with permission from Hugh Huxley, Brandeis University.] (b) Schematic of myofibrillar filament structure, longitudinal view (after Aidley, 1998). The unit of myofibril between two Z-lines is known as a sarcomere; sarcomeres are typically somewhat more than 2 Pm long. The end-to-end organization of the sarcomeres is responsible for the striations that gives this type of muscle its name. Transient cyclic binding of the myosin crossbridges to the thin actin filaments produces sarcomere shortening. (c) Transverse view of the filament organization of the sarcomere at different positions along its length. The M-line is seen to be caused by crosslinks between the thick filaments. The Z-lines and M-lines are actually disc shaped, since the myofibril is cylindrical.

The thick filaments, about 11 nm in diameter, are largely myosin II. The myosin molecules terminate in a series of crossbridging heads, which can link to the thin filaments; each thick filament has several hundred heads. The thin filaments, about 5 nm in diameter, are primarily actin and attach at the Z-line. From the transverse views

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in Figure 9.33c, it can be seen that the fibers are organized in a hexagonal array, in which there are twice as many actin filaments as there are myosin filaments. Contraction of the myofibril occurs when the sarcomere actively shortens in response to an increase in sarcoplasmic calcium. Shortening requires ATP as well as calcium ions. It begins with the binding of a calcium ion to a calcium-binding protein, troponin C, which is a component of the myofibrillar protein complex, troponin. When the muscle is at rest, contraction is inhibited by a second thin filament protein, tropomyosin, which blocks the myosin binding sites on the actin, inhibiting any interaction between the two contractile proteins. The blocking activity of tropomyosin is a result of its interaction with two other components of troponin, known as troponin I and troponin T. Binding of calcium to troponin C causes a conformational change in the troponin complex, which allows the tropomyosin to move away from the myosin binding site, unblocking it. Then, the latent crossbridges on the myosin filament are free to bind to the actin and generate a force that pulls the thin filaments toward the M-line (Fig. 9.33b). The thin filaments slide between the thick filaments; hence, this is known as the sliding filament mechanism. The force that is produced depends on the calcium concentration in the sarcoplasm (Fig. 9.34). Troponin and tropomyosin also affect the rate at which myosin can hydrolyze ATP to generate that force.

Figure 9.34. Dependence of force generated in a myofibril on sarcoplasmic calcium concentration (Heiny, 2001).

A great deal of molecular biology research has been devoted to unraveling the mechanism by which the myosin filaments attach to and pull the actin filaments. A requirement of this mechanism is that it be cyclic; it is not sufficient for the crossbridges simply to connect to the actin and pull, because they could never pull the thin filament beyond their own location on the thick filament. Indeed, the distance between crossbridges is only 0.6% of the change in sarcomere length that corresponds to the normal operating range of muscle. Therefore, they must bind to the thin filament, pull it a short distance relative to the myosin (the working stroke), disconnect, and then

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revert to their original conformation (the recovery stroke) in preparation for another working stroke. This sequence of events is known as the crossbridge cycle; we now describe its mechanism. Reversible binding between the two proteins is the responsibility of a globular region at one end of the myosin molecule known as the S1 head (Fig. 9.35a). The head has two functional regions: a catalytic region that has ATPase activity and is adjacent to the actin binding sites on the head, and a lever arm between the catalytic region and another portion of the myosin molecule, the S2 region, that links the head to the thick filament. A hinge at the S1–S2 junction allows the two regions to rotate relative to each other, and there is also a hinge between the catalytic region and the lever arm. Through the hinges, the lever arm translates rotation in the S1 head into directed movement of the actin filament relative to the myosin. Relative motion between the thick and thin filaments is accomplished as shown in Figure 9.35b. It is convenient to start at the point in the cycle (1) at which the S1 head binds to the actin filament; at this point, the ATP binding site is occupied by ADP and phosphate from a prior ATP hydrolysis, much of whose energy has not yet been expended. In the figure, a marker on the actin indicates the binding site. Myosin–actin binding catalyzes the release of the phosphate ion, generating a strained state (2) containing much of the energy from the earlier hydrolysis. This strain is released by a rotation of the catalytic region of the S1 head that decreases the angle between the lever arm and the S2 link (3). This is the working stroke of the system. Since the head is firmly fixed to the thin filament, this conformational change causes the thick and thin filaments to become more highly interdigitated (recall Fig. 9.33b), by about 5 nm/stroke in skeletal muscle under a light load. The relative motion resulting from a single working stroke is known as the working distance of the motor system. Upon completion of the working stroke, the ADP molecule dissociates from the catalytic site (4). On the basis of optical trap experiments and Monte Carlo simulations, Capitanio et al. (2006) recently proposed that the working stroke in skeletal muscle consists of two steps rather than one: the first step, of 3.4–5.2 nm, accompanies strain release as described above; the second step, of 1.0–1.3 nm, accompanies the subsequent loss of ADP.

The catalytic site subsequently binds a molecule of ATP (5). ATP binding weakens the actin–myosin bond, and the S1 head dissociates from the actin filament. Subsequent hydrolysis of the ATP causes the head and lever arm to rotate, increasing the angle between the S1 and S2 regions (6). This is the recovery stroke. During recovery, the products of hydrolysis remain associated with the catalytic core, and then the cycle repeats. Howard (2001) has noted a satisfying reciprocity between the mechanical and chemical aspects of the crossbridge cycle: myosin–actin binding (a mechanical step) is required for phosphate release (a chemical step), while ATP binding (a chemical step) is required for dissociation of the myosin–actin bond (a mechanical step). Howard's

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Figure 9.35. Mechanism of crossbridge function in muscle. (a) The S1 head of myosin. The hinges allow relative rotation within the head and between the head and the S2 link to the myosin filament. (b) The crossbridge cycle (keyed to the steps in the text). ATP is designated by a filled circle in the ATP binding site (see (a)), ADP by an open circle, and phosphate by Pi. There are two S1 heads at the end of each S2 link (muscle myosin is a dimer), but only one of these interacts with the actin filament during a single crossbridge cycle, and the other is omitted for clarity. The rotation of the catalytic region (see (a)) relative to the lever arm is similarly omitted. The arrows describing each transition are in the direction that accomplishes contraction; many of these steps are in fact reversible. Each crossbridge cycle uses one molecule of ATP.

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text is an excellent source of additional information regarding motor proteins and muscle mechanics. The S1 heads on the myosin filament operate independently, each head contributing about five strokes/sec. A typical thick filament has hundreds of heads, so traction and sliding are continuous, and the sarcomere contracts smoothly, even though, at any given time, a sizeable fraction of the S1 heads are unbound to actin (State 6 in Fig. 9.35b) and are not contributing to the contractile process. Under a light load, when the sarcomere is contracting rapidly, the amount of relative motion between a given head and the thin filament to which it reversibly binds can be substantial between strokes; the actin binding sites for two successive cycles of a single head can be 200–400 nm apart. Owing to the cooperative action of multiple heads, a sarcomere can contract at a typical rate of 15 μm/s. The rate of contraction of a muscle consisting of thousands of sarcomeres in series is correspondingly larger.

PROBLEMS: CHAPTER 9 1.

The electrolyte concentrations in the axoplasm are 14 mM Na, 140 mM K, and 3.5 mM Cl; the external concentrations are 142, 4, and 103 mM, respectively. The axoplasm is 90 mV negative with respect to the surroundings at 37ºC. Are any of the ions at equilibrium? Assume ideal solutions.

2.

Consider the electrical analog of the axolemma. Neglect the leak pathway and capacitive effects. The axoplasmic and external ionic concentrations, and the cell potential, are those in Table 9.1. Assume T = 37ºC and ideal solutions. (a) What is the ratio of the sodium conductance to the potassium conductance at rest? (b) An action potential rapidly depolarizes the cell by 90 mV, so that the axoplasmic potential is the same as that of the surroundings. The axoplasmic composition does not change in this short a time. What did the conductance ratio change to?

3.

Assume a propagating action potential increases the cell potential from a resting value of –70 mV to a peak value of 50 mV, and that all the current is carried by 2 sodium up to that point. If 1.5 pmols/cm of Na enters the axon during this time period, what is the capacitance of the axolemma per unit area?

4.

In ion substitution experiments, the composition of the solution outside an isolated cell or collection of cells is altered to learn more about the role of each ion in the transport process. In some studies of nerve function, choline, a cation whose permeability in the axolemma is low, is used. Two experiments are carried out. In one, half of the extracellular Na is replaced by choline; in the other, half of the extracellular K is replaced. Which change has the greater effect on resting potential, and why?

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5.

An ion substitution experiment is carried out on a nerve axon, in which some of the extracellular sodium is replaced by an equimolar amount of potassium. (a) What is the effect of the substitution on the resting potential of the axon? Why? (b) What is the effect of the substitution on the degree of hyperpolarization of the cell during the latter phase of the action potential? Why?

6.

Figure 9.9b shows the sodium and potassium current densities across the squid axolemma at 18.5ºC during the propagation of an action potential. The diameter of the axon is 475 μm. How much do the concentrations of sodium and potassium in the axoplasm change during a single action potential? Neglect the readjustment of ionic concentrations by the Na-K pump.

7.

Construct the state diagram for a four-gate channel, assuming all gates are distinct. Show how the state diagram collapses to Figure 9.13 if three of the gates are identical.

8.

Construct the state diagram for the potassium channel analogous to Figure 9.13 by assuming all of the gates are identical.

9.

Complete the state diagram in Figure 9.14 by inserting the rate constants for the rest of the steps. Explain why the first rate constant for closure is 4G/T.

10. Explain in words why the loop currents in electrotonic transmission depolarize the postsynaptic neuron. 11. Explain in words why the sodium–potassium exchange pump has a hyperpolarizing effect on the muscle fiber membrane. 12. The acetylcholine-gated cation channel at the neuromuscular junction shows little selectivity, and passes Na, K, and Ca with comparable ease. Yet when the channels open during neuromuscular transmission, the flux through the postsynaptic membrane consists almost entirely of Na ions. Why not K? Why not Ca? 13. The EMF and conductance of the channel pathway in Figure 9.28a reflects the departure of both Na and K from equilibrium across the endplate membrane. (a) Model the channel pathway by two ionic circuits in parallel (one for Na and one for K) and express the open circuit potential of the pathway in terms of the conductances and Nernst potentials of the two ions. (b) Assuming that the conductances of the AChR channel for Na and K are the same (recall the channel is relatively nonselective between the two), estimate the open-circuit potential of the channel pathway, using the Nernst potentials of the two ions given in Table 9.1. (c) Relate EEPP to the open-circuit potential found in (b). Explain your answer.

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14. Derive Eq. (9.24). 15. Estimate the maximum rate of contraction of a skeletal muscle that is 8 cm long.

10 EPITHELIAL TRANSPORT

INTRODUCTION In many tissues, cells lie side by side in single or multilayered sheets. Examples include the lining of the blood vessels, the kidney and organs of the gastrointestinal tract, the outermost and innermost layers of the cornea of the eye, and the secretory surfaces of the choroid plexus in the brain. These endothelial and epithelial layers have classically been credited with two primary functions: to serve as a barrier between the phases they separate, and to mediate the exchange of solutes and solvent between those phases. Since the first of these functions is a special case of the second, it is only a modest exaggeration to say that epithelia exist to regulate transport. Within the classification of biological tissues, endothelial layers are regarded as a subclass within epithelia. Endothelial cells form monolayers and are largely associated with the heart, blood vessels, and lymphatics, while epithelia can form single or multiple layers and line the internal and external surfaces of a large variety of organs. Because of their role as regulators of transport between the phases they separate, epithelia differ in some important ways from the single cells discussed in earlier chapters. Probably the most important difference is topological. In considering single-cell transport, the primary barrier is the cell membrane, which separates the interior of the cell from its surroundings. Epithelial cells, on the other hand, interact with two external environments, one at each of the two surfaces of the epithelial sheet. Thus, while matter can go only into or out of single cells, it can cross epithelia; the transport barrier can be the entire cell layer. Accordingly, most models of epithelial transport include at least three compartments: the two extracellular phases and an intracellular phase. These models — and the epithelia they represent — exhibit properties absent from single cells. Furthermore, since epithelia form sheets of cells, they can line the interior of hollow organs like the eye or stomach, or line tubes that carry biological materials such as food undergoing digestion, or urine. The first section of this chapter begins with a review of the nomenclature of epithelial organization and transport, and the classification of epithelial tissues according to their electrical resistance. Some of the more common features of the transepithelial pathways are then described. Several important consequences of epithelial barM.H. Friedman, Principles and Models of Biological Transport, DOI: 10.1007/978-0-387-79240-8_10, © Springer Science+Business Media, LLC 2008

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rier structure — the existence of multiple transport paths across the cell layer, the coupling of transepithelial water flow to active ion transport, and the effect of transepithelial solvent flow on the driving forces across the barrier — are also discussed. Electrical analogs of these cell layers are summarized next. In all of these treatments, the phases adjacent to the cell layer are assumed to be uniform in the plane of the layer. With this as background, the second section of the chapter is a review of the various transport functions that epithelia and endothelia perform, each illustrated by organs that carry out that function. An appendix describes transport through tubular epithelia, where the composition of the fluid inside the tube changes along its length because of transport across the tube walls. The equations derived here will also be useful in the following chapter, since they apply as well to transport within capillaries.

10.1. ORGANIZATION OF EPITHELIAL AND ENDOTHELIAL CELL LAYERS AND SOME CONSEQUENCES Since the most striking differences between transport in epithelial layers and in single cells arise from the more complex organization of the former, it is reasonable to begin with a brief review of epithelial morphology.

10.1.1. The Organization of Epithelial and Endothelial Cell Layers Figure 10.1ab is a sketch of a monolayer epithelium. This tissue is a collection of cells, joined together by junctions at one face, which is generally the apical surface of the layer. When a cavity, such as the inside of the stomach, is lined by an epithelial layer, the apical surface usually faces the cavity, and is termed the mucosal, or luminal, surface of the epithelium. The cavity, or the interior of a tubular structure surrounded by epithelium, is called the lumen. The remaining surfaces of the cells are the basolateral surfaces of the epithelium. Many epithelia separate lumens from capillary beds. The basal side of the epithelium faces the capillary bed and is also called the serosal, or abluminal, side. Capillaries and larger blood vessels are an exception to this rule, since blood fills the lumen; “lumenal” and “abluminal” are the terms most commonly used to distinguish the two faces of the endothelial lining of these vessels (Fig. 10.1d). The apical and basolateral surfaces of epithelial cells can have very different functions and correspondingly different transport properties, reflecting a different mix of integral membrane proteins within the bilayer. The lipid compositions of the apical and basolateral bilayers also differ. These differences lend to epithelial layers a polarity that most isolated cells lack. It is because of this polarity that epithelia can move species in a preferred direction across the layer, and can create and maintain a difference between the compositions (and, in many cases, electrical potentials) of the mucosal and serosal fluids. The means by which epithelia retain their polarity as their membrane proteins recycle through the cell will be discussed later in this chapter.

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Figure 10.1. A simplified diagram of a monolayer epithelium. (a) View from the mucosal side. (b) Section through the cells. Here and in the figures to follow, the space between the cells is enlarged for clarity. The cells are much more closely packed in real epithelia, their surfaces are more irregular, and the path between them is more tortuous. The cells are supported at their basal surfaces by a basement membrane, not shown. The basement membrane also bounds the serosal end of the paracellular (a.k.a. intercellular) channel. Other junctions between the cells, described in the text, are not shown. (c) The equivalent resistances of the pathways across the epithelium. Ric is the resistance of the intercellular channel; the meaning of the other subscripts is obvious. (d) Endothelium lining a capillary. [Adapted from Guyton and Hall (2000).]

Figure 10.1d is a sketch of an endothelial monolayer lining the inside of an capillary. While epithelial cells can be flat, or cuboid, or columnar like those in Figure 10.1a, endothelial cells are uniformly flat. Similarly, as noted earlier, epithelial tissues can be multilayered (Fig. 10.2), but endothelial cells form only monolayers. One face

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Figure 10.2. A realistically drawn section through the multilayered epithelium of the cornea. The shape of the cells varies from flat to columnar. Several transport paths across the epithelium are shown. Reprinted with permission from Hogan et al. (1971). Copyright © 1971, W.B. Saunders.

of the cell is exposed to flowing blood or lymph, and the other face adheres to underlying tissue. The interfaces between adjacent cells contain junctional proteins that hold the monolayer together. As illustrated in the preceding figures, transport across epithelium and endothelium proceeds through multiple pathways. Matter can cross the layer by passing through or between the cells. In epithelial layers in particular, transcellular fluxes (paths which traverse the cell interior) can cross the basolateral membranes either at the base of the cell, or laterally into the paracellular space. The paracellular pathway in parallel with the transcellular path is often called the shunt pathway, because it bypasses the cells. As shown in Figure 10.2, the number of possible paths increases rapidly with the number of cell layers in the epithelium. The epithelial cells are encircled at their apical surface by tight junctions, which hold them to one another. The tight junction is a major contributor to the resistance of the shunt pathway to passive diffusion of ions and larger molecules, though its permeability varies considerably among epithelia. A relatively impermeable shunt is necessary to maintain concentration differences across the cell layer; otherwise, such differences would be dissipated by paracellular fluxes from the high concentration side to the low concentration side. The tight junction is also a barrier to integral protein migration in the bilayer, and helps maintain the differences in protein composition (and, accordingly, differences in function) between the apical and basolateral portions of the

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cell membrane, thereby preserving the polarity of the cell layer. Two other types of cell–cell junctions are desmosomes, which “spot weld” adjacent cells to one another and give the epithelium mechanical strength; and adherens junctions, common in endothelial monolayers, which attach through the cell membrane to actin filaments in the cytoplasm. The areas of contact of these latter junctions can be small, and none of them impede transport as much as the tight junction does. The lateral membranes of epithelial cells also contain gap junctions that allow them to communicate with their neighbors. The tight junctions also allow epithelia to secrete synthesized protein selectively into one of the bounding phases. Secretory vesicles generally empty into the mucosal phase, and the junctions prevent the secretion from diffusing to the serosal side. The selectivity of the vesicles for the mucosal face is an example of epithelial polarity. The cells and paracellular path are bounded at the basal surface of the cell layer by a basement membrane on which the epithelium rests. The transport resistance of the basement membrane is often negligible, and it is frequently omitted from models of epithelial transport. We shall see that an important exception arises when the basement membrane supports a significant difference in concentration or pressure between the paracellular and serosal fluids.

10.1.2. The Pathways across Leaky and Tight Epithelia Epithelia can be classified as “leaky” or “tight,” depending on the size of the contribution of the paracellular pathway to the transepithelial conductance. In leaky epithelia, the paracellular conductance is greater — often much greater — than the transcellular value, even though the paracellular clefts between the cells occupy less than 1% of the epithelial surface area. In these tissues, the epithelial resistance is low, generally less 2 than 100 ohm-cm . Examples of leaky epithelia are those of the small intestine, gallbladder, proximal tubule of the kidney, and choroid plexus. The paracellular path is more resistive in tight epithelia, whose resistances range from several hundred to tens 2 of thousands of ohm-cm . Tight epithelia line the colon, urinary bladder, distal tubule of the kidney, and the sweat and salivary ducts. The corneal epithelium shown in Figure 10.2 is also tight. Table 10.1 lists a number of leaky and tight epithelia, demonstrating the considerable variation in their resistances. Table 10.1. The Resistances of Some Rabbit Epithelia Tissue Kidney, proximal tubule Gallbladder Small intestine, ileum Colon Urinary bladder Cornea

Resistance, :-cm2 7 20 100 286 5–10 K 9.1 K

Reference Lutz et al. (1973) Henin et al. (1977) Frizzell and Schultz (1972) Schultz et al. (1977) Lewis et al. (1976) Klyce and Wong (1977)

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The relative importance of the paracellular and transcellular transport pathways is different for the two classes of epithelia: the former pathway is naturally of greater importance to leaky epithelia, while the latter plays a greater role in tight epithelia. We will discuss the two pathways in turn, beginning with the shunt. The Shunt Pathway. The shunt pathway consists of two resistances in series: the tight junctions and the intercellular (i.e., paracellular) channels between the lateral membranes of adjacent cells. This series resistance path is in parallel with the transcellular transport route (Fig. 10.1c). The conductance of two paths in parallel is controlled by the more highly conducting path, which in leaky epithelia is the shunt. The conductance of the shunt, which consists of two transport resistances in series, is controlled by the barrier that has the greater resistance. The junctional resistance is almost always substantially more than the resistance of the intercellular channel. Thus, the junction is the most important determinant of the electrical behavior of leaky epithelia. Accordingly, many studies of transport in such tissues have focused on the intercellular junctions. In the electron microscope, viewed in the plane of the cell membranes, the tight junction appears as a series of sites along the intercellular channel at which the cell membranes of adjacent cells are in very close contact with one another. This is the result of strong interactions between integral membrane proteins in the plasma membranes of the two cells; these proteins include claudins, occludin, and other junctional adhesion molecules. Viewed perpendicular to the membranes, using freeze–fracture electron microscopy, it is seen than these adhesion points are in strands, as though the two cell membranes are stitched together (see Fig. 10.3). This network fills the apical end of the intercellular channels. The equivalent pore size of the junctional complex has been estimated from permeability measurements in rabbit gallbladder to be 0.4– 0.5 nm. The effects of pH and divalent cations on the ionic permeability of the tight junction suggest that the strands possess a negative charge, presumably due to acidic residues on the junctional proteins. In rabbit gallbladder, the equivalent pore size of the junction is comparable to ionic dimensions, and steric and viscous effects (Chap. 7) influence the cation selectivity of the shunt. Moreno and Diamond (1975) showed that, when these were taken into account, the selectivity sequence of the junction reflected electric field effects like those described in Chapter 4. They assumed that the observed ratio of the permeability of the ith alkali ion to that of sodium was the product of two ratios, the first representing electrostatic effects, and the second the effects of pore size: a  ki ¬­ Q*i ki ž . ž ­ kNa žžŸ kNa ®­­ Q*Na

(10.1)

Equation (7.23), with si equated to the unhydrated radius of the ith ion, was used to calculate Qi*. Table 10.2 presents the experimental permeability ratios obtained by Moreno and Diamond, and the “corrected” ratios, (ki /kNa)c, which were calculated by dividing each experimental ratio by the corresponding Q*-ratio. The corrected ratios

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Figure 10.3. Sketch of a tight junction. Reprinted with permission from Alberts et al. (2002). Copyright © 2002, Garland Science.

follow an Eisenman sequence, so electric field effects are probably present. Furthermore, when the pH of the bath was lowered to suppress electrostatic effects. the corrected ratios approached the ratios of the mobilities of the ions in free solution.

Table 10.2. Experimental and Corrected Permeability Ratios in Rabbit Gallbladder (Moreno and Diamond, 1975) Ion

Experimental ratio ki /kNa

Corrected ratio (ki /kNa)c

Li Na K Rb Cs

0.90 1 1.92 1.46 0.58

0.64 1 3.07 2.92 1.79

Although the tight junction appears to form a continuous band between adjacent cells, it is possible that the conformation of the junctional complex that forms

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when adhesion molecules from adjacent cells interact may itself contain a small gap that would constitute a “paracellular channel” (Wong and Goodenough, 1999). Simon et al. (1999) have presented evidence for such a channel mediating magnesium transport in the kidney. It may be relevant that the equivalent pore sizes of the tight junction, obtained in earlier work, are not so different from the estimated diameters of the more well-characterized ion channels that cross the plasma membrane. Perhaps in some cases the equivalent pore size of the tight junction is, in fact, the dimension of a real channel. The shunt is a major route for passive ion and water transport across leaky epithelia, and also for large polar nonelectrolytes, which find it difficult to cross the epithelial cell membranes. However, the shunt is not the primary path for all nonelectrolytes. In addition, there is evidence in many epithelia for a substantial transcellular water flux; see, for instance, the models of epithelial water transport later in this section. The shunt paths in tight and leaky epithelia differ substantially in resistance, but they do share some qualitative features. For instance, they are both cation selective under normal conditions, presumably because of their negative charges, and they both pass potassium more readily than sodium. There appears to be a relationship between the junctional resistance and the number of strands of junctional protein in the intercellular gap (fewer in leaky epithelia, more in the tighter tissues), which applies across the entire range of epithelial resistances. This supports the notion that the structures of the tight junctions in leaky and tight epithelia are not fundamentally different. The Transcellular Pathway and the Control of Sodium Flux. The relative importance of the transcellular path is greater in tight epithelia than in leaky ones, though it plays an important role in both. A significant feature of the transcellular path is its ability to control the transepithelial sodium flux; in the kidney, this maintains the normal concentration of sodium in the extracellular fluid. The direction of the sodium flux is from the apical side of the epithelium to the basal side. Sodium passively enters the cell through highly selective channels in the apical membrane, as well as through secondary transport systems of which it is the cosolute, and is actively transported across the basolateral membranes by the Na–K exchange pump (Fig. 10.4). The control of sodium throughput is centered in the apical membrane. In the steady state, the rate of sodium extrusion from the cell by the basolateral pump is equal to the rate of passive entry of the ion across the apical and basolateral membranes (Fig. 10.4). The basolateral influx is recycled to the serosal environment, so the transcellular flux is simply equal to the rate of sodium entry across the apical membrane. When the apical sodium conductance is raised, the intracellular sodium level rises, and the rate of extrusion of sodium through the basolateral membranes increases as well. The size of the increase in intracellular sodium depends on the sensitivity of the pump rate to the sodium activity in the cell; the more sensitive the pump rate, the smaller the increase in intracellular sodium that is needed to reestablish the steady state. One would like the pump rate to be rather sensitive to substrate level, so that the internal composition of the cell does not vary markedly when the apical conductance is altered by hormones or other agents. Indeed, the Na–K pumps in sodiumtransporting epithelia normally operate far from saturation.

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Figure 10.4. The transcellular sodium pathway. Open circles in the cell membrane represent the Na–K exchange pump; the filled circle represents the secondary transport systems in the apical membrane for which sodium is the cosolute. Dashed lines represent the passive entry of sodium across the basolateral membranes; this material is recycled to the serosal side by the Na–K pump. Passive basolateral sodium flux is often a minor contributor that is ignored in models.

Similarly, stimulation of the Na–K pump by a variety of agents induces an increase in the apical conductance. The steady-state condition requires that sodium enter the cell faster if it is pumped out faster; this is accomplished by a reduction in intracellular sodium level, which increases the driving force for sodium entry across the apical membrane. The extent of this reduction depends on the apical conductance: the greater the conductance, the smaller the change in intracellular sodium level needed to reestablish the steady state. This change is even less if there is an accompanying increase in the conductance of the apical membrane. As described in Chapter 8, the increase in conductance that accompanies a rise in pump rate allows the cell layer to modulate transcellular sodium transport while minimizing changes in cytoplasmic composition and the osmotic stresses that such changes might induce. An important facilitator of sodium passage across the apical membranes of transporting epithelia is the well-studied epithelial sodium channel (ENaC). This channel is highly selective for sodium; its sodium conductance is twenty times that for potassium. It also conducts lighter ions such as hydrogen and lithium, suggesting that cations traverse the channel without their hydration clouds. It does not appear to be gated by any of the stimuli described in Chapter 4; accordingly, when the apical conductance is determined by this channel, it reflects primarily the number density of channel proteins that are present. The number density, in turn, is determined by the kinetics of protein insertion into, and removal from, the membrane (Snyder, 2002). The single-channel conductance of the protein increases when the intracellular sodium

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level is lowered; this could explain the increase in apical conductance described earlier. The sodium sensitivity of the single-channel conductance also keeps the intracellular sodium level from rising dangerously if the Na–K pump is compromised. The ENaC channel is also an important mediator of the effects of hormones such as antidiuretic hormone (ADH) and aldosterone, both of which promote increases in sodium flux into the cells by signaling the recruitment of more channel protein molecules into the apical membrane. By increasing the transcellular sodium flux in the kidney, these hormones maintain the body's store of extracellular sodium at an appropriate level, which in turn helps regulate blood pressure. The hormone atrial natriuretic peptide (ANP) acts in the opposite direction, reducing ENaC conductance. As noted above, sodium also crosses the apical membrane via a multitude of secondary transport systems. Several of these will be described in the examples presented later in this chapter.

