Protein Chromatography: Process Development and Scale-Up

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Giorgio Carta and Alois Jungbauer Protein Chromatography

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Giorgio Carta and Alois Jungbauer

Protein Chromatography Process Development and Scale-Up

The Authors Giorgio Carta University of Virginia Department of Chemical Engineering 102 Engineers’ Way Charlottesville, VA 22904-4741 USA Alois Jungbauer University of Natural Resources and Applied Life Sciences (BOKO) Department of Biotechnology Muthgasse 18 1190 Vienna Austria

All books published by Wiley-VCH are carefully produced. Nevertheless, authors, editors, and publisher do not warrant the information contained in these books, including this book, to be free of errors. Readers are advised to keep in mind that statements, data, illustrations, procedural details or other items may inadvertently be inaccurate. Library of Congress Card No.: applied for British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliograﬁe; detailed bibliographic data are available on the Internet at . © 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim All rights reserved (including those of translation into other languages). No part of this book may be reproduced in any form – by photoprinting, microﬁlm, or any other means – nor transmitted or translated into a machine language without written permission from the publishers. Registered names, trademarks, etc. used in this book, even when not speciﬁcally marked as such, are not to be considered unprotected by law. Cover Design: Formgeber, Eppelheim Typesetting: Toppan Best-set Premedia Limited, Hong Kong Printing and Binding: betz-druck GmbH, Darmstadt Printed in the Federal Republic of Germany Printed on acid-free paper ISBN: 978-3-527-31819-3

V

Contents Preface IX Nomenclature 1 1.1 1.2 1.2.1 1.2.1.1 1.2.1.2 1.2.1.3 1.2.2 1.2.2.1 1.2.2.2 1.2.2.3 1.2.2.4 1.2.2.5 1.2.2.6 1.2.2.7 1.2.2.8 1.3 1.3.1 1.3.2 1.3.3 1.3.4 1.3.5 1.3.5.1 1.3.5.2 1.3.5.3 1.4

XIII

Downstream Processing of Biotechnology Products 1 Introduction 1 Bioproducts and their Contaminants 2 Biomolecules: Chemistry and Structure 2 Proteins 2 Oligonucleotides and Polynucleotides 15 Endotoxins 16 Biomolecules: Physiochemical Properties 19 UV Absorbance 19 Size 21 Charge 24 Hydrophobicity 27 Solubility 29 Stability 32 Viscosity 33 Diffusivity 36 Bioprocesses 37 Expression Systems 37 Host Cells Composition 40 Culture Media 41 Components of the Culture Broth 43 Product Quality Requirements 43 Types of Impurities 43 Regulatory Aspects and Validation 45 Purity Requirements 47 Role of Chromatography in Downstream Processing 49 References 54

Protein Chromatography: Process Development and Scale-Up. Giorgio Carta and Alois Jungbauer © 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 978-3-527-31819-3

VI

Contents

2 2.1 2.2 2.3 2.3.1 2.3.2 2.3.3 2.3.4 2.4 2.5 2.5.1 2.5.2 2.5.3 2.5.4

Introduction to Protein Chromatography 57 Introduction 57 Basic Principles and Deﬁnitions 57 Modes of Operation 61 Elution Chromatography 63 Frontal Analysis 64 Displacement Chromatography 65 Simulated Moving Bed Separators (SMB) 67 Performance Factors 69 Separation Performance Metrics 74 Column Efﬁciency 74 Chromatographic Resolution 78 Dynamic Binding Capacity 80 Scaling Relationships 81 References 83

3 3.1 3.2 3.2.1 3.2.2 3.2.3 3.2.4 3.2.5 3.2.6 3.3 3.4 3.4.1 3.4.1.1 3.4.1.2 3.4.1.3 3.4.2 3.4.3 3.4.4

Chromatography Media 85 Introduction 85 Interaction Types and Chemistry 86 Steric Interaction 86 Hydrophobic Interaction 87 Electrostatic Interaction 94 Complexation 97 Biospeciﬁc Interaction 99 Mixed Mode Interaction 103 Buffers and Mobile Phases 105 Physical Structure and Properties 108 Base Matrices 109 Natural Carbohydrate Polymers 109 Synthetic Polymers 111 Inorganic Materials 112 Porosity, Pore Size, and Surface Area 113 Particle Size and Particle Size Distribution 119 Mechanical and Flow Properties 119 References 122

4 4.1 4.2 4.2.1 4.2.2 4.2.3 4.2.4 4.3

Laboratory and Process Columns and Equipment Introduction 125 Laboratory-scale Systems 126 Pumps 128 Buffer Mixers 130 Monitors 132 System Volumes 134 Process Columns and Equipment 135

125

Contents

4.3.1 4.3.2 4.3.3

Columns 135 Systems 140 Column Packing References 143

5 5.1 5.2 5.3

Adsorption Equilibria 145 Introduction 145 Single Component Systems 147 Multi-component Systems 157 References 160

6 6.1 6.2 6.2.1 6.2.2 6.2.3 6.2.4 6.2.5 6.3 6.3.1 6.3.2 6.3.2.1 6.3.2.2 6.3.2.3 6.3.2.4 6.3.2.5 6.3.2.6 6.3.3 6.3.4

Adsorption Kinetics 161 Introduction 161 Rate Mechanisms 161 External Mass Transfer 163 Pore Diffusion 165 Diffusion in the Adsorbed Phase 170 Intra-particle Convection 173 Kinetic Resistance to Binding 178 Batch Adsorption Kinetics 179 Rate Equations 181 Analytical Solutions 183 External Mass Transfer Control 184 Solid Diffusion Control 184 Pore Diffusion Control 186 Binding Kinetics Control 187 LDF Solution 187 Combined Mass Transfer Resistances 188 Experimental Veriﬁcation of Transport Mechanisms 190 Multi-component Protein Adsorption Kinetics 195 References 197

7 7.1 7.2 7.2.1 7.2.2 7.3 7.4 7.5 7.5.1 7.5.2

Dynamics of Chromatography Columns 201 Introduction 201 Conservation Equations 201 Boundary Conditions 203 Dimensionless System 203 Local Equilibrium Dynamics 205 Multi-component Systems 217 Displacement Development 227 Prediction of the Isotachic Train 228 Transient Development 234 References 235

141

VII

VIII

Contents

8 8.1 8.2 8.3 8.3.1 8.3.2 8.3.3 8.3.4

Effects of Dispersion and Adsorption Kinetics on Column Performance 237 Introduction 237 Empirical Characterization of Column Efﬁciency 238 Modeling and Prediction of Column Efﬁciency 246 Plate Model 246 Rate Models with Linear Isotherms 249 Rate Models with Non-Linear Isotherms 258 Rate Models for Competitive Adsorption Systems 270 References 274

9 9.1 9.2 9.3 9.4 9.5 9.6

Gradient Elution Chromatography 277 Introduction 277 General Theory for Gradient Elution with Linear Isotherms 279 LGE Relationships for Ion Exchange Chromatography 286 LGE Relationships for RPC and HIC 295 Separations with pH Gradients 299 Modeling Gradient Elution with Non-linear Isotherms 304 References 307

10 10.1 10.2 10.3 10.3.1 10.3.2 10.3.3 10.3.4 10.4 10.5

Design of Chromatographic Processes 309 Introduction 309 Chromatographic Process Steps and Constraints Design for Capture 313 Wash Step 315 Elution Step 315 CIP Step 316 Equilibration Step 316 Design for Chromatographic Resolution 321 SMB Design 327 References 338 Index

341

311

IX

Preface Chromatography has become an essential unit operation in the production of biopharmaceuticals. This method facilitates the processing of the complex mixtures encountered in this industry using readily available stationary phases and equipment suitable for large-scale sanitary operation. Moreover, its practice as a process puriﬁcation tool is recognized by regulatory agencies so that chromatography is an integral part of essentially all licensed biopharmaceutical processes. An in-depth understanding of the process is desirable and is increasingly being sought by regulatory agencies. As a result, chemists, engineers, and life scientists working in this ﬁeld need to become familiar with the theory and practice of process chromatography. While, in general, the theory of chromatography is well established for small molecule separations, the design and scale-up of chromatography units for biopharmaceutical puriﬁcation remain largely empirical. Thus, optimum designs often remain elusive. On one hand, engineers, while possessing a strong foundation in transport phenomena and unit operations, often have a limited understanding of biomolecular properties. On the other, biochemists and biologists often have a limited understanding of the key scale-up relationships and models needed for optimum design. In an effort to address this dichotomy, in 2000 we started a new short course at BOKU in Vienna, Austria, with the principal aim of merging the theory and practice of biochromatography to achieve optimum design and scale-up of process units. Our goal was to help educate engineers who understand the biophysical properties of proteins and other bio-macromolecules and can implement this understanding in the bioprocess setting; and life scientists who understand transport phenomena and engineering models and who can apply these tools to the design of process units. Since 2000, the course, which has been open to both industrial and academic participants, has been held annually both in Vienna and at the University of Virginia, in Charlottesville, Virginia, USA. The course has both theoretical and practical, hands-on components. The participants learn the fundamentals of protein production, their structural and biophysical properties, and the varied nature of their contaminants. In the lectures, they learn the theory of chromatography, the properties of stationary phases, how to describe the equilibrium and kinetic factors that govern process performance, and how to achieve a proper balance of separation efﬁciency and productivity. In the laboraProtein Chromatography: Process Development and Scale-Up. Giorgio Carta and Alois Jungbauer © 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 978-3-527-31819-3

X

Preface

tory, they learn to pack columns which are useful as scale-down models, plan experiments to identify critical parameters, and use advanced chromatography workstations to measure the critical physiochemical properties needed to model retention and band broadening in different types of chromatographic operations. Ultimately, the participants complete a design exercise, in which they are asked to design an optimized column on the basis of the laboratory measurements and theories learned during the course. This book is based on the same framework. After teaching the course for more than ten times and after discussions with several hundred participants with very broad ranges of educational backgrounds and job functions, we now have a better understanding of the main difﬁculties that are encountered in understanding protein chromatography from both theoretical and practical viewpoints. Therefore, following the spirit of the course, we begin with a chapter on the biochemical and biophysical properties of proteins and their contaminants. We focus on the properties that are relevant for chromatography such as size, surface charge and hydrophobicity, solution viscosity, and diffusivity and on how to preserve biological activity. In Chapter 2, we provide a succinct, general introduction to chromatography identifying the key factors that are important for design and scale-up. This allows the reader who is not familiar with chromatography to put the various issues discussed in Chapters 3 to 10 into proper context. Chapter 3 addresses the chemistry and structure of many different stationary phases while Chapter 4 discusses laboratory and process columns and equipment. Both of these chapters are limited in scope to familiarizing the reader with examples of commercially available materials and equipment. No attempt has been made to provide comprehensive coverage, in large part because the ﬁeld is rapidly expanding and new media and equipment are constantly being introduced. The mechanical design of equipment has also been omitted, since separation scientists and engineers in the biopharmaceutical production setting are rarely required to undertake this task. Chapters 5 to 9 are structured to acquaint the reader with theory and models to design and scale-up chromatography units. Emphasis has been placed on phenomenological models whose parameters can be determined using suitable experimental studies. Many speciﬁc numerical examples are provided to illustrate the application of these models to the analysis of laboratory data and to the prediction of column performance. A great deal of emphasis has been placed on describing transport in the stationary phase, since adsorption kinetics is often limiting in industrial applications of biochromatography. Thus, Chapter 6 provides a detailed coverage of mass transfer effects and their relationship to the structure of the stationary phase. Chapter 7 explores the dynamic behaviour of chromatography columns to establish a link between equilibrium properties, which are described in Chapter 5, and column behaviour. Chapter 8 discusses how equilibrium and rate factors combine to determine column performance and how to model band broadening for practical conditions. Chapter 9 focuses on gradient elution chromatography. We chose to devote a separate chapter to this mode of operation, since, in our experience, it is frequently less well understood despite its major importance in the practice of biochromatography. Finally, Chapter 10 is designed

Preface

in hopes of bringing everything together and providing guidance for the optimum design of process units. Although most of the emphasis is on conventional, batch chromatography processes, we conclude with an overview of continuous or semicontinuous multicolumn systems that are attracting increasing interest for biopharmaceutical applications. It should be noted that the main intent of this book is not to address de novo process development – rather, the main focus is on the optimal design and scale-up of columns for a process whose steps have already been deﬁned. Nevertheless, understanding these concepts will also aid the scientist who is involved in early process development to identify process steps that are scalable and can be efﬁciently translated from the laboratory to the manufacturing suite. We are convinced that proper application of theory combined with adequate experiments is instrumental to the successful application of biochromatography on a large scale. We would be happy, of course, if the book encouraged some of the readers to attend our course and learn about the practical, laboratory aspects that accompany the theory. The book also provides extensive references to original literature, textbooks, and books on chromatography, for those seeking greater detail. We have endeavored to make the notation consistent throughout the book and to check the correctness of the mathematical equations. Notwithstanding these efforts, we strongly suspect that there may still be some inconsistencies. We would be very grateful to readers who inform us of any such issues so that they can be remedied. Finally, we would like to thank our students who, over the years, have helped us to develop and teach the laboratory and discussion sessions used in our short courses, which could not have been held without their input and enthusiastic support. We would particularly like to thank our students Timothy Pabst, Emily Schirmer, Jamie Harrington, Melani Stone, Jeremy Siebenmann-Lucas, Theresa Bankston, Yinying Tao, Robert Deitcher, and Ernie Perez-Almodovar at the University of Virginia and Tina Paril, Kerstin Buhlert, Rene Überbacher, Anne Tscheliesnig, Alfred Zoechling, and Christine Machold at BOKU and our colleague Rainer Hahn for their support. We also thank all the participants who have attended our courses and who have provided very valuable feedback and have shared with us much of their practical experience. Giorgio Carta Charlottesville, Virginia, USA

Alois Jungbauer Vienna, Austria

XI

wwwwwww

XIII

Nomenclature a A Aexternal Ai Ainternal As b B B0 c –c C Cf CF CM C0 Cs C* CV dc dp dpore D0 DL De De,b Ds

coefﬁcient in dimensionless van Deemter equation (2.4, 2.5, 8.50) or isotherm parameter coefﬁcient in van Deemter equation (8.49), m surface area outside particles per unit column volume (3.11), m2/m3 combined equilibrium parameter for retention in IEC (9.24), RPC (9.36) and HIC (9.42), variable units surface area inside particles per unit column volume (3.10), m2/m3 asymmetry factor (8.12) coefﬁcient in dimensionless van Deemter equation (2.4, 2.5, 8.50) or isotherm parameter coefﬁcient in van Deemter equation (8.49), m2/s hydraulic permeability (= ηLu/∆P), m2 protein concentration in pore liquid, kg/m3, or coefﬁcient in dimensionless van Deemter equation (2.4, 2.5, 8.50). average concentration in pore liquid, kg/m3 protein concentration in mobile phase, kg/m3, or coefﬁcient in van Deemter equation (8.49), s peak compression factor in linear gradient elution (9.15, 9.28, 9.40) concentration in feed, kg/m3 mobile phase modiﬁer concentration in IEC and HIC, M initial concentration, kg/m3 protein concentration in mobile phase at particle surface, kg/m3 mobile phase protein concentration in equilibrium with stationary phase, kg/m3 number of column volumes of mobile phase passed through column column diameter, m particle diameter, m pore diameter, m molecular diffusivity in mobile phase, m2/s axial dispersion coefﬁcient (see Equations 7.1 and 8.46), m2/s effective pore diffusivity (6.9), m2/s effective diffusivity in mobile phase (8.46), m2/s effective adsorbed-phase or surface diffusivity (see 6.14), m2/s

Protein Chromatography: Process Development and Scale-Up. Giorgio Carta and Alois Jungbauer © 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 978-3-527-31819-3

XIV

Nomenclature

˜e D DBC ED EBC F Fp h H H(t) I J J(n, nτ1) k ka kb kc k′ – k′ kf K KD Ke L L0 Lcri m M Mj Mr n N p P ∆P q

convection-enhanced effective intraparticle diffusivity (6.20), m2/s dynamic binding capacity or amount of protein held in column at a speciﬁed percentage of breakthrough (see 2.15, 8.59, 10.5), kg/m3 eddy diffusivity (8.47), m2/s equilibrium binding capacity or amount of protein held in column at equilibrium with feed, kg/m3 fractional approach to equilibrium (= 〈qˆ 〉/qˆ*) ratio of intraparticle and column superﬁcial velocities (6.15 and 6.18) reduced HETP (= H/dp, see Equations 2.6 and 8.50) height equivalent to a theoretical plate (HETP, see Equations 2.3 and 8.4), m unit step function ionic strength (1.16), mol/m3 mass transfer ﬂux, kg/m2⋅s J-function (item B in Table 8.1 or Equations 8.38 and 8.39) rate coefﬁcient for LDF model with adsorbed phase concentration driving force (8.30), s−1 second order adsorption rate constant (Equations 6.22 and 8.62), m3/kg⋅s Boltzmann constant (= 1.38 × 10−23 joule/K) rate coefﬁcient for LDF model with mobile phase concentration driving force (8.30a), s−1 retention factor (= φqˆF/CF or = φm for the linear isotherm case Equation 2.10) average retention factor (2.13) ﬁlm or boundary layer mass transfer coefﬁcient (6.1 and 6.3–6.5), m/s adsorption equilibrium constant (e.g. Equation 5.7), m3/kg distribution coefﬁcient (1.11 and 3.15) equilibrium constant for ion exchange (5.15) length of packed column in SMB separator (10.48), m length of packed chromatographic column or zone length in SMB separator, m uncompressed column length (see Equation 10.18), m critical, compressed column length (see Equations 10.18 and 10.19), m linear isotherm slope or Henry constant (e.g. Equation 5.5) (= qˆ*/C) mobile phase modiﬁer or amount injected, kg ﬂow rate ratio in zone j of SMB separator (10.45) molecular mass number of transfer units (Table 8.2) number of plates (2.3 and 8.5) switch time for SMB separator (10.48), s productivity (10.1), kg/m3⋅s column pressure drop, Pa adsorbed protein concentration, kg/m3

Nomenclature

qF qm qmax q0 –q qˆ 〈qˆ 〉 qˆ*, q* Q r rh rm rp rpore –r p R Rs S tb tC tF tG tmax tR ts tsh T u us uj j uSMB

v v′ vc vsh V

adsorbed protein concentration in equilibrium with feed, kg/m3 maximum protein adsorption capacity (e.g. Equation 5.4), kg/m3 maximum protein adsorption capacity (in SD or SMA models (5.21 and 5.23)), kg/m3 concentration of charged ligands in the stationary phase (e.g. see Equation 5.17), mol/m3 adsorbed protein concentration averaged over particle volume (see Equation 6.29), kg/m3 total protein concentration in stationary phase including amounts adsorbed and held in the pores (6.24), kg/m3 total concentration in stationary phase averaged over particle volume (6.29), kg/m3 adsorbed concentrations in equilibrium with mobile phase, kg/m3 volumetric ﬂow rate, m3/s particle radial coordinate, m hydrodynamic radius (e.g. see Equation 1.8), m molecule radius, m particle radius, m pore radius, m volume-average particle radius (3.16), m separation factor isotherm parameter (5.7); R = 1 for a linear isotherm, R = 0 for a rectangular isotherm chromatographic resolution (2.9 and 10.22 or 10.30) sensitivity coefﬁcient for retention in RPC and HIC (3.6 and 3.8 or 9.37 and 9.42), or column cross sectional area, m2 breakthrough time (see Figure 8.13), s total cycle time (see Figure 10.3), s duration of feed injection, s parameter in EMG function (8.14), s, or duration of gradient, s time elapsed from injection at peak maximum, s retention time (see Equation 7.22), s time required for separation (2.14), s time at which shock emerges from column, s temperature, K superﬁcial mobile phase velocity (4.1), m/s adsorbent superﬁcial velocity in SMB separator (see Figure 10.8), m/s superﬁcial mobile phase velocity in zone j of TMB-equivalent to SMB separator (see Figure 10.8), m/s superﬁcial mobile phase velocity in zone j of actual SMB separator (10.49), m/s interstitial velocity of mobile phase (= u/ε, Equation 4.2), m/s reduced velocity (= vdp/D0, Equations 2.7 and 8.51) chromatographic velocity for simple waves (7.28), m/s shock velocity (7.30), m/s liquid phase volume, m3

XV

Nomenclature

XVI

Vb Vc VF V0 Vp VR w w0 W X Y z

mobile phase volume passed through column at breakthrough, m3 column volume, m3 feed volume loaded to column, m3 column extraparticle void volume (= εVc), m3 volume of adsorbent particles, m3 retention volume, m3 solubility in solution, kg/m3 solubility in pure water, kg/m3 baseline width of pulse response peak (Figure 8.1), s or m3 dimensionless protein concentration in mobile phase (7.12) dimensionless protein concentration in stationary phase (7.12) protein effective charge (5.17) or column axial coordinate, m

Greek Symbols

α β ε δ δ ij δ(t) ∆ ε εp ε0 εt ε φ γ g η ηE [η] ϕ λD λcri λm µ0 µ1 ρ

selectivity ( = kB′ kA′ ) gradient slope (9.6) mM/s or mM/m3, or safety margin for SMB separator (10.46) stagnant ﬁlm or boundary layer thickness (6.2), m SMB separator parameter (10.63) delta function peak width at half-peak height (Figure 8.1), s or m3 extraparticle void fraction (4.3) intraparticle void fraction (see Figure 2.7) extraparticle void fraction of uncompressed bed (see Example 10.2) total column void fraction (2.1) power input per unit mass in an agitated tank (see Equations 6.6 and 6.7), m2/s3 ratio of stationary and mobile phase volumes in column (= (1 − ε)/ε)) normalized gradient slope (= βL/v = βV0/Q , see Equation 9.11), mM or shear rate, s−1 mobile phase viscosity, Pa·s elution recovery yield, (see Equation 10.2) intrinsic viscosity (1.24), ml/g volume fraction of organic modiﬁer in RPC Debye length (3.9), m critical bed compression factor (= (L0 − Lcri)/L0, see Example 10.2) ratio of protein and pore radii (= rm/rpore) zeroth moment of pulse response peak (8.1), kg⋅s/m3 or kg ﬁrst moment of pulse response peak (8.2), s or m3 density of mobile phase, kg/m3

Nomenclature

σ σG τ τa τG τp τ1 ψp ζ

steric hindrance parameter in SMA model (5.22) or standard deviation of pulse response peak (8.3), s or m3 parameter in EMG function (8.14) dimensionless time (= εvt/L, see Equation 7.13) or shear stress (1.22) time constant for afﬁnity binding, s parameter in EMG function (8.14), s tortuosity factor for intraparticle diffusion (see 6.9) dimensionless time (= (vt/L − 1)CF/φqF at column exit, see Equation 7.17) hindrance parameter for pore diffusion (6.10 and 6.11) dimensionless column length (7.13)

Dimensionless Transport Parameters

Bi PeL Pep Re Sc Sh St nﬁlm npore nsolid

Biot number (= rpkf/De) Peclet number based on column length (= vL/DL) intraparticle Peclet number (see Equation 6.21) Reynolds number (= ρudp/η) Schmidt number (= η/ρD0) Sherwood number (= kfdp/D0) Stanton number (= (1 − ε)kL/us) number of transfer units for ﬁlm mass transfer (= 3φkfL/vrp, see Table 8.2) number of transfer units for pore diffusion ( = 15φDe L vrp2 , see Table 8.2) number of transfer units for solid diffusion ( = 15φDs qF L vrp2 CF, see Table 8.2)

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1

1 Downstream Processing of Biotechnology Products 1.1 Introduction

Biological products are important for many applications including biotransformations, diagnostics, research and development, and in the food, pharmaceutical, and cosmetics industries. For certain applications, biological products can be used as crude extracts with little or no puriﬁcation. However, biopharmaceuticals typically require exceptional purity, making downstream processing a critical component of the overall process. From the regulatory viewpoint, the production process itself deﬁnes the biopharmaceutical product rendering proper deﬁnition of effective and efﬁcient downstream processing steps crucial early in process development. Currently, proteins are the most important biopharmaceuticals. The history of their development as industrial products goes back more than half a century. Blood plasma fractionation was the ﬁrst full-scale biopharmaceutical industry with a current annual production in the 100-ton scale [1, 2]. Precipitation with organic solvents has been and continues to be the principal puriﬁcation tool in plasma fractionation, although, recently, chromatographic separation processes have also been integrated into this industry. Anti-venom antibodies and other anti-toxins extracted from animal sources are additional examples of early biopharmaceuticals, also puriﬁed by a combination of precipitation, ﬁltration and chromatography. In contrast, current biopharmaceuticals are almost exclusively produced by recombinant DNA technology. Chromatography and membrane ﬁltration serve as the main tools for puriﬁcation for these products. Figure 1.1 shows the 2006 market share of various biopharmaceuticals. Approximately one-third are antibodies or antibody fragments [3], nearly 20% are erythropoietins, and 14% are insulins. The rest are enzymes, growth factors and cytokines [3]. Although many non-proteinaceous biomolecules such as plasmids, viruses or complex polysaccharides are currently being developed, it is likely that proteins will continue to dominate as biopharmaceuticals. Proteins are well tolerated, can be highly potent, and often posses a long half-life after administration, making them effective therapeutics. Some of these properties also make proteins potentially useful in cosmetics, although applications in this ﬁeld are complicated in part by the US and European legal frameworks that do not allow the use of pharProtein Chromatography: Process Development and Scale-Up. Giorgio Carta and Alois Jungbauer © 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 978-3-527-31819-3

2

1 Downstream Processing of Biotechnology Products

Figure 1.1 Biopharmaceuticals market share in 2006. Approximately 160 protein therapeutics have gained approval in the USA and EU. Data from La Merie Business Intelligence (www. lamerie.com).

macologically active compounds in cosmetics. Currently only a few proteins are used in this area. The most prominent one is the botulinum toxin, Botox®, used for skin care [4]. This and similar compounds are exclusively administered by physicians and thus are not considered to be cosmetics.

1.2 Bioproducts and their Contaminants

This chapter gives an overview of the chemical and biophysical properties of proteins and their main contaminants such as DNA and endotoxins. The description is not comprehensive; only properties relevant to chromatographic puriﬁcation will be considered. A detailed description of the chemistry of proteins and DNA is outside the scope of this book and can be found in a number of excellent biochemistry or molecular biology texts [5, 6]. 1.2.1 Biomolecules: Chemistry and Structure 1.2.1.1 Proteins Proteins constitute a large class of amphoteric biopolymers with molecular masses ranging from 5 to 20 000 kDa, which are based on amino acids as building blocks. There are enormous variations in structure and properties within this class. Insulin, for example, a peptide with molecular mass of 5808 Da, has a relatively simple and well-deﬁned structure. On the other hand, human van Willebrand factor, a large multimeric glycoprotein with a molecular mass of 20 000 kDa, has an extremely complex structure consisting of up to 80 subunits, each of which is 250 kDa in mass. Most proteins have a molecular mass well within these two extremes, typically between 15 and 200 kDa. Proteins are generally rather compact

1.2 Bioproducts and their Contaminants

molecules, yet they ﬂexible enough to undergo substantial conformational change in different environments, at interphases, upon binding of substrates or upon adsorption on surfaces. Proteins are highly structured molecules and their structure is generally critical to their biological function. This structure is organized into four different levels: primary, secondary, tertiary, and quaternary. The primary structure is determined by the amino acid sequence, the secondary structure by the folding of the polypeptide chain and the tertiary structure is deﬁned by the association of multiple secondary structure domains. Finally, the quaternary structure is deﬁned by the association of multiple folded polypeptide chains. The ﬁnal result is a complex three-dimensional superstructure linked by various intra- and intermolecular interactions. Often non-amino acid elements are incorporated into a protein. Wellknown examples include prosthetic groups in enzymes and iron-carrying heme groups in oxygen transport or storage proteins such as hemoglobin or myoglobin. Primary Structure The building blocks of proteins are amino acids. During biosynthesis, following transcription and translation, these molecules are linked together via peptide bonds to form a polypeptide chain in a sequence that is uniquely determined by the genetic code. The general structure of amino acids and the formation of a peptide bond are shown in Figure 1.2. The order in which the amino acids are arranged in the polypeptide chain deﬁnes the protein’s primary structure. Note that although amino acids are chiral molecules with L- and Disomers, only the L-isomer is found in natural proteins. The 20 amino acids naturally found in proteins are listed in Table 1.1 In typical proteins, the average molecular mass of the amino acid components is 109 Da. Thus, the approximate molecular mass of a protein can be easily estimated from the number of amino acids in the polypeptide chain. The peptide bond formed when amino acids are linked together has partial double bond character and is thus planar. This structure restricts rotation in the peptide chain making free rotation possible only in two out of three bonds. As a consequence, unique structures are formed depending on the particular sequence of amino acids. Certain conformations are not allowed owing to the restricted rotation, while others are energetically favored owing to the formation of hydrogen bonds and other intramolecular interactions. The amino acid side chains can be charged, polar, or hydrophobic (see Table 1.1), thereby determining the biophysical properties of a protein. The charged groups are acids and bases of differing strength or pKa. Thus, these groups will determine the net charge of the protein

Figure 1.2

General structure of amino acids and formation of a peptide bond.

3

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1 Downstream Processing of Biotechnology Products

Table 1.1 The proteinogenic amino acids, including three- and one-letter codes, the structure of their

R-group, relative abundance in E. coli, molecular mass, and pKa of the R-group. Note that proline is a cyclic imino acid and its structure is shown in its entirety. Name

3-letter code

1-letter code

R-group

Abundance in E. coli (%)

Molecular mass

-CH3

13.0

89

6.0

117

4.6

115

pKa of R-group

Hydrophobic R-groups Alanine

Ala

A

Valine

Val

V

CH3 -CH CH3

Proline

Pro

P

O -

CH2

O-C-CH NH2

CH2 CH2

Leucine

Leu

L

-CH2 -CH-CH3 CH3

7.8

131

Isoleucine

Ile

I

-CH-CH2 -CH3 CH3

4.4

131

Methionine

Met

M

-CH2-CH2-S-CH3

3.8

149

Phenylalanine

Phe

F

3.0

165

Tryptophan

Trp

W

1.0

204

-CH2 -

-CH2 -

NH

Polar but uncharged R-groups Glycine

Gly

G

-H

7.8

75

Serine

Ser

S

-CH2OH

6.0

105

Threonine

Thr

T

-CH-CH3

4.6

119

1.8

121

11.4

132

10.8

146

2.2

181

OH Cysteine

Cys

C

-CH2-SH

Asparagine

Asn

N

-CH2 -C-NH 2

8.5

O

Glutamine

Gln

Q

Tyrosine

Tyr

Y

-CH2 -CH2 -C-NH 2 O -CH2 -

-OH

10.0

1.2 Bioproducts and their Contaminants Table 1.1

5

Continued

Name

3-letter code

1-letter code

R-group

Abundance in E. coli (%)

Molecular mass

pKa of R-group

Acidic R-groups (negatively charged at pH∼6)

O-

Aspartic acid

Asp

D

-CH2 -C O

Glutamic acid

Glu

E

-CH2 -CH2 -C O O

9.9

133

3.7

12.8

147

4.2

Basic R-groups (positively charged at pH∼6) -CH2-CH2-CH2-CH2-NH2+ Lysine Lys K

7.0

146

10.5

Histidine

0.7

155

6.1

5.3

174

12.5

His

H

-CH2 -C

CH +

HN

NH C H

Arginine

Figure 1.3

Arg

R

-CH2 -CH2 -CH2 -NH-C-NH2 NH2 +

Formation of a disulﬁde bond upon oxidation of two cysteines.

as a function of pH. Hydrophobic side chains, on the other hand, determine the hydrophobic character of the primary structure, which plays a substantial role in determining the pattern of folding of the polypeptide chain. The amino acids cysteine and proline play particular roles. Free cysteine molecules can undergo an oxidation reaction to form disulﬁde bonds or bridges yielding cystine as shown in Figure 1.3. When cysteines form part of a polypeptide chain, these bridges can be either intramolecular (within the same polypeptide chain) or intermolecular to

6

1 Downstream Processing of Biotechnology Products

link different polypeptide chains. On one hand, these bridges contribute to the stabilization of a protein’s folded structure and on the other they can lead to the formation of covalently bonded multimeric protein structures. The formation of disulﬁde bridges is generally reversible. Bonds formed in an oxidative environment can be broken under reducing conditions thus destabilizing the protein’s folded structure and disrupting associated forms. This property is utilized, for example, in high-resolution analytical protein separation methods such as SDS polyacrylamide gel electrophoresis (SDS-PAGE) which are often carried out under reducing conditions. In this case, the resultant loss of structure and the elimination of associated forms allow the precise determination of the protein’s molecular mass. Covalent chromatography utilizing the reversible formation of disulﬁde bonds between a protein’s cysteine residues and sulfhydryl ligands bound to a surface [7] is also based on the reversible nature of these bonds and has been applied to the separation of IgG heavy and light chains. Proline also plays a special role in deﬁning protein structure. Proline is a cyclic imino acid and can exist in cis and trans forms. In turn, these forms inﬂuence the conformation of the polypeptide chain. In free solution, these isomeric forms are in equilibrium. However, in a polypeptide, the interconversion of these isomeric forms is often slow and can be the rate-limiting step in the establishment of folded protein structures. Secondary Structure The polypeptide chains found in proteins do not form knots or rings and are not β-branched. However, these chains can form α-helices, βsheets, and loops which deﬁne the protein’s secondary structure. α-Helices consist of a spiral arrangement of the polypeptide chain comprising 3.6 amino acid residues per turn. The helix is stabilized by intramolecular hydrogen bonds and may be hydrophobic, amphipathic or hydrophilic in character, dependent on the particular sequence of amino acids in the primary structure. Examples of such helices are given in Figure 1.4. In each case the character of the α-helix can be predicted by placing each amino acid residue in a spiral at 100 degree intervals so that there will be 3.6 residues per turn. As seen in Figure 1.4, for citrate synthase, the hydrophobic residues are dominant and uniformly distrib-

Figure 1.4 Schematic structures of hydrophobic, amphipatic, hydrophilic protein helices. Hydrophobic amino acid residues are shown in light gray, polar in white, and charged in dark gray. Based on data in [8].

1.2 Bioproducts and their Contaminants

uted so that the α-helix will be hydrophobic. In the last case, troponin C, the charged residues are dominant but also uniformly distributed so that the resulting helix will be hydrophilic. Finally, in alcohol dehydrogenase the hydrophobic and charged residues are non-uniformly distributed resulting in an amphipathic helix that is hydrophilic on one side and hydrophobic on the other. β-Sheets are very stable secondary structure elements that also occur as a result of hydrogen bonding. Although one hydrogen bond makes up a free energy of bonding (∆G) of only about 1 kJ mol−1, the large number of such bonds in β-sheets makes them highly stable. As seen in Figure 1.5, β-sheets have a planar structure, which can be parallel, anti-parallel, or mixed depending on the directional alignment of the polypeptide chains that form these structures. Formation of β-sheets is often observed during irreversible protein aggregation. Due to the strong intermolecular forces in these structures, vigorous denaturing agents are needed to disrupt the resulting aggregates. Urea, a strong hydrogen bond breaker, can be used for this purpose. However, the high concentrations of urea needed to disrupt the hydrogen bonding will often result in a complete destabilization and unfolding of the whole protein structure. Amyloid proteins and ﬁbers contain a large number of β-sheets which explains in part the properties of these classes of aggregation-prone proteins. Loops are very ﬂexible parts of the protein and often connect other secondary structure elements with each other. For example, loops often connect the portions of a polypeptide chain that form anti-parallel areas of parallel β-sheets or form the links between different α-helical and β-sheet domains. Several types of loops have been described such as α and ω types. Loops also play a critical role in the artiﬁcial fusion of different proteins as in the case of single chain antibodies. These artiﬁcial antibodies are connected by loops that signiﬁcantly contribute to the stability of the protein. The relative number of secondary structure elements present in a protein can be measured by several spectroscopic methods including circular dichroism (CD) and infrared spectroscopy. CD-spectroscopy is based on the anisotropic nature of the protein. In circularly polarized light, the electric ﬁeld vector has a constant length, but rotates about its propagation direction. Hence during propagation the light forms a helix in space. If this is a left-handed helix, the light is referred to as left circularly polarized, and vice versa for a right-handed helix. Due to the interaction with the molecule, the electric ﬁeld vector of the light traces out an elliptical path during propagation. At a given wavelength the difference between the absorbance of left circularly polarized (AL) and right circularly polarized (AR) light is ∆A = AL − AR

(1.1)

Although ∆A is the absorption measured, the results are usually reported in degrees of ellipticity [θ]. Molar circular dichroism (ε) and molar ellipticity, [θ], are readily interconverted by the equation

[θ ] = 3298.2 ⋅ ∆ε

(1.2)

A wavelength scan is used to show the content of the secondary structure of a protein and is an essential measure of integrity. It is often used either to follow

7

Figure 1.5

Schematic structure of parallel (left) and anti-parallel (right) β-sheets in proteins.

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1 Downstream Processing of Biotechnology Products

1.2 Bioproducts and their Contaminants

Figure 1.6

CD-Spectrum of native, refolded and unfolded α-lactalbumin.

protein refolding or to conﬁrm the native structure of a protein (Figure 1.6). Different algorithms have been applied to determine the content of secondary structure elements based on these measurements and quantiﬁcation is highly dependent on the particular algorithm used. Although CD-spectroscopy is not sufﬁciently sensitive to trace residual unfolded protein in a protein preparation, the method is well suited to and accepted for the study of thermally- or chemicallyinduced unfolding in proteins. Attenuated total reﬂectance Fourier transform infrared (ATR FT-IR) spectroscopy is also used to study conformational changes in the 3D-structure of a protein in situ. A change in the secondary structure elements can be assessed with ATR FT-IR even in suspensions and turbid solutions. The amide I band in the spectral region from 1600 to 1700 cm−1 is used to evaluate structural changes (Figure 1.7). As in the case of CD, application of certain algorithms leads to the determination of the content of the secondary structure of a protein, although, again this is highly dependent on the algorithm applied. An advantage of the method is that the structure can be determined when the protein is adsorbed. Tertiary Structure The tertiary structure is formed when elements of the secondary structure (α-helices, β-sheets, and loops) are folded together in a three-dimensional arrangement. Hydrophobic interactions and disulﬁde bridges are primarily responsible for the stabilization of the tertiary structure as exempliﬁed by the packing of amphipatic α-helices into a four-helix bundle. In this structure, the hydrophobic residues are tightly packed in its core, shielded from the surrounding

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Figure 1.7 Infrared spectrum of the amide I band of a protein. The shift of the amide I band of BSA upon adsorption to the matrix during HIC with an increasing concentration

Phe

of ammonium sulfate is shown on the right, indicating a signiﬁcant change in secondary structure content. Reproduced from [9].

Tyr

Trp

Relative fluorescence

10

260

280

300

320

340

300

320

340

360

300

320

340

360

380

400

420

440

Emission wavelength, nm

Figure 1.8 [10].

Relative ﬂuorescence of the amino acids, Phe, Tyr and Trp. Based on data from

aqueous environment, while the polar and charged residues remain exposed on its surface. Fluorescence spectroscopy provides information about the location of the highly hydrophobic residues, tryptophan, phenylalanine and tyrosine in such folded structures. As shown in Figure 1.8, these residues have characteristic ﬂuorescence spectra, which vary with their position in the protein structure. When these residues are exposed at the protein surface, the ﬂuorescence maximum shifts providing an indication that unfolding has occurred. Thus, the extent of unfolding can be calculated when the ﬂuorescence spectra of native and unfolded forms are known. Quaternary Structure The quaternary structure is established when two or more polypeptide chains are associated to form a superstructure, which, in many cases, is essential for the biological function. One of the best-known examples is hemo-

1.2 Bioproducts and their Contaminants

Figure 1.9 Left: retention of native and fully water–acetonitrile mixture. Right: separation folded α-lactalbumin on a Vydac C4 reversed of folding intermediates of α-lactalbumin phase column containing 5 µm particles with using the same column and conditions. a pore size of 30 nm. The mobile phase was a

globin, which consists of four polypeptide units held together by hydrogen bonding and hydrophobic interactions. In this case, the ﬂexibility of the quaternary structure in response to oxygen binding is critical for oxygen uptake and release in the lung and capillary environments. Antibodies are another example of proteins with quaternary structures. These molecules consist of four polypeptide chains (two light and two heavy) linked together by disulﬁde bridges. The resulting structure is generally quite stable, allowing antibodies to circulate freely in plasma. Folding Although individual steps in the folding pathway can be extremely rapid, the overall process of protein folding can be relatively slow. For instance the helixcoil transition and the diffusion-limited collapse of proteins occur on time scales in the order of microseconds. On the other hand, the cis-transproly-peptidyl isomerization is a slow reaction occurring over time scales of up to several hours. As a result, in some instances folding and the chromatography method used occur over similar time periods so that structural rearrangements can take place during separation. When folding processes are particularly slow, chromatography can be used to resolve intermediate folding variants. For example, as shown in Figure 1.9, partially unfolded proteins show different retention in reversed phase chromatography, which can be used either to analyze protein solutions during an industrial refolding process or for the preparative separation of partially unfolded forms. Protein structures are classiﬁed into several hierarchies which include protein families and superfamilies. Dayhoff [11] introduced the term ‘protein superfamily’ in 1974. Currently, the term ‘folds’ is more commonly used to describe broad classes of protein structures. Table 1.2 shows the relative abundances of protein folds found in the PIR-International Protein Sequence Database; an excellent description of the structural hierarchies of proteins can be found on the web site: http://supfam.mrc-lmb.cam.ac.uk/SUPERFAMILY/description.html Proteins have been classiﬁed into classes and folds so that common origins and evolutionary patterns can be identiﬁed. However, it should be noted that even

11

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Table 1.2 Classes of folds found in protein databases [12].

Class of protein fold

Relative abundance

All alpha All beta Alpha and beta with mainly parallel β-sheets (α/β) Alpha and beta with mainly anti-parallel β-sheets with segregated α- and β-regions (α + β) Multi domain Membrane and cell surface proteins Small proteins (dominated by cofactors or disulﬁde bridges)

20–30% 10–20% 15–25% 20–30% <10% <10% 5–15%

proteins belonging to the same class may behave differently since even the substitution of a single amino acid can result in large variations in biophysical properties. Post-Translational Modiﬁcations Post-translational modiﬁcations are often critical to a protein’s biological function and can dramatically impact downstream processing. Post-translational modiﬁcations occur after the primary structure is formed and are highly cell speciﬁc. They can also vary with the physiological status of the cell. The latter can vary, for example, when the cells are deprived of a certain nutrient or ﬁnd themselves in a low oxygen environment. Further modiﬁcations can also occur following expression. As a result, many protein-based biopharmaceuticals are highly heterogeneous and their biological and pharmacological activity is often greatly inﬂuenced by the production process. Difﬁculties encountered in fully characterizing the corresponding broad range of molecular diversity often require that protein pharmaceuticals be deﬁned by the process by which they are produced rather than as uniquely deﬁned molecular entities. While considerable effort is being devoted to developing so-called ‘well-characterized biologicals’, for which molecular qualities, rather than processing deﬁne the product, current regulations continue to deﬁne biologicals strictly by their manufacturing process. Post-translational modiﬁcations often represent a productivity bottleneck. At high expression rates, post-translational modiﬁcations are often altered or become incomplete when the cell’s ability to perform these transformations lags behind the protein translation machinery. The result is the expression of additional protein variants, with potentially varying biological activity, stability, and biophysical properties such as solubility, charge, hydrophobicity, and size. Thus, improved fermentation or cell culture titers often have to be balanced against the increased heterogeneity of the product formed. More than 200 types of post-translational modiﬁcations have been described in the literature. Table 1.3 summarizes the most relevant ones. The individual molecular entities produced by such modiﬁcations are called isoforms. Two especially relevant post-translational modiﬁcations are glycosylation and deamidation since both produce changes that can inﬂuence the chromatographic properties of the protein. Thus, chromatography can be used as a tool to separate

1.2 Bioproducts and their Contaminants

13

Examples of post-translational modiﬁcations of proteins. Data from http://www.expasy.ch/tools/ ﬁndmod/ﬁndmod_masses.html.

Table 1.3

Modiﬁcation

Characteristics

Average mass difference

Glycosylation

O-linked oligosaccharides bound to Ser and Thr; N-linked oligosaccharide bound to Asp

Varies with number of sugar moieties; up to several thousands

Phosphorylation

Ser, Thr and Tyr, a phosphoester is formed, typical modiﬁcation of allosteric proteins involved in regulation (signal transduction)

79.9799

Sulfation

Addition of sulfate to Arg and Tyr, a C-O-SO3 bond

80.0642

Amidation

Addition of -NH2 to C-terminus

−0.9847

Acetylation

Addition of CH3CO- to N- or C-terminus

42

Hydroxylation

Addition of -OH to Lys, Pro and Phe

16

Cyclization

Formation of Pyroglutamate at N-terminal Glu

−0.9847

Complexation of metals

Cys-CH2-S-Fe complexes in Ferrodoxins. Selenium-complexes with Cys and Met Copper-complexes with backbone of peptide bond

Halogenation

Iodination and bromination of Tyr (3-chloro, 3-bromo)

34, 78

Desmosin formation

Desmosin is formed by condensation of Lys, frequent in elastin

−58

γ-Carboxylation

In prothrombin and blood coagulation factor VII

44.0098

Hydroxyproline

Hydroxyproline formation in collagen responsible for mechanical stability

15.9994

Adenylation

Tyr residue of glutamine synthetase is adenylated

209

Methylation

Addition of methyl group to Asp, Gln, His, Lys und Arg of ﬂagella protein

14.0269

Deamidation

Asn und Gln are susceptible; both biological and processing deamidation are observed

0.9847

the corresponding isoforms. Glycosylation is the addition of carbohydrate molecules, either simple sugars or complex oligosaccharides, to the protein molecule. Glycosylation renders the protein more hydrophilic and thus more soluble. Additionally, however, since the terminal carbohydrates of such oligosaccharides are often neuraminic acids (generally known as sialic acid) which are negatively charged above pH 3, glycosylation also inﬂuences the net charge and isoelectric point of the protein. As a result, chromatographic separations based on the different charge of the glycovariants are possible. As an example, Figure 1.10 shows the isoelectric focusing (IEF) gel separation of recombinant human erythropoietin (rhEPO – currently the second largest seller in the biopharmaceutical industry). As can be seen in this ﬁgure, the starting material

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1 Downstream Processing of Biotechnology Products

Figure 1.10 Isoelectric focusing (IEF) of 0.015 M; (4) 0.03 M; (5) 0.06 M; (6) 0.15 M; rhEPO. Fractions obtained by DEAE–Sephacel (7) 0.35 M and (8) 1 M NaCl. Reproduced chromatography: (1) starting material; (2) from [13] with permission. unadsorbed material; (3) material eluted with

contains multiple variants with isoelectric points between 3.5 and 5.5. Loading the starting material on an anion exchange column and eluting with increasing salt concentrations results in eluted fractions that have substantially reduced heterogeneity. Later eluting fractions contain more acidic variants with lower isoelectric points. These variants are more negatively charged and elute only at higher salt concentrations from the positively charged anion exchanger. rhEPO is highly glycosylated and the glycovariants have different bioactivity. Thus, control of the glycosylation pattern and, in some cases, separation of certain undesirable variants is needed to maintain a consistent product quality. Deamidation can also have dramatic effects both on bioactivity and chromatographic behavior. Deamidation involves the chemical transformation of asparagine and glutamine, which are uncharged polar amino acids, into aspartic acid and glutamic acid respectively, both of which are negatively charged at pH values above 4. Deamidation of asparagine residues is observed more frequently than that of glutamines, but the process is highly dependent on the location of these residues in the protein structure. Surface exposed residues tend to be most affected, while those buried within the protein core are usually partially protected. Deamidation is generally facilitated by higher pH values and higher temperatures and occurs via the mechanism illustrated in Figure 1.11. In this process, an amino group is cleaved off from asparagine forming an L-cyclic imide intermediate. This intermediate is generally unstable and is further converted into L-aspartyl and L-iso-aspartyl peptides. Both introduce negative charge and lower the isoelectric point of the protein. It should be noted that the unstable L-cyclic amide can also undergo

1.2 Bioproducts and their Contaminants O C

NH2 R

C H2 N H

C H

C

N H

C H

C O

O

L-Asparaginyl peptide D-Aspartyl peptide NH3

O H2C

C N

N H

C H

C H2

C H2

D-cyclic Imide

C O

L-cyclic Imide

D-Isoaspartyl peptide O C

OH

O

CH2 N H

C H

R C

N H

C H

O

L-Aspartyl peptide Figure 1.11

C O

N H

H2C

C

C H

C

R N H OH

C H

C O

O

L-Isospartyl peptide

Mechanism of deamidation of proteins for asparagine residues.

racemization forming a D-cyclic amide, which is further converted into D-aspartyl and D-isoaspartyl peptide. The net result is the introduction of D-amino acids into a protein. Removal of deamidated variants is often an important task since these variants can have different bioactivity and their removal is a challenge for downstream processing. Separation by ion-exchange chromatography is possible but often difﬁcult since the net charge difference between native and deamidated forms can be small, resulting is low selectivity. 1.2.1.2 Oligonucleotides and Polynucleotides Oligonucleotides and polynucleotides are either contaminants or may constitute the product. For example, in the production of plasmid DNA for gene therapy applications, genomic DNA is a contaminant [14]. Conversely, in the production of protein pharmaceuticals, both genomic and plasmid DNA are contaminants.

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Polynucleotides are present in the cell either as deoxyribonucleic acid (DNA) or as ribonucleic acid (RNA). DNA or RNA encode genetic information. In humans, animals or plants DNA is the genetic material, while RNA is transcribed from it. In some other organisms such as RNA viruses, RNA is the genetic material and, in reverse fashion, the DNA is transcribed from it. The building blocks of these molecules are nucleotides, which, in turn, are composed of a phosphate group, a sugar group, and a nitrogenous nucleoside group. Nucleotides are thus rather hydrophilic and negatively charged because of the acidic phosphate group. In DNA, the nucleotides are arranged in a double-stranded helical structure held together by weak hydrogen bonds between pairs of nucleotides. The molecule resembles a twisted ‘ladder’, where the sides are formed by the sugar and phosphate moieties, while the ‘rungs’ are formed by the nucleoside bases joined in pairs with hydrogen bonds. There are four nucleotides in DNA, each containing a different nucleoside base: adenine (A), guanine (G), cytosine (C), or thymine (T). Base pairs form naturally only between A and T and between C and G so that the base sequence of each single strand of DNA can be simply deduced from that of its partner strand. RNA is similar to DNA in structure but contains ribose instead of deoxyribose. There are several classes of RNA molecules including messenger RNA, transfer RNA and ribosomal RNA. They play a crucial role in protein synthesis and other cell activities. miRNAs are global regulators of gene expression. miRNAs are noncoding double-stranded RNA molecules comprising 19 to 22 nucleotides that regulate gene expression at the post-transcriptional level by forming a conserved single-stranded structure and showing antisense complementarity that was identiﬁed initially in the nematode Caenorhabditis elegans. DNA and RNA are chemically very stable molecules unless DNAse or RNAse enzymes are present. In presence of these ubiquitous enzymes the polynucleotides are rapidly degraded. Polynucleotides are also very sensitive to mechanical shear. Upon cell lysis, DNA and RNA are released into the culture supernatant and dramatically alter the viscosity of fermentation broths as a result of their size and ﬁlamentous structure. Genomic DNA present in the nucleus of eukaryotic organisms is always associated with very basic proteins known as histones. Plasmid DNA, on the other hand, is present in the cytoplasm of prokaryotic organisms and is histone-free but exists in different physical forms including supercoiled, circular, linear, and aggregated as illustrated in Figure 1.12. These forms differ in size providing a basis for separation by gel electrophoresis or by size exclusion chromatography. Polynucleotides are negatively charged over a wide range of pH due to the exposed phosphate groups. Thus, they are strongly bound by positively charged surfaces. As a result, their removal in downstream processing is conveniently and efﬁciently carried out with anion-exchange resins or with positively charged membranes. 1.2.1.3 Endotoxins Endotoxins, also known as pyrogens, are components of the cell wall of Gramnegative bacteria. They are continuously excreted by bacteria and are ubiquitous

1.2 Bioproducts and their Contaminants

A

B

C

Figure 1.12 Different physical forms of plasmid DNA. (A) The linear strand is twisted to a supercoil; the supercoiled form has the highest transformation efﬁciency and is the predominant form in therapeutic plasmids.

D

E

(B) When one strand is nicked then an open circular form is generated (C) and with the cleavage of the double strand the linear form is generated. (D) Two circular forms generate a catanane or a (E) concatemer.

in bioprocessing. Endotoxins are extremely toxic when they enter the bloodstream and humans are among the most endotoxin-sensitive organisms. Thus their almost complete removal from the ﬁnished product is required. As shown in Figure 1.13, endotoxins are lipopolysaccharides comprising a lipid A moiety, a core region, and an O- or S-antigen. The lipid A moiety is the most conserved component and is found in all endotoxins. This is also the part of the molecule responsible for toxicity. The O- or S-antigen is highly variable and strain speciﬁc. The size and structure also depends on the growth conditions. Endotoxins target the immune responsive cells such as macrophages, monocytes, endothelial cells, neutrophils and granulocytes. They induce the expression of interleukins, tumor necrosis factor, colony-stimulating factor, leukotrienes and oxygen radicals in these cells. As a consequence of the presence of endotoxins in the bloodstream, the patient develops tissue inﬂammation and fever, drop in blood pressure, shock, palpitations, a decrease in vessel permeability, respiratory complications, and even death. The same symptoms occur with severe bacterial infection, so-called septic shock. Severe hepatic toxicity and hematological disorders have been observed to occur in humans in response to as little as 8 ng of endotoxins per kg body weight. In contrast, endotoxins are much less toxic to many animals. For example, the LD50 is as high as 200–400 µg/animal in mice. For parenteral biopharmaceuticals the threshold level for intravenous applications is 5 endotoxin units (EU) per kg body weight per hour. EU deﬁnes the biological activity of endotoxins with 1 EU corresponding to 100 pg of the EC-5 standard endotoxin or 120 pg of the endotoxin derived from the E. coli strain O111:B4. The detection of endotoxin is difﬁcult and is carried out using bioassays. In the past rabbits have been used for this purpose. This time-consuming test has been replaced by the so-called limulus amoebocyte lysate (LAL) test, which uses the hemolymph of the horseshoe crab. LAL coagulates in the presence of minute amounts of endotoxins (see Figure 1.14) forming the basis for assays with endotoxin detection limits as low as of 10 pg ml−1. General guidelines are described in the United States Pharmacopeia (USP) in Chapter 79 on pharmaceutical compounding and sterile preparations (CSP). Table 1.4 provides a summary of the typical endotoxin content of various solutions. Endotoxins are present in large concentration in protein solutions derived

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Gal Gal NGa

NGa

NGc

n = 4-40

Gal Gal NGa

NGa

NGc

Glc

NGc

Gal

Glc

Gal

Hep

Hep

O OH P O

O Hep

OO O H3N+

P

O

P O

O

O

KDO

KDO

OO

KDO

+

NH3

P O

O

O O HO

O P

O O O

O O

HO

O

O

NH O

O

O O

O

OH

OOO O P O O O P

NH O

O HO

Figure 1.13

O

Chemical structure of endotoxins. Reproduced from [15] with permission.

from bacterial fermentations, but can also be present as adventitious agents in many other systems. In the industrial production of pharmaceuticals for parenteral use, special care is used to prevent endotoxin contamination. For example, endotoxin-free water used in the preparation of culture media and chromatography buffers, is re-

1.2 Bioproducts and their Contaminants

19

Figure 1.14 Coagulation test using Amoebocyte lysate for detection of endotoxins. The lysate forms a gel in the presence of endotoxins. Reproduced with permission of Associates of Cape Cod, Inc. Table 1.4

Typical endotoxin concentrations in various solutions of crude and puriﬁed proteins [15].

Protein source

Solution

Endotoxin (EU ml−1)

Proteins from high-cell-density culture of E. coli TG:pλFGFB

Supernatant after homogenization

>>1 000 000

Proteins from shaking-ﬂask culture of E. coli

Culture ﬁltrate

70 000–500 000

Murine IgG1 from cell culture

Culture ﬁltrate

97

Whey processed from milk of local supermarket

Supernatant after acid milk precipitation

9900

Commercial preparation of BSA

Reconstituted lyophisate at a concentration of 1 mg ml−1

50 (Supplier I) 0.5 (Supplier II)

circulated at high temperature in order to avoid bacterial growth and the consequent formation of endotoxins. Although endotoxins are heat stable, they are destroyed at alkaline pH. Thus, cleaning processing equipment, tanks, membranes, and chromatography media with a sodium hydroxide solution is generally required to assure complete removal of these contaminants. 1.2.2 Biomolecules: Physiochemical Properties 1.2.2.1 UV Absorbance The concentration of a protein in solution is often quantiﬁed by UV absorbance which is primarily due to absorption by the aromatic amino acids tyrosine, tryptophan, and phenylalanine and the disulﬁde bridges. The wavelength absorbance maxima and corresponding extinction coefﬁcients for these components are summarized in Table 1.5. Because of the strong absorbance of tryptophan, absorption maxima for proteins are typically around 280 nm and this wavelength is most frequently used for quantitative determinations. According to the Lambert–Beer law, the absorbance of a protein solution at a given wavelength deﬁned as

20

1 Downstream Processing of Biotechnology Products Table 1.5 Absorbance characteristics of aromatic amino acids and disulﬁde bridges.

Amino acid

λmax (nm)

e mmax (M−1 cm−1)

−1 −1 e m280 (M cm )

Tryptophan Tyrosine Phenylalanine Disulﬁde bridge

280 275 258

5500 1490 200

5600 1400 134

Table 1.6 Representative values of the speciﬁc absorbance of proteins at 280 nm in a cuvette of 1-cm length

at a concentration of 1 mg ml−1 and the molar extinction coefﬁcient. Molar extinction coefﬁcients from [16]. Protein

Immunoglobulin Ga)

Molecular mass

Number of amino acids Trp-Tyr-Cys

Mass extinction coefﬁcient E1280 cm (ml mg−1 cm−1)

155 000

Varies with subclass and individual antibody

≈1.4

Molar extinction coefﬁcient e m280 (M−1 cm−1)

Chymotrypsinogen

50 600

8-4-5

2.0

50 600

Lysozyme (hen egg white)

14 314

6-3-4

2.73

37 900

β-Lactoglobulin

18 285

2-4-2

0.95

17 400

Ovalbumin (chicken)

42 750

3-10-1

0.74

32 000

Bovine serum albumin

66 269

2-20-17

0.68

45 000

Human serum albumin

66 470

1-18-17

0.58

39 800

a)

May vary with recombinant IgG, when variable domains contain an excess of aromatic amino acids.

I (1.3) I0 is linearly related to the molar concentration of the analyte, c, by the following equation: A = − log

A = ε m lc

(1.4)

where I0 is the incident light, I is the light transmitted through the solution, l is the length of the light path through the solution and, εm is the speciﬁc molar absorbance or extinction coefﬁcient. The validity of Equation 1.4 is generally limited to relatively dilute solutions and short light paths, for which A is less than 2. At higher values, the ratio of transmitted and incident light becomes too small to permit a precise determination. Thus, quantitative determinations of concentrated protein solutions require dilution or very short light paths. As shown in Table 1.6, the speciﬁc absorbance of typical proteins varies with the relative content of the aromatic amino acids Trp and Try and, to a lesser extent, of the disulﬁde bridges. Since the relative content varies for different proteins, an empirical determination is needed for exact quantitative determinations.

