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.
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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
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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
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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 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 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 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 zi dni.
(1.9)
dG \ z i dni 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 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 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 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 P i dni 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 P i dni .
(1.19)
i
The Gibbs–Duhem equation is obtained by subtracting Eq. (1.19) from (1.18): SdT VdP ni dP i 0 .
(1.20)
i
The sum in Eq. (1.20) can be broken into two sums: ni dP i ni dPi d\ ni zi . i
i
i
For electrically neutral solutions, ni zi 0 , i
so ni dP i ni dPi , i
i
and the electrochemical potential in the Gibbs–Duhem equation can be replaced by the chemical potential: SdT VdP 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 ¬ sP ®
T , ni
sP ¬ s\ . i zi 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 ¬ sP ®
T , ni
¯ s ¡ sG ¬ ° ¡ °. sP ¡ 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 ¬ sP ®
T , ni
s sni
sG ¬ ¯ ¡ ° ¡ 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 ¬ . sP ®T , n 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 ni (Vi dP dPic ) 0 .
(1.27)
i
The coefficient of dP in Eq. (1.27) is 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
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 ni dPic 0 .
(1.31)
i 1
Substituting Eq. (1.30) into (1.31) and rearranging: k
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 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 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
ni dPic VRT 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 i i . 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 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, 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: vi Ri l 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 dni , i 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 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 i dn * , v * ®
(1.60a)
v ¬ dni i dn * . v * ®
(1.60b)
— and for reactants:
Substituting Eqs. (1.60) into (1.59), dn * ¬ ¯ dG P i (vi ) P i vi ° , ¡ p °± v * ® ¡¢ r
(1.61)
where r and 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: P i vi 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: ¬ ¬ vi zi \ vi zi \ viVi P viVi P ® p ® p r r ¬ vi RT ln ai vi RT ln ai p ® r ¬ vi Pi0 vi Pi0 0 . p ® r
(1.63)
Let us consider each pair separately. The first pair can be rewritten as ¬ \ vi zi vi zi . p ® 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 viVi viVi . p ® 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, vi ln ai 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 ln aivi 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 UAX. 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 s , 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 RT 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 . 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 1 d ln J s ¬ . RT dx d ln cs ® cs dx ® The Teorell equation then becomes d ln J s ¬ dcs J s U s RT 1 . d ln cs ® dx
(2.13)
Define an augmented diffusion coefficient Ds* by d ln J s ¬ . D*s Ds 1 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 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 2 2 2 0 sx sy sz ® 2.
Cylindrical coordinates (r =radial coordinate, z =longitudinal coordinate, no azimuthal variation) Ds d dcs ¬ a) r-variation only: 0 r r dr dr ® b) r- and z-variation, different diffusion coefficients in r- and z-directions: Dsr s scs ¬ s2c Dsz 2s 0 r r sr sr ® sz
3. Spherical coordinates, r-variation only:
Ds d 2 dcs ¬ 0 r r 2 dr 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: 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 I i 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 . 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 U i cij , V j Ui cij , cations
anions
j
and N is the total concentration of the jth phase, defined as n
N j 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 ¬ '\ g a ln N , 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 . 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 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
¬ rU dc1 U 2 dc2 d\ RT 1 dx dx . dx z rU1c1 U 2 c2 ®
(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
i 1 n
i 1
n ¬ Ji dci d\ n ci zi , RT Ui dx dx i 1 ® i 1
(2.48)
n ¬ J i zi dc d\ n ci . RT zi i Ui dx dx i 1 ® 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
dx dx 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:
z
i
dci d d0 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:
Ji dN , RT Ui dx
(2.52)
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 '\ ¬ U i zi ¸ ci . dx 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: Ui ciII Ui ciI RT cations anions . '\ ln | Ui ciI 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 c c II K k ® Na
RT . ln K kNa ¬ I I cNa cK 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:
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.
64
CH. 2: FREE DIFFUSION
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 ¸ fu exp(Dvob ) . Ds K 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 ] ¬ 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-
PRINCIPLES AND MODELS OF BIOLOGICAL TRANSPORT
99
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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CH. 3: THE CELL
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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101
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 ) c dxdt n p dxdt n p dxdt . 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 ) c dxdt n p dxdt n p dxdt . 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 st st ® st 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 n c 2 u® J ¬ 1 and n n c . Substituting this into Eq. (3.14), 2 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 , dc ® 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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CH. 3: THE CELL
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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CH. 3: THE CELL
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 ¬ E . 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 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 NC 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 N T ¢¡ 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 ('\) ¡¡1 exp °° . ® NT RT RT ® ¢¡ ±°
(4.23)
Taking the reciprocal of both sides, rearranging, and taking the logarithm, N ¬ 'G 0 nᑠ'\ ln T 1 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 1 is linear in '\, with a slope from which 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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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 I Na ('\ ) J Na ('\ ENa ) N T ¡¡1 exp ° . ® RT RT ® ¢¡ ±°
(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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CH. 4: FACILITATED DIFFUSION: CHANNELS AND CARRIERS
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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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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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 ¬ 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 1 1¬ II I . 2 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 I 1 . * J A Vm 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 ¬ II K X exp RT ® 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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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.
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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 vi Pi 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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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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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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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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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 UX + 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 UY + ADP + Pi .
(5.3b)
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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 UX .
(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 UX .
(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-
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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 UY .
(5.6)
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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.
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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 ln . v '\ B 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
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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 ln Ca Q Na ln Na '\ '\ p 0 , I I aNa RT ® 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 aNa aCa,min aCa II exp . aNa ® 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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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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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.
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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 UAX + 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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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 UNa • E1 ~ P + ADP,
(5.15b)
Na • E1 ~ P UNa • E2 – P,
(5.15c)
II
(5.15d)
E2 – P + K UK • E2 – P,
(5.15e)
K • E2 – P UK • E2 + Pi ,
(5.15f)
Na • E2 – P UE2 – P + Na , II
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K • E2 + ATP UK • 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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CH. 5: ACTIVE TRANSPORT
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
PRINCIPLES AND MODELS OF BIOLOGICAL TRANSPORT
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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 AT