10.1.3. Transport in a Parallel Path System We now move to a discussion of several features of epithelial transport that are a direct consequence of the organization of this tissue. In this section, we deal with the first of these: the presence of circulation in parallel path structures, and its implications for the prediction of transepithelial solute flux. Because of the number of transport barriers in even the simplest epithelium or endothelial monolayer, many transport coefficients must be measured to characterize fully the transport process. It is difficult to make all the required measurements, if only because the transcellular and paracellular routes cannot be physically separated for individual study. If only ions cross the barrier, a variety of clever electrophysiological techniques, such as the use of intracellular electrodes and ion substitution experiments, can be used to isolate and measure many of the individual resistances to transepithelial flux. However, when solute– solvent coupling is significant, and a nonequilibrium thermodynamic formulation is required, the number of transport coefficients that must be measured increases further, and often exceeds the number of experiments that can be performed on the tissue. This problem persists even if the transcellular and paracellular paths are each modeled as a single barrier and only one solute crosses the cell layer. In this simplest situation, solute and water cross the epithelium via two parallel paths. Six phenomenological coefficients are needed to define the transport behavior of the system, three for each path. To evaluate these coefficients, six independent transport measurements must be made on the tissue. But there are only two fluxes to measure (Jv and Js ), and each of these depends linearly on the same two driving forces ('P and 'S). Naturally, this problem is worse for a multilayer epithelium: the pair of paths becomes a network, and the number of unknown coefficients increases further. Small wonder, then, that the transport properties of epithelia have often been characterized as though the cell layer were a simple (i.e., single path) membrane, having one hydraulic conductivity and, for each solute, one reflection coefficient and one permeability. Such global transport coefficients do not provide much information about the individual transepithelial paths. Furthermore, if these coefficients are used to predict the

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transepithelial solute flux when there is a concentration gradient across the tissue, the prediction will generally be wrong. To see how this error arises, we begin by reviewing the techniques that are commonly used to measure the phenomenological transport coefficients, and how these measured quantities are used to predict fluxes. Then we apply these considerations to a parallel-path membrane. Measurement of Global Phenomenological Coefficients and Prediction of Fluxes. Consider a simple membrane permeable to a single uncharged solute and water. All solutions are ideal and dilute. Transport across the membrane is governed by the Kedem–Katchalsky equations derived in Chapter 6:

J v  L p 'P  VL p 'S 

(6.32a)

J s  J v (1  V)cs Z'S ,

(6.36)

where 'S = RT'cs. The three transport coefficients required by these equations are commonly found in the following way: The hydraulic conductivity is often obtained from a measurement of the volume flow generated by a hydrostatic pressure difference in the absence of a concentration difference; that is, J ¬ L pa  žž v ­­­ , žŸ 'P ® 'S0

(10.2)

where the prime denotes an experimental, global, quantity. One method of obtaining the reflection coefficient is to measure the volume flow induced by an osmotic pressure difference in the absence of a hydrostatic pressure difference. The computation uses the experimental hydraulic conductivity:  J ¬­ V a  žžž v ­­ . žŸ L pa 'S ®­­ 'P  0

(10.3)

The experimental permeability, Zc, is usually obtained from the flux of radiolabeled solute at zero volume flow. These experimental coefficients are then substituted into the Kedem–Katchalsky equations to predict the fluxes across the barrier for other values of the driving forces 'P and 'S:

J v(pred)  L pa 'P  VaL pa 'S ,

(10.4)

J s(pred)  J v(pred) (1  Va)cs Za 'S .

(10.5)

This procedure will give the correct fluxes if the membrane is simple, because the experimental transport properties are in fact the true transport properties of the membrane. For instance, if Jv given by Eq. (6.32a) is substituted into the right-hand side of Eq. (10.2), one obtains L pa  L p .

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Application to a Parallel-Path Membrane. The situation is more complex when parallel paths are present. Consider a membrane possessing only two kinds of paths, whose properties are denoted by the subscripts “1” and “2.” The fluxes across the membrane are the sum of those through each path:

Jv = Jv1 + Jv2 = Lp1'P – V1Lp1 'S + Lp2'P – V2Lp2 'S

(10.6)

J s  J v1 (1  V1 )cs Z1'S J v 2 (1  V2 )cs Z2 'S .

(10.7)

Note that the driving forces are the same for each parallel path. Now assume that the parallel-path membrane is characterized in exactly the same way as the simple membrane considered earlier. Equations (10.6) and (10.7) can be used to express the experimental transport coefficients in terms of the true coefficients of the individual pathways, normalized to the area of the composite membrane. Substituting Eq. (10.6) into (10.2),

L pa  L p1 L p 2 .

(10.8)

Similarly, substituting Eqs. (10.6) and (10.8) into (10.3), Va 

V1 L p1 V2 L p 2 L p1 L p 2

(10.9)

.

When radiolabeled tracers are used to measure the solute permeability, the hydrostatic pressure and concentration of the abundant (i.e., unlabeled) solute are the same on both sides of the membrane. The osmotic pressure difference caused by the tracer concentration difference is negligible. Therefore, the only flux through either path is the diffusion of tracer down its concentration gradient. The experimental tracer permeability is easily shown to be the sum of the permeabilities of the two paths: Zc = Z1 + Z2.

(10.10)

By substituting Eqs. (10.8)–(10.10) into (10.4) and (10.5), the predicted fluxes can be written in terms of the transport coefficients of each path, and can be compared with the true fluxes given by Eqs. (10.6) and (10.7). When this is done, it is found that J v(pred)  J v , but J s(pred)  J s 

L p1 L p 2 (V1  V2 )2 cs 'S L p1 L p 2

.

(10.11)

Equation (10.11) can be rewritten as J s  J s(pred) H s ,

(10.12)

where Hs is the flux error. When 'Sis positive, Js and Hs are also positive (except when the two reflection coefficients are the same, and the error is zero); thus, the solute flux

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across this parallel-path membrane will always be underestimated if the membrane is assumed to be simple (and V1 z V2 ). The Solute Flux Error for a Membrane Containing Multiple Parallel Paths: Membrane Heteroreflectivity. The error term in Eq. (10.12) can be derived for any number of parallel paths:   (œ Vi L pi )2 ¯° H s  ¡¡ œ Vi2 L pi  cs 'S , œ L pi °±° ¡¢

(10.13)

where the sum is taken over all the paths, indexed by i. For a given driving force, the error term is proportional to the bracketed function, which is a property of the membrane and can be written as follows:

[...]  hL pa ,

(10.14)

where h  œ Vi2 O i  (œ Vi O i )2 , and the weighting factor Oi w Lpi /Lpc. The quantity h is the variance of the weighted reflection coefficients of the individual paths, and has been termed the heteroreflectivity of the membrane (Friedman, 1976). The error in predicted solute flux is directly proportional to this property of a parallel-path membrane. If the heteroreflectivity of the membrane is zero — that is, if the reflection coefficients of all paths are the same — then Hs is also zero and the barrier can be regarded as simple, irrespective of the relative hydraulic conductivities {Oi} and solute permeabilities {Zi /Zc} of the several paths. Equation (10.13) shows how the error in predicted solute flux depends on the transport properties of each path, but it is not very practical, because these properties are not known. The properties that are known are the experimental (i.e., primed) transport coefficients. These, though, are insufficient to determine Hs ; however, they can provide an upper bound on this quantity (Friedman, 1976):

H s ,max  L pa Va(1  Va)cs 'S .

(10.15)

Equation (10.15) can be used to estimate whether, for a particular barrier/solute combination, parallel path structure can have an important effect on the correctness of the predicted solute flux. Where does the “extra” solute flux come from? What transport process have we ignored by treating the heteroreflective membrane as though it were simple? To answer these questions, consider an experiment (Fig. 10.5) in which an osmotic pressure difference across a membrane is balanced by a hydrostatic pressure difference, so that the volume flux is zero. What is the solute flux? For a simple membrane, the answer is given directly by the Kedem–Katchalsky equation (6.36) for solute flux:

J s J 0  Z'S . v

(10.16)

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CH. 10: EPITHELIAL TRANSPORT

Figure 10.5. Two experiments at zero volume flow. (a) Simple membrane. The single pore in the membrane represents the one kind of path across it. There is no volume flow through this path. (b) Heteroreflective membrane. The upper pore represents a path with a low reflection coefficient; the lower pore has a higher reflection coefficient. The volume fluxes (solid arrows) through the two pores are equal and opposite; the convected solute fluxes, indicated by dashed arrows, are not.

However, for a heteroreflective membrane, the picture is not so simple. To begin with, the hydrostatic pressure difference needed to balance the osmotic pressure difference is not the same for all paths, since their reflection coefficients differ [recall Eq. (6.38)]. Yet the paths are in parallel, so the same hydrostatic pressure difference must act across all of them. Thus, in the experiment in Figure 10.5b, the applied pressure cannot abolish the separate volume fluxes through each of the individual paths, but I II only the sum of these fluxes. If S > S , then the volume flux through the paths with the higher reflection coefficient will go from Phase II to Phase I, and an equal volume flux will pass through the more open paths from Phase I to Phase II, in the same direction as the diffusive solute flux. Since the volume flux from Phase I to Phase II passes through the paths that have the lower reflection coefficient, it will convect more solute than will the equal volume flux from Phase II to Phase I (recall that the convected solute flux is proportional to 1 – V). Thus, the effect of the circulating volume flow is a net convective contribution to the solute flux in the same direction as the diffusive component. This contribution, which persists when Jv z 0, is unrecognized when the membrane is regarded as simple. Some Special Cases. We close by applying the analysis to a few special cases: When solute and solvent cross the membrane through separate pathways (one pathway per species, for simplicity), the reflection coefficient of the solvent pathway is unity because solute cannot be convected through it, and the solute permeability of the pathway is zero. Similarly, the hydraulic conductivity of the solute pathway is zero, so O= 1 for the solvent pathway. The quanitity h, and the error in solute flux, are zero, by Eq. (10.14). This is a reasonable result, since a circulating volume flow cannot be established through the solute pathway.

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If the membrane is not permeable to solute, the reflection coefficients of all paths are identically unity, so there is no variance among them, and hence no solute flux error. Naturally, the solute flux is zero in this case. If the membrane does not pass solvent, the experimental hydraulic conductivity is zero, and so is the solute flux error. There can be no circulating volume flow if solvent cannot cross the membrane.

10.1.4. Coupling of Transepithelial Water Flow to Active Ion Transport Among the species that cross epithelia and endothelia, water is one of the most important. In many cases, the driving force for water flow is a favorable chemical potential gradient, resulting from a difference between the osmotic and/or hydrostatic pressures at the two faces of the cell layer. In the case of osmotically driven flow, the water flux is directed toward the phase with the higher osmolarity, and thus tends to diminish the osmotic pressure difference that drives it. If the cell layer is sufficiently permeable to water, a significant osmotic pressure difference cannot be maintained across it, and any solute flux is accompanied by a flow of solvent. The solvent is said to “follow” the solute across the membrane. The driving force for the solvent flow is its chemical potential gradient. Because the hydraulic conductivity of the cell layer is high, a very small osmotic pressure difference is sufficient to generate the water flow that accompanies the solute flux across the cells. In addition to this mechanism, epithelia are also capable of transporting water from their mucosal side to their serosal side, in the absence of — or against — a gradient in the chemical potential of the solvent. This transport mechanism, which is a consequence of the active extrusion of sodium into the paracellular space by the lateral cell membranes, is intimately tied to the structure of the cell layer and is the primary topic of this subsection. We refer to this transport as “coupled” rather than “active” because the water flux is entirely passive, driven by its own chemical potential gradient. The solvent does not participate directly in the Na–K pump that generates that gradient. The first model of coupled water transport was proposed by Curran (1960) to explain the observed water flow across the intestinal epithelium of the rat. This model is a good starting point for our discussion because the equations describing its behavior are relatively simple, and it affords an intuitive feel for how the process might work. The Curran Model. The key structural feature of this model is the presence of two barriers in series, as shown in Figure 10.6. Barrier 1 contains an active transport system that pumps solute from Phase I to Phase II. All phases are well mixed and a steady state has been achieved. Phases I and III are effectively infinite in extent, so their composition is unaffected by the fluxes across the array. For simplicity, we will examine the behavior of this system when the concentraI III a I III a tions and pressures in the two outer phases are identical: S = S = S , P = P = P . Thus, the solvent chemical potential is the same in Phases I and III. One sure way to determine whether the membrane system defined above can move water from one side to the other is to write the transport equations and solve for

406

CH. 10: EPITHELIAL TRANSPORT

Figure 10.6. The Curran model of transepithelial water flow. For simplicity, only a single uncharged solute, which is also the substrate of the pump in Barrier 1, is assumed to be present. Osmotic components of the flow are indicated by appropriately sized open arrows, and hydraulic components by solid arrows. The volume flows across each barrier are the sum of the osmotic and hydraulic terms, and are indicated by thin arrows.

the solvent flux. The passive fluxes of solute and volume across each barrier are given by the Kedem–Katchalsky equations. Since the barriers are in series, the solute and solvent fluxes across each of them are the same in the steady state. The passive solute p flux across Barrier 2 (Js2 ) equals the sum of the passive and active solute fluxes across Barrier 1 (see Fig. 10.6). The transport equations that describe the system are J v  L p1 ( P a  P II )  L p1V1 (Sa  SII )  L p 2 ( P II  P a )  L p 2 V2 (SII  Sa ) .

(10.17a)

J s  J v cs (1  V1 ) Z1 (Sa  SII ) J sa1  J v cs (1  V2 ) Z2 (SII  Sa ) .

(10.17b)

The four Eqs. (10.17) contain four unknowns: the volume and solute fluxes, and the hydrostatic and osmotic pressures in Phase II. The solution of these equations to show how the solvent flux depends on the pump rate is left as an exercise. Here, instead of continuing with the solution of Eqs. (10.17), we will make some qualitative assumptions about the transport properties of the two barriers, and then proceed more intuitively. These assumptions are the ones implied by Curran in his model. They are: (1) Lp2 > Lp1 and (2) Lp2V2 < Lp1V1. These inequalities can both be sat-

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isfied if V2 is small enough relative to V1. How might such a system produce a solvent flux when the chemical potential of the solvent is the same in the two bounding phases? Assume that the pump is turned off for some time, and is then restarted after the hydrostatic and osmotic pressures in Phase II have become equal to those in Phases I and III. When the pump is turned on again, the solute concentration in Phase II begins to rise above the ambient value. The osmotic pressure difference that develops promotes flow across both barriers into the central compartment. The driving force across each barrier is the same, because the conditions in Phases I and III are identical. Since the LpV product of Barrier 1 is greater than that of Barrier 2, most of the osmotic flow into Phase II enters across Barrier 1, as shown in the lower panel of Figure 10.6. The volume of Phase II is fixed, so the hydrostatic pressure in this compartment rises above the ambient value, causing a hydraulic outflow that balances the inwardly directed osmotic flow. The hydrostatic pressure difference across both barriers is the same. Since Lp2 > Lp1, most of the hydraulic flow leaves Phase II across Barrier 2, as shown in Figure 10.6. Upon adding the components of the volume flow across each barrier, we find a net flow from Phase I to Phase II resulting from the high rate of osmosis across Barrier 1, and a net flow from Phase II to Phase III resulting from the sizeable hydraulic flow across Barrier 2. These net flows are necessarily equal in the steady state and result in the transport of solvent from Phase I to Phase III. In the Curran model, and in all but one of the models that follow, only solute is actively transported. The flux of solvent across the epithelium is entirely passive. The driving forces for solvent flow are produced by the solute pump. No one has yet discovered a primary active transport system in which water is the substrate. The Standing Gradient Model. A widely accepted model of pump-driven water transport is called the standing-gradient model and is shown, more or less in the form originally presented by Diamond and Bossert (1967), in Figure 10.7. It can be regarded as an approximate mapping of the Curran two-barrier model onto the structure of an epithelial layer. The solute pump that drives the water flow is the Na–K exchange pump in the lateral membranes of the cells. These membranes must therefore be Barrier 1, so the interior of the epithelial cell is Phase I and the intercellular space is Phase II. Phase I communicates with the mucosal fluid via passive pathways for solute and solvent in the apical membranes. The possibility of some solvent entry into the cell across the basal membrane (“short-circuiting” the water transport system) cannot be excluded. The pump causes the intercellular space to become hypertonic, like Phase II in the Curran model, and fluid enters it by osmosis from the cells. Because the lateral membrane area is large, the degree of hypertonicity needed to drive the observed transepithelial water flux into the intercellular space is small; in the proximal tubule of the kidney (see below), it may be only 2–3 mOsm/l. Solute and water pass down the intercellular channel and into the serosal fluid. The water flow is driven by a small hydrostatic pressure gradient directed along the

408

CH. 10: EPITHELIAL TRANSPORT

Figure 10.7. The original presentation of the standing-gradient model of epithelial water transport (Diamond and Bossert, 1967). The degree of shading in the channel indicates the solute concentration, which decreases in the serosal direction; this is the “standing gradient.” The solute pumps, indicated by white arrows, are concentrated at the junctional end of the channel. The dashed line represents the entry of water across the basal membrane of the cells; this material is recycled to the serosal side via the intercellular channel. The mapping of the phases and barriers of the Curran model onto the epithelium is also shown.

length of the channel and decreasing toward the open end; osmosis in the channel is negligible. The serosal fluid is equivalent to Phase III in the Curran model, and the terminal portion of the channel can be regarded as Barrier 2. Compared to molecular dimensions, the channel is very wide, so the reflection coefficient of “Barrier 2” is zero and its hydraulic conductivity is large. Thus, flow down the channel and into the serosal fluid is purely hydraulic, similar to the Curran model. Indeed, the two inequalities satisfied by the transport coefficients of the barriers in the Curran model are also satisfied by the properties of the “barriers” in the standing-gradient model. Although there are similarities between the Curran model and the standing-gradient model, there are also differences between them: 1. In the original standing-gradient model, the single solute was assumed to cross the lateral membranes only by active transport. This restriction does not apply to Barrier I in the Curran model. 2. The intercellular space, which is the counterpart of Phase II in the Curran model, is not well mixed, since it contains the standing solute concentration gradient for which the model is named. The concentration in the gap

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between the cells varies with distance in the epithelial thickness direction and is defined by an ordinary differential equation that includes diffusion, convection, and a source term due to the pumps in the lateral cell membranes: Di ciaa vcia  ci v a

2J a 0, r

(10.18a)

where Di is the solute diffusion coefficient, ci is solute concentration in a the channel, v is solvent velocity in the channel, J is the rate of solute injection into the intercellular space per unit area of lateral membrane, and r is the radius of the channel, whose cross-section was assumed to be circular. The primes denote differentiation with respect to distance in the epithelial thickness direction. Equation (10.18a) is derived in the Appendix to this chapter. The corresponding equation for solvent velocity is 2Jv  va  0 , r

(10.18b)

where Jv is the passive osmotic volume flow across a unit area of lateral membrane. The derivation of Eq. (10.18b) is left as an exercise. 3. In the standing-gradient model, there is no discrete “Barrier 2.” In the Curran model, an important function of Barrier 2 is to restrict solute flow, a so that the osmotic pressure in Phase II can exceed S and drive solvent into Phase II across Barrier 1. This function is served in the standing-gradient model by the distributed diffusional resistance along the length of the intercellular channel. Since the first analysis of the standing-gradient model by Diamond and Bossert (1967), considerable effort has been made to improve the model by relaxing some of the earlier assumptions. Some of the criticisms of this model are (Wright and Loo, 2000): 1. The Na–K pumps in the basolateral membrane of water-transporting epithelia are uniformly distributed over the membrane rather than being concentrated at the apical end of the intercellular channel as stipulated in the model. 2. In the original presentation of the standing-gradient model, the apical junction was assumed to be impermeable to both solute and water. Subsequent experiments have shown that some of the transported fluid enters the intercellular space through the junctions, but the importance of this contribution is uncertain. A similar leakiness of the junction to sodium could dissipate the intercellular standing gradient to some degree. 3. Epithelial hydraulic conductivities, obtained from measurements of osmotically driven volume flow, are 100 to 1000 times smaller than the conductivities required by the standing-gradient model. This difference has

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CH. 10: EPITHELIAL TRANSPORT

been attributed to unstirred layers adjacent to the epithelium in the osmotic experiments. Even when the epithelial preparation is flat, the thickness of these layers cannot easily be reduced by stirring to less than 50– 100 μm without damaging the tissue. It is generally agreed that unstirred layers do cause the measured hydraulic conductivity to be less than that of the epithelium alone; at issue is whether they can be responsible for an underestimate of the size needed to reconcile the model with the osmotic flow measurements. 4. Aquaporin channels have not been found in the plasma membranes of water-absorbing intestinal epithelia, so, at least for these cells, it is not clear how water enters the cell enroute to the intercellular channel. These deficiencies have motivated three kinds of responses. The most straightforward has been to “patch” the model (which the authors frankly acknowledged was simplified) by adding the missing components. For instance, Diamond and Bossert, in their original paper, examined the effect on effluent tonicity of the distribution of solute pumps in the intercellular channel. A second approach has been to create new models of coupled water transport that, by their geometry or assumptions, are distinct from the standing-gradient model. However different from the standing gradient they may be, however, these models retain the basic notion of the Curran and DiamondBossert models: that a pump generates an osmotic pressure difference that "pulls" water across a rate limiting membrane. The rate limiting membrane is Barrier I in the Curran model, and the lateral cell membranes in the standing-gradient model. The Sodium Recirculation Model. A recent example of the second approach, developed by Larsen and colleagues to describe water flow across intestinal epithelium, is the sodium recirculation model illustrated in Figure 10.8. In their first paper, Larsen et al. (2000) assumed that the water flux was coupled to the flux of only a single solute representing sodium ion; subsequently, the authors expanded their model (Larsen et al., 2002) to distinguish among the fluxes of sodium, chloride, and glucose. We will discuss the earlier model and compare it with those described above, which also assumed a single solute. The recirculation model differs from the standing-gradient model in several important ways. There are barriers at both ends of the intercellular channel (the tight junction at the apical end and the basement membrane at the basal end) which are permeable to both solute and water; solute crosses the barriers by convection and diffusion. The solute flux is described by Eq. (7.30), with )c = 1 – V and )d = 1. Most important, the intercellular channel is assumed to be well mixed; there is no concentration gradient along its length. Finally, the fluxes across the apical and basal cell membranes are included in the model, and the net influx of solute through the basal cell membrane plays an important role; hence the authors' reference to “solute recirculation.” Larsen et al. estimate that 60–80% of the substrate of the paracellular pumps enters the intestinal epithelium from the serosal side. Exchanges between the cells and the ambient media were absent from the Diamond–Bossert model, which was limited to the transport processes taking place within the intercellular channel.

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Figure 10.8. Recirculation model of intestinal water transport (Larsen et al., 2000). The anatomy is presented similarly to the standing gradient anatomy in Figure 10.7; an important difference is the inclusion in this model of the epithelial basement membrane at the end of the intercellular channel. As before, solute pumps are indicated by white arrows, and solvent fluxes by thin arrows; the pumps are drawn as though they are uniformly distributed along the intercellular channel, but in this model the channel and cell interiors are assumed to be well mixed, so the distribution of the pumps is irrelevant. Passive solute fluxes are indicated by bold arrows whose size reflects the relative size of the flux. Numerical values of the solvent and solute fluxes, normalized to the net flux of each species across the epithelium, are also presented.

Indeed, the recirculation model shares more features with the Curran model, with Phase II being the intercellular space. Solute is pumped into this space, making it hypertonic. Water enters the space osmotically, from the cells and through the tight junc–5 tions; the reflection coefficient of the basement membrane is very low (10 in their baseline model), so the driving force for osmosis from the serosal space is negligible. The intercellular hydrostatic pressure exceeds the serosal pressure by a small amount (5 mm Hg in the baseline model}, but this is sufficient to drive water through the basement membrane, whose hydraulic conductivity is two orders of magnitude higher than that of any other barrier in the system. The inequalities stipulated by Curran are therefore met, with Barrier 2 being the basement membrane, and Barrier 1 representing the several barriers across which water enters the intercellular space. The Cotransporter Hypothesis. A third response to the limitations of the original osmotically based models of epithelial water transport is to propose an entirely different mechanism, as has been done by Wright and Loo (2000). We shall see later in this chapter that sugar absorption in the intestine is mediated by a sodium–glucose cotransporter, SGLT-1, in the apical membrane of the intestinal epithelial cell. The transporter is driven by the transmembrane gradient in sodium electrochemical potential that is produced by the Na–K pump in the basolateral membrane.

412

CH. 10: EPITHELIAL TRANSPORT

Wright and Loo propose that this transporter carries water into the cell as well as glucose; thus, water crosses the cell layer because SGLT-1 pushes water into the cell from the mucosal side, and not because (as in the models described earlier) osmotic forces pull it out of the cell en route to the serosal surface. Their hypothesis is said to explain (Loo et al., 2002) (1) the substantial amount of water that crosses the intestinal epithelia in spite of the apparent absence of aquaporin molecules from their cell membranes, and (2) how diarrhea is reduced by glucose administration. The route by which water leaves the cell is unspecified; perhaps other facilitated or active transport systems are involved. Controversies are important stimulators of scientific progress, and this is only one of several such issues related to water transport (Reuss and Hirst, 2002).

10.1.5. The Effect of Unstirred Layers on Transepithelial Diffusion and Osmosis: Concentration Polarization Two kinds of unstirred layer effects reduce the flux across sheets of cells. As discussed in Chapter 2, stagnant layers adjacent to a diffusion barrier act like series resistances, reducing the driving force for passive transport across the barrier itself. A second kind of effect is seen when there is solvent flow across the barrier. When this is the case, solute is convected toward the barrier at one side, and swept away at the other. In the presence of unstirred layers, this causes the concentration at the upstream face of the barrier to rise, and that at the downstream face to decrease. This is known as concentration polarization. Concentration polarization generates an osmotic driving force that opposes solvent flow, reducing the solvent flux across the barrier. Two examples of concentration polarization are illustrated in Figure 10.9. In Figure 10.9a, the concentration profile, cs(x), in the neighborhood of a barrier is shown for an experiment in which the bulk concentrations are the same on both sides of the barrier and a specified volume flow, Jv, is driven across the barrier by a hydrostatic pressure difference. For simplicity of analysis, we will assume for the time being that the barrier is impermeable to the solute. The concentration profiles in the unstirred layers are described by the onedimensional convective diffusion equation (2.29). The fluid velocity in Phase I, directed toward the barrier, is Jv. Since the membrane is impermeable to solute, the steady-state solute flux at the upstream face of the membrane is zero. Thus, at this face (and at any parallel plane to its left), the solute concentration gradient generates a driving force for diffusion toward the left that exactly balances convection toward the right: Ds

dcs  J v cs . dx I

(10.19) I

Rearranging, and integrating from cs(–G) = csb to cs(0) = cs ,

Ds ln

csI  JvG . csbI

(10.20)

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Figure 10.9. Concentration polarization caused by unstirred layers of thickness G. (a) Volume flow-driven: 'csb = 0, 'P > 0. The hydrostatic pressure difference drives a volume flow from Phase I to Phase II. The barrier need not be permeable to solute. An osmotic pressure difference RT (csI  csII ) , which opposes 'P and reduces the volume flow, develops across the barrier. (b) Concentration difference-driven: 'csb > 0, 'P = 0. The osmotic pressure difference RT'cs drives a volume flow from Phase II to Phase I that is less than it would have been had the unstirred layers been thinner or absent.

Exponentiating Eq. (10.20), csI JG  exp v , I Ds csb

(10.21)

where the exponent is the relevant Peclet number (Chap. 7). Thus, the concentration of solute at the membrane surface is greater than that in the bulk solution by a multiplicative factor that depends on the volume flow, the solute diffusion coefficient in the unstirred layer, and the unstirred layer thickness. Note that the two concentrations are the same if either the volume flow or the thickness of the unstirred layer is zero. A similar, but opposite, effect is seen at the downstream face of the barrier. The osmotic pressure difference RT(csI – csII) resulting from concentration polarization opposes Jv and must be compensated for by raising the hydrostatic pressure difference across the barrier. The analysis of the system depicted in Figure 10.9a when the barrier is permeable to the solute is left as an exercise. In general, the rate of solvent flow through the barrier is not specified (as it was above), but rather is determined by the boundary conditions (pressures, concentrations) in the bulk solutions outside the unstirred layers. The effect of unstirred layers on purely osmotic water flow was modeled by Pedley (1983), under the assumption that the barrier was impermeable to solute. This situation is shown in Figure 10.9b. In this case, the solvent flow is not specified; rather, it is determined by the osmotic pres-

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sure difference across the barrier. The concentration profiles in both diffusion layers reflect the balance of convection and diffusion discussed above. Pedley solved for the ratio, J, of the solvent flux in the presence of unstirred layers to that in their absence. Since all flow in this system is osmotic ('P = 0), J is also the ratio of the true solute concentration difference across the cell layer, to the difference between the concentrations in the bulk solutions outside the diffusion layers: J

csI  csII . csbI  csbII

(10.22)

Pedley found that J  TII eEJ  TI eEJ II

where Ti 

(10.23)

I

L p RT (csb  csb )G csbi and E  ; Lp is the hydraulic conductivity of the II I csb  csb Ds

barrier. The quantity EJ is proportional to the product of the hydraulic conductivity of the cell layer and the true concentration difference across it; thus, it is proportional to the osmotic volume flux. Equation (10.23) can be manipulated to show that EJ increases toward a T-dependent asymptote as E becomes arbitrarily large. This means that, in the presence of unstirred layers, the osmotic flux cannot exceed a limiting value that depends on the bulk concentrations, no matter how leaky to water the membrane is. Unstirred layer effects have almost always been analyzed using one-dimensional flux equations. It is assumed that the only important spatial coordinate is the distance from the membrane surface, and that there are no gradients in any other direction. This simplification is valid if any spatial variations (in the plane of the membrane) in membrane transport properties occur over distances that are small relative to G. It is not clear that this requirement is always met for epithelia in which paracellular fluxes are important, since unstirred layer thicknesses and the spacing between paracellular paths (i.e., the width of a cell) are comparable. Hence, the models described above are most appropriate when stirring is poor, and the effective unstirred layer thickness is perhaps 100 μm or more. Of course, it is under these conditions that unstirred layer effects are most important, so the one-dimensional theory is most applicable where it is most needed.