1.2 Bioproducts and their Contaminants

Alternatively, the molar absorption coefﬁcient can be estimated with relative accuracy as the linear combination of the individual contributions of the Trp and Tyr residues and of the disulﬁde bridges according to the following equation:

ε m280 (M−1cm −1 ) = 5500 × nTrp + 1490 × nTyr + 125 × nSS

(1.5)

where nTrp, nTyr, and nSS are the numbers of its Trp, Tyr residues and disulﬁde bonds, respectively. It should be noted that nucleic acids have an absorbance maximum at 260 nm and can interfere substantially with protein determinations at 280 nm. Thus, when nucleic acids are simultaneous present in solution, corrections must be made in order to determine protein concentration from absorbance values at 280 nm. The peptide groups of proteins absorb light in the ‘far-UV’ range (180–230 nm) and very high absorbance values are observed in this region even for very dilute conditions. As a result, detection in analytical chromatography is often carried out at 218 nm, where absorbance is about 100 times greater. Proteins with additional chromophores either absorb in the near-UV or visible wavelength range. Typical examples are the iron-containing proteins such as hemoglobin, myoglobin and transferrin which are red in color, or Cu-Zn superoxide dismutase which is green. Nucleic acids show strong absorbance in the 240–275 nm region due to the π-π* transitions of the pyrimidine and purine nucleoside rings. Polymeric DNA and RNA absorb over a broad range with a maximum near 260 nm. The speciﬁc mass −1 −1 extinction coefﬁcient of DNA E1260 cm is 20 (ml mg cm ). The purity of DNA is estimated by the ratio of absorbance at 260 and 280 nm. For pure double-stranded DNA and RNA the ratio E260/E280 is between 1.8 and 2.0. The measurements are more reliable at alkaline pH. In contrast to proteins, the absorbance of nucleic acids is fairly sensitive to pH, and decreases at lower pH values [17]. 1.2.2.2 Size Solutions and suspensions found in downstream processing of biotechnology products contain molecules and particles with a broad range of sizes as illustrated in Table 1.7. Globular proteins are in the range of 3–10 nm, while nucleic acids can be much larger. Therapeutic plasmids are in the range of 100 nm. Virus and virus-like particles are in the range of 50 nm to 400 nm, while cells are in the micrometer range. While cells and cell debris are easily separated by centrifugation due to their high sedimentation velocity (Table 1.7), proteins and nucleic acids require more sophisticated methods such as chromatography and membrane ﬁltration. Separation of proteins by ultracentrifugation is only carried out for analytical purposes since extremely high rotation rates (as high as 50 000 rpm) are needed. The sizes given in Table 1.7 are for folded globular proteins. In this state, native protein structures are quite dense (mass density ∼1.4 g cm−3) and are spherical or ellipsoid in shape. However, denatured, ﬁbrous, rod, or disk shaped proteins deviate from these compact shapes. In these cases, the size of the proteins and other macromolecules is often described by other parameters which include the radius of gyration, rg, the hydrodynamic radius, rh, the radius established by rotat-

21

22

1 Downstream Processing of Biotechnology Products Table 1.7 Categories of bioproducts and their sizes.

Sedimentation velocity (cm h−1)

Category

Example

Mr (Da)

Size

Small molecules

Amino acids Sugars Antibiotics

60–200 200–600 300–1000

0.5 nm 0.5 nm 1 – nm

Macro molecules

Proteins Nucleic acids

103–106 103–1010

3–10 nm 2–1000 nm

<10−6

Particles

Viruses Bacteria Yeast cells Animal cells

50–500 nm 1 µm 4 µm 10 µm

<10−3 0.02 0.4 2

Figure 1.15 Lysozyme with a size of 2.6 × 4.5 nm is an ellipsoid shaped molecule. The molecular mass is 14.7 kDa, the mass density is 1.37 g/cm3. rm is the equivalent

radius of a sphere with the same mass and particle speciﬁc volume as lysozyme. rr is the radius established by rotating the protein about its geometric center.

ing the protein about its geometric center, rr, and the radius, equivalent to a sphere with the same mass and density as the actual molecule, rm.. Figure 1.15 illustrates these four different parameters for lysozyme. Note that the ﬁrst two, rh and rg, can be obtained from direct biophysical measurements, while the last two, rm and rr, can only be inferred from a knowledge of the actual protein structure. The radius of gyration can be measured by static light scattering. This is often carried out in conjunction with size exclusion chromatography (SEC) thus enabling protein mixtures to be analyzed. A general relationship exists between the radius of gyration and the amount of light scattered, which is directly proportional to the product of the weight-average molar mass and the protein concentration. Accordingly,

1.2 Bioproducts and their Contaminants

k ⋅c 1 = + 2A2c R (θ ) M wP (θ )

(1.6)

where R(θ) is the excess intensity of scattered light at a certain angle (θ), c is the sample concentration, Mw the weight-average molar mass, A2 the second viral coefﬁcient, k is an optical parameter equal to 4π n 2 (dn dc )2 ( λ 04NA ). n is the solvent refractive index and dn/dc is the refractive index increment, NA is Avogadro’s number, λ0 is the wavelength of scattered light in vacuum. The function P(θ) describes the angular dependence of scattered light. The expansion of 1/P(θ ) to ﬁrst order gives:

()

()

2 θ θ 1  16π  2 r ⋅ sin2 =1+  + f 4 sin 4 +… 2  3λ  g P (θ ) 2 2

(1.7)

At low angles the angular dependence of light scattering depends only on the mean square radius rg2 (alternatively known as the radius of gyration) and is independent of molecular conformation or branching. The hydrodynamic radius can be related to the protein translational diffusion coefﬁcient, D0, using the Stokes–Einstein equation: rh =

kbT 6πηD0

(1.8)

where kb is the Boltzmann constant, T is the absolute temperature, and η is the solution viscosity. Accordingly, rh represents the radius of a sphere with the same diffusion coefﬁcient as the actual protein. D0 can be conveniently determined by dynamic light scattering (DLS) also in conjunction with SEC in the case of mixtures. DLS is based on the ﬂuctuations or Brownian motion of a molecule, which in turn cause ﬂuctuations in the intensity of scattered light. The corresponding signal change with time can be described by an autocorrelation function. For small angles or Q-values the correlation function C(t) can be expressed by a single exponential term that allows the determination of D0 from the following equation: C (t ) = A1 + A2e −2DQ

2t

(1.9)

where A1 and A2 are the baseline at inﬁnite delay and the amplitude at zero delay of the correlation function, respectively. Tanford [18] has shown that the hydrodynamic radius of a globular protein can be related to its molecular mass, Mr, by a simple relationship. For practical calculations, the following equation provides reasonable values: 1

rh ≈ 0.081 × ( Mr )3

(1.10)

where rh is in nm. An alternative, commonly-used method for the determination of protein size is size exclusion chromatography (SEC). Molecules of different sizes do not all penetrate the pores of a SEC medium to the same degree thus leading to varying retention in the column. The SEC column can be calibrated with protein standards of known molecular mass allowing the size of an unknown protein to be estimated from its retention.

23

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1 Downstream Processing of Biotechnology Products

Figure 1.16 Calibration of a size exclusion column (Superdex 75, GE Healthcare, Uppsala, Sweden) with a set of reference proteins (molecular mass in parenthesis): (1) thyroglobulin (669000), ﬁbrinogen (340000), glucose oxidase (160000), IgG (160000), bovine serum albumin (66430), hemoglobin, (64500), trisosephophate

isomerase (53200), ovalbumin (45000), lectin (35000), carbonic anhydrase (29000), subtilisin (27000), chymotrypsinogen (25000), myoglobin (17000), calmodulin (16800), ribonuclease A (13700), ribonuclease S (13700), cytochrome c (13600), ubiquitin (8600), and Pep6His (1839). The symbol Kav is used in lieu of KD.

An example is shown in Figure 1.16. The distribution coefﬁcient, KD is deﬁned as follows: KD =

VR − V0 Vt − V0

(1.11)

where VR is the retention volume, Vt the total column volume, and V0 the extraparticle void volume. The latter is determined empirically from the retention of a compound sufﬁciently large to be completely excluded from the pores of the chromatography matrix. Blue Dextran, a 2000-kDa molecular mass dextran labeled with a blue dye, is often used for this purpose. When KD is plotted versus the logarithm of molecular mass an almost linear relationship is obtained for standard proteins. Other methods for the determination of the molecular size of proteins are SDSpolyacrylamide gel electrophoresis (SDS-PAGE) which provides information about the molecular mass, ultracentrifugation which provides information regarding the hydrodynamic radius, and other scattering techniques such as small angle X-ray scattering (SAXS). 1.2.2.3 Charge Proteins are amphoteric molecules with both negative and positive charges, which stem from the side chains of acidic and basic amino acids (Table 1.1) and from the amino and carboxyl terminus of each polypeptide chain. The latter have pKa values

1.2 Bioproducts and their Contaminants

around 8.0 and 3.1, respectively. Modiﬁcation of amino acid side chains may substantially contribute to the charge of a protein. Important examples are glycosylation with sialic acid which occurs, for example, at N-glycosylation sites in antibodies or erythropoietin, and deamidation of asparagine and glutamine residues. Both post-translational modiﬁcations make proteins more acidic and thus more highly negatively charged. In many cases they also affect the in-vivo half-life of the protein, so that their control can be an important goal in downstream processing. The net charge of a protein depends on the number of ionizable amino acid residues and their pKa values. The protonation of these residues changes with pH according to the following equations for acidic and basic residues, respectively: Ka = Ka =

[R − ][H + ] [RH ]

[R ][H + ] [RH + ]

(1.12) (1.13)

where the brackets indicate thermodynamic activities. In logarithmic form, we obtain: log log

[R − ]

[RH ]

= pH − pK a

[R ] = pH − pK a RH [ +]

(1.14) (1.15)

where p indicates −log10. From these equations it is obvious that acidic residues are completely deprotonated and thus negatively charged at pH values that are two units higher than their pKa. Conversely, basic amino acids are completely protonated and thus positively charged at pH values that are two units below their pKa. Based on the pKa values shown in Table 1.1, we can see that in practice, at the neutral pH values typically encountered in bioprocessing, all acidic residues in protein are negatively charged while all basic residues are positively charged. Histidine however, is an exception to this rule. Its pKa is near neutral; thus, under typical processing conditions, this residue will be charged to an extent that depends on the exact value of pH. At a particular pH, known as the isoelectric point or pI, the protein net charge becomes zero with an exact balance of positively and negatively charged residues. Knowing the pKa values of the side chains and the primary sequence the net charge and the theoretical isoelectric point can be calculated from Equations 1.12 and 1.13. An example is shown in Figure 1.17 for lysozyme. The calculation is only approximate because activity coefﬁcients were neglected and the pKa values were assumed to be equal to those of the free amino acids. This is likely incorrect since the microenvironment where the individual residues are actually found in the protein structure has a signiﬁcant effect. Nevertheless, the agreement between the theoretical net charge and that determined experimentally as a function of pH is remarkable. The more signiﬁcant deviations in this case occur for native lysozyme at low pH, but largely disappear when this protein is denatured, suggesting that the discrepancy arises because some of the acidic residues may be

25

1 Downstream Processing of Biotechnology Products

20 15

Net Charge

10 5 0 Theoretical

-5

Native Lyo data (Kuehner et al., 1999)

-10 -15

Denatured Lyo data (Tanford and Roxby, 1972)

0

2

4

6

8

10

12

14

pH Figure 1.17 Net charge of lysozyme in denatured and native forms compared to theoretical calculation. Data from [19, 20].

12

10

Relative abundance (%)

26

8

6

4

2

0

2

3

4

5

6

7

8

9

10

11

Isoelectric point, pI

Figure 1.18

Distribution of isoelectric points of proteins. Data from [21].

partially buried in the folded structure. As shown in Figure 1.17, the pI of lysozyme is around 11 and this is in agreement with IEF measurements. Around neutral pHs, this protein has a high net positive charge with a plateau region where the charge is only slightly affected by pH. Such conditions would be conducive to a robust adsorption process for the capture of lysozyme using a cation exchanger.

1.2 Bioproducts and their Contaminants

Figure 1.19 Distribution of positively charged (red) and negatively charged (blue) residues on the surface of lysozyme and human serum albumin.

A histogram showing the distribution of the pIs of many common proteins is given in Figure 1.18. As can be seen in this graph, the majority of the proteins have a slightly acidic isoelectric point and this is indeed found for the proteins present in many microorganisms such as E. coli. As a result, it is often easier to purify alkaline proteins that can be adsorbed on cation exchangers, because most of the host cell proteins are unlikely to be retained and will pass through these resins. Many monoclonal antibodies have high isoelectric points, allowing the development of platform puriﬁcation processes using cation exchangers. A ﬁnal important consideration with regard to protein charge is the spatial distribution of the charged residues. An example illustrating the location of positive and negative charges on the surface of lysozyme or human serum albumin at neutral pH is shown in Figure 1.19. A consequence of this heterogeneous spatial distribution is that frequently the net charge of the protein is not sufﬁcient to determine whether the protein will bind or not to an oppositely charged surface. For example, as a result of the localized concentration of positively-charged residues, it is possible for a protein to bind to a cation exchanger at pHs well above the protein pI, where the net charge is highly negative, or, vice versa, for a protein to bind to an anion exchanger at pHs well below the protein pI, because of localized negatively-charged residues. 1.2.2.4 Hydrophobicity The hydrophobicity of a protein is determined by the side chains of its non-polar amino acids. Although the term hydrophobicity is commonly used, a precise deﬁnition is difﬁcult and is extensively debated. The transfer of an apolar compound into a polar liquid such as water is associated with heat and quantiﬁed as free energy. The hydrophobic effect is strongest when entropic effects are dominant. Hydrophobic effects increase with the surface tension of water which is due to the

27

28

1 Downstream Processing of Biotechnology Products

Figure 1.20 Calculated hydrophobicity of human Cu-Zn superoxide dismutase calculated according to different hydrophobicity scales: Hopp and Wood [22] (left) and Kyte and Doolittle [23] (right).

attraction between the molecules in the liquid caused by various intermolecular forces. Hydrophobic effects are thus mainly due to the strong hydrogen bonds in water, while van der Waals forces generally play a minor role. The hydrophobicity of proteins can be theoretically calculated from the transfer energy of amino acids from an apolar solvent into water (see Figure 1.20). In a peptide chain the α-amino group and the carboxyl group are absent because they have reacted to form peptide bonds. Thus the free energy of transfer of amino acids does not totally reﬂect the hydrophobicity of a protein and strongly depends on the hydrophobicity scales which are used for calculating the hydrophobicity of the protein. The distribution of surface-exposed hydrophobic residues in proteins is not homogenous. This is illustrated for lysozyme in Figure 1.21. The density and distribution of these residues at the surface of a proteins is the basis for hydrophobic interaction chromatography (HIC) where surface hydrophobic residues interact with a mildly hydrophobic matrix. Since incorrect folding may lead to variations in the number of surface exposed-hydrophobic residues, HIC may be used as a tool to separate native proteins from misfolded isoforms. Another approach to measuring the hydrophobicity of a protein is by measuring its retention in a chromatography column packed with a hydrophobic medium. In this case, if the protein does not unfold, the retention is related to the number of hydrophobic amino acid side chains exposed at the surface [24]. The hydrophobic-

1.2 Bioproducts and their Contaminants

Figure 1.21 Distribution of hydrophobic clusters at the surface of lysozyme (left) and human serum albumin (right) at neutral pH. Yellow and light blue indicate hydrophobic and hydrophilic patches, respectively.

ity obtained by this method is relative and depends on the applied methodology, so it is useful only for ranking purposes. 1.2.2.5 Solubility Solubility is often a critical consideration in downstream processing, since it can vary dramatically with pH, ionic strength, and salt type. Predicting the solubility of a protein in aqueous media from its structure is difﬁcult and empirical measurements are usually needed. Protein solubility varies dramatically. Some proteins, e.g. Cu-Zn superoxide dismutase, have solubility as high as 400 mg ml−1 while others, e.g. recombinant interferon-γ, are soluble at concentrations less than 10 mg ml−1. In general, protein solubility is lowest at the isoelectric point, where the net charge is zero, but varies with ionic strength, which is deﬁned as follows

I=

1 n ∑ c j z2j 2 j =1

(1.16)

where cj is the concentration of ion j and zj its charge. The solubility of βlactoglobulin as a function of salt concentration and pH, is shown as an example in Figure 1.22. In general, salts at low concentrations increase the solubility of a protein, a process referred to as ‘salting in’. Conversely, at high concentrations salts reduce protein solubility, which is referred to as ‘salting out’. The magnitude of these effects is however highly dependent on the type of salt, as shown for example in Figure 1.23 for carboxyhemoglobin. Protein solubility trends can be described by the extended form of the Debye– Hückel theory. Accordingly, we have:

29

1 Downstream Processing of Biotechnology Products

2.8 0.02 N 2.4 Solubility, mg nitrogen per cc

30

0.005 N 2.0 0.010 N 1.6

1.2

0.8

0.4 0.001 N 0 4.8

5.0

5.2

pH

5.4

5.6

5.8

Figure 1.22 Solubility of β-lactoglobulin as a function of pH at four different concentrations of sodium chloride. Reproduced from [25].

log

w 0.5 ⋅ z1 ⋅ z2 I = − κ sI w0 1+ A I

(1.17)

where w is the protein solubility in the actual solution, w0 the solubility of the protein in water, z1 and z2 the salt charges, and κs and A are salt- and proteinspeciﬁc empirical parameters. At high ionic strengths, Equation 1.17 reduces to the following log-linear relationship: log

w = β − κ sI w0

(1.18)

which is shown for various proteins in Figure 1.24. The effect of the type of salt on protein solubility was formally described for the ﬁrst time by the Hofmeister [28] who ranked the anions and cations according to their ability to precipitate proteins, which is generally known as the Hofmeister or lyotropic series: Anions : SO24− > Cl − > Br − > NO3− > ClO−4 > I− > SCN− Cations : Mg2 + > Li + > Na+ > K + > NH+4

1.2 Bioproducts and their Contaminants 1.4 NaCl

1.2

KCl 1.0

Log S/S′

0.8

MgS

O4

0.6

(N

0.4

H

4) 2S

O

4

0.2

SO K2

0

4

–0.2

0

1.0

2.0 Ionic strength, m

3.0

4.0

Figure 1.23 Solubility of carboxyhemoglobin in aqueous solution with different electrolytes at 25 °C. S and S0 are used in lieu of w and w0. Reproduced from [26].

Pseudoglobulin

Log of solubility, grams/Iiter

0.50

Serum albumin C

0

Myoglobin –0.50

Fibrinogen –1.00

Hemoglobin 0

Figure 1.24

2

4 6 Ionic strength, m

8

10

The solubility of proteins in ammonium sulfate solutions. Reproduced from [27].

31

32

1 Downstream Processing of Biotechnology Products

A simple interpretation of this series is that certain ions bind free water decreasing the ability of the protein to remain in solution. Interestingly the salts in the Hofmeister series also correlate with the so-called Jones–Doyle B-coefﬁcient and the entropy of hydration so that both appear to be related to the effects of salts on the structure of water. Finally, it should be noted that in practice, the selection of salts for use in downstream processing depends not only on the Hofmeister series, but also on factors such as price, availability, biocompatibility, and disposal costs. 1.2.2.6 Stability Two different types of stability need to be considered for proteins: the conformational or thermodynamic stability, and the kinetic or colloidal stability. The conformational stability of a protein is described by the free energy ∆G of the equilibrium between native and the unfolded states. The transition of the native folded form, N, into the unfolded form, U, is described by the following quasichemical reaction: k1

→ N←  U k −1

(1.19)

where k1 and k−1 are rate constants. The corresponding equilibrium constant Keq = [U]/[N] is usually very low in aqueous solution, since protein folding is generally thermodynamically favored as a result of the concentration of the hydrophobic residues in the protein core. The corresponding ∆G is given by the following equation ∆G = −RT ln K eq

(1.20)

Representative values are given in Table 1.8 along with the corresponding ‘melting temperature’, which is deﬁned as the temperature at which half of the protein is in the unfolded state. Kosmotropic (or cosmotropic) salts and polyols such as sorbitol or sucrose stabilize proteins while chaotropic salts or urea at higher concentrations have a destabilizing effect on protein conformation. Kinetic stability, on the other hand, can be described by the following equation: k1

k2  → N← →A  U  k−1

(1.21)

which shows a further kinetically-driven step from the unfolded state to an irreversibly aggregated state A. Proteins with a high k2 exhibit a low kinetic stability. The overall stability thus depends on both thermodynamic and kinetic effects. It is possible, for example, for an added salt to decrease kinetic stability, while enhancing overall stability as a result of thermodynamic effects. However, this effect is often difﬁcult to predict, so that it in practice, overall stability and shelf-life are measured empirically [30].

1.2 Bioproducts and their Contaminants Table 1.8

Thermodynamic stabilities of proteins. Data for chymotrypsinogen from [29].

Protein

Conditions

Horse Cytochrome c at pH 6 and 25 °C

0M 2M 4M 6M

Hen egg white Lysozyme at pH 3.0

24 °C 40 °C 55 °C 75 °C

Bovine chymotrypsinogen at melting temperature and pH 2.0

0 M sorbitol 0.5 M sorbitol 1.0 M sorbitol

urea Urea Urea Urea

Free energy of the unfolding reaction ∆G, kcal mol−1

Melting temperature °C

31.3 22.3 14.2 3.2

n.a. n.a. n.a. n.a.

41.0 30.4 14.7 −5.9

n.a. n.a. n.a. n.a.

0.015 0.146 0.235

42.9 44.9 44.2

1.2.2.7 Viscosity Many of the solutions and suspensions encountered in bioprocessing are highly viscous. This is especially true for fermentation broths that contain DNA and for highly concentrated protein solutions. In general, the viscosity, η, is related to the shear stress, τ, and the shear rate, γ , by the following equation:

τ = η × γ

(1.22)

For Newtonian ﬂuids, η is a constant and the relationship between shear stress and shear rate is linear. For non-Newtonian ﬂuids, however, η varies with shear rate and the relationship is non-linear. For example, the behavior of pseudoplastic ﬂuids is described by the following equation:

τ = K × (γ )n

(1.23)

where K and n are the consistency and ﬂow index, respectively. For highly concentrated protein solutions and for many culture supernatants, n is smaller than unity, indicating that the apparent viscosity, η = τ γ , decreases with increasing shear rate. The ranges of shear rates for the various solutions and suspensions encountered in bioprocessing are shown in Table 1.9. Typical viscosities encountered in bioprocessing are shown in Table 1.10. In general, cell culture supernatants have viscosities lower than 10 mPa s, while cell homogenates are much more viscous with η-values of up to 40 mPa s. The greatest contribution to the viscosity of raw fermentation broths is DNA. Fortunately, however, both genomic and plasmid DNA are very sensitive to shear and are often mechanically degraded early on in the downstream process. DNAse enzymes, either naturally occurring or added intentionally, also help to degrade these molecules, thereby reducing viscosity.

33

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1 Downstream Processing of Biotechnology Products Table 1.9 Typical shear rates encountered in bioprocessing.

Operation

γ (s−1)

Expanded bed Packed bed Stirred tank High pressure homogenizer

<10 <103 102–103 106

Table 1.10

Viscosities of various ﬂuids at 20 °C.

Liquid

Apparent viscosity mPa s

Water Glycerol Ethanol Acetonitrile Clariﬁed cell culture supernatant Blood E. coli homogenate E. coli broth Penicillium chrysogenum fermentation broth Heinz ketchup

1 1070 1.20 0.34 <5 10 <40 <20 40 000 50 000–70 000

The intrinsic viscosity [η] is a measure of the contribution of a solute to the viscosity of a solution and is deﬁned as:

[η ] = lim c →0

η − η0 η0c

(1.24)

where η0 is the viscosity in the absence of the solute and c the solute concentration. At a semi-dilute limit, η can be described as a function of c by the following polynomial expression:

η − η0 2 3 = [η ] c + k1 [η ] c 2 + k2 [η ] c 3 + … η0

(1.25)

At very high protein concentrations, however, semi-empirical models are needed [31]. As an example, Figure 1.25 shows the relative viscosity (η/η0) of IgG solutions as a function of IgG concentration. For concentrations lower than about 100 mg ml−1 the data conform approximately to Equation 1.25. However, at higher concentrations the viscosity increases exponentially [31]. Table 1.11 lists the intrinsic viscosities of representative biomolecules. As can be surmised from these data, the intrinsic viscosity depends on the shape of the molecule. For instance, rod-shaped proteins have a higher intrinsic viscosity than

1.2 Bioproducts and their Contaminants

Relative viscosity ( / 0)

100 80 60 40 20 0 0

50

100

150

200

250

300

c (mg/ml) Figure 1.25 Viscosity of human, bovine, and pig IgG solutions as a function of IgG concentration. Data from [31].

Table 1.11 Intrinsic viscosities of various biologically important macromolecules in dilute solutions [32].

[η] (ml g−1)

Shape

Substance

Molecular mass

Globular

Ribonuclease Serum albumin Ribosomes (E. coli) Bushy stunt virus

13 680 67 500 900 000 10 700 000

3.4 3.7 8.1 3.4

Random coils (unfolded proteins)

Insulin Ribonuclease Serum albumin Myosin subunit

2970 13 680 68 000 197 000

6.1 16 52 93

Rods

Fibrinogen Myosin Calf thymus DNA

330 000 440 000 15 000 000

27 217 >10 000

globular forms. An empirical relationship between [η] and molecular mass Mr is given by the Mark–Houwink equation:

[η ] = K (Mr )a

(1.26)

where a is a parameter related to the ‘stiffness’ of the polymer chains. Theoretically, a = 2 for rigid rods, 1 for coils, and 0 for hard spheres. Empirically, however, values of a = 0.6, 0.7, and 0.5 have been found for BSA, ovalbumin, and lysozyme, respectively. Literature data (e.g. [33]) suggests a general relationship between intrinsic viscosity and the number of amino acid residues, which can be expressed as follows

[η ] = 0.732 n 0.656

(1.27)

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1 Downstream Processing of Biotechnology Products

with [η] in ml g−1. Accordingly, larger proteins and protein aggregates have higher intrinsic viscosity than smaller proteins and monomeric forms. 1.2.2.8 Diffusivity The molecular diffusion coefﬁcient or diffusivity in solution, D0, is a function of the size of the solute, the viscosity of the solution, and temperature. As previously noted, the Stokes–Einstein equation describes this relationship as:

D 0η k = b 6π rh T

(1.28)

where kb is the Boltzmann constant and rh is the solute hydrodynamic radius. The diffusivities encountered in bioprocessing range widely from 1 × 10−5 cm2 s−1 for salts and other small molecules to 1 × 10−9 cm2 s−1 for large biomolecules such as DNA. Protein diffusivities in dilute aqueous solution are generally in the range of 10−6 to 10−7 cm2 s−1. Table 1.12 provides a summary of typical diffusivities in dilute solutions at room temperature. In general, protein diffusivities are 10–100 times lower than those of small molecules. Plasmids have even smaller values by a factor of as much as 1000. Figure 1.26 illustrates the effect of molecular mass on the diffusivity in dilute aqueous solution at room temperature. Tyn and Gusek [34] have provided the following correlation for globular proteins: D0η 9.2 ⋅ 10 −8 = T (Mr )1 3

(1.29)

where D0 is in cm2 s−1, η in mPa s, T in K, and Mr in Da, which has an accuracy of ±10%. Several approaches are available for the experimental determination of diffusivity and are reviewed, for example, by Cussler [35]. Commonly used approaches for

Table 1.12

Diffusivities in dilute solution at room temperature.

Solute

Solvent

Viscosity, η (mPa s)

Diffusivity, D0 (10−5 cm2 s−1)

Benzoic acid Valine Sucrose Water Water Water Ribonuclease (Mr = 14 kDa) Albumin (Mr = 65 kDa) IgG (Mr = 165 kDa) pDNA (Mr = 3234 kDa)

Water Water Water Ethanol Glycol Glycerol Water Water Water Water

1 1 1 1.1 20 >120 1 1 1 1

1.00 0.83 0.53 1.24 0.18 0.013 0.12 0.060 0.037 0.004

1.3 Bioprocesses

1 10,000

Catalase

IgG

BSA

Ovalbumin

-Lactoglobulin

10

Chymotrypsin A Chymotrypsinogen

Ribonuclease

Diffusivity (10 -7 cm2/s)

100

100,000

Molecular mass, Mr

Figure 1.26 Diffusivity of globular proteins in dilute aqueous solution at room temperature. Data from [34].

proteins include dynamic light scattering (see Section 1.1.2.2), diffusion cells, Taylor dispersion-based methods, and microinterferometry.

1.3 Bioprocesses

This chapter discusses commonly used expression systems and the general structure of downstream processes needed to achieve the desired product purity. Special emphasis is placed on the production of recombinant proteins by fermentation and cell culture, which play a major role in industrial biotechnology. 1.3.1 Expression Systems

Many different expression systems have been developed for recombinant proteins, ranging from very simple bacteria to plants and animals. However, the number of host cells actually used in the industrial production of biopharmaceutical proteins is quite limited. The most popular bacterial strain is E. coli, BL21, which is used for the production of proteins whose biological activity does not require posttranslational modiﬁcations. Protein expression in E. coli can occur in three different ways. The protein can be secreted into the periplasm, which is the space between cell membrane and cell wall; it can be expressed in the cytoplasm as soluble protein; or it can accumulate in the cell as inclusion bodies. Each system is effective for certain proteins. The cytoplasm of E. coli is a strongly reducing environment which hinders the formation disulﬁde bridges, whereas the

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1 Downstream Processing of Biotechnology Products

Figure 1.27 Left: Scanning electron micrograph of E. coli cells over-expressing a recombinant protein as inclusion bodies. Right: SDS-PAGE under non-reducing

conditions. Lane 1: culture supernatant; lane 2: homogenate; lane 3: soluble fraction; lanes 4–7: supernatants from the wash steps; lanes 8–9: insoluble fraction containing the product.

periplasm offers a more oxidizing environment which allows the formation of these bridges thus facilitating folding. Some proteins which are toxic to the cells or possess a short half-life in the cytoplasm or periplasm have been successfully produced as inclusion bodies. In this case, the protein forms aggregates that cannot be attacked by proteases. On the other hand, although often not fully denatured, proteins expressed as inclusion bodies are generally not in their native conformation. Thus, a refolding procedure is typically required to generate a fully active form. Although refolding can be costly, on balance this approach is frequently economically viable, since expression levels in the inclusion body system are extremely high and simple washing procedures can be used to remove most of the host cell proteins resulting in relatively high initial purities. Figure 1.27 shows an example of E. coli cells over-expressing a protein as inclusion bodies and the corresponding SDS-PAGE analysis at various stages in the process. The yeast cells Saccharomyces cerevisae and Pichia pastoris have been successfully used for over-expression of various recombinant proteins including insulin and albumin. Saccharomyces cerevisiae is also used for the production of the hepatitis surface antigen. However, mammalian cells are used to produce the majority of biopharmaceutical proteins. Although mammalian cell culture is generally more complex, these cells can carry out complex post-translational modiﬁcations, such as glycosylation, which are often critical to proper biological and pharmacological activity. Chinese Hamster Ovary (CHO) cells are the most commonly used mammalian expression system, especially for recombinant antibodies. The human cell line PerC6 has been developed more recently and is also used for production of

1.3 Bioprocesses

some recombinant proteins. CHO and PerC6 are able to over-express antibodies in concentrations as high as 15 mg ml−1. Such high product titers are achieved mainly because the expression of proteins in these systems is generally independent of growth. As a result, the cells can be maintained in a bioreactor for long periods in a viable productive state and can be cultivated to very high densities, up to 108 cells per ml. Other mammalian cell lines used in recombinant protein production include baby hamster kidney cells (BHK), Vero cells, and Madin–Darbyn canine kidney cells (MDCK). Several coagulation factors that require γ-carboxylation are produced in BHK. Vero cells, derived from monkey kidneys, and MDCK cells are used for the production of vaccines. Insect cells such as cells from pupal ovarian tissue of the fall armyworm Spodoptera frugiperda (Sf9), have also been proposed for overexpression of recombinant proteins. These cells can be infected with insect viruses such as the Autograph California nuclear polyhedrosis virus also known as baculo virus. Although easy to handle, so far insect cell systems have not been licensed for the industrial production of biopharmaceutical proteins. In the past, transgenic animals were considered to be excellent production systems, since proteins can be secreted into the milk with titers up to 5 mg ml−1. A decade ago such titers could not be achieved with mammalian cell culture. Thus, at that time these expression systems were often preferred. Modern advances in cell culture, however, have made it possible to routinely achieve even higher titers. Transgenic systems require the very tedious procedure of developing offspring with the appropriate expression characteristics. Puriﬁcation of proteins from milk can also be cumbersome, especially when removal of casein by acid-induced precipitation is not compatible with the protein product. Furthermore, as an ‘open production system’, transgenic animals may be prone to safety issues and external contamination. The advantages of cultivation of mammalian cells in closed bioreactor systems together with titers higher than those achieved in transgenic milk and with simpler downstream processing are the reasons why cell culture is generally favored in the biopharmaceutical industry. Plants and plant cells are also potential candidates for expression of proteins. Although different from that in mammalian cells, post-translational modiﬁcations are also possible in these organisms. However, expression in leaves, stems, or seeds presents signiﬁcant recovery and puriﬁcation challenges since the tissue or seeds must be ground or pressed and extracted which yields very complex mixtures. An interesting expression system is rhizo-secretion, where the protein is secreted into a cultivation ﬂuid from hairy roots. Plant cells, on the other hand, present fewer downstream processing difﬁculties as these cells grow in very simple media. Another interesting system is the so-called ‘olesin technology’. In this case, lipids are accumulated in plant cells in the form of small droplets. Proteins, known as olesins can be concentrated as a monolayer on the surface of these droplets. After fusing a recombinant protein to the olesins, primary recovery of the target protein can be achieved by the relatively simple process of harvesting the oil droplets.

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1 Downstream Processing of Biotechnology Products

1.3.2 Host Cells Composition

The composition of the host cells has important effects on downstream processing when the product is expressed intracellularly, since in this case the cellular components are the major impurities. However, host cell components are also found extensively as impurities in secreted products, since cell lysis always occurs to some extent during fermentation. In fact, in some cases, cultivation procedures that yield high titers, such as those used for antibody production, result in partial lysis of the cells, which in turn causes contamination of the product with host cell components. An overview of the composition and physical characteristics of the major host cells is given in Tables 1.13 and 1.14. As can be seen from these tables, mammalian cells contain more protein but less nucleic acid than bacteria. Bacteria and yeast also contain numerous cell wall components. These components are frequently insoluble and are efﬁciently removed during early processing steps. These early steps are signiﬁcantly affected by the cell density and viscosity of the broth. Extremely high cell densities can be obtained for yeast, particularly for P. pastoris, for which cell densities up to 400 g of cells per liter have been reported. Such suspensions are extremely difﬁcult to clarify; the suspension must often be diluted in order to centrifuge the broth. Nucleic acids are present in the form of DNA and various forms of RNA. These compounds are responsible for the high viscosity of the broth but are often rapidly degraded by mechanical shear or endogenous nucleases. A ﬁnal consideration is the mechanical stability of the host cells. Bacteria and yeast cells are generally mechanically very stable and shear rates above 106 s−1 are necessary to break these cells. Such high shear rates can only be attained by using special equipment such high-pressure homogenizers or French presses. By comparison, mammalian cells are much weaker. The burst force needed to destroy a yeast cell is in the range of 90 µN whereas the burst force for mammalian cells is in the range of 2–4 µN. The burst force increases with culture length in a batch

Table 1.13

Composition of common host cells for expression of recombinant proteins.

Organism

Composition (% dry weight)

Cell count per ml

Dry mass/wet mass (mg ml−1)

20/100 80/400a) 130/400b) 100/400 0.17 to 1.7/1 to 10

Protein

Nucleic acids

Lipids

E. coli Yeast

50 50

45 10

1 6

1011 1010

Filamentous fungi Mammalian cells

50 75

3 12

10 Up to 10

109 107–108

a)

For a high density culture of P. pastoris grown on glucose medium and b) grown on methanol.

1.3 Bioprocesses Table 1.14 Composition of single cells for expression of recombinant proteins.

Component

Amount per HeLa cell

Amount per E. coli cell

Total DNA

15 pga)

0.17 pgb)

Total RNA

30 pg

Total protein

300 pg (5 × 10 molecules with an approx. Mr of 40 000)

0.2 pg (3 × 106 molecules with an approx. Mr of 40 000)

Dry weight

400 pg

0.4 pg

Wet weight

2500 pg

2 pg

Diameter

18 µm

0.5 × 3 µm

Volume

4 × 10−9 cm3

0.10 pg 7

a)

HeLa cells are hypotetraploid and contain four copies of each chromosome. The DNA content of normal cells is approx. 5 pg/cell. b) A fast growing E. coli cell contains on average a four-fold repetition of its genome. The weight of each genomic DNA molecule is about 0.0044 pg.

culture, which is consistent with the observation that older cells are more difﬁcult to disrupt. 1.3.3 Culture Media

Modern biopharmaceuticals are commonly produced with so-called deﬁned media whose components are chemically characterized. In the past, yeast, meat, and soy extracts, produced by proteolytic degradation and extraction, were commonly used for the cultivation of bacteria and yeast cells. The standardization of such raw material was extremely difﬁcult resulting in substantial batch to batch variations. Similarly, until recently it was common to supplement cultivation media for mammalian cells with fetal calf serum, in concentrations up to 10%. In addition to added complexity and cost, such supplements can introduce undesirable adventitious agents, such as prions, which can signiﬁcantly increase the challenges of downstream processing. Although testing for such agents may still be required, the use of deﬁned media greatly simpliﬁes downstream processing. Media for the industrial cultivation of bacteria are usually very simple and provide the essential sources of carbon, nitrogen and phosphate. Examples are given in Table 1.15. Sometimes a cocktail of trace elements is added, but frequently the trace elements present in the water are sufﬁcient. When the fermentation pH is controlled by the addition of NaOH, conductivities as high as 40 mS cm−1 can be reached by the end of the cultivation period. Such high conductivities can interfere with downstream processing operations such as ion exchange, therefore dilution or diaﬁltration steps are required. This difﬁculty

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1 Downstream Processing of Biotechnology Products Table 1.15

Composition of deﬁned culture media for the cultivation of E. coli.

Compound

Concentration (mg ml−1)

Glucose Na2HPO4 KH2PO4 (NH3)2PO4 MgSO4.7H2O CaCl2 FeSO4.7H2O

1.0 16.4 1.5 2.0 0.2 0.01 0.0005

may be circumvented by using ammonia to control the pH, which typically results in lower conductivity of the culture supernatant. In addition to the salts and sugar, which are required for cell metabolism, production of recombinant proteins in bacteria typically requires the addition of an inducer, most commonly isopropyl-β-D-thiogalactopyranoside (IPTG), which is used to activate protein expression when a certain cell density is reached. The use of natural compounds as inducers is advantageous since such species are readily degraded in the culture and are not signiﬁcant impurities. On the other hand, detergents or oils added to the culture as anti-foaming agents, although present in relatively small amounts, can affect downstream processes since they tend to foul membranes and chromatography matrices. Culture media for yeast are similar to those used for E. coli. Methanol is frequently used as the inducer for systems based on the alcohol oxidase promoter (AOX) expression system. Mammalian cell culture, on the other hand, requires much more complex media including glucose as the carbon source, amino acids, vitamins, inorganic salts, fatty acids, nucleotides, pyruvate, and butyrate. This basal medium is supplemented with proteins for oxygen transport, hormones, and growth factors. Oxygen transport proteins such as transferrin contain bound iron. In order to create a totally protein-free medium these proteins are often replaced by iron chelators such as ferric citrate, ferric iminodiaectic acid, ferric ammonium citrate and tropolone (2-hydroxy-2,4,6-cycloheptatrien-1-one). However, these compounds can also interfere with downstream processing. In addition, under slightly acidic conditions, ferric citrate forms a gel, which is difﬁcult to separate from proteins and other biomacromolecules. Several other additives may be present in cell culture media: pH indicators such as phenol red, added to laboratory scale culture media often bind to ion exchange resins and are best avoided for large-scale cultivation. However, polymers, such as poly(propylene glycol) or poly(ethylene glycol), are often needed in concentrations of up to 0.02% to protect the cells from shear stress. The pH in mammalian cell culture is typically regulated by the addition of CO2, although high density cultures may require addition of NaOH. Final conductivities of less the 17 mS cm−1 are typical, making direct capture by ion exchange easier than capture from yeast and E. coli homogenates.

1.3 Bioprocesses

1.3.4 Components of the Culture Broth

In general, prior to harvest, the culture broth contains the following components: intact cells, debris from lysed cells, intracellular host cell components, unused media components, compounds secreted by the cell, and enzymatically or chemically converted media components. Oxygen is depleted since during primary recovery the oxygen supply is shut down and the residual dissolved oxygen is rapidly consumed. The low oxygen content can induce necrosis and cells may rapidly die and lyse during this phase. Some cell types begin autolysis after just 30 min without oxygen. As a result, depending on the cell type rapid separation of the cells from the broth supernatant is necessary to maintain a low level of host cell impurities. The high concentrations of CO2 that may be produced by the residual cells in the culture broth will fragment and shift the pH towards the acidic region. CO2 is much more soluble in aqueous solutions than oxygen, so that substantial concentrations may be present. Dissolved CO2 is rapidly liberated when the pH is adjusted for downstream processing and forms bubbles which may then enter chromatography columns and disrupt the packing. Intracellular host cell components appearing as impurities in a culture supernatant can be estimated from the following equation:

(1 − Fraction of live cells ) × (Cell count ) × ( Amount of intracelllular component per cell ) where the concentration of intracellular components can be estimated from Tables 1.13 and 1.14. Thus, from a downstream processing perspective high cell viability is desirable; this not always possible however. For example, in high-titer antibody production by cell culture, the cells often lyse in the ﬁnal stage of cultivation. As a result, supernatants from mammalian cells cultures grown using deﬁned media will contain host cell proteins in the range of 1–3 mg ml−1 (1000 to 3000 ppm). These levels must be reduced to less then 100 ppm in the ﬁnal product. 1.3.5 Product Quality Requirements

The manufacture of biological products for pharmaceutical applications must follow general guidelines that have been established by the regulatory framework. Three keywords summarize the principal product quality requirements: purity, potency, and consistency. Industrially, these requirements must be met by processes that are economically viable and bring products to the market rapidly. Downstream processes must be designed to obtain sufﬁcient purity while maintaining the potency or pharmacological activity in a consistent manner. 1.3.5.1 Types of Impurities Purity requirements for biopharmaceuticals vary depending on the particular application. Thus, it is not possible to specify absolute values. However, an important distinction can be made among the various impurities which are frequently

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1 Downstream Processing of Biotechnology Products

categorized as critical or non-critical. A non-critical impurity is an inert compound without biological relevance. This can be, for instance, residual PEG from an extraction process or a harmless host component such as a lipid. On the other hand, endotoxins or growth factors secreted into the culture supernatant are examples of critical impurities, since they exert adverse biological activity. These impurities need to be traced throughout the process. Impurities stem either from the cultivation process or from materials added for processing. Examples of processrelated impurities are solubility enhancers, redox-buffers, enzyme inhibitors, and compounds leached from ﬁlters or chromatography media. Monomers which leak from the polymeric materials that come into contact with the product are sometimes of particular concern. Extensive testing and documentation of their removal is generally mandatory. The term adventitious agent is used to describe potentially infectious impurities that have not been added intentionally and are not essential to the process but are typically extremely hazardous. They may enter the process as a result of contaminated raw materials or cells. Examples of adventitious agents are viruses, virus-like particles, and prions such as transforming spongiform encephalitic agents. Certain toxic chemicals may also be considered to be adventitious agents. Finally, bioburden, originating from microbial contamination from the air or personnel or from inadequately cleaned equipment, can also have serious effects and must be carefully monitored and controlled. Table 1.16 summarizes the measures required to demonstrate the removal of adventitious agents and other impurities. This demonstration is usually carried out in scaled down experiments, because it would obviously be counterproductive to intentionally contaminate the production plant with an adventitious agent. For these determinations, also known as spiking experiments, a bolus of an adventitious agent, e.g. a virus, is added to the raw feed stream entering a puriﬁcation process step. The virus titer before puriﬁcation a′ = log10(Feed titer) and after puriﬁcation a′′ = log10(Harvest titer) are determined and the log-virus reduction (LVR) factor is calculated as follows: LVR = log10 (Feed titer ) − log10 (Harvest titer ) = a ′ − a ′′

Table 1.16

(1.30)

Measures required to demonstrate the removal of adventitious agents and

impurities. Measures

Adventitious agents

Other impurity

Spiking experiments to demonstrate clearance Starting material preferably free of contaminating agents Clearance measured at each step Control of ﬁnal product

Yes Yes No Yes

No No Yes Yes

1.3 Bioprocesses Table 1.17 Typical virus clearance values of murine leukemia virus for puriﬁcation of a recombinant antibody. Data from [36].

Step

LRV

Capture Low pH Cation exchange chromatography Anion exchange chromatography Hydrophobic interaction chromatography

4.23 >4.4 2.13 4.98 4.19

In order to account for the effect of volume changes (e.g. a mere 1 : 10 dilution results in a LVR of 1), the following individual reduction factor, Ri, is also calculated: Ri = LVR − log10

V′ V ′′

(1.31)

where V′ and V″ are the feed and harvest volumes, respectively. Finally, the LVR of the individual process steps are added together to arrive at a cumulative LVR for the entire process. Table 1.17 illustrates a typical virus clearance scheme for an antibody puriﬁcation process. Although almost all mammalian cells are infected with viruses making the validation of viral clearance obligatory, efforts are frequently made to omit these procedures by utilizing platform processes that have demonstrated clearance efﬁciency. 1.3.5.2 Regulatory Aspects and Validation Biopharmaceutical product quality and process validation are subject to regulations by the individual governments. In the United States, the legal framework is published in the Code of Federal Regulations 21 (21 CFR), Subchapter F Biologics. The US Food and Drug Administration (US FDA) is responsible for its implementation. The US code deﬁned biological products as follow: ‘Biological product means any virus, therapeutic serum, toxin, antitoxin, or analogous product applicable to the prevention, treatment or cure of diseases or injuries of man’. Plasmids, cells and tissues are not explicitly mentioned, but are considered to be part of this deﬁnition. In the European Union (EU), the legal framework is still under the sovereignty of the individual member states, although an EU-wide umbrella organization, the European Medical Agency (EMEA), has been founded, with the goal of harmonizing the EU regulatory structure. The guidelines for medicinal products derived by biotechnology as set out by the Commission of the European Community, deﬁnes biologics as products derived from:

1) human blood, other human body ﬂuids or human tissue 2) animal blood 3) microorganisms or components of microorganisms

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1 Downstream Processing of Biotechnology Products

4) animals or microorganisms for active or passive immunization 5) monoclonal antibodies 6) recombinant DNA products The existence of multiple regulatory frameworks adds complexity to the biopharmaceutical industry. Thus, the International Conference on Harmonization (ICH) has been established to develop a common international regulatory framework. ICH brings together the regulatory authorities of Europe, Japan and the United States, as well as experts from the pharmaceutical industry in these three regions to discuss scientiﬁc and technical aspects of product registration. The purpose is to make recommendations on ways of achieving greater harmonization in the interpretation and application of technical guidelines and the requirements for product registration. While progress is being made in the harmonization process, the regulatory framework of each country remains in effect. Over the years, the Center for Biologics Evaluation and Research (CBER) of the US FDA has issued detailed recommendations for the manufacture of biologicals in a series of ‘Points to Consider’ papers. The ﬁrst of these papers was released in 1983 with the title ‘Interferon Test Procedures: Points to Consider in the Production and Testing of Interferon Intended for Investigational Use in Humans’. Subsequent papers include: 1) Points to Consider in the Production and Testing of New Drugs and Biologicals Produced by Recombinant DNA Technology – 4/10/1985 2) Guidelines on Validation of the Limulus Amebocyte Lysate Test as an EndProduct Endotoxin Test For Human and Animal Parenteral Drugs, Biological Products and Medical Devices – 12/1987 3) Points to Consider in the Manufacture and Testing of Therapeutic Products for Human Use Derived from Transgenic Animals – 1995 4) Points to Consider in the Manufacture and Testing of Monoclonal Antibody Products for Human Use – 2/28/1997 5) Guidance for Industry: Monoclonal Antibodies Used as Reagents in Drug Manufacturing – 3/29/2001 A complete list can be found on the ofﬁcial website of CBER: http://www.fda.gov/ Cber/guidelines.htm#95. Validation is a critical aspect of biopharmaceutical process development. According to existing ICH deﬁnitions, critical parameters that may inﬂuence product quality need to be validated. After validation, a standard operating protocol (SOP) is established, which describes the process and the allowed variations. Critical operational parameters are deﬁned as a limited sub-set of process parameters that signiﬁcantly affect critical product quality attributes when varied outside a meaningful, narrow (or difﬁcult to control) operational range. Consider, for example, the operation of a chromatographic puriﬁcation step. As shown in Table 1.18 this operation will require the deﬁnition of a number of operating conditions as inputs, which in turn will result in certain performance characteristics, deﬁned as outputs. In order to validate the process, the input parameters must be varied over suitable ranges and the output measured. The critical parameters are then deﬁned

1.3 Bioprocesses

47

Table 1.18 Examples of input and output parameters in a chromatographic separation process for proteins.

Parameter Input parameter

pH

Temperature

Ionic strength

Load

Flow rate

Column height (residence time)

Output parameter (Performance)

Purity

Concentration

Stability

Yield

DNA content

Host cell protein content

Endotoxin content

Critical and Key Acceptable Limits (a)

Set Point

Operating Range (b) Non-Key Acceptable Limits

Figure 1.28

Deﬁnition of operation ranges for critical and non-critical parameters.

based on these experiments, which are usually carried out on a small scale. Suitably narrow operational ranges are established for these parameters as well as for non-critical parameters. Since the latter do not affect critical product quality attributes, their ranges will normally be broader than those for critical parameters (see Figure 1.28). 1.3.5.3 Purity Requirements It is difﬁcult to describe absolute purity requirements since they depend on the intended use of the biopharmaceutical, the dose, the risk–beneﬁt ratio, etc. Table 1.19 provides only approximate values which are meant to serve as general guidelines. Aggregates are an important concern for many biopharmaceutical proteins. It has been shown that aggregates can induce immune reactions or cause other side effects. Moreover, aggregates may constitute seeds for precipitation and reducing the shelf life of a product. As a result, controls on the percentage of dimers, oligomers and higher aggregate forms are often required. The leakage of ligands or

Back pressure

48

1 Downstream Processing of Biotechnology Products General guidelines for purity, consistency, and potency of protein biopharmaceuticals.

Table 1.19

Criteria

Requirement

Purity Speciﬁc protein content Dimer/oligomer content Ligand leakage Virus content DNA content Endotoxin content Prion content

>99.9% <1.0% Usually < 1 ppm Absence with a probability of <10−9 <10 pg/dose 5 EU kg−1 h−1 Absence with a probability of <10−9

Consistency Microheterogeneity Impurities

Permitted, but consistent Permitted, but consistent

Potency Folding Mutations Processing

Correctly folded Correctly expressed, no mutations Correctly processed

other leachable chemicals from chromatographic media and membranes is also an important concern since these materials can be immunogenic and toxic. Viral contamination is obviously a critical issue as in the past it has been responsible for many iatrogenic diseases, such as those occurring because of contaminated blood products. Since absolutely complete removal of these adventitious agents is not possible, limits are often established on the basis of a risk–beneﬁt analysis. For example, the World Health Organization (WHO) accepts for a vaccine one adverse case in 109 applications – hence, a probability value of 10−9 is suggested in Table 1.19. In practice, the difﬁculty of achieving such low probabilities is, of course, dependent on the dose. For example, the commonly set level of 10 pg of DNA per dose is relatively easy to achieve for a Hepatitis B vaccine, where one dose may contain only about 12 µg of protein. On the other hand, such a requirement may be rather difﬁcult to meet in the case of recombinant antibodies, where one dose can consist of as much as 500 mg of protein. As previously mentioned, most biopharmaceuticals are not fully deﬁned individual molecular entities – rather, they consist of a large number of similar isoforms or variants (some recombinant antibody products contain as many as 200 identiﬁable variants). Since the biological and pharmacological activity can vary dramatically among different isoforms, it is important to maintain the distribution of these variants within established acceptable ranges. Because of the complexity of bioproduction systems, similar consistency must also be maintained for the impurity proﬁles as determined from analytical assays, in order to assure product safety. Finally, test systems must be established to control the potency in vitro and, where necessary, in vivo.

1.4 Role of Chromatography in Downstream Processing

1.4 Role of Chromatography in Downstream Processing

Chromatography is the principal tool used for the puriﬁcation of biopharmaceuticals. This can be explained by certain advantages of chromatography over other unit operations. First, chromatography provides very high separation efﬁciencies, which allow the resolution of complex mixtures with very similar molecular properties. Properly designed chromatography columns can have the separation efﬁciency of hundreds or even thousands of theoretical plates. By comparison, extraction and membrane ﬁltration are usually limited to only a few stages. Second, chromatography columns packed with high capacity adsorbents are ideal for capturing molecules from the dilute solutions encountered in bioprocessing. In such systems, there is efﬁcient contact between large volumes of solution and small amounts of the adsorbent packed into a column, resulting in either rapid concentration of the product or the nearly complete removal of contaminants present in small concentrations. By comparison, liquid–liquid extraction systems typically require similar volumes of the two phases in order to function properly, so that concentration is not very feasible. A further advantage is that chromatography can be undertaken in an almost closed system and the stationary phase can be easily regenerated. Finally, chromatographic methods are well established in many practical biopharmaceutical manufacturing processes and suitable equipment and packing materials are readily available. A perceived disadvantage of chromatography is the difﬁculty of scale-up within the constraints of the biopharmaceutical industry. However, as will be shown in the remaining chapters of this book, proper application of engineering tools in combination with adequate measurements allows the design of optimum columns for large-scale applications. Indeed, as shown recently by Kelley [2], chromatographic puriﬁcation processes are suitable and technically and economically viable for protein puriﬁcation on scales as high as 20 tons per year. Although there are no products which are currently made on such a large scale, the popularity of biopharmaceuticals is increasing rapidly so that such scales can be envisioned in the future. Figure 1.29 illustrates the structure of a generic process for the recovery and puriﬁcation of biologicals produced by microbial fermentation or animal cell culture. The initial steps where cells are separated are often referred to as primary recovery. This step requires different strategies dependent on whether the product is secreted into the culture medium or expressed in the cell, either as inclusion bodies, in soluble form in the cytosol or periplasm, or anchored to the membrane. Generally, chromatography plays a minor role in these initial steps, which are focused on the removal of suspended solids such as cells or cell debris. Sedimentation, centrifugation, deep bed ﬁltration, and microﬁltration or combinations thereof are normally used for these early steps. However, chromatography, implemented through the use of ﬂuidized or expanded beds can also be used for the direct capture of secreted proteins from cell culture supernatants. In these systems the liquid ﬂows upwards through an initially settled bed of dense adsorbent particles. Above a certain ﬂow velocity the bed expands and the

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1 Downstream Processing of Biotechnology Products

particles become ﬂuidized allowing free passage of cells and other suspended matter while the product is directly captured by the adsorbent. This approach can be effective for dilute suspensions, but since bed expansion is directly inﬂuenced by the feed density and viscosity, the operation tends to be critically affected by variations in the composition of the broth. In practice, the high viscosity and cell density encountered in modern fermentation technology (up to 400 mg ml−1 wet cell mass for P. pastoris or 200 mg ml−1 for E. coli) make it difﬁcult to implement this approach reliably on an industrial scale. An alternative possibility for early capture without clariﬁcation is to use adsorption beds packed with large particles, sometimes referred to as ‘big beads’. If the particles are larger than about 400 µm in diameter, the interparticle spaces are sufﬁciently large to allow passage of cells and cell debris. Although the efﬁciency of capture is reduced by the diffusional limitations that accompany the larger particle diameter, the ensuing reduction in the number of processing steps can provide overall economic and operational advantages. Unlike expanded beds, packed bed processes are not very sensitive to feed viscosity so that reliable operation with large diameter beads can be achieved even with viscous feedstocks. As seen in Figure 1.29, following primary recovery, the general downstream processing scheme consists of successive capture, puriﬁcation and polishing steps, each comprising one or more operations. With only a handful of exceptions, current industrial processes for biopharmaceuticals almost exclusively employ chromatography for these three critical steps. Figure 1.30 shows an example of the puriﬁcation of a recombinant antibody. In this process, capture is realized using a selective adsorbent comprising of Staphylococcal Protein A immobilized in porous beads. This highly selective ligand

Fermentation broth

Cell removal Supernatant (Product secreted)

Cell (intracellular product)

Cell disintegration

Capture

Product as inclusion body

Product soluble

Product membrane associated

Extraction

Extraction

Extraction

Refolding/Oxidation

Fluidized bed “Big Beads”

Purification

Polishing

Formulation

Figure 1.29 Generalized downstream processing ﬂow sheet for puriﬁcation of proteins starting with the unclariﬁed fermentation broth.

1.4 Role of Chromatography in Downstream Processing

Clarified culture Staphylococcal Protein A Incubation at low pH lon-exchange Virus filtration Hydrophobic interaction Ultra/diafiltraion Bulk product Figure 1.30

Generalized ﬂow scheme for the puriﬁcation of a recombinant antibody.

allows the direct loading of the clariﬁed culture broth onto the capture column, and then selectively binds the antibody. In the subsequent steps, puriﬁcation and polishing are conducted with ion exchange and hydrophobic interaction columns to remove host cell proteins and aberrant protein variants. Note that intermediate, non-chromatographic, steps are also included. Firstly, incubation at low pH and then a ‘virus ﬁltration’ step are implemented for viral clearance. Secondly, an ultra/ diaﬁltration step is included for buffer exchange and ﬁnal formulation. In addition to puriﬁcation, chromatography is also employed in other bioprocesses. An important example is the use of chromatography to facilitate refolding of solubilized protein, which sometimes creates a bottleneck in industrial processes. Without simultaneous separation, misfolding and aggregation in particular, compete with the correct folding pathway. Aggregation may originate both from non-speciﬁc (hydrophobic) interactions of the predominantly unfolded polypeptide chains as well as from incorrect interactions of partially structured folding intermediates. As shown in Figure 1.31, aggregation reactions are second- (or higher) order processes, whereas correct folding is generally determined by ﬁrstorder reactions [37]. In practice, refolding conditions (e.g. denaturant concentration) are adjusted so that the equilibrium distribution favors the formation of native protein (i.e. k2 >> k3). The formation of intermediates is generally very fast, so that k1 can be neglected. For the case where k3→0, we effectively have competing ﬁrst and second (or higher) order reactions. Under these conditions, refolding in a batch system is described by the following equations: d [U ] = − (k2 [U ] + k4 [U ]n ) dt

(1.32)

d [N ] = k2 [U ] dt

(1.33)

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1 Downstream Processing of Biotechnology Products

1st order k1

k2

U

I

N k3 k4

≥ 2nd order A

Figure 1.31 Simpliﬁed reaction scheme for protein refolding with aggregation of intermediates.

where the brackets denote concentrations, k2 is the net rate constant of folding, k4 the net rate constant of aggregation, and n is the reaction order. An analytical solution of these equations is available for n = 2 and is given by the following equation [37]: Y (t ) =

{

}

k2 [U ]0 k4 ln 1 + (1 − e − k2t ) k2 [U ]0 k4

(1.34)

where Y is the yield of the refolding reaction and [U]0 is the initial concentration of unfolded protein. As time approaches inﬁnity, the ﬁnal yield of native protein is then given by the following equation: Y (t → ∞ ) =

k2 [U ]0 k 4  ln  1 +  k2  [U ]0 k 4 

(1.35)

This result suggests that dilution (i.e. low [U]0) is a simple and effective way of ensuring high refolding yields. While this is effective and widely used in practice, the ensuing large solution volumes complicate further downstream processing and increase cost. Refolding in chromatographic columns also known as matrixassisted refolding, can be a valuable refolding alternative to reduce the need for extensive dilution. The underlying mechanism leading to improved folding in chromatographic columns is not completely understood and may depend on the speciﬁc nature of the protein and the selected conditions. However, the effects can be dramatic as shown for example in Figure 1.32. In this case, refolding was conducted by separating the denaturing agent (urea) from the unfolded protein using size exclusion chromatography and allowing refolding to occur within the column. This resulted in a greater yield of folded protein compared to a simple dilution process. The apparent aggregation rate constant in this case was about 30 times smaller compared to that for the dilution process (Table 1.20). One possibility to explain this result is that aggregation may be inhibited within the matrix pores by

1.4 Role of Chromatography in Downstream Processing

Yield of refolded protein (%)

100 90

matrix-assisted refolding

80

batch-dilution (1:25) second order aggregation model

70 60 50 40 30 20 10 0 0

1

2

3

4

5

6

7

Initial concentration of denatured protein (mg/ml)

Figure 1.32 Refolding yield of a protein by batch dilution and with matrix-assisted refolding using size exclusion chromatography. Reproduced from [38].

Table 1.20 Refolding and aggregation rate constants of a protein in refolding by batch dilution and matrix-assisted refolding using size exclusion chromatography. Data from [38].

Rate constants

Batch-dilution Matrix-assisted refolding by SEC

Folding k2 (min−1)

Aggregation k4 (ml mg−1min−1)

0.0012 0.0012

0.3 0.01

steric hindrance which allows a greater portion of the protein to follow the path toward correct folding. In the example given in Figure 1.32 the unfolded protein was passed over a size exclusion column and the denaturant was slowly removed. Comparison of kinetic constants between conventional refolding by dilution into a refolding buffer and matrix-assisted refolding conﬁrms that aggregation is suppressed in the column (Table 1.20). SEC-promoted refolding is also possible in continuous processes, which can also include a recycling system for aggregated protein. Yield and productivity of a continuous refolding systems using pressurized continuous annular chromatography (P-CAC) considering initial protein concentration, residence time and recycling rate were extensively studied using α-lactalbumin as the model protein [39]. Also

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countercurrent chromatography systems such as the simulated moving bed (SMB) can be used for continuous matrix-assisted refolding. Ion exchange, afﬁnity adsorption and hydrophobic interaction have also been used to facilitate refolding. A method based on the adsorption of the unfolded protein onto an ion exchange resin was introduced by Creighton [40]. Further improvements of this method include the application of more sophisticated buffer during loading and elution. The methods can be also executed in a continuous manner. The surface contact can initiate refolding. It has often been observed that ﬁnal refolding takes place after the protein has been eluted from the column. Immobilization of denatured proteins on a solid support is often limited by the ﬂexibility to regain the native conﬁguration because of multi-point interactions with the matrix. Introduction of an N- or C-terminal poly-histidine-tag allowed the reversible one-point immobilization of the denatured protein on a solid support based on immobilized metal afﬁnity chromatography (IMAC). Refolding can be achieved by a simple buffer exchange in a stepwise or gradient manner. The use of hydrophobic interaction chromatography (HIC) was described for the refolding of lysozyme, BSA, α-amylase and recombinant γ-interferon [41]. Immobilized folding catalysts and artiﬁcial chaperones have also been suggested as refolding aids. Mimicking in vivo folding systems was a further step in improving the yield of in vitro refolding. The chaperones or compounds mimicking chaperones are immobilized on a chromatography matrix. The protein solution is passed through such columns. The folded proteins are slightly retarded and the denaturant is exchanged. The immobilized chaperones prevent aggregation. Thus refolding can be achieved at higher concentrations or yield. It has to be borne in mind that a chaperone acts in a stoichiometric manner. Thus large amounts of chaperone protein are necessary to avoid aggregation.

References 1 Buchacher, A., and Iberer, G. (2006) Biotechnol. J., 1, 148. 2 Kelley, B. (2007) Biotechnol. Progr., 23, 995. 3 Aggarwal, S. (2008) Nat. Biotechnol., 26, 1227. 4 Rinaldi, A. (2008) EMBO Rep., 9, 1073. 5 Berg, J., Tymoczko, J., and Stryer, L. (2006) Biochemistry, Palgrave Macmillan. 6 Voet, D.J., and Voet, J.G. (2006) Biochemistry, Textbook and Student Solutions Manual, John Wiley & Sons, Inc., New York. 7 Egorov, T.A., Svenson, A., Ryden, L., and Carlsson, J. (1975) Proc. Natl Acad. USA, 72, 3029. 8 Branden, C., and Tooze, J. (1991) Introduction to Protein Structure, Garland Publishing, Inc., New York.

9 Ueberbacher, R., Haimer, E., Hahn, R., and Jungbauer, A. (2008) J. Chromatogr. A, 1198, 154. 10 Schmid, F.X. (1997) Optical spectroscopy to characterize protein conformation, in Protein Structure A Practical Approach, 2nd edn (ed. T.E. Creighton), IRL Press, Oxford, pp. 261–297. 11 Dayhoff, M.O. (1974) Fed. Proc., 33, 2314. 12 Nötling, B. (1999) Protein Folding Kinetics, Springer, Berlin. 13 Gokana, A., Winchenne, J.J., BenGhanem, A., Ahaded, A., Cartron, J.P., and Lambin, P. (1997) J. Chromatogr. A, 791, 109. 14 Kneuer, C. (2005) DNA Pharmaceuticals (ed. S. Martin), Wiley-VCH Verlag GmbH, Weinheim.