10.1.6. Electrical Analogs of Cell Layers Networks of electrical analogs like those in Chapter 7 can be constructed to represent the transport pathways in epithelial cell layers. These equivalent circuits have been used to interpret the electrical properties of epithelia in terms of transcellular and paracellular conductances and EMFs. Several epithelial analogs are shown in Figure 10.10.

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Figure 10.10. Electrical analogs of epithelia. (a) An epithelium modeled as a single barrier (the Thevenin equivalent circuit). (b) The transcellular path across a tight epithelium; \c is the intracellular potential. (c) A leaky epithelium bathed by identical solutions on both sides. Separate equivalent circuits are used to describe the flux of each ion across the basolateral membrane, which contains a rheogenic pump. (d) Electrical analog of rabbit urinary bladder epithelium including membrane capacitance. [Modified from Lewis and Hanrahan (1990).]

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CH. 10: EPITHELIAL TRANSPORT

Figure 10.11. The current–voltage relation of a linear epithelium.

Since the current–voltage relations of each of the individual circuits that make up the networks in Figures 10.10a–c (and Fig. 10.10d in the steady state) are linear, each of these networks is also linear. The current–voltage relation of a linear (i.e., ohmic) epithelium is shown in Figure 10.11. The slope (i.e., the overall conductance of the tissue) and intercepts (i.e., the open-circuit potential and short-circuit current) of the line are functions of the EMFs and conductances of all the pathways in the tissue. When the current–voltage relation of an epithelium is found experimentally to be linear, it is often written in the Thevenin form (Fig. 10.10a): I = Gepi('\– Eepi).

(10.24)

In this equivalent circuit, I is the transepithelial current density, Gepi is the Thevenin conductance of the epithelium, and Eepi is the potential across the epithelium at open circuit (I = 0), the Thevenin EMF. The Thevenin conductance and EMF are convenient for describing the electrical properties of ohmic tissues or membranes, in the absence of a detailed understanding of the underlying transport mechanisms. An equation similar to Eq. (10.24) can be written to describe the relation between the transcellular current and the transepithelial potential (Fig. 10.10b). In this case, the Thevenin EMF is the epithelial potential at which the transcellular current is zero, and the Thevenin conductance is the overall conductance of the transcellular pathway. Both of these empirical quantities depend on the properties of the apical and basolateral membranes. For many epithelia, plots of transepithelial current vs. transepithelial potential are quite linear over the range of experimental interest. When the current–voltage relation of any of the epithelial membranes is markedly nonlinear (for instance, if a ratedetermining permeability is strongly potential dependent), it is likely that the current– voltage relation of the epithelium will be nonlinear too. For such epithelia, Eq. (10.24)

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417

no longer applies over the entire potential range. The slope conductance of a nonlinear epithelium at a given potential is defined as the slope of the current–voltage curve at that potential: s Gepi 

dI . d ('\ )

(10.25)

Since the conductance of a nonlinear epithelium depends on the epithelial potential, these tissues behave to some extent as rectifiers. The equation for the tangent to the current–voltage curve, at a point ('\c,Ic), is: s I  I a Gepi ('\  '\ a) .

(10.26)

Often, nonlinear epithelia appear linear over limited ranges of potential and current, and Thevenin quantities valid over such a range can be obtained for the tissue. Their values apply only to the portion of the current–voltage curve they represent. As the voltage range over which the curve is approximated becomes narrower, the Thevenin representation of the epithelium's electrical properties approaches Eq. (10.26), which is called the slope formalism. The slope formalism is illustrated in Figure 10.12, which also shows that the Thevenin quantities obtained from the tangent to the curve depend on the value of '\cat which they are determined.

Figure 10.12. The slope and chord formalisms. The dashed and solid straight lines are tangent to different points on the current-voltage curve and are defined by different Thevenin quantities. The slope of the dotted line is the chord conductance at %\  (%\ )1a .

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An alternative formulation, the chord formalism, requires that the open-circuit poc tential of the epithelium be known. The chord conductance, Gepi , is the slope of a straight line connecting a selected point on the current–voltage curve to the open-circuit (I = 0) point on the curve (Fig. 10.12). The equation of this line can be written in the same form as Eq. (10.26): c I  Gepi ('\  PD ) ,

(10.27)

Ia . When the epithelium is '\ a  PD nonlinear, the chord conductance depends on the point ('\c,Ic) at which it is evaluated. The slope and chord conductances are identical for an epithelium that is ohmic, but generally differ if it is nonlinear. The limitations on the Thevenin equivalents that result from a nonlinear relation between potential and current apply to single membranes as well as cell layers. They also hold for the active or passive transport of a single ionic species, if the current carried by the ion is not a linear function of the membrane potential. For a single ionic species moving passively through a membrane, the Thevenin EMF from the slope formalism equals the Nernst potential (which is the zero-current potential for a single ion) only when the transport path is ohmic. A nonlinear relation between pump current and membrane potential is usually represented by an equivalent circuit in which the “conductance” of the active transport system is potential dependent. The model of urinary bladder epithelium shown in Figure 10.10d includes the capacitances of the apical and basolateral membranes. Normally, capacitance effects are unimportant in nonexcitable tissues; this model was developed to simulate an experiment in which the epithelium was exposed to a (nonphysiological) square current pulse. Membrane capacitance does affect — and thus can be estimated from — the variation in transepithelial potential that follows a sudden change in current. Since the capacitance depends on membrane area (1–2 μF/cm2), measurements of the total capacitance of a cell preparation can under certain circumstances be used to estimate the true apical or basolateral membrane areas. Lewis and Hanrahan (1990) used this approach to estimate the apical membrane surface area in the rabbit urinary bladder.

c where PD is the open-circuit potential and Gepi 

10.2. EXAMPLES OF EPITHELIAL AND ENDOTHELIAL FUNCTION The variety of epithelia and their functions is considerable, and no attempt is made here to review the behavior of all epithelial tissues. Rather, we will select some examples that illustrate the roles that epithelia play in living systems. Our discussion will be organized in terms of three specific transport functions performed by cell layers: absorption, secretion. and filtration. Transcellular absorption, the uptake of materials at the mucosal surface of the epithelium and their transport to the serosal side, will be illustrated using three examples:

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the absorption of sugars by the small intestine, and the reabsorption of water and sodium in the proximal and distal tubules of the kidney. The regulation of vascular endothelial secretion by mechanical stresses will be reviewed briefly, followed by a description of glandular secretion. The molecular mechanisms of the secretory process were described in Chapter 3. Like absorption, epithelial secretion is decidedly directional, a consequence of the polarity of the cell layer. The mechanisms by which the polarity of the apical and basolateral membranes is maintained will also be discussed. Filtration will be illustrated by the ultrafiltration of plasma in the glomerulus of the kidney, and by Starling's Law for fluid exchange between capillary and interstitium in the microcirculation.

10.2.1. Absorption Absorption of Sugars in the Small Intestine. The principal function of the small intestine is to absorb nutrients from the digestive tract into the blood. Stretched out, the human small intestine is about 6 m long, tapering from a diameter of 5 cm at its origin to 2 or 3 cm, where it terminates at the large intestine. 2 A smooth tapered tube of this size has a surface area of 8000 cm . The rate of nutrient absorption across such an area would not be sufficient to meet the body's needs. Accordingly, the surface area of the intestine is increased many hundred times by the 2 folds and fingerlike projections illustrated in Figure 10.13, to 250 m , the surface area of a tennis court. This convoluted apical surface of the epithelial layer across which transport takes place is known as the intestinal brush border. Absorption across the intestinal epithelial cells, called enterocytes, is from the mucosal side to the serosal. Many mechanisms are used to accomplish this, depending on the polarity and size of the molecule or ion to be transported. Channels or less restrictive free-diffusion paths are employed by electrolytes and water-soluble vitamins; some lipoproteins and other large molecules are thought to enter the cell by pinocytosis or cross it by transcytosis; and there are numerous passive and active carrier-based transport systems present as well, moving sugars, amino acids, bile salts, nucleotides, and small ions across the cell membranes. Simple sugars, primarily glucose, are among the most important molecules absorbed from the intestinal lumen. These simple sugars are produced in the intestine by hydrolytic enzymes in the brush border that degrade complex sugars in the gut, like lactose and sucrose, to simpler sugars like glucose, fructose, and galactose. The absorption of sugars by the small intestine is an excellent illustration of how epithelia perform their absorptive functions, since it involves four of the transport mechanisms discussed in earlier chapters: passive carriers and channels, and primary and secondary active transport. The transport systems involved in sugar absorption are shown in Figure 10.14. The primary active process that drives the intestinal absorption of glucose and galactose is the Na–K exchange pump in the basolateral membrane of the cell. In the steady state, the pump-generated influx of potassium and efflux of sodium must be matched

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Figure 10.13. (a; facing page) Increasingly magnified views of the small intestine. [Reprinted with permission from Moog (1981). Copyright © 1981, Scientific American.] (b) Scanning electron micrograph showing villi. The image is comparable to the region in Panel 2 of (a) enclosed by dashed lines. [Reprinted with permission from Poley and Klein (1983). Copyright © 1983, Lippincott Williams & Wilkins.]

by oppositely directed passive fluxes. Part of the passive sodium flux into the cell enters through channels in the cell membrane; however, some of the sodium gradient is dissipated through a sugar symport system in the apical microvilli (Fig. 10.13a, Panel 5) that drives glucose and other simple sugars into the cell. The coupling coefficient of this sodium-dependent hexose transporter (SGLT-1) is probably two. Cotransport of sugars into the cell raises their intracellular concentration above that in the interstitial fluid which bathes the serosal surface of the cells. Sugar passively crosses the basolateral membranes of the epithelium, primarily via the GLUT2 carrier, and enters the interstitium, from which it passes into capillary blood in the villus (Fig. 10.13a, Panel 3). The sodium levels in the enterocyte and its surroundings are relatively stable, so the rate of transport of sugar through the hexose transporter depends primarily on the substrate concentrations in the intestinal lumen and the cell. The effect of changes in external sugar levels on intestinal uptake can be easily calculated in the steady state by equating the substrate fluxes across the apical and basolateral membranes, including consumption within the cell if desired. The setup of this simple application of compartmental analysis is included in the problem set.

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Figure 10.14. Cation-driven intestinal absorption of sugars (S).

The absorptive mechanism for amino acids is thought to be similar to that for sugars. The uptake of sugars by the intestinal epithelium can be reduced by increasing the concentration of certain amino acids in the mucosal fluid. Similarly, amino acid uptake is inhibited by elevating the mucosal glucose level. Two explanations of this mutual cross-inhibition have been offered; they are not mutually exclusive. One explanation is that certain carriers in the brush border are capable of transporting some sugars and some amino acids. If this is the case, then cross-inhibition is a form of competitive inhibition (Chap. 4). The other explanation is that increasing the amount of (for instance) a transported amino acid in the mucosal fluid increases the rate of symport of that solute into the cell, and hence the rate of sodium entry via the amino acid cotransporter. This dissipates the sodium concentration gradient across the apical membrane, reducing the rate at which sodium enters the cell via the sugar symport system. Put another way, cross-inhibition may reflect competition between the two kinds of solute for the same carrier, or between two kinds of carrier for the same cosolute. Transport of Water and Sodium in the Proximal Tubule of the Kidney. Two important functions of the kidney are: the maintenance of the composition of the extracellular fluid, in which the cells of the body live (the extracellular fluid is often termed the internal environment of the cells, and the maintenance of its composition is known as homeostasis); and the formation of urine, a hypertonic concentrate of the soluble waste products of metabolism. The kidney controls the ion concentrations, osmotic pressure, and pH of the extracellular fluid, and also regulates the fluid balance of the body. All of these functions involve epithelial transport processes. Transport in the kidney is centered in the nephron, whose anatomy and orientation in the kidney are shown in Figure 10.15. A single kidney contains more than a million of these transport units. Briefly, the nephron works as follows (refer to Fig. 10.15c):

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Figure 10.15. Increasingly magnified views of the kidney. The wedge in (a) is magnified in (b). Two nephrons are shown in (b): a cortical nephron whose glomerulus lies in the outer cortex, and a deeper juxtamedullary nephron. They are identified by the letters C and JM at their proximal tubules and loops of Henle. The flow direction in the cortical nephron is indicated by arrows. A single nephron is shown in (c). Panel a reprinted with permission from Guyton and Hall (2000). Copyright © 2000, Elsevier. Panel b reprinted with permission from Dorland's Illustrated Medical Dictionary, 1974. Copyright © 1974, W.B. Saunders. Panel c reprinted with permission from Guyton (1981). Copyright © 1981, W.B. Saunders.

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blood enters the nephron via an afferent arteriole, and passes into the glomerulus. There, some of it is filtered across the glomerular capillary wall into Bowman's capsule (also called Bowman's space). Glomerular filtration will be discussed later in this chapter. The pores in the glomerular filter are much larger than ionic dimensions, so the concentrations of ions and small molecules in the cell-free filtrate are nearly the same as those in plasma. The filtrate passes into the proximal tubule of the nephron. As it moves through the various tubular segments of the nephron, en route to the collecting tubule and urinary tract, solutes are selectively reabsorbed from it, into the peritubular capillaries that surround the nephron. The transport systems in the nephron, and their relative importance, differ among species and can vary within a single tubule. These variations are noted only briefly here, our object being to give a more general sense of how a variety of transport mechanisms are employed by the kidney to perform its functions. Over 90% of the sodium, glucose, calcium, chloride, and bicarbonate in the glomerular filtrate, and over 80% of the potassium, is reabsorbed into the capillaries. Ninety-nine percent of the water in the filtrate is also reabsorbed, so fluid is conserved while solute content is regulated. What remains of the filtrate by the time it exits from the collecting tubule is urine. By means of the reabsorption process, the kidney removes wastes and foreign solutes (e.g., drugs) from the blood and regulates the composition of the plasma. Since capillary blood readily exchanges solutes with the extracellular fluid, the internal environment of the cells is also maintained. The cells that line the lumen of the proximal tubule are cuboidal, with a microvillous brush border on the luminal side and an infolded (invaginated) basal membrane (also known as the peritubular membrane), which increase considerably the surface area of the cell. Sixty to seventy percent of the water in the glomerular filtrate is reabsorbed by this segment of the nephron. A similar fraction of the solutes in the filtrate, including all of the glucose and amino acids, are also reabsorbed, so the osmotic pressure of the fluid in the tubule remains similar to that of plasma. The changes in tubule fluid composition and transepithelial potential along the length of the proximal tubule are shown in Figure 10.16. The several routes for sodium transport into, out of, and across the tubule epithelium are shown in Figure 10.17a. The amount of sodium that enters the glomerular filtrate in 24 hours is more than one hundred times the daily intake of this ion, so almost all of it must be reabsorbed. Sixty-five percent of the sodium reabsorbed by the kidney travels across the walls of the proximal tubules. A major contributor to the reabsorption of sodium by the proximal tubule is the Na–K pump in the basolateral membranes of the epithelium, which transports sodium out of the cell. The mechanisms of transcellular sodium transport, driven by the pump, are similar to those described earlier in reference to the transcellular pathway, except that sodium enters the proximal tubule cells primarily via an Na–H antiporter in the apical membrane rather than Na-selective channels, which are less important in the proximal tubule. Glucose and amino acid transport systems similar to those in the intestine provide additional routes by which sodium can enter the cells, particularly in the early part of the tubule;

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Figure 10.16. (a) Concentration and (b) potential profiles along the length of the proximal tubule. PD is the potential at the mucosal surface relative to the serosal surface of the cell layer. Glucose and amino acids are reabsorbed by symport systems like those in the intestine. Panel a reprinted with permission from Guyton and Hall (2000). Copyright © 2000, Elsevier. Panel b reprinted with permission from Rector (1983). Copyright © 1983, American Physiological Society.

Figure 10.17. (a) Sodium and (b) water transport paths in the epithelium of the proximal tubule.

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the contribution of these symport systems to sodium reabsorption decreases along the length of the tubule as the concentration of the organic substrates in the luminal fluid falls (Fig. 10.16). Sodium also enters the cells through symport systems that carry lactate and other anions across the apical membrane. There is also a purely passive route for sodium transport from one side of the epithelium to the other: the intercellular shunt. This route is more important in the late portion of the tubule, where the symport systems in the apical membrane are short on substrate. Both the diffusive and convective components of the paracellular sodium flux vary along the length of the tubule, and so can the direction of the flux. The potential difference between the tubule fluid and the interstitium changes sign between the beginning and the end of this segment (Fig. 10.16). In the early proximal tubule, the small (2–4 mV) transepithelial potential opposes the reabsorption of sodium through the shunt (the lumen potential is more negative); in the late tubule, it favors it. The convective component of the paracellular sodium flux depends ultimately on the hydrostatic and osmotic pressures at the epithelial surfaces, since these, along with any coupled water transport, determine the paracellular water flow. It has been shown experimentally that increases in interstitial hydrostatic pressure reduce the serosallydirected paracellular sodium flux. If the interstitial pressure is high enough, the transjunctional sodium flux can actually be driven toward the lumen. This could reflect a reversal in the water flow through the junction (see below). Alternatively, the increased pressure might alter the structure of the junction, making it more permeable to the ion. There is good evidence that the contents of the intercellular space are hypertonic. If this is the case, a driving force does exist for transjunctional sodium diffusion toward the lumen. This driving force would be greater if, as in the standinggradient model, the hypertonicity is maximized just below the junction. The proximal tubule has a large flow conductivity, so a sizeable osmotic pressure difference cannot be maintained across it. Since the osmolarity of the interstitium is uniform along the length of the tubule, so also must be the tonicity of the fluid in the tubule. It follows that the osmolarity of the mix of solutes and solvent that crosses the cell layer into the interstitium must be nearly identical to that in the tubule. Furthermore, any alteration in the rate of solute reabsorption has a parallel effect on water reabsorption. As noted earlier, water is said to “follow” sodium (and the accompanying anions that maintain electroneutrality) across the epithelium. The several routes for water transport across the proximal tubule are shown in Figure 10.17b. The Na–K pump in the lateral membranes is thought to augment water reabsorption by generating a hypertonic region between the cells. This is a primary hypothesis of the standing-gradient model, but it is also consistent with the Curran and recirculation models, which show that coupled water flow can be produced even if the intercellular space is well mixed, as long as there is a barrier with the appropriate transport properties (high Lp, low V) at the basal end. The epithelial basement membrane could be such a barrier. Most of the water flux does appear to enter the interstitium from the intercellular space, which is consistent with — but does not require — coupled water transport. In a recent review, Weinstein (2003) cites earlier estimates, subsequently confirmed experimentally, that more than half of the water transport across the proximal tubule epithelium is coupled to active ion transport.

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The relative importance of the transcellular and transjunctional routes in water transport across the proximal tubule depends on the pressure in the interstitium, to which the transjunctional flow is more sensitive. Low interstitial pressure favors transjunctional flow into the intercellular space. However, if the interstitial pressure is too high, transjunctional flow is reduced and can be reversed. Experimental evidence supports the notion that, under physiological conditions, water flows into the intercellular space through the both routes, but the normal balance between the two is uncertain. The transcellular route to the intercellular space relies on aquaporin-1 water channels in the apical brush border and basolateral membranes of the tubule (along with aquaporin-4 in the basolateral membranes of some species). These channel proteins, besides providing an alternative route to the intercellular space, also provide an exit route to the interstitium through the basal membrane, bypassing the channel between the cells. In man, the glomerular filtration rate, and hence the flow rate through the proximal tubule, varies by about 33% during the day (Koopman et al., 1989), yet the fraction of the entering sodium and water that is reabsorbed in the tubule remains remarkably stable. This glomerulotubular balance implies that the reabsorption rate must respond in some way to the tubular flow rate. It is exhibited by isolated tubules, thus excluding the role of hormones or surrounding structures in this response. Du et al. (2004) have proposed that the tubular epithelia sense the flow past their apical surface as a torque in their microvilli. All else being equal, the torque is proportional to the flow rate in the tubule. The torque could then be the input to a transduction process that ultimately modulates the activity of the rate-limiting Na–H exchanger in the apical membrane. Transport of Water and Sodium in the Distal Tubule of the Kidney. The cells of the distal tubule are similar to the proximal tubule cells, except that they lack a brush border. Like the proximal tubule, the distal tubule reabsorbs sodium and water from the tubule fluid. The distal tubule epithelium plays an important role in maintaining normal potassium levels by secreting this ion into the tubule lumen, thereby contributing about 70% of the potassium found in urine. Some of the transport systems in the lining of the distal tubule are shown in Figure 10.18. The filtrate enters the distal tubule after passing through the ascending limb of the loop of Henle (Fig. 10.15c), a segment that actively reabsorbs ions but has limited water permeability. The water permeability of the initial segment of the distal tubule is similarly low, but that of the rest of the tubule is larger, and is increased in some species by antidiuretic hormone (ADH, vasopressin), which is believed to act by inserting aquaporin channels in the cell membrane (recall Chap. 8). Because of the substantial removal of ions from the luminal fluid during its passage through the ascending limb and the early segment of the distal tubule, the osmotic driving force for water flow out of the remainder of the distal tubule is large, and under the action of ADH, enough water is reabsorbed across the wall of the tubule to raise the tonicity of the fluid leaving this segment to a value not far below that of plasma.

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Figure 10.18. Some aspects of transport in the late distal tubule. This is essentially the model used by Chang and Fujita (1999); see below.

The intercellular junctions in the tight distal tubule are less permeable to electrolytes than those in the leaky proximal tubule. As a consequence, the shunt pathway is less available for ion transport in this portion of the nephron, particularly in the early part of the tubule. However, sodium is absorbed via the transcellular route. The ion enters the cells through an Na–Cl symport system as well as sodium-selective ENaC channels, and is pumped out by the exchange pump in the basolateral membranes. The symporter is more important than the channels in the early portion of the tubule, and the two routes are comparable in the late portion. Though most of the potassium that is pumped into the cell by the Na–K pump returns to the interstitium through channels in the basolateral membrane, some enters the tubule lumen via channels in the apical cell membrane and a K–Cl symporter. The ion and water fluxes across the distal tubule are coupled to some extent and are subject to hormonal and pharmacologic control. Diuretics, such as amiloride, block sodium entry through the ENaC channels in the apical membrane of the epithelium. This reduces the rate of sodium extrusion by the basolateral pumps into the interstitial fluid, and the water flux is also diminished. More water remains in the tubule, and more urine is formed. The hormone aldosterone, which regulates salt and water metabolism, has an opposite effect on the sodium flux. By increasing the conductance of the apical membrane, it enhances sodium reabsorption. Chang and Fujita (1999, 2001) have modeled the transport of water and ions in the distal tubule of the rat. Their first paper dealt with only Na, K, and Cl, and is sufficient to illustrate the modeling principles they used; the second paper includes buffering ions (HCO3, NH4 and PO4) and is accordingly more complex. The transcellular transport processes in the earlier model included all of those in Figure 10.18. The junction at the apical surface of the cell was assumed to be permeable to all three ions, and there was no intercellular compartment since the intercellular space was assumed

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to communicate freely with the serosal phase. They also assumed that both the luminal fluid and the intracellular compartment contained fixed concentrations of nondiffusible charge. The equations they solved are as follows: — one mass balance for each ion in the lumen, equivalent to Eq. (10.18a), with an adjustment to include migration of the ion under an axially directed electric field; — the mass balance equation for the solvent, Eq. (10.18b); The equations in Chang and Fujita differ somewhat from Eqs. (10.18). They defined the movement of species down the lumen in terms of mass flows, F; for solutes, Fi = Sr2vci mols/sec, and for solvent, Fv = Sr2v cc/sec. They also defined fluxes towards the serosal surface as positive, while they are defined as negative in Eqs. (10.18).

— a condition that the mass flow of impermeant (subscript “I”) in the lumen is constant along its length: dFI /dx = 0; — electroneutrality in the lumen (superscript “l”) at any axial location, x: l l cNa ( x ) cKl ( x )  cCl ( x ) zI cIl ( x )  0 ;

— the steady-state condition, applied to each of the four species in the intracellular compartment. This is the only compartment in the model because the entire resistance in the paracellular pathway is in the junction. For each of the four species, the net mass flow entering the cell across the apical membrane equals the net mass flow exiting across the basolateral membrane. — electroneutrality of the cell interior. This equation is identical to the equation for electroneutrality in the lumen, except that intracellular concentrations are used. The interior is assumed to be well mixed. These eleven equations are complemented by another set of equations for the fluxes. To summarize: — volume flows across each of the three barriers (apical membrane, basolateral membrane, and apical junction) were assumed to be driven only by osmosis, and it was further assumed that all reflection coefficients and all osmotic coefficients were unity. — passive ion fluxes through the junction and the channels in the cell membranes were described by the constant field equation, Eq. (2.55), using permeability instead of mobility. — a complex rate equation was derived to relate the fluxes through the Na– Cl and K–Cl cotransporters to the concentrations of their substrates at both sides of the apical membrane. The rate of each cotransporter depended on the concentrations of both substrates (i.e., cation and chloride) on both sides of the membrane and the dissociation constants of the cation and chloride binding sites.

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— the rate of the Na–K exchange pump was given by a nonlinear function of the intracellular sodium level. Chang and Fujita parameterized their model using experimental data, and then showed that it was able to describe the distal tubule response to several drugs and other interventions. This sort of demonstration is essential to build confidence in any model that contains a large number of fitted parameters. The early and late distal tubules were both simulated. The transcellular fluxes predicted by the model were in the same direction in both cases, and the arrows are so directed in Figure 10.18. The sodium and chloride fluxes through the junction were directed into the lumen throughout the tubule, while the potassium flux reversed sign, directed luminally in the early tubule and serosally in the late tubule.

10.2.2. Secretion The process of secretion, by which cells discharge the products of biosynthesis into the extracellular milieu, was described in Chapter 3. Some examples of secretion by epithelia will be presented below. Vascular Endothelium. The endothelia that line the inside of blood vessels are capable of secreting a large variety of synthesized chemicals, such as prostacyclin and nitric oxide, into the vessel lumen. Endothelial secretion may be regulated, at least in part, by ion channels in the luminal membrane that have been shown to respond to the shear stress and tensile stretch to which the cells are subjected by the flowing blood and pulsatile blood pressure. These channels include stretch- and shearactivated cation channels that conduct divalent ions, and potassium-selective channels that have been shown to open upon exposure to shear stresses in the physiological range (ca. 10 dynes/cm2). The mechanism by which the mechanical stress acting on the endothelial cell membrane is translated into channel opening is unclear, but it is believed that the cytoskeleton plays a role in force transmission to the transport protein. More uncertain is the mechanism by which channel opening might prompt secretion by these cells. Since the Nernst potential for potassium across the cell membrane is more negative than the cell potential, the activation of mechanically sensitive potassium channels causes hyperpolarization of the cell. Hyperpolarization increases the driving force for calcium entry, which might be further enhanced if the divalent ion channels also open. This would cause an increase in intracellular calcium and stimulate secretion. Acinar Cells. Acinar cells are found in secretory organs such as the pancreas and the lacrimal and salivary glands. In contrast to vascular endothelia, these cells produce a solution, comprised of a synthesized protein (usually an enzyme) and fluid. The cells generate the fluid component of the secretion by inducing a transepithelial water flow from the interstitial fluid at the serosal surface of the cells to the lumen of the secretory duct that they line. The means by which this is done is shown schematically in Figure 10.19.

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Figure 10.19. Transport mechanisms in acinar cell secretion. (a) The cell at rest. X represents the substrates of basolateral sodium-driven secondary transport systems in addition to the Na–K–2Cl transporter. At any instant, most of the apical chloride channels and basolateral potassium channels are closed. (b) The stimulated cell. (c) The operating point of the Na–K–2Cl transporter in both cases; note that it is close to static head at rest.