References 15 Petsch, D., and Anspach, F.B. (2000) J. Biotechnol., 76, 97. 16 Mach, H., Middaugh, C.R., and Lewis, R.V. (1992) Anal. Biochem., 200, 74. 17 Wilﬁnger, W.W., Mackey, K., and Chomczynski, P. (1997) Biotechniques, 22, 474. 18 Tanford, C. (1976) Biochemistry, 15, 3884. 19 Kuehner, D.E., Engmann, J., Fergg, F., Wernick, M., Blanch, H.W., and Prausnitz, J.M. (1999) J. Phys. Chem. B, 103, 1368. 20 Tanford, C., and Roxby, R. (1972) Biochemistry, 11, 2192. 21 Righetti, P.G., and Caravaggio, T. (1976) J. Chromatogr. A, 127, 1. 22 Hopp, T.P., and Woods, K.R. (1981) Proc. Natl Acad. Sci. USA, 78, 3824. 23 Kyte, J., and Doolittle, R.F. (1982) J. Mol. Biol., 157, 105. 24 Mahn, A., Lienqueo, M.E., and Salgado, J.C. (2009) J. Chromatogr. A, 1216, 1838. 25 Fox, S., and Foster, J.S. (1957) Introduction to Protein Chemistry, John Wiley & Sons, Inc., New York. 26 Green, A. (1932) J. Biol. Chem., 95, 47. 27 Cohn, E., and Edsall, J.T. (1943) Proteins, Amino Acids, and Peptides, Academic Press, New York. 28 Hofmeister, F. (1888) Arch Exp. Pathol. Pharmakol., 24, 247.

29 Gekko, K., and Timasheff, S.N. (1981) Biochemistry, 20, 4677. 30 Ahrer, K., Buchacher, A., Iberer, G., and Jungbauer, A. (2006) J. Biochem. Biophys. Methods, 66, 73. 31 Monkos, K., and Turczynski, B. (1999) Int. J. Biol. Macromol, 26, 155. 32 Sibileva, M.A., Zatyaeva, A.A., and Matveeva, N.I. (2001) Mol. Biol., 35, 73. 33 Reisner, A.H., and Rowe, J. (1969) Nature, 222, 558. 34 Tyn, M.T., and Gusek, T.W. (1990) Biotechnol. Bioeng., 35, 327. 35 Cussler, E.L. (1997) Diffusion – Mass Transfer in Fluid Systems, 2nd edn, Cambridge University Press, Cambridge. 36 Li, Y., and Galperina, O. (2005) PDA Viral Safety Workshop, Langen, Germany. 37 Kiefhaber, T., Rudolph, R., Kohler, H.H., and Buchner (1991) J., Biotechnol., 9, 825. 38 Schlegl, R., Necina, R., and Jungbauer, A. (2005) Chem. Eng. Technol., 28, 1375. 39 Schlegl, R., Iberer, G., Machold, C., Necina, R., and Jungbauer, A. (2003) J. Chromatogr. A, 1009, 119. 40 Creighton, T.E. (1990) United States Patent No. 4977248. 41 Geng, X., and Chang, X. (1992) J. Chromatogr. A, 599, 185.

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2 Introduction to Protein Chromatography 2.1 Introduction

The scope of this chapter is to provide an introduction to the terminology, basis for separation, and modes of operation of chromatography. We will consider chromatography as a special case of a general class of operations known as percolation processes. In the broadest sense, percolation processes include operations that are often identiﬁed as differential or elution chromatography, adsorption, ion exchange, countercurrent adsorption, and even deep-bed ﬁltration and leaching. In each of these cases a ﬂuid phase is contacted by a solid phase with a distribution of one or more species between the two phases. The basic operating principles and the mathematical description of these processes are virtually the same even if the physiochemical interactions of solutes between the two phases, the objective of the operation, and the mechanical implementation of the various processes are quite different.

2.2 Basic Principles and Deﬁnitions

We deﬁne chromatography as a process that employs a ﬁxed-bed of a soluteinteracting material, known as the stationary phase, to separate a mixture of components that are carried through the bed by a ﬂuid phase, known as the mobile phase. Chromatography includes a variety of interaction mechanisms and modes of operation for the separation of mixtures, but in all cases the mixture of components that are to be separated must be partitioned differently between the stationary phase and the mobile phase. In the conventional and better-known implementation of chromatography, the feed mixture is supplied to one end of a ﬁxed bed column and individual components are separated in time over the length of the bed as a result of their different partitioning between phases. The separated components are eventually recovered at the bed exit, as they elute with different residence times.

Protein Chromatography: Process Development and Scale-Up. Giorgio Carta and Alois Jungbauer © 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 978-3-527-31819-3

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2 Introduction to Protein Chromatography

(a)

(b)

Feed

Mobile phase

Mobile phase

Mobile phase

Time 0

Time 10

Time 20

Time 30

Figure 2.1 Schematics of conventional column chromatography (a) and continuous annular chromatography (b). In both cases the process is two dimensional. In the ﬁrst

Fixed feed inlet

Mobile phase

case the dimensions are column length and time and in the second, they are column length and angular position.

Note that based on this deﬁnition, chromatography is a two dimensional process. Typically, the bed length and time are the two dimensions in which the separation is carried out. Accordingly, we can associate the components with their retention times. In this case the process is intrinsically a batch operation. On the other hand, a continuous operation is also possible, in general, if two spatial coordinates are used instead of bed length and time. As long as the two phases are kept in relative motion, a steady state chromatographic separation is developed in the two dimensions. Accordingly, each component will elute at a characteristic position in the two-dimensional system. Examples of batch and continuous chromatographic operations are shown in Figure 2.1, for classical column chromatography and for an equivalent continuous annular chromatography system. The ﬁrst case uses a ﬁxed bed column and the separation is obtained in bed length and time coordinates. The feed components that interact more strongly with the stationary phase are retarded and are eluted later while those that interact more weakly travel more quickly and are eluted sooner. The second case uses an annular bed of the stationary phase which rotates around its axis with a continuous crosscurrent ﬂow of the mobile phase. If the feed mixture is supplied continuously at a point that remains ﬁxed in space, the separation is obtained in bed length and angular coordinates. Each component will travel along a helical path with the more strongly interacting species traveling farther from the feed point and the more weakly interacting species traveling a shorter distance before leaving the column.

2.2 Basic Principles and Deﬁnitions

The deﬁnition of chromatography given above goes beyond the frequent deﬁnition of chromatography as a differential migration process, where each component travels at a different, constant speed through the chromatographic column. Increased generality is needed particularly to describe the chromatography of biopolymers. In this case, the interaction with the stationary phase is, in many instances, so strong as to be considered essentially irreversible for some components and very weak or even absent for others. For these conditions, differential migration of the strongly bound components cannot occur without changing thermodynamic conditions or altering the composition of the mobile phase. Accordingly, afﬁnity adsorption which is typically dominated by very strong adsorption, is one type of chromatography. Because of their two-dimensional nature, both column chromatography and continuous annular chromatography (CAC) are capable of multi-component separations. Countercurrent adsorption systems are however, fundamentally different. As shown in Figure 2.2, such systems involve the countercurrent ﬂow of the stationary phase material, an adsorbent for example, and the mobile phase. This can be accomplished either with a true ‘moving bed’ system, with the actual counterﬂow of the two phases, or, more practically, with a ‘simulated countercurrent’ system. In the latter case, an arrangement of packed columns is operated by periodic switching of the beds using a system of valves, in a direction opposite to that

(a)

(b) Weakly adsorbed species (“lights”)

Feed mixture

Fresh adsorbent

Strongly adsorbed species (“heavies”)

Figure 2.2 Schematics of true moving bed (a) and simulated countercurrent (b) adsorptive systems. Both are examples of countercurrent separators. Dashed lines represent ﬂow of the adsorbent. The simulated countercurrent system mimics

Weakly adsorbed species (“lights”)

Feed mixture

Fresh adsorbent bed

Strongly adsorbed species (“heavies”)

counterﬂow through the periodic switching of packed beds as indicated by the arrows. In both cases, interphase mass transfer occurs at each point between the two countercurrent ﬂowing phases.

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2 Introduction to Protein Chromatography

of the ﬂuid phase ﬂow. In either case, movement of the feed components in the system depends on the relative velocity of transport in the two countercurrent ﬂowing phases. In general, components that are more strongly adsorbed will travel in the direction of the ﬂow of the adsorbent, while those that are more weakly adsorbed, will travel in the opposite direction. To achieve these conditions, at steady state the feed components will be split in two fractions: one containing the more strongly adsorbed species (the ‘heavies’ by analogy to a distillation process) which leave the column with the adsorbent at the ﬂuid feed end, and the other containing the more weakly adsorbed species (the ‘lights’) which leave the column with the ﬂuid phase at the opposite end. Since the heavies leave the column with the adsorbent, a desorption step is generally needed to recover them. Although limited to binary splits (i.e. multi-component separations require multiple units) countercurrent separators maximize mass transfer efﬁciency and serve as the basis for simulated moving bed processes (SMB). Thus, although not strictly ‘chromatography’, countercurrent separators are also discussed here as they play important roles in bioprocessing. The processes described above, whether chromatographic or countercurrent, are general with respect to the nature of the interactions of solutes between the two phases that comprise the chromatographic system. In most cases, interphase distribution is based on attractive molecular interactions between the feed components and functional groups present on the surface of the stationary phase. This includes the favorable electrostatic interaction between charged molecules and oppositely charged ionogenic groups in ion-exchange chromatography (IEC), van der Waals forces between non-polar solutes and surfaces functionalized with hydrophobic ligands in hydrophobic chromatography (RPC and HIC), the formation of a coordination complex with metal ions immobilized on a resin in metal ion interaction chromatography (MIC), and the biospeciﬁc binding of an antigen to a surfacebound antibody in biospeciﬁc interaction or afﬁnity chromatography (BIC). One exception is size-exclusion chromatography where, in principle, interactions with the stationary phase are purely steric, leading to partial exclusion of the larger molecules from the stationary phase. Table 2.1 provides a brief summary of the various branches of modern liquid chromatography as they apply to biopolymer separations, deﬁned by the type of stationary phase ligate. Chapter 3 provides a detailed discussion of the available materials and their properties. As seen from Table 2.1, a different mobile phase chemistry and composition is required for each case. With the exception of SEC, this composition is typically modulated by the addition of a ‘mobile phase modiﬁer’ in order to control selectivity or desorb strongly adsorbed species. The modiﬁer is typically a water-miscible organic solvent in RPC, a salt such as ammonium sulfate in HIC, a monovalent counter ion in IEC, a competitively adsorbed species in MIC, and pH in BIC. In each case, the purpose of the modiﬁer is to modulate the attractive interactions with the stationary phase ligate. The selectivity of the different chemistries will vary and more than one type of chemistry will often be used in a multi-step downstream process in order to achieve robust separation and removal of impurities by exploiting orthogonal

2.3 Modes of Operation Branches of chromatography deﬁned by the nature of the stationary phase ligate and mobile phase composition.

Table 2.1

Stationary phase ligate

Mobile phase

Branch

Acronym

None

Aqueous

Size exclusion

SEC

Hydrophobic

Normally a water–organic solvent mixture. Elution with increasing % of organic solvent

Reversed phase

RPC

Mildly hydrophobic

Normally an aqueous solution with Hydrophobic high salt concentration. Elution with interaction decreasing salt concentration

HIC

Charged

Normally an aqueous solution with low salt concentration. Elution by increasing salt concentration

Ion exchange

IEC

Metal chelate

Aqueous. Desorption with a competitively bound species

Metal interaction

MIC

Biospeciﬁca)

Aqueous. Desorption with pH

Biospeciﬁc interaction/afﬁnity

BIC

a)

Pseudo-afﬁnity ligates also exist that mimic biospeciﬁc interactions with synthetic groups.

interaction types. Thus, for example, IEC and HIC are often used sequentially to help resolve complex mixtures. There are also mixed modes stationary phases, in which two or more interactions are combined and can provide advantages in terms of selectivity. Although the molecular forces involved are different in each branch, engineering models describing equilibrium and transport in these systems are formally very similar, allowing a uniﬁed treatment of chromatography as a unit operation. Moreover, in many instances, the same mobile and stationary phase chemistries can be used in different ways to effect separations in either classical column chromatography or countercurrent separation systems.

2.3 Modes of Operation

Tiselius ﬁrst deﬁned the classical modes of operation of chromatography in 1943 [1]: elution chromatography, frontal chromatography, and displacement development. More than half a century later, the classiﬁcation is still useful since most practical column chromatography processes will fall into one of these categories. Basic features of these techniques are illustrated in Figure 2.3. Simulated moving bed (SMB) is discussed later as a countercurrent separation process. In Figure 2.3, the left-hand side represents the column after introduction of the feed and during the

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2 Introduction to Protein Chromatography

(a)

A+B

Isocratic

M

M M

A

B Time or volume

B+M M

Gradient M

A+M M B

A

Time or volume

(b)

A+B

A+B

A+B

B

CF,B A CF,A

A M Time or volume

(c)

A+B

D

D

CD

D B

B A

A

M Time or volume

Figure 2.3 The three classical chromatographic methods according to Tiselius: (a) elution chromatography; (b) frontal analysis; (c) displacement development. The left-hand side represents the column immediately after introduction of the feed

and during the course of the separation. The right-hand side shows the corresponding efﬂuent concentration proﬁles. A and B are the feed components. M is the mobile phase modiﬁer and D the displacer. Adapted from [2].

2.3 Modes of Operation

course of the separation. The right-hand-side shows the corresponding efﬂuent concentration proﬁles. As shown, elution, frontal analysis and displacement development might seem unrelated at ﬁrst sight since they yield very different results. For instance, isocratic elution is traditionally associated with a dilution of the separated products, frontal chromatography only allows the recovery of a single pure component, and displacement may under certain conditions, yield products which are both separated and concentrated. These differences are however, much less striking from other points of view. First, all three modes share the same operational features: they are carried out with columns and use the same equipment. Second, all three can frequently be implemented with the same stationary phase. Third, the same equilibrium and dispersive factors affect each, although the relative importance of each varies. Finally, the same engineering models and mathematical analyses may describe all three operations. The salient features of each operation are discussed below. In all three techniques, the ﬁrst step is to supply the feed to be separated to the column. The process that follows results in different types of separations and is determined by the properties of the mobile phase modiﬁer. 2.3.1 Elution Chromatography

In elution chromatography, the mobile phase modiﬁer that follows the feed has an afﬁnity for the stationary phase that is lower than that of any of the feed components. Thus, the modiﬁer travels ahead of the feed components which migrate through the column at a rate that is dependent on their afﬁnity for the adsorbent at the modiﬁer concentration that prevails in the column. There are two variations of elution chromatography: isocratic elution, where the modiﬁer concentration remains constant throughout the separation; and gradient elution, where the concentration of the modiﬁer is varied over time. In either case, the less strongly adsorbed modiﬁer travels ahead of the feed components. Thus, their interactions with the stationary phase are affected by the local concentration of the modiﬁer in the mobile phase at each point in the column and at each point in time. Isocratic elution is often the method of choice for analysis. In this case very small feed injections are used and the amount of material adsorbed is very small compared with the adsorbent capacity. The adsorption equilibrium is thus essentially linear (e.g. follows the Henry’s law limit). As a result, each component travels through the column independently of other species. This behavior allows the use of pure component standards in quantitative analysis and permits each component to be uniquely associated with a retention time. On the other hand, since linearity of the adsorption equilibrium is restricted to low loadings of the stationary phase, usage of the adsorbent capacity can be inefﬁcient. At higher loadings, the isotherm becomes non-linear, retention becomes concentration dependent, and the adsorbed components interact with each other. As a result, the retention time is not a constant even when the composition of the mobile phase is ﬁxed and the ensuing chromatographic peaks are skewed by the non-linear nature of the adsorption equilibrium.

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An additional source of inefﬁciency of isocratic elution is encountered when the feed mixture contains components with widely differing adsorptivities. In this case, weakly retained species are eluted prematurely with little separation, while strongly adsorbed species are eluted very late with the consumption of large volumes of mobile phase and long separation times. A ﬁnal issue is the case where separation is very sensitive to the precise composition of the mobile phase. This is especially true for biopolymers where chromatographic retention is often an extremely strong function of mobile phase composition. In these cases, the robustness of isocratic elution is limited. Despite these inefﬁciencies however, the technique is also still frequently used in preparative applications, especially for low molecular weight compounds. On the other hand, gradient elution is more suitable for the separation of components which have widely different afﬁnity for the adsorbent and in cases where the separation is highly sensitive to the mobile phase composition. In this case, the elution time of the more strongly retained species can be reduced with an appropriate variation in the concentration of the mobile phase modiﬁer. Since band spreading is directly dependent on the residence time, product dilution is reduced and productivity is increased. In addition, in the case of biopolymer chromatography, gradient or stepwise elution (sometimes called ‘step gradient’) is the only practically feasible elution method. These molecules are frequently found to be adsorbed very strongly over a certain range of mobile phase composition and completely unretained elsewhere, with the transition often occurring over an extremely narrow range making it virtually impossible to obtain reproducible isocratic separations. By operating in gradient mode, the column becomes exposed to a range of mobile phase compositions that bracket adsorbed and desorbed states eliminating the requirements for extremely precise control of the modiﬁer concentration. Accordingly, the separation is not monitored as a function time; rather it is timed relative to the modiﬁer concentration emerging from the column. One of the characteristics of elution chromatography, whether isocratic or gradient, is that for a given stationary phase and mobile phase ﬂow rate, the separation generally increases with column length. However, the separated products are increasingly diluted as the column is lengthened. 2.3.2 Frontal Analysis

In frontal analysis the feed mixture dissolved in an appropriate mobile phase is continuously supplied to the column for conditions where the mixture components are strongly and sometimes competitively adsorbed. In this case, the feed solutes are partially separated in a series of fronts, with the least strongly adsorbed species forming a pure component band. The partially separated bands are followed upstream by the advancing feed front, which eventually saturates the column. The technique is best suited to the removal of strongly adsorbed impurities from an unretained product of interest. In this case a large amount of feed can be processed before the impurities begin to break through from the column.

2.3 Modes of Operation

When this point is reached, the bed is washed to remove the desired product from the interstitial voids, and the adsorbent is regenerated. The method can only provide a single component in pure form, but avoids product dilution almost completely. In fact, as a result of competitive binding, the more weakly adsorbed component can actually be concentrated downstream of the feed front. This can be advantageous, but can also result in operational difﬁculties when the product is prone to aggregation or has otherwise limited solubility. Since frontal analysis occurs during the feed loading step on any chromatographic mode, awareness of the possibility of this concentration effect is critical. Although frontal analysis can only give a single pure component, multi-component separations can be obtained with multiple processing steps in the form of either a series of frontal chromatography steps, or a combination of elution and displacement separations, or other separation techniques. In either case, an efﬁcient utilization of the adsorbent can be obtained. Chromatographic capture is, of course, a type of frontal analysis. Here removal of the product from the liquid solvent is the principal concern, rather than separation of feed components. Nonetheless, even during chromatographic capture some level of separation can occur. First, the captured product is separated from unretained impurities that ﬂow through the column unadsorbed. Secondly, related feed impurities can be partially resolved as a result of competitive binding. 2.3.3 Displacement Chromatography

In displacement chromatography the column is partially loaded with the feed mixture as in frontal chromatography. The feed supply is then stopped and replaced with a constant supply of a mobile phase containing a modiﬁer species that is adsorbed more strongly than any of the feed components. In this case the term displacer is used instead of modiﬁer. For these conditions, advancement of the displacer front through the column results in desorption of the feed components and their concentration downstream of the displacer front. In turn, the displaced components are competitively readsorbed downstream of the displacer front. As in frontal analysis, the less strongly adsorbed species tend to migrate faster, ahead of the more strongly bound species. In this case, if the chromatographic bed is sufﬁciently long, the feed components eventually become distributed into a pattern of adjacent pure component bands, where each upstream component acts as a displacer for each component immediately downstream. When this occurs, all components, including the displacer, will move through the column at the same velocity with the component bands forming a so-called isotachic pattern or isotachic train. Ideally, the separated species will emerge from the column as adjacent rectangular bands in order of increasing afﬁnity for the adsorbent. In practice, of course, dispersion will be present and some mixing of components between the adjacent bands will occur, requiring some recycling of the mixed bands. With this technique multi-component separation and under certain conditions, simultaneous separation and concentration of compounds can be

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2 Introduction to Protein Chromatography

achieved. Additionally, displacement development is capable of separating compounds that exhibit very little difference in afﬁnity for the adsorbent and allows the use of high feed loadings with an efﬁcient utilization of the adsorbent capacity. Finally, displacement development has the advantage that unknown impurities are often concentrated at the interface between adjacent pure product bands. Thus, separating these species from the fractions collected can be simpliﬁed. Unlike elution chromatography, however, once the isotachic pattern is attained, the separation is not improved further by increasing the column length. In this case, greater resolution can only be attained by reducing the ﬂow rate and reducing mixing or increasing the mass transfer efﬁciency of the stationary phase (e.g. by reducing particle size). Although the principles of displacement development are well known and the technique has been successfully applied to inorganic and organic low-molecular weight separations, its application to biopolymer separation is not straightforward and it is currently conﬁned to laboratory studies. One of the major problems in translating the technique to production-scale applications is associated with the difﬁculties in ﬁnding suitable biopolymer displacers which are non-toxic, available at a reasonable cost, and can be efﬁciently desorbed to allow reuse and recycling of the stationary phase. A practical example is shown in Figure 2.4 for the separation of a mixture of cytochrome c and lysozyme using the cation exchanger SP-Sepharose-FF by linear gradient elution (Figure 2.4a) and displacement development (Figure 2.4b). The same column and ﬂow rate was used in both processes. In the ﬁrst, separation was obtained using a linear gradient in the salt concentration while in the second separation was obtained using a polyelectrolyte (Nalcolyte) as the displacer. Note 60

50

45

(a)

Lysozyme

Conductivity

20

(b)

Lysozyme

40

50

30 30 25 20 20

15 Cytochrome c

30

Displacer (Nalcolyte)

10 20

5 10

Conductivity (mS/cm)

Conductivity

Conductivity (mS/cm)

Cytochrome c

35

Concentration (mg/mL)

40

40

mAU

66

10

15

0

10 0

5

10

15

20

Column volumes

Figure 2.4 Chromatographic separation of a 1 mg/ml cytochrome c/lysozyme mixture using a 1 × 10 cm SP-Sepharose-FF cation exchange column and a mobile phase ﬂow rate of 0.5 ml/min. (a) Analytical separation by linear salt gradient elution (0.1 ml feed volume, 100–500 mM NaCl gradient in 20

0

0 0

5

10

15

20

Column volumes

CV). (b) Preparative separation by displacement development using 10 mg/ml Nalcolyte 8105 as the displacer (225-ml feed volume). Note that in b the conductivity rises ahead of the displacer front because of displacement of sodium ions from the resin [3].

2.3 Modes of Operation

that linear gradient elution also resolves various impurities, which are seen as separate peaks. Although not shown explicitly, these impurities accumulate at the band boundaries in displacement development. As can be seen in Figure 2.4, while the product peaks are greatly diluted in gradient elution, the product bands are highly concentrated in displacement development, about 12 times for the more weakly retained cytochrome c and about 20 times for the more strongly retained lysozyme for this example. 2.3.4 Simulated Moving Bed Separators (SMB)

The general scheme of a countercurrent adsorption process was shown in Figure 2.2a. This scheme is useful for simple capture applications or to remove impurities with a selective adsorbent and can be conveniently implemented with a series of ﬁxed-bed columns as shown in Figure 2.2b. In his case, the bottom bed which receives the feed, approaches saturation ﬁrst. When this occurs, the bed is removed and all other beds in the series are moved up one position in a direction opposite to that of ﬂuid ﬂow. The bed closest to the efﬂuent point is then replaced by a fresh adsorbent bed and the cycle is repeated. Countercurrent ﬂow maximizes the efﬁciency of adsorbent utilization and even a few beds in series effectively approach the ideal true moving bed behavior without the complexities associated with the movement of particles. Binary separations can also be obtained by combining together multiple simulated countercurrent adsorption units in a process known as simulated moving bed (SMB). The SMB concept was originally used in a process developed and licensed by UOP under the name ‘Sorbex’ [4, 5]. Although the Sorbex process was originally applied to hydrocarbon separations, extensive industrial applications have been developed for sugars and amino acids, and, more recently, for ﬁne chemicals, especially chiral separations [6–8]. Interest in applications to biopolymers is increasing [9]. The original Sorbex process was implemented with a single rotary valve to switch the beds in position at periodic time intervals. However, using multiple individual valves to control the distribution of ﬂows is often more practical, especially on a smaller scale. The basic principle of operation is illustrated in Figure 2.5a with reference to an equivalent true countercurrent moving bed (TMB) system, which comprises four countercurrent adsorption columns or ‘zones’. The feed containing components A and B is supplied between zones II and III. The least strongly adsorbed species, A, is recovered in the rafﬁnate stream between zones III and IV, while the more strongly adsorbed species, B, is recovered in the extract stream between zones I and II. The adsorbent is re-circulated from the bottom of zone I to the top of zone IV. A desorbent or eluent make-up stream is added to the ﬂuid recycled from zone IV and the combined stream is fed to the bottom of zone I. The main purpose of each zone is as follows: zone III adsorbs B while letting A pass through, zone II desorbs A while adsorbing B, zone I desorbs B allowing the adsorbent to be recycled, zone IV adsorbs A. Practical implementation is obtained with an SMB

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2 Introduction to Protein Chromatography

(a)

(b) Desorbent

Zone I

Extract (B-rich)

Zone IV Raffinate (A-rich)

Zone III Feed (A+B)

Zone II

Adsorbent recycle

Mobile phase recycle

68

Zone IV

Direction of bed rotation simulated by periodic port switching

Zone II

Extract (B-rich)

Zone I Raffinate (A-rich)

Zone III

Feed (A+B)

Desorbent

Figure 2.5 Schematics of countercurrent adsorption processes for binary separations. (a) True moving bed system (TMB); (b) simulated moving bed system (SMB). A and B are the more weakly and more strongly bound feed components, respectively.

system using multiple ﬁxed beds and a set of valves as shown in Figure 2.5b. Since as few as one bed per zone will often provide a good approach to the ideal behavior of a true moving bed, practical implementation is feasible without excessive mechanical complexity. Conceptually, the process can be understood by the simple analogy shown in Figure 2.6. Here the adsorbent is represented by the conveyor belt and the components to be separated by fast moving hares, representing the weakly adsorbed component A, and slowly moving turtles, representing the more strongly adsorbed species B. If a mixture of hares and turtles is dropped in the middle of the conveyor belt and the belt speed is adjusted so that it is intermediate between the speeds of the hares and turtles, the two will be separated. Thus, in the analogy, zone II is needed to remove any hares that are carried to the left, leaving only turtles while zone III is needed to remove any turtles that are carried to the right, leaving only hares. In the physical system, proper selection of operating conditions to obtain the desired separation is attained by considering adsorption and ﬂuid ﬂow in each of the four zones. Methodologies for the selection of these conditions are discussed in Chapter 10. Beside continuous operation, the advantages of SMB are greater productivity, lower eluent consumption, and less product dilution than is possible with conventional isocratic elution chromatography. These advantages stem from two factors. The ﬁrst is the countercurrent action, which maximizes the driving force for mass transfer and, thereby, the efﬁciency of adsorbent utilization. The second is the internal recycling of partially resolved products. The disadvantages are equipment and operational complexity, limited robustness, which requires delicate control of

2.4 Performance Factors

A+B

B

A

Zone II Figure 2.6 SMB analogy. Hares and turtles represent the more weakly adsorbed (fast moving) and the more strongly adsorbed (slow moving) feed components. The conveyor belt represents the movement of the

Zone III adsorbent. To obtain separation of hares and turtles, the speed of the belt is adjusted to be intermediate between that of the hares and the turtles.

stationary and mobile phase properties, and applicability to binary splits only. Additionally, regulatory concerns are often cited as disadvantages because of the added complexities of SMB system. In general, the advantages will outweigh the disadvantages for very difﬁcult separations or when the cost of adsorbent and eluent is very high. Hybrid systems which integrate elements of classical chromatographic operations with the SMB concept have recently been developed for application to bioseparations. Such systems include gradient SMB, which allows further productivity improvement compared to isocratic SMB, and the multicolumn countercurrent solvent gradient puriﬁcation process (MSGP), which enables three-fraction separations to be carried out by integrating batch gradient and continuous elements in a single multicolumn process [10–13].

2.4 Performance Factors

We shall now consider the factors that affect the design and performance of chromatographic operations. In general, the process will involve interphase transport of solutes and ﬂuid ﬂow through a typically porous granular medium as shown

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2 Introduction to Protein Chromatography

Extraparticle porosity,

Intraparticle porosity, p

Figure 2.7 Schematic of a packed chromatographic columns showing extra-particle and intra-particle porosities.

in Figure 2.7. There are at least three relevant porosities: the extra-particle porosity ε, the intra-particle porosity εp, and the total column porosity, εt. The latter is related to intra-particle and extra-particle porosities by the following equation:

ε t = ε + (1 − ε ) ε p

(2.1)

For randomly packed beds of spherical or irregular particles, ε varies over a relatively narrow range (0.3–0.4 is typical) and is largely independent of size for particles with similar shape and mechanical properties. Much higher ε values are obtained for so-called continuous bed structures, such as monoliths, where the porosity is deﬁned by an organized structure built in situ rather than generated by the random packing of particles. The intra-particle porosity, εp, on the other hand, depends on the particle structure and varies broadly from near zero for pellicular stationary phases that contain no intra-particle pores, to 90% or even higher for low-density gels, such as agarose or cross-linked dextran. In general, only a fraction of the intra-particle porosity is accessible by macromolecular solutes and this fraction depends on the ratio of molecule and pore sizes. Thus, the accessible intraparticle porosity is sometimes deﬁned for each solute. Chapter 3 provides an overview of the available stationary phases and their properties. In general, for typical chromatography columns ﬂow of the mobile phase is largely restricted to the extra-particle porosity. Two velocities can thus be deﬁned: the superﬁcial velocity, u, which is equal to the volumetric ﬂow rate divided by the cross-sectional area of the column and the interstitial velocity, v. The latter is related to u by the equation u v= (2.2) ε and represents the average ﬂuid velocity in the extra-particle space. In most cases, for columns used in bioprocess applications, ﬂow conditions are laminar. Thus,

2.4 Performance Factors

the Reynolds number, deﬁned by Re = ρudp/η where dp is the particle diameter and ρ and η are the density and viscosity of the mobile phase, respectively, is always much lower than 1 [14]. In most practical cases, ﬂuid ﬂow is typically absent within the intra-particle pores, which are normally much smaller than the extra-particle space. Thus, solute transport within the particles is normally dependent on diffusion. There are exceptions however, such as in cases where intra-particle convection or perfusion, which is driven by the pressure gradient along the column axis, occurs to sufﬁcient extent to inﬂuence the overall rate of mass transfer within the particles. In practice, this requires very large intra-particle pores (>100 nm), small particle sizes (<20 µm) and high ﬂow rates of the mobile phase. Chapter 6 provides a quantitative analysis of intra-particle convection and its effects on intra-particle transport. The separation performance of a chromatographic column will, in general, depend on factors that can be broadly grouped in two categories: equilibrium and dispersive factors. Equilibrium factors include adsorption equilibrium, which determines the distribution between stationary and mobile phases, and any other process that occurs on time scales that are short compared to the time scale of chromatographic separation. Examples of the latter include ionic dissociation and, in some cases, intermolecular association. Dispersive factors, sometimes also referred to as ‘rate factors’, are discussed in detail in Chapter 6 and their effects on process performance in Chapter 8. These factors include mass transfer resistances, kinetic resistance to binding, and mobile phase dispersion effects. The latter include axial diffusion, hydrodynamic dispersion, and non-uniform ﬂow. Aside from these factors which are dependent on the solutes being separated, the adsorbent material and the column hardware, effects external to the column must also be considered, including mixing devices, bubble traps, valves, detectors and monitors, and product collection systems. Equilibrium factors usually determine the best achievable separation and are generally associated with the term ideal chromatography, which is disscussed in Chapter 7. This limiting behavior is approximated by using very small adsorbent particles and very low ﬂow rates, but is seldom attained in practice for process-scale operations, especially with biopolymers, for which dispersive factors are usually very important. These factors determine the difference between ‘ideal’ and ‘real’ chromatographic performance and will typically force a compromise of separation efﬁciency and productivity for economically optimum designs. Figure 2.8 illustrates the location of the relevant dispersive factors and their effects on a dilute isocratic elution separation as an example. In this case, under ideal conditions, that is, with uniform ﬂow and no mass transfer or kinetic resistance to binding, and no axial diffusion or dispersion, the product bands will have the same width and height as the feed. Dispersive factors will broaden these bands causing dilution and loss of resolution but without generally affecting the average retention times. The latter are determined primarily by equilibrium factors, which, in turn, are dependent on thermodynamics and are thus unaffected by particle size or ﬂuid ﬂow.

71

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2 Introduction to Protein Chromatography

A+B

Axial diffusion and hydrodynamic dispersion Mass transfer and binding kinetics

Time

Flow nonuniformity

A

B

“Ideal” “Real”

Time

Figure 2.8 Location and effects of dispersive factors contributing to band broadening in chromatographic separations illustrated for the case of isocratic elution. Under dilute

ideal conditions the product bands will have the same width and height as the feed. Dispersive factors broaden the bands causing dilution and loss of resolution.

Non-uniformity of ﬂow generally occurs at the column level while mass transfer and axial dispersion effects occur at the particle level. Thus, the former is more difﬁcult to predict on scale-up while the latter can often be predicted accurately based on experiments using laboratory-scale columns. Mass transfer is mainly associated with resistance to diffusional transport in the stationary phase. Thus, it depends heavily on the intra-particle structure and the mechanism of diffusion as well as the molecular size of the adsorbed species and the viscosity of the mobile phase. Axial dispersion is the result of two contributions: axial diffusion and hydrodynamic dispersion. The ﬁrst is associated with axial solute concentration gradients and is usually unimportant in most types of liquid chromatography. The second is associated with the splitting and mixing of streamlines in the extraparticle space and can be signiﬁcant depending on the operating conditions. There are multiple causes of non-uniformity of ﬂow in both process- and laboratory-scale columns including poor column header design, lack of homogeneity in the density of the packing material, by-pass along the column wall, and viscous ﬁngering. Column design and packing of the stationary phase are discussed in Chapter 4. Viscous ﬁngering occurs as a result of viscosity gradients in the axial column direction. This occurs, for example, when a viscous feed slug is followed by a less viscous mobile phase causing hydrodynamic instability which results in component bands having a sharp front followed by a jagged pattern with solute ﬁngers protruding rearward. The opposite behavior occurs when a less viscous

2.4 Performance Factors

solution is followed by a more viscous ﬂuid. Density gradients can also inﬂuence viscous ﬁngering, but these effects are usually quite small. In practice, the effects of viscous ﬁngering are more pronounced at lower ﬂow rates and with more viscous solutions, which are often encountered in SEC. In this case, low ﬂow rates and higher protein concentrations are often required to achieve separation and reasonable productivity. Viscous ﬁngering effects are also important during protein desorption from a high capacity adsorbent particularly where this is carried out at low ﬂow rates and as a result of a rapid change in the composition of the mobile phase to cause immediate desorption. A practical example is shown in Figure 2.9. In this case, a high capacity anion exchange

(a)

(b)

Figure 2.9 Magnetic resonance images of ﬂow patterns in a 1 × 8 cm UNOsphere Q anion exchange column from Bio-Rad Laboratories operated at a ﬂow rate of 38 cm/h. The color scale represents the relative concentration of the magnetic contrast agent with red being the most concentrated and blue being the least and corresponds approximately to the salt proﬁle in the column. (a) Protein-free column

(c)

subjected to a 500-mM step change in salt concentration. (b) Albumin-saturated column subjected to a 500-mM salt step after 4 ml. (c) Same as in (b) but after 6 ml. Top and bottom images represent vertical and horizontal cross-sections, respectively, the latter at the position indicated by the white line. Note that ﬂow is almost uniform in the absence of protein and highly unstable when the protein is present (source: ref. [15]).

73

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2 Introduction to Protein Chromatography

column saturated with albumin was eluted with a salt step. Internal ﬂow proﬁles were visualized by magnetic resonance imaging using MagneVist which is a magnetic contrast agent that is added to the salt eluent. Since MagneVist is a small molecule that is not adsorbed, it highlights the shape of the salt front in the column. In the absence of protein, the ﬂow is fairly uniform and stable (Figure 2.9a). However, when the high concentration salt solution enters the proteinsaturated column, the desorbed protein accumulates in the mobile phase in very high concentration as a result of the high adsorption capacity of the resin, thereby generating a large viscosity gradient in the direction of ﬂow. The conditions shown in Figure 2.9b and c are produced by an unstable and highly non-uniform ﬂow of the mobile phase. In turn, this causes tailing in the protein elution peak. These effects can be moderated by eluting the protein with a salt gradient, rather than a step, or eluting at a lower salt concentration to produce more gradual desorption.

2.5 Separation Performance Metrics 2.5.1 Column Efﬁciency

The efﬁciency of a chromatographic column is deﬁned as the degree to which ideal chromatography conditions are approached. This is conveniently expressed in terms of the height equivalent to a theoretical plate (HETP), H, or the corresponding plate number, N. The two quantities are related to each other by the equation: N=

L H

(2.3)

where L is the column length. The dispersive factors discussed in the previous section collectively determine the magnitude of H and N for a given column. Accordingly, ideal behavior is approached when H → 0 or N → ∞. Since the dispersive factors are in part dependent on the molecular properties of the solute, the HETP is generally different for different components in the same column and varies with ﬂow rate. The HETP is also dependent on equilibrium factors, since interactions with the stationary phase affect band broadening. However, this dependence is usually less signiﬁcant so that the HETP principally measures the inﬂuence of dispersive factors on chromatographic performance relative to the components of interest and the deﬁned operating conditions. The experimental measurement and detailed prediction of the HETP for a given column are discussed in detail in Chapter 8. However, there is a general empirical relationship for columns where ﬂow is substantially uniform and intra-particle mass transfer occurs by ordinary pore diffusion. In this case the relationship is expressed in terms of the reduced HETP, h, and the reduced velocity, v′, and is given in [16, 17] by the following equations:

2.5 Separation Performance Metrics

h=

b + a ( v ′ )0.33 + cv ′ v′

(2.4)

for porous particles and by: h=

b + a ( v ′ )0.33 + c ( v ′ )0.66 v′

(2.5)

for pellicular stationary phases. This relationship is sometimes referred to as the generalized van Deemter curve. The reduced quantities in these expressions are deﬁned as follows: h=

H dp

(2.6)

v′ =

vdp D0

(2.7)

and

where dp is the particle diameter, v is the interstitial velocity, and D0 is the solute molecular diffusivity in the mobile phase. As an approximation, the parameters b, a, and c in Equations 2.4 and 2.5 can be considered to be universal constants, independent of the solute, mobile phase, and stationary phase. The ﬁrst of these parameters takes into account axial diffusion, the second hydrodynamic dispersion, and the third mass transfer effects. Typical values are b = 2, a = 1, and c = 0.05, but these values are only rough approximations and should be used only for preliminary estimations. In practice, these values vary by a factor of 2 or more. Thus, precise estimates require detailed considerations of the axial dispersion and transport mechanisms as discussed in Chapters 6 and 8. The general trends predicted by Equations 2.4 and 2.5 are shown in Figure 2.10 along with the approximate ranges of reduced velocities normally encountered in various types of chromatography. The dominant dispersive factor in each range is also shown. It can be seen that the three major contributions to the HETP are important over different ranges of reduced velocities. In turn, since v′ is a function of molecular diffusivity, the HETP and hence the column efﬁciency, will be different for different solutes in the same column and, in general, will be dependent on the ﬂow rate. The relevant ranges of reduced velocities for each type of chromatography depend on the combination of allowable ﬂow velocities, particle sizes, and molecular properties that are characteristic of each. These ranges and the dominant contributions to the reduced HETP for each are summarized in Table 2.2. As seen in Figure 2.10 and Table 2.2, axial diffusion is typically dominant in gas chromatography (GC). Thus, in this case the column efﬁciency is expected to increase as the ﬂow rate increases. On the other hand, for small molecules and high performance liquid chromatography (HPLC), the dominant contribution is hydrodynamic dispersion. The latter is relatively insensitive to ﬂow rate and molecular diffusivity (note the 0.33 power dependence on v′ in Equations 2.4 and 2.5).

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2 Introduction to Protein Chromatography

1000

Reduced HETP, h=H/d p

76

Range for GC

Range for HPLC of small molecules

Range for macromolecules

Mass transfer controlling

100 Axial diffusion controlling

10

Porous particles Hydrodynamic dispersion controlling

1 0.1

1

10

Pellicular

100

1000

10000

Reduced velocity, v'=vd p/D0 Figure 2.10 Generalized van Deemter curves for well-packed columns characterized by uniform ﬂow. The curve for porous particles is valid for pore diffusion while the curve for pellicular particles assumes external resistance only. The dominant contribution to reduced HETP and the approximate range of

reduced velocities encountered in practical applications are shown. The curves are used for rough estimates only. Precise estimates require detailed consideration of axial dispersion and mass transfer mechanisms. Adapted from [18].

Table 2.2 Practical ranges of reduced velocities and dominant dispersion factors for different types of chromatography.

Type

Range of v′

Dominant dispersive factor

Gas chromatography (GC) Small molecules and analytical HPLC Macromolecules in monoliths Macromolecules in porous particles

0.1–1 1–100 1–10 100–10 000

Axial diffusion (b-term) Hydrodynamic dispersion (a-term) Hydrodynamic dispersion (a-term) Mass transfer (c-term)

Thus, for these conditions, different solutes will exhibit similar HETP and the column efﬁciency will be relatively insensitive to ﬂow. Finally, in the case of proteins and other macromolecules, mass transfer in the stationary phase is typically dominant due to the porous particles used in preparative and process scale applications. For these conditions, in dimensional form, Equation 2.4 yields:

2.5 Separation Performance Metrics

H ≈c

vd 2p D0

(2.8)

Thus for preparative/process protein chromatography the HETP can be expected to increase linearly with ﬂow rate, to increase with the square of the particle size, and, in the case of porous particles with suitably large pores, to decrease in proportion to the protein molecular diffusivity. Example 2.1 provides a quantitative illustration.

Example 2.1 equation

Estimation of HETP from the generalized van Deemter

Case 1. Estimate H for gas chromatography with dp = 100 µm, v = 1 cm/s. Solution – Gas phase diffusivities at room temperature and 1 bar pressure are typically around 0.1 cm2/s [14]. Thus, v′ = 1 × 0.01/0.1 = 0.1 and h = 2/0.1 + 0.10.33 + 0.05 × 0.1 = 20 + 0.47 + 0.005 ∼ 20 or H = 0.20 cm, showing nearly complete axial diffusion control. Case 2. Estimate H for HPLC of small molecules with dp = 5 µm, v = 0.1 cm/s. Solution – Liquid phase diffusivities of small molecules in non-viscous solvents at room temperature are typically around 1 × 10−5 cm2/s [14]. Thus, v′ = 0.1 × 0.0005/1 × 10−5 = 5 and h = 2/5 + 50.33 + 0.05 × 5 = 0.4 + 1.7 + 0.25 = 2.4 or H = 0.0012 cm, showing nearly complete hydrodynamic dispersion control. Case 3. Estimate H for process chromatography of a 150-kDa molecular mass protein with dp = 100 µm, v = 0.1 cm/s. Solution – In this case, assuming dilute conditions with mobile phase viscosity of 1 mPa·s (1 cp), D0 ∼ 4 × 10−7 cm2/s (cf. Figure 1.26 and [19]), v′ = 0.1 × 0.01/4 × 10−7 = 2500 and h = 2/2500 + 25000.33 + 0.05 × 2500 = 0.0008 + 13 + 125 = 138 or H = 1.4 cm, showing nearly complete mass transfer control. Note that under the same conditions for a small molecule such as salt, D0 ∼ 1 × 10−5 cm2/s, v′ = 100, and h = 0.02 + 4.6 + 5 = 9.6, showing a much greater effect of hydrodynamic dispersion.

For all three cases in Example 2.1, we have assumed that the columns are well packed and with substantially uniform ﬂow. Accordingly, the HETP values estimated are predicted to be independent of column length. However, this is not necessarily the case for columns where large deviations from plug ﬂow occur with correspondingly greater values of the a-term [20–22]. Additionally, when mass transfer is dominant, the speciﬁc characteristics of the stationary phase can dramatically affect column efﬁciency. For example, for protein mass transfer in small pores, hindered or restricted diffusion can occur, severely reducing the rate

77

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2 Introduction to Protein Chromatography

of mass transfer, thereby resulting in a much larger c-term than assumed. In other cases, where intra-particle mass transfer is enhanced by other transport mechanisms, the c-term may be smaller. Nonetheless, despite these limitations, Equations 2.4 and 2.5 are useful. They deﬁne the critical parameters that need to be controlled and can be varied to attain the desired efﬁciency. Additionally, they provide the basis for the rational scale-up of chromatography columns that meet speciﬁed pressure constraints. 2.5.2 Chromatographic Resolution

A common measure of the chromatographic separation achieved between two components is the chromatographic resolution, Rs. A more general deﬁnition is given in Chapter 10. The following is limited to the case of elution chromatography under trace conditions and with small pulse feed injections, but is sufﬁcient to address many practical cases. With reference to Figure 2.11, the chromatographic resolution between A and B, where B is the more strongly retained component, is deﬁned as [16]: Rs =

tR ,B − tR , A 1 (WA + WB ) 2

(2.9)

tR,A and tR,B are the elution times of the two components and WA and WB are the corresponding baseline widths. Since, for a constant volumetric ﬂow rate, time and tR,B - tR,A

A

B

WA

WB

Time

Figure 2.11 Chromatographic parameters the midpoints of the peaks to the baseline. used for the calculation of resolution of two The calculated resolution is Rs ∼ 1 for the case adjacent peaks. The baseline widths WA and shown in this ﬁgure. WB are measured by extending the tangents to

2.5 Separation Performance Metrics

eluted volume are proportional to each other, the same calculation can be used done using volume-based rather than time-based quantities. Generally, the two components are completely separated when Rs ∼ 1.5. In practice, Rs = 1, corresponding to approximately 98% separation (see Figure 2.11) is considered adequate for many purposes. Lower values in concert with recycling strategies may, however, be the economical optimum. For the special case of isocractic elution with a linear isotherm, that is, in the limit where partitioning between phases is described by Henry’s law, the elution time of each component is given by: L tR ,i = (1 + ki′) v

(2.10)

where ki′ = (1 − ε ) mi ε is the retention factor and mi is the distribution coefﬁcient of component i. Since mi depends only on thermodynamics and mobile phase composition, ki′ is expected to be independent of ﬂow. In turn, Rs is related to column efﬁciency by the following approximate relationship: Rs =

1 α − 1 k′ L 1 α − 1 k′ = N 2 α + 1 1 + k′ H 2 α + 1 1 + k′

(2.11)

α=

kB′ k A′

(2.12)

where:

is the selectivity (equal to the ratio of the Henry’s law isotherm slopes) and k′ =

k A′ + kB′ 2

(2.13)

is the average retention factor. Equation 2.11 is approximately valid only when A and B are closely related species, but shows that, in general, the separation can be affected in three ways. Firstly, it is affected by the selectivity. When α increases the resolution increases with the largest percentage changes occurring for values of α close to 1. Secondly, the resolution is affected by the average retention factor. At large k ′ values, however, the term k ′ (1 + k ′ ) approaches unity, so that the effect of this variable will eventually disappear. Thirdly, the resolution is affected by the number of plates or, equivalently, for a given column length, by the HETP. Rs increases in proportion to N . Peak spreading, however, also varies in proportion to N , so that increasing peak separation will generally result in greater dilution. The separation time, ts, is also affected and is given approximately by the equation:

(

L 2  L 2  t s = (1 + kB′ )  1 +  = (1 + kB′ ) 1 +  v L H v N

)

(2.14)

Example 2.2 illustrates simple applications of these equations to obtain useful scaling relationships.

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2 Introduction to Protein Chromatography

Example 2.2 Estimation of the plate number and column length requirements Consider an isocratic elution separation of two proteins A and B each of ca. 150 kDa molecular mass and k A′ = 2 and kB′ = 4, respectively. What are the plate number and the column length requirement to yield Rs = 1 with small injections under trace conditions if the particle size is 100 µm and the interstitial velocity is 0.1 cm/s? Solution – For the conditions in this example, Equations 2.11, 2.12 and 2.13 yield the following values: k′ =

2+4 4 = 3, α = = 2 2 2 2

(

)

2 + 1 1+ 3 2 α + 1 1+ k ′   N =  2Rs = 2 × 1 = 64  2−1 3 α − 1 k ′  For the conditions of Example 2.2, we obtain v′ = 2500 and H is estimated to be 1.4 cm (cf. Example 2.1, Case 3). Thus, the required column length is 2 90 (1+ 4) 1+ = 5625 s. L = 64 × 1.4 = 90 cm, and the separation time is ts = 0.1 64 A shorter column would require smaller particles or lower ﬂow rates.

(

)

2.5.3 Dynamic Binding Capacity

The performance of packed columns used in capture applications with a selective adsorbent is frequently characterized by the dynamic binding capacity (DBC). The DBC is deﬁned for frontal analysis as the amount of protein held in the column before the efﬂuent concentration reaches a speciﬁed percentage of the protein feed concentration, CF. A value of 10% for CF is typical and the DBC is usually normalized by the column volume, Vc. If the protein leakage prior to breakthrough is neglected, the DBC can be expressed by the following equation: DBC10% =

CFQtb,10% CFVb,10% = Vc Vc

(2.15)

where Q is the volumetric ﬂow rate of the protein load and tb,10% and Vb,10% are the breakthrough times and load volumes at which the efﬂuent reaches 10% of CF, respectively. The DBC is generally related to the equilibrium binding capacity, EBC, but is inﬂuenced by dispersive factors, approaching the EBC only for conditions where the column has inﬁnite efﬁciency. The EBC, also referred to as the static capacity, depends only on thermodynamics and protein feed concentration and is thus expected to be independent of ﬂow and column length. The DBC, on the other hand, will vary with column efﬁciency and length.

2.5 Separation Performance Metrics

In general, the relationship between DBC and EBC is complicated and is dependent on the exact nature of the dispersive mechanisms. However, in cases where protein binding is very favorable and the binding capacity is high compared to the protein concentration in solution, this relationship is given approximately by a unique function of the plate number, or:

( )

DBC L = f (N ) = f EBC H

(2.16)

The exact functional forms of this dependence, appropriate for different dispersive mechanisms are discussed in Chapter 8. However, even without knowledge of the exact functional dependence, this relationship is useful and provides a practical criterion for the design of equivalent columns based on scaling arguments. For proteins and other biopolymers under conditions suitable for preparative and process scale applications, mass transfer is usually the controlling dispersive factor. Thus, H increases linearly with ﬂow rate. The impact of DBC on productivity is discussed in Chapter 10. 2.5.4 Scaling Relationships

General scaling relationships suitable for empirically-guided design and scale-up of protein chromatography columns can be deduced based on the efﬁciency and separation performance criteria deﬁned in the previous sections. A general treatment of optimum column design is presented in Chapter 10. The following is limited to isocratic elution under trace conditions and capture with a selective adsorbent. Gradient elution separations can be scaled in a similar manner with the gradient slope as an added degree of freedom as discussed in Chapter 9. As seen above, for these conditions, both resolution and dynamic binding capacity can be kept constant by keeping the plate number constant. In the case of protein chromatography, v′ is large and H is determined almost completely by intra-particle mass transfer. In this case, scaling at constant resolution and dynamic binding capacity requires that we keep D0L = constant d 2pu

(2.17)

In general, the problem of scaling chromatography columns for these purposes is subject to constraints that include: (i) the allowable column pressure, and (ii) the desired separation time. The column pressure can be quite large for laboratoryscale applications, even if glass columns are used. However, at the process scale, bioprocess columns are often limited to only a few bars of pressure (see Chapter 4). For the laminar ﬂow conditions encountered in these columns, the pressure is related to ﬂow by Darcy’s law. Accordingly, we have: ∆P =

ηuL B0

(2.18)

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2 Introduction to Protein Chromatography

where η is the liquid viscosity and B0 is the hydraulic permeability. For rigid stationary phases, B0 is a constant independent of ﬂow but typically proportional to the square of the particle diameter. Thus, ∆P ∝

uL d 2p

(2.19)

For these conditions, scaling at constant pressure requires:

ηuL = constant d 2p

(2.20)

Note that according to Equation 2.14, the separation time will change for constant L/H in proportion to the terms L/v. Example 2.3 demonstrates applications of these scaling relationships.

Example 2.3 Scaling protein chromatography to meet the constraints of pressure and separation time Case 1. The design of a process-scale column based on the linear scale up of laboratory experiments results in an estimated pressure that is too high for the available column hardware. Change the column aspect ratio to attain the same separation at lower pressure with the same throughput. Solution – Increasing the column diameter by a factor of 2, reduces the superﬁcial velocity u by a factor of 4. From Equation 2.17, maintaining resolution and DBC requires that L is reduced by a factor 4. In turn, based on Equation 2.20, the column pressure is reduced by a factor of 16. Column volume and separation time remain the same. Case 2. The design of a process-scale column based on the linear scale up of laboratory experiments results in a separation time that is too long and results in unacceptable product degradation. Switch to smaller particles maintaining the same column pressure and throughput. Solution – Based on Equation 2.17, decreasing the particle diameter by a factor of 2 reduces the column length, L, by a factor of 4. In turn, if the column is operated at the same velocity and has the same diameter, based on Equation 2.20, the column pressure remains the same but the column volume and the separation time are reduced by a factor 4. The resolution and DBC remain the same if the properties of the smaller particles are the same. Case 3. It is desired to scale up a chromatographic separation obtained with a laboratory column and a ﬂow rate of 2 ml/min to a column operated at a ﬂow rate of 2 L/min while maintaining the same resolution, DBC, and pressure as the laboratory column. Laboratory scale conditions were: dp = 50 µm, column diameter dc = 1 cm, column length L = 10 cm. The larger scale separation will use 90 µm diameter particles.

References

Solution – The superﬁcial velocity in the laboratory scale column is 2.6 cm/min. From Equations 2.17 and 2.20, constant resolution, DBC, and pressure can be maintained on scale-up if: ulab = uprocess Llab  d p,lab  = Lprocess  d p,process 

2

Using these relationships yields the following values for the process-scale column: diameter = 40.3 cm, length, 32.4 cm. The separation time is increased by a factor of 32.4/10 = 3.24.

It must be noted that the use of the scaling relationships illustrated in Example 2.3 assumes that adsorption equilibrium and mass transfer mechanisms remain the same as a function of particle size. This will occur when the chemistry and structure of the stationary phase do not change. We also assumed that the particles are rigid. The design of compressible beds is discussed in Chapter 10 and is complicated by the fact that column packing and hydraulic permeability will vary with column diameter. Temperature and viscosity have also been assumed to be constant in the scale-up. Temperature can obviously affect chromatographic retention and EBC, and both temperature and viscosity affect D0 and column pressure. Thus, care must be taken to address these potential variations. Sometimes process development is carried out on the laboratory scale at room temperature, while the production process is undertaken at lower temperatures. This can result in lower column efﬁciencies and higher operating pressures than expected. For protein chromatography, both factors can be compensated for by using the general scaling relationship discussed above. A further consideration is the availability of columns that can be packed reliably with large diameter-to-length aspect ratios, which might be required to meet pressure constraints. In practice, column depths smaller than 5–10 cm become impractical in large diameter columns and can result in substantial non-uniformity of ﬂow thus invalidating the underlying assumption that dispersion is controlled by mass transfer. Finally, it should be noted that these relationships are only suitable for preliminary estimates. In practice, experimental measurements on an intermediate scale are usually needed to determine ﬂow properties and compressibility of the stationary phase and validate the results prior to the ﬁnal design.

References 1 Tiselius, A. (1943) Kolloid Z., 105, 101. 2 LeVan, M.D., and Carta, G. (2007) Adsorption and ion exchange, in Perry’s Chemical Engineers’ Handbook, Section

16, 8th edn (ed. D.W. Green), McGraw-Hill, New York. 3 Bloomingburg, G.F. (1992) Separation of proteins by continuous annular chromatography. PhD Dissertation.

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2 Introduction to Protein Chromatography

4 5 6

7

8

9 10

11 12

13 14

University of Virginia, Charlottesville, Virginia, USA. Broughton, D.B., and Gerhold, C.G. (1961) US Patent No. 2,985,589. Broughton, D.B. (1985) Sep. Sci. Technol., 19, 723. Broughton, D.B., Neuzil, R.W., Pharis, J.M., and Brearley, C.S. (1970) Chem. Eng. Progr., 66, 70. Juza, M., Mazzotti, M., and Morbidelli, M. (2000) Trends Biotechnol., 18, 108. Kinkel, J.N., Schulte, M., Nicoud, R., and Charton, F. (1995) Proceedings of Chiral Europe ‘95 Symposium, Spring Innovation Limited, Stockport, UK. Gottschlich, N., and Kasche, V. (1997) J. Chromatogr. A, 765, 201. Ludenmann-Hombourger, O., Nicoud, R., and Bailly, M. (2000) Sep. Sci. Technol., 35, 1829. Li, P., Xiu, G., and Rodrigues, A.E. (2007) AIChE J., 53, 2419. Strohlein, G., Aumann, L., Mazzotti, M., and Morbidelli, M. (2006) J. Chromatogr. A, 1126, 338. Aumann, L., and Morbidelli, M. (2007) Biotechnol. Bioeng., 98, 1043. Bird, R.B., Stewart, W.E., and Lightfoot, E.N. (1960) Transport Phenomena, John Wiley & Sons, Inc., New York.

15 Hunter, A.K. (2002) Protein Mass Transfer in Heterogeneous Gels and Chromatography Media. PhD Dissertation. University of Virginia, Charlottesville, Virginia, USA. 16 Giddings, J.C. (1965) Dynamics of Chromatography, Part I – Principles and Theory, Marcel Dekker, New York. 17 Snyder, L.R., and Kirkland, J.J. (1979) Introduction to Modern Liquid Chromatography, John Wiley & Sons, Inc., New York. 18 Carta, G., Ubiera, A.R., and Pabst, T.M. (2005) Chem. Eng. Technol., 28, 1252. 19 Tyn, Y., and Gusek, T.W. (1990) Biotechnol. Bioeng., 35, 327. 20 Nicoud, R.M., and Perrut, M. (1991) Hydrodynamics of chromatography columns, in Chromatographic and Membrane Processes in Biotechnology (eds C.A. Costa and J.S. Cabral), Kluwer Academic Publishers, Dordrecht, The Netherlands, pp. 53–83. 21 Miyabe, K., and Guiochon, G. (1999) J. Chromatogr. A, 857, 69. 22 Broyles, B.S., Shalliker, R.A., and Guiochon, G. (2001) J. Chromatogr. A, 917, 1.

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3 Chromatography Media 3.1 Introduction

A number of different interaction principles are employed for chromatography of biomolecules as indicated in Table 2.1. These interactions include size exclusion chromatography (SEC), also known as gel ﬁltration or gel permeation chromatography, normal phase chromatography (NPC), reversed phase chromatography (RPC), hydrophobic interaction chromatography (HIC), ion-exchange chromatography (IEC), metal chelate chromatography (MIC), and biospeciﬁc interaction chromatography (BIC). The chemistry of the stationary phase also determines the nature of the mobile phase. As a result, optimization of both phases is required. In addition the optimization of protein chromatography is constrained by the limited stability of proteins in certain environments, such as extreme pH, salt conditions or in certain organic solvents. The principle governing SEC is molecular size, which results in partial exclusion from the pores. NPC employs a stationary phase that is more hydrophilic than the mobile phase, which is typically a mixture of organic solvents. Thus, application of NPC to biomolecules is limited to peptides and other small molecules that exhibit solubility in such solvents. In contrast, RPC employs a stationary phase that is more hydrophobic than the mobile phase. The latter consists of a hydroorganic mixture in which certain proteins are soluble and stable. Retention in RPC is achieved through discrete interactions between non-polar ligands and hydrophobic patches accessible on the surface of the protein or peptide. RPC was originally introduced in 1950 by Howard and Martin [1] and for some time widely used in paper chromatography [2]. It regained popularity with the development of modern silica-based stationary phases with bonded alkyl ligands. Hydrophobic interaction chromatography (HIC) also employs a hydrophobic stationary phase but the density and hydrophobic character of the ligand are much smaller than in RPC. In this case, aqueous solutions of kosmotropic salts are used as the mobile phase. However, in both RPC and HIC, proteins are bound when the surface tension in the mobile phase is high and eluted in mobile phases where the surface tension is low.

Protein Chromatography: Process Development and Scale-Up. Giorgio Carta and Alois Jungbauer © 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 978-3-527-31819-3

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3.2 Interaction Types and Chemistry

Chromatography media can be categorized according to the interaction type and chemistry or according to the physical structure of the stationary phase. For preparative separation of biomolecules porous materials are usually preferred because of their higher binding capacity. Thus, interactions between the protein molecules and the stationary phase ligands take place mainly within the porous network of a support structure. Non-porous particles, where interactions are limited to the outer surface, are used primarily for analytical separations. 3.2.1 Steric Interaction

The steric exclusion of large molecules from the porous network of a chromatography particle is employed in size exclusion chromatography. The term ‘gel ﬁltration’ [3] is also used, although it should be avoided since the separation is based on the differential migration of different solutes through the column rather than on a ﬁltration. SEC normally employs porous chromatography particles such as those illustrated in Figure 3.1. Molecules of different size access a different fraction of the total porosity in the particle. Thus, they are differently retained in the column and elute at different times with larger molecules eluting ﬁrst and smaller molecules eluting last. A distribution coefﬁcient KD is deﬁned for each solute according to Equation 1.11. Experimental results for proteins often show a linear-log relationship between KD and molecular mass: K D = a + b log Mr

(3.1)

where a and b are empirical parameters. KD can also be expressed as a function of the radius of gyration (see Section 1.2.2.2) or of the hydrodynamic radius (Equation 1.8) often yielding a better description, because both are more closely related to the actual size and shape of the protein molecule. Because of the linear-log relationship expressed by Equation 3.1, the selectivity of SEC is not very high. Thus, the method is mainly used for the following four purposes: determination

Figure 3.1 Electron micrograph of a porous particle (Monobeads, GE Healthcare, Uppsala, Sweden) used for separation of proteins.