Figure 10.19a shows an acinar cell at rest. As in the intestine, the Na–K exchange pump in the basolateral membrane creates a sodium activity gradient that favors sodium reentry into the cell and can drive secondary active transport. In these cells, the sodium gradient drives an electrically neutral cotransport system in the basolateral membrane that brings in one potassium ion and two chloride ions with each cosolute sodium ion. In the steady state, the rate at which sodium is extruded by the pump equals the rate at which it enters the cell through the Na–K–2Cl cotransporter and other secondary transport systems of which it is the cosolute. The potassium and chloride ions that are actively transported into the cell leave through ion-selective channels in the cell membrane. The potassium channels, some fraction of which are calcium activated, are found predominantly in the basolateral membranes of the cell, and the chloride channels are concentrated in the apical face.

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The chloride channels are activated by calcium or cAMP-mediated phosphorylation (the latter are the CFTR channels of Chap. 4). In the steady state, the intracellular concentrations of potassium and chloride are such that the rates at which they passively diffuse out of the cell are identical to the rates at which they are actively imported. At rest, the passive chloride permeability of the apical membrane is low, so the intracellular chloride level rises. This increases the head against which the cotransporter must operate, lowering the transport rate. Thus, in the resting state, the transcellular flux of chloride is small (Fig. 10.19c). Stimulation of the gland (Fig. 10.19b) via calcium or the cAMP/PKA pathway (recall Chap. 8) signals the apical chloride channels to open. Chloride rushes into the lumen of the duct, and the cotransport rate increases as the intracellular chloride level falls. The result is an overall increase in transcellular chloride flux. The additional potassium flux through the cotransporter leaves the cell through the basolateral channels; when calcium activates the secretory process, calcium-activated potassium channels that open at the same time also play a role in facilitating the exit of potassium from the cell. The maintenance of electroneutrality in the interstitial and duct fluids requires a compensating current to balance the negatively charged transepithelial chloride flux. This is accomplished by a passive paracellular sodium flux into the duct. To maintain osmotic equilibrium across the cell layer, the neutral flux of sodium and chloride is accompanied by enough water, passing passively between the cells, to make the secretion isosmotic to the interstitial fluid. Thus, water follows the chloride in these epithelia as it did sodium in the proximal tubule. If the CFTR chloride channels synthesized in the secretory cells are miscoded because of a genetic defect, they do not insert in the cell membrane, the secretory function of the cells is lost, and cystic fibrosis ensues, in which (among other symptoms) the airways become clogged with mucus that is too dry to be removed by ciliary beating. Cholera toxin causes an excess of intracellular cAMP in intestinal secretory epithelia, leading to overactivation of their chloride channels, oversecretion into the intestinal lumen, and dehydration. Maintenance of Epithelial Polarity. Although some proteins synthesized by epithelial cells are exocytosed into the surrounding media, most proteins are used by the cell, and many of these function as transport proteins in the cell membrane. The transport of newly synthesized proteins to the cell border can be regarded as an internal secretion. To maintain epithelial cell polarity, it is necessary that many of these proteins find their way preferentially to an appropriate region (e.g., apical or basolateral) of the plasma membrane. In the preceding illustrations, we have seen that such is the case for the Na–K pump, the GLUT2 carrier, the ENaC channel, the CFTR channel, and many others. The several mechanisms by which this requirement is met are described below and cartooned in Figure 10.20. Different cells use different mechanisms, as do different proteins in the same cell.

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Figure 10.20. Epithelial mechanisms for maintaining polarity. (a) Differential retention. (b) Default/transcytosis. The proteins denoted by solid circles are intended for the apical membrane, while those denoted by open circles are intended for the basolateral membrane. All proteins are first transported to the latter membrane, which is the default. (c) Sorting tag. Adapted from Putnam (1998).

1. Differential retention. In this mechanism, vesicles or secretory granules (we will henceforth use the term “vesicle” to signify both) transport newly synthesized protein to all portions of the cell membrane, but the protein is preferentially retained by the membrane region(s) for which it is intended. Proteins that find their way to the “wrong” membrane are rapidly endocytosed and returned to the cell interior. For instance, for kidney cells in culture, the residence time of the membrane Na–K ATPase is 1–2 h in the apical membrane, but more than 36 h in the basolateral membrane (which is where it “belongs”). The steady-state concentration of the transport protein is considerably higher in those portions of the cell membrane in which it is preferentially retained.

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The greater resistance of the basolateral ATPase to endocytosis is believed to be due to its interaction with actin-binding proteins, such as ankyrin and spectrin, that are involved in the interaction between the cortical cytoskeleton and the cell membrane. Colocalization of ankyrin, fodrin and Na–K ATPase at the basolateral membranes of renal epithelia has been demonstrated experimentally (Nelson et al., 1990; Nelson, 1993), suggesting that these molecules may form a protein complex that stabilizes the ATPase in the cell membrane. A similar localization of the three molecules has been found at the apical membranes of retinal pigment epithelium and choroid plexus.

2. Default/transcytosis. In this method, all vesicles are transported to one region (the “default” region) of the cell membrane, where they are sorted; the vesicle contents intended for other regions are recovered by endocytosis and transported intracellularly to the correct site. This is a form of transcytosis in which the contents of the transporting vesicle are moved from one membrane domain to a different one, rather than from one external milieu to another. This mechanism has been observed in cells of liver, kidney, and intestine. In Figure 10.20, sorting of the proteins occurs at the default membrane; in other models, portions of the default membrane containing unsorted proteins endocytose into a vesicle, in which sorting subsequently takes place. 3. Sorting tag. Here, proteins are tagged at the Golgi apparatus with a sorting signal designating their target destination, and are transported directly to the intended site. Targeted transport to both the apical and basolateral membranes has been demonstrated in cultured kidney cells.

10.2.3. Filtration Glomerular Filtration. The organization of the glomerular wall is shown in Figure 10.21. It consists of a thick basement membrane that supports a layer of capillary endothelial cells on the side facing the blood, and a layer of epithelial cells on the other side, which line Bowman's space. The gaps between the endothelial cells, called fenestra, are large (ca. 100 nm). The basement membrane is comprised of collagen and proteoglycan, and is also quite porous. Even the epithelial layer is unusually loose, with intercellular gaps of 20–50 nm, and without tight junctions. These porous structures cause the hydraulic conductivity of the glomerulus to be more than 100 times that of common capillary beds. In man, about 150–180 liters of fluid cross the walls of the glomerular capillaries each day. The driving force for this flow is [cf. Eq. (6.43a)] 'P – 6Vi'Si , where 'designates the difference between the values of hydrostatic and osmotic pressure in the capillary and in Bowman's space, and the sum includes all solutes. Since the openings in the glomerular capillary wall are relatively large, the reflection coefficients of small solutes are near zero. Thus, the effective osmotic pressure difference acting across the capillary wall is due almost entirely to large plasma pro-

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435

Figure 10.21. Organization of the glomerular wall.

teins. These proteins have a reflection coefficient near unity, are too large to cross the capillary wall, and are absent from the filtrate; therefore, the driving force can be rewritten as I

II

I

P – P - 6plasma proteins Si , where Phase I is the inside of the capillary and Phase II is Bowman's space. The last term in this expression is often called the colloid osmotic, or oncotic, pressure in the capillary. The hydrostatic pressure difference exceeds the opposing oncotic pressure, and water crosses into Bowman's space by ultrafiltration. The driving force for filtration varies along the length of the capillary. The capillary hydrostatic pressure decreases from the inlet arteriolar value because of the frictional losses that accompany blood flow through the glomerulus, and the capillary oncotic pressure increases as filtration causes the proteins in the blood to become more concentrated. These effects can be incorporated into a mathematical description of glomerular filtration by coupling the equation for local ultrafiltration flux with the Poiseuille equation [Eq. (7.8), which describes the pressure drop along the capillary] and conservation equations for plasma and plasma protein in the capillary (Papenfuss and Gross, 1978; Fung, 1990). The decrease in driving force causes the filtration rate to diminish toward the far end of the capillary. The mean driving force for filtration, averaged over the length of the capillary, has been estimated at between 10 and 25 torr. Nominal ranges of the terms in the driving force are: capillary pressure = 45–60 torr, Bowman's pressure = 10–18 torr, oncotic pressure = 25–35 torr. The single-nephron glomerular filtration rate (SNGFR) is the product of the hydraulic conductivity of the glomerular wall, the area available for filtration in each glomerulus, and the mean driving force: SNGFR = L p A P I  P II  6 plasma proteins SIi ,

(10.28)

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CH. 10: EPITHELIAL TRANSPORT

where the angular brackets denote the mean. The LpA product is sometimes called the ultrafiltration coefficient of the glomerulus. Although it does not appear in the equation, the capillary flow rate also affects SNGFR, because it determines the increase in capillary oncotic pressure that accompanies filtration; the derivation of this relationship is left as an exercise. Roughly 20% of the plasma entering the kidney enters the filtrate. The overall ultrafiltration coefficient of an individual's kidneys can be written as LpATOT, where ATOT is the individual's total glomerular area. This quantity can be estimated by summing Eq. (10.28) over all glomeruli. It is assumed that the mean driving force is the same for all nephrons. The total filtration rate (TFR) replaces SNGFR in the equation, and ATOT replaces A. The calculation of LpATOT for a normal adult is left as an exercise. The pores in the glomerular wall hardly retard the convection of small solutes in the plasma crossing the barrier. The total rate at which such solutes enter Bowman's space in a given individual is simply the product of the total filtration rate and the solute concentration in the plasma: Qi  TFR q cip (small solutes) ,

(10.29)

where the notation is self-evident. The renal clearance is an approximate measure of the ease with which solutes can enter Bowman's space. It is inversely proportional to the time constant for the removal of a solute from the plasma by the kidney. All else being equal, solutes that cross the glomerular barrier with more difficulty are retained in the plasma longer and exhibit lower clearances. The ease with which solutes can enter Bowman's space decreases with increasing molecular size. We observed earlier that the pathways across the capillary wall are narrow enough to exclude most large plasma proteins from the filtrate. Inulin, whose molecular weight is 5200 and whose radius is 1.5 nm, passes easily. Measurements of the clearance of uncharged dextrans by rat kidney showed that the capillary wall retards the flux of solutes as small as 2.1 nm in radius. Dextrans whose radii exceeded 4.4 nm did not cross the barrier. Values around 5 nm have been obtained for the equivalent capillary pore radius in human glomeruli, also using dextrans. Based on the size of the gaps in the cellular layers that line the glomerulus, it would seem that a significant fraction of the resistance to the passage of large molecules must arise from the basement membrane. As noted above, solutes more than about 2 nm in radius, though permeable, are retarded in their passage through the membrane. When retardation occurs in a filtration process, it is referred to as sieving. Albumin, with a radius of 2.5 nm, is sieved by the glomerular membrane. The most commonly used measure of membrane sieving is the nominal sieving coefficient, Tia , defined for the glomerulus as the concentration of the ith solute in the filtrate, cif, divided by that in the plasma, cip. The nominal sieving coefficient is unity when the membrane passes the solute with ease, zero for an impermeant, and between zero and unity when the solute is retarded by the membrane. The concentration of any solute in the filtrate is equal to the total rate at which it crosses the membrane, divided by the total filtration rate: cif = Qi /TFR. Therefore,

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437

from the definition of the nominal sieving coefficient, the total rate at which the solute enters Bowman's space is Qi  TFR q cip Tia

(10.30)

For small solutes, Tia | 1, and Eq. (10.29) is recovered.

Figure 10.22. Concentration profile ci(x) for solute transport across the glomerular capillary wall in the presence of sieving. Compare Figure 10.9a.

When the membrane retards the passage of a permeable solute, the nominal sieving coefficient overestimates the true sieving coefficient of the membrane, because of concentration polarization. The concentration profile (Fig. 10.22) in this case is similar to Figure 10.9a. The solution that is actually sieved by the capillary wall is the solution that is immediately adjacent to it, and the concentration of solute at the upstream face (cim) is greater than that in the plasma (cip). To find the relation between the two concentrations, Daniels et al. (1992) solved the one-dimensional convective diffusion equation for this case, setting the net solute flux to the upstream face equal to the solute flux entering Bowman's space. The equality of the two fluxes can be written Di

dci J v ci ( x )  J v cif . dx

(10.31)

Setting ci(–G) = cip, using the definition of Tia , integrating, and rearranging, cim  (1  Tia )e J v G / Di Tia . cip I

(10.32) I

The concentrations cim and cip are equivalent to cs and csb in Figure 10.9a and Eq. (10.21). Indeed, Eq. (10.21) is seen to be a special case of Eq. (10.32); when the bar-

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CH. 10: EPITHELIAL TRANSPORT

rier is impermeable, Tia = 0. It is a simple matter to solve for the true sieving coefficient, Tia  cif /cim: Ti  Tia

cip cim



Tia . (1  Tia )e J v G / Di Tia

(10.33)

Daniels et al. found that neglect of concentration polarization caused a 30% overestimate of the sieving coefficients of rat glomerular basement membrane for albumin and immunoglobulin G (IgG). The true coefficients for the two molecules were 0.13 and 0.05, respectively. The permeability of the glomerular wall to native albumin is about one-twentieth of its permeability to uncharged dextrans of similar size. Albumin carries a negative charge, so this difference in permeability is evidence that the filtration process includes passage through a negatively charged region that electrostatically opposes albumin transport. Indeed, when the negatively charged groups on native albumin are neutralized, the glomerular permeability of the protein increases 100-fold. This negatively charged region could arise from proteoglycans in the basement membrane, or glycocalyx originating from the endothelial cells. Anionic proteins have also been proposed as an explanation (Lumsden and Silverman, 1990). Further evidence for the role of electrostatic exclusion is the observation that sulfated dextrans pass through the capillary wall more slowly than uncharged dextrans of the same size, while cationic dextrans are removed from the blood more rapidly. Deen et al. (1983) estimate that the fixed charge density in the filtering pores of the rat glomerulus is more than 100 meq/1, which is quite substantial. Water Flow across the Capillary Wall. The function of most capillaries is different from that of the glomerulus. The primary purpose of the glomerulus is to generate a filtrate, free of cells and large proteins, from which solvent and selected solutes can be reabsorbed into the blood. For the ultrafiltering glomerulus, fluid flow and convection are more important than diffusive transport. The purpose of most capillaries is to nourish the tissue in which they are embedded, and to remove waste products; considering transport across the capillary wall, solute diffusion is much more important. The diffusional transport of dissolved gases across the capillary endothelium and in the surrounding tissue will be considered in the next chapter. Yet there is also fluid flow across the capillary wall, driven by the same forces that drive filtration in the glomerulus. The volume flow across the capillary wall is defined by Starling's Law, a rephrasing of the expression for Jv given in Chapter 6: I

II

I

II

Jv = Lp[P – P – (So – So )], 2

(10.34) k

where Jv is the transcapillary volume flux, in cc/s-cm capillary surface area, and So is the oncotic pressure of the kth phase; Phase I is generally taken to be the inside of the capillary and Phase II is the interstitial fluid outside. Starling noted that the hydrostatic and oncotic pressure differences across the wall, averaged along the length of the capillary, were very close to one another, so that the net filtration from the blood to the tissue is very small. Guyton and Hall (2000) calculate a difference of less than

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439

2% between 'P and 'So. Because this difference is so small, the net volume flow across all the capillaries in the body is only 2 ml/min, compared to a volumetric blood flow rate of about 5 l/min. The transcapillary flow is recovered in the lymphatics and returned to the circulation. Equation (10.34) ignores the complexity of transcapillary flow, describing filtration as though the water passes through a homogenous barrier. In reality, water crosses the capillary wall through intercellular clefts, which are not uniform along their length. Furthermore, the entrance to the cleft from the capillary side is filled with glycocalyx originating from the cells adjacent to the gap. Hu and Weinbaum (1999) reexamined Starling's Law in light of these heterogeneities. They propose that the driving force for filtration, 'P – 'So, should not be measured between the plasma and interstitium, but between the plasma and the back side of the glycocalyx, since the glycocalyx plug is the only structure along the filtration path capable of having a sizable sieving coefficient for small proteins. Their model of capillary filtration includes other structural features absent from the Starling model, and a more detailed description of the flow field in the intercellular cleft. The local driving force for hydraulic flow varies along the length of the capillary as the luminal hydrostatic pressure falls from its arteriolar value to that at the venules. In contrast to the glomerulus, the volume flow out of the capillary is not sufficient to cause a substantial variation in luminal oncotic pressure along its length. Since the net filtration out of the entire capillary is essentially zero, this means that plasma leaves the capillaries near their arterial end and reenters the microvessels near their venous end. The quantity of plasma that filters out of, and returns to, the capillaries is perhaps ten times the net loss from the capillaries to the lymphatics.

PROBLEMS: CHAPTER 10 1.

Typical values of the resistances of the pathways through the epithelia of the salamander (Necturus) gallbladder and the toad urinary bladder are, respectively, 2 2 Ra = 1800 and 12500 :-cm , Rb =140 and 1000 :-cm , Rj = 120 and 100000 :2 2 cm , and Ric = 50 and 130 :-cm , where the notation is that used in Figure 10.1c (Wills et al., 1996). (a) Calculate the overall resistance of each epithelium. (b) State whether each epithelium is leaky or tight, and explain how you reached your conclusion. (c) Identify, for each epithelium, the single resistance that controls the overall resistance of the cell layer, and explain why this is so.

2.

An epithelial cell layer has ENaC channels in its apical surface and a Na–K exchange pump in its basolateral membrane. The apical permeability is ka, and the a serosally directed sodium flux through the pump is given by J = a + b(cic – cs), where cic is the intracellular sodium concentration and cs is the sodium concentration at the serosal face. Denote the sodium concentration at the mucosal face by cm. Show that the change in intracellular sodium concentration accompanying a

440

CH. 10: EPITHELIAL TRANSPORT

given increase in apical permeability is less when the pump rate is more sensitive to substrate concentration in the cell. 3.

An epithelial layer can be modeled as two membranes in series. Assume such an epithelium is placed between two baths (Phases I and II) that contain the same I single nonelectrolyte solute. The solute concentration in Phase I is c and that in II Phase II is c . The hydrostatic pressure is the same in the two baths. The membrane facing Phase I is permeable to water but not solute; the membrane facing Phase II is permeable to both. Call the phase between the two membranes, which would ordinarily represent the intracellular compartment, Phase III, and assume it is well stirred. (a) Write a set of equations that can be solved for the steady-state solute concentration in Phase III as a function of the properties of the bounding phases and the transport coefficients of the two membranes. (b) Identify the unknowns in the equations to show that their number is equal to the number of equations.

4.

A transporting epithelium is capable of producing only one type of carrier for a particular solute, A, and the same number of carriers are in both the apical and basolateral membranes. The concentration of A in one bath is cAI, and the concentration in the other bath is zero. Write an expression for the intracellular concenI tration of A in terms of cA , and Vm and K of the carrier.

5.

(a) Solve Eqs. (10.17) to show how the water flux in the Curran model depends on solute pump rate. (b) Are the two inequalities among the hydraulic conductivities and reflection coefficients of the two barriers that define the Curran model necessary for coupled water transport to take place? Justify your answer. (c) Does the Curran model still “work” if (as in the standing-gradient model) there is no passive solute flux across Barrier 1? Justify your answer.

6.

Derive Eq. (10.18b). Hint: the sum of the channel flow into the differential element of the channel in Figure A10.1, and the osmotic volume flow into the element across the lateral membranes, must be equal to the volume flow out of the element.

7.

Rederive Eqs. (10.18a) for a slit-shaped channel whose cross-section in the plane of the membrane is h q l, where l >> h.

8.

Explain in words how the recirculation model of intestinal water transport works. In the baseline model shown in Figure 10.8, the reflection coefficient of the tight –5 junction was set equal to 0.55, and that of the basement membrane was 10 .

PRINCIPLES AND MODELS OF BIOLOGICAL TRANSPORT

9.

441

Use the analysis of concentration polarization at an impermeable membrane to derive an expression for the difference between the apparent and true Lp's of such a membrane, when the flow is pressure driven and the concentration of solute is the same on both sides.

10. Repeat the analysis of concentration polarization for the case in which the membrane has a solute permeability, ZContinue to assume that the volume flow is specified. 11. An impermeable membrane is bounded by two 100-mM sucrose solutions that are very well stirred, so the unstirred layers are negligible. When the pressure drop across the membrane is 100 mm Hg, the transmembrane volume flux is –4 2 equal to 6 q 10 cc/cm -s. The stirrer on the upstream side of the membrane begins to fail, and the unstirred layer at that surface begins to grow. Plot the pressure drop needed to maintain the volume flow as a function of unstirred layer thickness, between the original value of zero, and 100 Pm. The sucrose diffusion –6 2 coefficient at the experimental temperature of 25ºC is 5.5 q 10 cm /s. 12. Explain in words why there is an upper limit to the osmotic flow that can be generated by a bulk concentration difference across a semipermeable membrane when unstirred layers are present, no matter how large the hydraulic conductivity of the barrier is. 13. It has been claimed that the fractional error in the hydraulic conductivity of a membrane obtained from the concentration-driven experiment in Figure 10.9b can be reduced by reducing the concentration difference across the membrane. Confirm or disprove this assertion. Define the fractional error as the ratio (Lp,apparent – Lp, true)/Lp,true. 14. Consider the simplified electrical analog of a passive epithelium given by Figure 10.10b. The analog implies that the electrical properties of the cell layer are determined by the transport properties and Nernst potentials of a single ion, and that there is no paracellular pathway. The concentration of the ion on Side I is suddenly raised. Among the two conductances (G's) and two battery potentials (E's) in the analog, (a) Which, if any, change discontinuously, and why? (b) Which, if any, change smoothly, and why? (c) Which, if any, do not change, and why? 15. Express the Thevenin conductance and EMF of the epithelium in Figure 10.10b in terms of the battery EMFs and conductances of the apical and basolateral membranes. 16. Refer to Figure 10.14. Assume that the actively transported flux of a sugar, S, across the apical membrane via SGLT-1 is given by JS = LSS'PS + P, where LSS

442

CH. 10: EPITHELIAL TRANSPORT

and P are constants, and that the flux of S across the basolateral membrane via GLUT2 can be described by the simple carrier of Chapter 4 in the diffusion limit. These are the only transport pathways for S. S is consumed negligibly by the cell. Write an equation which can be solved for the intracellular concentration of S in the steady state, in terms of the mucosal and serosal concentrations of S, and the properties of the two transport systems. You need not solve the equation. Neglect the effect of pressure on the chemical potential and assume the system is isothermal. 17. What is the effect of eating a high-protein meal on sodium reabsorption in the proximal tubule? Explain your answer. 18. Explain in words why the osmolarity of the fluid in the proximal tubule remains nearly constant along the length of the segment. 19. Calculate the Peclet number (Pe = vma/DNa) for sodium ion in the proximal or distal tubule of the kidney, to determine whether diffusion may play a role in axial or radial transport within the tubule lumen. Use either tubule length or tubule radius, accordingly, as the characteristic length, a. If Pe >> 1; convection dominates; if Pe << 1, diffusion dominates; if Pe | 1, both convection and diffusion have to be taken into account. Find the mean fluid velocity in the tubule (vm), the tubule dimensions, and the sodium diffusion coefficient from the literature. 20. Derive an expression for the decrease in filtration rate along the length of a glomerular capillary that results from the increase in capillary oncotic pressure caused by the loss of fluid into Bowman's space. 21. The following values are measured from an experimental preparation of a rat I II glomerulus (I = capillary; II = Bowman's space): P = 49 mm Hg, P = 14 mm Hg, 'S = 17 mm Hg, SGFR (single glomerulus filtration rate) = 44 nl/min. Assume that the entire osmotic pressure difference, 'S, is due to albumin, which cannot cross the glomerular wall. The barrier area of a single glomerulus is esti2 mated to be 1 mm from histological measurements. What is the hydraulic conductivity of the glomerular wall? 22. The total glomerular filtration rate in a normal adult is about 180 l/day. Guyton I II and Hall (2000) estimate that, in man, typical values of P , P , and 'S (using the notation in the preivous problem) are 60, 18, and 32 mm Hg, respectively. Using these values, estimate the overall ultrafiltration coefficient in man. 23. Explain in words why the nominal sieving coefficient overestimates the true coefficient. 24. Derive Eq. (10.32).

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443

25. The nominal sieving coefficient for albumin filtration through the glomerular basement membrane was found by Daniels et al. (1992) to be 0.164 when the –4 2 volume flow through the membrane was 1.2 q 10 cc/cm -s. The diffusion coef–7 2 ficient of albumin in their solutions was 6.7 q 10 cm /s. (a) What is the effective thickness of the unstirred layer upstream of the membrane? (b) What would the nominal and true sieving coefficients of albumin be if the flow rate were doubled? (c) If the true sieving coefficient of IgG under the original flow rate is 0.05, what is the nominal sieving coefficient of the immunoglobulin? –7 2 The diffusion coefficient of IgG is estimated to be 4.3 q 10 cm /s at the experimental temperature. 26. How did the atrial natriuretic peptide get its middle name?

APPENDIX 10.1. CONVECTION, DIFFUSION AND MASS ADDITION IN CHANNEL GEOMETRIES Among the transport processes described in this chapter, there are several instances in which there is an exchange of solute and solvent across the walls of a channel through which solution flows. Such is the case for the intercellular channel in the standinggradient model of coupled water transport, the several segments of the nephron, and the capillaries of the glomerulus and other tissues. Because of the fluxes across the channel walls, which can be passive or active, the composition and flow rate of the solution in the tubule lumen varies along its length. All of these situations can be treated similarly by using a mass balance in the luminal fluid to relate the changes in its composition and volume flow to the transport rates. As in earlier chapters, we will focus on the steady state behavior of such systems, since any variations in their performance can generally be regarded as quasisteady. The transport resistance of the channel wall is assumed to be large relative to the diffusional resistance in the lumen, so the contained fluid may be regarded as well mixed across the cross-section. This implies that the tubular composition and mean velocity are functions of only the distance, x, along the length of the channel. The equations that govern their variation are obtained by combining the one-dimensional convective diffusion equation (2.29) with mass balances on solute and solvent. Referring to Figure A10.1, a mass balance on the ith solute between x and x + dx gives AJ i ( x dx )  AJ i ( x ) dni ;

(A10.1)

the variables are defined in the figure caption. The convective diffusion equation relates the solute flux (down the channel) to the local concentration gradient and fluid velocity: J i  Di cia vci , where the prime denotes differentiation with respect to x.

(2.29)

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CH. 10: EPITHELIAL TRANSPORT

Figure A10.1. Fluxes in a differential length of a capillary, tubule, or intercellular channel. Ji(x) is the flux of the ith solute down the channel, in moles/s-cm2 channel cross-section. The concentration of solute is ci(x) and the solvent velocity is v(x). Over a differential length of channel, the rates of addition of solute and solvent to the fluid in the lumen are dni(x) mols/s and dnw(x) cc/s, respectively. The mass flow rate of the solute down the channel is AJi(x) mols/s, where A is the crosssection of the channel, generally assumed to be uniform.

Equation (2.29) is used to evaluate Ji(x + dx) – Ji(x): J i ( x dx )  J i ( x )  J ia( x )dx  (Di ciaa vcia ci v a)dx . Thus, the solute balance Eq. (A10.1) becomes A(Di ciaa vcia ci v a)dx  dni .

(A10.2)

In all of the models described here, the channel was assumed to be cylindrical, with radius r, and we will continue to make that simplification. Assume that the ith solute a 2 is injected into the channel at a rate equal to J mols/s-cm channel surface (note the a a 2 difference in unit area between Ji and J ). Thus, dni = 2SrJ dx; naturally, A = Sr . Equation (A10.2) becomes Di ciaa vcia  ci v a

2J a 0. r

(10.18a)

Similarly, a mass balance on volume gives 2Jv  va  0 , r

(10.18b) 2

where Jv is the volume flux across the lateral membrane, in cc/s-cm channel surface. These equations are then coupled with expressions that relate the radial fluxes into or out of the channel to the local luminal composition and the organization and properties of the channel wall. In application to transport down the kidney tubule, the most commonly used organization of the surrounding wall is similar to that in Figure 10.17: the models include a transcellular pathway in parallel with an paracellular pathway, and a transport path between the cell interior and the intercellular channel (Weinstein, 1994). Thus, the wall consists of two compartments — intracellular and intercellular

PRINCIPLES AND MODELS OF BIOLOGICAL TRANSPORT

445

— and five distinguishable barriers — the apical, lateral, and basal cell membranes, and the tight junction and basement membrane at the ends of the shunt. Notwithstanding the standing-gradient model, the intercellular compartment is most commonly treated as well mixed, as is the interior of the cell. In the steady state, the compositions of the two compartments are constant, and the number of mols of a given species entering a compartment in unit time must equal the number leaving the compartment. For instance, for the intracellular compartment, the rate at which a solute enters the cell across the apical membrane must equal the sum of the solute flows out of the cell across the lateral and basal cell membranes. The same is true for volume flow. If the boundary condition at the outside of the tubule lies beyond the basal membrane of the epithelium — for instance, at a capillary bed deeper in the tissue — the resistance of the interstitial tissue between the epithelium and capillaries must be included in the model. This can be done by augmenting the basal membrane and basement membrane resistances accordingly.