3.2 Interaction Types and Chemistry

UV280 [mAU]

60 50

M500-11, high viability M500-11, low viability M250-9, high viability

40

M250-9, low viability

7 kDa

4 3 kDa

30 20

IgG

1084 kDa 570

10

44 24 kDa

0 0

10

20

30

40

50

Retention time [min] Figure 3.2 Separation of aggregates from recombinant IgG by SEC using a Superdex 200 column from GE Healthcare. Column size, 30 × 1 cm I.D.; sample volume, 50 µl; ﬂow rate, 0.5 ml/min. Reproduced from [4].

of size or molecular mass, group separation according to the size of the molecule, desalting of protein solutions, and refolding of proteins. Several pharmacopeias recommend SEC for the determination of aggregate concentrations of protein pharmaceuticals. Figure 3.2 shows an example of SEC pertaining to the determination of aggregate content in a sample of recombinant antibodies obtained from two different clones with high and low productivity. In both cases aggregates elute earlier than the product and truncated forms. Although SEC is rarely used for separations in process chromatography due to low productivity, dilution of the product, and low selectivity, it is important for process control and can be used as a means of buffer exchange and as a ﬁnal step in the overall process to place the product in the desired formulation buffer. An important advantage is that SEC is generally non-denaturing since no adsorption occurs. Reference [5] provides a comprehensive list of commercially available size exclusion chromatography media for process application. 3.2.2 Hydrophobic Interaction

Shepard and Tiselius [6] introduced hydrophobic interaction chromatography (HIC) for separations using sulfate and phosphate solutions. In a follow-up study, Shaltiel and Erel [7] used the term hydrophobic chromatography or hydrophobic afﬁnity chromatography. Hofstee [8] described the method as hydrophobic adsorption chromatography, and ﬁnally Hjerten [9] called the method hydrophobic interaction chromatography. In HIC, hydrophobic ligands are immobilized on a backbone or base matrix. Interaction between the hydrophobic patches on the surface of the protein with these ligand results in protein retention. In contrast to

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Figure 3.3 bottom.

Examples of ligands used in RPC and HIC. Hydrophobicity increases from top to

HIC, where interactions are promoted by salt, interaction with RPC surfaces is typically modulated by water-miscible organic solvents due to the stronger hydrophobicity of the stationary phase. In this case, proteins are eluted by adding an organic modiﬁer such as acetonitrile, ethanol, methanol, or isopropanol. Figure 3.3 gives examples of ligands commonly used in RPC and HIC stationary phases. The ligand densities are different in RPC and HIC and are usually much higher in RPC resulting in greater hydrophobicity. Table 3.1 compares HIC and RPC with regard to their practical application. The hydrophobic interaction has been explained by van der Waals forces between the non-polar amino acid side chains and the HIC ligands. Water is a solvent with a high surface tension due to its hydrogen bonding capability making dissolution of non-polar substances thermodynamically unfavorable. Thus, the process is thought to occur via the formation of a cavity into which the solute can ﬁt. The same process is thought to occur at the surface of a hydrophobic stationary phase. The water is restructured leading to a positive entropy change. The whole process can be described by the change of free energy (∆G): ∆G = ∆H − T∆S

(3.2)

The restructuring of water leads to a positive entropy change and since cavity formation is thermodynamically unfavorable, the restructuring of water and minimization of the surface are the driving forces in hydrophobic interaction. In the case of isocratic elution, retention in hydrophobic chromatography can be expressed by the retention factor k′ according to Equation 2.10. Melander and

3.2 Interaction Types and Chemistry Table 3.1

General characteristics of HIC and RPC. HIC

RPC

Base matrix

Agarose, cross-linked Polymetaacrylate Polystyrene-divinylbenzene Cellulose Silica for HPLC use

Silica Polystyrene

Ligands

Short alkyl/aryl chains

Linear alkyl chains or aromatic

Ligand density

10–50 µmol/ml

>100 µmol/ml

Coupling

Ether linkage

Silanol groups; endcapping with chlorosilane

Buffers/eluents

Na2SO4, K2SO4, (NH4)2SO4, Na2HPO4, NaCl

Acetonitrile, ethanol, methanol, isopropanol

Additives

TFA, HFBA

Advantages

Maintenance of biological activity

High resolving power Coupling with MS

Disadvantages

Addition of salt to the sample

Detrimental to protein structure

Horvath [10] showed that ln k′ in HIC is linearly related to salt concentration as shown, for example, in Figure 3.4. Thus, there is a similarity between the salting out of proteins and HIC retention [11, 12], suggesting that the solvophobic theory, ﬁrst described by Green [13], can be used to interpret the shape of these curves. The following equation, obtained from the solvophobic theory describes solubility, w, over a broad range of salt concentrations: B( m 0.5 ) w ln   = + Λm − Ωσ m  w 0  RT [1 + C ( m 0.5 )]

(3.3)

where m is the molality of the salt solution, w is the protein solubility in the salt solution, w0 is the solubility in pure water, and B and C are constants in the Debye– Hückel equation (cf. Equation 1.17). Λ and σ are salting-in and salting-out coefﬁcients, respectively, and Ω is a lumped parameter describing cavity formation. Melander and Horvath adopted this theory to explain retention behavior in HIC, showing that the retention factor can be expressed by an equation analogous to Equation 3.3. Accordingly, we have: 1

ln k ′ = ln k0′ −

B ⋅m2

1

1 + C ⋅ m2

− Λ ⋅ m + Ωσ m

(3.4)

where k0′ is the retention factor in the absence of salt and B and C are empirical parameters as shown in Equation 3.3. The parameter σ in this equation is related to the so-called molal increment in surface tension deﬁned as

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3 Chromatography Media

γ −γ0 (3.5) m where γ is the surface tension at a certain salt concentration and γ0 is the surface tension of water. σ is an intrinsic property of a salt as summarized in Table 3.2. The hydrophobic contact area between solute and sorbent, the number of charged residues on the protein, the molal increment in surface tension of the salt, and repulsive forces of the ligand at zero salt molality contribute to the slope and the intercept of the straight line portion of the plot of ln k′ versus salt concentration shown in Figure 3.4. This plot is ﬂat at low ionic strength, but it becomes straight at higher ionic strength. For high salt concentrations Equation 3.5 can be simpliﬁed to the following: σ=

ln k ′ = ln k0′ + S ⋅ m

(3.6)

where S is an empirical parameter known as the sensitivity coefﬁcient. In some cases, at low salt concentration k′ decreases with increasing ionic strength and then increases for higher salt concentrations. Figure 3.5 provides an example of this behavior, which is dependent on the particular type of protein and stationary phase. Table 3.2 Molal surface tension increments of various salts. Data from [10].

Salt

Surface Tension Increment (σ), 10−3 dyne g cm−1 mol−1

Salt

Surface Tension Increment (σ),10−3 dyne g cm−1 mol−1

KSCN NaClO3 NH4I LiI KI NH4NO3 KClO3 NaI NaNO3 NH4Br LiNO3 LiBr KBr NaBr CSI NH4Cl KCIO4 FeSO4 LiCl NaCl CsNO3 CuSO4

0.45 0.55 0.74 0.79 0.84 0.85 0.86 1.02 1.06 1.14 1.16 1.26 1.31 1.32 1.39 1.39 1.40 1.55 1.63 1.64 1.57 1.82

K2-tartrate Ba (NO3)2 LiF Na2HPO4 NiSO4 MgSO4 MnSO4 CuSO4 (NH4)2SO4 ZnSO4 Na2-tartrate K2SO4 Na3PO4 Na2SO4 Li2SO4 FeCl3 BaCl2 K3-citrate MgCl2 CaCl2 K4FE(CN)6 K3FE(CN)6

1.96 2.00 2.00 2.02 2.10 2.10 2.10 2.15 2.16 2.27 2.35 2.58 2.66 2.73 2.78 2.78 2.93 3.12 3.16 3.66 3.9f 4.34

3.2 Interaction Types and Chemistry

Figure 3.4 Plot of protein retention versus salt concentration in isocratic HIC. The dotted lines are linear approximations of the behavior at high salt concentrations.

Figure 3.5 Plot of protein retention versus ionic strength in isocratic chromatography experiments. At low salt concentration either electrostatic repulsion or attraction is dominant and overrides the hydrophobic interaction.

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Equation 3.6 can be extended to a three-parameter expression [14] by considering electrostatic interactions generated by ﬁxed charges that may be present on the solid support. In this case, the retention factor is expressed as follows: ln k ′ = A − B ⋅ ln m + C ⋅ m

(3.7)

where A is a constant, and B and C are electrostatic and hydrophobic interaction parameters, respectively. B is a function of the effective charge of the protein and salt counter ion and indirectly, a function of the charge of the stationary phase. C is dependent on the hydrophobic contact area between the protein and stationary phase and on the molal increment in surface tension of the salt. The inﬂuence of different anions on hydrophobic interaction follows the wellknown Hofmeister series, which is discussed in Section 1.2.2.5. This series is in line with the ranking of salts according to the surface tension increment. Thus, ions that promote precipitation or salting-out also promote retention in HIC. However, the solubility of the salt also needs to be considered. For example, the most popular salt for HIC is ammonium sulfate. Not only it is kosmotropic, but it is soluble in water up to a concentration of 4.3 molar. Other salts, such as magnesium sulfate, while kosmotropic, do not have sufﬁcient solubility to cause protein retention. An important consideration that limits application of HIC especially at very high feed concentrations is its complex selectivity behavior and potentially destabilizing effects on proteins. The latter can lead to conformational changes and yield losses induced by contact with the hydrophobic surfaces. It is known that proteins undergo conformational changes when they encounter a strongly hydrophobic surface such as those used in RPC. However, despite presenting less harsh conditions than RPC, conformational changes have also been reported during the use of HIC [15, 16]. Various models have been developed to describe retention in RPC including the solvophobic theory [17] and partition theory [18]. An important difference between HIC and RPC is that the bonded alkyl chain ligands themselves undergo conformational changes as a function of the mobile phase composition [19]. As a result, the selectivity varies strongly with ligand density and type. For instance selectivity is very different for polyaromatic hydrocarbons and linear alkyl chain ligands reﬂecting the active role of the stationary phase [20–22]. It is still unclear which model is best suited to explain the retention mechanism. Recent studies indicate that retention is mainly governed by a partitioning process rather than adsorption [23]. More is known about retention mechanisms for small molecules in RPC [24–28]. Proteins have not been investigated to the same extent but general aspects of the retention mechanism for small solutes can be adapted to biomolecules. In general, retention increases dramatically with the molecular mass of the solute. A qualitative example is given in Figure 3.6. Although there is a smooth relationship between k′ and the volume fraction of the organic modiﬁer in the case of small molecules, the relationship between k′ and mobile phase composition is represented by very steep curves for larger peptides and proteins.

3.2 Interaction Types and Chemistry

Figure 3.6 Qualitative effect of the mobile phase modiﬁer in RPC on retention of small molecules, peptides, and proteins.

In general, the retention factor in RPC can be related to the mobile phase composition according to the following equation ln k ′ = ln k0′ − Sϕ

(3.8)

where k0′ is the retention factor in pure water as the mobile phase, S is the sensitivity coefﬁcient, and ϕ is the volume fraction of the organic modiﬁer. The steps used to implement HIC and RPC for protein puriﬁcation are shown in Figure 3.7. Typically, for HIC, the feed is ﬁrst dissolved in the high concentration salt buffer used for equilibration of the column in order to bind the protein. In some instances, initial binding does not occur and the protein passes through, such as when the feed contains high molarity of chaotropic agents such urea or guanidine hydrochloride. Removal of these agents may be needed for successful HIC separation. In other cases, proteins may precipitate at the high salt concentrations needed to promote retention. On the other hand, in RPC, preparation or conditioning of the feed is often not necessary since proteins present in aqueous samples are typically strongly bound to the RPC stationary phases. Nonetheless, better resolution is sometimes obtained when the sample is dissolved in the same mobile phase used to equilibrate the column. Elution in HIC is effected by a linear or stepwise decrease in salt concentration. However, more tightly bound contaminants may need to be removed by a regeneration step with cleaning agents such as alcohol, NaOH or detergents. Conversely, elution in RPC is effected by increasing the volume fraction of the organic modiﬁer in a linear or stepwise manner. In this case, regeneration can be effected by an increase in the organic modiﬁer of up to 95%. It should be noted that although unfolding or denaturing is a concern, the use of HIC is not limited to highly stable proteins. Table 3.3 gives

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Figure 3.7

Steps used to implement HIC (left) and RPC (right) for protein puriﬁcation.

Table 3.3 Examples of puriﬁcation of labile proteins with HIC.

Protein Human factor X Bispeciﬁc monoclonal antibodies Endotoxin removal Human monoclonal IgM Recombinant antibody fragments

Reference [29] [30] [31] [32] [33]

some examples of labile proteins of medical relevance that have been puriﬁed by HIC. Figure 3.8 shows an example demonstrating the separation power of HIC sorbents in the case of whey proteins. In this case, the whey feedstock was prepared from colostral milk by acidic precipitation [34]. Although a baseline separation of all whey proteins could not be achieved, different proteins eluted in different fractions as demonstrated by the corresponding SDS-PAGE analyses. 3.2.3 Electrostatic Interaction

Electrostatic interaction forms the basis for ion exchange chromatography (IEC). As discussed in Chapter 1, proteins have different charges because of the different content of acidic and basic amino acids and this charge varies with pH. Protein

3.2 Interaction Types and Chemistry

Figure 3.8 Separation of whey proteins on Phenyl Sepharose HP and SDS-PAGE of the corresponding fractions; M, marker; FT, ﬂow through; E1–E4, eluate 1–4; R, regenerate; S, original sample.

binding occurs reversibly to oppositely charged ligands immobilized on a support. Elution is effected by increasing the ionic strength and thus the concentration of salt ions, which in turn, compete with the protein molecules for the charged ligands. Negatively charged ion exchangers bind cations and are known as cation exchangers while positively charged exchangers bind anions and are known as anion exchangers. Examples of charged ligands commonly used in ion exchangers are shown in Figure 3.9. Ion-exchangers are also categorized as weak or strong. This distinction does not necessarily reﬂect the protein binding strength. Rather, it is based on the

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3 Chromatography Media

Figure 3.9 Examples of ligands used in ion exchangers and their categories according to the titration curve.

protonation behavior of the ionogenic ligands. Strong ion exchangers are charged at any practical pH (2 to 10) whereas the charge of weak ion exchangers depends on pH. Weak cation exchangers carry a negative charge at pHs greater than about 5 while weak anion exchangers are positively charged at pHs lower than about 9. Charge density and protonation characteristics of the ligands can be determined experimentally from direct titration experiments as described for example in [35]. The Debye length, λD, describes the dimensional scale over which ionic interactions occur and is given by:

λD =

εr ε 0RT 2F 2I

(3.9)

where εr is the relative permittivity, ε0 is the dielectric constant of vacuum, F is Faraday’s constant, and I is the ionic strength. Increasing the ionic strength reduces the Debye length and weakens the interaction. The net charge of a protein is dependent on the pH of the buffer system as shown, for example, in Figure 1.17. At a pH distant to the isoelectric point (pI) of the protein, binding to an IEC stationary phase is generally stronger than at a pH near the pI. However, even at the pI, despite the fact that the net charge is zero, proteins are sometimes bound to ion exchangers [36]. This is thought to occur because of the heterogenous distribution of charged residues on the protein surface as seen for example in Figure 1.19. In this case, the electrostatic interactions responsible for binding are established between speciﬁc patches containing charged residues and oppositely charge ligands on the surface of the stationary phase.

3.2 Interaction Types and Chemistry

IEC is used extensively because of its versatility, broad range of available stationary phases and buffer systems, high binding capacity and resolving power, and, perhaps most importantly, its ability to preserve the biological activity of the product. Detailed information regarding the use of ion exchangers for protein separations is given by Jansen [37]. Different models have been developed to explain the retention mechanism. One is the stoichiometric displacement (SD) model, which describes protein adsorption as a stoichiometric exchange of ions [36]. This model has now been reﬁned to the steric mass action (SMA) model [38] which takes into account the shielding of charges by the adsorbed protein. Another approach is to model retention by taking into account the electrostatic potential created by the charged ligands immobilized on the resin surface [39, 40]. Quantitative relationships derived from these approaches are described in Chapter 5. 3.2.4 Complexation

Porath et al. [41] introduced metal chelate afﬁnity chromatography (MIC) in 1975 based on the discovery that chelated metals such as Cu++, Zn++, Ni++, or Fe3+ immobilized on the surface of a solid support can bind proteins selectively. Ligands commonly used to immobilize the metals include iminodiacetic acid (IDA) and tris(carboxymethyl)ethylenediamine (TED) which are illustrated in Figure 3.10. In solution, metal complex stability constants for these ligands range from 107 to 1017 M. The stronger attachment with the hexacoordinate central metal ion in TED leads to greater steric hindrance for protein binding but to lower leakage of the metal compared to the IDA ligand.

Figure 3.10 Schematic representations of metal chelate ligands with a hexacoordinate central metal ion. Reproduced from [42] with permission.

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Proteins are thought to interact with the chelated metals via surface-exposed His, Trp, and amino groups. A single surface-exposed His is generally sufﬁcient to bind chelated Cu++. More than one His and particularly, multiple clustered His at the protein surface leads to very strong binding. General rules regarding the binding strength have been reported by Sulkowski [43] and are shown in Table 3.4. Chelating interactions are pH dependent and this can be exploited for loading and elution of proteins. For instance, at low pH, the metal chelate complex is disrupted and the bound protein is readily eluted. Alternatively, competing species, such as imidazole, can be used as mobile phase modiﬁers to effect desorption of the bound protein. The strong binding observed for clustered His residues forms the basis for the so-called His-tags, which are fused either to the N- or to the C-terminus of the protein as shown for example in Figure 3.11. His-tags are very commonly used for puriﬁcation of recombinant proteins, especially at the laboratory scale and can be used as a generic puriﬁcation tool with either Ni or Cu chelate adsorbents. Although the His-tag is an artiﬁcial domain in the protein, it rarely interferes with biological function. However, in cases where removal of the His-tag from the target protein after puriﬁcation is required, further processing may be necessary to obtain the ﬁnal product since enzymatic or chemical cleavage does not generate an authentic N-terminus. In the case shown in Figure 3.11 a GS residue would be left at the N-terminus of the target protein after cleavage with thrombin. Proteins containing metal ions present challenges for MIC because the proteinbound metal may be removed during the process and the protein deactivated. Similarly, other amino acids such as cysteine and naturally-occurring His-rich Table 3.4 Inﬂuence of exposed His and Trp residues on binding strength of proteins on metal chelate adsorbents according to Sulkowski [43].

Presence of His, or Trp exposed on the surface of protein

Metal ions providing adsorption

No His/Trp One His More than one His Clusters of His Several Trp, no His

– Cu++ Cu++ > Ni++ Cu++, Ni++, Zn++, Co++ Cu++

Figure 3.11 Schematic overview of the essential elements of a his-tag with a hexa his-tag, and a linker containing an enzymatic thrombin cleavage site.

3.2 Interaction Types and Chemistry

Figure 3.12 MIC puriﬁcation of B. amyloliquefaciens pyroglutamyl aminopeptidase (pGAP) produced in E. coli with an N-terminal his-tag (HT-pGAP, tag sequence: MEP(H)6L). HT-pGAP. Lane M, MWM; lane 1, cell

extract; lane 2, supernatant fraction of the cell extract; lane 3, ﬂow through fraction from the IMAC; lane 4, eluted HT-pGAP. Reproduced from [44] with permission.

regions in other proteins present as impurities may result in unwanted protein binding during MIC puriﬁcation. Since its initial development, numerous proteins and peptides have been puriﬁed using His-tags, and several therapeutic candidates that have been puriﬁed using this method are currently in clinical studies. An example illustrating the puriﬁcation potential of MIC is shown in Figure 3.12. The over-expressed protein in this example could be puriﬁed in a single step to virtual homogeneity. 3.2.5 Biospeciﬁc Interaction

In afﬁnity chromatography the separation occurs by a highly speciﬁc, reversible biospeciﬁc interaction (BIC) between a protein and a natural or synthetic ligand immobilized on a porous support. The main steps in BIC are: (i) column loading with a mixture containing the target molecule, (ii) washing to remove unbound species, (iii) elution of the target protein, and (iv) stripping and regeneration to remove non-speciﬁcally bound species. Proper design of an effective afﬁnity matrix is based on the selection of the appropriate support, the appropriate ligand, and the appropriate ligand immobilization strategy. An effective ligand associates speciﬁcally with its interaction partner with a high association rate under certain conditions and dissociates at a high dissociation rate under other conditions. The

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term ‘afﬁnity chromatography’ was coined by Cuatrecasas et al. in 1968 [45] in a report describing the BrCN activation of a natural polymer to immobilize afﬁnity ligands such as proteins or peptides onto a chromatography surface. Activation of the surface and immobilization without steric hindrance is critical for preparing afﬁnity adsorbents. In principle, any molecule capable of interacting speciﬁcally with a protein can be used for BIC. Figure 3.13 shows a classiﬁcation of afﬁnity ligands. Practical examples are shown in Table 3.5. While natural compounds are effective, synthetic ligands such as dyes or amino acids immobilized on a chro-

Figure 3.13

Overview of biosepeciﬁc ligands.

Table 3.5 Examples of selective and reversible complexes used in afﬁnity puriﬁcation systems.

Ligand

Puriﬁed product

Substrate Co-factor Inhibitor Carbohydrate Lectin Boronate Hormone Antibody, single chain antibody Hapten Bacterial immunoglobulin binding proteins Oligonucleotide Dye

Enzyme Enzyme Enzyme Lectin Glycoprotein Glycoprotein Receptor Antigen (protein etc.) Antibody Antibody DNA binding protein Enzyme, serum protein

3.2 Interaction Types and Chemistry

matography support can also act as afﬁnity ligands for selective puriﬁcation processes. The main advantage of BIC is the high selectivity of the interaction. This can be based on several principles that can be classiﬁed according to the ligand interaction with the target protein according to the following list. 1) A macromolecular ligand forms a cavity and recognizes the target protein; the prime example is immunoafﬁnity chromatography with immobilized antibodies. 2) A macromolecular ligand induces a conformational change and upon this induction a ﬁt between two proteins is generated; the most prominent example is the Staphylococcal protein A, which recognizes the Fc part of immunoglobulin G. 3) A small molecular ligand ﬁts into a cavity present in the target protein; the prime example is an enzyme inhibitor or a peptide mimicking the epitope of an antibody. 4) A small ligand nestles at the surface of a protein; this has often been observed with peptides and compounds from combinatorial libraries. In this case strong hydrogen bonding and β-sheet formation is often observed. The interaction strength varies dramatically with the type of ligand and protein. Afﬁnity constants as high as 106 to 109 M−1 have been observed although much lower values, as low as 103 M−1, are sometimes encountered in practice. An important concern is that very large binding constants may require harsh elution conditions, which in turn, may be detrimental to either the target protein and/or the ligand. The fusion protein concept has also been used for BIC. Examples of useful tags are given in Table 3.6. Despite the broad range of afﬁnity ligands available as discussed above, in practice, bacterial immunoglobulin-binding proteins are the most successful from the viewpoint of large-scale manufacturing. Table 3.7 provides example of such ligands, their origin, and their speciﬁcity and Figure 3.14 shows examples of the binding domains of three different ligands in this category. Staphylococcal protein A (SpA) is used as ligand for industrial-scale puriﬁcation of antibodies and is the most important afﬁnity ligand. This ligand contains ﬁve binding domains, each selective for the Fc-region of IgG subclasses 1, 2, and 4. Some antibodies also bind to Protein A via their light chain. Although Protein A contains ﬁve binding domains, in practice only between two and three are available for IgG binding due to steric hindrance. The afﬁnity constant of the SpA binding domains for human IgG is estimated to be of the order of 108 M−1 although it may vary substantially with recombinant and engineered antibody fragments. Currently three types of Protein A media are commercially available for industrial antibody puriﬁcation 1) Wild-type Protein A without the cell wall domain. 2) Recombinant Protein A without the cell wall domain, engineered in a manner that allows directed immobilization. 3) Engineered Protein A domain with high alkaline stability.

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3 Chromatography Media Table 3.6 Examples of proteins and peptides used as fusion partners for puriﬁcation of recombinant proteins. Adapted from [46].

Enzymes

Size

Fusion at N or/and C-terminus

β-Galactosidase

116 kDa

N,C

Glutathione-S-transferase

26 kDa

N

Chloramphenicol acetyl transferase

24 kDa

N

TrpE

27 kDa

N

Polypeptide-binding proteins IgG binding domain of Staphylococcal protein A IgG/albumin binding domain of Streptococcal protein G

14–31 kDa 28 kDa

N C

Carbohydrate-binding domains Maltose-binding protein Starch-binding domain Cellulose-binding domain

40 kDa 119 aa 111 aa

N C N

Biotin-binding domain

8 kDa

N

Antigenic epitopes RecA FLAGTM

144 aa 8 aa

C N

Charged amino acids Poly (Arg) Poly (Asp) Glutamate

5–15 aa 5–16 aa 1 aa

C C N

Metal chelating amino acids Poly (His) tails

1–9 aa

N,C

Other poly (amino acid) tails Poly (Phe) Poly (Cys)

11 aa 4 aa

N N

Avidin-biotin

Table 3.7 Main immunoglobulin-binding bacterial proteins and their afﬁnity characteristics.

Microbial protein

Origin

Speciﬁcity

Main antibody interaction

Protein Protein Protein Protein Protein

Staphylococcus aureus C and G Streptococci Peptococcus magnum A Streptococci Clostridium perfringens

Fc-region Fc-region Kappa chains Fc-region Kappa chains

IgG IgG IgM and IgG IgA IgM

A G L ARP P

3.2 Interaction Types and Chemistry

Figure 3.14 Immunoglobin binding domains of the bacterial immunoglobulin binding proteins Protein G, A, and L.

3.2.6 Mixed Mode Interaction

The terms mixed mode and multimodal interaction [47, 48] have been used to describe a broad range of cases where protein binding occurs as a result of a combination of electrostatic and hydrophobic effects or via a combination of positive and negative ligands. Such complex interactions can be more selective than pure HIC or IEC surfaces. Hydroxyapatite can also be regarded as a chromatography medium for mixed mode interactions. An early mixed mode chromatography medium for protein puriﬁcation was introduced by Baker under the name Bakerbond ABx [49] as a mixed mode ion exchanger designed for antibody puriﬁcation. For this material, low salt concentrations are required for protein binding as for regular IEC matrices. However, selectivity is much higher. More recently, several attempts have been made to develop selective mixed mode surfaces combining hydrophobic and electrostatic interactions allowing protein binding to occur under high salt conditions and elution at low salt concentrations or by altering the pH. The 2-mercapto-5-benzimidazolesulfonic ligand shown in Figure 3.15 offers such features and has been successfully applied to antibody puriﬁcation [47]. With these types of ligands, proteins can be adsorbed in the presence of salt at concentrations of up to 500 mM. Elution is achieved with low salt concentrations and low pH. Hydroxyapatite is a calcium phosphate mineral with the formula Ca10(PO4)6(OH)2 and a complex crystalline structure. It was introduced as a stationary phase for protein chromatography by Tiselius et al. in 1956 [50] and a large number of applications of hydroxyapatite have been described for protein and DNA puriﬁcation [51–53]. Hydroxyapatite synthesized according to the method of Tiselius forms unstable rectangular plate-shaped crystals with poor ﬂow properties. In order to overcome these disadvantages ceramic hydroxyapatite (CHT) materials

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Figure 3.15 A schematic representation of the native structure of 2-mercapto-5-benzimidazolesulfonic acid (MBISA; A) and the resin carrying MBISA which is covalently attached to the bead (MEP HyperCel; B). Reproduced from [48] with permission.

Figure 3.16 Scanning electron micrographs of ceramic hydroxyapatite CHT Type II (Bio-Rad Laboratories) at different magniﬁcations taken from [53].

were developed based on hexagonal columnar crystals agglomerated to form stable particles by sintering at high temperature. CHT has excellent ﬂow properties, high particle porosity and the surface properties of hydroxyapatite. Micrographs of CHT particles at different magniﬁcations are shown in Figure 3.16. Hydroxyapatite surfaces contain two types of functional groups: positively charged pairs of calcium ions (C-sites) and clusters of six negatively charged

3.3 Buffers and Mobile Phases Catalase 5

log MW

4

3

Transferrin BSA Conalbumin Ovalbumin Protein Pepsinogen Carbonic anhydrase a-Chymotripsin Trypsin inhibitor Trpsinogen Myoglobin Lysozyme a-Lactalbumin Insulin Ribonuclease A Cytochrome c Insulin chaine A Insulin chaine B Peptide Angiotensin III Angiotensin II Enkephalin

Phe Ile, Val Glu 2 Pro Ala

Tyr Met HIS Asp Ser Glv

Arg

1

Lys

10 To

Amino acid

Thr

20

30

Retention time (min)

Figure 3.17 Retention of amino acids, peptides and proteins on a 7.5 mm I.D × 100 mm ceramic hydroxyapatite column (CHT Type II, Bio-Rad Laboratories) operated at 1 ml/min with a linear sodium phosphate gradient from 1 to 400 mM over 40 min. Courtesy of T. Ogawa.

oxygen atoms, which are associated with triplets of crystalline phosphates (P-sites). Thus, different mechanisms of interaction with acidic and basic proteins as well as DNA can be observed. While α-amino and guanidinyl groups of proteins interact with phosphate ions, carboxyl and phosphate groups interact with calcium ions. The negatively charged backbone of plasmids due to the phosphate groups, interacts with the calcium ions but they are also repelled by the phosphate ions of hydroxyapatite. For plasmids, addition of sodium chloride enhances the binding capacity, likely because the hydroxyapatite phosphate ions are shielded [54]. Figure 3.17 summarizes the retention characteristics of different biomolecules on hydroxyapatite. Retention of proteins is usually stronger than that of peptides and small molecules such as amino acids. In general, retention of acidic proteins increases with decreasing pI while the opposite is true for basic proteins.

3.3 Buffers and Mobile Phases

The selection of the mobile phase is often extremely important for maximizing the capacity and selectivity of the stationary phase. For biospeciﬁc interaction the

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choice of buffers is ample because the interaction is dominated by the speciﬁcity of the ligand itself, whereas in other types of chromatography the composition of the mobile phase is critical. For instance, Protein A afﬁnity chromatography can be operated in the pH range from 6.5 to 8 and at salt concentrations from 25 to 250 mM. On the other hand, changes in composition over the same range when loading an antibody onto a cation or anion exchanger will have very large effects. The buffer itself does not only inﬂuence binding and elution but also the stability of the protein. Table 3.8 gives an overview of the mobile phase conditions used for different applications. A list of buffers and their pKa values is given in Table 3.9. The selection depends on the application, size of the operation and the nature of the product. For very large scale operations the choices are more limited, since cost, availability, and disposal options impose signiﬁcant constraints. Amino acids such as histidine with a pKa of 6.1 are also used, because it has been observed that these compounds are able to stabilize proteins. Arginine is used as a solubility enhancer whereas glycine is used as an elution buffer in afﬁnity chromatography. Buffers for ion exchange chromatography require special consideration since the buffering species can interact strongly with the ionogenic groups. In general, negatively charged buffering species are preferred for cation exchange while positively charged species are preferred for anion exchange in order to avoid large variations in pH caused by retention of the buffering species.

Table 3.8 Characteristic mobile phases used in protein chromatography.

Type of chromatography

Common adsorption conditions

Common elution conditions

Regeneration conditions

Size exclusion

Any ionic strength, any pH, low feed

Isocratic separation

Salt washing, dilute alkaline washing

Hydrophobic interaction

High ionic strength

Ionic strength decrease (ammonium sulfate)

Alkaline washing, organic acids, chaotropics, aqueous-miscible solvents

Reversed-phase

Hydro-organic mixtures

Organic modiﬁers: acetonitrile, methanol etc.

Solvents, glycols, urea, acidic solutions

Cation-exchange

Low ionic strength, pH below protein pI

Increase of ionic strength, increase pH

Acid/base washing, high conc. salts

Anion-exchange

Low ionic strength, pH above protein pI

Increase of ionic strength, increase pH

Acid/base washing, high conc. salts

Hydroxyapatite

Dilute phosphate buffers, pH 6.8

Increase phosphate buffer concentration

Sodium hydroxide

Afﬁnity

Close to physiological conditions except special cases

pH changes, ionic strength changes, competitive elution

Salt washing, speciﬁc displacement treatments

3.3 Buffers and Mobile Phases Table 3.9

pKa (25 °C)

107

Buffers used for protein chromatography. pH interval

Substance

Conc. (mM)

Remark

Buffer components for cation exchange chromatography 2.00

1.5–2.5

Maleic acid

20

Dicarboxylic acida)

2.35

2.0–2.8

Glycine

20

Elution buffer in afﬁnity chromatography

2.88

2.38–3.38

Malonic acid

20

Dicarboxylic acida)

3.13

2.63–3.63

Citric acid

20

Also elution buffer in afﬁnity chromatography

3.81

3.6–4.3

Lactic acid

50

3.75

3.8–4.3

Formic acid

50

4.21

4.3–4.8

Butanedioic acid

50

4.76

4.8–5.2

Acetic acid

50

5.68

5.0–6.0

Malonic acid

50

6.15

5.5–7.0

MES

20

7.20

6.7–7.6

Phosphate

50

7.20

6.7–7.7

MOPS

20

7.55

7.6–8.2

HEPES

50

7.00

7.5–8.5

HEPPS

Dicarboxylic acida)

Dicarboxylic acida)

50 b)

7.98

2.6–8.6

5,5-Diethylbarbituric acid

20

8.35

8.2–8.7

BICINE

50

8.5

8.0–10.0

Borate

20

Wide buffer range with Na acetate

Prepared from boric acid and Borax, slightly toxic

Buffer components for anion exchange chromatography 4.75

4.5–5.0

N-methyl piperazine

20

5.68

5.0–6.0

Piperazine

20

5.96

5.5–6.0

L-histidine

20

6.46

5.8–6.4

bis-Tris

20

6.80

6.4–7.3

bis-Tris propane

20

6.99

6.6–7.5

Imidazole

20

7.76

7.3–7.7

Triethanolamine

20

8.06

7.6–8.0

Tris

20

8.52

8.0–8.5

N-methyl-diethanolamine

50

8.88

8.4–8.8

Diethanolamine

20 at pH 8.4 50 at pH 8.8

Used for elution in metal chelate chromatography

108 Table 3.9

pKa (25 °C)

3 Chromatography Media

Continued pH interval

Substance

Conc. (mM)

8.64

8.5–9.0

1,3-diamino-propane

20

9.50

9.0–9.5

Ethanolamine

20

9.73

9.5–9.8

Piperazine

20

10.47

9.8–10.3

1,3-diamino- propane

20

11.12

10.6–11.6

Piperadine

20

12.33

11.8–12.0

Phosphate

20

Remark

a) Complexes bivalent cations such as Ca++ or Mg++. b) Not recommended because of its narcotic properties.

Figure 3.18 [55].

Categories of chromatography media according physical attributes taken from

3.4 Physical Structure and Properties

In this section we classify chromatography media according to physical attributes as shown in Figures 3.18 and 3.19. We ﬁrst distinguish between stationary phases based on particles or beads packed into a column and continuous stationary phases, also known as monoliths, which are formed in situ creating a network of pores or

3.4 Physical Structure and Properties

Figure 3.19 Comparison of membrane chromatography, beads, and monoliths for protein chromatography.

channels through which the ﬂuid ﬂows. Membranes used for adsorption of proteins can be also considered as monoliths with a very large aspect ratio. The membranes are often rolled around a central tube in multiple layers to increase the effective bed length. Beads can be non-porous, porous, or contain a solid core surrounded by a porous shell. In these materials, the pores can be completely open and ﬁlled with the mobile phase providing access to surface-bound ligands by diffusion. Alternatively, the pores can be partially or even completely ﬁlled either with a cross-linked gel or with grafted polymers, which provide sites for protein binding. 3.4.1 Base Matrices

Desirable features of a support matrix for protein chromatography include low non-speciﬁc adsorption, high mechanical strength, surface functionality for immobilization of ligands, absence of toxic leachables, and stability in solutions used for cleaning and sanitation. A variety of materials are available that meet these requirements and form the basis for modern chromatography media. 3.4.1.1 Natural Carbohydrate Polymers The ﬁrst application of a natural carbohydrate polymer to protein chromatography was reported by Peterson and Sober in 1956 [56]. They derivatized cellulose beads with ion-exchange groups. Three years later the ﬁrst dextran-based media, Sephadex G-25 and G-50, became commercially available [3]. Common to many natural polymers such as cellulose, agarose, dextran, and chitosan, is their hydrophilic character, which results in low non-speciﬁc adsorption. Examples of commonly used media based on natural polymers are shown in Table 3.10. Figure 3.20 shows a schematic representation of the structure of agarose and its chemical composition. Agarose polymers dissolved in a hot aqueous solution

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3 Chromatography Media Table 3.10

Examples of protein chromatography media based on natural carbohydrate

polymers. Matrix material

Physical shape

Example

Manufacturer

Cellulose

Fibers Particles Particles Particles

DE 32 Express-Ion D Sephacel Celluﬁne

Whatman Whatman GE Healthcare Millipore

Dextran

Spherical particles

Sephadex

GE Healthcare

Agarose

Spherical particles

Sepharose Sepharose FF Sepharose XL

GE Healthcare

Chitosan

Spherical particles

Figure 3.20 Microphotograph and schematic representation of the interior structural and chemical characteristics of agarose beads used for protein chromatography. Agarose polymers consisting of β-D galactose-3,6anhydro-α-L-galactose units form a gel in hot

Mitsubishi

aqueous solution which on cooling forms a macroporous network that can be chemically cross-linked to increase mechanical strength. The resulting pore size depends on the agarose content.

3.4 Physical Structure and Properties

form a gel upon cooling providing a macroporous network through which proteins can diffuse and which is resistant to alkaline conditions. Chemical cross-linking can be used to increase mechanical stability while hydroxyl groups in the agarose polymer provide sites for functionalization with suitable ligands. Fibrous cellulose is extremely hydrophilic but also difﬁcult to pack, limiting bed height to less than 20 cm. Cellulose beads are also commercially available but are less hydrophilic. The general features of chromatography media made from natural polymers are • • • • • •

Low solid densities (90–96% water) Limited mechanical strength Smaller pores (gels) Easy functionalization Resistance to CIP Low non-speciﬁc binding

3.4.1.2 Synthetic Polymers Synthetic polymers are also frequently used as base materials for protein chromatography. Table 3.11 provides examples of media used for protein chromatography based on synthetic polymers. Common to all synthetic polymers is their relative hydrophobicity. For example, the wetting angle of agarose beads is about 20° whereas poylmethacrylate has a wetting angle of 40°. Very hydrophobic materials, such as styrenic polymers, require coating with hydrophilic materials in order to prevent fouling and low protein recovery. An advantage of some synthetic polymers is their resistance to extreme chemical conditions such as pH or oxidizing environment and autoclaving. Table 3.11 Examples of protein chromatography media based on synthetic polymers.

Matrix material

Examples

Manufacturer

Acrylamido and vinyl co-polymers

UNOsphere Macroprep Bio-Gel

Bio-Rad Laboratories

Acrylic polymers

Toyopearl Fractogel

Tosoh Bioscience E. Merck

Poly(methacrylate)

CIM disks and tubes

BiaSeparations

Poly(styrene-divinyl benzene) co-polymers

Source, Resource POROS

GE Healthcare Applied Biosystems.

The common features of organic polymers as base materials for protein chromatography media are: • •

Higher solid densities (50–80% water) Large pore sizes possible (macroporous and gigaporous)

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• • • •

Greater mechanical strength Resistance to CIP Moderate or high non-speciﬁc binding Relatively difﬁcult to functionalize

3.4.1.3 Inorganic Materials Table 3.12 provides examples of protein chromatography media based on inorganic materials. Hydroxyapatite was described in Section 3.2.6. Silica is used extensively in the preparation of RPC media [57–59]. Exposed OH groups in this material can interact with proteins especially at higher pH values. Therefore capping the residual silanol groups is generally needed. Although silica is not stable under alkaline conditions, zirconium treatment of the surface allows the short-term use of buffered eluents at pHs as high as 9.0. Silica-matrices have also been coated with cellulose, polystyrene, dextran, agarose and poly(alkylaspartamide) to yield stable surfaces. Another method of stabilizing silica supports is to treat them with a zirconium salt followed by covalent bonding with a hydrophilic organo-silane. Porous glass is also used as a support matrix. For example, controlled pore glass (CPG) contains large pores and has excellent ﬂow properties, although it is not spherical in shape (see Figure 3.21). There are reactive hydroxyl groups on the surface of the glass which can be used to immobilize ligands. For instance, the Protein-A adsorbent ProSep A (Millipore) has been successfully applied to largescale therapeutic antibody puriﬁcation. The general features of inorganic supports for protein chromatography media are:

• • • • • •

Highest solid densities (30–60% water) Large pore sizes possible (macroporous and gigaporous) Rigidity May not be resistant to CIP (especially silica, glass) Moderate or high non-speciﬁc binding Relatively difﬁcult to functionalize

Table 3.12

Examples of protein chromatography media based on inorganic materials.

Base material

Trade name

Manufacturer

Hydroxyapatite [(Ca5(PO4)3OH)2] Silica

CHT Kromasil LiChrospher Chromolith Prosep A HyperDa)

Bio-Rad Laboratories Eka Chemicals AB Merck KgA Merck KgA Millipore Pall Biosciences YMC

Controlled pore glass Zirconia TiO2 a)

The pores are ﬁlled with a hydrophilic polymer.

3.4 Physical Structure and Properties

Figure 3.21 Scanning electron micrograph of controlled pore glass functionalized with staphylococcal protein A (Prosep A, Millipore).

3.4.2 Porosity, Pore Size, and Surface Area

In order to attain high binding capacity, adsorbent materials used in process applications are usually porous, such as those shown in Figure 3.1. The extraparticle and intra-particle porosity, ε and εp, deﬁned in Figure 2.7, and the pore diameter, dpore, determine the internal surface area available per unit volume of packed bed according to the following equation Ainternal =

4 (1 − ε ) ε p d pore

(3.10)

This area is available for protein binding as long as the pore size is sufﬁcient to allow diffusion of the protein molecules through the porous network. The external area which is in immediate contact with the ﬂowing mobile phase is also available for protein binding. This area is readily accessible by convective and diffusive transport through a boundary layer. The external area per unit volume of packed bed is dependent on the particle diameter, dp, according to the following equation Aexternal =

6 (1 − ε ) dp

(3.11)

The total area available for binding is of course the sum of the two. Figure 3.22 compares the areas calculated from these equations for the case of porous particles and monoliths. For particles used in preparative and industrial protein chromatography with diameters greater than 30 µm, the external surface area does not

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3 Chromatography Media

Figure 3.22 Examples of how external and internal areas contribute to the total area in a chromatography column packed with porous particles (left) or a monolith (right).

contribute signiﬁcantly to the total area. Only when the particle diameter is less than 10 µm does the external area become important. On the other hand, in the case of monoliths which are composed of small, non-porous micrometer-sized granules that make up their porous framework, only the external area is available. The intra-particle porosity of beaded stationary phases and the macroporosity of monoliths can be determined by nitrogen adsorption and mercury intrusion. Inverse size exclusion chromatography (iSEC) can also be used for packed beds. Mercury intrusion and nitrogen adsorption can be used only in a dry state. Thus, these methods cannot be used with soft particles and gels whose structure depends on hydration. Nitrogen adsorption provides the surface area without information on the pore size based on the amount of nitrogen adsorbed at liquid nitrogen temperatures. Combining this with the area occupied by the nitrogen molecule yields the total surface area of the solid. Before the surface area can be determined, the material must be stripped of any adsorbed molecules. The total area of a sample of porous material in m2 is calculated as follows: Atotal =

WmNA gas −18 10 Mr

(3.12)

where Wm is the weight of the adsorbed material, ¯ N Avogadro’s number, Mr the molecular mass of the adsorbed gas, and Agas the surface area occupied by an adsorbed gas molecule (0.162 nm2 for nitrogen [60]). The speciﬁc surface area is obtained by dividing Atotal by the sample mass. Finally, the packed bed density is used to relate it to the volume of packed bed.

3.4 Physical Structure and Properties

Mercury intrusion is useful for determining pore size and porosity of rigid materials. Intrusion of mercury into a pore requires pressure which depends on the pore radius, rpore, the surface tension, γHg, and the contact angle between the mercury and the surface of the solid, θ, according to the following equation: ∆P =

2γ Hg cosθ rpore

(3.13)

Since θ and γHg are generally known, the pore size distribution can be obtained from this equation by determining the incremental volume of mercury that penetrates into the particle under increasing pressures. The total porosity can then be obtained by measuring the total volume penetrated or by integrating the pore size distribution. Figure 3.23 shows an example of mercury intrusion for two different acrylate-based chromatography matrices. Inverse size exclusion chromatography (iSEC) is based on the chromatographic retention of non-adsorbing solutes of known size. Since it can be used with hydrated particles, the method can be employed to determine the effects of mobile phase composition such as salt concentration, pH, etc. and can be used with protein-loaded particles under actual process conditions. In the iSEC method, a set of inert standard compounds, usually dextrans, with know hydrodynamic 100 90 80

HW-65M (40–90 mm)

dV (%)

70 60 50 40 30 20 10 0 1000 100 90 80

5 4 3 2 100

5 4 3 2 10

543 2 1

543 2

0.1

543 2

0.01

543 2

0.001

543 2

543 2

0.1

543 2

0.01

543 2

0.001

HW-75F (30–60 mm)

dV (%)

70 60 50 40 30 20 10 0 1000

5 4 3 2 100

543 2

10

1

Pore diameter (µm) Figure 3.23 Example of the pore size distributions in two acrylate-based resins, Toyopearl HW-65 and HW-75 determined by mercury intrusion using an AUTOSCAm-60

porosimeter from Quanta Chrome. A mercury contact angle of 140° was assumed. Courtesy of Tosoh Bioscience, Stuttgart, Germany.

115

116

3 Chromatography Media

Figure 3.24 Partitioning of a spherical solute in a cylindrical pore. The region of thickness rm is inaccessible to a molecule of radius rm.

radius is injected into the column and the distribution coefﬁcient is determined for each standard according to Equation 1.11. A model describing partitioning of the standard compounds into the pores is then used to obtain the pore size. As a result, the outcome is dependent on the particular model used. In practice, the analysis is sensitive to the particular model used so that the results are frequently only qualitative. The ﬁrst step in iSEC analysis is to experimentally measure the distribution coefﬁcient KD from the retention volume, VR, of each probe according to the following equation: KD =

VR Vc − ε (1 − ε )

(3.14)

where ε is the extra-particle porosity determined, for example, with a substance that is completely excluded from the particles. A simple model relating KD to pore size assumes that the particles contain only cylindrical pores with uniform size as shown schematically in Figure 3.4. In such pores, assuming that the probe molecules are spherical with a radius rm, partitioning can be described by the following equation: rm   KD = ε p  1 −  rpore 

2

(3.15)

Thus using a linear plot of K D versus rm, εp and rpore can be determined from the intercept and slope. Figure 3.25 exempliﬁes this procedure for an agarose-based cation exchanger using dextran probes as standards. The results also demonstrate the effect of salt concentration on the accessible pore size. A certain pore size distribution can also be assumed for analysis of iSEC data. In this case, the experimentally determined KD values are related to individual pore sizes through an integral relationship where Equation 3.15 is assumed to be valid for each pore size. Unfortunately, the results are usually insensitive to the particular distribution assumed so that an exact determination is often impractical. Figure 3.26 shows an example for Protein A adsorbents based on the assumption of a Gaussian distribution of pore sizes. The average pore size and standard deviation are then used to ﬁt the data. As seen from this example, the accessible pore size is inﬂuenced by the adsorption of IgG.

3.4 Physical Structure and Properties

Figure 3.25 Determination of pore size for SP-Sepharose-FF (GE Healthcare) by iSEC using glucose and dextran standards of 4.4, 21.4, 66.7, 196, and 401 kDa molecular mass. The corresponding hydrodynamic radii used in the analysis are 0.4, 1.5, 3.0, 5.2, 8.6,

12.0 nm. Pore sizes and porosities determined from Equation 3.15 are rpore = 16.7 nm, εp = 0.76 at low salt concentration and rpore = 19.3 nm, εp = 0.81 at 500 mM NaCl. Data from [61].

Figure 3.26 Determination of pore size for Protein A adsorbents showing the effect of IgG adsorption. The average pore sizes shown in nm are obtained assuming a Gaussian pore size distribution. From [62].

117

118

3 Chromatography Media

Electron microscopy methods are also frequently employed to determine morphological characteristics of porous stationary phases. Scanning electron microscopy (SEM), is generally limited to dry materials, although the so-called environmental scanning microscopy (ESM) technique is emerging as an alternative for imaging wet particles. Transmission electron microscopy (TEM) can also be used as a highresolution imaging tool to determine the internal pore structure. In this case, the particles are embedded in a resin, sectioned, and stained with suitable contrast agents. Special dehydration procedures can be implemented which are thought to preserve the hydrate structure allowing application to soft gels. Figure 3.27 provides examples of SEM and TEM images of porous chromatography beads. Table 3.13 summarizes the porosities and pore sizes of some typical stationary phases in bead and monolith forms determined by the methods described above.

Figure 3.27 SEM (left) and TEM (right) images of UNOsphere S beads (Bio-Rad Laboratories). The TEM image, shown for an 80-nm section near the external surface of a

particle, was obtained after dehydration in an ethanol gradient, embedding in Spurr’s resin, and staining with a uranyl acetate, lead acetate mixture. Images from [63].

Table 3.13 Pore size and porosities for some representative stationary phases for protein chromatography.

Matrix

Manufacturer

Base material

Type

Intra-particle porosity

Approx. Pore diameter (nm)

Sepharose FF

GE Healthcare

Cross-linked agarose

Soft gel beads

0.85

30

TSK-HW

Tosoh Bioscience

Acrylate polymer

Rigid beads

0.50

100

UNOsphere

Bio-Rad

Acrylamido/vinyl co-polymer

Rigid beads

0.70

130

Source

GE Healthcare

Polystyrene-DVB

Rigid beads

0.50

200

POROS

Applied Biosystems

Polystyrene-DVB

Rigid beads

0.50

400

CIM a)

BIA Separations

Polymethacrylate

Represents the porosity of the porous network.

Monolith

a)

0.65

1500

3.4 Physical Structure and Properties

Proteins are usually smaller than 10 nm thus the listed materials are deﬁnitely suitable for protein chromatography. Monoliths offer large pore sizes and thus they are suitable for separation of larger biomolecules such as plasmid viruses and large proteins. 3.4.3 Particle Size and Particle Size Distribution

Chromatography particles can be either spherically shaped (e.g. see Figures 3.12 or 3.20) or irregular (e.g. Figure 3.21). As discussed in Chapter 2, particle size inﬂuences chromatographic resolution and dynamic binding capacity and thus certain sizes are preferred for different applications. Particles smaller than 10 µm are used for analytical high-resolution applications while particles of sizes between 30 and 100 µm are typical for large-scale puriﬁcation. Even larger particles can be used for capture applications, since in these cases the plate requirements are modest. Chromatography particles with a uniform size distribution (monodispersed) or with a distribution of particle sizes are also available. The particle size distribution (PSD) does not inﬂuence resolution and dynamic binding capacity greatly. However, it can have large effects on column pressure and packing quality. In all cases, ﬁnes should be avoided, since they can migrate with time and clog the column. When a discrete distribution of sizes is available, the average particle size, ¯r p and standard deviation, σp, can be calculated using the following equations: M

rP = ∑ f jrPj

(3.16)

j =1

σP =

M

∑f j =1

j

(rPj − rP )2

(3.17)

where M is the number of particle fractions in the distribution and fj is the volume fraction of particles of size rpj. Figure 3.28 shows the PSD for an agarose-based ion exchanger along with the average and standard deviation calculated from these equations. Note that other deﬁnitions of average particle size are sometimes used. However, the average expressed by Equation 3.16, which is equivalent to the socalled 4,3 average, is appropriate for describing diffusion-limited adsorption kinetics as discussed in Chapter 6. 3.4.4 Mechanical and Flow Properties

Materials ranging from very soft and elastic to completely rigid and brittle are available and used in practice. Some materials are compressible under the applied ﬂow whereas others are not. Although the particles are mechanically stable they are usually unable to resist abrasive forces and for this reason special care is required when mixing particle suspensions with mechanical agitators. In the labo-

119

120

3 Chromatography Media

Figure 3.28 Particle size distribution of SP-Sepharose-FF. Average and standard deviations are calculated from Equations 3.16 and 3.17. Data from [64].

ratory, magnetic stir bars should be avoided since they tend to grind the particles. In large-scale operations piston pumps should be avoided for the transport of particle slurries. The mechanical properties of the particles and their size determine the pressure ﬂow behavior. The ﬂow in columns packed with incompressible particles generally follows Darcy’s law under the laminar ﬂow conditions encountered in practice. Accordingly, we have ∆P =

ηuL B0

(3.18)

where B0 is the hydraulic permeability and L is the column length. The hydraulic permeability depends on the particle size and packing density. The Karman– Cozeny equation is commonly used to estimate B0 for rigid particles: B0 =

d 2p ε 3 2 150 (1 − ε )

(3.19)

This equation predicts that the pressure drop is inversely proportional to the square of the particle diameter, which is observed in practice. However, the numerical coefﬁcient of 150 is empirical and depends to some extent on the shape of the particles. Thus, B0 is probably best measured experimentally, rather than estimated from Equation 3.19. Figure 3.29 shows the speciﬁc hydraulic permeabilities, deﬁned as B0 d 2p , as a function of bed porosity for packed beds and monolithic columns. Normally, packed bed porosities are limited to maximum values of ε of around 0.45 since higher values lead to unstable beds that tend to collapse over time. Monoliths,

3.4 Physical Structure and Properties

Figure 3.29 Speciﬁc permeabilites of chromatography beds and monoliths compared to various permeability models as discussed in [65].

however, consist of a continuous network which allows much larger ε values with high speciﬁc permeability. On the other hand, the size of the granules that form the porous network in monoliths is quite small (1–6 µm), so that the actual permeabilities are low. For compressible soft beads the situation is different. In this case, as the ﬂow rate is increased, the bed becomes compressed and the hydraulic permeability declines. As a consequence, the column pressure increases in a non-linear manner up to a certain velocity known as critical velocity, ucrit, beyond which the pressure becomes extremely high. In general, the pressure ﬂow curve and the critical velocity depend both on the column diameter and on the initial bed height. Figure 3.30 shows data for two different agarose-based media with different agarose content [66]. As seen from these data, the softer material with lower agarose content (4FF) exhibits lower critical velocities and steeper pressure–ﬂow curves. Stickel and Fotopoulos [66] have developed an empirical relationship between critical velocity and the aspect ratio of the initially packed bed which enables the pressure–ﬂow curves for compressible media to be predicted. The relevant equations are presented in Chapter 10. Once the column has been packed and compressed, the headers are pushed against the compressed bed. At this point, as long as the pressure does exceed the packing value, a linear pressure–ﬂow curve relationship is obtained at lower ﬂow velocities.

121

3 Chromatography Media 300 (a) D = 20 cm

10

4.4

3.2

250

∆P (kPa)

200 150 100 50 0 0

2500

5000 7500 10000 12500 15000 uLo (cm2/h)

200 4.4

(b)

3.2

2.6

175 D = 30 cm 150 ∆P (kPa)

122

125 100 75 50 25 0 0

2000

4000 6000 uLo (cm2/h)

8000

Figure 3.30 Experimental and predicted pressure–ﬂow proﬁles for packing compressible media in PBS at 6 °C. Loss of wall support with increasing diameter is emphasized by normalizing for bed height differences in the

10000

abscissa. (a) Phenyl Sepharose 6FF: (䉬) 20 × 17.1; (䉭) 10 × 25.7; (䊏) 4.4 × 22.2; (䊊) 3.2 × 23.4. (b) Sepharose 4FF: (䉬) 30 × 19.6; (䉭) 4.4 × 14.2; (䊏) 3.2 × 30.3; (䊊) 2.6 × 36.7. Reproduced from [66].

References 1 Howard, G.A., and Martin, A.J.P. (1950) Biochem. J., 46, 532. 2 Ettre, L.S. (1981) J. Chromatogr. A, 220, 65. 3 Flodin, P. (1961) J. Chromatogr. A, 5, 103. 4 Grzeskowiak, J.K., Tscheliessnig, A., Toh, P.C., Chusainow, J., Lee, Y.Y., Wong, N., and Jungbauer, A. (2009) Protein Expr. Purif., 66, 58.

5 Jungbauer, A., and Feng, W. (2002) in HPLC of Biological Macromolecules (eds F. Regnier and K. Goodings), Marcel Dekker, New York, p. 281. 6 Shepard, C.C., and Tiselius, A. (1949) Discussion of the Faraday Society, Hazell Watson and Winey, London, p. 275. 7 Shaltiel, S., and Er-el, Z. (1973) Proc. Natl Acad. Sci. USA, 52, 778.

References 8 Hofstee, B.H.J. (1973) Anal. Biochem., 52, 430. 9 Hjerten, S. (1973) J. Chromatogr., 87, 325. 10 Melander, W.R., and Horvath, C. (1977) Arch. Biochem. Biophys., 183, 200. 11 Dixon, M., and Webb, E.C. (1961) Adv. Protein Chem, 16, 197. 12 Scopes, R.K. (1994) Protein Puriﬁcation – Principle and Practice, SpringerVerlag, New York. 13 Green, A.A., and Hughes, W.L. (1955) Methods in Enzymology, Academic Press Inc. 14 Melander, W.R., El Rassi, Z., and Horvath, C. (1989) J. Chromatogr., 469, 3. 15 Haimer, E., Tscheliessnig, A., Hahn, R., and Jungbauer, A. (2007) J. Chromatogr. A, 1139, 84. 16 Ueberbacher, R., Haimer, E., Hahn, R., and Jungbauer, A. (2008) J. Chromatogr. A, 1198–1199, 154. 17 Horvath, C., Melander, W., and Molnar, I. (1976) J. Chromatogr. A, 125, 129. 18 Dill, K.A. (1987) J. Phys. Chem., 91, 1980. 19 Sander, L.C., and Callis, J.B. (1983) Anal. Chem., 55, 1068. 20 Sander, L.C., and Wise, S.A. (1987) Anal. Chem., 59, 2309. 21 K.B., and Dorsey, J.G. (1989) Anal. Chem, 61, 930. 22 Sentell, K.B., and Dorsey, J.G. (1989) J. Chromatogr., 461, 193. 23 Dorsey, J.G., and Cooper, W.T. (1994) Anal. Chem., 66, 857A. 24 Carr, P.W., Lay, C.T., and Park, J.H. (1996) J. Chromatogr. A, 724, 1. 25 Tchapla, A., Heron, S., Lesellier, E., and Colin, H. (1993) J. Chromatogr. A, 656, 81. 26 Wilson, N.S., Dolan, J.W., Snyder, L.R., Carr, P.W., and Sander, L.C. (2002) J. Chromatogr. A, 961, 217. 27 Wilson, N.S., Nelson, M.D., Dolan, J.W., Snyder, L.R., and Carr, P.W. (2002) J. Chromatogr. A, 961, 195. 28 Wilson, N.S., Nelson, M.D., Dolan, J.W., Snyder, L.R., Wolcott, R.G., and Carr, P.W. (2002) J. Chromatogr. A, 961, 171. 29 Husi, H., and Walkinshaw, M.D. (2001) J. Chromatogr. B, 755, 367.

30 Manzke, O., Tesch, H., Diehl, V., and Bohlen, H. (1997) J. Immunol. Methods, 208, 65. 31 Wilson, M.J., Haggart, C.L., Gallagher, S.P., and Walsh, D. (2001) J. Biotechnol., 88, 67. 32 Tornoe, I., Titlestad, I.L., Kejling, K., Erb, K., Ditzel, H.J., and Jensenius, J.C. (1997) J. Immunol. Methods, 2005, 11. 33 O’Connor, K.C., Ghatak, S., and Stollar, B.D. (2000) Anal. Biochem., 278, 239. 34 Hahn, R., Schulz, P.M., Schaupp, C., and Jungbauer, A. (1998) J. Chromatogr. A, 795, 277. 35 Pabst, T.M., and Carta, G. (2007) J. Chromatogr. A, 1142, 19. 36 Kopaciewicz, W., Rounds, M.A., Fausnaugh, J., and Regnier, F.E. (1983) J. Chromatogr. A, 266, 3. 37 Janson, J.C., and Rydèn, L. (1998) Protein Puriﬁcation: Principles, High Resolution Methods and Applications, John Wiley & Sons, Inc., New York. 38 Gallant, S., Kundu, A., and Cramer, S.M. (1995) Biotechnol. Bioeng., 47, 355. 39 Stahlberg, J., Jonsson, B., and Horvath, C. (1991) Anal. Chem., 63, 1867. 40 Stahlberg, J., Jonsson, B., and Horvath, C. (1992) Anal. Chem., 64, 3118. 41 Porath, J., Carlsson, J., Olsson, I., and Belfrage, G. (1975) Nature, 258, 598. 42 Porath, J., and Olin, B. (1983) Biochemistry, 22, 1621. 43 Sulkowski, E. (1985) Trends Biotechnol., 3, 1. 44 Arnau, J., Lauritzen, C., Petersen, G.E., and Pedersen, J. (2006) Protein Expr. Purif., 48, 1. 45 Cuatrecasas, P., Wilchek, M., and Anﬁnsen, C.B. (1968) Proc. Natl Acad. Sci USA, 61, 636. 46 Einhauer, A., and Jungbauer, A. (2001) J. Chromatogr. A, 921, 25. 47 Brenac, V., Ravault, V., Santambien, P., and Boschetti, E. (2005) J. Chromatogr. B, 818, 61. 48 Girot, P., Averty, E., Flayeux, I., and Boschetti, E. (2004) J. Chromatogr. B, 808, 25. 49 Nau, D.R. (1986) Bio Chromatography, 1, 82–94. 50 Tiselius, A., Hjerten, S., and Levin, O. (1956) Arch. Biochem. Biophys., 65, 132.