Equations (10.18) can be generalized to describe transport in other geometries, such as slit-shaped intercellular channels or channels whose cross-section varies axially (i.e., along the length of the channel). The rates of mass addition can be specified a as a function of x, like J in the standing-gradient model, where the solute pumps were concentrated at the junctional end of the channel (Fig. 10.7), or they can depend on the compositions of the solutions inside and/or outside the channel. The equation for Jv in the standing-gradient model is of the latter kind; the osmotic water flux across the lateral cell membrane was assumed to be proportional to the difference between the local osmolarity in the channel [i.e., ci(x)] and the assumed uniform intracellular osmolarity. When the fluxes depend on the axially varying composition of the lumen fluid, they will naturally vary along the length of the tube. This was the case for the solvent and all of the ions in the Chang–Fujita (1999) model of the distal tubule. The transport properties and structure of the channel wall can also vary axially, either continuously or discretely; the latter is illustrated by the distinction between the late and early portions of the distal tubule. In capillary models, axial diffusion is often ignored, but axial variation in hydrostatic pressure is included. For a channel with a circular cross-section, the axial pressure gradient can be related to the fluid velocity through Poiseuille's equation (7.8), replacing 'P/a by –dP/dx = –Pc and q by Sr2v, where v is the mean velocity: Pa  

8Kv . r2

(A10.3)

In this case, the filtration velocity across the capillary wall is affected by axial gradients in both hydrostatic and osmotic pressure. This in turn indirectly affects the transport of permeable solutes across the wall.

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CH. 10: EPITHELIAL TRANSPORT

Some Additional Comments In the preceding analyses it has been assumed that neither solute nor solvent are synthesized or consumed within the tubule; this is a reasonable assumption for most ions and water, but not for species that participate in chemical reactions, such as hydrogen ion or buffers. An important constraint on the solution of the tubule transport equations is the requirement that all compartments and the lumen remain electrically neutral. Stated another way, the net current into (or out of) any compartment, summed over all pathways, must be zero. Accordingly, a net cation flux into or out of the lumen must be accompanied by an equal net anion flux. Axially varying transepithelial diffusion potentials develop to ensure that this is so. The anion in this case is usually chloride, which follows the cation, most commonly sodium, across the epithelial layer.

11 GAS TRANSPORT

INTRODUCTION Many of the transport principles and models described in earlier chapters apply to the passive transport of dissolved gases. From a living systems point of view, the most important of these gases is clearly oxygen, and the focus in this chapter is on oxygen transport, though certain aspects of the transport of its metabolic byproduct, carbon dioxide, are also described. The rate equations that describe the flux of dissolved gases are similar to those in Chapter 2 and are presented first. This is followed by an overview of the transport processes that bring oxygen from ambient air to the mitochondria of metabolizing cells. The transport overview is followed by descriptions of respiratory transport and the transport of oxygen to tissue in the microcirculation. Respiratory exchange in the lung is a diffusive process in which the phase at one side of the transport barrier is a gas mixture. The partial pressure of each gas is used as a measure of concentration, in both the gas phase and in solution. Almost all of the oxygen that enters the pulmonary capillaries binds to hemoglobin in red cells, and its delivery to the microcirculation is a convective process; the gas goes where the blood carries it. Similarly, carbon dioxide is convected from the microcirculation to the lungs, primarily as bicarbonate ion. These convective processes are in the domain of vascular and respiratory fluid mechanics, and are outside the scope of this text. Once bound oxygen reaches the microcirculation, it diffuses out of the blood and into the perfused tissue, and again falls within our purview. Most of this chapter deals with models of the tissue oxygenation process. Oxygen is consumed as it diffuses through the tissue, so we begin by extending our description of solute diffusion to include concurrent chemical reaction. We then continue with the classical Krogh model of oxygen transport from capillaries to tissue, and review some modifications to the model that have been made to improve its realism. Even so, the Krogh model omits many important features of the real transport process, including transport resistance inside the capillary, the heterogeneity of the perfusion and organization of the capilM.H. Friedman, Principles and Models of Biological Transport, DOI: 10.1007/978-0-387-79240-8_11, © Springer Science+Business Media, LLC 2008

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lary bed, and diffusive interactions between and among arterioles and capillaries. These are discussed in the remainder of the chapter.

11.1. PARTIAL PRESSURE AND THE EQUATIONS FOR GAS FLUX Gas concentrations in solution can be expressed in several ways. Throughout this text, concentration has been defined as mass per unit volume. For gases, rather than use mols or grams as a unit of mass, concentration is usually expressed in terms of ml of gas at standard temperature and pressure (STP: 0ºC, 760 mm Hg) per unit volume of solution. One mol of gas is the same as 22,414 ml at STP. Another very common measure of dissolved gas concentration in physiological systems is partial pressure. The meaning of partial pressure in gas mixtures is clear: the partial pressure of a gaseous species in an ideal gas mixture is the product of the mol fraction of the species and the pressure of the mixture. What does partial pressure mean when the gas is in solution? When a gas mixture is allowed to equilibrate with a solvent, the constituents of the mixture dissolve in the solvent to varying degrees. The partial pressure of a gaseous species in solution is, by definition, equal to the partial pressure of the species in a gas mixture that is in equilibrium with the solution. For instance, when plasma at 37ºC is allowed to equilibrate with air at one atmosphere, the concentration of oxygen dissolved in the liquid is cO2 = 0.48 ml O2 (at STP)/100 ml plasma. The mol fraction of oxygen in air is 0.21, so the partial pressure of oxygen in the gas phase is pO2 = 760 q 0.21 = 160 mm Hg. Thus, the partial pressure of oxygen in plasma is said to be 160 torr when the concentration of the dissolved gas is 0.48 ml O2/100 ml plasma. Put another way, the partial pressure of a gaseous species in solution is defined so that, when a liquid and a gas mixture, both containing that species, are at equilibrium, the partial pressure of the gas is the same in both phases. Partial pressure is most commonly measured in mm Hg, or the equivalent unit, torr. Since a common medical term for “pressure” is “tension” (as in hypertension = high blood pressure), the partial pressure of a dissolved gas is often referred to as its tension. When there are no chemical reactions between the gas (denoted by the subscript “g”) and the solvent, cg is found to be proportional to pg over a wide range of partial pressures. This relation is known as Henry's Law. The proportionality constant, Sg = cg/pg, is the solubility of the gas, often signified by D. Typical values of SO2 are 1.5 q –6 –6 10 M/torr in red blood cells and 1.34 q 10 M/torr in plasma (cited in Hellums et al., –6 1996), and 1.49 q 10 in frog muscle at 20ºC (Sten-Knudsen, 2002). The passive flux of a dissolved gas is proportional to its partial pressure gradient [cf. Fick's First Law of diffusion, Eq. (2.9)]:

J g  Dg

dpg dx

,

(11.1)

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where Dg is the diffusion coefficient of the gas. Similarly, the flux of a dissolved gas across a barrier is proportional to the difference between its partial pressures in the solutions at the two sides [cf. Eq. (2.20)]:

J g  kg 'pg ,

(11.2)

where kg is the permeability of the barrier to the gas. The units of the diffusion coefficient in Eq. (11.1), and the permeability in Eq. (11.2), are different from those in Chapter 2 and elsewhere in this text. This is because the driving force for the diffusion of a gas is expressed in terms of partial pressure rather than concentration. For reasons that will become clear later, Dg is often referred to as the Krogh diffusion coefficient. To express the diffusion coefficient or permeability of a gas in concentration units, it is necessary to divide Dg or kg by the solubility of the gas. In muscle, oxygen is transported not only by so-called Fickian diffusion, but also by a carrier protein, myoglobin. Each molecule of myoglobin can bind a single oxygen molecule, forming oxymyoglobin. Estimates of the importance of this mechanism of oxygen transport vary, depending on the chosen value of the myoglobin diffusion coefficient. Simulations of oxymyoglobin diffusion based on recently measured diffusion coefficients of the protein (Jurgens et al., 2000; Papadopoulos et al., 2001) suggest that myoglobin carries only about 4% of the oxygen consumed in maximally exercising skeletal muscle, and 2% in similarly loaded cardiac muscle. Whitely et al. (2002) used a more complex model and reached a similar conclusion, provided that the tissue pO2 was not too low, and so did Ji et al. (2006). Earlier estimates of the importance of myoglobin-facilitated oxygen transport are reviewed in Jurgens et al. The equations for oxygen transport in tissue are more complicated when the oxymyoglobin flux is included. Since the contribution of myoglobin to oxygen flux in muscle is now thought to be small under most conditions, and since the protein is absent from most tissues, oxymyoglobin transport is not dealt with in the discussion to follow. When we review earlier work that included oxymyoglobin transport, the myoglobin terms are removed from the equations, with an annotation that myoglobin diffusion was included in the original model.

11.2. OVERVIEW OF THE GAS TRANSPORT PROCESS The body obtains the oxygen it needs by inspiring (inhaling) air. After passing through the nasal or oral passages, and the pharynx and larynx, the inspired air enters the tubular trachea. The trachea bifurcates to form two daughter tubes, which are the main bronchi; each main bronchus feeds air to one of the lungs (see Fig. 11.1). The organization of the air-carrying tubes in the lung is tree-like, with the main bronchus as the trunk; each level of branching produces more numerous and finer passageways, terminating finally in the alveoli, where gas exchange with pulmonary capillary blood takes place. The primary purpose of the more than 20 generations of branching in the lung is to increase the surface area available for gas exchange, and this is accomplished exceedingly well: the lungs possess about 300 million alveoli, whose diameters range between 1/10 and 1/3 mm, and whose combined surface area has been estimated at between 30 and 100 square meters!

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Figure 11.1. The path of air through the respiratory system. Reprinted with permission from Dorland's Illustrated Medical Dictionary (1974). Copyright © 1974, W.B. Saunders.

Proceeding into the lung, the conduit dimensions and air velocities decrease with each branch, and the Peclet number (Chap. 7), which is proportional to the product of these quantities, is less in each generation. Thus, while convection is the dominant mechanism of gas transport in the upper airways, diffusion becomes comparable to convection around the seventh generation and is the dominant mechanism further downstream and in alveolar transport. The pulmonary circulation of oxygen-lean blood from the right side of the heart feeds a rich capillary network in the walls of the alveoli. The capillaries are about eight microns in diameter, and outnumber the alveoli; one estimate is that there are 280 billion alveolar capillaries in the lung. The barrier between the alveolar gas and the capillary blood is known as the respiratory membrane. Oxygen and carbon dioxide diffuse in opposite directions across the respiratory membrane; though some of the oxygen from the alveolus dissolves in the capillary plasma, by far the greater portion of the gas enters the red blood cells (erythrocytes) in the capillaries, and binds to hemoglobin. Oxygenated blood leaves the pulmonary network, enters the pulmonary veins, and then passes through the left side of the heart. The heart pumps this suspension of cells into the systemic arterial system.

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The arteries branch successively, with the daughter vessels narrower than the parent. When the vessel diameter reaches 10–15 μm, the vessels are termed arterioles. Further branching and structural changes lead to the capillaries, which are 4–9 μm in diameter. The capillaries join at their distal ends to form venules, which carry deoxygenated blood to the veins for reoxygenation in the lung. The arterioles, capillaries and venules constitute the microcirculation. Though blood flows through the three components of the microcirculation in series, there is considerable interaction among them with regard to gas transport (Pittman, 2005). Oxygen diffuses into the tissue from both arterioles and capillaries, and venular blood can act as an oxygen sink that reduces the oxygen tension of adjacent tissue. There are also “short circuits” called diffusive shunts between portions of the microcirculation that are close to one another but not connected: between arteriole and venule, between arteriole or venule and capillary, and between capillary and capillary. Shunting between these elements of the microcirculation involves diffusion through the tissue between them. Furthermore, capillaries branch and connect in anastomoses, further complicating transport analysis. These interactions among the components of the microcirculation compensate to some extent for differences between oxygen supply rates in the incoming arteriolar flows, and contribute to a more uniform supply of oxygen to the tissue. Initially, we will consider oxygen transport out of individual non-interacting capillaries; interactions such as those described above are addressed subsequently. Oxygen dissociates from hemoglobin in the erythrocyte, crosses the capillary wall, and enters the interstitial fluid that bathes the cells of the tissue being nourished. The loss of oxygen from the capillary causes the partial pressure of oxygen in the capillary blood to fall towards its venous value. Meanwhile, the oxygen in the interstitium crosses the plasma membranes of the target cells, diffuses through the cytosol, and finally reaches the mitochondrion, where it is used in metabolism and ATP synthesis. The variation of oxygen tension along the path taken by the gas, from ambient air to the mitochondria, is shown in Figure 11.2. An important feature of oxygen transport is illustrated in Figure 11.2: because the transport process is entirely passive, the partial pressure of the gas is less in each succeeding compartment. There is a pressure drop across each of the barriers it must cross, and there are many of these. The partial pressure of oxygen at the mitochondrion can be as low as 1–2 torr when the oxygen demand is high. The difference between the arterial oxygen tension and that in venous blood leaving a particular organ, multiplied by the rate of blood flow to that organ, yields the organ's oxygen consumption rate. Organs vary in their ability or need to extract oxygen from incoming arterial blood; though the arterial oxygen tension is essentially the same for almost all organs, the venous tensions can vary. They are usually 35 torr or more, but can be drawn down to lower values by the heart or brain, or by exercising skeletal muscle. Table 11.1 gives some data on oxygen consumption by several major organs and tissues.

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Table 11.1. Oxygen Consumption by Various Organs

Organ Skeletal muscle Liver Brain Heart Kidneys Skin a

Blood flow rate, ml/min 1200 1400 750 250 1100 500

Oxygen consumption, ml/min

Specific oxygen consumption rate, ml/min-100 g

Venous oxygen tension, torr

72 60 48 26.4 16.8 4.8

4 (at rest) 95 50 70 360 3

34a 43 33 23 56 60

Can be reduced to zero in heavy exercise.

Figure 11.2. Variation of oxygen tension from the ambient air to the mitochondria. Adapted with permission from Clemmer (1980). Copyright © 1980, Little, Brown.

The oxygen consumption by the entire body can be calculated similarly, by multiplying the difference between the arterial and mixed venous tensions by the cardiac output. This is known as the Fick equation. The arteriovenous tension difference corresponds to a concentration difference of 40–50 ml of oxygen per liter of blood; multiplying this by a typical cardiac output of 5–6 l/min, we obtain an oxygen consumption rate of ca. 250 ml/min.

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453

11.3. GAS EXCHANGE IN THE LUNG As noted earlier, gas transport in the upper airways is purely convective, and since such flows are the domain of fluid mechanics and are accordingly outside the scope of this text, our interest in the gas transport process will begin several generations short of the alveolus, where gas transport is dominated by diffusion. In these smallest branches, oxygen diffuses toward the alveoli under a partial pressure gradient that arises because of the passive transport of oxygen from the alveoli to the pulmonary capillary blood across the respiratory membrane. This interface between the alveolar gas and the capillary lumen consists of a number of barriers in series, including, in order, a fluid film inside the alveolus, the alveolar epithelium, its basement membrane, a small gap between the alveolar and capillary structures, the basement membrane of the capillary endothelium, and the capillary endothelial cells themselves. This entire series of resistances is only about 2 μm thick, so gas exchange is rapid. The major resistances that comprise the respiratory membrane are each assumed to obey Eq. (11.2), and since the resistances are in series, the flux across the respiratory membrane also obeys Eq. (11.2). The proof of this is left as an exercise. In most studies of lung physiology, transport in the entire lung is studied, and the two sides of Eq. (11.2) are multiplied by the total area available for transport to give:

J g  kg ,rm Arm ( pg ,alveolar   pgA ) ,

(11.3)

where Jg is the total transport rate of gas in the lung, directed toward the capillary, kg,rm is the permeability of the composite respiratory membrane, Arm is the total transport area, and the pressure driving force is the difference between the partial pressure of the gas in the alveolus and an appropriately averaged value in the pulmonary capillaries. The capillary partial pressure is related to the average gas concentration in the plasma by  pgA    cgA  / Sg . The product kg ,rm Arm is known as the diffusing capacity of the respiratory membrane. Since the permeability of the membrane depends on the nature of the gas, the diffusing capacity does too. A typical value for the diffusing capacity of oxygen in a resting man is 21 ml/min-mm Hg; this value can triple in exercise. Carbon dioxide crosses the respiratory membrane much more rapidly than oxygen; its diffusing capacity has been estimated at 400–450 ml/min-mm Hg at rest (Guyton and Hall, 2006). Although the partial pressure of oxygen in air is 160 mm Hg, it is only about 100– 110 mm Hg in the alveolus. The principal reasons for this difference are the rapid diffusion of carbon dioxide across the respiratory membrane from the blood to the alveolus, and humidification of the air by the lung tissue. The total pressure in the alveolus is the same as that in the atmosphere, and the alveolar oxygen is diluted by the added water vapor and CO2. Table 11.2 presents the compositions of atmospheric, humidified, and alveolar air.

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Table 11.2. Composition of the Mixed Gas in the Atmosphere, after Humidification, and in the Alveolus (Guyton and Hall, 2006) Gas

Atmospheric air

Nitrogen Oxygen Carbon dioxide Water vapor

78.6% 20.8 <0.05 0.5

Humidified air

Alveolar air

74.1% 19.7 <0.05 6.2

74.9% 13.6 5.3 6.2

The vapor pressure of water at 37ºC is 47 mm Hg.

Venous blood enters the pulmonary capillaries with an oxygen tension of ca. 40 mm Hg. The oxygenation of pulmonary capillary blood can be described using the model of tubule transport presented in the previous chapter. Assuming the capillary blood to be well mixed radially, neglecting axial oxygen diffusion, and in the absence of volume addition, Eq. (10.18a) becomes

v

dci 2 J a ,  dz r

(11.4)

where the notation is the same as in Chapter 10, except that the coordinate that defines distance along the capillary axis is denoted by z. In the present application, v is the mean blood velocity in the capillary; ci(z) becomes cA(z), the total (bound and dissolved) concentration of oxygen in the blood at any axial location; r is capillary raa dius; and J is the oxygen flux into the capillary from the alveolus. Using Eq. (11.3) for the oxygen flux, Ja 

Jg Arm

 kg ,rm ( pg ,alveolar  pA ) ,

(11.5)

where the local capillary oxygen tension pA depends on z. Eq. (11.4) becomes v

dc A 2k g ,rm ( pg ,alveolar  pA )  . dz r

(11.6)

It will be shown below [Eq. (11.20)] that the relation between the total concentration of oxygen in the blood (cA) and its partial pressure (pA) depends on the solubility of the gas in plasma and the concentration of hemoglobin in the blood. Both cA and pA increase with increasing z, so the driving force for oxygen uptake in the capillary, pg ,alveolar  pA , decreases as the oxygen content of the capillary blood rises. As blood proceeds down the capillary, its oxygen tension asymptotically approaches the level in the alveolus. Transport across the respiratory membrane is sufficiently rapid that, by the time the capillaries exit the lung, their oxygen tension has risen essentially to that in the alveolus. The oxygenated blood is diluted with some pulmonary blood that bypassed the alveoli, bringing the oxygen tension of the blood entering the left side of the heart to about 90–95 mm Hg.

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455

11.4. OXYGEN TRANSPORT IN THE BLOOD Most of the oxygen carried in the blood is bound to the iron-containing protein, hemoglobin, which is inside the red blood cells. Hemoglobin (Hb) makes up 90% of the dry weight of red blood cells. By transporting the gas as oxyhemoglobin (HbO2), the blood can carry many times more oxygen than can a like volume of cell-free plasma. Correspondingly, the cardiac output required to provide oxygen to the cells of the body is enormously reduced. The rate at which a vessel carries bound oxygen to target tissue is equal to the blood flow rate multiplied by the concentration of bound oxygen in the blood. For a 2 circular vessel of radius R, the blood flow rate equals SR v, where v is the average blood velocity. The concentration of bound oxygen is the product of the concentration of hemoglobin in the blood, the fraction of the hemoglobin sites that have oxygen bound to them, and the oxygen binding capacity of the protein, b. The concentration of hemoglobin in the blood equals the product of the volume fraction of erythrocytes in the blood (known as the hematocrit, H, and equal to about 40–45%) and the total (free and oxygenated) concentration of hemoglobin in them, cHb,c . The fraction of the hemoglobin sites that are occupied is termed the hemoglobin saturation, s. Accordingly, the rate of oxygen transport in the blood is SR 2 vHcHb,c sb. The concentration of hemoglobin in erythrocytes is cHb,c x 34 g Hb/100 ml cells, and b = 1.34 ml O2/g Hb. The calculation of oxygen transport rate from these values is left as an exercise. The saturation is a nonlinear function of oxygen tension, as shown in Figure 11.3. This equilibrium saturation curve is usually described by the Hill equation:

s

KpOn 2 1 KpOn 2

,

(11.7a)

where the Hill coefficient n = 2.5–2.7. The constant K depends on several environmental variables, including pH and temperature. Decreases in pH shift the saturation curve to the right; this is known as the Bohr effect. Increases in temperature also shift the curve to the right. A reduced blood pH can result when the perfused tissue is metabolically active and large quantities of carbon dioxide are produced, since CO2 forms carbonic acid in the plasma. When the saturation curve is shifted to the right, the equilibrium saturation for a given oxygen partial pressure is reduced, facilitating the release of oxygen from the hemoglobin, and thereby making more of the gas available to the active tissue. The Hill equation is often written in terms of p50, the oxygen tension at which s = n 0.5. By inspection, this is the case when Kp50  1. Solving for K, and substituting into Eq. (11.7a), n

 pO ¬­ žž 2 ­ žž p ­­­ Ÿ 50 ® s n  pO ¬­ ž 2 ­ 1 žž ­ žŸ p50 ®­­

(11.7b)

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Figure 11.3. Equilibrium saturation curve for hemoglobin at pH 7.4 and 37ºC. The value of p50 is about 27 torr. The concentration of bound oxygen in the blood, given by the saturation curve and the right-hand ordinate, is based on a hemoglobin concentration of 15 g/100 ml blood. The concentration of oxygen dissolved in plasma is given by the dashed line, whose slope is 0.003 ml O2/100 ml blood-torr.

Figure 11.3 is a convenient context for emphasizing the predominance of bound oxygen relative to that dissolved in the plasma. A second ordinate in the figure shows the oxygen content of 100 ml of blood as a function of oxygen tension, for normal hematocrit and hemoglobin levels. The concentration of bound oxygen follows the saturation curve. Also shown on the figure is the amount of oxygen dissolved in 100 ml of plasma, as a function of oxygen tension. It is clear that the oxygen-carrying capacity of the red blood cells far exceeds that of plasma.

11.5. TRANSPORT FROM RED BLOOD CELLS TO TISSUE Early models of the transport of oxygen from erythrocytes to target cells focused on the diffusion of the gas from individual capillaries into the surrounding tissue. Capillaries in mammalian skin or skeletal muscle are typically ca. 5 μm in diameter and about 1 mm long; each is lined by about 30 endothelial cells. There are many capillaries in the human body; according to one source, if all of them were laid end to end, they would circle the world 2½ times. The disc-shaped red blood cells, whose diameter is 8 μm, must deform to enter the capillaries, and squeeze through in single file. The average hematocrit in the capillaries is 8 to 20%, which is much less than the hematocrit in larger vessels. The numerous paths that oxygen takes to pass from the red blood cells to the tissue are schematized in Figure 11.4. Oxygen bound to hemoglobin must dissociate

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457

Figure 11.4. The routes of oxygen transport, from the inside of the red blood cell (RBC) to the interior of the tissue. The more common routes from the RBC to the capillary wall, and across the wall, are shown with heavier lines. Myoglobin-facilitated transport of oxygen in the tissue is not included.

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from the protein and dissolve in the cytosol of the erythrocyte before crossing the cell membrane. Dissolved oxygen diffuses to the cell border, driven by an intracellular pO2 gradient. A comparable amount is carried to the red cell membrane in the form of oxyhemoglobin. The oxyhemoglobin releases its cargo at the cell membrane, and the protein diffuses back into the bulk cytoplasm, where it binds oxygen once more. The dissociation of oxyhemoglobin at the cell membrane decreases the local concentration of the oxygenated form of the protein and increases the local concentration of deoxygenated hemoglobin, thereby setting up the driving forces for the diffusion of the two forms between the bulk cytoplasm and the plasma membrane. This hemoglobinfacilitated transport of oxygen from the bulk cytoplasm to the margin of the cell is very similar to the carrier systems described in Chapter 4. Oxygen is more soluble in lipid than in water, and the erythrocyte membrane offers little resistance to transport. The oxygen molecules then enter a stagnant film of plasma that surrounds the red blood cell. Since the erythrocyte is squeezing through the capillary, some of this film is at the capillary wall, and oxygen can diffuse directly to the plasma membrane of the capillary endothelium. This is the more common transport path at normal hematocrits. A minor fraction of the gas diffuses out of the film around the red cell, enters bulk phase plasma, and undergoes convective diffusion to the stagnant film at the capillary wall, within which it diffuses to the capillary endothelial surface. The oxygen must now cross the capillary endothelium. The primary route is transcellular, owing to the high solubility of oxygen in the endothelial cell membrane. As we saw in the previous chapter, the intercellular clefts between capillary endothelial cells provide a route for fluid flow between the lumen of the capillary and the interstitial space, so convective diffusion through these clefts could be an alternative route for transcapillary oxygen transport. Quantitative estimates by Fournier (2007) indicate that diffusion dominates convection in the clefts, and that the transcellular transport rate far exceeds that via the paracellular route. The primary reason for this is the substantially larger transport area presented by the cell membranes. Upon crossing the capillary endothelium, oxygen enters the interstitial space between the cells of the surrounding tissue, and it diffuses into these cells under a partial pressure gradient that results from the intracellular consumption of the gas. The transport of oxygen from capillary blood to surrounding tissue has been the subject of considerable analysis, dating from the Krogh tissue cylinder model, which was published in 1919 to describe the nutrition of skeletal muscle. We too will begin with the Krogh model, and follow it with a review of the many modifications to it that have since been made. The Krogh model describes transport in the tissue, so it is not surprising that many of these modifications focus on the tissue phase — the far right portion of Figure 11.4 — with scant attention to the processes taking place within the capillary lumen and wall. We then review more recent analyses that include these latter processes. In all of this work, the original concentric geometry of the Krogh cylinder is retained. There are, however, inevitable limitations associated with the tissue cylinder geometry, and this chapter will conclude with a synopsis of several approaches that have been developed to describe oxygen transport in more realistic geometries. But before

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we can engage in any of this discussion, we must first incorporate solute consumption into the basic equations of nonelectrolyte diffusion.

11.5.1. Reaction–Diffusion Processes Transport of oxygen in tissue is predominantly diffusional, driven by oxygen tension gradients just as the diffusion of other solutes is driven by gradients in activity or concentration. There are two important differences between the transport of oxygen in tissue and that of most other biological solutes. In most of the transport processes described in earlier chapters, solutes crossed one-dimensional barriers between largely homogeneous compartments such as the interstitium or the interior of a cell. Whatever roles these solutes played in the body — activating a biochemical process, maintaining an appropriate osmotic pressure, or serving as a metabolic substrate — they performed within those compartments. The transport process between the compartments was effectively decoupled from the function of the transported solute. The solute entered the compartment and then it did its job. None of the solutes were consumed or produced as they crossed the barrier, and indeed most of them were neither consumed nor produced within their recipient or originating compartments. Oxygen, on the other hand, performs its essential role in metabolism while it diffuses through tissue, and that role results in its being consumed as it is diffusing. The diffusive process and the consumption process are coupled. The mathematical description of diffusion developed in Chapter 2, which served us well when these processes were distinct, must be extended to describe the diffusion of a solute while it is participating in a chemical reaction. Furthermore, some models of oxygen transport within the capillary are best analyzed as unsteady transport processes. Thus, unsteadiness and chemical reaction will be included in the analysis to follow. Convection also takes place in the capillary, but it will be included by reference to the steady-state convective diffusion equation derived in Chapter 2. Consider the differential volume dxdydz shown in Figure 11.5. Solute enters the volume through the three faces at x, y, and z, and leaves through the three faces at x + dx, y + dy, and z + dz. The component of the solute flux vector in the ith direction is denoted by Ji. If the rate at which solute enters and is produced in the volume differs from the rate at which solute leaves and is consumed, then the concentration of solute in the volume will change. Written somewhat differently, Rate of change in solute content of the volume = Solute influx + Net solute production rate – Solute efflux,

(11.8)

where the net production rate equals the production rate minus the consumption rate. Written in term of the fluxes in Fig. 11.5,

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 sc ¬­ žž ­ dxdydz  J x ( x )dydz J y ( y)dxdz J z ( z )dxdy rdxdydz žŸ st ®­  [ J x ( x dx )dydz J y ( y dy)dxdz J z ( z dz )dxdy]  sJ ¬ sJ y sJ  rdxdydz  žžž x dxdydz dydxdz z dzdxdy­­­, sy sz Ÿž sx ®­

(11.9)

where c(x,y,z) is solute concentration and r(x,y,z) is the local net solute production rate in mols/time-volume. Each term in Eq. (11.9) has the units of mass/time.