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3 Chromatography Media 51 Giovannini, R., and Freitag, R. (2000) Bioseparation, 9, 359. 52 Giovannini, R., and Freitag, R. (2002) Biotechnol. Bioeng., 77, 445. 53 Jungbauer, A., Hahn, R., Deinhofer, K., and Luo, P. (2004) Biotechnol. Bioeng., 87, 364. 54 Grushka, E. (1972) Anal. Chem., 44, 1733. 55 Jungbauer, A. (2005) J. Chromatogr. A, 1065, 3. 56 Peterson, E.A., and Sober, H.A. (1956) J. Am. Chem. Soc., 78, 751. 57 Unger, K., Roumeliotis, P., Mueller, H., and Goetz, H. (1980) J. Chromatogr. A, 202, 3. 58 Unger, K., Schick-Kalb, J., and Krebs, K.F. (1973) J. Chromatogr., 83, 5.

59 Unger, K.K., and Janzen, R. (1986) J. Chromatogr. A, 373, 227. 60 Lowell, S., and Shields, J. (1991) Powder Surface Area and Porosity, Kluywer. 61 Ubiera, A.R., and Carta, G. (2006) Biotechnol. J., 1, 665. 62 Hahn, R., Bauerhansl, P., Shimahara, K., Wizniewski, C., Tscheliessnig, A., and Jungbauer, A. (2005) J. Chromatogr. A, 1093, 98. 63 Hunter, A.K., and Carta, G. (2000) J. Chromatogr. A, 897, 65. 64 Carta, G., and Ubiera, A. (2003) AIChE J., 49, 3066. 65 Martin, C., Coyne, J., and Carta, G. (2005) J. Chromatogr. A, 1069, 43. 66 Stickel, J.J., and Fotopoulos, A. (2001) Biotechnol. Prog., 17, 744.

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4 Laboratory and Process Columns and Equipment 4.1 Introduction

Very simple equipment was used in the early days of protein chromatography. The stationary phase was usually packed in a glass column equipped with a simple frit at the bottom and the mobile phase ﬂowed through the column under gravity. Efﬂuent fractions were collected manually, tested off-line for composition, and a chromatogram reconstructed from this information. High-resolution systems and sophisticated computer-controlled equipment capable of operating at high pressures up to 400 bar are available today. Chromatography workstations have been developed that integrate precise pumps, valves, and detectors, with validated data acquisition and storage. The required pressure and ﬂow rate determine the design and the different elements of a chromatography workstation. Pressure, itself principally a consequence of the particle size, ﬂow rate, and column length, is determined by the type of application. For high-resolution applications a large number of plates and, thus, small particles and/or long columns are needed which results in high pressures. On the other hand, when selectivity is high, a small number of plates is sufﬁcient and large particles and/or short columns can be used thus producing low pressure. As shown schematically in Figure 4.1, there are two different velocities in column chromatography: the superﬁcial velocity, u, and the interstitial velocity, v. The superﬁcial velocity is deﬁned as the volumetric ﬂow rate, Q, divided by the column cross-sectional area: u=

Q S

(4.1)

Because, in most cases, only the extra-particle space is available for ﬂow (see Figure 2.7) the actual ﬂuid velocity is higher than u and is approximated by the interstitial velocity, v, which is deﬁned as follows: v=

Q u = ε S×ε

Protein Chromatography: Process Development and Scale-Up. Giorgio Carta and Alois Jungbauer © 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 978-3-527-31819-3

(4.2)

126

4 Laboratory and Process Columns and Equipment

Figure 4.1

Interstitial velocity (v) and superﬁcial velocity (u) in column chromatography.

where

ε=

V0 Vc

(4.3)

V0 and Vc are the extra-particle and total column volumes, respectively. Typical values of ε are generally 0.3 to 0.4 for columns packed with particles, but higher values are obtained with monoliths. It should be noted that porous particles also contain liquid. This liquid is stagnant and is considered to be part of the stationary phase. The situation is more complex in perfusion chromatography where a part of the mobile phase ﬂows also through the gigapores of the perfusive particles (see Section 6.2.4). The superﬁcial mobile phase velocities used in protein chromatography vary from values in the range of 30 to 100 cm/h with column lengths between 30 and 100 cm for high-resolution application to values as high as 1000 cm/h and column lengths between 10 and 40 cm for capture applications. Column pressures are often limited to a few bars for large-scale columns, but can be up to 50 bar for laboratory-scale units.

4.2 Laboratory-scale Systems

Modern chromatographic systems suitable for laboratory or pilot-scale use include at least seven components: 1) Two pumps with mixer – a mixer is not needed in isocratic elution or stepwise elution when the elution buffer is not mixed in line but additional pumps are needed when buffers are mixed in-line from stock solutions.

4.2 Laboratory-scale Systems

Figure 4.2 Conﬁguration of a typical laboratory-scale workstation (AKTA Explorer from GE Healthcare). The unit includes two pumps with multiple inlets, a six-port sample injector valve with sample load pump, a

ﬂow-inverter valve, column selection valves, UV/VIS detector, conductivity and pH monitors, outlet selector valve, and a fraction collector. From the User Manual for the AKTA Explorer 10 and 100.

2) A feed or sample application system – either a valve with a calibrated loop or a separate feed pump. 3) A separation column ranging from ≈0.3 ml in the laboratory to as much as 1000 l or more for industrial scale use. 4) One or more detectors depending on the application. 5) A device for collection of fractions – in the simplest design this is a valve. 6) A computer control system for pumps and valves. 7) A recorder and/or electronic data acquisition system – certiﬁcation is required for industrial-scale production in a cGMP environment. These components are often integrated with sophisticated computer-control systems, which allow programming and execution of unattended runs to scout sequentially broad ranges of conditions and columns. Figure 4.2 shows schematically an example of such a unit. In general, desirable system properties are low dead or extra column volume, inert materials for any part that is comes in contact with product, accurate ﬂow, accurate in-line mixing of running buffers, and switching of the valves without interrupting ﬂow and generating pressure spikes.

127

128

4 Laboratory and Process Columns and Equipment Table 4.1 Summary of pumps used in protein chromatography in laboratory- and large-scale

systems. Pump type

Applications

Check valves

Pulsatile ﬂow

Syringe pump

For laboratory and high precision

+

−

Long-stroke piston pump in synchronized mode

For FPLC

−

−

Single piston

For laboratory and high precision

+

+

Dual piston

For laboratory and high precision

+

+

‘Fast’ ﬁll single piston

For laboratory and high precision

+

−

Peristaltic pumps

Laboratory to large scale, not suitable for pressures above 3 bar

−

+

Diaphragm pumps

Large scale up to 1000 m3/h

+

+

Rotary lobe pumps

Laboratory to large scale, up to 400 m3/h

−

−

Centrifugal pumps

Not suitable for protein chromatography

−

−

Long-stroke piston pumps

Reciprocating pumps

Figure 4.3

Schematic diagram of a single piston syringe pump.

4.2.1 Pumps

Positive displacement pumps are generally used to maintain an accurate ﬂow especially at high pressures. Table 4.1 lists examples of pumps used in protein chromatography. Syringe pumps (Figure 4.3) are characterized by very accurate, non-pulsatile ﬂow. However, the syringe volume must be sufﬁcient for a whole chromatography run. As a result, these pumps are ideally suited to capillary chromatography and

4.2 Laboratory-scale Systems

Figure 4.4

Schematic diagram of a dual long-stroke piston pump used in FPLC systems.

Figure 4.5 Schematic diagrams of tandem piston pumps used in AKTA systems from GE Healthcare. Four pump heads are used to generate gradients. Courtesy of GE Healthcare, Uppsala, Sweden.

micro LC. Dual long-stroke piston pumps operated in synchronized mode (Figure 4.4) have been used successfully in so-called FPLC (Fast Protein Liquid Chromatography) systems. In this case, while one piston delivers the liquid, the second ﬁlls up. When the ﬁrst piston is empty a valve is rapidly switched to the second pump with a momentary interruption of ﬂow. An advantage of syringe pumps is that no check valves are required. However, a pressure wave is generated during valve switching. Reciprocating piston pumps are usually installed in HPLC (High Performance Liquid Chromatography) systems, either as single or tandem units. These pumps require check valves, which may become fouled by biological ﬂuids. Additionally, these pumps generate pulsatile ﬂow which requires a pulse damper. Tandem reciprocating piston pumps are used in most modern laboratory-scale systems. An example is shown in Figure 4.5. Rotary lobe, diaphragm, and, sometimes, peristaltic pumps are used in large-scale systems and are discussed in Section 4.2. Pulse dampers are used to smooth out ﬂow pulsations and dampen pressure spikes. In the simplest construction, pulse dampers consist of a soft elastic tube, which expands with pressure thus compensating for the oscillating velocity. Diaphragm dampers consisting of an air- or liquid-ﬁlled cushion that is compressed

129

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4 Laboratory and Process Columns and Equipment

at high pressure and expands at low pressure, are also used. It should be noted that the system volume changes with ﬂow when a pulse damper is included. Corrections for this effect become important in the analysis of data when small columns are used. The solubility of air in aqueous solutions decreases with temperature. Thus, when temperature ﬂuctuations occur because of ambient changes or heat generated by mixing different solutions, air may be liberated from the mobile phase and bubbles may form. When these bubbles are introduced into the column they may either prevent uniform distribution of the liquid over the column diameter, or they may create cracks in the bed. At the laboratory-scale, these problems can be circumvented by degassing the mobile phase with vacuum or helium sparging. Inline vacuum degassers consisting of a tube that allows diffusion of air when a vacuum is applied are also used. 4.2.2 Buffer Mixers

Mixers are important in chromatography systems for the generation of gradients. Two buffers, rarely three, are blended in varying proportions to generate an appropriate gradient. This is accomplished using either low-pressure or high-pressure mixers. The former type uses a single pump with ratio inlet valves, while the latter use independent pumps with ﬂow rates adjusted over time. Both types require efﬁcient mixing in order to generate accurate gradients. Figure 4.6 shows an

Figure 4.6 Schematic diagram (left) and sketch (right) of mixer used in AKTA systems from GE Healthcare. Courtesy of GE Healthcare.

4.2 Laboratory-scale Systems

example of a mixer used in laboratory-scale units. It consists of a small tank or cell where two streams are introduced at the bottom and mixed by a magnetic stirrer. Upﬂow prevents accumulation of bubbles. With increasing ﬂow rate and higher viscosities the mixer volume must also increase in order to ensure complete mixing. A range of different mixer volumes may thus have to be used with workstations capable of operating over a very wide range of ﬂow rates. Gradient mixers can be modeled and designed with precision as demonstrated in [1]. Two particular aspects are important. The ﬁrst is when step changes in mobile composition are applied upstream of a mixer, the output is a sigmoidal function that depends on the characteristic mixing and residence times. As a rule of thumb, ﬁve mixer volumes are required to achieve virtually complete clearance of the mixer. However, larger volumes may be required if dead zones are present. The second aspect is when a gradient in composition is produced by adjusting the relative ﬂows of two pumps the real gradient differs from the ideal as a result of inaccuracies of the pumps at low ﬂow rates and mixing effects. In general, as shown in Figure 4.7, the accuracy of the gradient produced is greatest near the middle, which is also the ideal elution position when it is desired to elute a component under a linear gradient. Impingers and static mixers are also used for fast mixing. In the former type, jet streams from two nozzles are directed toward each other at a certain angle and a highly turbulent ﬂow is produced resulting in excellent mixing. This mixer is characterized by a minimal dead volume. However, high shear and gas bubbles can be formed in such a device. Static mixers with stationary vanes inside a tube are also used.

Figure 4.7 Accuracy of gradients generated by high pressure mixing and ideal peak position for elution in a linear gradient.

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4 Laboratory and Process Columns and Equipment

4.2.3 Monitors

The column outlet is normally continuously monitored most commonly by ultraviolet/visible absorbance (UV/VIS), conductivity, or pH. Refractive index (RI), ﬂuorescence, electrochemical, amperometric, radioactivity, light scattering, and mass spectrometry monitors are also used in laboratory-scale systems. In analytical SEC, light scattering and RI monitors are often used together for the absolute determinations of size and/or molecular mass. A simple two-detector method has been described by Wen et al. [2] for this purpose. UV detectors are effective because proteins exhibit a strong absorbance signal at 280 nm especially when they contain numerous tryptophan and tyrosine residues (see Section 1.2.2.1). Absorbance values can be converted to concentration using either the Lambert–Beer law or from an empirical calibration curve. The Lambert–Beer law is given by the following equation: A = log

()

I0 = ε m lc I

(4.4)

where A is the absorbance, εm the molar extinction coefﬁcient, l the optical path length, c the protein molar concentration, I0 is the intensity of the incident light, and I the intensity of the emergent light. The linear relationship expressed by Equation 4.4 between A and c is generally valid for A values less than 2. At higher values, UV/VIS detector readings generally become unreliable since the emergent light comprises a very small fraction of the incident light. Reducing the optical path length is helpful for concentrated protein solutions and cells with a length, l, of 1 to 2 mm are commonplace in preparative units. Conductivity monitors are frequently used to follow the salt concentration in the efﬂuent since aqueous salt solutions are the most commonly used mobile phases in protein chromatography (see Chapter 3). An example is shown in Figure 4.8. The conductivity, κ, is proportional to the distance between the electrodes, l, and inversely proportional to their area, A, and to the resistance offered by the solution passing through the cell according to the following equation:

Figure 4.8 Sweden.

Schematic diagram of a conductivity cell. Courtesy of GE Healthcare, Uppsala,

4.2 Laboratory-scale Systems

κ=

l R×A

(4.5)

In turn, Kohlrausch’s law relates the concentration of an electrolyte, C, to conductivity as follows:

κ = Λm × C

(4.6)

where the molar conductivity Λm is a function of temperature and solution viscosity and varies from salt to salt. For concentrated solutions, Λm also varies with salt concentration and this dependence can be described by the following empirical relationship: Λ m = Λ 0m − S C

(4.7)

where Λ is the molar conductivity at inﬁnite dilution and S is an empirical coefﬁcient dependent on the speciﬁc electrolyte. Figure 4.9 shows the conductivity of NaCl solutions at 18 °C and the corresponding molar conductivity in comparison with Equation 4.7. Note that there is a linear relationship between salt concentration and conductivity up to 1 M NaCl. In HIC higher salt concentrations are often used and this results in a non-linear relationship between conductivity and salt concentration. pH monitors are relatively slow in responding and in laboratory workstations such a detector can contribute signiﬁcantly to the overall dead volume. In 0 m

Figure 4.9 Conductivity and molar conductivity of sodium chloride solution at 18 °C compared to that calculated using Equations 4.6 and 4.7. Data from Lange’s Handbook of Chemistry.

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4 Laboratory and Process Columns and Equipment

laboratory-scale workstations this detector should be placed downstream of all other detectors. A drift in response is often observed over time making accurate in-line pH measurements difﬁcult. 4.2.4 System Volumes

The mixers, detectors, tubing and connectors contribute to delays and extra column band broadening that can be signiﬁcant in the analysis of laboratory data. Such non-ideal conditions can also impact high-resolution separations conducted on a larger scale. Thus, care must be taken to eliminate these effects as completely as possible. Under certain conditions, experimental data can be corrected as indicated schematically in Figure 4.10 to take into account extra-column effects. The characteristic parameters of the eluted peak are the apparent ﬁrst moment, µapparent, and 2 the apparent variance, σ apparent , which are calculated as described in Chapter 8. 2 Similar quantities, µextra and σ extra , can be determined for the system without the 2 column. Finally, the corrected values µcolumn and σ column which describe the actual column behavior can be calculated using the following equations:

µcolumn = µapparent − µextra

(4.8)

2 2 2 σ column = σ appraent − σ extra

(4.9)

These corrections can be signiﬁcant when scaling down to minimize the amount of stationary phase and protein required and are almost always needed at the laboratory scale, particularly with high-resolution and low-capacity stationary phases.

Figure 4.10

Corrections for extra-column effects to apparent peak ﬁrst moment and variance.

4.3 Process Columns and Equipment

For example, in the system illustrated in Figure 4.2, dead volumes are in the order of 1 ml, while changing from one buffer to another requires as much as 10 ml because of the exponential wash-out characteristics of the mixer, pump heads, and connecting tubes. Since columns in the range of 0.5 to 5 ml are often used on the laboratory scale, it is obvious that corrections for extra-column effects are generally needed. In process systems, bubble traps, discussed in Section 4.3.2, are often signiﬁcant contributors to extra-column band spreading while other extra-column volumes are usually negligible relative to the column volume.

4.3 Process Columns and Equipment 4.3.1 Columns

Process columns are usually large, wide columns made of stainless steel, glass, or plastics with total volumes as large as 1000 l and diameters up to 2 m. An example is shown in Figure 4.11. Such columns are equipped with headers and base plates to contain the packed bed within a cylindrical vessel. Modern designs allow automated movement of the header, although manually operated systems are also frequently used. The design of process columns has to take into account several requirements including: (i) the ability to ﬁll the column with the stationary phase during packing; (ii) the ability to distribute the mobile phase uniformly over the cross section of the packed bed during operation; and (iii) sanitary design allowing the column to be cleaned with no unswept areas. Distributing the ﬂow from the inlet pipe which is usually less than a few cm in diameter to column diameters of up to 2 m, is challenging and several designs including ‘fractal schemes’ have been proposed to accomplish this. An example which includes special bafﬂes to distribute the liquid over the bed is shown in Figure 4.12. Multiple inlet ports are sometimes used to facilitate distribution of the liquid over columns with large diameter-to-length aspect ratios. The chromatography particles are retained in the column either by a mesh or by a sintered frit made from polymeric materials or metals. Multilayer designs are also often used. In either case, the mesh size or diameter of pores in the frit depends on the diameter of the stationary phase particles and their particle size distribution and must be sufﬁciently small to retain the smallest particles. Early large-scale column designs did not include movable headers, which resulted in large voids at the top caused by bed compression during packing or operation. However, systems with adjustable headers that enable compressible media to be efﬁciently packed with the elimination of voids are now available. The header is usually mounted on a hollow spindle which allows it to be lowered and raised during packing. The header itself must be sealed against the wall. For this purpose, inﬂatable O-rings, which are expanded after the header has been ﬁxed in position, are sometimes used. Dynamic axial compression systems are also

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4 Laboratory and Process Columns and Equipment

Figure 4.11 Schematic diagram of a process-scale column (GAQuick Scale column from Millipore Inc. Bedford, MA, USA).

4.3 Process Columns and Equipment

Figure 4.12 Example of ﬂow distributor for process-scale columns (reproduced by permission of Millipore Inc. Bedford, MA, USA).

available. In this case, a preset pressure is maintained automatically compensating for any change in packed bed height that may occur during operation. There are also various column designs which facilitate the process of packing the stationary phase. In the simplest case a slurry of the particles suspended in a suitable packing buffer is rapidly poured or pumped into the open column with the header removed. Other systems incorporate valves that allow the slurry to be pumped directly into the column with the header in place. Hofmann [3], for example, designed a column header with a spray nozzle operated by a threeposition valve that can be used to pack and unpack the stationary phase by pumping the slurry into place until the desired degree of compression is attained. Figure 4.13 shows an example of such a design. Alternative designs with top headers actuated with electric or pneumatic motors are also available as shown for example in Figure 4.14. In this case, a calibrated amount of slurry is poured in. The top header is then inserted and lowered at a preset velocity, usually together with an applied ﬂow of the packing buffer. The header is stopped when the desired degree of bed compression is attained.

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4 Laboratory and Process Columns and Equipment

Figure 4.13 Pilot-scale Resolute® chromatography column equipped with packing nozzles at the header and bottom (Pall Corporation, USA).

Figure 4.14 Example of a chromatography column designed with a motorized top header: the InPlace™ column with packing motor (Bio-Rad Laboratories, Hercules, CA, USA).

4.3 Process Columns and Equipment

Figure 4.15 Schematic diagram of a radial chromatography column (left) and radial ﬂow monolith with stainless steel housing for protein chromatography (right) (BIA Separations, Ljubljana, Slovenia).

Radial ﬂow chromatography columns are also available. Such columns are mainly used for very soft stationary phases or for monolithic columns; in both cases the bed height must be short. In these columns the liquid does not ﬂow in the axial direction but in the radial direction, as shown for example in Figure 4.15. In this case, the annular geometry makes it possible to use large cross sections with very short bed depths, thus reducing the footprint of the unit. For such columns, the superﬁcial velocity changes in the radial direction, r, according to the following equation: u=

Q 2π rL

(4.10)

where Q is the ﬂow rate and L the height of the annular bed. The average velocity u ¯ is calculated as follows: u=

Q ln (rout r in ) ⋅ 2π L rout − rin

(4.11)

where rout and rin are the outer and inner radii. In industrial-scale columns the change of velocity in the radial direction can be neglected because the inner and outer radius often differ by less than 10% [4]. In extreme cases, the performance may change with scale, because the same average velocity may be generated, although the velocity-proﬁle in radial direction may be different. In the case of monoliths this is less crucial, since performance is unaffected by ﬂow over a wide range. Theoretical models for radial ﬂow chromatography columns are discussed in [5, 6].

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4 Laboratory and Process Columns and Equipment

4.3.2 Systems

An example of a process-scale chromatography system is shown schematically in Figure 4.16. Either one pump with multiple inlets or two pumps are used to generate step and gradient changes in mobile phase composition. Various detectors are located both downstream of the column, to monitor the efﬂuent composition, and upstream for feedback control of mobile phase composition. A system of valves allows the column to be bypassed and operated in the reverse direction. This capability is useful especially during packing to remove air bubbles when the column header is inserted or lowered. Rotary lobe pumps as illustrated schematically in Figure 4.17 or diaphragm pumps are commonly used in large-scale systems (see also Table 4.1). These positive displacement pumps are capable of ﬂow rates of up to 400 m3/h, can be steam sterilized in situ and are available in sanitary designs. They can be used to pump protein solutions because they generate relatively low shear. Peristaltic pumps are also frequently used because of their low cost. Additionally, the tube can be easily replaced and thus cleaning validation can be avoided when different products are handled with the same pump in a cGMP environment. Peristaltic pumps generate a pulsatile ﬂow requiring a pulse damper and ﬂow controllers when precision pumping is required since the ﬂow rate is not always proportional to the rotation rate and can vary with use as the tube ages. Moreover, they are not suitable for use under high pressures.

Figure 4.16 Conﬁguration of a typical process-scale chromatography skid (Bio-Rad Laboratories, Hercules, CA).

4.3 Process Columns and Equipment

Figure 4.17

Schematic diagram of a rotary lobe pump.

Gas bubbles are often a serious concern in large-scale systems as they can severely disrupt the packing materials in the columns. Such bubbles are often formed in response to temperature changes or by cavitating pumps. Degassing is not practical for large-scale applications. Thus, air sensors which halt the ﬂow when bubbles are detected, and bubble traps which remove bubbles, are placed in-line in front of the column. A bubble trap is a vessel with an inlet and outlet at the bottom that allows bubbles to accumulate at the top. From time to time the accumulated air is released by opening a port at the top of the vessel. To a certain extent bubble traps also act as pulse dampers because the accumulated air serves as a cushion. On the other hand, bubble traps can signiﬁcantly lower the efﬁciency of a chromatography system because they increase dead space (often by several liters) and contribute to band spreading. As shown in Figure 4.16, valves are often used to bypass the bubble trap when needed. 4.3.3 Column Packing

The method used to pack the stationary phase in the column depends on the mechanical properties of the chromatography beads and the available hardware. In some cases, either dry or slurry packing is possible. In practice, however, dry packing is limited to materials that do not collapse or shrink substantially when water is removed. In all other cases slurry packing is the method of choice. Slurry concentrations vary depending on the application but values around 50% (v/v) are

141

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4 Laboratory and Process Columns and Equipment

typical. For compressible media, a compression factor, deﬁned as the ratio of the settled and compressed packed bed height, is typically determined experimentally ideally giving the correct slurry volume for the desired ﬁnal bed height. Packing large diameter columns is difﬁcult however, since bed compression varies with column diameter and initial bed height as discussed in Section 3.4.4, and sometimes requires extensive experimentation. Abrasive forces must generally be avoided during column packing. Such forces are generated when the slurry is agitated or pumped, or when the column is compressed and the particles sheared off along the column wall. It is also important to keep the slurry agitated in order to avoid setting of the particles by gravity. Due to the often broad particle size distribution of many stationary phases, classiﬁcation of particles can occur during settling leading to a non-homogeneous bed. The large particles settle faster than the smaller ones leading to gradients in the packing density that result in higher operating pressures. After pouring or pumping the slurry into the column, the ﬂow of the packing buffer is established either by suction from the bottom or by pumping from the top. Suction packing is sometimes simpler, but limited to a pressure differential of 1 bar or less if dissolved gases are present. In either case, when the slurry is compacted, the ﬂow is stopped, and the header lowered until it touches the top of the bed. Sometimes mechanical compression is applied by further lowering the header. This should be carried out with care however, since mechanical compression can lead to severe channeling or rupture of the particles. Putting the header in place after the bed is consolidated is also sometimes challenging since air bubbles are easily introduced and remain trapped between the top of the bed and the header. In this case, reversing the ﬂow can sometimes remove the air bubbles. As an alternative to ﬂow packing and if a motorized header is available, the slurry can be compacted by lowering the header while allowing the packing buffer to ﬂow out the bottom. Systems that allow the slurry to be pumped in are also available as discussed in Section 4.2.1. In this case, the slurry is pumped into the column with a slurry pump to the desired degree of compaction in a preset column volume. Packing quality is typically determined by measuring the HETP and the peak asymmetry factor (see Section 8.2). For this purpose a pulse of an inert tracer, frequently salt, is injected into the column and its appearance at the column outlet recorded as a function of time. HETP and the asymmetry factor must fall within deﬁned ranges. Actual numbers depend on the application and the type of stationary phase. However, in general reduced HETP values, h = H/dp, in the range of 2 to 6 and asymmetry factors, As, in the range of 0.8 to 1.2 are desirable and indicate a reasonably homogeneous ﬂow and a well-packed column. Packing quality can also be assessed, for example during the cyclic operation of a column from the response to steps in salt concentration during regeneration or re-equilibration, since the pulse response can be obtained from the derivative of the response to step inputs. Additionally, visual indications can be achieved by injecting dyes when using columns with a transparent wall.

References

References 1 Kaltenbrunner, O., and Jungbauer, A. (1997) J. Chromatogr. A, 769, 37. 2 Wen, J., Arakawa, T., and Philo, J.S. (1996) Anal. Biochem., 240, 155. 3 Hofmann, M. (1998) J. Chromatogr. A, 796, 75. 4 Jungbauer, A., and Hahn, R. (2002) in Monolithic Materials: Preparation, Properties

and Applications (eds F. Svec, T. Tenikova, and Z. Deyl), Elsevier, New York, p. 561. 5 Huang, S.H., Lee, W.C., and Tsao, G.T. (1988) Chem. Eng. J., 38, 179. 6 Gu, T., Tsai, G.J., and Tsao, G.T. (1991) Chem. Eng. Sci., 46, 1279.

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145

5 Adsorption Equilibria 5.1 Introduction

In general, adsorption is deﬁned as the concentration of a species at a solid surface. Molecules interact with surfaces as the result of two types of forces: dispersion– repulsion forces (also known as London or van der Waals forces); and electrostatic forces, which exist as the result of a molecule or surface group having a permanent electric dipole or quadrupole moment or a net electric charge. Short-range repulsive forces are dominant very near the surface, but decrease rapidly as the distance from the surface increases, while attractive electrostatic forces are generally longer range. As a result, the concentration of molecules near a surface varies with the distance from it. Ideally, integrating over a sufﬁciently long distance from the surface provides the total surface excess of the molecule. Alternatively, a so-called ‘Gibbsian dividing surface’ can be deﬁned which extends a ﬁnite distance from the surface so that there is a uniform concentration of adsorbate molecules between this surface and that of the adsorbent. Such a description of adsorption is generally satisfactory for small molecule adsorbates, whether in the gas or liquid phase, and can be extended to multi-component systems in a thermodynamically consistent manner [1]. However, adsorption of proteins and other macromolecules is usually much more complicated and currently not amenable to precise theoretical treatment. Complicating factors include the following: 1) Proteins contain a heterogeneous distribution of charged and hydrophobic groups; some of these groups are located on the external surface, while others are partially or completely buried within the core (see Chapter 1). Thus, on one hand, protein adsorption frequently has a directional character, which makes it impossible to represent proteins as simple spheres for the purpose of predicting adsorption equilibria. On the other hand, protein adsorption is frequently ‘multivalent’, in the sense that the protein molecules interact simultaneously with multiple binding sites on the surface or through complex biospeciﬁc interactions involving the cooperative contributions of multiple amino acid residues.

Protein Chromatography: Process Development and Scale-Up. Giorgio Carta and Alois Jungbauer © 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 978-3-527-31819-3

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5 Adsorption Equilibria

2) Protein adsorption is often accompanied by local and sometimes global unfolding events; as a result, the structure of bound proteins may be different from that of the protein molecules in solution. 3) Broad ranges of ionic strength are often encountered in protein chromatography; as a result, the Debye length (see Equation 3.9), which can be taken as a measure of how far the electrostatic force ﬁeld extends from the adsorbent surface, varies from a fraction of the size of a protein molecule to many molecular diameters so that the dielectric properties of the protein affect the diffuse electrical double layer itself. Additionally, in the case of adsorption in particles with small pores, the diffuse electrical double layers can overlap, so that protein adsorption can effectively occur in a non-electrically neutral medium. This is especially important in soft matrices and polymer gels, where the mesh size is comparable in magnitude to the size of the protein molecules. 4) Protein molecules occur in a broad range of sizes (see Chapter 1), and in some instances have a tendency to self-associate; this can occur in solution (so that multiple molecular species are in fact present), but also in the stationary phase. As a result, attractive or repulsive interactions between adsorbed protein molecules can often be important. Additionally, exclusion of protein molecules from a fraction of the available surface can occur when the adsorbent has a distribution of pore sizes (see Chapter 3). 5) Protein adsorption is often quite slow because of diffusional resistances. Even with small particles and large pores, the rate of protein adsorption is sometimes very slow because of limitations in the binding kinetics. As a result, from a practical viewpoint, it is sometimes difﬁcult to experimentally determine adsorption equilibrium because unfolding, aggregation, or degradation (which can occur, for example, when proteases are present as impurities) take place before equilibrium is established. In these cases, a true adsorption equilibrium may not be established since the protein may undergo substantial molecular changes before reaching equilibrium with the surface. As a result of these complications, empirical or semi-empirical approaches are generally needed to describe protein adsorption equilibrium. Prediction of multicomponent protein adsorption is also challenging. Approaches such as the ideal adsorbed solution theory [1] suitable for small molecules often fail for proteins because of the large size and complex interactions. Thus, experimental determinations are almost always necessary. Experimentally, adsorption equilibria are frequently measured by suspending the particles in a protein solution and allowing sufﬁcient time for equilibrium to be attained. In this case, it is often easier to determine the ultimate binding capacity with a high protein solution concentration. However, determining protein binding at low concentration is often difﬁcult since the times needed to achieve equilibrium can become very long. Using small particles or crushing the original material is helpful in these cases. Automated, robotically-operated equipment is now available to measure adsorption equilibrium by such batch methods in 96-well plate format over broad ranges of protein concentration, mobile phase compositions, and for different stationary phases.

5.2 Single Component Systems

Chromatographic methods including frontal analysis, isocratic elution, and linear gradient elution are also used to determine adsorption equilibria (see Chapter 2). Compared to small molecules, the determination of protein adsorption equilibrium from frontal analysis is complicated by the fact that broad bands are often obtained as a result of strong diffusional limitations, while isocratic elution is sometimes difﬁcult to implement because of the extreme sensitivity of protein adsorption equilibrium to mobile phase composition. Gradient elution alleviates this difﬁculty to some extent as is discussed in Chapter 9. Adsorption equilibria can also be deduced from the tailing of elution curves, when deformation of the peak is caused by non-linearity of the isotherm, and broadening caused by mass transfer effects is limited. In practice, in the context of a biopharmaceutical process development effort, acquisition of adsorption equilibrium data is often difﬁcult because the amounts of protein available for testing are frequently very limited. As a result, care must be taken to ensure that precise data are obtained with minimal amounts of protein. Following empirical determinations of adsorption equilibria, the data are generally correlated using suitable adsorption isotherm expressions, which are reviewed in the following sections.

5.2 Single Component Systems

Single component systems are obviously not the norm in bio-chromatography applications where complex mixtures are usually encountered. Nonetheless, systems involving highly selective adsorbents, situations where only minor impurities are present, or cases where the adsorption equilibrium is linear may be considered to fall into this category. The concentration of adsorbed protein in the stationary phase at equilibrium with the mobile phase is expressed by the adsorption isotherm. In general, a linear relationship between adsorbed and free protein is expected in the low concentration range, although this may be difﬁcult to observe in practice under conditions where protein adsorption is highly favorable. In most cases, the relationship becomes non-linear at higher protein concentrations and levels off to a maximum capacity. The linear limit is dependent on the concentration of accessible binding sites and on the speciﬁc afﬁnity of the protein for these sites. Conversely, the maximum capacity is generally limited by the accessible surface area or by the concentration of binding sites. In addition to temperature, the mobile phase composition will frequently affect protein adsorption. Thus, adsorption isotherms are obtained while maintaining a constant mobile phase composition. Some interaction types are more sensitive to the mobile phase composition than others; ion-exchange and hydrophobic adsorption are very sensitive whereas biospeciﬁc interaction is often insensitive to salt as long as extremely high salt concentrations or extreme pHs are avoided. Figure 5.1 shows examples of protein adsorption isotherms for IEC, HIC, and BIC. As seen in Figure 5.1a for IEC, the adsorption isotherm is extremely favorable at low salt concentrations but becomes less favorable (indicated by a shallower isotherm) as the salt concentration is increased. As seen in Figure 5.1b, for a

147

5 Adsorption Equilibria 350

(b)

250 (a) 300 250

q (mg/ml)

q (mg/ml)

200 150 100

200 150

+

20 mM Na

Cytochrome c - pH 4.0 Lysozyme - pH 6.5 Myoglobin - pH 5.0 Albumin - pH 5.0 IgG - pH 5.0

100

+

50 mM Na 85 mM Na+ + 120 mM Na

50

50

0

0 0

0.5

1

1.5

2

2.5

0

3

0.5

1

1.5

2

2.5

3

3

3

C (mg/cm )

C (mg/cm ) 100

(d)

120 (c) 80

100 80

q (mg/ml)

q (mg/ml)

148

60 40

Ammonium sulfate 2.0 M 1.75 M 1.4 M

20 0 0

0.5

1

1.5

2

2.5

3

60

40

20

Ground particles Whole particles

0 3.5

3

C (mg/cm )

Figure 5.1 Examples of adsorption isotherms for ion exchange chromatography (a, b), hydrophobic interaction chromatography (c), and Protein A afﬁnity chromatography (d). The experimental data have been

0

0.5

1

1.5

2

2.5

3

3.5

3

C (mg/cm )

approximated by the Langmuir adsorption isotherm with parameters qm and K ﬁtted for each mobile phase composition and each protein. Adapted from [2].

constant salt concentration, different apparent capacities are obtained for different proteins as a result of different binding strengths and accessible surface areas. Additionally, repulsive or attractive interactions between the adsorbed molecules can affect the maximum amount absorbable on a given stationary phase. In the case of HIC, shown in Figure 5.1c, adsorption is promoted by increasing the concentration of a kosmotropic salt, ammonium sulfate in this particular example. The isotherm is highly favorable above 2 M ammonium sulfate but becomes more linear as the ammonium sulfate concentration is reduced. Finally, for BIC, exempliﬁed by IgG binding to a Protein A adsorbent in Figure 5.1d, the adsorption isotherm is extremely steep as a result of the very stable Protein A–IgG complex. Note that, in this example, similar capacities are obtained with whole or crushed particles, conﬁrming that diffusional resistances did not affect the ﬁnal results and

5.2 Single Component Systems

149

that the accessible surface area is independent of particle size. As discussed in Chapter 3, the isotherm in this case is only weakly dependent on salt concentration but can be shifted by changing the pH. A model used commonly to describe protein adsorption equilibrium is the Langmuir isotherm, which was originally developed for the adsorption of gases onto metal surfaces [3]. A simple derivation of this model assumes a stoichiometric association of an adsorbate molecule, P, with a surface-bound ligand, L, according to the following equation: k1   → PL P + L← 

(5.1)

k2

where k1 and k2 are forward and reverse rate constants for adsorption and desorption, respectively. Accordingly, the adsorption rate is described by the following equation: d [PL ] = k1 [P ][L ] − k2 [PL ] dt

(5.2)

which is subject to the following constraint:

[L ] = [L0 ] − [PL ]

(5.3)

where [L0] is the total concentration of surface-bound ligands. At equilibrium, d[PL]/dt = 0. In this case, combining Equations 5.2 and 5.3 yields: q=

qmKC 1 + KC

(5.4)

where q ∝ [PL] is the adsorbed concentration, qm ∝ [L0] the maximum adsorption capacity, C ∝ [P] the concentration in solution, and K = k1/k2 is the equilibrium constant for reaction 5.1. Note that q is normally expressed per unit volume or mass of stationary phase and can be deﬁned in molar or mass units. Figure 5.2

q = 100 mg/ml

K = 100 ml/mg K = 2 ml/mg

80

m

60

q = 50 mg/ml m

40

20

q = 10 mg/ml m

0 0

1

2

3

Concentration in liquid phase, C (mg/ml)

Concentration in solid phase, q (mg/ml)

Concentration in solid phase, q (mg/ml)

100 q 100

80

m

m

60

40

20 1/K 0 0

1

2

3

Concentration in liquid phase, C (mg/ml)

Figure 5.2 Langmuir adsorption isotherm with different qm and K values (left) and limiting behavior for low and high C values (right). M is the initial slope or Henry constant and 1/K is the liquid phase concentration in equilibrium with one-half of qm.

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5 Adsorption Equilibria

illustrates the effects of the parameters qm and K and the limiting behavior of the isotherm. As C → 0, the Langmuir isotherm approaches the linear limit: q ≈ qmKC = mC

(5.5)

where m = qmK is the initial slope of the isotherm, which is sometimes referred to as the Henry constant. On the other hand, as C → ∞, the isotherm approaches the maximum capacity q ≈ qm. The term 1/K represents the liquid phase concentration at which q is equal to one-half of the maximum capacity, qm. Although the underlying assumptions of the Langmuir isotherm are often not met physically in protein adsorption, this model still provides a reasonable description of adsorption equilibrium data for many systems as shown for instance in Figure 5.1. The model parameters, however, have to be determined by data ﬁtting at each mobile phase composition, this can be done either by non-linear regression or using linearized forms. For example, rearranging Equation 5.4 as follows: q = Kqm − Kq C

(5.6)

provides a linear relationship between q/C and q. The data can be graphed accordingly (often referred to as a Scatchard plot) to test conformity with the Langmuir model and the parameters determined by linear regression. An example is shown in Figure 5.3. For many practical conditions (see for example Figure 5.1), isotherms found in protein chromatography exhibit very large initial slopes, so that the determination of the K-value by regression of isotherm data can be rather uncertain. For process applications, it is useful to express the Langmuir isotherm in terms of the separation factor R, which is deﬁned as: R=

1 1 + KCref

(5.7)

35

50 qm = 30.6 mg/ml K = 4.52 ml/mg Reg. coeff. = 0.992

30 40

30

20

q/C (-)

q (mg/ml)

25

15

20

10 10 qm = 30.6±0.2 mg/ml K = 4.58±0.24 ml/mg

5 0

0

0

1

2

3

C (mg/ml)

4

5

20

22

24

26

28

30

32

C (mg/ml)

Figure 5.3 Example of a protein adsorption isotherm with Langmuir parameters determined by non-linear least square ﬁt (left) and corresponding Scatchard plot with parameters determined from linear regression (right).

5.2 Single Component Systems

where Cref is a reference concentration, normally taken as either the feed or the initial values encountered in the process. Accordingly, equilibrium is expressed by the following constant-separation factor isotherm: C Cref q = qref R + (1 − R )C Cref

(5.8)

where qref is the adsorbed concentration in equilibrium with Cref. Equation 5.8 is mathematically equivalent to the Langmuir isotherm for R ≤ 1. For a given value of K, R is obviously a process-dependent parameter that deﬁnes the relative curvature of the isotherm for a speciﬁc application. Consider, for instance, an adsorptive capture step where a protein is removed from a feed solution of concentration CF. If the feed solution is very dilute (Cref = CF → 0), KCF approaches zero and R approaches unity. For these conditions, Equation 5.8 yields q ∼ (qref/Cref)C = mC so that the isotherm can be considered to be linear. On the other hand, for a concentrated feed, KCF becomes much larger than unity so that R → 0. Under these conditions, Equation 5.8 yields q ∼ qref = qF, where qF is the adsorbed concentration in equilibrium with the feed, indicating that the isotherm can be considered to be nearly rectangular. In practice, the isotherm can be considered to be nearly linear when R > 0.9 − 1 and nearly rectangular when R < 0.1 Intermediate values of R between 1 and 0 correspond to increasing degrees of non-linearity becoming increasingly more favorable as R approaches zero. This distinction regarding the curvature of the isotherm is important since as discussed in Chapters 6, 7, and 8, the shape of the isotherm has profound effects on the dynamics of adsorption in batch and column systems. Other well known empirical isotherm models are the Freundlich [4], Temkin [5], Toth [6], and the Langmuir–Freudlich or Sips isotherm [7]. The Freundlich isotherm is given by the following equation: q = aC 1 b

(5.9)

where a and b are empirical constants with b > 1. This equation does not have the expected linear limit at low concentrations and does not indicate that a maximum adsorption capacity is attained as C → ∞. However, conformity with this model, indicated by a linear log-log plot of experimental data, is sometimes found and is taken as evidence that the adsorbent surface is heterogeneous. The Temkin isotherm has been proposed to describe adsorption on strongly heterogeneous surfaces and is given by: q = a × ln (1 + KC )

(5.10)

where a is an empirical constant and K is the equilibrium binding constant corresponding to the maximum binding energy. It has been used to describe binding of proteins to metal chelate adsorbents. The Toth isotherm is given by: q=

qmC 1

 1 + Cb  b  Kb 

(5.11)

151

5 Adsorption Equilibria 30

25

20

q (mg/ml)

152

15

10 Langmuir-Freundlich

5

Langmuir 0 0

1

2

3

C (mg/ml)

Figure 5.4 Comparison of Langmuir–Freudlich (b = 2) and Langmuir isotherm (b = 1) with K = 4 and qm = 30.

where Kb and b are empirical constants. This isotherm, often used to correlate adsorption data for highly heterogeneous surfaces, becomes linear in the dilute limit (C → 0) and reaches a maximum capacity qm as C → ∞. Finally, the LangmuirFreundlich isotherm is given by: qm (KC ) b 1 + (KC ) b

q=

(5.12)

where b is an empirical constant. When b = 1 this equation becomes identical to the Langmuir isotherm. Conversely, as can be seen in Figure 5.4, for values of b > 1, this equation describes a sigmoidal isotherm with an inﬂection point at low C-values and a plateau when C → ∞. The sigmoidal form has been interpreted as an indication of a cooperative adsorption process. A general model describing multilayer adsorption has been proposed by Brunauer, Emmett, and Teller and is known as the BET isotherm [8], which is given by the following equation: q=

qmKC  1 + KC − C   1 − C   Cs   Cs 

(5.13)

where K and Cs are constants and qm is the monolayer adsorption capacity. Although formally developed for gases, this isotherm can be also used for an empirical description of the multilayer adsorption of proteins. This can occur, for example, for proteins that have a tendency to associate or aggregate with surface-bound protein molecules. In general, when applied to the description of protein adsorption, the parameters in the model equations discussed above are of an empirical nature and depend

5.2 Single Component Systems

on the mobile phase composition (salt concentration, pH, concentration of organic modiﬁers etc.). Thus, these isotherms have value principally as correlating functions to describe the dependence of q on C at a given mobile phase composition. In the case of protein adsorption on ion exchangers it is possible however, to describe the effects of mobile phase composition by assuming that adsorption occurs through the stoichiometric exchange of ions. Kopaciewicz et al. [9] introduced the stoichiometric displacement model (SD) for this purpose. The underlying assumption is that protein adsorption occurs via the stoichiometric exchange with counter-ions. Accordingly, for the adsorption of a protein onto a cation exchanger with sodium as a counter-ion, we have: Pz + + z R − Na + R z−Pz + + z Na +

(5.14)

where Pz+ is the protein, assumed to have positive charge z and R− represents the charged ligands bound to the stationary phase. An equivalent expression can be written for the adsorption of a negatively charge protein Pz− on an ion exchange with an anion, for example, Cl−, replacing Na+. It should be noted that, in general, z is different from the net charge of the protein at a given pH, since not all charged amino acid residues can interact with the stationary phase. Thus, z is referred to as the effective charge or binding charge. An equilibrium constant for Equation 5.14 can be deﬁned as follows:

[Rz−P z + ][Na + ] z [P z + ][R −Na + ] z

Ke =

(5.15)

where, ideally, the terms in brackets represent thermodynamic activities. In practice, concentrations rather than activities are often used. In this case, applying a charge balance to the surface-bound charged ligands gives the following result:

[R −Na + ] + z [Rz−P z + ] = [R − ]0

(5.16)

−

where [R ]0 is the total concentration of protein-accessible ligands in the stationary phase. Combining Equations 5.15 and 5.16 yields the following result: q=

(q0 − zq )z C K e ( CNa + )

z

(5.17)

where q and C have been used to denote adsorbed and ﬂuid phase concentrations, respectively, and q0 has been used in lieu of [R−]0. At low q values, which are obtained at higher salt concentration and very low protein concentrations, Equation 5.17 reduces to the following linear relationship: q≈

K e q0z C = mC ( CNa+ )z

(5.18)

where the slope is given by: m=

K e q0z ( CNa+ )z

or in a linearized form, by:

(5.19)

153

154

5 Adsorption Equilibria

log m = log (K e q0z ) − z log CNa +

(5.20)

The ensuing log-log relationship between protein retention and salt concentration under linear isotherm conditions in frequently observed experimentally in ionexchange chromatography (see Chapter 3). The maximum protein binding capacity, attained at low salt and high protein concentration, is obtained from Equation 5.17 as follows: qmax =

q0 z

(5.21)

Obviously, qmax depends on the total concentration of protein-accessible, surfacebound charged ligands and on the number of ligands that are involved in the assumed stoichiometric exchange process. Practical application of the SD model requires an empirical determination of the parameters q0, Ke, and z with consistent molar units. A commonly used procedure is to determine z and the product K e q0z based on Equation 5.20 using m values that have been empirically determined from isocratic or gradient elution experiments at relatively high salt concentrations. q0 is then determined from Equation 5.21 by measuring qmax from adsorption equilibrium experiments at low salt concentrations. Alternatively, all three parameters can be determined by non-linear regression of isotherms obtained over a range of salt and protein concentrations. The steric mass action model (SMA), introduced by Brooks and Cramer [10], is an important reﬁnement of the SD model. Figure 5.5 illustrates the underlying concept. The key assumption is that because of the large footprint of the protein molecule, protein binding not only involves z-ligands through a counter-ion exchange process as in the SD model, but also results in the shielding or steric hindrance of a number σ of these ligands. Extension of the SD model to account for this effect results in the following isotherm expression: q=

K e [q0 − ( z + σ ) q ]z C ( CNa+ )z

(5.22)

Figure 5.5 Schematic drawing of the effects of steric hindrance on protein binding to the surface of a cation exchanger.

5.2 Single Component Systems

which includes the steric hindrance or shielding factor, σ, as an additional parameter. Obviously, when σ is zero, the SMA model reduces to Equation 5.17. The low loading limit (q → 0) of Equation 5.22, obtained at high salt or very low protein concentrations, is also the same as that of the SD model. Thus, the same log-log relationship between retention and salt concentration is expected for these conditions. The maximum protein binding capacity is, however, different, and is given by: qmax =

q0 z +σ

(5.23)

The model parameters q0, Ke, z, and σ can thus be determined in consistent molar units as discussed above for the SD model or by non-linear regression of isotherm data over suitably ample ranges of salt and protein concentration. Figures 5.6 and 5.7 show an example for cytochrome c adsorption on a cation exchanger. As shown in Figure 5.6, the isotherm data can be ﬁtted with similar accuracy using either the Langmuir model or the SMA model. However, while the former requires different parameter values at each salt concentration, the latter requires just four parameters to describe the effects of protein and salt concentration over very ample ranges. The low-loading behavior of the data is illustrated in Figure 5.7, which shows the initial slope of the isotherm as a function of Na+ concentration on linear and log-log scales. The slope of the log-log plot yields the effective charge, z. As can be seen in this ﬁgure, the relationship between the isotherm slope and the counter-ion concentration is an extremely steep curve, which results from the multipoint interaction of the protein with the charged ligands as demonstrated by the high z values.

Adsorbed protein concentration, q (mg/ml)

200

SMA model Langmuir model 0 mM NaCl

150

100

25

50

50

75 100 150

0 0.0

0.5

1.0

1.5

2.0

Protein concentration in solution, C (mg/ml)

Figure 5.6 Isotherm data for cytochrome-c adsorption on S-HyperD-M in a 10 mM Na2HPO4, pH 6.5 buffer as a function of added NaCl. Solid lines are calculated from the SMA model with q0 = 220 µmol/cm3 Ke = 0.2, = 5.7 and σ = 9.1. Reproduced from [11].

155

156

5 Adsorption Equilibria

Figure 5.7 Linear isotherm slope for cytochrome-c adsorption on S-HyperD-M in a 10 mM Na2HPO4, pH 6.5 buffer as a function of added Na+ concentration using linear (left)

and log-log scales (right). Solid lines are calculated from the SMA model with q0 = 220 µmol/cm3, Ke = 0.2, and z = 5.7.

In general, the z values obtained for different proteins are in the range of 3 to 12, with smaller values observed for smaller proteins and larger ones for large proteins such as IgM (Mr ∼ 950 kDa) for which values of z of around 12 have been reported. Two important considerations need to be made in this regard. The ﬁrst is that, as already noted, the effective charge z is generally smaller than the net charge of the protein, since charged residues distant from the surface are not likely to interact (see Chapter 1). In fact, because of the heterogeneous distribution of the charged residues on the protein surface it is certainly possible for a protein to bind to a cation exchanger above the pI (where the protein has a net negative charge) or to bind to an anion exchanger below the pI (where the protein has a net positive charge). In general, z changes with pH, but this effect can be more or less pronounced than the effect on the net charge. The second consideration concerns the dependence of z for a given protein under given conditions, on the nature of the adsorbent surface. To some extent, when the concentration of surface-bound charged ligands is high, the z value is independent of the exact nature of the stationary phase. On the other hand, when the charge density is low or the adsorbent is based on a cross-linked polymer gel, different z values are often found with different matrices since different structures can allow the protein to interact with a variable number of charged residues. Other models have also been used to describe protein adsorption on ion exchangers including colloidal models where the protein molecules are assumed to be

5.3 Multi-component Systems

charged spheres [12]. The potential of the electrostatic interactions between two particles and between particle and surface can be calculated using the Yukawa equation [13] assuming a speciﬁc interaction parameter and the Debye length (Equation 3.9). For the limiting case when the C → 0, the linear isotherm slope (or Henry constant) can be calculated. The uncertainty of this model stems from the interaction parameter, which cannot be calculated and requires certain experimental information which is difﬁcult to access. The model can be also extended to include van der Waals interactions, but then another parameter which is also difﬁcult to access, the so-called Hamaker constant, must be estimated. Colloidal models provide theoretical insight into the protein adsorption, but have limited practical value for quantitative predictions of adsorption isotherms especially in a saturation regime.

5.3 Multi-component Systems

Multi-component systems are, of course, encountered in practice. Fermentation broths contain many different components, at least several thousands. Moreover, protein products are often highly heterogeneous as a result of post-translational modiﬁcations (see Chapter 1) so that even nominally pure protein products are actually often complex mixtures of isoforms. Although, several examples of multicomponent isotherm models are available, predictions from single component data are frequently inaccurate. Experimental determination is also time consuming and complicated. For practical purposes, the complex situation can sometimes be described in a pragmatic manner by grouping together all the compounds with lower afﬁnity to the surface than the product into one group and grouping all compounds with higher afﬁnity into another. Accordingly, the description can be reduced to a three-component system. The extension of the Langmuir isotherm into a multi-component system with N-adsorbed species is straightforward as has been described by Butler and Ockrent [14] and Markham and Benton [15]. For a system with M adsorbed species, Equations 5.2 and 5.3 become: d [PL i ] = k1,i [Pi ][L ] − k2,i [PL for i = 1, M i ] dt

(5.24)

M

[L ] = [L0 ] − ∑ [PjL ]

(5.25)

j =1

where [L0] is the total concentration of surface-bound ligands in the stationary phase. Taking d[PiL]/dt = 0 and combining the two equations yields: qi =

qmK iCi M

1 + ∑K jC j j =1

(5.26)

157

158

5 Adsorption Equilibria

where, Ki = k1,i/k2,i is the adsorption equilibrium constant for component i (given by the ratio of the corresponding adsorption and desorption rate constants) and qi and Ci have been used in lieu of [PiL] and [Pi], respectively. Note that in this model, the maximum binding capacity qm ∝ [L0] is the same for all components, a condition required for thermodynamic consistency and sometimes observed for the adsorption of similarly sized proteins or isoforms that differ only slightly from each other. Reference [16] provides an application example. In practice, however, Equation 5.26 is also used with different values of qm for different components, as follows: qi =

qm ,iK iCi N

1 + ∑K jC j

(5.27)

j =1

While this is sometimes justiﬁed experimentally, such an assumption is prone to error when the ultimate binding capacities are very different. It should be noted that both Equations 5.26 and 5.27 predict a selectivity deﬁned by:

α i, j =

qi C j qm ,iK i = q j C i q m , jK j

(5.28)

that is independent of protein concentration for a given mobile phase composition. This is not always true in practice and can be used as a test of the validity of this model. Gu et al. [17] have developed a modiﬁed multi-component isotherm model taking into account differences in the binding capacity for the individual proteins in the mixture resulting from their different sizes. In their treatment, protein molecules of varying size are assumed to access a different fraction of the available surface-bound ligands as a result of size exclusion effects. Accordingly, Equations 5.24 and 5.25 can be replaced by the following equation: M   d [PL i ] = k1,i [Pi ]  [ L0,i ] − ∑ θ i , j [PjL ] − k2,i [PL for i = 1, M i ] dt   j =1

(5.29)

where [L0,i] is the concentration of surface-bound ligands accessible by component i and the θi,js are the so-called ‘discount factors’ deﬁned as follows: 1 for i = j or [ L0,i ] ≥ [ L0, j ] θi, j =  < 1 for [ L0,i ] < [ L0, j ]

(5.30)

The multi-component isotherm is then obtained by taking d[PiL]/dt = 0 and solving the resulting system of algebraic equations. For a two-component system, assuming that component 1 is excluded to a greater degree than component 2 (i.e. [L0,1] < [L0,2]), the following result is obtained [17]: q1 =

K 1C1 [ (1 + K 2C2 ) qm ,1 − θ1,2K 2C2qm ,2 ] 1 + K 1C1 + K 2C2 + (1 − θ1,2 )K 1K 2C1C2

(5.31)

q2 =

K 2C2 [ (1 + K 1C1 ) qm ,2 − K 1C1qm ,1 ] 1 + K 1C1 + K 2C2 + (1 − θ1,2 )K 1K 2C1C2

(5.32)

5.3 Multi-component Systems

Figure 5.8 Adsorption equilibrium for binary mixtures of γ-globulin and lysozyme obtained by Garke et al. [18] for Streamline SP in 100 mM acetate buffer at pH 5. The symbols correspond to different mass percentages of each protein in the initial mixture: ●, 100%; ⵧ, =80%; 䉱, 50%; 䉮, 20%. Solid lines are calculated from the extended multi-

component Langmuir isotherm (Equation 5.27) for γ-globulin and from the modiﬁed multi-component Langmuir model (Equations 5.31 and 5.32) for lysozyme using parameters obtained from single component data. Predictions for γ-globulin based on Equations 5.31 and 5.32 were similar to those shown. Reproduced from [18] with permission.

where θ1,2 = qm,1/qm,2 < 1 represents the ratio of binding capacities. Note that in this case the selectivity is not a constant and varies with composition according to:

α1,2 =

q1 C2 qm ,1K 1 1 = q2 C1 qm ,2K 2 1 + (1 − θ1,2 )K 1C1

(5.33)

As a result, isotherm crossover and selectivity reversal can occur as the solution composition is varied as discussed in [17]. As an example, Figure 5.8 shows binary protein adsorption equilibrium data obtained by Garke et al. [18] for two proteins of very different size and binding capacities on a cation exchanger. In this example, lysozyme (Mr ∼ 15 kDa) has a larger single component binding capacity and displaces γ-globulin (Mr ∼ 150 kDa), which has a lower single component binding capacity as a result of size exclusion, when present in high concentration. On the

159

160

5 Adsorption Equilibria

other hand, γ-globulin does not appear to have a large effect of the adsorption of lysozyme. In the case of protein adsorption on ion exchangers, the SMA model can also be extended to multi-component systems [19]. For example, the following equation is obtained for adsorption of a mixture on M proteins on a cation exchanger with Na+ as a counter-ion: zi

M   K e ,i q 0 − ∑ ( z j + σ j ) q j  j =1   qi = Ci zi C ( Na + )

(5.34)

where q0 is the concentration of protein-accessible charged ligands, assumed to be the same for all species present. This expression is implicit requiring a trial-anderror calculation of qi. According to the SMA model, when the effective charge is the same for all proteins, selectivity is also constant and independent of salt concentration. In this case, separation is possible only because of differences in the Ke-values. Conversely, the selectivity is composition and salt-concentration dependent when the effective charge is different for different proteins. In this case, in the dilute, linear isotherm limit, the selectivity is given by:

α i, j =

K e ,i q0zi z j − zi z (C + ) K e , j q0 j Na

(5.35)

which suggests that the selectivity for the protein with the higher effective charge will decrease as the salt concentration is increased.

References 1 Myers, A.L., and Prausnitz, J.M. (1965) Chem. Eng. Sci., 20, 549. 2 Bankston, T.E., Stone, M.C., and Carta, G. (2008) J. Chromatogr. A, 1188, 242. 3 Langmuir, I. (1918) J. Am. Chem. Soc., 40, 1361. 4 Freundlich, H. (1907) Phys. Chem., 57, 385. 5 Temkin, M.I. (1940) Z. Fiziceskoy Chimii, 14, 1153. 6 Toth, J. (1981) J. Coll. Interface Sci., 79, 85. 7 Sips, R. (1948) J. Chem. Phys., 16, 490. 8 Brunauer, S., Emmett, P.H., and Teller, E. (1938) J. Am. Chem. Soc., 60, 309. 9 Kopaciewicz, W., Rounds, M.A., Fausnaugh, J., and Regnier, F.E. (1983) J. Chromatogr. A, 266, 3. 10 Brooks, C.A., and Cramer, S.M. (1992) AIChE J., 38, 1969.

11 Lewus, R.K., and Carta, G. (1999) AIChE J., 45, 512. 12 Oberholzer, M.R., and Lenhoff, A.M. (1999) Langmuir, 15, 3905. 13 Rowlinson, J.S. (1989) Physica A, 156, 15. 14 Butler, J.A., and Ockrent, C. (1930) J. Phys. Chem., 34, 2841. 15 Markham, E.C., and Benton, A.F. (1931) J. Am. Chem. Soc., 53, 497. 16 Converse, A.O., and Girard, D.J. (1992) Biotechnol. Progr., 8, 587. 17 Gu, T., Tsai, G.J., and Tsao, G.T. (1991) AIChE J., 37, 1333. 18 Garke, G., Hartmann, R., Papamichael, N., Deckwer, W.D., and Anspach, F.B. (1999) Sep. Sci. Technol., 34, 2521. 19 Rege, K., Tugcu, N., and Cramer, S.M. (2003) Sep. Sci. Technol., 38, 1499.

161

6 Adsorption Kinetics 6.1 Introduction

This chapter discusses the physical phenomena that determine the rate at which proteins and other biopolymers are adsorbed onto chromatographic stationary phases. Engineering models developed to represent these phenomena in order to interpret experimental measurements and carry out design calculations are also discussed. The emphasis is primarily on mass transfer processes, which, in most cases, control the overall rate of protein adsorption. Rate limitations caused by slow binding kinetics occur in some cases such as immunoafﬁnity chromatography where an antibody binds to a surface-bound antigen. However, with the exception of some afﬁnity adsorbents, these situations are less common in process applications. As discussed in Chapter 2, mass transfer in the stationary phase is typically the controlling dispersion factor in protein chromatography columns using porous particles. However, the transport resistance in the mobile phase is also relevant when using pellicular particles or monoliths or when impractical mass transfer is enhanced by transport mechanisms other than pore diffusion.

6.2 Rate Mechanisms

We will consider the mechanisms through which a protein or other adsorbate is transported from the mobile phase into the stationary phase and is eventually adsorbed. In some cases, the same mechanisms are responsible for transport whether the protein is strongly or even irreversibly bound on the adsorbent surface or whether it merely diffuses in and out of the stationary phase without binding. Other mechanisms depend on whether or not adsorption takes place. In either case, however, protein molecules in the mobile phase usually encounter a series of resistances, which together determine the time scale over which adsorption and desorption will take place.

Protein Chromatography: Process Development and Scale-Up. Giorgio Carta and Alois Jungbauer © 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 978-3-527-31819-3

162

6 Adsorption Kinetics

Figure 6.1 Location of transport and kinetic resistances to protein adsorption in porous particles. (1) External mass transfer; (2) pore diffusion; (3) kinetic resistance to binding; and (4) solid or adsorbed phase diffusion. The sketch on the left shows particles in a

packed bed with surrounding streamlines for mobile phase ﬂow. The sketch on the right shows an approximation of the corresponding ﬁlm model where external transport is represented by diffusion through a stagnant ﬁlm.

Figure 6.1 illustrates the location of transport resistances as they are found in a typical packed bed of porous adsorbent particles. These resistances include: 1) 2) 3) 4)

External mass transfer in the ﬂuid surrounding the particle; pore diffusion in the liquid-ﬁlled intra-particle pores; kinetic resistance to binding at the liquid–adsorbent interface; and solid or adsorbed phase diffusion.

As discussed in Chapter 2, ﬂow of the mobile phase is largely conﬁned to the extra-particle void space and is illustrated schematically by the thicker lines in Figure 6.1. The time scale for ﬂuid transport over the length of one particle is usually short in process scale applications. For example, in a column packed with 100-µm particles with ε = 0.4 and operated at a superﬁcial velocity of 150 cm/h, the time scale for convective transport over a distance equal to the particle diameter is dp/v ∼ 0.1 s. This time scale is generally much shorter than the time scale for protein adsorption. As a result, the protein concentration, C, in the ﬂuid surrounding an individual particle at each instant in time may be considered uniform. Thus, the simpler picture sketched in Figure 6.1, where mass transfer in the external liquid is represented by diffusion across a stagnant ﬁlm of thickness δ, can be used to represent the system. Accordingly, protein transport is driven by the concentration difference C – Cs, where Cs is the protein concentration at the particle surface. Although this representation is far from the physical realities of ﬂow in a packed bed or the complex hydrodynamics of particles suspended in an agitated

6.2 Rate Mechanisms

vessel, the ﬁlm model is adequate for most practical purposes in both column and batch adsorption systems provided the ﬁlm thickness is either obtained empirically or is estimated from suitable engineering correlations. 6.2.1 External Mass Transfer

The external mass transfer resistance is represented by a ﬁlm mass transfer coefﬁcient, kf, deﬁned in such a way that the mass transfer ﬂux at the particle surface is: J = k f (C − C s )

(6.1)

Since for steady state diffusion over a thin stagnant ﬁlm of thickness δ we have J = (D0/δ )(C − Cs), the relationship between kf and δ is given by the following equation:

δ=

D0 kf

(6.2)

where D0 is the diffusivity in free solution. General engineering correlations for mass transfer coefﬁcients in packed adsorption beds, expressed in terms of Sherwood (Sh), Reynolds (Re), and Schmidt (Sc) numbers, are discussed in [1]. For protein chromatography, laminar ﬂow conditions prevail in virtually all practical situations and the following two equations from Wilson and Geankoplis [2] and Kataoka et al. [3], respectively, are recommended: Sh =

1.09 0.33 0.33 Re Sc ε

(6.3)

( ε ε)

(6.4)

Sh = 1.85

1−

0.33

Re 0.33Sc 0.33

where Sh = kfdp/D0, Re = ρudp/η, and Sc = η/ρD0. Although originally based on data primarily in the range of Re > 1, the following equation from Carberry [4] has also been used to predict protein mass transfer coefﬁcients by extrapolation to lower Re values: Sh =

(ε) 1.32

0.5

Re 0.5Sc 0.33

(6.5)

Figure 6.2 shows a plot of these equations for ε = 0.4. The experimental data sets used in developing these correlations do not include proteins or other macromolecular solutes. Thus, extrapolation, made possible by the dimensionless transport parameters included in these equations, is necessary. Data are scarce for proteins since, in practice, external mass transfer is rarely a controlling factor in protein chromatography columns. A very limited experimental data set by Fernandez et al. [5], where kf was measured directly for proteins under conditions where the external resistance was controlling, is included in Figure 6.2 and is seen to be in

163

6 Adsorption Kinetics 10

1

Sh/Sc0.33

164

0.1

Wilson and Geankoplis [2] Kataoka et al. [3] Carberry [4] Fernandez et al. data [5] Hansen and Mollerup data [6]

0.01 0.01

0.1

1

Re = udp/ Figure 6.2 Recommended correlations to predict mass transfer coefﬁcients in packed chromatography columns by Wilson and Geankoplis [2], Kataoka et al. [3], and Carberry [4] shown for ε = 0.4. The Fernandez et al. data [5] are based on direct measurements

with proteins for conditions where the external resistance is dominant. The Hansen and Mollerup data [6] are based on ﬁtting protein breakthrough curves with a two-ﬁlm resistance model.

reasonable agreement with the recommended literature correlations. The data of Hansen and Mollerup [6], obtained by ﬁtting breakthrough curves with a two-ﬁlm model to take into account combined external and intra-particle resistances, are also included in the ﬁgure. These data correspond to much lower rate coefﬁcients than predicted and limiting Sherwood numbers that are much lower than 2. Moreover they predict a 1.2 power dependence on the liquid ﬂow rate. Such behavior is not expected for creeping ﬂow in packed beds where the Sherwood number is expected to depend on a 1/3 power of ﬂuid velocity. This dependence is shown by Equations 6.3 and 6.4 and has been conﬁrmed experimentally by Sarfert and Etzel [7] for protein mass transfer in membrane adsorbers where the boundary layer resistance is controlling and there is no complicating effect of intra-particle diffusional resistances. Despite the uncertainties noted above, in practice, since external mass transfer is rarely controlling in preparative and process protein chromatography, estimates based on the recommended equations are often sufﬁciently accurate for process design calculations assuming a 1/3 power dependence of Sh on Re for low Re values and a dependence on Sc0.33 over the entire range. Mass transfer coefﬁcients for small particles in the size range of interest for protein chromatography applications (6–400 µm) are summarized in Figure 6.3 along with the correlations of Levins and Glastonbury [8] and Armenante and Kirwan [9] respectively, and given by:

6.2 Rate Mechanisms 10

(Sh-2)/Sc0.33

Levins and Glastonbury (1972) [8]

1

Armenante and Kirwan (1989) [9] Data from Armenante and Kirwan, 1989 [9] (small molecules)

0.1

Data from Fernandez and Carta, 1996 (proteins) [5] Data from Borst et al., 1997 (amino acids) [11]

0.01 0.01

0.1

1 4 0.33

( dp )

10

/

Figure 6.3 Experimental data and engineering correlations of Levins and Glastonbury [8] and Armenante and Kirwan [9] for external mass transfer coefﬁcients for

 ρ (ε d p4 ) Sh = 2 + 0.52 ×  η 

0.33

 ρ ( ε d p4 ) Sh = 2 + 0.50 ×  η 

0.33

  

0.52

  

0.62

small particles suspended in agitated contactors. ε is the agitator power input per unit mass of liquid.

Sc 0.33

(6.6)

Sc 0.33

(6.7)

where ε is the agitation power input per unit mass of liquid. The latter can be estimated using power number correlations for agitated vessels (e.g. see [10]) or measured experimentally. Both equations can be seen to ﬁt within the scatter of the original small molecule experimental data set that formed the basis for these equations. They also describe limited data available for proteins and for amino acids in ion exchange resins which are also included in this graph. 6.2.2 Pore Diffusion

Pore diffusion occurs in pores that are sufﬁciently large for the solute to diffuse without interacting with the force ﬁeld exerted by the pore wall. The latter can be due to electrostatic interactions in ion exchange or van der Waals forces in hydrophobic adsorption. In the gas-phase adsorption literature, this mechanism is referred to as macropore diffusion, with reference to the IUPAC classiﬁcation of macropores deﬁned as being larger than 50 nm. In liquid phase applications, such pores can be considered to be ﬁlled with the liquid phase with diffusional transport

165

166

6 Adsorption Kinetics

Example 6.1

Estimation of external mass transfer coefﬁcient

Estimate the ﬁlm mass transfer coefﬁcients and the equivalent ﬁlm thickness for the adsorption of a 150-kDa protein at 4 °C on 100-µm diameter particles if the liquid viscosity is 2.5 mPa s (2.5 cp) and the density is 1.0 g/cm3 for the following two cases. Case 1. The particles are packed in a chromatographic column with ε = 0.4 and operated at 300 cm/h superﬁcial velocity. Solution – The free solution diffusivity at 298 K and 1.0 cp solution viscosity is estimated to be 4 × 10−7 cm2/s. Thus, from Equation 1.29, at 277 K and 2.5 cp, 1.0 277 D0 4°C,2.5 cp = × × D0 25°C,1.0 cp = 1.5 × 10 −7 cm2 s. Consequently, Sc = 2.5 298 η/ρD0 = 0.025/(1.0 × 1.5 × 10−7) = 1.7 × 105. For these conditions, Re = ρudp/η = 1.0 × (300/3600) × 0.01/0.025 = 0.033. In turn, Equations 6.3–6.5 yield Sh = 47, 36, and 18, respectively. The corresponding kf -values are 7.0 × 10−4, 5.4 × 10−4, and 2.6 × 10−4 cm/s. Finally, from Equation 6.2, the equivalent stagnant ﬁlm thicknesses corresponding to these values are 2.1, 2.7, and 5.7 µm, respectively, all a small fraction of the particle diameter.

( )( )

Case 2. The particles are suspended in 100 cm3 of liquid in an unbafﬂed vessel agitated with a 4-cm diameter propeller at 300 rpm. Solution – ε can be estimated from the correlation in Figure 9.14 in [10], where NRe = Da2n ρ η is a Reynolds number that depends on the agitator diameter, Da, and the agitator speed, n˙ , in revolutions per second. Np = P Da5n 3 ρ is the power number where P is the agitator power. For our conditions, NRe = 42 × (300/60) × 1.0/0.025 = 3200, with corresponding Np values between 0.3 and 0.7, dependent on the particular type of propeller. Thus, ε = (0.5 ± 0.2) × 45 × (300 60 )3 × 1.0 100 = 640 ± 260 cm2 s3. Equation 6.6 yields Sh = 26 ± 2 or kf = (4.0 ± 0.2) × 10−4 cm2/s. Note that Sh is typically quite insensitive to ε so that a rough estimate is sufﬁcient.

occurring in the pore liquid, much as it would in free solution. In the case of biopolymers, however, the molecular size can be of the same order of magnitude as that of macropores resulting in hindered or restricted diffusion effects. Pore diffusion is typically expressed in terms of effective pore diffusivity, De, deﬁned such that the protein mass transfer ﬂux in the stationary phase is given by: J = −De∇c

(6.8)

where ∇c is the protein concentration gradient in the pore liquid. De is smaller than the free solution diffusivity and is commonly written as: De =

εp D 0 ψp τp

(6.9)

6.2 Rate Mechanisms

where εp and τp are the intra-particle porosity and the tortuosity factor, respectively, and ψp is a diffusional hindrance coefﬁcient. The εp-term in Equation 6.9 takes into account the limited space available for pore diffusion within the particle assuming that the void volume and open area fractions are the same. Chapter 3 discusses typical values for commercial protein chromatography matrices and approaches for its empirical determination. The τp-term takes into account the typical random orientation of the intra-particle pores, which gives an effective diffusion path that is longer than a straight trajectory. Although there are some correlations in the literature relating τp to εp (see for example [1]), they are highly empirical and prone to error. As a general rule, τp is smaller for larger values of εp. However, τp is best regarded as an empirical parameter to be measured experimentally, preferably using a non-adsorbed test solute. Values of τp between 1.5 and 4 are typical for protein chromatography matrices. Values that are greater than this range indicate severely restricted diffusion while smaller values usually indicate that mechanisms other than pore diffusion have inﬂuenced the measurement. Ideally, once τp is determined for a test solute, the same value can be applied to other molecules as well as other conditions. Alternative descriptions of pore diffusion based on network models which account explicitly for the connectivity of the pore structure, are also available (e.g. see [12–14]). Although these models have potentially predictive capability and can be useful in designing pore networks a priori, they are mathematically much more complex and require detailed knowledge of the pore structure which is seldom available. The diffusional hindrance coefﬁcient, ψp, is generally related to the ratio of protein and pore radii, λm = rm/rpore. Since this ratio can be considerably larger than zero for proteins and other biopolymers in practical chromatographic matrices, diffusional hindrance requires special consideration. In general, for non-adsorbing species, two factors contribute to ψp. The ﬁrst is purely steric and is associated with size exclusion, which arises from the fact that the centerline of the diffusing molecules cannot approach the pore wall at a distance closer than a molecular radius. The second is associated with viscous drag or hydrodynamic resistance. Approximate theories are available to describe these factors based on a colloidal representation of a protein molecule diffusing in an idealized cylindrical pore. Accordingly, the diffusional hindrance coefﬁcient is given by the following expressions adapted from [15] and [16], respectively:

(

9 ψ p = 1 + λm ln λm − 1.539λm 8

)

for λm < 0.2

(6.10)

ψ p = 0.865 (1 − λm )2 ( 1 − 2.1044 λm + 2.089λm3 − 0.984 λm5 ) for λm > 0.2 (6.11) The protein radius can be estimated from the Stokes–Einstein equation: rm =

kbT 6πηD0

(6.12)

where kb is the Boltzmann’s constant. A plot of Equations 6.10 and 6.11 is shown in Figure 6.4. It can be seen that in order to avoid excessive diffusional hindrance

167

6 Adsorption Kinetics

1

0.8

0.6 p

168

0.4

0.2

0 0

0.1

0.2

0.3

0.4

0.5

0.6

m

Figure 6.4 Diffusional hindrance coefﬁcient plotted as a function of the ratio of protein and pore radii. The solid line is calculated from Equations 6.10 and 6.11 over the respective ranges of validity.

(which occurs, for example, when ψp < 0.5), the pore radius needs to be about eight times the protein radius or λm < 0.125. Thus, for example, for a 150-kDa protein with rm ∼ 5 nm, the pore radius needed to avoid excessive hindrance is around 40 nm. Example 6.2 illustrates the application of the relevant equations.

Example 6.2

Estimation of effective pore diffusivity

Estimate the effective pore diffusivity for an antibody (Mr ∼ 150 kDa) for a dilute aqueous solution with η = 2.5 mPa s (2.5 cp) at 4 °C. The characteristics of the adsorbent particles are εp = 0.8, τp = 1.5, and rpore = 30 nm. Solution – From Example 6.1, the free solution diffusivity for these conditions is D0 ∼ 1.5 × 10−7 cm2/s. From Equation 6.12, rm = kbT/6πηD0 = (1.38 × 10−18 × 298)/(6π × 0.01 × 4 × 10−7) = 5.5 nm. Thus, λm = rm/rpore = 5.5/30 = 0.18. Finally, from Equation 6.10 we obtain ψp ∼ 0.38. Thus, De = 0.8 × 1.5 ×10−7 × 0.38/1.5 = 0.30 × 10−7 m2/s, which is about ﬁve times smaller that the diffusivity in free solution.

The estimation of De for more realistic conditions and when the protein is adsorbed along the pore wall is further complicated by several factors: 1) The pore size is not uniform and a distribution of sizes around the mean is typically found in real adsorbents. Thus, diffusion can be almost completely restricted in some smaller pores and nearly unhindered in others.

6.2 Rate Mechanisms

2) The pores are not completely connected and the distribution may include dead-end and unconnected pores. 3) Adsorbed protein along the pore wall may signiﬁcantly reduce the size of the pore available for diffusion. Thus, De for protein-free and protein-saturated particles may be different. 4) There may be electrostatic interactions between the pore wall and the protein even if the pore size is considerably larger than the size of the protein. The latter effects can be especially important in ion exchangers, when the ionic strength is low. Simpliﬁed theories describing partitioning of neutral and charged spherical colloidal particles in idealized cylindrical pores have been reported (see e.g. [17, 18]). A rigorous theory taking into account the actual protein and pore structures is not available. However, it is well established that the signiﬁcance of these effects depends on the thickness of the electrical double layer, which scales with the Debye length [19] (also see Chapter 3):

λD =

ε r ε 0RT 2F 2I

(6.13)

where F is Faraday’s constant, I the ionic strength, and εr and ε0 the relative permittivity and dielectric constant of a vacuum, respectively. For I = 0.02 M, we obtain λD ∼ 2.2 nm. As a result, since the diffuse double layer extends from the surface of a charged pore wall well beyond the Debye length, especially in spherical or cylindrical pores, it is apparent that the electrical potentials may overlap and inﬂuence protein partitioning and transport in the pore ﬂuid even in relatively large pores [20]. Despite these limitations, however, Equations 6.9 to 6.11 still provide useful predictions for many practically relevant cases. Table 6.1 shows, as an example, the De-values obtained for lysozyme in the agarose-based cation exchanger SPSepharose-FF. The measurements were obtained using various techniques under a broad range of conditions, including both adsorbing (low salt) and non-adsorbing

Table 6.1 Summary of effective pore diffusivities for lysozyme in SP-Sepharose-FFa) (source [21]).

Method/conditions

De (10−7 cm2/s)

De/D0

τp = εpψpD0/Deb)

Isocratic elution (unretained, 520 mM salt) Linear gradient elution (20 to 520 mM salt) Stirred batch adsorption (20 mM salt) Shallow bed adsorption (20 mM salt) Frontal analysis (20 mM salt) Confocal microscopy (20 mM salt) Average

3.1 2.6 2.9 2.6 2.3 2.6 2.7 ± 0.3

0.28 0.24 0.26 0.24 0.21 0.24 0.24 ± 0.02

1.4 1.7 1.5 1.7 1.9 1.7 1.6 ± 0.2

a) Based on εp = 0.8, rpore = 15 nm, D0 = 1.1 × 10−6 cm2/s. b) Based on Equations 6.9 through 6.12.

169

170

6 Adsorption Kinetics Table 6.2 Summary of effective pore diffusivities of different proteins in SP-Sepharose-FF, Butyl Sepharose 4FF, and MabSelect (source: [22]).

Adsorbent

Protein

Salt conc.(M)

pH

De(10−7 cm2/s)

De/D0

τpa)

SP-Sepharose-FF (rpore = 15 nm)

Cytochrome c (12.5 kDa) Lysozyme (15 kDa) Albumin (65 kDa) IgG (150 kDa)

0.02

4.0

3.8 ± 0.3

0.33

1.4

0.02

6.5

2.4 ± 0.2

0.22

1.8

0.02

5.0

0.44 ± 0.02

0.074

2.8

0.02

5.0

0.070

0.020

5.3

Butyl Sepharose 4FF (rpore = 23 nm)

Albumin (65 kDa)

2.0

6.8

1.8 ± 0.2

0.30

1.1

MabSelect (rpore = ∼ 23 nm)

IgG (150 kDa)

0.15

7.4

0.77 ± 0.07

0.19

1.1

a)

Based on Equations 6.9 through 6.12.

(high salt) cases as well as varying protein concentrations and loads. As can be seen from this table, the De values vary little with experimental conditions and are about four times smaller than the free solution diffusivity. The tortuosity factor calculated from these data is around 1.6 and is in the range expected for this highly porous matrix. Table 6.2 provides examples of effective pore diffusivities for a range of proteins and stationary phases, including the cation exchanger SP-Sepharose-FF, the hydrophobic interaction matrix Butyl Sepharose 4FF, and the Protein A adsorbent MabSelect, all from GE Healthcare. In SP-Sepharose-FF the smaller proteins exhibit De/D0 values in the range 0.22 to 0.33, with corresponding tortuosity factors in the expected range. However, for albumin and IgG, De/D0 decreases dramatically indicating severely restricted diffusion. This occurs because the molecular size (3.6 and 5.5 nm radius for albumin and IgG, respectively, based on Equation 6.12) begins to approach the size of the matrix pores. Albumin in Butyl Sepharose 4FF and IgG in MabSelect exhibit De values that are considerably higher than the corresponding values in SP-Sepharose-FF. This occurs principally because of the larger pore size of these chromatographic matrices, which results in lower diffusional hindrance. 6.2.3 Diffusion in the Adsorbed Phase

In the case of pore diffusion discussed above, transport occurs within the mobile phase contained inside the particle, where the solute concentration is generally similar in magnitude to the concentration in the external ﬂuid. A protein molecule

6.2 Rate Mechanisms

transported by pore diffusion may attach to the pore surface and detach many times along its path. In other cases, where the isotherm is highly favorable, attachment can be essentially permanent, but in both cases, only detached molecules undergo transport. In contrast, the terms solid diffusion and adsorbed phase diffusion are used interchangeably to denote transport processes where diffusion occurs in the adsorbed state within a phase that is distinct from the bulk pore liquid. Such processes include surface diffusion which is associated with movement along pore surfaces without detaching, micropore diffusion, associated with transport in cavities that are comparable in size to that of the diffusing molecule, and homogeneous diffusion, associated with transport in a pore ﬁlled with a liquid that is immiscible with the external ﬂuid or with diffusion of a charged molecule in an oppositely charged gel whose mesh size is such that the electrical double layers overlap. The physical origin is obviously different for each of these mechanisms. However, they all have in common the fact that the driving force for diffusion is expressed in terms of the adsorbed-phase concentration, q. Accordingly, the mass transfer ﬂux is expressed by: J = −Ds ∇q

(6.14)

where ∇q is the concentration gradient in the adsorbed phase and Ds is the effective adsorbed-phase diffusivity. Although the functional form is the same as Equation 6.8, there are major quantitative differences. Firstly, the Ds values are generally smaller than those of De since diffusion in the adsorbed phase is likely more restricted. Secondly, unlike pore diffusivities, which are typically independent of concentration, Ds values are often concentration dependent, since the concentration in the adsorbed phase can be much higher than that in solution. Finally, the magnitude of the diffusion ﬂux can be quite different depending on conditions. Since for a high capacity adsorbent q can be much larger than C, even if Ds is smaller then De, the mass transfer ﬂux in the adsorbed phase may be much larger than that resulting from pore diffusion. Surface and micropore diffusion mechanisms are well established for gas adsorption in many practical materials ranging from zeolites to activated carbon. Homogeneous diffusion of ions, including organic ions such as amino acids, in cross-linked ion exchange resins is also well known and details can be found in many standard adsorption and ion exchange reference books and handbooks (e.g. [1, 23, 24]). However, adsorbed-phase diffusion mechanisms for proteins are far less well understood. Much of the difﬁculty stems from the fact that mass transfer models based on vastly different transport mechanisms typically provide nearly identical descriptions of macroscopic adsorption rate data. Table 6.3 summarizes some of the available data for which the experimentally determined effective pore diffusivities were found to be substantially higher than the diffusivity in the free liquid phase. All of these cases involve proteins diffusing in ion exchangers comprising relatively soft ﬁbrous gels, either free standing (cellulose, Sephadex, chitosan) or incorporated into a rigid support matrix (HyperD) and mass transfer rates were measured for conditions where binding is strong (i.e. with a highly favorable isotherm) but reversible. In each case, the data could be described with

171

172

6 Adsorption Kinetics Table 6.3 Examples of protein mass transfer data exhibiting apparent effective pore diffusivities higher than the corresponding free solution diffusivities under favorable binding conditions.

System

Diffusion model

De (cm2/s)

Ds (cm2/s)

Apparent De/D0a)

Reference

DEAE cellulose/ovalbumin Basic chitosan/albumin Q-HyperD/albuminb) Q-HyperD/lactalbuminb) Sephadex A50/albumin Q-HyperD/albumin Q-HyperD/ovalbumin Q-HyperD/lactalbumin S-HyperD/lysozyme

Pore Parallel Parallel Parallel Solid Solid Solid Solid Solid

2.9 × 10−6 2.4 × 10−7 0.1 × 10−7 0.7 × 10−7 – – – – –

– 0.2 × 10−8 4.5 × 10−8 13 × 10−8 2.0 × 10−8 0.9 × 10−8 1.5 × 10−8 4.4 × 10−8 1.1 × 10−8

3.8 1.5 15c) 26c) 3.9 9.7 13 11 5.9

[25] [26] [27] [27] [28] [5] [5] [5] [29]

a) Based on De = (q/C)Ds at C = 1 mg/cm3. b) Based on data from isocratic elution with k′ between 0 and 5. c) Extrapolated.

either the pore or solid diffusion model, as well as with models accounting for parallel pore and solid-phase diffusion. Thus, the apparent De/D0 values merely represent the effective pore diffusivity required to ﬁt the adsorption rate data normalized by the free solution diffusivity. The fact that this ratio is substantially higher than unity suggests that mechanisms other than pore diffusion contribute to the overall mass transfer process. As shown in Table 6.3, the corresponding Ds values are typically of the order of 10−8 cm2/s. While smaller than typical De values (cf. Table 6.2), they result in relatively high mass transfer rates when the driving force ∇q is large, which occurs when the adsorption capacity is high. Similar results showing mass transfer rates that are substantially enhanced compared to those predictable on the basis of pore diffusion alone have also been reported for some proteins with both commercial and experimental dextran-grafted, agarose based ion exchangers [30–33]. Such enhancement has been attributed to the interaction of the protein with the oppositely charged dextran polymers that are incorporated into the supporting agarose structure. The prediction of Ds for proteins is obviously not straightforward, especially for composite stationary phases whose structure is heterogeneous. Limited success has been achieved for transport in charged polyacrylamide gels, which form clear, homogenous ﬁbrous networks with small mesh size (<10 nm) comparable to the size of the diffusing protein. In such a small mesh, the protein diffusional mobility is severely restricted [34, 35], but is still retained under conditions where proteins of charge opposite are favorably partitioned in the gel. For these cases, the combination of low Ds and high q gives comparatively high rates of mass transfer, especially at low protein concentrations, where the contribution of pore diffusion becomes very small. It has been shown that, for these cases, the Ds values deter-

6.2 Rate Mechanisms

mined for gels supported in small capillaries could be used to predict protein transport in both Q- and S-HyperD stationary phases obtained by ﬁlling the pores of a rigid support matrix (silica) with comparable gels [36, 37]. In general, however, empirical measurements with the actual stationary phase are almost always necessary to establish both the governing mechanisms and the magnitude of the relevant diffusional time constants. This is especially important to assess the functional dependence between mass transfer rates and protein concentration, which are typically important process variables. 6.2.4 Intra-particle Convection

So far, our discussion of intra-particle mass transfer mechanisms has been limited to diffusional processes, which are dominant in most practical cases. There are exceptions, however, when (i) the intra-particle pores are very large; (ii) the protein diffusivity in the mobile phase is very small; (iii) the particles are small; and (iv) the mobile phase velocity is high. In this case, intra-particle ﬂow of the mobile phase may occur to an extent sufﬁcient to affect the overall rate of mass transfer of slowly diffusing molecules. This effect has been referred to as intra-particle convection or perfusion. With reference to Figure 6.5, it can be seen that extra-particle and intra-particle ﬂow occurs within porous structures that, in general, are vastly different in size. As a rough approximation, the extra-particle spaces where most of the mobile phase ﬂow occurs are in the order of the size of the adsorbent particles, obviously much larger than the size of the intra-particle pores. In turn, the hydraulic permeability of the extra-particle space is usually much larger than the internal hydraulic permeability of the particles. Since the liquid will follow the path of least resist-

Figure 6.5 Location of extra-particle (1) and intra-particle ﬂow (2) for a large-pore, hydraulically permeable spherical particle in a chromatography column. The diagram on 1

the right represents the idealized ﬂow model of Happel [38] with rp r0 = (1− ε )3 .

173

174

6 Adsorption Kinetics

ance, it is obvious that intra-particle ﬂow will be in most cases a small fraction, Fp, of the total liquid ﬂow. Although small, however, this ﬂow may contribute signiﬁcantly to intra-particle transport in cases where intra-particle diffusion is exceedingly slow, as with large biopolymers. Determination of Fp is thus critical to assess the signiﬁcance of this potential contribution. Estimates of Fp can be obtained with one of the following ﬂow models. The ﬁrst assumes that the ratio of intra-particle and extra-particle ﬂows is equal to the ratio of intra-particle and extra-particle hydraulic permeabilities, B0,p and B0. Accordingly: Fp =

B0, p B0

(6.15)

Since ﬂow is laminar, the Karman–Cozeny equation (see Equations 3.18 and 3.19) is applicable and the extra-particle permeability is given by: B0 =

1 ε3 rp2 37.5 (1 − ε )2

(6.16)

If the adsorbent particles are considered to be equivalent to a bed of microparticles whose radius is three times the pore radius, we also have: B0, p =

ε p3 9 2 rpore 37.5 (1 − ε p )2

(6.17)

As a result we obtain: 2

ε p 3  1 − ε   rpore  Fp = 9     ε   1 − ε p   rp 

2

(6.18)

Since in all practical cases, Fp << 1, the intra-particle superﬁcial velocity, up is simply Fpu. The factor of 9 in Equation 6.18 will, of course, be different if different assumptions are made for the relationship between pore size and microparticle size, but this value will generally be an upper limit. The second model is based on an extension of the well-known free surface model described by Happel [38]. Accordingly, the ﬂow ﬁeld in a packed bed of spherical particles is represented as in Figure 6.5 by the ﬂow between two concentric spheres where the inner sphere is the particle and the outer sphere has a radius r0 such that: 1 rp = (1 − ε )3 r0

(6.19)

As a result, the extra-particle void fraction of the actual column and that of the unit cell in the free surface model are the same. For an impermeable particle, solving the Navier–Stokes equations for creeping ﬂow between rp and r0 provides a result that is nearly identical to that obtained using the Karman–Cozeny equation; that is, the pressure drop over one particle in the Happel model is the same as that predicted by Equation 6.16. Extension of the free surface model to the case of hydraulically permeable particles was carried out by Neale et al. [39] using the Brinkmann extension of Darcy’s

6.2 Rate Mechanisms

175

0.0005

0.0005

(b)

(a)

Particle surface

0.0004

Particle surface

0.0004

0.0003

z (cm)

z (cm)

0.0003 1E-11

0.0002

1E-9 5E-9

0.0002

1E-10

1E-8 2.5E-10

0.0001

0.0001

2.5E-8 5E-8

0

0 0

0.0001

0.0002

0.0003

0.0004

x (cm)

Figure 6.6 Intra-particle ﬂow streamlines predicted by the model of Neale et al. [39] for 8.8-µm diameter particles with ε = 0.33 for two different intra-particle permeabilities. (a) B0,p = 2.4 × 10−12 cm2; (b) B0,p = 2.4 × 10−10 cm2.

0.0005

0

0.0001

0.0002

0.0003

x (cm)

x and z are coordinates originating at the particle center perpendicular and parallel to the ﬂow direction, respectively. Based on data from [40].

law to describe intra-particle ﬂow. The expressions for the streamlines for the intra-particle and extra-particle ﬂow ﬁelds can be found in the original work of Neale et al. while expressions for the axial and radial components of the intraparticle velocity are given in Carta et al. [40]. As an example, Figure 6.6 shows predicted intra-particle streamlines for 8.8-µm particles packed with ε = 0.33 assuming either moderate (B0,p = 2.4 × 10−12 cm2) or very high intra-particle permeability (B0,p = 2.4 × 10−10 cm2). Based on Equation 6.17 for εp = 0.5, these values correspond to pore radii of 45 and 450 nm, respectively. As can be seen in Figure 6.6, even in the high permeability case, the streamlines are for the most part straight and vertical indicating that except for a small region near the boundary, the ﬂow velocity in the particle is predicted to be substantially uniform. Physically, this occurs because the longer intra-particle ﬂow paths near the vertical axis are almost exactly compensated by a larger pressure drop. The latter is the maximum between the north and south poles of the particles where the ﬂow path is longest. The net result is that the perfusion ﬂow in a homogenously permeable particle is essentially uniform. The effect of intra-particle permeability on the axial of intra-particle ﬂow velocity is shown in Figure 6.7 for 8.8-µm diameter particles with ε = 0.33. The solid lines are calculated from the Neale et al. model and the dashed line from Equation 6.18. Obviously, the latter approximate calculation sufﬁces except, perhaps, in cases where the intra-particle permeability is very high. The remaining factor to be considered is the effect of intra-particle convection on the overall rate of mass transfer of the protein in the stationary phase. Even for idealized particles with a perfectly homogeneous and isotropic pore structure, exact predictions are complicated since the predominantly axial convective

0.0004

0.0005

6 Adsorption Kinetics 1

1

0.9 0.1 0.8

Fp = up/u

176

0.7 0.01 0.6

0.001

0.5

10 -12

10 -11

10 -10

Intraparticle permeability, B0,p (cm2) Figure 6.7 Effect of intra-particle hydraulic permeability on the intra-particle ﬂow ratio for 8.8-µm diameter particles with ε = 0.33. The solid lines are calculated from the Neale et al. model. The dashed line is calculated using

Equations 6.15 and 6.16. Ω represents the ratio of pressure drop for permeable and impermeable particles. Based on data adapted from [40].

transport is superimposed to diffusion, which occurs predominately in the radial direction. A detailed analysis requires numerical calculations to handle the twodimensional transport problem and can be found in the work of Liapis and McCoy [41] for slab-shaped particles and in Meyers and Liapis [42] for particles represented by a network model. In practice, however, a simpler analysis based on the evaluation of the transfer function for simultaneous diffusion and convection provides a useful expression for a convection-enhanced effective intra-particle dif˜ e. Details of the procedure can be found in Carta et al. [40], Carta and fusivity, D Rodrigues [43], and Rodrigues et al. [44] and is analogous to that used in the chemical reaction engineering literature to approximate the effects of intra-particle convection on the effectiveness factor of a catalyst particle (e.g. see Rodrigues et al. [45]). The exact expression for spherical particles is a complicated function and is explained in Carta et al. [40]. A good approximation, based on an analogy between slabs and spheres [43] is given by the following equation: −1 1 1  D e (Pe p 6)  = − 3  tanh (Pe p 6) Pe p 6  De

(6.20)

where Pep is an intra-particle Peclet number deﬁned as: Pep =

u pdp Fpudp = De De

(6.21)

˜ e/De represents an enhancement factor relative to pore diffusion alone. The ratio D For practical calculations, Fp can be estimated from Equation 6.18 and De by the

6.2 Rate Mechanisms 10

~ Enhancement factor, D e/De

Exact solution for spheres, Carta et al. (1992)

Frey et al. (1993)

Eq. 6.20

1 1

10

100

Intraparticle Peclet number, Pe p =Fpudp/De

Figure 6.8 Enhancement factor describing the effect of intra-particle convection in spherical particles. The convection-enhanced ˜ e can be used for effective diffusivity D

approximate column efﬁciency and dynamic binding capacity calculations. Note that ˜ e/De ∼ Pep/18 at high Pep values. Data are D taken from Carta et al. [40] and Frey et al. [46].

˜ e value can then be used in methods described in Section 6.2.2. The resulting D lieu of De in column efﬁciency and dynamic binding capacity calculations. A plot of Equation 6.20 is shown in Figure 6.8 along with the exact solution for spheres and the nearly equivalent approximation by Frey et al. [46]. All three relationships predict that enhancing mass transfer rates by a factor of 2 over pore diffusion alone requires an intra-particle Peclet number, Pep, of around 30. At high Pep values, Equation 6.20 reaches a linear asymptote corresponding to a completely ˜ e = Fpudp/18. At this limit, intra-particle transconvection-dominated case where D port becomes proportional to the mobile phase ﬂow rate. Example 6.3 illustrates an application of the relevant equations. Practical large pore matrices designed to augment mass transfer by intra-particle convection typically comprise a bimodal pore network with large pores for convective ﬂow and much smaller pores to provide the surface area for protein binding. This can be accomplished, for example, with a particle architecture consisting of microparticles with smaller, purely diffusive pores aggregated to form a large-pore, hydraulically permeable network, which is characteristic of polymeric particles obtained by suspension polymerization with suitable porogens. This approach can be effective, provided the intra-particle Peclet number for the convective pores is large since the length scale over which the protein needs to diffuse is limited to the size of the constituent microparticles. This introduces a diffusional resistance even at high ﬂow rates, which reduces the efﬁciency (see e.g. [43]). Nonetheless, the column pressure is still determined primarily by the size of the particles since the main ﬂow is in the extra-particle space and can be expected to be much lower than that for a column packed with the unaggregated microparticles. Many experimental examples of intra-particle convection in protein chromatography can be

177

178

6 Adsorption Kinetics

Example 6.3 Estimation of enhancement of mass transfer by intra-particle convection Estimate the enhancement of mass transfer resulting from intra-particle convection for 20-µm particles with 400 nm pore size and 50% porosity packed into a column with ε = 0.35 operated at 100, 500 and 1000 cm/h if the protein free solution diffusivity is 4 × 10−7 cm2/s. Solution – Equations 6.16 and 6.17 with rp = 0.001 cm, rpore = 0.00002 cm, and εp = 0.5, yield B0 = 2.7 × 10−9 cm2 and B0,p = 4.8 × 10−11 cm2. Thus, Fp = 4.8 × 10−11/2.7 × 10−9 = 0.018. Assuming τp = 1.5, De is estimated to be 1.1 × 10−7 cm2/s, based on the methods described in Section 6.2.2. At 100 cm/h (u = 0.028 cm/s), ˜ e/De = 1.14, Pep = 0.018 × 0.028 × 0.002/1.1 × 10−7 = 9.0 and Equation 6.20 yields D showing little enhancement by intra-particle convection. At 500 and 1000 cm/h, ˜ e/De increases to 2.87 and 5.33, respectively, showing more signiﬁcant effects. D Note that if the particle diameter is increased to 100 µm, at 1000 cm/h Pep drops ˜ e/De to 1.49, showing that there would be a very modest effect of to 17.9 and D intra-particle convection with the larger particles even at high ﬂow rates. Reducing D0 will, instead, reduce De, thereby increasing Pep showing a greater effect of intra-particle convection for slower diffusing species.

easily found in the literature, including Afeyan et al. [47], Carta et al. [40], Frey et al. [46], Freitag et al. [48], Pfeiffer et al. [49], Gustavsson et al. [50], Nash and Chase [51], and Li et al. [52]. 6.2.5 Kinetic Resistance to Binding

The kinetics of protein binding to functionalized surfaces used for non-selective process chromatography applications is generally sufﬁciently faster than typical mass transfer rates for it not to be of concern. This includes most ion exchange and hydrophobic adsorption systems. Even with pellicular particles, monoliths, and membrane adsorbers, where diffusional resistances are limited to the ﬂowing ﬂuid phase, the binding kinetics resistance is often negligible and band broadening is controlled by other dispersion mechanisms, including ﬁlm resistance, axial dispersion, and ﬂow non-uniformity (e.g. see [53]). However, this is not always the case for secondary processes that occur on surfaces, such as unfolding or other molecular rearrangements, which can occur after binding over much longer time scales. The kinetics of protein binding to afﬁnity ligands can, however, be a signiﬁcant factor under certain conditions. In this case, the rate of binding can be expressed by the following equation [54]: q r = ka C (qm − q ) −   K 

(6.22)

6.3 Batch Adsorption Kinetics Table 6.4

179

Representative apparent kinetic and equilibrium constants for IgG binding to Protein A domains. τa (s)b)

Liganda)

IgG species

ka (M−1 s−1) K (M–1)

SpA ZZ-domains

Human

3.5 × 105

6.8 × 108

Unknown

8.0 × 10

Human Rabbit

SpA SpA Protein A B-domains

2.9 × 10

7

1.2 × 10

5.5 × 10

7

1.3 × 10 1.7 × 105

1.7 × 10 1.0 × 108

3 4 5

8

0.42

Method

Reference

Surface plasmon resonance

[55]

18

Acoustic wave guide

[56]

12

Porous SiO2 interferometry

[57]

Quartz crystal microbalance

[58]

1.2 0.9

a) SpA, Staphylococcus aureus protein A. b) τa, time constant for binding at 1 mg/ml protein.

where ka is a second order rate constant, qm the maximum binding capacity, and K is the association constant of the equilibrium reaction. At equilibrium, Equation 6.22 yields: q=

qmKC 1 + KC

(6.23)

which is consistent with the Langmuir isotherm (see Section 5.2). The magnitude of the rate constant for surface-bound ligands is difﬁcult to determine by conventional methods since diffusional resistances (that are often dominant), tend to skew experimental results based on adsorption rate measurements. Thus, some of the earlier literature reports rate constants that are much smaller than expected and were probably inﬂuenced by other rate limitations. A sample of more recent determinations for IgG binding to surface immobilized Protein A and protein domains is shown in Table 6.4. Although the apparent rate constants are obviously dependent on the particular system and IgG species, most ka values are in the range 104 to 106 M−1 s−1. At protein concentrations of 1 mg/cm3, these values correspond to the time scales for binding, τa = (kaC)−1, in the order of 0.1 to 10 s. By contrast, the time scale for pore diffusion of an antibody in a 100-µm particle is in the order of rp2 De or 250 s assuming De = 1 × 10−7 cm2/s. As a result, for process scale applications where relatively large particles are used, the kinetic resistance to binding is usually negligible. This may not be the case, however, for pellicular particles or monoliths where the time constants for mass transfer are often much shorter.

6.3 Batch Adsorption Kinetics

Batch adsorption is sometimes used in bioprocess applications and, more frequently, in laboratory studies. As a process, batch adsorption has the advantage

180

6 Adsorption Kinetics

(a) Protein solution (b) Wash (c) Eluent

Detector

Detector

Figure 6.9 Stirred batch (left) and shallow bed (right) adsorption systems for laboratory studies of protein adsorption rates.

that it can be used with particulate-containing, unclariﬁed broths. However, since the efﬁciency is limited to one theoretical plate, its usefulness is limited to cases where adsorption is highly selective. Additionally, desorption and reuse of the adsorbent particles requires settling or ﬁltration units. A useful approach employs batch adsorption to load the protein onto a selective adsorbent, which is then packed in a column for subsequent wash and elution steps. In laboratory studies, batch adsorption is often used to determine adsorption kinetics and compare different adsorbents. Scale-up parameters can also be obtained by comparing experimental adsorption and desorption rates with suitable models. At the laboratory scale batch adsorption systems often employ small stirred vessels with the adsorbent particles either suspended in the liquid (see Figure 6.9) or held in a rotating cage (for a review see [21] and references therein). High-throughput approaches based on 96-well plate formats have also been developed for kinetic measurements [59]. In these approaches, the protein concentration in the bulk liquid varies as a function of time and the overall adsorption rate is obtained from material balances. To minimize the error associated with the subtraction of two numbers of similar magnitude, the initial and ﬁnal protein concentrations (C0 and Cf ) must be adequately different. In practice values of Cf/C0 ∼ 0.5 are desirable, but the exact value depends on the precision of the protein concentration measurement. An alternative approach uses a shallow-bed of the adsorbent (see Figure 6.9). If the bed is sufﬁciently thin and is operated at an adequately high ﬂow rate, the protein concentration will vary only slightly across the bed length. The amount of protein adsorbed at each time can be found by quickly interrupting the ﬂow, rapidly washing out the interstitial liquid, and then eluting the adsorbed protein from the particles. The area of the ensuing elution peak is related to the amount adsorbed at each time by using a suitable calibration curve. Alternatively, the

6.3 Batch Adsorption Kinetics

amount adsorbed can be measured directly with radioactively labeled protein (e.g. with 125I) by determining the radioactivity in the shallow bed over time with a gamma counter (see [32]). The advantages of the shallow bed system are that adsorption rates are obtained directly, without having to depend on a material balance, for an essentially constant protein concentration, and for conditions of ﬂow similar to those encountered in an actual chromatography column (for a review see Hahn et al. [60] and references therein). Either technique can be used for multi-component adsorption rate measurements, although this usually requires off-line analyses. Spectroscopic techniques or radiotracer-based methods can sometimes be used to avoid these analyses. 6.3.1 Rate Equations

The description of batch adsorption is based on coupling differential material balances for the protein in the adsorbent particles with an overall balance for the protein in the liquid phase. In the following we assume that the particles are spherical with radius rp and uniform properties and that the liquid phase is perfectly mixed. In many cases, using the average particle size is sufﬁcient and the volume-average deﬁned by Equation 3.16 is recommended in the usual case where intra-particle diffusion is controlling. For cases where the adsorption isotherm is highly favorable, however, accounting for the particle size distribution in an explicit manner is necessary. In this case, the smaller particles in the distribution are completely saturated early on while the larger ones take a longer time, giving the impression of faster than expected initial rates for short times and much slower then expected rates for long periods. The particle conservation equations are summarized in Table 6.5 for the individual rate mechanisms discussed in Section 6.1 and for common combined mechanisms. In these equations, the variable qˆ represents the protein concentration in the adsorbent particles, which, in general, comprises both the adsorbed protein and the protein simply held in the pore liquid. Accordingly, ˆ q = q + ε pc

(6.24)

However, in most practical cases involving adsorption from relatively dilute solutions, q is much larger than the pore ﬂuid concentration. Thus, we obtain qˆ ∼ q. This also holds true for many chromatographic separations, except of course, for size exclusion chromatography where there is no adsorption, or for elution with very low k′-values. Note that both qˆ and q have been deﬁned per unit volume of adsorbent particle. This quantity can be related to units of column or packed bed volume simply multiplying it by (1 − ε), where ε is the extra-particle void fraction. As shown in Table 6.5, each rate mechanism gives rise to a different mathematical form (A–D). Additionally, combinations of diffusion ﬂuxes in parallel (E) or series (F) give rise to more complex equations that involve the adsorption isotherm q*(C), discussed in Chapter 5, and its slope dq*/dC, when equilibrium is assumed

181

182

6 Adsorption Kinetics

Table 6.5 Rate equations describing protein adsorption in spherical adsorbent particles.

Controlling mechanism

Rate equation

A

Film resistance

dqˆ 3k f (C − C s ), qˆ = qˆ * (Cs) = rp dt

B

Pore diffusion

q 1 ∂ ∂ˆ ∂c Der 2 = ∂t r 2 ∂r ∂r (∂c/∂r)r=0 = 0, (De ∂c ∂r )r =rp = k f (C − c r =rp ) or (c )r =rp = C for no external resistance

C

Solid diffusion

q 1 ∂ ∂ˆ ∂q Ds r 2 = ∂t r 2 ∂r ∂r (∂q/∂r)r=0 = 0, (Ds ∂q ∂ r )r =rp = k f ( C − C s ) or (q )r =rp = q* (C s ) for no external resistance

D

Kinetic resistance

dqˆ q = ka C (qm − q ) −  dt Ka  

E

Parallel pore and solid diffusion assuming local equilibrium between pore and adsorbed phase

∂qˆ 1 ∂ = ∂t r 2 ∂r (∂c/∂r)r=0 =

F

Diffusion in bi-disperse particles consisting of aggregates of spherical microparticles of radius rs assuming no external resistance

(

)

(

)

(

)

r 2 D + D dq * ∂c  e s  dc ∂r  0, [(De + Ds dq * dc i ) ∂c ∂r ]r =rp = k f (C − c r =rp ) or (c )r =rp = C for no external resistance

∂q 1 ∂  2 ∂q  , (∂q/∂ρ) ( ( ) = ρ =0 = 0, q )ρ =rs = q * c  Ds ρ  ∂t ρ2 ∂ρ  ∂ρ  q (r , t ) =

3 rs 2 ρ qdρ rs3 ∫0

(

)

q 1 ∂ ∂ˆ ∂c , (∂c/∂r)r=0 = 0, (c )r =rp = C Der 2 = ∂t r 2 ∂r ∂r G

dqˆ = k (qˆ* − qˆ ) dt

LDF model

between the pore liquid and the adsorbent surface. An additional empirical model, known as the Linear Driving Force (LDF) approximation, is also included in Table 6.5 (item G) where the mass transfer rate is represented by the following equation: d ˆ q = k (ˆ q* − ˆ q) dt

(6.25)

Here 〈ˆq 〉 is the average concentration in the particle, k is a ﬁrst order rate constant, and ˆq * is the concentration in the particle in equilibrium with the liquid. In practice, most isotherm measurements do not distinguish between protein adsorbed and that held in the pores. Thus, q* is often used in lieu of ˆq *. Good agreement between the LDF approximation and more rigorous diffusion models is obtained if k = 15Ds rp2. This is equivalent to representing the intra-particle mass transfer resistance as that encountered for solid diffusion through a ﬁlm of thickness equal

6.3 Batch Adsorption Kinetics

to one-ﬁfth of the particle radius, which gives the following result for the mass transfer ﬂux: J=

5Ds (ˆ q* − ˆ q) rp

(6.26)

Although empirical, this simple rate model, originally developed by Glueckauf [61], is adequate for many practical purposes, greatly simplifying the mathematical complexity as seen in Section 6.3.2.5. Description of batch adsorption behavior with variable external protein concentration requires coupling the proper rate equation with a material balance for the external liquid given by the following equation: Vp d ˆ dC q =− dt V dt

(6.27)

C = C0 where C0 is the initial protein concentration and V and Vp are the volumes of liquid and adsorbent particles, respectively. Most protein chromatography matrices are usually kept saturated with water. Thus, Vp is inclusive of any liquid held in the pores. Integrating Equation 6.27 gives the following result: ˆ q =

V (C 0 − C ) Vp

(6.28)

where: rp

ˆ q = q + εpc =

3 ( q + ε pc ) r 2dr rp3 ∫0

(6.29)

is the total concentration in the stationary phase averaged over the particle volume. For cases where the particle size distribution is broad and needs to be considered explicitly, the rate equation as written for each individual particle size and ˆq in Equation 6.27 or 6.28 is replaced by the equivalent quantity properly averaged over all particles (see [33, 62]). In most cases, solution of these equations requires numerical calculations obtained by separating the particle conservation equations by ﬁnite differences. Orthogonal collocation schemes have also proven effective (e.g. see [63–66]). Analytical solutions are generally available when the adsorption isotherm is either linear or rectangular and the most relevant cases are presented in the next section. Other cases may be considered to follow intermediate trends. The rectangular isotherm case is especially relevant for protein adsorption with both non-selective and selective adsorbents used in frontal analysis and capture applications as discussed in Chapter 5. 6.3.2 Analytical Solutions

This section presents analytical solutions in dimensionless form. F = 〈ˆq 〉/ˆq * is the fractional approach to equilibrium and is replaced by ¯q/q* when the adsorption

183

6 Adsorption Kinetics 1.0

R=0

0.2 0.4

0.8

0.6 0.8 R=1

0.6

F

184

0.4

0.2

0.0 0.0

1.0

2.0

3.0

4.0

(3kft/rp)(C0/q*)

Figure 6.10 Solution for batch adsorption in an inﬁnite bath with Langmuir isotherm and ﬁlm mass transfer control. Adapted from [1].

capacity is high. Where appropriate, the isotherm is assumed to follow the Langmuir model (Equation 5.4) or the constant separation factor isotherm (Equation 5.8), whose curvature is described by the separation factor: R=

1 1 + KC 0

(6.30)

The linear isotherm limit R = 1 is approached when KC0 << 1 while the rectangular isotherm limit is approached when KC0 >> 1. Solutions are presented for both constant external concentration, approximated when V/Vp → ∞ (inﬁnite bath), and for ﬁnite values of V/Vp ( ﬁnite bath). 6.3.2.1 External Mass Transfer Control The solution for external ﬁlm mass control (item A in Table 6.5) is given in [1] for the Langmuir isotherm and is shown in Figure 6.10 for an inﬁnite bath. As can be seen in this ﬁgure, when the isotherm is nearly rectangular, F increases linearly with time and is given by:

F=

3k f C 0 rp ˆ q*

(6.31)

This behavior is observed with pellicular particles where external mass transfer is the only resistance and with particles where solid diffusion enhances the rate of mass transfer and the protein concentration is low. 6.3.2.2 Solid Diffusion Control When solid diffusion is controlling (item C in Table 6.5) the solution with constant Ds for batch adsorption in an inﬁnite bath is:

6.3 Batch Adsorption Kinetics

F =1−

6 ∞ 1 ∑ exp ( −n2π 2τDs ) π 2 n =1 n 2

(6.32)

where τDs = Ds t rp2 is a dimensionless time. Approximations useful for numerical computations are as follows: F =6

( ) τ Ds π

0.5

− 3τDs

(6.33a)

for F < 0.8 and: F = {1 − exp [π 2 ( −τDs + 0.96τD2s − 2.92τD3s )]}

0.5

(6.33b)

over the entire range. The analytical solution for a ﬁnite bath is only available for the linear isotherm case and is given by the following equation: exp ( − pn2τDs ) 2 n = 1 9 Λ (1 − Λ ) + (1 − Λ ) pn ∞

F = 1 − 6∑

(6.34)

where Λ = Vpˆq ∞/VC0 is the amount of protein adsorbed at equilibrium divided by the original amount of protein in the solutions and the pns are the positive roots of the equation tan pn 3 = pn 3 + (1 Λ − 1) pn2 Representative curves for various values of Λ are given in Figure 6.11 which shows that the difference between the inﬁnite and ﬁnite bath solutions is signiﬁcant for Λ > 0.1. 1.0 0.9 0.8

0.7 0.5 0.3 =0

F

0.6

0.4

0.2

0.0 0.0

0.1

0.2

0.3

0.4

0.5

0.6

2 1/2

(Dst/rp )

Figure 6.11 Solution for batch adsorption in a ﬁnite bath with linear isotherm and solid diffusion control. Adapted from [1].

185

6 Adsorption Kinetics 1.0 0.1 0.8

0.3

0.5 0.7 R=1

0.6

F

186

0.4

0.2

0.0 0.0

0.1

0.2

0.3 2

[(Det/rp )/(qmK)]

0.4

0.5

0.6

1/2

Figure 6.12 Solution for batch adsorption in an inﬁnite bath with Langmuir isotherm and pore diffusion control. Adapted from [1].

6.3.2.3 Pore Diffusion Control The solutions for batch adsorption with pore diffusion control (item B in Table 6.5) and a linear isotherm are identical to Equations 6.31 and 6.34 provided the dimensionless time τDe = De t mrp2 replaces τ Ds . m = ˆq */C is the slope of the linear isotherm. The general case with a non-linear isotherm requires a numerical solution. Examples of numerically computed uptake curves are shown in Figure 6.12 for different values of R. Note that the uptake curves become increasingly steeper as the curvature of the isotherm increases from linear (R = 1) to nearly rectangular (R = 0.1). Analytical solutions are also available for pore diffusion in the case of the rectangular isotherm and also include the external ﬁlm resistance. These solutions are given by: 2 C 0 De t 1 1 1 = − F − (1 − F )3 2 qm rp 2 3 2

(6.35)

for an inﬁnite bath and by:

( )

1 C 0 De t = 1− I 2 − I1 qm rp2 Bi

(6.36) 3

(

)

( )

I1 =

 λ 3 + η3  λ + 1   1 1  −1 2η − λ 2−λ  − tan n −1 ln  3 tan   + λ 3 6λΛ  λ + 1  λ + η   λΛ 3  λ 3 

I2 =

1  λ 3 + η3  ln   3Λ  λ 3 + 1 

η = (1 − F )1 3

6.3 Batch Adsorption Kinetics

λ=

( )

Λ=

Vp qm VC 0

13 1 −1 Λ

for a ﬁnite bath system. The dimensionless group: Bi =

k f rp De

(6.37)

is the Biot number, which determines the relative importance of external ﬁlm and pore diffusional resistances. In most practical cases, when pore diffusion controls intra-particle transport, Bi > 10, indicating that the external resistance is negligible. These equations are based on the assumption that the protein forms an inﬁnitely sharp concentration proﬁle as it diffuses into the particle. Thus, at each time point until saturation, the protein is localized in an outer shell of the particle with a protein-free core that shrinks over time. This corresponds to the well-known shrinking core model. In practice, although derived for the rectangular isotherm limit, Equations 6.35 and 6.36 will also provide a good approximation of the ﬁnite values of R, provided R is less than about 0.1. 6.3.2.4 Binding Kinetics Control The solution for the case where the binding kinetics the overall rate is readily derived from the corresponding rate equation (item D in Table 6.5), which is an ordinary differential equation. The following solution is obtained for an inﬁnite bath system:

(

F = 1 − exp −

ka t 1−R

)

(6.38)

6.3.2.5 LDF Solution As in the case of binding kinetics control, solutions for the LDF rate equation (item G in Table 6.5) are straightforward since the adsorption kinetics are described by an ordinary differential equation. For an inﬁnite bath system, the solution is simply:

Ds t   F = 1 − exp ( −kt ) = 1 − exp  −15 2   rp 

(6.39)

Figure 6.13 provides an instructive comparison of the analytical solutions for different rate models for an inﬁnite bath system for favorable binding conditions. Each solution has been plotted against a dimensionless time, τD, which is different for each mechanism. The numerical solution for pore diffusion with a Langmuir isotherm and R = 0.05 is also shown and is almost indistinguishable from the shrinking core solution, which assumes R = 0. Comparing the various solutions, it is obvious that while the mathematical forms are all very different, the general shape of the uptake curves is quite similar. This is good news and bad news. It is

187

6 Adsorption Kinetics 1.0

0.8

0.6

Solid diffusion,

D

=

Ds t rp2

F

188

Pore diffusion (R=0.05), 0.4 Pore diffusion (R=0), LDF approximation,

0.2

Binding kinetics,

D

D

D

1 De t C0 0.6 rp2 q* 1 De t C 0 = 0.6 rp2 q* D

=

= kt

= (1 R)ka t

0.0 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

D

Figure 6.13 Solution for batch adsorption in an inﬁnite bath for different rate mechanisms. The dimensionless time τD is deﬁned in the annotation for each mechanism.

good news for the design engineer, since even a very simple model provides a reasonable description of the uptake kinetics. For example, a simple equivalence between pore and solid diffusion models is found by comparing the dimensionless times. Accordingly: Ds ∼

1 C0 De 0.6 q *

(6.40)

allowing the LDF approximation to be used to describe pore diffusion in lieu of the more complex partial differential equation form of the rigorous model. On the other hand, such equivalence is bad news for the scientist who is interested in establishing the mass transfer mechanism by comparing macroscopic rate data from different models since the ability to discriminate on this basis is quite limited. Of course, despite the qualitative similarity of the uptake curves, the dependence of the rate on the physical properties of the adsorption system varies widely. For both solid and pore diffusion, adsorption times scale vary with the square of the particle size. However, when the isotherm is highly favorable, the adsorption time will be almost independent of protein concentration for solid diffusion, but proportional to protein concentration for pore diffusion. The binding kinetic model, on the other hand, has no dependence on particle size. Thus, experimental variations may be used to discern the rate-controlling mechanism. 6.3.2.6 Combined Mass Transfer Resistances Exact analytical solutions for combined resistances (e.g. pore and solid diffusion in parallel or series) are generally available only for the linear isotherm case (R = 1) and for some rate mechanisms in the case of the rectangular isotherm limit (R = 0). These solutions can be found in standard adsorption texts and handbooks

6.3 Batch Adsorption Kinetics

(see e.g. [1, 23]). In many cases the LDF model provides a useful approximation for practical calculations involving favorable isotherms by expressing the rate coefﬁcient as follows: Parallel pore and solid diffusion (item E in Table 6.5) k=

15Ds 1 15De C0 + rp2 0.6 rp2 q *

(6.41)

Pore and solid diffusion in bi-disperse particles (item F in Table 6.5) rp2 q * r2 1 = s + 0.6 k 15Ds 15De C 0

(6.42)

External ﬁlm and intra-particle diffusion in series 1 1 rp q * = + k0 k 3k f C 0

(6.43)

In Equation 6.43, k0 is an overall LDF rate coefﬁcient for use in Equation 6.25 with k calculated from either Equation 6.41 or 6.42, depending on whether the intra-particle resistances are in parallel or series, respectively. Examples 6.4 and 6.5 provide illustrative scenarios for the equations above.

Example 6.4

Estimate the effective diffusivity from batch adsorption data

Figure 6.14 shows batch adsorption data from [62] for 1 mg/ml αchymotrypsinogen in a 20 mM sodium phosphate buffer at pH 6.5 on SP-Sepharose-FF in a shallow bed operated at 1200 cm/h and in a 100-ml vessel stirred at 300 rpm with 0.16 ml of particles. The adsorption isotherm is described approximately by q = 235 × 100 × C/(1 + 100 × C), with both q and C expressed in mg/ml. Determine the effective pore diffusivity De assuming that the external resistance is negligible. Solution – At C0 = 1 mg/ml, q* = 233 mg/ml and R = 1/(1 + 100 × 1) = 0.01. Thus, approximation with a rectangular isotherm is adequate. As seen in Figure 6.14, Equation 6.35 with Bi = ∞ provides an approximate ﬁt of the shallow bed data with De = 3 × 10−7 cm2/s. However, the adsorption rate is underestimated for short periods of time and overestimated for long periods. A much better ﬁt is obtained when the particle size distribution (PSD) is taken into account. The solution for this case, assuming a rectangular isotherm is given by the following expression [62]: 2 M   1  q  rp  De t C 0     π 1 F= = ∑ f j 1−  + cos  + cos−1 1− 12   2 (6.44)     rp, j  rp qm    qm j =1   2  3 3   where q is the adsorbed concentration averaged over all particles, ¯r p is the average particle radius, fj is the volume fraction of particles of radius rp,j, and M the number of fractions in the PSD. 3

189

6 Adsorption Kinetics

For adsorption in the stirred vessel we have Λ = (0.16 × 235)/(100 × 1) = 0.376. Thus, at equilibrium, approximately 38% of the protein initially in solution is adsorbed. The ﬁt of the data with Equation 6.36 for De = 3 × 10−7 cm2/s is only approximate and is again greatly improved by taking into account the PSD. The solution for this case with ﬁnite Λ is more complex and can be found in [62]. Solutions for the rectangular isotherm taking into account the PSD and the external resistance can be found in [33]. 250

250

200

200

150

150

q= (mg/mL)

q= (mg/mL)

190

100

100

(a) SP-FF, Shallow bed PSD Avg. rp

50

50

(b) SP-FF, Stirred-batch PSD Avg. r p

Experimental

Experimental 0 0

2000

4000

6000

0 0

Time (sec) Figure 6.14 Batch adsorption of 1 mg/ml α-chymotrypsinogen on SP-Sepharose-FF in (a) a shallow bed and (b) a stirred vessel with V = 100 ml, Vp = 0.16 ml. The dashed lines are calculated using Equations 6.35

2000

4000

6000

Time (sec) and 6.36 for (a) and (b), respectively, using the 4,3 average particle radius (48 µm) and De = 3 × 10−7 cm2/s. The solid lines take into account the particle size distribution. Reproduced from [62].