Figure 11.5. Differential volume used to derive the expression for solute conservation in a reaction-diffusion system. The three fluxes entering the volume are shown with heavy arrows, and one of the three fluxes leaving the volume is shown with a lighter arrow.

Substitute Fick's First Law [Eq. (2.9)] into the right-hand side of Eq. (11.9), component by component. For generality, we allow the diffusion coefficient Ds to vary spatially and with the direction of diffusion:  sc ¬ s sc ¬ žžž ­­­ dxdydz  rdxdydz  žžžDsx ­­­ dxdydz Ÿ st ® sx Ÿ sx ® s sc ¬ s sc ¬  žžDsy ­­­ dydxdz  žžDsz ­­­ dzdxdy , ž ž sy Ÿ sy ®­ sz Ÿ sz ®

(11.10)

where Dsi is the diffusion coefficient for transport in the ith direction. Dividing through by dxdydz, rearranging, and returning to vector notation,  sc ¬­ žž ­  ‹<( Ds ‹c) r , žŸ st ®­

(11.11)

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where Ds takes on the appropriate value for each of the three spatial derivatives defined by ‹c. When the diffusion coefficient is the same in all directions — the system is termed isotropic — Eq. (11.11) simplifies to  sc ¬ 2 žžž ­­­  Ds ‹ c r , Ÿ st ®

(11.12)

2

where ‹ is the Laplacian operator. In the steady state and in the absence of net solute production, Eqs. (11.11) and (11.12) reduce to Fick's Second Law [Eqs. (2.27) and (2.28)].

11.5.2. The Krogh Tissue Cylinder The basic Krogh cylinder geometry (Krogh, 1919) is shown in Figure 11.6. Each capillary, of radius R1, nourishes a surrounding cylinder of tissue. The radius of the cylinder is R2. All of the oxygen consumed by the tissue cylinder comes from its concentric capillary, so each tissue cylinder/capillary unit functions independently. This implies that the rate of transport of oxygen to a tissue or organ equals the transport rate for a single unit, multiplied by the number of units in the tissue or organ.

Figure 11.6. Krogh cylinder geometry. Although the Krogh model is pictured in three dimensions in (a), the geometry (b) in which the mathematical problem is solved does not involve the axial coordinate.

The original assumptions used to compute the oxygen tension profile in the tissue surrounding the capillary are: 1. Steady state; the tissue oxygen tension, p, depends on only position. 2. The oxygen consumption rate, m, is the same everywhere in the tissue, and is independent of p; that is, the metabolic rate is zeroth order in oxygen.

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3. Axial gradients of oxygen tension are absent. Thus, all diffusion in the tissue is in the radial direction, and p is a function of only the radial coordinate, r. 4. The blood in the capillary is homogeneous and well mixed radially; that is, its oxygen tension, pA, is independent of r (r b R1). Applying Eq. (11.12) under these assumptions, the left-hand side of the equation becomes zero, and the right-hand side yields D d ž dp ¬­ žr ­  m , r dr žŸ dr ®­

(11.13)

where p has replaced c, and D is the Krogh diffusion coefficient of oxygen in the tissue, equal to DO2 SO2 ; the units of D are mols/s-cm-torr. Furthermore, –m has replaced the production rate r because m is the rate of consumption of the gas. The oxygen tension in the tissue is axisymmetric and depends on only the distance, r, from the capillary axis (Fig. 11.6b). The boundary conditions are 1. Since all the oxygen that leaves the capillary is consumed in the tissue cylinder, no oxygen leaves the cylinder: dp  0 (r  R2 ) . dr

(11.14a)

2. The oxygen tension is continuous at the capillary–tissue interface:

p  pA (r  R1 ) .

(11.14b)

Subject to these boundary conditions, the solution of Eq. (11.13), in the tissue region R1 b r b R2, is p(r )  pA

m ž r 2  R12 ¬­ R22 m r ­ ln ž D žžŸ 4 ®­­ 2 D R1

( R1 b r b R2 ) .

(11.15)

The average tension in the tissue cylinder can be obtained by integrating p over the area between R1 and R2:

p  pA

mR24 R 3m 2 ( R2  R12 )  ln 2 . 2 2 16 D 4 D( R2  R1 ) R1

(11.16)

11.5.3. Modifications to the Basic Krogh Model Obviously, there are many differences between the Krogh model and the real tissue oxygenation process. These differences have prompted a fair number of modifications to the original model. Almost invariably, the resulting equations are not analytically soluble. Some examples are described below.

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1. The resistance of the capillary wall to oxygen transport has been included by changing the boundary condition at r = R1. The boundary condition given by Eq. (11.14b) implies that the resistance of the capillary wall to oxygen diffusion is negligibly small; if it is not, then Eq. (11.14b) must be replaced by the following expression, which equates the oxygen flux into the tissue (left-hand side) to that across the wall (right-hand side): D

dp  k ( p A  p) dr

(r  R1 ) ,

(11.17)

where k denotes the permeability of the capillary wall, in units consistent with D. 2. The dependence of tissue metabolism on available oxygen has been included by letting m depend on p. Two functions that have been used to describe m(p) are the Michaelis–Menten form introduced in Chapter 3, m  Ap /( B p) , where A and B are constants, and a simple linear relationship, m = Ap + B. The former expression provides a convenient standard for testing the assumption that m is constant. The pressure distribution is first calculated under the assumption that m takes on a constant value equal to A. If the lowest tension in the tissue is found to be even a few multiples of B, then m = A is a good assumption throughout the region. The oxygen consumption rate is within 20% of A if p t 4B. Reported values of B vary from about ½ torr to a few torr. 3. Axial variation in arterial oxygen tension has been included by performing a mass balance within the artery analogous to the analysis of tubule transport presented in Chapter 10. In the simplest modification, axial diffusion in the tissue is ignored, so the solution is the same as Eq. (11.15), except that the left-hand side is p(r,z) and the first term on the right-hand side is pA(z). The equation that defines the axial variation of pA is derived from an oxygen balance over a differential length of capillary, dz (Fig. 11.7). The oxygen flux across the capillary wall in the original Krogh solution (in which capillary wall resistance is neglected) is obtained from the pressure gradient at the capillary-tissue interface [Eq. (11.1)]:  dp ¬ m J out  D žž ­­­ ( R22  R12 ) .  žŸ dr ® 2 R r  R1 1

(11.18)

2

The units of Jout are mass/cm of lateral capillary surface-time. The lateral area of the differential length of capillary dz is 2SR1dz, so the rate at which oxygen leaves the capillary element in Fig. 11.7 is: dnout  Sm( R22  R12 )dz .

(11.19)

Equation (11.19) shows that the rate at which oxygen leaves a segment of capillary (the left-hand side) equals the rate at which it is consumed in the corresponding Krogh tissue volume (the right-hand side).

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Figure 11.7. Oxygen balance over a differential length of capillary.

The loss of oxygen from the capillary is accompanied by a change in pA. If the mean blood velocity in the capillary is v, then the blood flow rate is SR12 v and the rate at which oxygen enters the differential length is SR12 vc A , where cA is the (z-dependent) concentration of oxygen in the blood. The oxygen in the blood is in two forms: dissolved and bound. The concentration of dissolved oxygen is simply SpA, where S is the solubility of the gas. Assume that the hematocrit is such that saturated blood binds N ml of oxygen per liter. Then the concentration of bound oxygen is Ns, where s, the hemoglobin saturation, is a function of pA. The total concentration of oxygen at any point in the capillary is: cA(z) = SpA(z) + Ns[pA(z)]

(11.20)

Blood-borne oxygen leaves the differential length of capillary at a rate equal to  ¬ dc SR12 v žžc A A dz­­­ . In the steady state, the rate at which oxygen enters the differential žŸ dz ® length of capillary must equal the rate at which it leaves, either in the exiting blood or through the capillary wall:    dc ¬ ¯ SR12 vc A  SR12 v ¡c A žž A ­­­ dz ° dnout , ¡ Ÿž dz ® °± ¢ or

(11.21)  dc ¬ SR12 v žž A ­­­ dz  dnout . žŸ dz ®

The latter form of Eq. (11.21) can also be derived directly from Eq. (A10.2) by setting Di = 0 (no axial diffusion in the capillary) and vc = 0 (water flow across the capillary

PRINCIPLES AND MODELS OF BIOLOGICAL TRANSPORT

465

wall can be neglected). The sign convention used here for n is opposite to that in Chapter 10. The axial gradient of the oxygen concentration in the capillary is obtained by substituting Eq. (11.19) into (11.21): dc A m( R22  R12 )  . dz R12 v

(11.22)

It remains to write the left-hand side of Eq. (11.22) in terms of oxygen tension, which is the dependent variable of the problem. From Eq. (11.20),  ds ¬­ dpA dc A dp ­  S A N žžž . žŸ dpA ®­­ dz dz dz

(11.23)

In general, the second term on the right-hand side of Eq. (11.23) is nonlinear, because of the shape of the saturation curve. If, however, the slope of the saturation curve is reasonably constant over the tension range of interest, then Eq. (11.22) can be transformed into an equation that is linear in pA: dpA m( R22  R12 )  , dz R12 vS '

(11.24)

where Sc = S + N(ds/dpA). The boundary condition on pA is pA(0) = pAo,

(11.25)

where pAo is normally taken to be the arterial oxygen tension. At any axial location, the lowest oxygen tension in the tissue occurs at the outer boundary of the cylinder, r = R2. At any radial location, the lowest tension is found at the distal end of the capillary, z = L, where the capillary tension is least. Thus, the lowest tension anywhere in the tissue occurs at z = L, r = R2, a site known as the “lethal” or “anoxic” corner. Figure 11.8 shows the location of the lethal corner on a surface map of p(r,z). 4. Longitudinal diffusion in the tissue has been included in some numerical solutions; p in these analyses is a function of r and the axial coordinate, z. The boundary conditions given earlier still hold, except that the ordinary derivative of p is replaced by a partial derivative. The governing differential equation (11.13) is modified by the addition of a second term, Dw2p/wz2, to the left-hand side. Simulations of oxygen transport in cylindrical geometries patterned after the Krogh cylinder have uniformly shown that axial diffusion has a minor influence on tissue oxygen profiles. 5. Organized tissue, such as muscle, might be expected to exhibit different resistances to oxygen diffusion in the r- and z-directions. Thus, in some models, the isotropic diffusion coefficient, D, is replaced in the oxygen transport equation by the radial and axial diffusivities, Dr and Dz..

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Figure 11.8. Variation of capillary and tissue oxygen tensions along two capillaries. The capillaries are 60 Pm apart (R2 = 30 Pm), and r is measured into the tissue from the capillary axis. The capillary tension falls from 90 to 30 torr between z = 0 and z = L. The surface that describes the variation of pressure with distance along and from the capillary is shaded. The lethal corner is seen to be the site of the lowest oxygen tension.

When modifications 2, 4, and 5 are made to the basic Krogh model, the equation for oxygen transport in the tissue becomes rather more complex: Dr s ž sp ¬­ s2 p žžr ­­ Dz 2  m( p) r sr Ÿ sr ® sz

( R1 b r b R2 , 0 b z b L ) .

(11.26)

Other sophistications include a nonuniform oxygen diffusion coefficient, D(r); the inclusion of additional species, such as carbon dioxide, glucose and lactic acid; and the inclusion of myoglobin-facilitated oxygen flux. An interesting modification of the Krogh model that has been explored by a number of authors is a change in the shape of the tissue region to one that is space filling. Two of these geometries, which have been applied to muscle oxygenation, are shown in Figure 11.9. In the hexagonal array (Fig. 11.9a), there are equal numbers of capillaries and muscle fibers. In a third geometry (not shown), there are twice as many capillaries as fibers, and the plane perpendicular to the axes of the capillaries and fibers (i.e., the plane of Fig. 11.9a) is paved with equilateral triangles rather than hexagons. The capillary axes pass through the centroids of the triangles and the fiber axes pass through their vertices. This last geometry was used to test the sensitivity of the Krogh solution to the shape of the region (Groebe, 1992). The Krogh solution was obtained for a tissue cylinder whose cross-sectional area was the same as the area of one of the triangles; this solution was compared with the numerical solution for the triangular region. The average oxygen tensions in the two regions differed by 5% or less over a wide range of

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467

Figure 11.9. Two space-filling capillary-muscle fiber geometries. (a) Hexagonal array (Hudson and Cater, 1964). The axes of cylindrical muscle fibers pass through alternate vertices of the hexagons, marked with solid circles. The surface of one of the fibers is indicated by the dashed circle. The diffusion equation is solved in the plane perpendicular to the capillary axis, as in Figure 11.6b. (b) Capillary sheet geometry (Bailey, 1967). Blood flow is in the z-direction. The problem is solved in the plane of the figure. There are no gradients normal to this plane; the capillary and tissue are modeled as infinite sheets. Bailey used this geometry to treat cocurrent and countercurrent flows. In cocurrent flow, the flow in every capillary is in the direction of increasing z, and each transport unit functions independently, as in the Krogh model. In countercurrent flow, the flow in the evennumbered capillaries is in the direction of increasing z, and the flow in the odd-numbered capillaries is in the opposite direction. When flow is countercurrent, oxygen crosses the boundaries of the transport units, and diffusive shunting takes place from the oxygen-rich inlets of one set of capillaries to the oxygen-lean outlets of the adjacent set, as shown in the figure. It will be seen later in this chapter that this model anticipates current thinking regarding the interaction among neighboring microvessels.

consumption rates. This agreement suggests that, notwithstanding the gaps between the cylinders, the Krogh geometry works as well as the space-filling geometries if the capillaries are coaligned and reasonably well organized spatially. A nearly space-filling geometry can be obtained by packing the tissue space with coaxial Krogh cylinders of varying radii. Turek et al. (1991) examined such a model, in which the Krogh radii followed a log normal distribution, truncated at the low end so that the smallest cylinder radius exceeded the capillary radius, and at the high end so that the cylinders were not unrealistically large. The capillary radius was assumed to be the same for all cylinders. Such a distribution is well suited to describe tissue in which the capillary spacing is nonuniform.

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A log normal distribution is one in which the logarithm of the variable, R2 in this case, is normally distributed. When the distribution is not truncated, the variable, Y, ranges from 0 (ln Y = –f) to f.

11.5.4. Beyond the Krogh Cylinder Notwithstanding the many modifications that have been make to the original Krogh model, it does not incorporate many important features of microcirculatory oxygen transport. Indeed, its predictions do not agree well with experimental data in skeletal muscle, the tissue it was originally developed to represent (Honig et al., 1971). We discuss here two important improvements in capillary perfusion modeling that go beyond those described in the previous subsection: the incorporation of transport processes within the capillary, and the use of more realistic capillary bed geometries. Transport within the Capillary. In all of the analyses described in the previous subsection, the initial assumption of the Krogh model that the interior of the capillary is homogeneous and well mixed radially was retained. Only relatively recently have the processes that take place inside the vessel been included in oxygen transport models. A useful point of entry into intraluminal transport modeling is a review by Hellums et al. (1996), who point out that the resistance to oxygen transport within the capillary is at least as important as that in the tissue. Their review also gives an historical perspective on attempts to model oxygen transport in capillaries. Initial models (e.g., Reneau et al., 1969) of intracapillary transport regarded the blood as a homogeneous continuum, but one in which radial diffusion was driven by radial gradients in partial pressure. Axial convection was also included, while axial diffusion, though initially included as well, was shown to be negligible. The starting equation for analyzing intracapillary transport subject to the these assumptions is obtained by adding the convective term from Eq. (2.30b) to Eq. (11.12):  ¬ žž sc ­­  Ds ‹2 c  v<‹c r žŸ st ®­

(11.27)

Applied to the concentration of dissolved oxygen in the capillary, cA(r,z), and assuming that a steady state has been reached, Eq. (11.27) becomes   1 s  sc A ¬ s 2 c A ¯ ­ žr °  v sc A r  0 , Dp ¡ ¡ r sr žžŸ sr ®­­ sz 2 ° sz ¢ ±

(11.28)

where Dp is the diffusivity of oxygen in plasma. The local rate of production of dissolved oxygen per unit volume of capillary lumen, r, is equal to the local rate of dissociation of oxyhemoglobin. To express r in terms of hemoglobin dynamics, a mass balance on oxyhemoglobin is performed analogous to the mass balance on total capillary oxygen that led to Eq. (11.21). The equation required here is obtained by replacing cA by cHbO2 in Eq. (11.21). After division by SR12 dz ,

PRINCIPLES AND MODELS OF BIOLOGICAL TRANSPORT

v

scHbO2



sz

469

1 dnout . SR12 dz

(11.29)

Here, dnout is the rate at which oxyhemoglobin leaves the differential volume of blood, just as it represented, in Eq. (11.21), the rate at which oxygen was lost. Whereas oxygen was lost by diffusion into the tissue, oxyhemoglobin is “lost” by dissociation to hemoglobin and oxygen. Thus, the right-hand side of Eq. (11.29) is equal to the rate at which oxyhemoglobin dissociates, per unit volume of capillary. This rate is also the rate of production of unbound oxygen per unit volume, which is precisely r. Substituting into Eq. (11.28),   1 s  sc A ¬ s 2 c A ¯ sc ­­ žžr °  v sc A  v HbO2  0 . Dp ¡ ¡ r sr žŸ sr ®­ sz 2 ° sz sz ¢ ±

(11.30)

The last derivative in Eq. (11.30) can be written in terms of the axial oxygen tension gradient: scHbO2 sz



dcHbO2 spA , dpA sz

(11.31)

where pA(r,z) is the oxygen tension in the capillary. Since the concentration of oxyhemoglobin equals the total concentration of the protein, cHb, multiplied by the saturation, dcHbO2 dp A

 cHb

ds , dpA

(11.32)

where ds/dpA is the slope of the saturation curve. We now do a series of simple substitutions and manipulations: replace cA in Eq. (11.30) by SpA; use Eqs. (11.31) and (11.32) to express the axial gradient of HbO2 concentration in terms of the axial gradient of oxygen tension; divide all terms by the solubility, S; and group the terms in vspA / sz . This gives the working differential equation for capillary tension:   1 s  sp A ¬ s 2 p A ¯  ¬ ­­ žžr °  v sp A žž1 cHb ds ­­­  0 Dp ¡ 2 ¡ ­ ° ž S dp A ®­ sz žŸž ¢ r sr Ÿ sr ® sz ±

(r b R1 , 0 b z b L ) . (11.33)

A corresponding equation for diffusion and consumption in the tissue transport, such as Eq. (11.26), must be solved simultaneously with Eq. (11.33), subject to the appropriate boundary and inlet conditions. These are, for the capillary: a. Specified uniform inlet tension: pA(r,0) = pAo; sp A b. Zero axial tension gradient at the exit: (r , L )  0 ; sz sp A (0, z )  0 ; c. Axial symmetry within the capillary: sr d. Continuity of pressure at the capillary–tissue interface: pA(R1,z) = p(R1,z); e. Continuity of flux at the capillary-tissue interface:

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Dp

sp A sp ( R1 , z )  Dr ( R1 , z ) sr sr

The boundary conditions on the tissue equation, in addition to the continuity conditions above, are: zero axial tension gradient at z = 0 and z = L (R1 b r b R2), and zero radial tension gradient at r = R2 (0 b z b L). It is perhaps superfluous to note that these equations (and the others that follow) must be solved numerically. The inclusion of discrete red cells in later models increased the fidelity of the modeling, but also its complexity. The cells are generally modeled as a train of evenly spaced erythrocytes separated by regions of the capillary occupied by only plasma. Even if the cells are regarded as static — the train is stationary — the concentration of available oxygen varies periodically along the length of the capillary, causing axial variations in tissue oxygen tension that drive axial oxygen fluxes. Groebe (1995) examined the importance of axial gradients of this kind using a geometric model that consisted of three concentric layers: a cylindrical capillary containing a stationary train of erythrocytes; a surrounding cylinder (termed the carrier-free region to distinguish it from the myoglobin-containing tissue) that represented the region extending from the plasma film inside the capillary through the interstitium; and a concentric outer tissue cylinder (Fig. 11.10). He superimposed on this geometry a transport model in which oxygen diffused through the carrier-free region only where the red cell was present, but the flux in the tissue was assumed to be the same everywhere along the length of the capillary. The reduction in the area available for transport through the carrier-free region resulted in a 3–4 mm Hg increase in the difference in oxygen tension between the red cell and the interface between the carrier-free region and the tissue.

Figure 11.10. Groebe model of oxygen transport from a train of red blood cells. Arrows indicate regions where oxygen flux is present. Transport in the carrier-free region takes place only adjacent to a red cell. Adapted from Groebe (1995).

In more recent models, the erythrocyte train is moving and the nonuniformity in tissue oxygen flux that results from the particulate nature of the oxygen source moves with it. In Vadapalli et al. (2002), the oxygen source was a train of erythrocytes, deformed to a realistic “parachute” shape, passing through the capillary. The remainder of the capillary was filled with plasma, and the capillary lumen was surrounded by concentric regions representing the capillary wall, interstitium, and tissue.

PRINCIPLES AND MODELS OF BIOLOGICAL TRANSPORT

471

The first step in the transport process occurs in the erythrocyte, where oxygen dissociates from the hemoglobin and diffuses toward the red cell wall. Viewed from the perspective of the red cell, the process is inherently unsteady, since the hemoglobin saturation in the cell continues to fall as the erythrocyte moves down the capillary; no steady hemoglobin saturation is reached. As expressed in Eq. (11.34), the rate of change of oxygen tension at any point in the cell (the left-hand side of the equation) is the sum of (1) the accumulation (or loss) resulting from diffusion inside the cell (the first term on the right) and (2) a source term describing the rate of dissociation of the gas from local oxyhemoglobin. n     1 s  sp ¬ s 2 p ¯  pc ¬­ ¯° spc ¡ c­ c° ž ž ¡  Dc Sc kc cHb,c ¡ s  (1  s)žž ­­ ° , Sc r 2 ° ¡ ­­ žž žŸ p50 ®­ ° st ¡¢ ¢ r sr Ÿ s r ® s z a ± ±

(11.34)

where the subscript “c” is used to distinguish properties inside the red cell from those in the tissue or plasma, kc is the rate constant for hemoglobin dissociation, and cHb,c is the hemoglobin concentration in the red cell. A variety of rate equations for hemoglobin dissociation have been proposed (Popel, 1989); the one used here, from Vadapalli et al. (2002), implies that the local rate of dissociation is proportional to the difference between the local oxygen tension and the tension that would be in equilibrium with the local hemoglobin saturation. A second equation, similar to Eq. (11.34), describes the rate of change of the hemoglobin saturation, s. The coordinate system is referred to the red cell and moves with it down the capillary; the axial coordinate is accordingly denoted zc, which differs from the axial coordinate z that is tied to the tissue. After oxygen crosses the red cell membrane by simple diffusion, it diffuses across four phases in series: the plasma layer, capillary wall, interstitium, and tissue. These barriers are all axially symmetric in Krogh-type geometries and transport across them can be described by the unsteady version of Eq. (11.12) in cylindrical coordinates: sc j st

 Sj

  1 s  sp ¬ s 2 p ¯ sp žr ­ ° rj , j  1,!,4 ,  Dj Sj ¡ ¡ r sr žžŸ sr ®­­ sz 2 ° st ¢ ±

(11.35)

where the production rate, rj, is in concentration units. The subscript “j” indexes the local oxygen concentration, oxygen solubility, diffusion coefficient, and production rate in the four phases. The production rate is zero in the plasma layer and interstitium; rj equals the negative of the oxygen consumption rates in the capillary wall and tissue. The equation for oxygen transport in the tissue is more complicated if myoglobin facilitation is included, as it was in Vadapalli et al. Characteristic oxygen diffusion coefficients in several phases of interest at 37ºC are presented in Table 11.3. The z-coordinate in Eq. (11.35) is fixed in the physical space of the tissue, while the zc-coordinate in Eq. (11.34) is tied to the red blood cell, which is moving along the capillary at some velocity v. We want to solve these equations in a single coordinate system, and it is convenient to use the red cell coordinates for this purpose. If we fix

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Table 11.3. Diffusion Coefficients at 37ºC of Oxygen in Several Phases Relevant to Capillary Transport (Lightfoot and Duca, 2000) Solvent Water Plasma Whole blood Red cell cytoplasm

Diffusion coefficient, cm2/s 3.0 q 10–5 2.0 q 10–5 1.4 q 10–5 0.95 q 10–5

the red cell coordinates, the red cell is stationary and the tissue appears to be moving upstream (i.e., in the –zc direction) with a speed v. The difference between — and equivalence of — these two frames of reference is shown in Figure 11.11.

Figure 11.11. Equivalence of tissue-based and erythrocyte-based frames of reference. Panels (a) and (b) show a red cell at a tissue location z0 marked with a filled circle. The center of the red cell at zc = 0 is marked with a cross. The distance along the wall between the filled circle and the open circle is the distance vcdt traveled by the cell in time dt. In (a), the tissue is fixed and the red cell moves to the right. In (b), the red cell is fixed and the tissue moves to the left. Panel (c) shows the location of the red cell after time dt. The outcome is the same irrespective of the frame of reference.

In the red cell coordinate system, oxygen is carried to the left in the “moving” tissue, introducing a convection term [see Eq. (11.27)] into Eq. (11.35), which becomes, in the red cell frame of reference, Sj

  1 s  sp ¬ s 2 p ¯ sp ° rj  vS j sp ,  Dj S j ¡ žžžr ­­­ 2° ¡ st sz ' ¢ r s r Ÿ s r ® sz ' ±

(11.36)

PRINCIPLES AND MODELS OF BIOLOGICAL TRANSPORT

473

where c j  S j p has been used. The full set of differential equations that describe the oxygenation of tissue and include transport resistances and kinetic processes within the capillary must be integrated numerically and do not yield closed form solutions. The boundary conditions used in the numerical solutions usually prescribe the conditions at the inlet to the capillary, and require continuity of oxygen tension p and oxysp at the interfaces between the concentric regions. gen flux D j sr Transport within the capillary is often expressed in terms of mass transfer coefficients. Let kl be the mass transfer coefficient that describes the transport of oxygen from the red cell interior to the inner surface of the capillary wall. The oxygen flux is given by J O2  kl ( pc  picw ) ,

(11.37)

where pc is the equilibrium partial pressure corresponding to the hemoglobin saturation in the red cell, and picw is the partial pressure of oxygen at the inner surface of the capillary wall. The coefficient kl can be regarded as the permeability of the intracapil2 lary transport barrier, in mols/cm -s-torr. A similar coefficient can be used to describe transport from the interior of the red cell to the adjacent plasma layer. This description of the transport process can be extended to include the transport resistance of the capillary wall itself. Let the permeability of the capillary wall be kw. It is easy to show that the flux across the entire path from red cell to interstitium is given by J O2  K ( pc  pocw )

(11.38)

k w kl and pocw is the oxygen tension at the interface between the outer k w kl surface of the capillary wall and the interstitium. Models of oxygen transport within capillaries can be used to understand not only the oxygenation of target tissue such as skeletal muscle, but also the oxygenation of capillary blood in the lung. Whitely et al. (2001) have examined pulmonary oxygenation using such a model; they find that oxygen convection in the plasma does not play a major role in pulmonary oxygen transport, and that hemoglobin binding is sufficiently fast that the hemoglobin–oxygen reaction is essentially at equilibrium. Even as oxygen moves from the alveolus to the cell, carbon dioxide moves in the opposite direction. The most important differences between the transport paths for the two gases are more chemical than physical. The diffusion of both gases across the respiratory membrane and through tissue follow similar laws, but oxygen and carbon dioxide interact differently with the components of the red cell as they are carried from the alveolar capillaries to the interstitium or vice versa. As we have already seen, oxygen diffuses into the red blood cell across the erythrocyte membrane and is carried as oxyhemoglobin. Oxyhemoglobin dissociates in the microcirculation, and oxygen is released; the release rate is modulated in metabolically active tissue by the Bohr efwhere K 

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fect. The mechanisms are different for carbon dioxide. As with oxygen, only a limited amount of the CO2 that enters the blood from the tissue dissolves in the plasma; most of it diffuses into the red cells. There, it is either bound to hemoglobin or converted to protons and bicarbonate by the enzyme carbonic anhydrase: H 2 O CO2 l H + HCO3 .