6.3.3 Experimental Veriﬁcation of Transport Mechanisms

Although empirically derived rate coefﬁcients can be used for design purposes, the determination of the actual mass transfer mechanism is not only an important scientiﬁc endeavor but is also useful, in practice, to explain the relationship between mass transfer rates and operating conditions since different mechanisms generally result in a different dependence on protein concentration and adsorbent properties. As can be seen in Figure 6.13, when adsorption is highly favorable, macroscopic uptake rates are insensitive to the nature of the controlling rate mechanism. The differences are even smaller when the isotherm is linear. However, when the isotherm is highly favorable there is a much greater depend-

6.3 Batch Adsorption Kinetics

Example 6.5

Estimation of Biot number

Estimate the Biot number for the conditions shown in Example 6.4. Solution – The Biot number determines the relative importance of external and intra-particle mass transfer resistances. For the case of pore diffusion, Bi is given by Equation 6.37. For α-chymotrypsinogen (Mr ∼ 26 kDa), D0 = 9.3 × 10−7 cm2/s and Sc = 11 000 assuming η = 1.0 cp. For the shallow bed experiment, Re = 1 × (1200/3600) × 0.0096/0.01 = 0.32. For these conditions taking ε = 0.3, Equations 6.3–6.5 yield Sh between 20 and 54. The corresponding kf-values are between 0.002 and 0.005 cm/s. Thus, Bi = kfrp/De is estimated to be between 32 and 83 using the average particle radius and De = 3 × 10−7 cm2/s from Example 6.4. The large values of Bi indicate that the external resistance is negligible. For adsorption in a stirred vessel, following Example 6.1, Case 2, ε ∼ 640 cm2 s3. Using this value with the average particle size, Equation 6.6 yields Sh = 17 or kf = 0.0017 cm/s. The corresponding Bi = 27 is still sufﬁciently large for the external resistance to be considered negligible.

ence on the shapes of the intra-particle concentration proﬁles during transient adsorption. As an example, Figure 6.15 shows uptake curves and the corresponding concentration proﬁles in the particles calculated for batch uptake with pore and solid diffusion mechanisms. The uptake curves are nearly identical. However, since the isotherm is very favorable for these simulations (R < 0.1), the pore diffusion mechanism yields sharp fronts while solid diffusion yields smooth proﬁles. Signiﬁcant advances have been made in recent years with the introduction of microscopic techniques for the visualization of intra-particle protein concentration proﬁles, especially for ion exchangers, where mass transfer phenomena are typically less well understood. Images of lysozyme adsorption in a SP-Sepharose-FF particle obtained by confocal scanning laser microscopy (CSLM) are shown in Figure 6.16 from the work of Dziennik et al. [68]. The CSLM technique, originally introduced by Ljunglof and Hjorth [69] for protein adsorption studies, collects the light emitted by a ﬂuorescently-labeled protein tracer from a thin optical section of the particle. Assuming that the ﬂuorescently-labeled protein tracer and the unlabeled protein behave identically, the ﬂuorescence intensity is directly related to the total protein concentration along the radial coordinate. Fluorescence attenuation effects skew the experimental proﬁles, which can be seen in Figure 6.16, and can be corrected [70]. However, even without correction, the trends in Figure 6.16 are clear. At 6 mM ionic strength, when binding is very strong, the proﬁle is consistent with the shrinking core behavior predicted by pore diffusion alone. Conversely, at 100 mM ionic strength, although the adsorption isotherm remains quite favorable (see [68]), the proﬁles appear to become much smoother suggesting a change in the dominant transport mechanism.

191

6 Adsorption Kinetics

200

(a)

3

q (mg/cm )

150

100

Pore diffusion

50

Diffusion in the adsorbed phase 0 0

500

1000

1500

2000

Time (s) 200

(b) 150 3

q (mg/cm )

192

100

50

0 0

0.2

0.4

0.6

Distance from interface, 1-r/r

0.8

1

p

Figure 6.15 Comparison batch adsorption in spherical particles calculated for pore and solid diffusion mechanisms. (a) Batch uptake curves; (b) concentration proﬁles in the particle at 100, 250, 500, 1000, 1500, and 2000 s. Reproduced from [67].

Comparable results can be obtained by refractive index-based microscopy as shown in Figure 6.17 from the work of Stone and Carta [71]. This approach takes advantage of the refractive index gradient that is established in the particle during transient protein adsorption. When light traverses a particle partially saturated with protein, light rays are concentrated at the refractive interface between the retreating protein-free core and the advancing protein-saturated shell. When the interface is sharp, this optical phenomenon is manifested by a bright ring marking the position of the adsorption front near the particle equatorial section, which is visible with an ordinary optical microscope. For pore diffusion alone, with a rectangular isotherm and without external mass transfer resistance, the position of the adsorption front, rs, is expected to obey the equation [23]:

6.3 Batch Adsorption Kinetics

Figure 6.16 Images (A) and (B) and digitized ﬂuorescence intensity proﬁles (C) for adsorption of lysozyme containing trace amounts of lysozyme labeled with the Cy5 ﬂuorophor in SP-Sepharose-FF particles. Ionic

strength of (A) 6 mM and (B) 100 mM. Reproduced from [68]) with permission. Copyright 2003 National Academy of Sciences, USA.

1

0.8

f( ) s

0.6

s

s

, f( )

s

0.4

0.2

0

0

10

20

30

40

50

t (min) Figure 6.17 Images for the adsorption of 2 mg/ml lysozyme at 20 mM ionic strength in a 115-µm SP-Sepharose-FF particle obtained by optical microscopy. The bright light ring marks the position of the adsorption front. Images are shown for 0 to 40 min at 5-min

intervals. The ﬁgure on the right shows the dimensionless position of the protein adsorption front and shrinking core model prediction for De = 2.6 × 10−7 cm2/s. Data and images from [71].

193

6 Adsorption Kinetics

2ρs3 − 3ρs2 + 1 =

6DeC 0 t qmrp2

(6.45)

where ρs = rs/rp. Accordingly, De can be determined from the slope of f ( ρs ) = 2ρs3 − 3ρs2 + 1 versus time, which as can be seen in Figure 6.17, is linear. The corresponding value of De = 2.7 × 10−7 cm2/s is in agreement with other determinations. Experimental evidence for an adsorbed phase diffusion mechanism has been obtained for protein transport in oppositely charged polyacrylamide gels supported in quartz capillaries with a square cross-section and a representative result is shown in Figure 6.18 from the work of Russell et al. [35]. The images correspond to the adsorption of 1 mg/ml myoglobin on a positively charged polyacrylamide gel at pH 9.6. At this pH, myoglobin is negatively charged and partitions very favorably into the gel. Since the protein is colored and the supporting capillary has a square cross-section, images representing the total protein concentration in the gel can be obtained with an ordinary optical microscope. Despite the favorable nature of the isotherm, as shown in Figure 6.18, the concentration proﬁles in the gel are smooth and consistent with a solid diffusion mechanism. When plotted versus the Boltzmann transformation variable, z t , these proﬁles collapse to a single curve as expected for a Fickian diffusion mechanism [72]. The Ds value obtained varied somewhat with concentration but averaged

1.0

Experimental Predicted 0.8

0.6

q/q*

194

8h 4h

0.4

2h 1h 0.2

0.5 h 0.05 h

0.0 1.0

0.8

0.6

0.4

0.2

0.0

Distance into gel (mm) Figure 6.18 Images and digitized with a square cross-section. Dashed lines are concentration proﬁles for the adsorption of calculated for a solid diffusion mechanism 1 mg/ml myoglobin at 50 mM ionic strength with Ds = 1.3 × 10−7 cm2/s. Reproduced and pH 9.6 in a cationic polyacrylamide gel from [35]. supported in a 100-µm-lumen quartz capillary

6.3 Batch Adsorption Kinetics

at about 1.3 × 10−7 cm2/s. This behavior suggests that while the protein is very favorably partitioned in the gel, it retains diffusional mobility. This combination gives a high rate of mass transfer, which is consistent with experimental observations for HyperD adsorbents based on similar gels supported in rigid pore particles. 6.3.4 Multi-component Protein Adsorption Kinetics

Limited data are available in the literature for multi-component adsorption kinetics under competitive binding conditions. If the adsorption isotherm is linear, predictions are straightforward since each component will behave independently. However, at higher loadings competitive adsorption complicates the kinetic behavior. Even for ordinary pore diffusion, where the effective pore diffusivities are known with reasonable certainty, predictions are not straightforward, primarily because of the difﬁculties encountered in describing multi-component adsorption equilibrium as discussed in Chapter 5. In general, detailed numerical simulations are required (e.g. see [73] and [74]). In many cases, however, protein adsorption is highly favorable, and the adsorption isotherm for each pure component is approximately rectangular. This situation results in an instructive limiting case, which is shown schematically in Figure 6.19 for the simultaneous adsorption of two

t < tc

t > tc

B

B qm,B

A c'B

rB

rA

qm,B

qA*

A

c'B

qA*

CA,CB

CA,CB

qB *

qB*

rp

rA

rp

Figure 6.19 Diagram of intra-particle concentration proﬁles for the case of two-component adsorption with rectangular isotherms and B displacing A. Thick and thin lines represent the q and c proﬁles, respectively, on different scales. Reproduced from [75].

195

196

6 Adsorption Kinetics

proteins A and B. Both are assumed to have a highly favorable isotherm, but A is preferentially adsorbed. If the adsorbent is exposed to a mixture of the two proteins and mass transfer is controlled by pore diffusion, the B-component diffuses into the particle and is concentrated in a spherical shell ahead of the A-adsorption front. For longer periods of time, pure B ﬁlls the inner shrinking core while A continues to diffuse into the particle eventually displacing all of the B that was adsorbed in excess of the equilibrium quantity. For an inﬁnite bath system with constant external protein concentrations and rectangular pure component isotherms, the equations describing the dimensionless positions of the adsorption fronts ρA = rA/rp and ρB = rB/rp are as follows: 2ρ3A − 3ρ2A + 1 =

6De , AC A t q*A rp2

2ρB3 − 3ρB2 + 1 = (1 + β )

qm ,B − qB* 6De , AC A t qm ,B qA*rp2

1  −1  1− 1  c B′  ρA  = 1 +  1 + 1 1  β CB  −   ρA ρB 

(6.46)

(6.47)

(6.48)

for t < tc, and: 2ρ3A − 3ρ2A + 1 =

6De , AC A t q*A rp2

1 cB′ =1+ β CB

(6.49)

(6.50)

for t > tc, where tc =

β=

q*A rp2 1 qm ,B − qB* 1 + β 6De , AC A qm ,B

q *A De ,BCB (qm ,B − qB* )De , AC A

(6.51)

(6.52)

In these equations, the qm values are the pure component adsorption capacities while the q* values are the equilibrium capacities for the mixture. c B′ is the pore ﬂuid concentration of B immediately ahead of the A-adsorption front. More complicated relationships are available for the case where ﬁlm mass transfer is important and for a ﬁnite bath system and can be found in [75]. An experimental example from this work is shown in Figure 6.20 for the simultaneous adsorption of lysozyme and cytochrome c on SP-Sepharose-FF. Both proteins have essentially rectangular isotherms under these conditions, but at equilibrium lysozyme displaces cytochrome c completely. However, during transient adsorption, a substantial amount of cytochrome c is adsorbed consistent with Figure 6.19. Over time, the cytochrome c adsorbed in excess of the equilibrium quantity is desorbed by

References 200

300

Cytochrome c Lysozyme SC Model

250

150

q (mg/ml)

q (mg/ml)

200

150

100

100

50 Cytochrome c

50

Lysozyme

0

0 0

1

2

3

C (mg/ml)

Figure 6.20 Pure component adsorption isotherms for lysozyme and cytochrome c at 20 mM ionic strength pH 6.5 on SP-SepharoseFF (left) and batch uptake curves for a mixture containing 0.5 mg/ml lysozyme and 1.5 mg/ml cytochrome c. The traces are

0

1000

2000

3000

4000

5000

Time (s)

predicted from the dual shrinking core model (right). Note that the more weakly bound protein, cytochrome c, is ﬁrst bound in the interior of the particle and then displaced by the advancing lysozyme front. Reproduced from [75].

the advancing lysozyme front. Model predictions based on the dual shrinking core model discussed above are also shown and are in good agreement with the experimental behavior. Obviously, similar effects can be expected in a chromatography column during the feed load step, when adsorbed concentrations can temporarily exceed equilibrium values as a result of diffusional mass transfer limitations.

References 1 LeVan, M.D., and Carta, G. (2007) Adsorption and ion exchange, Section 16, in Perry’s Chemical Engineers’ Handbook, 8th edn (ed. D.W. Green), McGraw-Hill, New York. 2 Wilson, E.J., and Geankoplis, C.J. (1966) Ind. Eng. Chem. Fundam., 5, 9. 3 Kataoka, T., Yoshida, H., and Ueyama, K. (1972) J. Chem. Eng. Jpn., 5, 132. 4 Carberry, J.J. (1960) AIChE J., 6, 460. 5 Fernandez, M.A., and Carta, G. (1996) J. Chromatogr. A, 646, 169. 6 Hansen, E., and Mollerup, J. (1998) J. Chromatogr. A, 827, 259. 7 Sarfert, F.T., and Etzel, M.R. (1997) J. Chromatogr. A, 764, 3. 8 Levins, D.M., and Glastonbury, J.R. (1972) Trans. Inst. Chem. Eng., 50, 132.

9 Armenante, P., and Kirwan, D.J. (1989) Chem. Eng. Sci., 44, 2781. 10 McCabe, W.L., Smith, J.C., and Harriott, P. (1985) Unit Operations of Chemical Engineering, 4th edn, McGraw-Hill, New York. 11 Borst, C.L., Grzegorczyk, D.S., Strand, S.J., and Carta, G. (1997) Reac. Polym., 32, 25. 12 Meyers, J.J., Crosser, O.K., and Liapis, A.I. (2001) J. Biochem. Biophys. Methods, 49, 123. 13 Meyers, J.J., Crosser, O.K., and Liapis, A.I. (2001) J. Chromatogr. A, 908, 35. 14 Bryntesson, L.M. (2002) J. Chromatogr. A, 945, 103. 15 Brenner, H., and Gaydos, L.J. (1977) J. Coll. Int. Sci., 58, 312.

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6 Adsorption Kinetics 16 Anderson, J.L., and Quinn, J.A. (1974) Biophys. J., 14, 130. 17 Smith, F.G., and Deen, W.M. (1980) J. Coll. Int. Sci., 78, 444. 18 Smith, F.G., and Deen, W.M. (1983) J. Coll. Int. Sci., 91, 571. 19 Probstein, R.F. (1994) Physicochemical Hydrodynamics: An Introduction, 2nd edn, John Wiley & Sons, Inc., New York. 20 Yang, Y., Waltz, J., and Pintauro, P. (1995) J. Chem. Soc. Faraday Trans., 91, 2827. 21 Carta, G., Ubiera, A.R., and Pabst, T.M. (2005) Chem. Eng. Technol., 28, 1252. 22 Bankston, T.E., Stone, M.C., and Carta, G. (2008) J. Chromatogr. A., 1188, 242. 23 Ruthven, D.M. (1984) Principles of Adsorption and Adsorption Processes, John Wiley & Sons, Inc., New York. 24 Helfferich, F. (1962) Ion Exchange, McGraw-Hill, New York. 25 Shiosaki, A., Goto, M., and Hirose, T. (1994) J. Chromatogr. A, 679, 1. 26 Yoshida, H., Yoshikawa, M., and Kataoka, T. (1994) AIChE J., 40, 2034. 27 Farnan, D., Frey, D.D., and Horvath, Cs. (2002) J. Chromatogr. A, 959, 65. 28 Tsou, H.S., and Graham, E.E. (1985) AIChE J., 31, 1959. 29 Weaver, L.E., and Carta, G. (1996) Biotechnol. Prog., 12, 342. 30 Thommes, J. (1999) Biotechnol. Bioeng., 62, 358. 31 Hubbuch, J., Linden, T., Knieps, E., Ljunglöf, A., Thömmes, J., and Kula, M.R. (2003) J. Chromatogr. A., 1021, 93. 32 Ubiera, A.R., and Carta, G. (2006) Biotechnol. J., 1, 665. 33 Stone, M.C., and Carta, G. (2007) J. Chromatogr. A., 1146, 202. 34 Tong, J., and Anderson, J.L. (1996) Biophys. J., 70, 1505. 35 Russell, S.M., and Carta, G. (2005) Ind. Eng. Chem. Res., 44, 8213. 36 Lewus, R.K., and Carta, G. (2001) Ind. Eng. Chem. Res, 40, 1548. 37 Russell, S.M., Belcher, E.B., and Carta, G. (2003) AIChE J., 49, 1168. 38 Happel, J. (1958) AIChE J., 4, 197. 39 Neale, G., Epstein, N., and Nader, W. (1973) Chem. Eng. Sci., 28, 1865. 40 Carta, G., Gregory, M.E., Kirwan, D.J., and Massaldi, H.A. (1992) Sep. Technol., 2, 6.

41 Liapis, A.I., and McCoy, M.A. (1992) J. Chromatogr. A, 599, 87. 42 Meyers, J.J., and Liapis, A.I. (1998) J. Chromatogr. A, 827, 197. 43 Carta, G., and Rodrigues, A.E. (1993) Chem Eng. Sci., 48, 3927. 44 Rodrigues, A.E., Lu, Z.P., Loureiro, J.M., and Carta, G. (1993) J. Chromatogr., 653, 189. 45 Rodrigues, A.E., Ahn, B.J., and Zoulalian, A. (1982) AIChE J., 28, 541. 46 Frey, D.D., Schweinheim, E., and Horvath, Cs. (1993) Biotechnol. Progr., 9, 273. 47 Afeyan, N.B., Gordon, N.F., Mazsaroff, I., Varady, L., Fulton, S.P., Yang, Y.B., and Regnier, F.E. (1992) J. Chromatogr. A, 519, 1. 48 Freitag, R., Frey, D., and Horvath, Cs. (1994) J. Chromatogr. A, 686, 165. 49 Pfeiffer, J.F., Chen, J.C., and Hsu, J.T. (1996) AIChE J., 42, 932. 50 Gustavsson, P.E., Mosbach, K., and Larsson, K. (1997) J. Chromatogr. A, 776, 197. 51 Nash, D.C., and Chase, H.A. (1998) J. Chromatogr. A, 807, 185. 52 Li, Q.L., Grandmaison, E.W., Goosen, M.F.A., and Taylor, D. (2000) AIChE J., 46, 1927. 53 Gebauer, K.H., Thommes, J., and Kula, M.R. (1997) Chem. Eng. Sci., 52, 405. 54 Chase, H.A. (1984) J. Chromatogr., 297, 179. 55 Jendeber, L., Persson, B., Andersson, R., Karlsson, R., Uhlen, M., and Nilsson, B. (1995) J. Mol. Recognit., 8, 270. 56 Saha, K., Bender, F., and Gizeli, E. (2003) Anal. Chem., 75, 835. 57 Schwartz, M.P., Alvarez, S.D., and Sailor, M.J. (2007) Anal. Chem., 79, 327. 58 Mitomo, H., Shigenatsu, H., Kobatake, E., Furusawa, H., and Okahata, Y. (2007) J. Mol. Recognit., 20, 83. 59 Lacki, K.M., Bergander, T., Grönberg, A., and Öberg, K. (2008) High throughput process development: Determination of dynamic binding capacity using microtiter plates ﬁlled with chromatography resins. ACS BIOT Web-based Symposium, Jan. 60 Hahn, R., Tscheliessnig, A., Zöchling, A., and Jungbauer, A. (2005) Chem. Eng. Technol., 28, 1241.

References 61 Glueckauf, E. (1955) Trans. Faraday Soc., 51, 1540. 62 Carta, G., and Ubiera, A. (2003) AIChE J., 49, 3066. 63 Liapis, A.I., and Rippin, D.W.T. (1978) Chem. Eng. Sci., 33, 593. 64 Santacesaria, E., Morbidelli, A., Servida, A., Storti, G., and Carra, S. (1982) Ind. Eng. Chem. Proc. Des. Dev., 21, 446. 65 Saunders, M.S., Vierow, J.B., and Carta, G. (1989) AIChE J., 35, 53. 66 Whitley, R.D., van Cott, K.E., Berninger, J.A., and Wang, N.H.L. (1991) AIChE J., 37, 555. 67 Schirmer, E.B., and Carta, G. (2007) AIChE J., 53, 1472. 68 Dziennik, S.R., Belcher, E.B., Barker, G.A., DeBergalis, M.J., Fernandez, S.E.,

69 70 71 72

73 74 75

and Lenhoff, A.M. (2003) Proc. Natl Acad. Sci. USA, 100, 420. Ljunglöf, A., and Hjorth, R. (1996) J. Chromatogr. A, 743, 75. Yang, K., Shi, Q.H., and Sun, Y. (2006) J. Chromatogr. A, 1136, 19. Stone, M.C., and Carta, G. (2007) J. Chromatogr. A, 1160, 206. Cussler, E.L. (1997) Diffusion–Mass Transfer in Liquid Systems, 2nd edn, Cambridge University Press, Cambridge, UK. Lewus, R.K., and Carta, G. (1999) AIChE J., 45, 512. Gallant, S.R. (2004) J. Chromatogr. A, 1028, 189. Martin, C., Iberer, G., Ubiera, A., and Carta, G. (2005) J. Chromatogr. A, 1079, 105.

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7 Dynamics of Chromatography Columns 7.1 Introduction

In this chapter we address the dynamics of chromatographic columns subjected to various inputs. In principle, the treatment is applicable to virtually all transient chromatography applications, independent of operating mode and of the physical basis for separation. The fundamental basis is the so-called axially dispersed plug ﬂow model. In this model, ﬂow across the column diameter is assumed to be uniform and any deviation from ideal ﬂow is included in a Fickian, axial dispersion term. The approach is suitable for the description of real columns provided deviations from uniform ﬂow are not extreme. Where large variations in mobile phase velocity across the column diameter occur, detailed multidimensional ﬂow models are more appropriate. In practice, however, such conditions are not desirable for industrial chromatographic applications and even large-diameter columns if properly designed and packed, will often exhibit substantially uniform ﬂow (see Chapter 4). It is instructive to consider the solution of the conservation equations that arise from the axially dispersed plug ﬂow model in two steps. In this chapter, we neglect axial dispersion and assume that local equilibrium exists between the mobile phase and the stationary phase at each time and point within the chromatography column. Such ideal chromatography conditions are approximated in well packed columns at low ﬂow rates and containing small particles. Later, in Chapter 8, we consider dispersion effects by taking into account axial dispersion and adsorption kinetics, the latter being especially important in protein chromatography.

7.2 Conservation Equations

We begin with a simpliﬁed model of chromatography in which we assume plugﬂow of the mobile phase through a uniformly packed sorption bed. Deviations from this assumption occur in most real chromatography columns. Yet in most Protein Chromatography: Process Development and Scale-Up. Giorgio Carta and Alois Jungbauer © 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 978-3-527-31819-3

202

7 Dynamics of Chromatography Columns

cases the plug ﬂow analysis is sufﬁcient even for these systems if a suitable axial dispersion term is included. With reference to Figures 2.7 and 4.1, in a system with M components we can write the following equations and boundary conditions for each species i: Mass balances

ε

∂C i ∂ qˆi ∂C ∂2C + (1 − ε ) + ε v i = εDL 2i ∂t ∂t ∂z ∂z

(7.1)

Rate equations ∂ ˆ qi = f i (C j, q *j ), ∂t

j = 1, 2, … , M

(7.2)

Boundary conditions z = 0: Ci = CiF + z = L:

DL ∂Ci v ∂z

∂C i =0 ∂z

(7.3)

In these equations, 〈qˆ i〉 is the particle-average adsorbate concentration given by the following expression: rp

ˆ qi = qi + ε p ci =

3 (qi + ε pc i ) r 2dr rp3 ∫0

(7.4)

Note that 〈qˆ i〉 includes both the adsorbed molecules, with particle average concentration − qi, as well as the molecules that are simply held in the pore ﬂuid, with particle-average concentration − ci. As discussed in Chapter 6, in many cases where adsorption is favorable and the solution is dilute, as a good approximation we have qˆ i ∼ qi and 〈qˆ i〉 ∼ − qi (see Section 6.3.1). DL in Equation 7.1 is an axial dispersion coefﬁcient, which is used to describe deviations from plug ﬂow and axial mixing in the column. Equation 7.3 represents the so-called Danckwerts boundary conditions, which are appropriate for suitably long columns [1, 2]. If axial dispersion is neglected, i.e. DL = 0, Equations 7.1 to 7.3 simplify to the following expressions:

ε

∂C i ∂ qˆi ∂C + (1 − ε ) + εv i = 0 ∂z ∂t ∂t

z = 0: Ci = CiF

(7.5) (7.6)

The rate equations, Equation 7.2, are based on an appropriate representation of transport and kinetic processes within and outside the adsorbent particles as discussed in Chapter 6. Table 6.5 provides a summary of these equations for commonly encountered rate processes. In the general case, mass balances and rate equations require the use of the multi-component isotherm, q *j = q *j (C1, C2, … , CM ), to relate the adsorbed and liquid phase concentrations at the particle–ﬂuid interface.

7.2 Conservation Equations

7.2.1 Boundary Conditions

The mathematical description given by the equations outlined in the previous section is quite general and applies to all modes of operation of chromatography with an adequate selection of the column entrance conditions. The entrance conditions are given below for the classical modes of operation: Elution chromatography Isocratic–analytical (pulse injections) CiF =

Mi δ (t ) , i = 1, 2, … , M Q

(7.7)

Isocratic–preparative (ﬁnite injection) CiF = CFeed ,i, i = 1, 2, … , M for 0 < t < tFeed CiF = 0, i = 1, 2, … , M for t > tFeed

(7.8)

Gradient–analytical (linear) CiF =

Mi δ (t ) , i = 1, 2, … , M − 1 Q

(7.9)

CMF = CM0 + β t = modulator concentration

Frontal analysis CiF = CFeed ,i × H (t ) , i = 1, 2, … , M

(7.10)

Displacement development CiF = CFeed ,i, i = 2, 3, … , M

for 0 < t < tFeed

C = 0, i = 2, 3, … , M , C = CD, for t > tFeed F i

F 1

(7.11)

In these equations, Mi is the amount of component i injected, Q is the volumetric ﬂow rate, β is the gradient slope, and δ(t) and H(t) the delta function (representing a pulse of inﬁnitesimal duration and inﬁnite height) and the unit step function, respectively. The delta function input is approximated in analytical chromatography and permits simpliﬁcation of the mathematical analysis. However, solutions obtained for a delta function input are also suitable in many practical cases, even under preparative or process conditions with ﬁnite feed injections provided the actual feed volume is a fraction of the eluted peak volume. 7.2.2 Dimensionless System

Before proceeding to the analysis of the dynamics of chromatography, it is useful to introduce dimensionless expressions of the column mass balances since much

203

204

7 Dynamics of Chromatography Columns

of the adsorption literature is expressed in this form. The following dimensionless variables are deﬁned [3]: ˆ Ci q , Yi = i Cref qref

(7.12)

ε vt ut z = , ζ= L L L

(7.13)

Xi =

τ=

Xi and Yi are dimensionless concentrations normalized by reference values, Cref and qref. The latter are normally taken to be the feed or the initial values, dependent on the application. The variables τ and ζ are dimensionless time and column axial coordinates, respectively, with τ equal to the number of column volumes of mobile phase passed through the column. This quantity is also expressed by CV = tQ/Vc, where Vc is the column volume. With the introduction of these dimensionless variables, the mass balance equations become: kref ′

1 ∂2 X i ∂Yi ∂X i 1 ∂X i + = + ∂τ ∂τ ε ∂ζ εPeL ∂ζ 2

(7.14)

where kref ′ is a reference retention factor deﬁned by the following equation: kref ′ =

1 − ε qref ε Cref

(7.15)

PeL =

vL DL

(7.16)

and

is a Peclet number based on the column length. When PeL → ∞, axial dispersion is negligible. In this case, it is convenient to redeﬁne the dimensionless time as follows [3]:

τ vt −ζ −ζ ε τ1 = = L kref kref ′ ′

(7.17)

so that the material balances become simply ∂Yi ∂X i + =0 ∂τ 1 ∂ζ

(7.18)

The variable τ1 is a dimensionless time, which can be viewed as a throughput parameter [3]. If τ1 = 1, at the column outlet (ζ = 1) we have: vt −1 L =1 kref ′ or

(7.19)

7.3 Local Equilibrium Dynamics

L t = (1 + kref ′ ) v

(7.20)

For isocratic elution with a linear isotherm, Equation 7.20 corresponds to the elution time of a component with k ′ = kref ′ (cf. Equation 2.10). Alternatively, for frontal loading of a clean column, with favorably adsorbed components, if Cref is deﬁned as the feed concentration and qref is the corresponding adsorbed equilibrium concentration, Equation 7.20 corresponds to the time of passage of the adsorption front through the column under ideal chromatography conditions. In this case, τ1 = 1 when the column has been supplied with an amount of feed equal to that required to obtain complete saturation of the adsorbent at equilibrium with the feed concentration.

7.3 Local Equilibrium Dynamics

The simplest analysis of the dynamics of chromatography columns is based on the assumptions of non-dispersed plug ﬂow and local interphase equilibrium. This is referred to as the local equilibrium theory or the ideal model of chromatography. In this model we neglect axial dispersion and assume that all rate factors are sufﬁciently fast so that mobile and stationary phases can be considered to be in thermodynamic equilibrium. The only effect considered is thus the coupling of ﬂow and adsorption equilibrium with appropriate boundary conditions. Solutes in a chromatographic column can be present either within the adsorbent particles (adsorbed or in the stagnant pore ﬂuid) or in the mobile phase in the extra-particle space (see Section 4.1). While in the mobile phase, these solutes move down the column at the interstitial velocity, v. Conversely, while in the particles, on average they have zero velocity in the axial direction. An exception is the case of chromatography with large-pore permeable supports, where some convective ﬂow occurs within the particles. However, as seen in Chapter 6, even with highly permeable particles, intra-particle ﬂow is a small fraction of the total. Thus, this assumption is also valid as an approximation for these cases. If we consider, for example, isocratic elution with a linear isotherm, we can envision that each solute molecule will spend some time in the stationary phase and then diffuse back out in a series of random steps. If transport in and out of the particle is fast, the velocity at which the solute will move down the column will be proportional to the fraction of that solute which at each time is found in the mobile phase. This velocity is referred to as the chromatographic velocity and is given by: vci = v × ( fraction of solute i in mobile phase) Of course, the randomness of the diffusion process will cause spreading of the solute molecules around the front deﬁned by this velocity, which has to be interpreted as an average. The fraction of solute i in the mobile phase is

205

206

7 Dynamics of Chromatography Columns

εCi εCi = εCi + (1 − ε ) qˆi εCi + (1 − ε ) miCi where mi is the slope of the linear isotherm describing partitioning between the two phases. From the two equations above we obtain the following result: vci =

v 1− ε 1+ mi ε

(7.21)

Accordingly, the elution time of a component under linear isocratic elution conditions is given by: tR ,i =

(

)

L L L 1− ε = 1+ mi = (1 + ki′) vci v v ε

(7.22)

which is consistent with Equation 2.10. The general expression for the chromatographic velocity with an arbitrary isotherm can be obtained from Equation 7.5, valid for DL = 0. For local equilibrium conditions, this equation becomes:

ε

∂C i ∂ˆ qi* ∂C i + (1 − ε ) + εv =0 ∂t ∂t ∂z

(7.23)

Since the partial derivative ∂qˆi*/∂t can be written as follows: ∂qˆi* dqˆi* ∂Ci = ∂t dCi ∂t

(7.24)

Equation 7.23 yields:  1 − ε dqˆi*  ∂Ci ∂C i  1 + ε dC  ∂t + v ∂z = 0 i  

(7.25)

We can now consider the dependence of the solute mobile phase concentration on t and z. This will be a function of the form Ci = Ci(t, z) with differential: dCi =

∂C i ∂C dt + i dz ∂t ∂z

(7.26)

For given initial and boundary conditions, this function can be mapped on the z-t plane, which is referred to as the physical plane. By setting dCi = 0 in Equation 7.26 we obtain: ∂Ci dz ∂Ci + =0 ∂t dt ∂z

(7.27)

Finally, by comparing Equations 7.25 and 7.27, we recognize that dCi = 0 along the lines on the physical plane having a slope given by the following equation:

7.3 Local Equilibrium Dynamics

vci =

dz = dt

v 1+

1 − ε dqˆi* ε dCi

=

v 1+φ

dqˆi* dCi

(7.28)

where φ = (1 − ε)/ε is the ratio of stationary and mobile phases in the column. These lines are the characteristics and their slope, vci, is the characteristic velocity. For a linear isotherm, where dqˆi*/dCi = mi , Equation 7.28 coincides with Equation 7.21. In general, however, dqˆi*/dCi is not constant. Thus, the characteristic velocity, υci, varies with concentration. In the dimensionless system, the equivalent of Equation 7.28 is given by the following equation

υci =

dζ  dYi*  = dτ 1  dX i 

−1

(7.29)

where υci is a dimensionless characteristic velocity.

Example 7.1

Linear isocratic elution

The cross-hatched area represents the zone where the two components are mixed. Complete separation occurs where the A-characteristics starting at t = 0 cross the B-characteristics starting at t = tF. From geometry, it is possible to z determine that this occurs when tR,B − tR,A = (kB′ − k A′ ) = tF . Thus, we have v vtF Lmin = = 10.6 cm kB′ − k A′ Note that, in practice, a longer column will be needed to overcome dispersion effects that will cause the bands to broaden resulting in loss of resolution and dilution. If the column is 20 cm long, the outlet proﬁles, which, under ideal conditions will have the same width as the feed, tF, will be centered at tR,A + tF/2 = 2 5.1 + 2.5 = 27.6 min and tR,B + tF/2 = 34.6 + 2.5 37.1 min, for A and B respectively. Thus, the separation between peak centers will be tR,B − tR,A irrespective of tF. Two components A and B are to be separated in a 1.0-cm diameter chromatographic column with ε = 0.4 operated at a ﬂow rate 1.0 ml/min. The adsorption isotherm is linear with mA = 2.4 and mB = 2.9. Determine the minimum column length required to attain complete separation of a 5.0-ml injection, assuming local equilibrium with no axial dispersion. Solution – For these conditions, u = Q/(π × 0.52) = 1.27 cm/min, v = u/ε = 3.18 cm/min, φ = 1.5, tF = 5.0/1.0 = 5 min, k A′ = 1.5 × 2 = 3, and kB′ = 1.5 × 3 = 4.5. Accordingly, the chromatographic velocities are vcA = 3.18/(1 + 3) = 0.796 cm/min and vcA = 3.18/(1 + 4.5) = 0.579 cm/min for A and B, respectively. The characteristics are shown in Figure 7.1 along with inlet and outlet concentration proﬁles, in the top and bottom graphs, respectively.

207

Inlet Conc, Ci

7 Dynamics of Chromatography Columns

tF 0 A B

Axial position, z (cm)

5

10

15

20

tR,B

tR,A Outlet conc., Ci

208

tF

A B

0

10

20

tF

30

40

Time, t (min) Figure 7.1 Characteristics for isocratic ideal conditions and are centered at elution chromatography of two components tR,A + tF/2 and tR,B + tF/2. Complete with a linear isotherm according to Example separation occurs when z = 10.6 cm. 7.1. The outlet concentrations are shown for

In Example 7.1, because of the linear isotherm assumption, the characteristics for each component have constant slope, independent of adsorbate concentration. Thus, discontinuities occurring at the column boundary where the concentration changes from zero to the feed value and back to zero propagated unchanged through the column. However, when the isotherm is non-linear, the slope of the characteristics is concentration dependent. Consider, for example, a column subject to a step change in feed concentration for a single component system adsorbed with a favorable isotherm, i.e. an isotherm that is downwardly concave. In this case, the characteristics emanating from the time axis at z = 0 (representing

7.3 Local Equilibrium Dynamics

the column entrance) and those emanating from the z-axis at t = 0 (representing the initial state of the column) are able to cross. This results in a shock front that travels through the column at a velocity, vsh, accompanied by discontinuity in concentration. It can be shown that vsh is intermediate between the velocities of the crossing characteristics. The general expression for vsh is obtained from a material balance across the shock, which yields the following result: v

v sh = 1+φ

(ˆ qi* )′ − (ˆ qi* )′′ ′ (Ci ) − (Ci )′′

(7.30)

or, in the dimensionless system:

Example 7.2

Frontal loading of a column with a Langmuir isotherm

Its value depends on the concentration C. Two families of parallel characteristics are plotted in Figure 7.2. Those emanating from the z-axis, where C = 0, have a slope of: v c (0 ) =

v 25.5 = = 0.0705 cm min 1+ φqmK 1+ 1.5 × 120 × 2

while those emanating from the t-axis, where C = CF, have a slope of: vc (CF ) =

v 25.5 = = 0.621 cm s qmK 120 × 2 1+ φ 1 + 1 5 . (1+ KCF )2 (1+ 2 × 1)2

Since the upstream characteristic vc(CF) is higher than the downstream velocity vc(0), a shock front occurs. Upstream of the shock we have C′ = CF = 1 mg/ml and (q*)′ = qmKCF/(1 + KCF) = 120 × 2 × 1/(1 + 2 × 1) = 80 mg/ml. Downstream, we have C″ = (q*)″ = 0. Thus, from Equation 7.30, the shock velocity is given by the quantity: v sh =

v 25.5 = = 0.210 cm min q* (CF ) 80 1+ φ 1+ 1.5 CF 1

and the time at which the shock emerges from the column is tsh =

L 20 = = 95.0 min v sh 0.210

An initially clean 20-cm long column with 1.0-cm diameter and ε = 0.4 is loaded with a 1 mg/ml protein solution and eluted at 8 ml/min. Determine the breakthrough proﬁle under ideal chromatography conditions if the protein adsorption isotherm is given by: q=

qmKC 1+ KC

(7.32)

209

7 Dynamics of Chromatography Columns

with qm = 120 mg/ml and K = 2 ml/mg. Solution – For these conditions we have u = 10.2 cm/min, v = 25.5 cm/min, and φ = 1.5. Since the adsorption capacity is high and the solution dilute, we can assume qˆ ∼ q. Thus, the isotherm slope is: dqˆ * qmK = (7.33) dC (1+ KC )2 From Equation 7.28, the characteristic velocity is given by:

Inlet Conc, Cin

v c (C ) =

v qmK 1+ φ (1+ KC )2

(7.34)

CF

0

vc(C=0) vc(C=CF) vsh(0->CF)

Axial position, z (cm)

5

10

15

20

tSh CF

Cout

210

0

20

40

60

80

100

120

140

Time, t (min) Figure 7.2 Characteristics for frontal loading of a single component with a Langmuir isotherm according to Example 7.2. The outlet proﬁle shown for ideal conditions consists of a shock at t = 95.0 min.

7.3 Local Equilibrium Dynamics

υ sh =

( X i )′ − ( X i )′′

(7.31)

(Yi )′ − (Yi )′′

In these equations ′ and ″ refer to the conditions upstream and downstream of the shock front, respectively. Note that the proﬁle in Example 7.2 remains shock, independent of column length and velocity. In fact, even a less than perfect step input, for example with a more gradual rise from 0 to CF, will converge to a shock front after some time, since each progressively higher concentration supplied to the column will move

Example 7.3

Elution of a saturated column with a Langmuir isotherm

Inlet Conc, Cin

A 20-cm long column with 1.0-cm diameter, ε = 0.4, and initially saturated with a 1 mg/ml protein solution is eluted isocratically at 8 ml/min. The protein

C0

CF=0 0

Axial position, z (cm)

5

10 vc(C=0) vc(C) 15

vc(C=C0) 20 t(C=0)

t(C=C0)

Cout

C0

0

0

40

80

120

160

200

240

280

320

Time, t (min)

Figure 7.3 Characteristics for elution of a single component with a Langmuir isotherm according to Example 7.3.

211

212

7 Dynamics of Chromatography Columns

adsorption isotherm is the same as in Example 7.2. Determine the elution proﬁle under ideal chromatography conditions. Solution – Representative characteristics with slopes calculated from Equation 7.34 are shown in Figure 7.3. They can be seen to fan out from those with maximum slope vc(C0) = 0.621 cm/min, to those with minimum slope vc(C = 0) = 0.0705 cm/min. The elution proﬁle is thus a gradual wave starting at t = L/vc(C0) = 32.2 min, where C = C0, and ending at t = L/vc(0) = 284 min, where C = 0. Between these times, the concentration varies smoothly and can be calculated from Equation 7.34, which gives the following result: 1

1  φq K  2  C =  m  − 1 K  vt L − 1   

(7.35)

faster through column and catch up with the lower concentrations that have preceded it. In an experimental system, on the other hand, breakthrough will occur earlier than predicted under ideal conditions as a result of dispersion and kinetic resistances. The experimental breakthrough curve will thus be an S-shaped function, centered around tsh, but with a slope that will depend on the magnitude of band broadening effects. As seen from Equation 7.35, the elution proﬁle in Example 7.3 becomes increasingly broader as L increases and v decreases. Note that the accompanying tailing behavior is due exclusively to the interaction of ﬂow and adsorption equilibrium and is independent of any dispersive effects. The experimental elution curve is expected to follow a similar trend, with some additional broadening due to axial dispersion and kinetic resistances. In practice, of course, the tailing in the elution curve is reduced or eliminated by changing the mobile phase composition for elution. This is especially effective for proteins whose binding is frequently very sensitive to mobile phase composition. In this case, protein adsorption can be highly favorable for some conditions and essentially absent for others. This may not be possible, however, during the wash step of a chromatographic column, implemented to elute weakly and non-speciﬁcally bound contaminants, prior to desorbing the captured product. If these contaminants exhibit favorable binding, the wash step may require many column volumes because of the gradual wave behavior seen in this example.

Example 7.4

Isocratic elution with a preparative injection

The column in Example 7.2 is subjected to a 320-ml injection with CF = 1 mg/ ml and then eluted isocratically. Operating conditions and the isotherm are the same as in Example 7.2. Determine the elution proﬁles for different column lengths up to 20 cm.

7.3 Local Equilibrium Dynamics

Inlet Conc.

Solution – The duration of the feed injection is tF = 320/8 = 40 min. Representative characteristics are shown in Figure 7.4. The characteristics emanating from the z-axis (C = 0) and those emanating from the t-axis for t > tF have slope vc(0) = 0.0705 cm/min, while those emanating from the t-axis for 0 < t < tF, where C = CF, have slope vc(CF) = 0.621 cm/min. Since the upstream characteristics have higher vc, and the families of intersecting characteristics have initially con-

CF

0

vsh

Axial position, z (cm)

5

10

vc(0)

vc(C)

vc(0)

15

Cout

20

CF

L = 6 cm

Cout

0 CF

L = 12.7 cm

Cout

0 CF

L = 20 cm

Cmax 0 0

40

80

120

160

200

240

280

320

Time, t (min) Figure 7.4 Characteristics for the isocratic elution of a single component injection with a Langmuir isotherm according to Example 7.4.

213

214

7 Dynamics of Chromatography Columns

stant slopes, the solution for short times follows a shock front with constant velocity is: v sh =

v 25.5 = = 0.210 cm min q* (CF ) 80 1+ φ 1+ 1.5 CF 1

as in Example 7.2. Thus, for a short column, the elution proﬁle consists of a shock to CF followed by a plateau and then by a gradual wave from CF to 0, as in Example 7.3. For longer times, the shock path intersects the fan of characteristics emanating from t = tF. Thus, the shock velocity decreases from this point on and each point on the shock path must satisfy the following two equations:

( ) dz dt

() z t

= v sh (Cmax ) = sh

= vc (Cmax ) = sh

v q* (Cmax ) 1+ φ Cmax

v dq* 1+ φ dC

(7.36)

( )

(7.37)

Cmax

Simultaneous solution of these equations yields the following result [3, 4]: 1

 φqmK 1  2  C =  − 1 for t > tsh  K  v (t − tF ) L − 1   

(7.38)

where: 2

vtF L   L  tsh = tF + 1+ φqmK  1−    v φqm CF  

Cmax

1 = K

vtF L φqm CF vtF L 1− φqm CF

(7.39)

(7.40)

which is plotted in Figure 7.4 at L = 6, 12.7, and 20 cm. As in Example 7.3, the peak tailing is due exclusively to the non-linear nature of the isotherm. This effect is referred to as concentration overload, since it is associated with adsorbate concentrations sufﬁciently high to cause adsorption outside the linear Henry’s law limit. The experimental peak is expected to follow the general shape suggested by the local equilibrium model. The initial shock will be smoothed out by dispersive effects, but will remain fairly sharp. The tail will be somewhat broader than predicted. However, even with relatively low column efﬁciency, the experimental peak shapes will typically be quite similar to those shown in Figure 7.4.

7.3 Local Equilibrium Dynamics

The procedure illustrated in Examples 7.2 to 7.4 is general with regard to the nature of the adsorption isotherm. However, the results are different for each case. For example, if the isotherm is unfavorable, that is, dq*/dC > 0 over the range of compositions of interest, the breakthrough curve corresponding to Example 7.2 would be a gradual wave, the elution curve corresponding to Example 7.3 would be a shock, and the overloaded chromatographic peaks in Example 7.4 will exhibit fronting instead of tailing. However, isotherms that contain inﬂection points over the range of concentrations that are of interest require special consideration. An example is given below.

Example 7.5 isotherm.

Frontal loading of a column with a Langmuir–Freundlich

Repeat the calculations in Example 7.3 for the case where isotherm is described by the Langmuir–Freudlich model given by the equation (cf. Equation 5.12): qm (KC ) 1+ (KC )b b

q* =

(7.41)

with qm = 40, K = 2, and b = 2. Solution – The derivative of the isotherm is given by the following equation: dq* qmbK bC b−1 = dC 1+ (KC )b 2

(7.42)

A plot of the isotherm and its derivative are shown in Figure 7.5. Since the isotherm exhibits an inﬂection point, the characteristic velocity ﬁrst decreases 40

CF= 1

dq*dC

0.8

CF,q*(CF)

Cout

q, dq*dC

30

20

0.6 0.4

Cs

10 0.2

Cs,q*(Cs)

0 0

0.5

0 1

C

1.5

2

0

10

20

30

40

50

Time, t (min)

Figure 7.5 Langmuir–Freundlich isotherm and its derivative (left) and breakthrough curve (right) according to Example 7.5. The tangent to the isotherm that passes through point CF, q*(CF), gives the outlet concentration Cs.

215

216

7 Dynamics of Chromatography Columns

and then increases as C increases from 0 to the feed value of 1 mg/ml. As a result, the breakthrough curve is comprised of a gradual wave up to an intermediate concentration Cs, followed by a shock from Cs to CF. The gradual wave and the shock are described by Equations 7.28 and 7.30, respectively. In our case, these equations yield: v c (C ) =

v sh (C s → CF ) =

v dq* 1+ φ dC

= 1+ φ

v qmbK bC b−1 2

b 1+ (KC ) 

(7.43)

v v = q* (CF ) − q* (C s ) ( ) ( * − q C q KC s )b 1+ (KC s )b  F m 1+ φ 1 + φ CF − C s CF − C s (7.44)

where q*(CF) = 26.7 is the adsorbed concentration in equilibrium with the feed. Cs is determined by recognizing that the characteristic and shock velocities must be the same at C = Cs. Accordingly, we have:

( ) dq* dC

Cs

=

q* (CF ) − q* (C s ) CF − C s

(7.45)

or qmbK bC sb−1 b 2

1+ (KC s ) 

q* (CF ) − qm (KC s ) 1+ (KC s )  CF − C s b

=

b

(7.46)

For the parameters of Example 7.5, Equation 7.46 yields Cs = 0.225 and q*(Cs) = 3.67. Thus, the shock occurs at ts = L/vsh = 20/0.560 = 35.7 min. A plot of the corresponding outlet concentration proﬁle is given in Figure 7.5. The graphical interpretation of Equation 7.45 is also shown in this ﬁgure. Cs is determined by drawing the tangent to the adsorption isotherm that passes through the point CF,q*(CF). This is sometimes known as the ‘Golden rule’ [5, 6]. Note that for the case analogous to Example 7.3, where a saturated column is eluted isocratically, the outlet proﬁle is a shock from the initial concentration C0 to a new intermediate concentration Cs followed by a gradual wave to 0. The new intermediate concentration for this case is found by drawing the tangent to the adsorption isotherm that passes through the point C = 0, q = 0. It should be noted that isotherms of the type described by Equation 7.41 with b > 1 are often found in biological systems, for example in the case of cooperative binding in allosteric enzymes (e.g. see the hemoglobin oxygen saturation curve). They also sometimes occur in heterogeneous adsorption systems. The consequences for the column outlet proﬁle are breakthrough within 1 CV, which is completely independent of any dispersive factor. These factors will, of course, broaden the gradual wave further and smooth the shock transition somewhat, leaving, however, a general pattern similar to that predicted under ideal conditions.

7.4 Multi-component Systems

7.4 Multi-component Systems

Multi-component systems under local equilibrium conditions involving M competitively adsorbed components require the solution of a set of M material balances (7.5 or 7.18) coupled with M equilibrium relationships. The method of characteristics described in Section 7.2 is easily extended to two-component systems based on the theory of coherence and can be treated either graphically or analytically. Higher order systems involving more than two absorbable species are usually best handled using matrix methods [7–10]. In the special case where the multi-component adsorption equilibrium is represented by a constant separation factor isotherm, the so-called h-transformation of Helfferich and Klein provides a useful framework to obtain local equilibrium solutions for an M-component system, which are not subject to the condition of coherence [11, 12]. General treatments of multi-component local equilibrium adsorption dynamics can be found in several standard adsorption texts and handbooks [3, 11, 13, 14]. Only a limited treatment is provided here sufﬁcient to address frontal analysis and displacement development. We also consider the interactions of buffer components with columns containing weak acid or weak base groups, since the corresponding dynamic behavior is especially relevant in many practical protein separation processes. The theory of coherence considers only stable waves. For each stable wave in a column the characteristic velocity is the same for all components. Thus, for a simple or gradual wave, the following equalities must be satisﬁed: v c1 = v c 2 = = v c M

(7.47)

or, equivalently: dqˆ1* dqˆ2* dqˆ* = = = M dC1 dC2 dCM

(7.48)

Similarly, for shock waves we have: v sh1 = v sh2 = = v shM

(7.49)

∆qˆ1* ∆qˆ2* ∆qˆ* = = = M ∆C1 ∆C2 ∆CM

(7.50)

or

where ∆ represents the difference between concentrations upstream and downstream of the shock front. We initially limit our attention to a two-component system. There is no restriction with regard to the isotherm, although we will require that the isotherm makes sense from a physical viewpoint. Thus, for example, for a mixture of A and B, we will assume that the isotherm correctly represents the situation where increasing the solution concentration of A while keeping B constant will result in an increase in the amount of A adsorbed, and vice versa. For simplicity, we also assume that

217

218

7 Dynamics of Chromatography Columns

the adsorption capacity is high; thus, we have qˆ i ∼ qi. Finally, we will restrict the analysis to constant initial and ﬁnal states. Accordingly, we will assume that the initial composition of the column is uniform and that the feed composition remains constant following a step change. Such conditions represent, for example, frontal analysis where an initially clean column is fed with a mixture of absorbable components. For a binary mixture of A and B, the equilibrium relationships are given by the following equations: q *A = q A* (C A, CB )

(7.51)

qB* = qB* (C A, CB )

(7.52)

We consider ﬁrst simple waves for which Equation 7.48 gives the result: ∂q *A ∂q A* dCB ∂qB* dC A ∂qB* + = + ∂C A ∂CB dC A ∂C A dCB ∂CB

(7.53)

which can also be written as: ∂q *A 2  ∂q A* ∂qB*  ∂q * λ + λ− B =0 −  ∂C B ∂C A  ∂C A ∂ C B 

(7.54)

where λ = dCB/dCA. The partial derivatives are evaluated from Equations 7.51 and 7.52. The solution is obtained ﬁrst on the hodograph, a plane where CB is plotted versus CA. Initial and ﬁnal states are represented by points on the hodograph that are connected by coherent paths. Any intermediate states that are established in the column as a result of any particular input change can be determined by following these paths or trajectories. Coherent paths are determined from Equation 7.54 starting with any arbitrary point with coordinates CA, CB and determining the slopes λ of the coherent paths passing through that point by solving this equation. This equation is quadratic in λ, thus for each point there are two solutions and hence two coherent paths. In the remainder, we will assume that B is more strongly adsorbed than A. In that case, ∂qB*/∂CB > ∂q*A /∂C A . Since both ∂q *A /∂CB and ∂qB*/∂C A are negative Equation 7.54 has two real solutions, one positive and one negative, λ+ and λ−, corresponding to the coherent paths on the hodograph with positive and negative slopes, respectively. The corresponding characteristic velocities are obtained from the following equations: vc + =

v v =  ∂q*A ∂qA*   ∂qB* 1 ∂qB*  + + 1+ φ λ+  1 + φ    ∂ C A ∂ CB   ∂ C A λ + ∂ CB 

vc − =

v v =  ∂qA* ∂qA*   ∂qB* 1 ∂qB*  + + 1+φ λ−  1 + φ    ∂ C A ∂ CB   ∂ C A λ − ∂ CB 

(7.55)

(7.56)

7.4 Multi-component Systems

The proper path from the initial to the ﬁnal state is selected on physical grounds by considering that the characteristic velocity must decrease as we move from the initial state to the ﬁnal state. If a path of simple waves that satisﬁes this requirement does not exist, shocks are formed. In that case, by analogy we deﬁne coherent shock paths based on Equation 7.50. For our two-component system, this equation yields the following result: q *A (C A′ , CB′ ) − q A* (C A′′ , CB′′ ) qB* (C A′ , CB′ ) − qB* (C A′′ , CB′′′ ) = C A′ − C A′′ CB′ − CB′′

(7.57)

As for simple waves, starting from an arbitrary point C A′ , CB′ the direction of the coherent shock paths Λ± = dCB/dCA passing through that point are found by solving Equation 7.57 for CB′′ for a given increment in C A′′ . Shock velocities can then be determined from the following equation: v sh =

v q* (C ′ , C ′ ) − q*A (C A′′, CB′′) 1+φ A A B C A′ − C A′′

=

v

(7.58)

q* (C ′ , C ′ ) − qB* (C A′′ , CB′′) 1+φ B A B CB′ − CB′′

using the multi-component isotherm expressions to calculate the q* values. Once the proper, physically consistent coherent paths are determined, the last step is to reconstruct the concentration proﬁles by assigning time and distance coordinates to each CA, CB pair along the proper path.

Example 7.6 Frontal loading of a binary mixture with a multi-component Langmuir isotherm Determine the frontal analysis behavior of a binary mixture with C FA = CBF = 1mg ml applied to an initially-clean 20-cm long column with v = 25.5 cm/min and φ = 1.5. The adsorption isotherm is given by the following equation: qi* =

qmK iC i 1 + K AC A + K B C B

(7.59)

with qm = 120 mg/ml, KA = 1 ml/mg, and KB = 2 ml/mg. Solution – The four partial derivatives of Equation 7.59 are obtained form the following equations: qmK i (1+ K jC j ) ∂qi* = ∂C i (1+ K AC A + K BCB )2

(7.60)

qmK iK jC i ∂qi* =− ∂C j (1+ K AC A + K BCB )2

(7.61)

with i and j equal to A and B. Substituting these derivatives in Equation 7.54 and simplifying we obtain the following result: −K AK BC Aλ 2 + [K A (1+ K BCB ) − K B (1+ K AC A )] λ + K AK BCB = 0

(7.62)

219

7 Dynamics of Chromatography Columns

When CB = 0, the trivial root of Equation 7.54 is λ = 0. Conversely, when CA = 0, the trivial root is λ = ∞. Thus, the axes of the hodograph are coherent paths. All other cases involving ﬁnite values of CA and CB result in two real roots, one positive and one negative. It can be shown (see e.g. [13]) that for this isotherm type the corresponding coherent paths are straight lines. Further, these lines coincide with the paths followed by coherent shocks passing through the same point. This is a general property of the multi-component Langmuir isotherm, but is not true, in general for other isotherm forms. The hodograph and the corresponding derivatives and root values are shown in Figure 7.6a and Table 7.1, respectively. The coherent paths passing through states I and III intersect at points II and IV, which are intermediate states. However, neither path I → II → III nor I → IV → III are physically possible simple waves since, for one vcII+ > vcI + and vcIII− > vcII− while for the other vcIV− > vcI − and vcIII+ > vcIV+ . Thus, shocks are formed between the intermediate state and both initial and ﬁnal states. The corresponding shock velocities can be calculated using Equation 7.58 with the q values shown in Table 7.1 and are given by the following values: v sh (I, II ) = 0.388 v sh (I, IV ) = 0.101 v sh (II, III ) = 0.208 (a) 1.5

(b)

2 A B

Ci

1.5

III

II III

1 0.5

1

I

0 0

20

40

CB

tII

60

80

100

tIII

Time (min) (c)

0.5

1.2 1

III

A B

0.8

IV Ci

220

0.6 0.4

0

IV

0.2

II

I

I

0 0

0

0.5

1

1.5

2

tIII+

tIV+ 100

tIV-

200

tI-

Time (min)

CA Figure 7.6 Coherent paths and outlet concentration proﬁles for Example 7.6. In (b) a clean column is loaded with a mixture containing 1 mg/ml each of A and B. In (c) a clean column initially saturated with a mixture containing 1 mg/ml each of A and B is eluted isocratically.

300

7.4 Multi-component Systems

221

v sh (III, IV ) = 0.554 all in cm/min. It is apparent that the physically correct path is I → II → III, which yields shock velocities that decrease from the initial state I to the ﬁnal state III. The corresponding concentration proﬁles at the column outlet are shown in Figure 7.6b. The more weakly bound component A is concentrated ahead of the mixture front. tII = L/vsh(I, II) and tIII = L/vsh(II, III) are 51.6 and 71.4 min, respectively. In practice the concentration of roll up of the more weakly bound component can be advantageous, but can also cause problems for example during capture, when weakly-bound contaminants are concentrated above their solubility in the mobile phase. It should be noted that if states I and III in the example above are reversed, that is, the column is initially saturated with a mixture containing 1 mg/ml each of A and B and is eluted isocratically, the simple wave path III → IV → I becomes physically possible since vcIV− > vcI − and vcIII+ > vcIV+ . Thus intermediate state IV is established in the column and is separated from the ﬁnal initial states by gradual waves. The ensuing concentration proﬁles at the column outlet are shown in Figure 7.6c. The times tIII + = L vc + (III) , tIV + = L vc + (IV ) , tIV − = L vc − (IV ) , and tIV − = L vc − (I) are 13.6, 98.9, 137 and 283 min, respectively.

Slopes of coherent paths and characteristic velocities for Example 7.6. C and q values are expressed in mg/ml and vc values in cm/min.

Table 7.1

State

CA

CB

λ+

λ−

∂qA/∂CA

∂qA/∂CB

v c+

v c-

qA

qB

I (initial) III (ﬁnal) II (intermediate) IV (intermediate)

0 1 1.78 0

0 1 0 0.219

0 0.781 0 0.781

∞ −1.28 −1.28 ∞

120 22.5 15.5 83.4

0 −15.0 −55.3 0

0.141 1.48 1.05 0.202

0.071 0.401 0.195 0.146

0 30 76.8 0

0 60 0 36.6

A useful application of the method of characteristics is the prediction of pH transitions that occur in columns packed with stationary phases that contain either weak acid or weak base functional groups. The functional groups found in strong cation exchangers such as sulfonic acid (S) or sulfopropyl acid (SP) can be considered to be completely dissociated and thus negatively charged at most pH values of practical interest in protein chromatography applications. The same is true for strong anion exchangers whose quaternary ammonium ion (Q) functional groups are completely protonated and thus positively charged at practical pHs. As discussed in Chapter 3, most applications in protein ion exchange chromatography use buffered mobile phases with buffers normally chosen so that the buffering species are either neutral or have the same charge as the stationary phase. Thus acidic buffering species like acetate, phosphate, or MES are used with cation exchangers while basic buffering species such as TRIS or ethanolamine are used

222

7 Dynamics of Chromatography Columns

with anion exchangers. In practice, the counter-ions that accompany the buffering species, for example Na+ with acidic buffers and Cl− with basic buffers, are present in large excess compared to the concentrations of H+ or OH− normally encountered at the moderate pHs used in protein chromatography. As a result, the functional groups of resins are almost always completely in the form of the buffer counter-ion even when the buffer concentration is relatively low. Under these conditions, changes in mobile phase composition at the column entrance propagate rapidly through the column and exit unretained in less than 1 CV. This is not the case, however, for resins that contain either weak acid or weak base groups either alone or in combination with strong acid or base groups. The pK values of such weak groups are often in the range of 4–7 for weak cation exchangers and 6–9 for weak anion exchangers. Thus, their charge varies with pH over the ranges often used for protein chromatography. Since dissociation of weak acid or base groups is accompanied by the release of protons and by the uptake of equivalent quantities counter-ions, even with fairly well buffered mobile phases signiﬁcant excursions in pH may occur as a result of changes in salt concentration. The ensuing pH transitions are retained and propagate through the columns in a manner that depends on the interactions of buffer and stationary phase components. The magnitude of the pH transitions can be quite large, may damage pH-sensitive proteins, and may require very long column equilibration times [15]. In other cases, such transitions may be exploited to intentionally generate pH gradients that result in effective separations (see e.g. [16, 17]). In practice, because of the small size of buffering species and their counter-ions, these waves are determined primarily by equilibrium effects. Thus, the local equilibrium assumption is particularly appropriate. Its application is demonstrated in Example 7.7 for a resin containing weak acid groups.