(11.39)

A fraction of the bicarbonate ions enter the plasma through the Cl–HCO3 exchanger in the red cell membrane. Bicarbonate in the plasma, and that remaining in the red blood cells, is carried through the venous system to the lung. The protons are taken up by recently deoxygenated hemoglobin [An interesting model of combined oxygen and carbon dioxide transport in tissue, including the kinetics of the Cl–HCO3 exchanger and the mutual influence of the two gases on hemoglobin binding, can be found in Huang and Hellums (1994).] In the alveolar capillaries, the process is reversed: bicarbonate enters the red cell through the exchanger; hemoglobin releases carbon dioxide and protons as it is oxygenated; Eq. (11.39) proceeds in reverse, catalyzed by the anhydrase; and the dissolved carbon dioxide diffuses out of the red cell toward the respiratory membrane. Seventy percent of the carbon dioxide will have traveled from the capillaries in the form of bicarbonate ion. Just as carbon dioxide from tissue enhances the release of oxygen in the capillaries through the Bohr effect, oxygen from the alveoli shifts the saturation curve for carbon dioxide binding to the right, facilitating its release from the hemoglobin carrier. This is known as the Haldane effect. The complex interaction among the fluxes of oxygen, carbon dioxide, hydrogen ion and bicarbonate from red cell to tissue has recently been modeled by Dash and Bassingthwaighte (2006). A concentric model was used, consisting of a cylinder representing the red cell, surrounded in turn by annular regions representing plasma, interstitium and tissue. The model also includes the competitive binding of the two gases by hemoglobin, myoglobin-facilitated transport in the tissue, and axial variation in transport rates. Capillary Bed Heterogeneity. When the tissue and its capillary bed are modeled as a collection of straight parallel cylinders in a concentric geometry, no account is taken of the heterogeneity of the real microvasculature. This heterogeneity exists at three levels: the capillary blood, the geometry of the individual capillary, and the topology of the microcirculation. 1. The flow velocity and hematocrit, and therefore the red cell flux, are not the same in every capillary. Furthermore, the red blood cells can lose significant amounts of oxygen across the walls of the arterioles that precede the capillaries (Pittman, 2000). The amount of oxygen that is lost depends on the time spent by the erythrocytes in the arteriolar network and the oxygen tension in adjacent tissue, and neither of these are the same for all paths through the network. As a result, there is a distribution of oxygen tensions and oxygen delivery rates in the arteriolar blood entering each capillary bed.

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475

Figure 11.12. A more realistic picture of a capillary bed. Arrows indicate the direction of blood flow. Reprinted with permission from Berne and Levy (1977). Copyright © 1977, Elsevier.

2. The lengths and diameters of the capillaries in a given bed are not all the same, and in many beds the capillaries are not nearly straight. 3. The capillary bed is not organized in as formal a fashion as Figures 11.6 or 11.9 would suggest. This can be seen from Figure 11.12, which is a more realistic representation of a capillary network. Even when capillaries are aligned more or less parallel to one another, there are numerous connecting segments, known as anastomoses, between them. Clearly, the diffusion paths in real tissue are not as simple as they are in the Krogh model and its extensions. In concentric models based on the Krogh cylinder, each tissue cylinder with its axial capillary is an independent transport unit; there is no interaction between the transport processes in adjacent cylinders. However, in real tissue, the rate at which oxygen is provided to the cells by a given microvessel is strongly dependent on neighboring vessels: their location and orientation, their oxygen saturation, and the rate at which they are delivering oxygen to adjacent tissue. The implications of tissue heterogeneity have been explored in more recent work by solving the transport equations for networks of interacting and interdependent microcirculatory vessels, including arterioles and venules as well as capillaries. Early models of this kind laid the capillaries out on a highly structured three dimensional lattice, like those shown in Figure 11.13. The tissue was thought of as being perfused by these lattice-like units rather than by individual capillaries.

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Figure 11.13. Two lattice models of the microcirculation. (a) Grunewald (1968) model. Arrows indicate the direction of blood flow. (b) Metzger (1973) model of capillaries in brain. The tetrahedral transport unit is outlined and the flow directions within it are also shown. [Reprinted with permission from Metzger (1973). Copyright © 1973, Plenum.

More recently, Secomb and Hsu (1994) modeled oxygen transport in a microcirculatory volume (Fig. 11.14) that was based on images of the capillary bed in a hamster check pouch retractor muscle. The model included two arterioles and several capillaries. They used the steady-state form of Eq. (11.12) in Cartesian coordinates, with c replaced by Sp and r replaced by –m:  s2 p s2 p s2 p ¬ Ds S žžž 2 2 2 ­­­  m . sy sz ®­ Ÿ sx

(11.40)

Equation (11.13) is the counterpart of Eq. (11.40) in cylindrical coordinates. The authors neglected the small amount of oxygen dissolved in the plasma, in which case Eqs. (11.21) and (11.23) yield the following mass balance for intravascular oxygen: QN 2

dn ds   out , d[ d[

(11.41)

where the blood flow rate, Q = SR1 v, and [ is a coordinate along the length of the vessel, which is not assumed to be aligned with any of the Cartesian coordinates. The resistance to oxygen transport in the capillary was described by a mass transfer coefficient, as in Eq. (11.37). To relate the oxygen flux out of the capillary [defined by Eq. (11.37)] to the rate of oxygen loss per unit capillary length (equal to dnout/d[), we

PRINCIPLES AND MODELS OF BIOLOGICAL TRANSPORT

477

simply multiply the flux by the area available for transport out of a differential length of capillary: dnout  J O2 < 2 SR1d [ .

(11.42)

Figure 11.14. Microcirculatory volume analyzed by Secomb and Hsu (1994). The element is based on the capillary bed in the hamster cheek pouch retractor muscle and contains 12 capillaries and two arterioles, one of which (A2) is branched. The unbranched arteriole A1 supplies capillaries C2 and C3. The flow in C2 is to the right and the flow in all other capillaries is to the left. Several capillaries are seen to be deflected around the arterioles. Reprinted with permission from Secomb and Hsu (1994). Copyright © 1994, American Physiological Society.

Secomb and Hsu solved for the oxygen fluxes among the arterioles, capillaries, and tissue for the geometry in Figure 11.14 using realistic parameter values and a novel Green's function approach (Hsu and Secomb, 1989). Figure 11.15 shows how oxygen is transferred from the incoming arterioles and capillaries to the tissue, in the resting state and in exercise. At rest, the oxygen flux from the arterioles into the tissue is 86% of the oxygen consumption rate. A fraction of this flux serves tissue metabolism, while the rest diffuses into more distant and less well oxygenated capillaries. Oxygen also diffuses into the tissue from oxygen-rich capillaries, satisfying the remaining metabolic demand. Capillary-derived oxygen can also diffuse through the tissue into more distal, less saturated capillaries, much like the diffusive shunting described earlier. This oxygen is then carried downstream by the receiving capillary to

478

CH. 11: GAS TRANSPORT

Figure 11.15. Oxygen fluxes among the arteriolar, capillary, and tissue compartments for the microcirculatory volume shown in Figure 11.14. The volume is schematized to consist of a single arteriole flowing into the region from the upper left and forming an anastomosis with a capillary entering from the lower right. Solid lines denote convection in blood; dashed lines denote diffusion in tissue; the jagged line corresponds to tissue consumption. Fluxes are normalized in each case to the tissue consumption rate within the volume, so the value on the jagged line is unity. (a) Resting tissue. (b) Exercising tissue; the oxygen consumption rate and arteriolar flow rate equal ten times the resting level. Adapted from Secomb and Hsu (1994).

nourish tissue more distant from the arterioles. In this way, the tissue serves as an oxygen conduit, allowing arterioles and oxygen-rich capillaries to oxygenate by diffusion capillaries to which they need not be physically connected, thereby improving the distribution of oxygen throughout the tissue. This computational result has been con-

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479

firmed in experiments: the oxygen tension in a capillary does not necessarily decrease continuously along its length, but can exhibit local increases where it passes near an arteriole (Ellsworth and Pittman, 1990). The heterogeneities in flow rates, hematocrits, and capillary dimensions mentioned earlier can be superimposed on the capillary network. At branch points (nodes) in the network, both the blood flow and the volume flux of red cells entering the node must equal those exiting; put another way, 3

œQ  0 , i

(11.43a)

i 1

3

œQ H i

i

0

(11.43b)

i 1

where, for each of the three capillary segments meeting at the node, Qi is the flow rate into the node and Hi is the corresponding hematocrit. These conservation laws are combined with an oxygen transport equation in the tissue and a mass balance in the capillary. In an analysis of a capillary bed consisting of anastomosed and tortuous capillaries (Fig. 11.16), Goldman and Popel (2000) used a tissue transport equation similar to that used by Secomb and Hsu, except that the

Figure 11.16. A selection of the capillary networks analyzed by Goldman and Popel (2000). (a–c) Straight capillaries with connecting anastomoses; SA3 has twice the number of anastomoses as SA1 and SA2. (d) Sinusoidally varying (i.e., tortuous) vessels with anastomoses. The authors also calculated oxygen tension distributions and distal capillary tension for straight unbranched (i.e., without anastomoses) cocurrent and countercurrent capillaries, unbranched tortuous capillaries, and parallel unbranched capillaries whose axes were placed randomly. Reprinted with permission from Goldman and Popel (2000). Copyright © 2000, Elsevier.

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Laplacian operator in Eq. (11.12) was retained to allow the tissue flux to be calculated without regard to any specific coordinate system (Goldman and Popel included myoglobin facilitation in their tissue transport equation, but that process is absent from what follows, as explained earlier): Ds S‹2 p  m  0

(11.44)

To derive the mass balance equation, substitute Eq. (11.23) into (11.21), replacing dz by d[, divide by d[, and rearrange:

SR12 vS a

dpA dnout 0, d[ d[

(11.45)

where Sc is defined after Eq. (11.24). In an asymmetric geometry, the flux out of the capillary can vary circumferentially, so we define an average flux out of the capillary, J , such that dnout  2SR1 Jd [ ,

(11.46)

2S

1 where J ([)  J ([, T)d T 2 S ¨0 Using Eq. (11.46) and dividing by SR1, Eq. (11.45) becomes R1 vS a

dpA 2J  0 . d[

(11.47)

In the complex geometry treated by Goldman and Popel, J([,T) was given by a permeability condition equivalent to Eq. (11.17). Their simulations demonstrated that countercurrent flow, and a combination of anastomoses and capillary tortuosity, both improved oxygen transport considerably relative to earlier models in which the flow was cocurrent and the vessels were straight and independent.

PROBLEMS: CHAPTER 11 1.

Use the Peclet number to explain why oxygen is convected from the lung to the microcirculation, while diffusion is the primary mechanism by which the gas reaches the cells of the perfused tissue.

2.

Although the respiratory membrane consists of multiple barriers in series, gas transport across it can be characterized by a single permeability. Show why this is so, and express the permeability of the respiratory membrane, kg,rm, in terms of the permeabilities of the six resistances in series that it represents.

3.

Assume that the alveolar oxygen tension is 100 torr and that the average oxygen tension along the length of a pulmonary capillary is (40 + 100)/2 = 70 torr.

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(a) Calculate the number of moles of oxygen that cross the respiratory membrane each minute, based on the driving force implied by these two pressures. (b) Compare this number with the known oxygen consumption rate of a resting male, and explain any discrepancy between the two numbers. Hint: Sketch the variation of pO2 along the length of the capillary and compare with the estimated average oxygen tension. 4.

The oxygen extraction ratio is the fraction of the arterial oxygen that is removed from the blood during its passage through the capillaries. Estimate this quantity from the information given in this chapter.

5.

The saturation curve for myoglobin can be described by Eq. (11.7b) with n = 1 and p50 = 5.3 torr. (a) Sketch the myoglobin saturation curve on the same scale as that for hemoglobin in Figure 11.3. (b) Assume you are about to be transfused with either of two blood substitutes, one based on hemoglobin and one based on myoglobin. Both substitutes contain the same concentration of oxygen binding sites. Which substitute would you request, and why?

6.

Blood flows through a capillary 6 μm in diameter with a mean velocity of 0.3 mm/s. The hematocrit is 45% and the oxygen saturation is 50%. What is the oxygen transport rate through the capillary?

7.

What hematocrit was assumed in the construction of Figure 11.3?

8.

The solubility of oxygen in plasma is 1.34 q 10 mols/cc-torr and is about 10% higher in red blood cells, so the effective oxygen solubility in blood at 40% he–9 matocrit is about 1.4 q 10 mols/cc-torr. At that hematocrit, the concentration of hemoglobin (Hb) in blood is 2 mM. The oxygen binding capacity of hemoglobin is 4 mol O2 /mol Hb. What fraction of the total oxygen content of blood is bound to hemoglobin when the oxygen partial pressure is 30 torr. Use n = 2.7 in the Hill equation.

9.

A slab of skeletal muscle 0.5 mm thick (the nominal length of a capillary) is perfused by a collection of 0.5-mm long straight parallel capillaries that traverse the 2 slab; there are 500 capillaries/mm slab surface. The capillaries are 2.5 μm in diameter and the mean blood velocity in the capillaries is 2.5 mm/s. The oxygen tension of blood entering the capillary is 100 torr. The oxygen consumption rate in the tissue is 40 cc O2/100 cc tissue-min. (a) Is there sufficient oxygen in the incoming blood to meet the oxygen demand of the tissue?

–9

482

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(b) If the answer to (a) is yes, what is the oxygen tension at the distal end of the capillary? (c) If the answer to (a) is no, what blood flow rate is needed to meet the tissue demand? 10. Demonstrate that Eq. (11.15) satisfies Eqs. (11.13) and (11.14). 11. Derive Eq. (11.15) from (11.13) and (11.14). To start, multiply both sides of (11.13) by r and integrate with respect to r. 12. A capillary bed is modeled by an array of Krogh cylinders whose radii is 30 μm. The oxygen tension at a point along the capillary is 90 mm Hg, while that at the corresponding outer edge of the cylinder is 10 mm Hg. The capillary radius is 10 –6 2 μm and the diffusion coefficient of oxygen in the tissue is 6.3 q 10 cm /s. What is the metabolic consumption rate of the tissue? 13. Although the Krogh cylinder was developed to describe transport in muscle, Lightfoot and Duca (2000) have reported some numbers for cerebral tissue. The capillary radius is 3 μm; the capillary spacing, equated to the cylinder radius, is –5 30 μm; the oxygen diffusion coefficient in the tissue is estimated at 1.5 q 10 2 cm /s; and the oxygen consumption rate is 0.0372 mM/s. The capillary blood flow velocity is 400 μm/s, and a nominal capillary length is 180 μm. If the oxygen tension at the inlet to the capillary is 95 torr, what is the tension at the outlet? 14. The use of ml (or cc) as a unit of mass for a gas can lead to confusion when transport calculations are being made. The units of other variables can also be different from those used here. In these less familiar units, the values of D –5 2 and m determined by Krogh for frog muscle at 20ºC are 1.4 q 10 cm /min-atm –4 –1 and 5 q 10 min , respectively. Consider a Krogh cylinder 10 Pm in diameter with a capillary oxygen tension of 75 mm Hg. (a) Plot the oxygen tension at the outer edge of the cylinder vs. R2, for values of R2 between 50 and 250 μm. (b) Assume the frog muscle is working very hard, so that its oxygen consumption rate is ten times the normal value. Construct the same plot as in (a). (c) What is unusual in the second figure and what limitation of the Krogh model does it illustrate? 15. The solution for the oxygen tension profile in the Krogh cylinder is given by Eq. (11.15). Consider a point at radius r within the cylinder, and sketch the variation of p(r) with the radius of the cylinder, R2. Does anything unphysiological happen as R2 gets large? What failure of the Krogh model does this reflect? 16. Derive Eq. (11.16) from (11.15).

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17. Explain why the variation of pA with z in Figure 11.8 is concave down even though the oxygen consumption rate is the same throughout the tissue. 18. Sketch the layout of the capillaries and muscle fibers on the triangular array described at Figure 11.9. Draw the equivalent tissue cylinder surrounding one of the capillaries. Would you expect the average oxygen tension in the cylinder to be greater or less than the average tension in one of the triangles? Explain your answer. 19. Based on the properties of cerebral tissue capillary beds given in Problem 13, use the Peclet number to estimate the importance of axial diffusion in the capillary relative to convection. 20. Derive Eq. (11.33) from the equations that precede it, following the steps specified in the text. 21. Demonstrate that the dissociation rate in Eq. (11.34) is proportional to the difference between the current oxygen tension and the tension that would be in equilibrium with the current hemoglobin saturation. 22. Derive Eq. (11.38). 23. Referring to Figure 11.15, when muscle is exercising and blood flow increases in proportion to oxygen consumption, the arteriolar oxygen flux into the tissue drops from the resting value of 86% of tissue consumption to 32%, and the flux from tissue to capillaries drops from 46 to 5%. Explain in physical terms why this is so.

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INDEX

A Abluminal side of epithelia, 392 Absolute reaction rate theory, 143–146 Accelerative exchange diffusion, 173 Accumulation ratio in active transport, 197, 219–220 Acetyl coenzyme A, 80 Acetylcholine receptor (AChR) channel, 120, 123–124 kinetics, 152–154 Acetylcholine receptor (AChR) in neuromuscular transmission, 370–378 Acidification systems and pH regulation, 324 Acinar cells, secretion by, 430–432 Actin, 77, 99, 384 Actin-myosin binding, 385-387 Action potential, 330–331 electrical aspects, 333–349 in muscle, 370–372, 376–378 molecular aspects, 349–359 propagating, in nerve, 341, 343–347 Activation and deactivation equilibria in active transport, 213 Activation and inactivation of channels, 113, 122, 125–131, 349–351 Activation energy, 113, 145 Activation in scalar active transport, 191 Activation of ATPase, 201 Activation parameters in nerve, 338–339 Activation ratio, 214 in scalar active transport, 206, 214–215 Active exchange with a closed compartment, 52–53 Active transport, 57, 185–223 classification, 188–189 coupling of transepithelial water flow to, 405–412 definition, 248–250 effect on membrane potential, 369–370 identification, 190 kinetics, 205–220 of calcium, 307–309 primary, 188, 192–193, 257 primary scalar, 213–215 rheogenic, 207 scalar, 191–199 (see also Cotransport) cyclic representation, 192 of sodium and potassium, 202–204 secondary, 193–199

secondary, 188 (see also Cotransport) cyclic representation, 197 tertiary, 199 cyclic representation, 200 vectorial, 191, 199–202 cyclic representation, 201 of sodium and potassium, 216 Activity coefficient, 12–13, 32 in biological solutions, 12–14, 17–19 Activity, solute, 12 Adaptin, 92 Adenosine diphosphate (see ADP) Adenosine triphosphate (see ATP) Adenylyl cyclase, 303 Adherens junctions, 395 ADP, 80–81, 96 Affinity of a chemical reaction, 199, 215–216, 248 of ATP hydrolysis, 187 of binding, 191 Affinity-type cotransport, 211 Agonist stimulation of transport activity, 305 Aldosterone, 306, 400, 428 Alkali ions, channel selectivity toward, 114–115, 117 Allosteric effects in excitation–contraction coupling, 381 Allosteric response in ion channels, 121 Alternating access model of carrier transport, 139 Amphipathic proteins, 70–71 Anaerobic glycolysis, 79 Anastomoses, 479 Animal cell (see Cell, animal) Anisotropy and vectorial active transport, 199 Antegrade propagation, 345 Antidiuretic hormone (ADH), 400, 427 (see also Cyclic-AMP) in transport regulation, 301, 304–306 Antiport (see Countertransport) Apical surface of epithelium, 392, 398–399 Aquaporin-2-loaded vesicles, 304–305 Aquaporins, 133–136 Association–dissociation process in channel transport, 115 ATP, 77, 97, 105 binding in muscle, 385–386 in active transport, 187 synthesis, 78–84

497

498 ATPase in active transport, 80–81, 201, 216 P-type, 188–189 ATP-sensitive potassium channels, 132 ATP synthase, 80–81 nonequilibrium thermodynamic description, 253 Atrial natriuretic peptide (ANP), 400 Autohydrolysis, 303, 305 Automodulation, 366 Axolemma, 335-336 Axon, 330, 334 Axoplasm, 331

B Ball-and-chain model of gating, 121 Ball-and-chain model of sodium channel inactivation, 357, 359 Barrier heights in channel transport, 150 Barriers in lipid bilayer, 269–270 Basal surface of epithelia, 392 Basement membrane, 395 Basolateral surfaces of epithelia, 392, 398–399 Behn equation for ionic electrodiffusion, 48 ß-cells regulating glucose levels, 315–316 Biased random walk in chemotaxis, 101 Bidomain model of conduction in cardiac tissue, 376 Bi-ionic solution, diffusion potential, 51–52 Binary diffusion coefficient, 32, 34 Binding cooperativity in active transport, 211 Binding of ions to macromolecules and activity coefficient, 13 application of specific ion electrodes, 18 Binding sites in saturable channels, 147 on carriers, 112 Bipolar orientation of water dipoles in aquaporins, 136 Bohr effect, 455 Boundary conditions for electrodiffusion, 48–49 rate of change and transport quasisteadiness, 42 Brownian motion, 30 Brownian ratchet mechanism of ATP synthase, 82–83 Brush border, intestinal, 419

C Cable theory, 343–347 Calcium activation of potassium channels, 128–129 and compartmental analysis, 309–315 and muscle fiber conductance, 369 compartmental modeling of dynamics, 313–315 in excitation–contraction coupling, 378–380 release in muscle excitation, 380–382 role in neurotransmission, 360 Calcium-activated potassium channels, 132 Calcium ATPases, 307

INDEX Calcium-based regulation of transport activity, 307–315 Calcium channels, 132 Calcium-dependent secretion, 96 Calcium-induced calcium release (CICR), 381 Calcium release channels, 121–122, 380–381 Calcium signaling, 307–309 Calmodulin in calcium transport, 308 Calsequestrin, 378 Capacitance of membrane, 294 Capillaries and water transport, 438–439 Capillary bed heterogeneity, 474–480 Capillary, gas transport within, 468–474 Carbon dioxide transport in blood, 474 Cardiac muscle, 367 conduction in, 376 Cargo receptor proteins, 92 Carrier(s) activation in cotransport, 209-211 binding multiple substrates, 169–170 conservation of, 208 differences from channel systems, 139–141 in countertransport, 194-195 mechanisms, 111–141 models, 137–139, 162–170 Carrier–substrate complex, 169 Carrier transport inhibited, kinetic representation, 178 inhibition, 175–179 kinetic and cyclic representations, 163 kinetic models, 161–164 kinetics, 141–143 rates, 140 Caspr, 348 Catabolism, 78 Caveolae, 91 Caveolin, 91 Cell, animal structure, 68–78 osmotic effects on, 245–246 overview, 67–68 Cell membrane(s), 67 (see also Lipid bilayer) composition and structure, 68–75 heterogeneity, 68–73 mosaic model, 71–73 polarization, 73 transport routes across, 72, 265–270 Cell migration, 100–101 Cell pH, regulation, 322–325 Cell potential, 333 and ionic permeability, 57–59 Cellular motility, 100–107 Cellular organelles (see Organelles) Cell volume, regulation, 317–322 Centrosome, 78, 105 Channel block, 174–175 Channel geometries, convection, diffusion and mass addition in, 443–445

INDEX Channel(s) (see also Pore) activation, 113, 122, 125-131 calcium-dependent potassium, 128-129 classification, 131-137 differences from carrier systems, 139–141 energy profiles in, 127-131 ion selectivity, 72, 114-115, 140 mechanically activated, 129-130 monovalent cation, table, 134 saturation, 146–151 sensory, 130-131 stochastic properties, 158–161 structure and models, 120–124 voltage-activated sodium, 154–158 Channel transport, models, 141–143 Chaperone proteins, 88 Charged macromolecules in Donnan equilibrium, 19–20 Charged membranes, 59–61 Chemical equilibrium, 21–24 Chemical potential, 1–14, 31–32, 43, 231 solute, dependence on composition, 7–11 solute, dependence on pressure, 6–7 solvent, 232–233 Chemical potential difference, 233 Chemical potential gradient, 30-31 Chemical potential in secondary active transport, 198 Chemoattractant/chemokine, 101–105 Chemotactic velocity, 104 Chemotaxis, 101–105 Chloride and muscle fiber conductance, 369–370 and secretion by acinar cells, 431–432 conductance in nerve, 362 exchange, 171 Chloride–bicarbonate exchanger, 322-324 Chloride channels, 132–133 Cholera toxin, 432 Cholesterol, 70–71 Chord conductance of epithelia, 418 Chord formalism, epithelia, 418 Cilia, 105–107 Circulation across a parallel-path membrane, 402–403 Cis stimulation of pump, 195 Citric acid cycle, 80 Clathrin, 92 Claudins, 396 Coated pits, 92, 94 Colloid osmotic pressure, 435 Compartmental analysis, 43, 166, 300–301, 309–315 nonequilibrium binding in, 327–328 Competitive inhibition, 140, 175, 177–179 Composition, dependence of chemical potential on, 7–11 Compound action potential, 346 Concentration polarization, 412–414 Conductance effect of membrane potential, Hodgkin-Huxley model, 337–340

499 effect on membrane potential, 291–292 membrane, 290, 293 of epithelia, 414–416 single channel, 131, 151–152 single channel, from current fluctuations, 158–161 Conduction velocity in nerve, 333–335 Conjugate forces and fluxes, 186, 225–226, 234, 248 Connexin, 136 Constant-field equation, 53–57 application to nerve, 333 Constitutive activity, 91 Constitutive secretion, 96 Contraction and relaxation in excitation–contraction coupling, 382 Convection, 36 Convective diffusion, 41, 282–283, 443–445 Convective diffusion equation, 41, 277 Cooperativity, 113, 125, 131 Cosolute, 188, 193 Cotransport, 140, 193 energetics, 196 features, 194–196 kinetic representation, 210 kinetics, 206–208 of glucose, by intestine, 212–213 stimulation, 195 Cotransporter, 169, 172–174 Cotransporter hypothesis of transepithelial water flow, 411–412 Cotransport rate, 210 effect of membrane potential, 209 Countercurrent models of oxygen transport, 467 Countertransport, 139, 140, 194–195, 197–199 cyclic representation, 194 Countertransporter, 169 Coupling in nonequilibrium thermodynamics, 250–252 of transepithelial water flow to active ion transport, 405–412 Coupling coefficient/ratio, 52 in active transport, 212 in secondary active transport, 195 Coupling parameter in active transport, 215–216 Crossbridge cycle, 385–386 Cross-inhibition in intestinal transport, 422 Cross-phenomenological coefficients, 228 Cuprophan, transport coefficients for sucrose and water, 257–258 Curie theorem, 199 Curran model of water flow across epithelia, 405–407 Current density, 44 Current–voltage relations in epithelia, 416–418 Cyclic AMP system, 301–306 Cystic fibrosis, 432 Cystic fibrosis transmembrane conductance regulator (CFTR) channel, 122, 125, 432 Cytochrome oxidase, 80 Cytoplasm, 75

500 Cytoskeleton, 73, 76–77, 101 and mechanically activated channels, 129–130 cortical, 77 Cytosol, 69, 75–76, 85 Cytosolic calcium concentration, 307, 309 total vs. free, 313 Cytosolic protons in pH regulation, 324

INDEX Donnan potential, 21–22 Donnan ratio, 21–22 Driving forces (see also Conjugate forces and fluxes) in electrodiffusion equation, 44 in free diffusion, 30–31 Dynamin, 92, 99 Dynein, 99–100, 105–107

E D Deactivation in active transport, 192–193 Debye-Hückel theory, 12–13 Degree of coupling, 250–251 Delayed rectifier, 342 Dephosphorylation, 82 Depolarization and channel activation, 128–129 in neuromuscular transmission, 375–377 of axon membrane, 335, 341–342 Desmosomes, 395 Diacylglycerol, 308 Differential retention of proteins, for maintenance of epithelial polarity, 433–434 Diffusing capacity of the respiratory membrane, 453 Diffusion convective, 41, 443–445 free, 29–61 hindered, 272–276 nonelectrolyte, 43–44, 266–267 passive, 29 through pores, 282–283 transbilayer, 267–269 Diffusion coefficient, 32, 37, 84–86, 266, 460–461 and unstirred layers, 36 effect of solute size, 38–39 Diffusion equation, 40–41 Diffusion layer (see Unstirred layers) Diffusion limit of carrier transport, 164 Diffusion potential, 45 (see also Open-circuit potential) of bi-ionic system, 51–52 Diffusive shunting in oxygen transport, 467 Diffusive water permeability, 283–284 Dihydropyridine receptors (DHPRs), 381 Dipole, 70 Dipole moment in aquaporins, 136 Dipoles in ion channels, 120–121, 123 Dissipation function, 226 in passive transport, 234–235, 247 Dissociation equilibrium constant in active transport, 206 Distal tubule, transport in, 427–430 Diuretics, mechanism, 428 Domains of ion channels, 120 Donnan effect, 35, 74 in channels, 115 suppression, 21 Donnan equilibrium, 14, 19–21, 48–49, 59–60 Donnan osmotic pressure, 244–245, 319

Effective diffusion coefficient in cytoplasm, 84 Efficacy, of flux coupling in active transport, 251–252 Efficiency, of flux coupling in active transport, 252 Eisenman selectivity sequences, 117 Electrical analogs (see Equivalent circuits) Electrical distance between wells, 145 Electrical mobility of ions, 51 Electrical potential gradient, 30 Electrochemical potential, 2–5, 43 definition, 3–5 difference and electrokinetic phenomena, 247 units, 11 Electrodiffusion equation, 29, 43–46 boundary conditions, 48–49 integration, 47–49 limitations, 61 special cases, 49–57 Electrogenic active transport (see Rheogenic active transport) Electrokinetic phenomena, 247 Electrolytes free diffusion, 29, 43–61 passive transport of, nonequilibrium thermodynamic description, 247 Electromotive force (EMF) of a rheogenic pump, 293 Electroneutrality condition, 21, 44, 47, 54 in charged membranes, 60–61 Electroosmosis, 247 Electrostatic forces in channels, 115 Electrostatic potential, contribution to electrochemical potential, 5 Electrostatic potential difference in nonequilibrium thermodynamics, 247 Electrotonic transmission, 366–367 Endocytosis, 91–95 Endoplasmic reticulum (ER), 77–78 and calcium transport, 307–309 and cytosolic calcium, 314–315 Endosomes, 78, 92 Endothelium structure of, 392–395 transport pathways across, 394 Energetics of ion selectivity, 116–119 Energy barrier models of channel block, 176 of ion channels, 143–151 of the lipid bilayer, 269–270 Energy of binding, 116, 118 Energy of hydration, 116, 118–119

INDEX Energy profile, for ion channel transport, 122, 125 Enterocytes, 419 Epithelia classification as leaky or tight, 395 Curran model of water flow across, 405–407 electrical analogs, 414–418 hydraulic conductivity, 409-410 maintenance of polarity, 432–434 resistance, table, 395 secretion by, 430–434 sodium transport across, 398–400 structure, 392–395 transport in parallel-path system, 400–405 transport pathways across, 394–400 water transport across, 405–412 Epithelial sodium channel (ENaC), 132, 399–400 Equilibrium closeness to, in biological systems, 258–259 conditions, 4, 5, 14–15, 21–24 Equilibrium constant, 24 in active transport, 211, 213 in carrier transport, 167–168 Equivalent circuits, 289–294 for epithelia, 414–416 Erythrocytes monosaccharide transport in, 165–166 nonelectrolyte permeability, 35 Eukaryotic cell, 68 Exchange diffusion, 170 Exchange flux, 173, 234 Exchange rate of carrier, 173 Exchangers, 140, 170-174 in the erythrocyte, 170–172 serotonin transporter, 172-173 Excitable cells, 329–387 Excitation–contraction coupling, 378–382 Excitation–secretion coupling, 360-361 Excitatory postsynaptic potential (EPSP), 362, 364 Excitatory synapse, 362 Exocytosis, 95–96

F F0 unit of ATP synthase, 80, 99 Facilitated transport hallmarks, 112–113 inhibition, 174–179 kinetics, 141–174 Faraday number, 3 Fast sodium channels, 342 Fast synaptic transmission, 362 Faxen's function for diffusion through a pore, 275 Feedback effects in transport systems, 315–317 Fenestra, 434 Fick equation, 452 Fick's first law of diffusion, 32, 227, 448–449 for nonideal solutions, 32–33 Fick's second law of diffusion (see Diffusion equation) Field strengths of binding sites and ion selectivity, 116–119

501 First law of thermodynamics, 2 Flagella, 105–107 Flow conductivity (see Hydraulic conductivity) Flow, definition, 311 Flux definition, 30, 311 in active transport, 186–187 in free diffusion, 30–31 Flux equations for primary scalar active transport, 214–215 Focal adhesions, 101 Free diffusion, 29–61, 112 flux in, 30–31 of dissolved gas, 447 of electrolytes, 43–61 of nonelectrolytes, 30–43 Free energy change in secondary active transport, 196 contribution of pressure, 7 Gibbs, 3–4 in chemical equilibrium, 22–23 Frequency spectrum of current through channels, 159 Frictional coefficients, 229–230 Frictional drag, 229 Fusion pore, 95

G Gap junctions, 136–137 Gas transport in the lung, 453–454 within the capillary, 468–474 Gates in channels, 121–122 regulation, 122–131 Gating charges, 127–128 Gating currents, 126, 354 Gating kinetics, 142–143, 151–159, 351–353 GDP and transport regulation, 303–306 Generalized conductance of a membrane, 228 Generalized resistance of a membrane, 227 Gibbs-Donnan equilibrium (see Donnan equilibrium) Gibbs-Duhem equation, 5–6, 9 Gibbs equation, 2–3 Gibbs free energy, 3–4 Glomerular filtration, 434–439 Glomerulotubular balance, 427 Glucose carriers, 315–316 Glucose transport in erythrocytes, 165 GLUT1, 316 GLUT2, 316 Glycocalyx, 74 Goldman-Hodgkin-Katz constant-field equation, 55–56 Golgi apparatus, 78 G proteins, 88, 96, 100, 303 in slow synaptic transmission, 362 in transport regulation, 303, 306 Gramicidin channel, model, 120, 123

502 Groebe model of oxygen transport from capillaries, 470 GTP-binding proteins (see G proteins) Guanosine diphosphate (see GDP)

H Hagen-Poiseuille equation, 272 Haldane effect, 474 Hemichannels, 136 Hemoglobin transport, 455–458 Hemolysis, 246 Henry's Law, 448 Heteroexchange, 170–171 Heterogeneity of microvasculature, 474–480 Heteroreflectivity of a parallel-path membrane, 403 Heterosynaptic modulation, 365–366 Hill equation, 455 Hindered convection, 279 Hindered diffusion, 272–276 Hindrance factors, 283 Hodgkin-Huxley model of the action potential, 337–359 Homeostasis, 42 Hormonal regulation of transport, 306–307 Hormones, as messengers, 301 Hourglass model of aquaporin channel, 135 Hydrated radius of ions, table, 114 Hydration shells, 70, 115 Hydraulic conductivity, 235–236 (see also Phenomenological coefficients) measurement, 401 of a cylindrical pore, 272–273 Hydraulic flow, 237 Hydrogen bonds, 70 Hydrogen–potassium exchange pump, 189 Hydrophilic/hydrophobic interactions, in cell membranes, 70, 72 Hydrostatic pressure, 8 difference, 234 in cells, 238-239 Hypertonic solutions, 243 Hypotonic solutions, 243

I Impermeant charge, 20–21 Inactivation gate, 355, 357–358 Inactivation of channels (see Activation and inactivation of channels) Independence in ion transport, 143, 146 Inhibition of active transport, 205, 422 of carrier transport, 175–179 of channel transport, 125, 174–175 of facilitated transport, 113 Inhibitory postsynaptic potentials (IPSPs), 363 Inhibitory synapse, 362–363 Inositol 1,4,5-triphosphate (IP3), 308

INDEX Insulin exocytosis, 95 feedback in regulation of secretion, 315–317 Integral conductance of membrane, 228 Integral driving force, 31 Integral membrane proteins, 73 Integral resistance of membrane, 227 Integrins, 100 Intercalated discs, 376 Intercellular shunt, 426 Internal concentration in energy wells, 145 Internal energy, 2 Internal leak in pumps, 251 Interphase equilibrium, definition, 4–5 Intestinal absorption, 419–422 Intestine, structure, 419–421 Intracapillary transport, 468–474 Intracellular diffusion, 84–87 coefficient, 84–85 Intracellular diffusional resistance, 37–38 Intracellular electroneutrality, 318 Intracellular environment, regulation by transport, 301 Intracellular transport, 68, 84–100 Inward rectifier potassium channels, 132, 146, 174 Ion channels (see Channels) Ion electrodes, specific, 17 Ion equilibrium across membranes, 14–21 Ion flux across a membrane, 55–56 in electrodiffusion, 48–49 Ionic conductance, 294 Ionic current, 44, 50 Ionic permeability, 56 and cell potential, 57–59 Ionic strength of solution, 12 Ionophores, 120 Ionotropic receptors, 362 Ions diffusion potential in a bi-ionic solution, 51–52 electrical mobility, table, 51 transport in proximal tubule, 424–430 Ion selectivity of channels, 114–119, 131 energetics, 116–119 Irreversible thermodynamics, 225–264 Isotonic solutions, 243

J Junctions, in epithelia, 392–394, 396–398

K Kedem-Katchalsky equations, 235–237, 401 for a multicomponent system, 246 for concentrated solutions, 260 numerical illustration, 237 Kidney action of ADH on, 304 structure, 423

INDEX Kinesins, 96–99 Krebs cycle, 80 Krogh diffusion coefficient, 449 Krogh tissue cylinder, 458, 461–462 limitations, 468 modifications, 462–468

L Lamellipodia, 100–101 Lattice models of oxygen transport, 476 Law of mass conservation, 40–41 Leaks in active transport (see Pump-leak systems) Lecithin, chemical structure, 71 Level flow in active transport, 189, 209 Ligand-activated channels, 122 Ligand-gated calcium channels, 132 Ligand-gated cation channels, 133 Ligand-gated chloride channels, 133 Linear nonequilibrium thermodynamics, 225, 227-229 Lipid bilayer, 71 energy profile, 269–270 models of transport across, 265–270 Lipid rafts, 73 Locomotion, 100–107 Loop currents, 343–344 Loop of Henle, transport in, 427 Luminal side of epithelium, 392 Lung, gas exchange in, 453–454 Lysosomes, 78, 92

M Mass addition in channel geometries, 443–445 Mass transfer coefficient, 473 Maximum transport rate in carrier systems, 166 Mechanically gated cation and chloride channels, 132 Mechanosensitive channels, 129–130 Mediated diffusion/transport, 111–184 features, 112–113 Membrane(s) charged, 59–61 conductance, 50–51 ion flux across, 55-56 ionic equilibrium across, 14–21 Membrane action potential, 335–337 excitation, 340–343 Membrane conductance, in electrical analogs, 290, 293 Membrane current, 15–16 Membrane noise analysis, 158–161 Membrane permeability, 33–34 nonelectrolyte, 35 Membrane permittivity, 47 Membrane potential, 15–17, 47–48, 50, 55–56, 74–75, 332 (see also Open-circuit potential) and channel conductance, 125–128 dependence of conductance on, 337–340 influence on voltage-sensitive channels, 157 Membrane proteins, 70–71

503 Membrane sieving, 436–438 Metabolic coupling, 186–187, 215–216, 250 (see also Active transport) Metabolism and ATP production, 78–84 Metabotropic receptors, 362 Michaelis-Menten equation, 104, 305, 313 Microtubules, 105–106 Mitochondrion, 77–80 transport of proteins into, 88–90 Molality and molarity, 242 Monosaccharide transport in erythrocytes, 165–166 Motile cells, 100 Motor proteins, 99 nonequilibrium thermodynamics, 252–253 Mucosal surface of epithelium, 392 Multi-ion transport, electrical analog, 290–293 Muscle classification, 367-368 contraction, 382–387 structure, 368-369 Muscle fiber active transport in, 368–370 excitation, 370–382 resting state, 368–370 Myelinated neuron, 347–349 Myofibrils, 368, 382 Myoglobin, 449 Myosin II, 99, 383

N Neck linker, 96 Negative coupling (see Countertransport) Nephron structure, 423 transport in, 422, 424–430 variation of flow and fluid composition along, 425 Nernst equation, 15 Nernst equilibrium, 14–15 Nernst-Planck equation (see Electrodiffusion equation) Nernst potential, 15–17, 50, 52, 55, 57, 294, 332 dependence on activity ratio, 17–19 in resting muscle fiber, 368–369 Nerve potential dependent channels in, 337–340 sodium channel densities, table, 348 Neuromodulators, 365–367 Neuromuscular junction, 370-371 Neuromuscular transmission, 370–378 Neuron(s) radii and conduction velocities, table, 346 resting state, 331–333 structure, 330 Neurotransmitter-gated channels, 132 Neurotransmitters, 331, 360–367 action, 363–365 chemical structures, 365 release, 360–364

504 Node of Ranvier, 347 Noise analysis, 158–161 Nominal sieving coefficient, 436–438 Noncompetitive inhibition, 179 Nonconjugate forces and fluxes, coupling of, 250–252 Nonelectrolytes diffusion across lipid bilayer, 266-267 diffusion compared to electrolyte diffusion, 43–44 free diffusion, 30–43 multiple, passive transport of, 246 permeability, 29 transport through pores, 273 Nonequilibrium binding in compartmental analysis, 327–328 Nonequilibrium thermodynamics information content, 259 limitations, 230, 254–260 of active transport, 248–253 of motor proteins and ATP synthase, 252–253 Nonideal solutions, free diffusion in, 32–34 Nonlinearity of electrodiffusion, 47 Nonselective channels, 131 Nuclear envelope, 77 Nuclear export signal, 88 Nuclear import receptor, 87 Nuclear localization signal, 87 Nuclear pore complex, 69, 77, 87 Nucleus, 69, 75, 77 transport of proteins into, 87–88

O Obligatory exchange/transport, 171, 173-174, 251 Occluded state in carrier transport system, 169 Occludin, 396 Occupancy fraction, in chemotaxis, 101 Occupancy state, 150–151 Ohm's Law, 38, 50, 293 Oncotic pressure, 435–436 Onsager reciprocal relation, 229, 231 Open circuit, 45–46 Open-circuit potential, 47–48 (see also Membrane potential) from equivalent circuits, 293–294 of a bi-ionic system, 51–52 Organelles, 75, 77–78 Osmolarity, 242, 245 Osmometer, 239–240 Osmosis, 238–239 in presence of unstirred layers, 412–414 Osmotic coefficient, 242–243 Osmotic equilibrium, 245 Osmotic flows, 317 Osmotic pressure, 234–235, 238–243, 245–246 Osmotic pressure difference, 234 Osmotic water permeability, 283 Ouabain, effect on the sodium–potassium pump, 205

INDEX Oxidative phosphorylation, 80–81 Oxygen consumption by various organs, table, 452 Oxygen pathway from red cell to tissue, 456–458 Oxygen tension, variation along gas transport path, 452, 465–466, 471–472 Oxygen transport in blood, 455–459 in capillaries, 456–459, 464–465 in microcirculation, 476–479 in tissue, 459–461 Oxyhemoglobin, 456, 458, 469 Oxymyoglobin, 449

P P-type ATPases, 188–189 Pacemaker fibers, 376 Paracellular channel, 398 Paracellular conductance, 395 Parallel-path membrane, transport across, 400–405 Partial molar free energy, 4 Partial molar volume, 7 Partial pressure, 448–449 Partition coefficient, 266–268 Passive exchange with a closed compartment, 52–53 Passive flux of dissolved gas, 448–449 Passive transport, nonequilibrium thermodynamic description, 231–247 Patch clamp technique, 151–158 Peclet number, 282–283 Peripheral membrane proteins, 73 Permeability, 33–34, (see also Phenomenological coefficients) apparent, due to unstirred layers, 36, 38 cell membrane, effect of nonpolar solute size, 266 diffusive and osmotic, of water, 283–284 effect of temperature, 39–40, 113, 266–267 in free diffusion, 33–35 in nonequilibrium thermodynamics, 235–236, 255 measurement, 401 nonelectrolyte, cell membrane, table, 35 of a cylindrical pore, 273–276 of nonelectrolytes, 29 ratio in transport through pores, 283, 287–288 relation to mobility, 57–58 pH electrode, 17 pH, regulation of cellular, 322–325 Phagocytosis, 92 Pharmacokinetic modeling, 312 Phenomenological coefficients, 227–229 concentration dependence, 257–258 in a multicomponent system, 246 measurement, 401 of a parallel-path membrane, 402 Phosphatidylethanolamine, 73 Phosphodiesterase, 305 Phospholipase C-ß, 308

INDEX Phospholipids, 70–71 in cell membranes, 68 Phosphorylation affinity, 187 and channel activation, 122, 125 in nonequilibrium thermodynamics, 253 Ping-pong transport, 171, 173–174 Pinocytic vesicles, 91 Pinocytosis, 91 Planck solution for open-circuit potential, 47 Plasma membrane (see Cell membranes) Plasma membrane calcium pump, 189 P-loop, 120, 123–124 Poiseuille equation, 435 Poisson equation, 47 Polarity in epithelia, 392 maintenance of, 432-434 Polarization, in cell migration, 100–101 Pore combined diffusion and convection through, 282–283 hydraulic conductivity, 272–273 ion selectivity, 120–121 saturation of, 148–149 Pore radius/size, 271 and solute permeability, 34-35, 39 equivalent, 276–278, 287–288 from permeability data, 273–276 from reflection coefficients, 279–282 in facilitated transport, 118-119 in human erythrocytes, 285 in the glomerulus, 276–277 of rabbit gallbladder junctions, 396–397 Pore transport, models, 270–288 of charged solutes, 280–282 single-file, 283–287 Positive coupling (see Cotransport) Post-Albers scheme, 202–204, 216 Potassium channels, 120–121, 123, 132 kinetics and gates, in nerve, 353-354 structure, 126–127 Potassium-chloride cotransporter, 324 Potassium conductance, in nerve, 349-351 in volume regulation, 320 Potassium-hydrogen antiporter, 324 Potassium leak channels, 292, 320 Potential barrier models (see Energy barrier models) Pressure, dependence of chemical potential on, 6–7, 9 Primary active transport, 188 closeness to equilibrium, 257 Primary scalar active transport, 192–193, 213–215 flux equations, 214–215 Protein/import receptor complex, 87–88 Protein kinase A (PKA), 304 Protein kinases, 304 and channel activation, 122 Protein/lipid ratio in cell membranes, 68, 70 Protein phosphatases, 305

505 Protein transport, intracellular, 87–90 Proximal tubule, transport in, 422–427 P-type ATPases, 188–189 Pulmonary circulation, 450 Pump-leak systems, 190, 216–219, 251 Pump rate, effect on cell volume, 319–320 Pump, rheogenic, 293 Pump selectivity, 204–205 Purkinje fibers, 376 Pyruvate, 78

Q Quasisteadiness, 42-43, 162

R Rate constants for channel activation, 128 in carrier transport, 167–168 in cotransport, 211 Reaction-diffusion processes in oxygen transport, 459–461 Receptor-mediated endocytosis, 92 Reciprocal stimulation in secondary active transport, 195 Recovery stroke, muscle, 385 Rectification by asymmetric carrier systems, 168 by channels, 146 Recycling in vesicular transport, 93 Reflection coefficient, 235–236 (see also Phenomenological coefficients) measurement, 401 of a parallel path membrane, 404-405 of a pore, 279–282 physical significance, 237–241 Regulated secretion, 96 Regulation by transport, 299, 317–325 Regulation of sodium transport across epithelia, 398–400 Renal clearance, 436–438 Respiratory enzyme complexes, 80 Respiratory membrane, 450 Resting potential (see also Open-circuit potential) of an axon, 333–336 Retrograde propagation, 345 Reversal potential of an exchanger, 198 of a rheogenic pump, 293 of the acetylcholine receptor channel, 372-373 Reverse osmosis, 241 Reversible receptor-ligand binding, 93–94 Rheogenic active transport, modeling of, 207 Ribosomes, 78 Ryanodine receptors (RyRs), 381

S S1 head in myosin, 385–387

506 Saltatory conduction, 347 Salt bridges, 52 Sarcomere, 383 Sarcoplasmic/endoplasmic reticulum calcium (SERCA) pump, 189, 307, 378–380 Sarcoplasmic reticulum, 368 Saturability of facilitated transport, 113 of pump associated carriers, 205 Saturation curve of hemoglobin, 455, 456 Saturation of carrier transport, 146–151 Scalar active transport, 191–199 Schwann cells, 347 Secondary active transport, 188 Secondary scalar transport, 193–199, 206–208 equations, 208–209 Second law of thermodynamics, 2–3 Second messenger systems, receptor-mediated, 301–306 Secretion, 96–97 by epithelia, 430–434 Secretory granules, 95 Selectivity in facilitated transport, 112–113 of channels, 114–119, 131 of transport ATPases, 204–206 Selectivity filter, 119–121, 131 Semipermeable membrane, 238 Sensory gated channels, 130–131 Serosal surface of epithelia, 392 Serotonin transporter, 172–173 Shaker potassium channel, 354 Short-circuit current, 45–46 Shunt pathway, in epithelia, 394, 396–398 Shunts in active transport, 218–220 Sieving by a pore, 281–282 in membranes, 241 Signal transmission in nerve, 330–331, 333–335 Single channel conductance, 131 Single-file transport, 143, 272, 283–287 Single-nephron glomerular filtration rate (SNGFR), 435–436 SI units, 11 Sliding filament mechanism, in muscle, 384 Slippage, in active transport, 217–218 Slope conductance of epithelia, 417 Slope formalism, epithelia, 417 Slow synaptic transmission, 362 Smooth muscle, activation, 367-368 Sodium conductance, in nerve, 351–353 transport across epithelia, 398–400 Sodium-bicarbonate–chloride exchanger, 323 Sodium–calcium exchanger, 198, 307 Sodium channel kinetics, 355–359 Sodium channels, 132 Sodium chloride solutions, activity coefficient, 13

INDEX Sodium-dependent hexose transporter (SGLT-1), 209-213, 411–412, 421 Sodium–glucose transporter (see Sodium-dependent hexose transporter) Sodium–hydrogen exchanger, 322-323 Sodium–potassium–chloride cotransporter, 196–197 in cell volume regulation, 322 Sodium–potassium exchange pump, 188–189, 202205, 318–322, 331–332 nature of, 216 Sodium reabsorption in kidney, 306–307 Sodium recirculation model of water flow across epithelia, 410–411 Sodium transport in distal tubule, 427–430 in proximal tubule, 422–427 Solubility, gas, 448 Solute activity, 12 chemical potential, 10 diffusion coefficient, 32 mobility, 30 Solute flux, 30, 36, 235–236 Solute flux error in a parallel-path membrane, 403–404 Solute permeability (see Permeability) Solute radius, 271 and solute permeability, 39 Solute retardation, intracellular, 85–86 Solution theory, applications of, 38–40 Solvent chemical potential, 8, 232–233 Solvent drag, 237, 249 Solvent viscosity and solute diffusion coefficient, 39 Soma, 331 Sorting tag, for maintenance of epithelial polarity, 434 Space charge, 47 Space filling capillary-muscle fiber geometries, 466–468 Specific ion electrodes, 17 Standard concentration/state in equilibrium thermodynamics, 10 Standard free energy change, 24 Standing-gradient model of water flow across epithelia, 407–410 Starling's Law, 240, 438–439 Static head in active transport, 187, 189, 198, 217–218 Steady state assumption in active transport, 208 in carrier transport, 163 justification, in biological systems, 41-43 Steady-state diffusion equation, 40 Steric effects in pore transport, 241, 274, 279 on channel selectivity, 116–119 Stoichiometric coefficient, 22 Stokes-Einstein equation, 38–39, 85–86, 115 Straight phenomenological coefficients, 228 Streaming effects, 247 Stretch-activated channels, 320–321 Structured water, influence on transport compartment in cytosol, 190

INDEX Subconductance levels in channel kinetics, 354 Substrate activation by, 125 binding in scalar active transport, 206 nonequilibrium uptake and release of, by carriers, 168–169 Subunits of ion channels, 120 Sucrose solutions, transport coefficients across Cuprophan, 257–258 Sugars, intestinal absorption, 419–422 Surface charge on cell membrane, 74–75 Symmetry of carrier transport systems, 164 Symport (see Cotransport) Synapse/Synaptic transmission, 359–367

507 Transport rate carrier mediated, 163–164 in facilitated transport, 112–113 in secondary scalar transport, 208-209 quasisteady, 42–43 Transport regulation by second messenger systems, 301–306 Transport systems activity, 299 feedback in, 315–317 Trans stimulation of pump, 195 Transverse tubules, 368 Tropomyosin, 384 Troponin C, 308, 384 Two-gate model of carrier transport, 139

T Temperature coefficient of facilitated transport, 113 Tension, gas (see Partial pressure) Teorell equation, 30–31 integration, 31–35 Teorell-Meyer-Sievers (TMS) model of transport across a charged membrane, 59–60 Tertiary active transport, 199 Theophylline, 305 Thermodynamic equilibrium, condition for, 4 Thermodynamics First Law of, 2 Second Law of, 2–3 Thevenin conductance and EMF, of epithelia, 416–418 Thevenin equivalent circuit, 292 Tight junctions in epithelia, 394–395 Time scales in biological transport, 42–43 Tonicity, 243 Tortuosity, of pores, 273 Tracer flux in erythrocytes, 165 Transbilayer diffusion, 267–269 Transcellular absorption, 418–430 Transcellular current in epithelia transport, 416 Transcellular pathway and sodium transport, 398–400 Transcytosis, 100, 434 Trans effects in carrier transport, 177 in facilitated transport, 113 Transition rate between states, 98 Transmembrane current density, 48 Transmembrane potential, effect on carrier transport, 207 Transmembrane potential difference, 45 Transmembrane receptors, in chemotaxis, 101 Transport activity regulated by aldosterone, 306–307 regulated by calcium, 307–315 regulation, 299–315 Transport coefficients (see Phenomenological coefficients) Transport, gas, 449–452 Transport mediators, 111–112

U Ultrafiltration, 241 in the glomerulus, 436 Ultrafiltration coefficient, 436 Unidirectional flux in tracer experiments, 165 Unitary channel conductance, 151, 158 of ion-selective channels, 131 Unitary current, 151 Unsteady diffusion equation, 84 Unstirred layers and free diffusion, 36–38 effect on osmotic water flow, 413-414 Uphill transport, 185–186, 188

V Valence-selective channels, 131 Valinomycin as an electrochemical voltage clamp, 292 transport mechanism, 139 van 't Hoff's Law, 9 Vascular endothelium, secretion by, 430 Vasopressin (see Antidiuretic hormone) Vectorial active transport, 191, 199–202, 216 Velocity-type cotransport, 211 Vesicles, 78, 90 secretory, 96 Vesicular transport, 90–100 Viscous effects in pore transport, 274–276, 280 Void fraction, effect on permeability, 34 Voltage-activated sodium channel, 154–158 Voltage clamp, 142, 151 electrochemical, 292 Voltage-gated calcium channels, 132 Voltage-gated chloride channels, 133 Voltage-gated delayed rectifier potassium channels, 132 Voltage-gated fast transient potassium channels, 132 Voltage-gated slow M-type potassium channels, 132 Voltage-gated sodium channels, 132 Voltage sensor, 126–127, 155

508 Volume flux, 234, 236, 238 Volume-regulated anion channels, 321

W Water, chemical potential, 8 Water transport across capillary wall, 438–439 across epithelia, 405–412 and aquaporins, 133–136 in distal tubule, 427–430 in proximal tubule, 422–427

INDEX Well-mixed assumption, and compartmental analysis, 310 Well-stirred assumption, and unstirred layers, 37 Work in thermodynamics, 2 Working stroke, muscle, 384

Z Zeta potentials, 74–75

Figure 4.6. (a) Stereo side view of the Kv1.2 potassium channel. A possibly regulatory β subunit is docked to the cytoplasmic end of the channel. Each subunit is a different color. The TM region spans the cell membrane and is the integral protein component of the channel complex. The T1 domains of the subunits form a tetrameric complex within the cytoplasm that lies on the axis of the pore, so the ions that exit the cell through the pore enter the channel through side portals. Negative charges at the rims of the portals contribute to the cation selectivity of the channel. [Reprinted with permission from Long et al. (2005a). Copyright © 2005, AAAS.] (b) Proposed voltage sensor. The S4–S5 linkers of three of the four subunits are shown in red; the tipping of the linker toward the cytoplasm upon closure can be seen clearly. The S6 segments are shown in blue. Left panel: open state. Right panel: closed state. [Reprinted with permission from Long et al. (2005b). Copyright © 2005, AAAS.]

Figure 9.11. Fluorescent micrograph of rat optic nerve labeled for potassium channels (blue), Caspr (red) and sodium channels (green). Caspr is a protein that marks the edge of the Schwann cell. The strong green signal highlights the highly concentrated sodium channels at the nodes of Ranvier, and the more diffuse blue signal demonstrates the presence of potassium channels in the internodal axolemma. Reprinted with permission from Rasband and Shrager (2000). Copyright © 2000, Blackwell Publishing.

Figure 9.31. Architecture of the SERCA ATPase (Toyoshima et al., 2000). The ATP binding and phosphorylation sites are marked. The structure was determined while two calcium ions were bound within the protein, which is in the Ca2E1 state. Toyoshima's group subsequently published (Toyoshima and Nomura, 2002) an analogous study of the pump in the calcium-free E2 state, detailing the substantial conformational changes that accompany the transition to that state from the E1 state depicted here. These changes also explain the reduction in the calcium affinity of the ATPase that accompanies the transition. See also recent structural studies by Olesen et al. (2004). Reprinted with permission from Toyoshima et al. (2000). Copyright © 2000, Macmillan Publishers.