Example 7.7 Local equilibrium prediction of pH transients in weak acid ion exchange columns subject to changes in salt concentration A 10-cm long column with ε = 0.31, packed a cation exchange resin containing weak acid groups and operated at u = 5.09 cm/min is subject to 0 to 0.5 M NaCl steps up and down. The mobile phase is buffered with 0.02 M sodium acetate adjusted to pH 5.5 with acetic acid. The resin has εp = 0.76 and contains 1.03 mol/l of weak acid groups, based on the solid volume. Determine the pH changes and the time needed to attain equilibrium following each salt step. Solution – A complete analysis of this problem can be found in [18] and only a simpliﬁed version is presented here. Useful references for these problems are [19–21]. The solution requires a description of both mobile phase and resinphase dissociations, both of which are assumed to be at equilibrium. For the acetate buffer in the mobile phase, the solution equilibrium is described by

7.4 Multi-component Systems

CH3COOH CH3COO − + H+

(7.63)

with Ka =

C CH3COO− CH+ C CH3COOH

(7.64)

The acetate ion concentration is easily obtained from Equation 7.64 and is given by C CH3COO− =

K aC A K a + C H+

(7.65)

where C A = CCH3COOH + CCH3COO− is the total concentration of acetate species in solution. The concentrations of the charged species in solution are bound by the electro-neutrality condition, which is expressed by the following equation: CNa+ + CH+ = CCH3COO − + COH− + CCl −

(7.66)

K aC A K + w + C Cl − K a + C H+ C H+

(7.67)

or CNa+ + CH+ =

where K w = CH+ COH− . For typical conditions where the difference CNa+ − C Cl − is much greater than either CH+ or Equation 7.67 yields the following result:  CA  C H+ = K a  −1  CNa+ − CCl − 

(7.68)

The dissociation of the resin’s weak acid groups is described by: RH R− + H+

(7.69)

qR− qH+ qRH

(7.70)

where K=

is an apparent dissociation constant and the qis represent resin-phase concentrations. As pointed out by Helfferich [19], the pH in the resin phase is related to the solution composition by: qH+ =

C H+ q + CNa+ Na

(7.71)

Combining Equations 7.70 and 7.71 we obtain K=

qR− qNa+ CH+

(qR − qR− ) CNa+

(7.72)

where qR = qRH + qR− is the total concentration of weak acid groups in the resin. In practice, qNa+ ∼ qR− . Thus, Equation 7.72 can be rewritten as follows:

223

224

7 Dynamics of Chromatography Columns

K=

2 qNa + C H+ (qR − qNa+ ) CNa+

(7.73)

The ﬁnal result describing the sodium uptake isotherm, obtained by combining Equations 7.71 and 7.73, is given by the following equation: 2 KC +  1  KC +  KC +  (7.74) qNa+ =  − Na +  Na  + 4qR Na   C H+  2  CH+ CH+  The dynamic behavior of the column is described by material balances of the form of Equation 7.23 written for total acetate (CA), chloride ion ( C Cl − ), and sodium ion (CNa+ ) in conjunction with Equations 7.68 and 7.74. With the assumption that there is no adsorption of chloride or acetate species, the corresponding characteristics and shock velocities are as follows:

vc A = vcCl =

1 1+ φεP

(7.75)

vcNa =

v dq +   1+ φ εP + (1− εP ) Na  dC  Na+ 

v shNa =

v q′ + − qNa ′′ +   1+ φ εP + (1− εP ) Na CNa ′ + − CNa ′′ +  

(7.76)

(7.77)

A diagram of the hodograph showing CCl − versus CNa + and of the corresponding states in the column is shown in Figure 7.7. Points I and III represent the NaClfree starting acetate buffer and the acetate buffer containing 0.5 M NaCl, respec-

II

CClIII

CClI

I

CNaI

III

III

Feed State CNaIII, CClIII, CBIII

Slow Wave (Na+ exchange)

II

Intermediate State CNaII, CClIII, CBIII

Fast Wave (no Na+ exchange)

I

Initial State CNaI, CClI, CBI

IV

CNaIII

Figure 7.7 Hodograph and corresponding states during salt steps for Example 7.7. Points I and III represent the NaCl-free and high salt buffers, respectively. Point II and

IV represent intermediate states during positive and negative salt steps, respectively. Reproduced from [18].

7.4 Multi-component Systems

tively. CA is assumed to be the same as is the pH for both buffers. For each point there are two coherent paths. Since Cl− is not adsorbed, one of the two paths has zero slope. The other must satisfy the following equality: vcCl = vcNa

(7.78)

As a result, the coherent path from I to III passes through an intermediate state represented by point II in Figure 7.7. Equation 7.78 yields I II qNa + = qNa+

(7.79)

Thus, this path corresponds to a fast wave with no exchange of sodium between the stationary phase and the mobile phase. Considering Equation 7.74, for this wave we also have I

 KCNa+   KCNa+   C  =  C  H+ H+

II

(7.80)

which shows that, as long as K is constant, an increase in sodium from state I to state II must be accompanied by an increased concentration of H+. This increase temporarily lowers the pH of the solution. Based on Equation 7.74, it is clear that state II emerges from the column in ε(1 + φεp) column volumes. Then, the rest of the solution follows the path from II to III, along which both CCl − and CA are constant. This path consists of either a plateau at II followed by a simple wave with characteristic velocity vcNa =

v    ∂q +  1+ φ ε p + (1− ε p )  Na    ∂CNa+  C II ,C II   Cl A 

(7.81)

if vcNa decreases from II to III or alternatively, if vcNa increases from II to III, the path consists of a plateau at II followed by a shock from II to III with shock velocity v shNa =

v   III  qII + − qNa +   1+ φ ε p + (1− ε p )  Na II III   CNa+ − CNa+  C II ,CII   A   Cl −

(7.82)

CNa+ and pH proﬁles calculated using these equations and assuming that pKa = 4.76 for acetic and that the pK of the weak acid groups of the resin is 5.2 are shown in Figure 7.8, where CV = ut/L. For the salt step up (Figure 7.8a), the intermediate state that satisﬁes Equation II 7.78 can be derived from Equations 7.68 and 7.80 and is CNa + = 0.5042 M . The + II Na adsorption isotherm is plotted from this value of CNa+ to the feed value + III CNa + = 0.52 M in Figure 7.8a and exhibits an inﬂection point. Thus, the Na concentration proﬁle consists of a shock from 0.5042 to 0.5097 M, followed by a gradual wave to 0.52 M. As can be seen in this ﬁgure, the pH drops from the

225

7 Dynamics of Chromatography Columns

initial and ﬁnal value of 5.5 to pHII = 4.098 and returns to the equilibrium value only after 7.05 CV. The proﬁles for the salt down step (Figure 7.8b) are completely different in shape. In this case, the intermediate Na+ concentration fol+ IV lowing passage of the fast wave is CNa + = 0.0234 M . The Na adsorption isotherm I from this value to the feed value CNa + = 0.02 M is shown Figure 7.8b and is upwardly concave. Since the Na+ concentration increases during this transition, the resulting wave is a shock that emerges from the column in 4.12 CV. The ensuing pH proﬁle rises from 5.5 to 6.85 before returning to the feed value. (a) 0.7

(b) 0.7 III

0.5 Shock

0.4 0.3 0.2

0.5

Shock

0.4 0.3 0.2

II

0.1 0.5

IV

0.6

qNa (M)

qNa (M)

0.6

I

0.1 0.505

0.51

0.515

0.52

0.525

0.0195

0.021

6.0

0.6

7.0

0.5

pH CNa

II

0 2

4

6

8

10

CNa (M)

0.3

6.5 6.0

0.2

4.5

5.5 0.1

4.0 12

CV +

0.4

pH

0.2

pH

5.0

0.3

pH CNa

IV

5.5 0.4

0

0.024

0.6

0.5

0.1

0.0225

CNa (M)

CNa (M)

CNa (M)

226

Figure 7.8 Na adsorption isotherms and outlet proﬁles during salt steps for Example 7.7. (a) and (b) correspond to 0–0.5 M positive and negative NaCl steps with

0 0

2

4

6

8

10

5.0 12

CV

0.02 M sodium acetate at pH 5.5. Model predictions assume pK = 5.2. The thin lines in the top graphs connect the shock transitions.

Comparison of model predictions from Example 7.7 for an actual experimental system and various buffers are shown in Figure 7.9 taken from [18] using UNOsphere S to create conditions similar to those described in Example 7.7. Actual predictions require that activity coefﬁcients be taken into account. Moreover, a proper description of the resin’s dissociation equilibrium requires multiple dissociation constants. Details are in the original reference. As seen in this ﬁgure, the dynamic behavior varies dramatically from buffer to buffer and is predicted with remarkable accuracy by the local equilibrium model and requires no adjustment of the parameters. For a constant initial and ﬁnal Na+ concentration, the shortest pH transitions are obtained with the MES buffer. This buffer is zwitterionic and has the greatest buffering capacity over this range of pH and Na+ concentrations. The longest transitions are associated with the phosphate buffer,

7.5 Displacement Development 7

0.7 0.6

5

0.2

0

10

20

30

40

+

5.5 0.3 5

0.2

4.5

0.1 0

[Na ] (M)

+

[Na ] (M)

0.3

6

0.4

4.5

0.1

4 50

0

0

10

20

CV

30

7

7

0.7

(c) 0.6

0.6

5

0.2

4.5

0.1 20

30

40

4 50

CV Figure 7.9 Experimental and predicted pH and sodium concentration proﬁles for positive and negative 0–0.5 M NaCl steps with UNOsphere S using buffers containing 0.02 M

5.5

+

+

0.3

6

0.4

pH

5.5

pH

0.4

6.5

0.5

6

[Na ] (M)

0.5

[Na ] (M)

(d) 6.5

10

4 50

40

CV

0.7

0 0

pH

pH

5.5

6.5

0.5

6

0.4

(b)

0.6

6.5

0.5

7

0.7

(a)

0.3 5

0.2

4.5

0.1 0

0

10

20

30

40 50

4 60 70 80

CV Na+ at pH 5.5. (a) Acetate, (b) citrate, (c) MES, and (d) phosphate. (䊊) Experimental Na+, (ⵧ) experimental pH, (- - - -) model Na+, (—) model pH. Reproduced from [18].

whose pKa is farthest from the 5.5 operating pH. Interestingly, the wave shapes with this buffer are opposite to those predicted and seen experimentally for acetate, namely a shock for the up step and a gradual wave for the down step.

7.5 Displacement Development

The general operating principles of displacement chromatography were illustrated in Chapter 2. In a typical process, the feed mixture is ﬁrst loaded onto a column under conditions were the feed components are strongly and competitively adsorbed. Following introduction of the feed, the column is supplied with a mobile phase containing a displacer component selected to have a higher afﬁnity for the stationary phase than that of any of the feed components. Competitive binding drives the feed components downstream of the displacer front where they compete for adsorptions sites. Under ideal conditions, when the multi-component adsorp-

227

228

7 Dynamics of Chromatography Columns

tion isotherm has a regular behavior (e.g. as described by the multi-component Langmuir, Equation 5.26), the feed components are eventually separated into adjacent individual bands that travel through the column at a constant, uniform speed. The ﬁnal steady state proﬁle is referred to as the isotachic train since all bands, including the displacer, travel through the column at the same rate. Following elution of the displaced components, the column is regenerated to remove the strongly bound displacer and re-equilibrated with the initial mobile phase. Horvath [22] provides a useful general reference including a historical perspective for liquid chromatography applications. Prediction of the isotachic train in displacement chromatography is generally straightforward when pure component bands are formed and is discussed below. This is not always the case, however, when the adsorption equilibrium selectivity is composition dependent. Such cases can lead to mixed bands corresponding to the formation of adsorption azeotropes in the column. Consider, for example, a hypothetical case where in a binary mixture the pure component isotherms cross. One component is favored at low concentrations, while the other is favored at higher concentrations. Since displacement development usually starts with low feed concentrations and ends with concentrated bands in the isotachic train, azeotropic compositions, where the selectivity is 1, can be formed in the column. In practice this can occur, for example, in the separation of molecules that differ considerably in size. At low adsorbent loadings, the larger molecules will often be more strongly retained. However, because of size exclusion effects, the smaller molecule can actually have greater capacity and, thus, a higher apparent selectivity at high adsorbent loadings. General theoretical considerations for liquid chromatography in this regard can be found in [23]. An experimental example of adsorption azeotropes is given in [24] for the separation of α-aminobutyric acid–isoleucine mixtures by displacement development with cation exchange resins. Note that, in general, prediction of adsorption azeotropes in displacement development requires knowledge of the full multi-component isotherm. In fact, even systems whose single component isotherms do not cross can exhibit azeotropic behavior during displacement when the multi-component behavior is highly non-ideal. Unlike constant pressure binary vapor–liquid equilibrium azeotropes, where there the azeotropic composition is unique, azeotropic bands of different composition may be attained in a binary displacement separation by varying the displacer concentration. Thus, in a way, the displacer concentration plays a role analogous to that of pressure in distillation allowing azeotropic mixtures to be resolved by shifting the position of the azeotrope. 7.5.1 Prediction of the Isotachic Train

The general theory of displacement chromatography under local equilibrium conditions is found in the classical works of Helfferich and Klein [11] and Rhee and Amundson [25]. Prediction of the isotachic train for these conditions is derived from the limiting, steady-state solution provided by these theories. We consider

7.5 Displacement Development

ﬁrst an adsorption system with favorable single component isotherms leading to pure component bands. In a system with M adsorbable components where 1 is the displacer and M is the feed component with lowest afﬁnity for the stationary phase, there will be M waves. The ﬁrst is the displacer front and Mth is the front between component M and the carrier ﬂuid initially present in the column. For these conditions, all of the waves in the column are shocks of equal velocity. Thus: v sh ,1 = v sh ,2 = = v sh ,M

(7.83)

or from Equation 7.30: qˆ1* qˆ2* qˆ* = I = = MI D C1 C 2 CM

(7.84)

where C1D is the displacer concentration, C iI is the concentration of feed component i in the isotachic train, and qˆi* is the corresponding equilibrium adsorbed concentration. The graphical interpretation of Equation 7.84 is shown in Figure 7.10. The operating line which passes through the origin and has a slope of qˆ1*/C1D , intersects the pure component isotherms at C2I and C3I , which are the product concentrations in the isotachic train. The latter are independent of the original concentrations at which the feed mixture was loaded onto the column. Thus, for a given system the products can be more concentrated or dilute relative to the feed value, dependent on the magnitude of the displacer concentration. For favorable isotherm shapes, the condition necessary to attain displacement of component i is: qˆ1*  dqˆi*  < C1D  dCi  C i →0

(7.85)

120

Stationary phase concentration, qi*

Operating line, slope q1*/C1D 100

80 Displacer, comp. 1 60 2 40

3

20 C3I

C2I

C1D

0 0

0.5

1

1.5

2

2.5

Mobile phase concentration, Ci

Figure 7.10 Schematic showing the intersection of the operating line with the pure component isotherms for a two-component separation. Adapted from [3].

229

7 Dynamics of Chromatography Columns

Example 7.8

Prediction of isotachic train with Langmuir isotherms

Determine the isotachic train for the separation of two components by displacement development assuming that equilibrium is described by the multi-component Langmuir isotherm (Equation 5.26) with qm = 120, K1 = 2, K2 = 1, and K3 = 0.667 if the feed concentrations are C 2F = C 3F = 0.5 and the displacer concentration is C1D = 2 . The volume of feed is equal to 20 column volumes and ε = 0.4. Solution – From Equation 7.84 we obtain the following equalities: qmK 1 q K q K = m 2I = m 3I D 1+ K 1C1 1+ K 2C 2 1+ K 3C 3 In turn, these equalities yield the following result for each displaced feed component: C iI = C1D −

1  Ki  1− Ki  K1 

(7.86)

The values calculated from Equation 7.86 for this problem are C 2I = 1.5 and C 3I = 1.0 . The displacer front elutes at t = L/vsh,1 or, in terms of column volumes at CV = tu L = ε + (1− ε ) q1*/C1D = 0.4 + (1− 0.4) × 96 2 = 29.2 . The product concentration proﬁles are obtained from material balances. In terms of column CF volumes, the duration of elution of each component band is CVi = iI × CVF, Ci which gives 10 and 6.67 column volumes for components 2 and 3, respectively. Predicted proﬁles are illustrated in Figure 7.11. 2.5 C1D

2

1

C2I

1.5 Ci

230

2

C3I

1

3

0.5

0 0

10

20

30

40

CV

Figure 7.11 Isotachic train for displacement development separation of a two-component mixture according to Example 7.8.

7.5 Displacement Development

Components not meeting this requirement are not displaced and elute ahead of the isotachic train. In Example 7.8 we have considered a situation where the adsorption isotherm parameters are constant. This occurs when temperature and mobile phase composition are uniform. Protein binding on ion exchangers, however, is generally accompanied by a stoichiometric exchange of counter-ions, which in turn, affects the protein adsorption equilibrium. In many practical applications the protein concentration is sufﬁciently low that the protein is a small fraction of the total ion concentration in solution. For these conditions, changes in counter-ion concentration caused by protein binding are negligible. In displacement development, however, the protein and displacer concentrations can be high enough to cause signiﬁcant changes in counter-ion concentration which, in turn, can affect the separation. A broad range of displacers can be used for protein ion exchange including both charged polymers (e.g. [26]) and small molecules [27, 28]. When used in sufﬁciently high concentration, these displacers will also produce changes in counter-ion concentration, which propagate through the column and affect the development of the separation. A model to predict these effects has been developed by Cramer and co-workers [29–31] based on the steric mass action (SMA) model discussed in Chapter 5. The following development is equivalent to that of Cramer and co-workers and is derived by taking into account the co-ion behavior in the column in a manner analogous to Example 7.7. Considering as an illustrative example, a cation exchanger with sodium as the counter-ion, protein binding can be described by the following equation (see Equation 5.22 and associated discussion): qi  q − ( z i + σ i ) qi  = K e ,i  0  Ci CNa +  

zi

(7.87)

where q0 is the total concentration of functional groups in the ion exchanger, Ke,i the equilibrium constant for the protein–Na+ exchange, and zi and σi the protein effective charge and steric hindrance factor, respectively. In this analysis, Equation 7.87 is assumed to apply to each protein (i = 2, 3, ··· M) in the feed as well as to the displacer (i = 1). Electro-neutrality requires that the following equation be satisﬁed: M

CNa + + ∑ ziCi = C 0

(7.88)

i =1

where C0 is the total co-ion equivalent concentration. Since the co-ions are negatively charged, they travel unretained through the column. Thus, any change in C0 at the column entrance leaves the column in the void volume ε tVc. The ﬁrst step in ﬁnding the solution is to determine the displacer front. Since the displacer is strongly bound, this front is a shock whose velocity is given by Equation 7.30 where in our case the concentration ratio is calculated from the following equation:  q − ( z1 + σ 1 ) q1*  q1* = K e ,1  0  D D C1 CNa +  

z1

(7.89)

231

232

7 Dynamics of Chromatography Columns + D CNa + is the Na concentration in the displacer. In general, solution of this equation for q1* requires a trial and error or graphical procedure. The co-ion concentration in the displacer C 0D propagates as an unretained front through the column and is the same at every point in the column up to this front. Thus, from Equation 7.88, downstream of the isotachic train, the Na+ concentration is C 0D − z1C1D . The next step is to determine the concentration, CiI , of each protein in the pure component bands formed in the isotachic train. Each of these must simultaneously satisfy the following equations:

qi* q1* = CiI C1D

(7.90)

 q − ( zi + σ i ) qi*  qi* = K e ,i  0  i CiI CNa +  

zi

(7.91)

i I D CNa + + z iC i = C 0

(7.92)

+

i Na +

where C is the Na concentration in the band of component i. The ﬁrst of these equations is the same as Equation 7.84; the second is the SMA model; and the third is the electro-neutrality condition. Simultaneous solution of these equations yields the result: 1

 1 q1*  zi q0 − C  D  K e ,i C1  D 0

CiI =

1

(7.93)

 1 q1*  zi ( zi + σ i ) D − zi  D C1  K e ,i C1  q1*

The ﬁnal step is to calculate the elution width of each band. By material balance, in terms of column volumes, this can be calculated as: CVi =

C iF CVF C iI

(7.94)

where CVF is the number of column volumes of feed applied in the feed load step. Note that introduction of the feed also results in a fast propagating Na+ wave that accompanies the co-ion present in the feed. The corresponding Na+ concentration is equal to C 0F . Also note that the Na+ concentration will be different in each band i and the amount adsorbed will depend on each particular value of CNa + . As discussed by Brooks and Cramer [29], since this dependence is different for each protein, the order of displacement can change with displacer concentration as a result of the induced salt steps. Additionally, as also noted by these authors, if the pure component isotherms are plotted at the Na+ concentration at which they are actually displaced, the isotachic concentrations are still located at the intersects of the operating line having the slope q1D C1D. Obviously, the same approach applies to anion exchangers, where the co-ions are positively charged species and the counter-ions are negative.

7.5 Displacement Development

Example 7.9 Isotachic train for protein ion exchange displacement chromatography using the Steric Mass Action (SMA) model A mixture of two proteins with SMA parameters as shown in Table 7.2, is separated by displacement development on a cation exchange column with εt = 0.7 and q0 = 567 mmol/l resin when 0.65 column volumes of feed are applied. The column is initially equilibrated with a ‘carrier’ mobile phase containing 75 mM Na+. Determine the composition of the isotachic train. Table 7.2 SMA parameters and compositions for Example 7.9. Parameter values are taken

from [29].

Displacer

Composition no.

zi

σi

Ke,i

1

64

130

Ci (mM)

C Na+ (mM)

C0 (mM)

5.45 × 1044

0.75

75

123

−2

Feed

2 3

6.0 4.8

53.6 49.2

1.06 × 10 9.22 × 10−3

0.88 0.74

75 75

83.8

Carrier

–

1

–

–

–

75

75

Solution – From Equation 7.88, introduction of the feed, whose components are strongly bound, results in a Na+ wave with CNa+ = 83.8 mM traveling with the unretained co-ion. The elution volume of this band is equal to the feed volume and is thus 0.65 CV. From Equation 7.89, for the conditions in this example we have:  567 − (64 + 130 ) q1*  q1* = 5.45 × 10 44   75 C1D  

64

Solving this equation by trial and error or graphically with C1D = 0.75 mM gives q1*/C1D = 4.01 . The displacer front travels at velocity v sh,1 = v/(1+ φq1*/C1D ) and q* reaches the column outlet in ε + (1− ε ) 1D = 0.7 + (1− 0.7) × 4.01 = 1.90 column C1 volumes. The composition and duration of the displaced bands is calculated from Equations 7.92 to 7.93 and the results are as follows: Comp. #

C iI (mM)

i C Na + (mM)

qi* (mM)

CVi

2

1.059

116.6

4.25

0.381

3

0.656

119.8

2.63

0.517

233

7 Dynamics of Chromatography Columns 1.6

140 C0

1.4

120 Na+

1.2

100 2 80

0.8 60

1 0.6

3

CNa+, C0 (mM)

1.0

Ci (mM)

234

40

0.4

20

0.2 0.0

0 0

0.5

1

1.5

2

2.5

CV

Figure 7.12

Isotachic train for Example 7.9.

The concentration proﬁle at the column outlet corresponding to Example 7.9 is shown in Figure 7.12. Experimental results obtained by Jayaraman et al. [26] for the separation of a mixture of α-chymotrypsinogen and cytochrome c under the conditions described in this example with an 8-µm diameter strong cation exchangers in a 10 × 0.5 cm column eluted at 0.1 ml/min (u ∼ 30 cm/h), are in good agreement with these theoretical predictions. As expected, however, broader bands with mixed zones are obtained experimentally with larger particles or higher ﬂow rates as a result of dispersive mass transfer effects (see e.g. [31]).

7.5.2 Transient Development

The determination of transient proﬁles during displacement development requires a solution of the system of coupled material balance equations and equilibrium relationships. Analytical solutions can be obtained using the method of Rhee et al. [32], which is based on the fundamental differential equation of the Riemmann’s problem, or by the method of Helfferich and Klein [19], which is based on the h-transformation to orthogonalize the system of mass balance equations. The latter is more general and does not require coherence. Thus, it permits a prediction of non-coherent waves that exist temporarily during the development of the displacement train without resorting to ﬁnite difference approximations (see e.g. [12]). Applications of these methods to displacement development are illustrated in Rhee and Amundson [25] and Helfferich and Klein [11]. A concise illustration of the use of the h-transformation to determine the transient development of the displacement train, including a numerical example can be found in [3]. Other examples relevant to liquid chromatography are given in [33]. The reader is referred to these standard texts and classical literature references for details

References

regarding the theory and application of these methods. While useful for many systems, both methods are, however, restricted to the special case of constant separation factor isotherms; that is, adsorption systems with the multi-component Langmuir isotherm or equivalent ion exchange systems governed by the mass action law. Such systems are equivalent to an M-component adsorption problem being analogous to a stoichiometric ion exchange problem with M + 1 components. Other cases, involving non-constant separation factors isotherms require, in general, numerical solutions which can be obtained by ﬁnite difference methods. Useful references including optimization by computer modeling are Jen and Pinto [34], Katti and Guiochon [35], and Phillips et al. [36].

References 1 Danckwerts, P.V. (1953) Chem. Eng. Sci., 2, 2. 2 Wen, C.Y., and Fan, L.T. (1975) Models for Flow Systems and Chemical Reactors, Marcel Dekker, New York. 3 LeVan, M.D., and Carta, G. (2007) Adsorption and ion exchange, Section 16, in Perry’s Chemical Engineers’ Handbook, 8th edn (ed. D.W. Green), McGraw-Hill. 4 Golshan-Shirazi, S., and Guiochon, G. (1988) Anal. Chem., 60, 2364. 5 Frey, D.D. (1990) Chem. Eng. Sci., 45, 131. 6 Golden, F. (1973) Theory of Fixed-bed Performance for Ion Exchange Accompanied by Chemical Reaction. PhD Thesis, University of California, Berkeley. 7 Jeffrey, A., and Taniuti, T. (1964) Nonlinear Wave Propagation, Academic Press, New York. 8 Jacob, P., and Tondeur, D. (1981) Chem. Eng. J., 22, 187. 9 Davis, M.M., and LeVan, M.D. (1987) AIChE J., 33, 470. 10 Rhee, H.K., Aris, R., and Amundson, N.R. (1986) First-Order Partial Differential Equations: Volume 1. Theory and Application of Single Equations; Volume 2. Theory and Application of Hyperbolic Systems of Quasi-Linear Equations, Prentice Hall, Englewood Cliffs, NJ. 11 Helfferich, F., and Klein, G. (1970) Multicomponent Chromatography, Marcel Dekker, New York. 12 Helfferich, F. (1991) Chem. Eng. Sci., 46, 3320.

13 Ruthven, D.M. (1984) Principles of Adsorption and Adsorption Processes, John Wiley & Sons, Inc., New York. 14 Guiochon, G., Shirazi, D.G., Felinger, A., and Katti, A.M. (2006) in Fundamentals of Preparative and Nonlinear Chromatography, 2nd edn, Elsevier Academic Press, Amsterdam. 15 Ghose, S., McNerney, T.M., and Hubbard, B. (2002) Biotechnol. Progr., 18, 530. 16 Pabst, T.M., Antos, D., Carta, G., Ramasubramanyan, N., and Hunter, A.K. (2008) J. Chromatogr. A, 1181, 83. 17 Pabst, T.M., Carta, G., Ramasubramanyan, N., Hunter, A.K., Mensah, P., and Gustafson, M.E. (2008) Biotechnol. Progr., 24, 1096. 18 Pabst, T.M., and Carta, G. (2007) J. Chromatogr. A, 1142, 19. 19 Helfferich, F. (1962) Ion Exchange, McGraw-Hill, New York. 20 Bennett, B.J., and Helfferich, F.G. (1984) pH Waves in Anion Exchange Columns, in Ion Exchange Technology (eds D. Naden and M. Streat), Horwood, Chichester, UK., pp. 322–330. 21 Soto Pérez, J., and Frey, D.D. (2005) Biotechnol. Progr., 21, 902. 22 Horvath, Cs. (1985) Displacement chromatography: yesterday, today, and tomorrow, in The Science of Chromatography (ed. F. Bruner), Elsevier, Amsterdam. 23 Antia, F., and Horvath, Cs. (1991) J. Chromatogr. A, 556, 119.

235

236

7 Dynamics of Chromatography Columns 24 Carta, G., and Dinerman, A. (1994) AIChE J., 40, 1618. 25 Rhee, H.K., and Amundoson, N.R. (1982) AIChE J., 28, 423. 26 Jayaraman, G., Gadam, S.D., and Cramer, S.M. (1993) J. Chromatogr. A, 630, 53. 27 Kundu, A., and Cramer, S.M. (1997) Anal. Biochem., 248, 111. 28 Kundu, A., Barnthouse, K.A., and Cramer, S.M. (1997) Biotechnol. Bioeng., 56, 119. 29 Brooks, C.A., and Cramer, S.M. (1992) AIChE J., 38, 1969. 30 Gadam, S.D., Gallant, S.R., and Cramer, S.M. (1995) AIChE J., 41, 1676.

31 Natarajan, C.V., and Cramer, S.M. (1999) AIChE J., 45, 27. 32 Rhee, H.K., Aris, R., and Amundson, N.R. (1970) Phil. Trans. Royal Soc. London, 267A, 419. 33 Antia, F., and Horvath, Cs. (1985) AIChE J., 31, 400. 34 Jen, S.C.D., and Pinto, N.G. (1992) J. Chromatogr., 590, 47. 35 Katti, A.M., and Guiochon, G. (1988) J. Chromatogr., 449, 24. 36 Phillips, M.W., Subramanian, G., and Cramer, S.M. (1988) J. Chromatogr., 454, 1.

237

8 Effects of Dispersion and Adsorption Kinetics on Column Performance 8.1 Introduction

This chapter discusses the effects of dispersion and adsorption kinetics on the performance of protein chromatography columns. The reference case is that of ideal chromatography, which is described by the local equilibrium theory. In this limiting case the column separation performance is at its maximum and is determined exclusively by the interaction of ﬂow and adsorption equilibrium. In real chromatography columns, however, the combined effects of dispersion and adsorption rate limitations reduce the performance. This is manifested by bands that are broader than predicted by the local equilibrium theory, but that are generally centered around the same location. Accordingly, we use the term column efﬁciency to denote the degree to which an actual column approaches the limit of ideal chromatography. As discussed in Chapter 2, the relative importance of the various bandbroadening factors generally depends on the molecules being separated and the particular operating conditions. For biopolymers, in particular, these factors are heavily inﬂuenced by ﬂow rate. Thus, the column efﬁciency cannot be regarded as a constant property but can be quite different for different molecules and change signiﬁcantly with operating conditions. Quantiﬁcation of band-broadening factors and their relationship to biomolecular properties and to the physiochemical characteristics of the stationary phase are thus especially critical for engineering protein chromatography columns. Protein adsorption kinetics is generally controlling in process columns and the relevant fundamental relationships were presented in Chapter 6. Its description in terms of suitable rate equations is thus incorporated into the general conservation equations presented in Section 7.1 to arrive at a complete description of column behavior and provide the means for designing and optimizing bioprocess systems.

Protein Chromatography: Process Development and Scale-Up. Giorgio Carta and Alois Lungbauer © 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 978-3-527-31819-3

238

8 Effects of Dispersion and Adsorption Kinetics on Column Performance

8.2 Empirical Characterization of Column Efﬁciency

A useful approach for the empirical characterization of column efﬁciency is based on the determination of the height equivalent to a theoretical plate (HETP), H, or the corresponding plate number, N. Both were introduced in Chapter 2 as general measures of column efﬁciency. As previously discussed, the reduced HETP, h = H/dp is conveniently deﬁned in order to compare stationary phases with different particle sizes. This quantity is generally a function of reduced velocity, v′ = vdp/D0, which determines the relative importance of dispersive and mass transfer kinetics effects. As shown in Chapter 2, v′ is typically low for small molecules and small particles, while it can be quite large for proteins and other biopolymers in process chromatography. Thus, the HETP for different solutes can be vastly different in the same column under identical operating conditions. As a result, when used for the empirical characterization of column efﬁciency, H will often measure different contributions to column efﬁciency dependent on the test solutes used for the measurement. The empirical determination of the HETP for a given column is based on the dynamic response to a pulse injection of a test solute under isocratic conditions. The response will be a sharp peak when the column efﬁciency is high and a broad peak when the column efﬁciency is low. Two requirements must be met for a proper determination of HETP as a direct measure of column efﬁciency. The ﬁrst is that the injection must be sufﬁciently small to be treated as a delta function. In this regard, the upper limit occurs when the injected volume becomes a substantial fraction of the eluted peak volume. The lower limit is generally determined by the sensitivity of the detector, which must provide a response peak with adequately high signal-to-noise ratio and low drift. The second requirement is that the adsorption equilibrium, relating qˆ to C, must be strictly linear. As discussed in Chapter 7, even small deviations from linearity result in tailing or fronting peaks. This type of band broadening has, of course, nothing to do with column efﬁciency and is determined solely by the interplay of ﬂow and thermodynamics. Ensuring that adsorption occurs with a linear isotherm is often not difﬁcult when the test solute is a small molecule and can be achieved by selecting a suitable mobile phase composition. On the other hand, when operating isocratically, this is not always easy when using a protein as the test solute, since protein binding is often extremely sensitive to the exact composition of the mobile phase. Moreover, secondary effects such as association or unfolding which are caused in part, by the complex and heterogeneous nature of these molecules will often complicate the experimental measurements. To overcome these problems it is sometimes possible to measure the HETP for non-binding conditions, where the protein merely diffuses in and out of the particle pores. Whether the resulting measurements are representative of the mass transfer processes when binding occurs will depend on the nature of the transport mechanisms. For example, as discussed in Chapter 6, if pore diffusion is dominant, the same mechanism will control transport whether or not protein

8.2 Empirical Characterization of Column Efﬁciency

binding occurs. Conversely, if diffusion in the adsorbed phase either dominates or occurs in parallel to pore diffusion, HETP measurements under non-binding conditions will not be representative of what will happen when protein binding occurs. Proper calculation of the HETP from the pulse injection response peak generally requires evaluation of the following statistical moments [1]: ∞

µ0 = ∫ Cdt

(8.1)

0

∞

µ1 =

1 Ctdt µ0 ∫0

σ2 =

1 C ( t − µ1 )2dt µ0 ∫0

(8.2)

∞

(8.3)

where C is the peak proﬁle as a function of time at the column outlet. The physical meanings of the moments are as follows. µ0 is the peak area, proportional to the amount injected, µ1 is the mean elution time, and σ2 is the variance of the peak. The HETP is then calculated from the following equation: H=

σ 2L µ12

(8.4)

where L is the column length. In turn, the plate number is given by the relationship: N=

L µ12 = H σ2

(8.5)

Equivalent expressions can be written for the moments using the eluted volume, V, or the number of column volumes, CV, in place of time, t. When this is done, the moments will have different values and units, but H will, of course, be the same. Higher moments can also be calculated and used to deﬁne the asymmetry of the peak. The peak skew is deﬁned as follows [2]: ∞

Skew =

1 C (t − µ1 )3dt µ0σ 3 ∫0

(8.6)

A peak that is symmetrical around µ1 will have a skew of 0. Conversely, a peak with skew greater than 0.7 is highly asymmetrical. The calculation of the moments for actual experimental peaks requires considerable care. Useful references are [1–5]. It is usually not difﬁcult to obtain accurate values of µ0 and µ1 even for fairly asymmetrical and noisy peaks. However, the calculation of σ2 is often difﬁcult for tailing peaks since the term (t − µ1)2 in Equation 8.3 tends to exacerbate any errors at long retention times. Drift of the detector signal causing an apparent shift of the baseline is a typical experimental problem. Additionally, the relationship between detector signal and C is sometimes not linear. Suitable corrections are needed in order to obtain meaningful values of the

239

240

8 Effects of Dispersion and Adsorption Kinetics on Column Performance

HETP if these effects are signiﬁcant. The potential for error is obviously even greater for the calculation of peak skew and the calculation of higher moments from experimental chromatograms is often impossible with any degree of precision. As discussed in Chapter 4, a further aspect potentially affecting the experimental determination of the moments is the extra-column contributions to band broadening that are associated with injector, detector, and any connections between these and the column. For a linear system, it has been shown both theoretically and experimentally [2, 6, 7] that contributions to the ﬁrst moment and variance of the pulse response caused by multiple processes occurring in series are additive. Thus, the apparent values obtained from an experimental chromatogram are related to the true values for the column alone by the following equations:

( µ1 )app = µ1 + ( µ1 )inj + ( µ1 )con + ( µ1 )det = µ1 + ( µ1 )extra

(8.7)

(σ 2 )app = σ 2 + (σ 2 )inj + (σ 2 )con + (σ 2 )det = σ 2 + (σ 2 )extra

(8.8)

where subscripts app, con, det, and extra refer to the contributions of the injector, connecting tubing, detector, and total extra-column effects, respectively. (µ)extra and (σ2)extra can easily be determined independently by making suitably small pulse injections with a bypass line allowing the actual values to be determined by subtraction. Note that the relative importance of extra-column contributions to the moments can vary considerably with retention of the test solute and with ﬂow. Extra-column effects are sometimes negligible for larger scale process columns. However, they are frequently important in the laboratory and increase in relevance with column efﬁciency. In some instances, when the column efﬁciency is very high, such as with monoliths, extra-column factors are dominant creating special challenges for the accurate determination of HETP [8, 9]. Under certain conditions, nearly symmetrical peaks that are approximately Gaussian are obtained in response to pulse injections. In this case, the HETP can be calculated from the following approximate relationships [1]: H=

L  ∆ 2   5.54  t max 

H=

L  W 2   16  t max 

(8.10)

µ0  2 H = 2π  L  C max t max 

(8.11)

(8.9)

where tmax is the elution time of the peak maximum (which coincides with µ1 when the peak is exactly symmetrical), ∆ is the peak width measured at mid height, W is the baseline width, and Cmax is the peak maximum. The numerical coefﬁcients in these equations are based on the properties of the normal probability Gaussian function, which are illustrated in Figure 8.1. Equations 8.9 to 8.11 are generally much easier to use than the more delicate moment-based calculations. In particular, the ‘area method’, based on Equation 8.11, is used in some chromatographic

8.2 Empirical Characterization of Column Efﬁciency 1.2 Tangents to inflection points 1.0

C/Cmax

0.8

Area =

(2 )1/2

0.6 = 2.35 0.4

0.2 tmax = µ1 0.0

t W=4

Figure 8.1 Properties of Gaussian peaks used for the approximate calculation of HETP from Equations 8.9 to 8.11. Adapted from [10].

workstations, since it is easy to implement when the response peak is available in digital form. However, the accuracy of these equations is dependent on how closely the experimental peak is approximated by a Gaussian curve. This can be tested t C by plotting the relative fractional recovery ∫ dt versus t on a graph with µ 0 0 probability-linear coordinates. If the plot is linear, Gaussian behavior is conﬁrmed and Equations 8.9 to 8.11 are usually adequate. This will generally be the case when the skew of the experimental peak is less than about 0.7. At higher values, the moment calculations should be used in lieu of the simpler equations. An example of the potential errors incurred in HETP calculations by different methods is shown in Figure 8.2. The pulse response peaks shown were simulated for a 20-cm column at different ﬂow rates of the mobile phase. The model used for these calculations [11] assumes a linear isotherm with intra-particle pore diffusion being the only rate factor and the conditions simulated are typical of those encountered in protein chromatography. The peaks become increasingly broad as the velocity of the mobile phase increases. Peak tailing also increases dramatically. Thus, the peak maximum shifts relative to the peak mean elution time, which, as can be seen in Figure 8.2b, is correctly measured by µ1. Since the isotherm is linear, this quantity is expected to be constant in CV units and independent of mobile phase velocity. On the other hand, for these conditions, where intra-particle mass transfer is controlling, the HETP is expected to increase linearly with ﬂow rate. As shown in Figure 8.2c, this trend is correctly predicted when the HETP is calculated using the moment method. However, for this example, calculations based on the width

241

8 Effects of Dispersion and Adsorption Kinetics on Column Performance 0.06

1

(a) CV

0.05

0.5

0.04

First moment Peak maximum

0 5

0.03

HETP (cm)

C

242

0.02 0.01 0 0.5

1

1.5

2

2.5

3

Moment method Midheight width

4 3 2 1 0

0

(b)

(c) 0

200

400

600

800 1000 1200

u (cm/h)

CV Figure 8.2 Isocratic pulse response peaks (a) obtained by simulation assuming pore diffusion control and corresponding peak parameters (b and c). The HETP was calculated from the moment method (Equations 8.1–8.4) and from Equation 8.9

using the peak maximum and the width at mid-height. Peaks and calculated parameters are shown at 50, 100, 200, 300, 500, and 1000 cm/h for a 20-cm column. Data from [18].

at mid height, Equation 8.9, are accurate for the peaks shown only up to about 300 cm/h and deviate signiﬁcantly at higher ﬂow rates. At high values, the HETP curve calculated using this method ﬂattens out to give the false impression that the column efﬁciency has become independent of ﬂow rate. This is not the case, of course, when intra-particle diffusion is controlling. As shown in Figure 8.2a, in fact the peaks continue to become broader and more asymmetric as the ﬂow rate is increased. Other than peak skew, there is no universally accepted way of deﬁning the peak asymmetry with simple methods. A practical metric is the asymmetry factor [1, 2] given by the following expression: As =

b a

(8.12)

with a and b deﬁned according to Figure 8.3. For tailing peaks, such as the one shown in this ﬁgure, As > 1, while fronting peaks are deﬁned by As < 1. The following empirical equation has been derived [10] to describe the relationship between As (which is easily determined) and peak skew (whose determination can be challenging) based on data in [1]: 0.19   Skew = 0.51 + As − 1  

−1

(8.13)

8.2 Empirical Characterization of Column Efﬁciency 1.2

1.0

C/Cmax

0.8

0.6

0.4

0.2 a

b

0.0

Figure 8.3

10% of peak height

t Parameters used for the deﬁnition of the peak asymmetry factor, As = b/a.

This relationship predicts a peak skew < 0.7 when As < 1.25. Such peaks are generally desirable for efﬁcient chromatographic separations. Even for strictly linear isotherms, tailing peaks are observed in practice for a variety of reasons, including slow mass transfer in the stationary phase, binding on multiple adsorption sites with different afﬁnity, skewed distribution of the particle sizes, and non-uniformities of ﬂow across the column radius [12–15]. Under certain conditions the latter can also cause peak fronting [16], although these conditions are less common. A useful empirical description of tailing peaks is the so-called exponentially modiﬁed Gaussian (EMG) function, deﬁned by the following relationship [17]: ∞

C=

µ0 t − tG − t ′  t ′  exp  −  dt ′ − σ Gτ G 2π ∫0   σ G 2  τ G 

(8.14)

The EMG function depends on the three parameters tG, σG, and τG and represents the convolution of a Gaussian curve with mean elution time tG and standard deviation σG with an exponential decay having time constant τG. The main advantage is that the parameters representing peak elution time, breadth, and skew can be calculated directly from the following relationships without having to calculate the moments by integration:

µ1 = tG + τ G H=

(8.15)

σ L (σ + τ ) L = µ12 (tG + τ G )2 2

Skew =

2 G

2 G

(8.16)

2 (τ G σ G )3 2 1 + (τ G σ G ) 

32

(8.17)

243

8 Effects of Dispersion and Adsorption Kinetics on Column Performance

For EMG peaks, the skew increases with the ratio, τG/σG. When this ratio is unity, the peak skew is 0.707. Fitting the experimental peak with the EMG function is sometimes effective when the direct calculation of HETP by the moment method is inaccurate because of baseline drift or noise. Using this approach, the HETP and skew are easily calculated from Equations 8.16 and 8.17 after determining the EMG parameters by regression.

Example 8.1

Calculation of HETP from experimental pulse response peak

The experimental data in Figure 8.4 were reported in [18] for 100-µL injections of 2 g/l lysozyme (Mr ∼ 15 kDa) in a 1-cm diameter, 10-cm long SP-Sepharose-FF column (∼100 µm particle size) with a mobile phase containing 1.5 M NaCl in 10 mM Na2HPO4 buffer adjusted to pH 6.5 with phosphoric acid and ﬂowing at a rate of 4 ml/min. The data are reported as mAU versus eluted volume in ml. Determine the HETP. Solution – We assume that the mAU values are proportional to protein concentration and will neglect the extra-column effects. The injected volume is 0.1 ml, which is a very small fraction of the width of the response peak in units of volume. Thus, the injection may be treated as inﬁnitesimal. The peak parameter values and HETP calculated according to various methods are as follows: 160 Experimental 140

EMG Gaussian

120 100

mAU

244

80 60 40 20 0 0

5

10

15

V (mL)

Figure 8.4 Comparison of experimental, Gaussian, and EMG peaks for Example 8.1. The experimental data are from [18] and are for a 100-µL lysozyme injection onto a

1 × 10 cm SP-Sepharose-FF column eluted at a ﬂow rate of 4 ml/min. The Gaussian and EMG peaks shown have the same mean and variance as the experimental data.

8.2 Empirical Characterization of Column Efﬁciency

Moment method (Equations 8.1 to 8.4)

Mid-height width method (Equation 8.9)

Baseline width method (Equation 8.10)

Area method (Equation 8.11)

µ1 = 6.79 ml σ2 = 2.82 ml2 H = 0.612 cm

tmax = 6.07 ml ∆ = 3.29 ml H = 0.530 cm

tmax = 6.07 ml W = 5.74 ml H = 0.559 cm

µ0 = 505 mAU × ml tmax = 6.07 ml Cmax = 142 mAU H = 0.546 cm

In Example 8.1, the statistical moments were obtained by numerical integration using the trapezoidal rule. tmax, ∆, W, and Cmax were obtained graphically. As seen in Figure 8.4, the peak is fairly asymmetrical with a = 2.3 and b = 3.9 ml. Thus, the asymmetry factor is As = 3.9/2.3 = 1.7. As a result, the three methods based on a Gaussian approximation underestimate the HETP by as much as 13%. Even greater error will result for peaks with greater asymmetry. Figure 8.4 also shows calculated Gaussian and EMG peaks that have the same µ1 and σ2 as the experimental peak. The EMG peak was calculated with tG = 5.50, τG = 1.29, and σG = 1.08 ml. It can be seen that the Gaussian peak provides a poor ﬁt of the data, even though the ﬁrst and second moment are exactly the same as the experimental peak. On the other hand, the EMG function provides a much better ﬁt. Note that the reduced velocity for this experiment is around 2800, based on ε = 0.3. For these conditions, assuming that the column is reasonably well packed and that ﬂow is uniform, band broadening is completely controlled by the diffusional resistance in the particle pores and the HETP is expected to increase linearly with ﬂow rate. In biopharmaceutical manufacturing, HETP measurements frequently comprise part of the validation protocols aimed at determining the reproducibility of column packing and its integrity during successive cycles of use. In order for the HETP to be representative of packing quality and uniformity of ﬂow, measurements must be made for conditions where intra-particle mass transfer is not limiting. Based on Figure 2.10, this requires v′ to be of the order of 50 or less. Using salt as a tracer, which can be conveniently monitored with a conductivity detector, since D0 ∼ 1 × 10−5 cm2/s, with 100 µm particles, such values of v′ are obtained at superﬁcial velocities of less than 50 to 100 cm/h. At higher ﬂow rates, even salt will experience diffusional resistances in particles of this size. Breakthrough curves obtained in frontal analysis can also be used to determine the HETP directly provided the adsorption isotherm is linear, since for a linear system the pulse response is the derivative of the response to a step input. Equivalent peaks can thus be obtained by differentiating experimental breakthrough curves. This is sometimes convenient since the response to salt steps is available from manufacturing runs allowing packing integrity to be monitored without having to resort to separate pulse injections. To avoid the error introduced by taking the numerical derivative of a noisy detector signal, a useful approach is to ﬁt the experimental breakthrough curve with a smooth function and then use the

245

246

8 Effects of Dispersion and Adsorption Kinetics on Column Performance

analytical derivative of the ﬁtted function to determine the equivalent pulse response peak. If the isotherm is not linear or the pulse injection is not isocratic or is not inﬁnitesimal, it is still possible to determine a meaningful HETP. However, doing so requires a model to distinguish band broadening caused by dispersive factors from that caused by thermodynamics. A useful approach using gradient elution data is presented in Chapter 9. Other approaches rely on ﬁtting frontal analysis data with the equations that follow in this chapter.

8.3 Modeling and Prediction of Column Efﬁciency 8.3.1 Plate Model

The classical plate model of Martin and Synge [19] provides a useful conceptual understanding of band broadening effects. Conceptually, the plate model can be represented as in Figure 8.5, where the chromatographic column is treated as a series of equilibrium stages or plates. The stationary phase is distributed equally among the stages while the mobile phase ﬂows from stage to stage with volumetric ﬂow rate Q. Adsorption equilibrium is assumed at each stage. Following for example Villermaux [20], material balances for each plate j yield:

Q, C0

C1

C2

1

...

CN-1

2

CN

N

Mobile phase volume, V m

Stationary phase volume, V s

Cj-1

Cj

j Figure 8.5 Conceptual view of the equilibrium plate model. The stationary and mobile phase volumes are distributed evenly in N mixed stages. The mobile phase ﬂows from stage to stage.

8.3 Modeling and Prediction of Column Efﬁciency

QC j −1 = QC j + Vm

dC j dqˆ j + Vs , dt dt

j = 1, 2, … , N

(8.18)

where Vm and Vs are the volumes of mobile and stationary phases in each stage. These quantities are given by Vm = εVc/N and Vs = (1 − ε)Vc/N, respectively, where Vc is the column volume. With the assumption of linear equilibrium in each stage (qˆj = mCj), we obtain the following system of ordinary differential equations: C j −1 = C j +

tR dC j , N dt

j = 1, 2, … , N

(8.19)

where: tR =

(

)

εVc 1−ε L 1+ m = (1 + k ′ ) ε Q v

(8.20)

Mathematically, the plate model can also be derived from the general conservation equations (7.1) neglecting axial dispersion, assuming local equilibrium, and discretizing the axial derivative by backwards ﬁnite differences. Accordingly, ∂C/∂z = (Cj − C j−1)/∆z where ∆z = L/N resulting in a discretized form identical to Equation 8.19. For a pulse injection of a quantity Mi of the test solute, Equation 8.19 is subject to the condition: C0 =

M δ (t ) Q

(8.21)

The solution subject to this condition is easily found in the Laplace domain and is given by: G (s ) =

(

C N stR = 1+ 0 C N

)

−N

(8.22)

where G(s) is the transfer function. Inversion of this function yields the time domain solution: C N (t ) =

−1

M Q NN  t N t exp  −N   tR (N − 1)!  tR  tR 

(8.23)

This is a Poisson distribution and is generally an asymmetric function of time. When N is large, Equation 8.23 is closely approximated by: C N (t ) ∼

Mi Q tR

2

N  N t  exp  −  − 1    2π  2 tR 

(8.24)

which is a Gaussian distribution. Calculated curves based on these two equations are shown in Figure 8.6 for different values of N. It is obvious that as N increases the exact solution becomes narrower and increasingly symmetrical around tR eventually approaching the Gaussian limit.

247

8 Effects of Dispersion and Adsorption Kinetics on Column Performance 3 Poisson

N = 50

Gaussian

2

CN x (tRQ/Mi)

248

N = 25

1 N=5

0 0

0.5

1

1.5

2

t/tR Figure 8.6 Chromatographic response curves calculated from the equilibrium plate model with a linear isotherm.

The statistical moments of the concentration proﬁle CN(t) are of interest and are easily obtained directly from the transfer function with the van der Laan theorem [21]: ∞

µk =

∫C

(t ) t k dt

N

0 ∞

∫C

N

(t ) dt

k  ∂ G (s )  = ( −1)k   ∂s k  s →0

(8.25)

0

The results for the ﬁrst and second central moments are given by the following equations: ∞

µ1 =

∫C

N

( t ) tdt

0 ∞

∫ C N (t ) dt

∂G ( s )  = − = tR  ∂s  s →0

(8.26)

0

∞

σ = 2

∫C

N

(t ) ( t − µ1 )2dt

0

∞

∫C

N

( t ) dt

2 t2  ∂ G(s )  = − µ12 = R 2  N  ∂s  s → 0

(8.27)

0

Thus, the ﬁrst moment of the elution proﬁle equals tR, and coincides with the peak maximum when N is large. The standard deviation of the response peak is:

8.3 Modeling and Prediction of Column Efﬁciency

σ=

tR N

(8.28)

so that the quantity: H=

σ 2L σ 2L L = 2 = µ12 tR N

(8.29)

is the HETP. While the equilibrium plate model provides a useful framework for modeling linear chromatography and explains quantitatively the link between the deﬁnition of HETP and column efﬁciency, it is usually inadequate for the description of tailing peaks or to predict adsorption rate limitations, especially when the isotherm is highly non-linear. One of the problems is that it is assumed intrinsically that the HETP is the same for all components. This is not the case, as we have seen, in bioprocess applications when molecules of different size and properties are separated. Nevertheless, the backwards ﬁnite difference discretization is still useful as a numerical description of dispersion. In his case, the assumption of local equilibrium is relaxed and each dqˆij dt term in Equation 8.18 is expressed by a suitable rate equation such as those introduced in Chapter 6. The resulting differential equations are then solved numerically. Since the backwards discretization introduces numerical dispersion, the number of discretization points, N, can be used in an empirical manner to describe the physical axial dispersion phenomena expected in a real chromatographic column. [22, 23]. 8.3.2 Rate Models with Linear Isotherms

Analytical solution of the general conservation equations describing the axially dispersed plug ﬂow model (Equation 7.1) is possible for a variety of rate mechanisms described in Chapter 6 provided the adsorption isotherm is linear. In most cases, the solution is easily found in the Laplace domain, which allows the calculation of the statistical moments of the pulse response using the van der Laan theorem. If the solution is desired in the time domain, it can be obtained by inversion of the transfer function. Closed form solutions are available in some cases, while others require numerical integration. We begin by presenting the analytical solutions obtained by neglecting axial dispersion and assuming that the adsorption kinetics is described by the linear driving force (LDF), model. Accordingly, the rate equation is expressed by: q ∂ ˆ q* − ˆ q) = k (ˆ ∂t

(8.30)

where k is the LDF rate coefﬁcient based on an adsorbed phase concentration driving force. Alternatively, the rate equation can be expressed as: ∂ ˆ q = kc (C − C * ) ∂t

(8.30a)

249

250

8 Effects of Dispersion and Adsorption Kinetics on Column Performance Table 8.1 Analytical solutions for the column response to pulse, step, and periodic injections with linear isotherm and LDF approximation.

A

Input

Solution

Reference

Pulse

C τ F n − nτ1 − n e e I1(2n τ 1 ) = CF τ1

[24]

or

C τF = CF 2 π

with τ F =

n exp  −n ( τ 1 − 1)  for n > 5 (τ 1 )1 4 τ1 2

vtF 1−ε , k′ = m Lk ′ ε n,

B

C = J (n , nτ 1 ) = 1 − e − nτ1 ∫ e − ξI 0(2 nξ ) dξ CF 0

Step

or

[25, 26]

C 1 ≈ [1 − erf ( n − nτ 1 )] for n > 20, τ1 < 1 CF 2 C 1 ≈ [1 + erf ( nτ 1 − n )] for n > 20, τ1 > 1 CF 2

C

Periodic injections of length tF with cycle time tc

C 2 ∞ 1 j 2n   = γ + ∑  exp  − 2 sin ( jπγ )  j + ω 2  CF π j =1  j

[11]

t tF L  jω n    − − × cos 2 jπ  −  tC 2tC vtC  j 2 + ω 2    with γ =

tF n vtc , ω= tC 2π k ′ L

where kc is an LDF rate coefﬁcient based on a liquid phase concentration driving force. For a linear isotherm, since C* = 〈qˆ〉/m, taking kc = mk renders the two equations exactly equivalent. Thus, identical results will be obtained with either form. (This is not the case, however, when the isotherm is non-linear. With a favorable isotherm, Equation 8.30 is usually preferred when intra-particle mass transfer is the dominant resistance). Table 8.1 provides the analytical solutions for the linear isotherm case for three different column inputs: pulse injection, step input, and periodic injections of ﬁnite duration. The solutions are presented in dimensionless form where: vt −1 τ1 = L k′ n=

(1 − ε ) kc L εv

(8.31) (8.32)

τ1 is the dimensionless time and n is the number of transfer units available in the column. τF = vtF/Lk′ is the loading factor representing the feed load in dimensionless form.

8.3 Modeling and Prediction of Column Efﬁciency 0.4 1/(4 )1/2

(C/CF)/( Fn1/2)

0.3

0.2

0.1

n=5 10 20 100

0 0

1

2

3

1

Figure 8.7 Pulse response based on the LDF rate model with linear isotherm according to item A in Table 8.1. Adapted from [10].

The pulse response, item A in Table 8.1, is shown graphically in Figure 8.7. Note that for n > 20 the peak maximum occurs for τ1 ∼ 1, which corresponds to L t max = tR = (1 + k ′ ). The peak height and the standard deviation are then given by: v C max τ F = CF 2

n π

σ = (1.7 ln 2 )

(8.33)

L L L t max − k′ v ∼ v = v n n2 n2

t max −

(8.34)

As a result, the following equations are obtained for HETP and the plate number:

( ) ( ) L 1 + k′ n N = =( H k′ ) 2 H=

k ′ 2 2L k ′ 2 2ε v σ 2L = = 2 µ1 1 + k′ n 1 + k ′ (1 − ε ) kc

(8.35)

2

(8.36)

(1 +k ′k ′ ) approaches unity. In 2

Note that if the retention factor is high, the term

this case the plate number, N, and the number of transfer units, n, are simply related by: N=

n 2

(8.37)

251

8 Effects of Dispersion and Adsorption Kinetics on Column Performance

Example 8.2 Comparison of LDF model for pulse response with experimental data Compare the experimental results of Example 8.1 with the LDF model, assuming ε = 0.3. Solution – For these conditions, u = 4/(π × 0.52) = 5.09 and v = 5.09/0.3 = 16.9 cm/min. From the moment calculations in Example 8.1 we obtained µ1 = 6.79 ml. Thus, the retention factor is k′ = (µ1 − V0)/V0 = (6.79 − 2.36)/2.36 = 1.88, where V0 = ε Vc = 2.36 ml is the column void volume. The duration of the feed pulse is tF = VF/Q = 0.1/4 = 0.025 min. Thus, τF = vtF/k′L = 0.0226. From Example 8.1, for the experimental peak we have N = 10/0.612 = 16.3. Thus, k′ 2 from Equation 8.36, n = 2N = 13.9. A graph of the solution (item A in 1+ k ′ Table 8.1) is shown in Figure 8.8 indicating fairly good agreement with the experimental result. The corresponding value for the LDF rate coefﬁcient is kc = 10 min−1. Note that in order to make this comparison, the experimental mAU values are converted to C/CF using the absorbance of the feed sample at the same wavelength.

( )

0.03 Experimental LDF Model 0.02

C/CF

252

0.01

0 0

5

10

15

V (mL)

Figure 8.8 Comparison of experimental data and LDF model prediction for a pulse injection of lysozyme onto a 1 × 10 cm SP-Sepharose-FF column eluted at a ﬂow rate of 4 ml/min according to Example 8.2.

The response to a step input, item B in Table 8.1, is given by the so-called J function, which is available in graphical and tabulated forms in various references [10, 24]. A series approximation suitable for practical calculations is given by the following [27]:

8.3 Modeling and Prediction of Column Efﬁciency 1

C/CF

0.8

n=1

0.5

2 5 10

0.3

20 50 100 0 0

0.5

1

1.5

2

1

Figure 8.9 Plot of the J function representing the breakthrough curve with linear isotherm and LDF rate model.

g k ⋅ (nτ 1 )k (k !)2 k =0 ∞

J ( n , nτ 1 ) = 1 − e − n τ 1 ∑

(8.38)

where: g 0 = 1 − e −n g k = kg k −1 − n k e − n for k > 0 Alternatively, when n > 20, the following approximation can be used [28]: 1 [1 − erf ( n − nτ1 )], with τ1 < 1 2 1 J (n, nτ 1 ) = [1 + erf ( nτ 1 − n )], with τ 1 > 1 2 J ( n , nτ 1 ) =

(8.39)

A plot of the J function is shown in Figure 8.9. As n increases, the breakthrough curve becomes increasingly steeper, approaching the vertical line at τ1 = 1 when n → ∞. Note that when n is large (> 20), the breakthrough curve becomes symmetrical with the mid-point C/CF = 0.5 occurring at τ1 = 1. Additionally, note that the J function also represents the response to a negative concentration step. In fact, provided the isotherm is linear we have the general result (C − C0)/ (CF − C0) = J(n, nτ1), where C0 and CF are arbitrary initial and feed concentrations, respectively. Finally, the steady state solution for periodic injections of pulses of arbitrary duration tF followed by isocratic elution for a time tE with a cycle time tC = tF + tE can be predicted from item C in Table 8.1. Steady state is achieved after repeated injections and elution. However, by choosing a total cycle time sufﬁciently larger than tF, the same solution can also be used to predict the response to a single

253

8 Effects of Dispersion and Adsorption Kinetics on Column Performance 1

0.8

2.0 0.6

C/CF

254

1.0

0.4

0.5 0.2 F

= 0.05

0 0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

1

Figure 8.10 Elution curves calculated from item C in Table 8.1 with linear isotherm and LDF model for n = 40 and different values of the loading factor τF.

injection of arbitrary duration. An example is given in Figure 8.10 for n = 40 with different values of the loading factor. As τF increases, the peak maximum is shifted and increases in magnitude until C/CF = 1. At higher values of τF, the elution proﬁle consists of a breakthrough curve with midpoint at τ1 = 1 followed by a symmetrical elution curve with midpoint at τ1 = 1 + τF. Conditions where τF is a signiﬁcant fraction of the total peak volume are referred to as volume overloaded. Resolution of adjacent components is reduced when this occurs since the bands are broader than would be obtained with inﬁnitesimal pulse injections. The degree to which this occurs is a function of band spreading and can be predicted with these equations. Predictions of column behavior with other rate mechanisms, is obtained by combining the equation in Table 6.5 with Equations 7.1 to 7.4. With a linear isotherm, analytical solutions can be obtained for different input conditions in the Laplace domain with relatively little effort. For example, the case of pore diffusion with ﬁlm resistance and axial dispersion (Equation 7.1 and item B in Table 6.5) yields the following transfer function [29]: G(s ) =

C

C z = 0

vL 4DL  vL  = exp  − 1 + 2 [s + M (s )]  D D v 2 2  L  L

(8.40)

where: sm     sinh  rp   s ˆ q 3 (1 − ε ) k f   De  M (s ) = =  1 − Bi sm sm sm rp C      rp cosh  rp + (Bi − 1) sinh  rp      De De  De   (8.41)

8.3 Modeling and Prediction of Column Efﬁciency

is the transfer function for the particle. The time domain solution is obtained by inversion of the Laplace transform [12, 30, 31]. However, in most cases the analytical expression is complex and involves inﬁnite integrals, which are difﬁcult to evaluate [32–35]. A rapidly converging series solution is available for the case of pore diffusion with ﬁlm resistance but neglecting axial dispersion, which is often a good approximation for protein chromatography (see [11]). Other cases are usually more easily handled by solving the original partial differential equations numerically by the numerical method of characteristics [36], ﬁnite differences [22, 23], or orthogonal collocation approaches [37–39]. On the other hand, the moments of the response curve can be evaluated easily using the van der Laan theorem (Equation 8.25), directly from the transfer function. The following result is obtained from Equations 8.40 and 8.41:

(

)

µ1 =

1−ε L L 1+ m = (1 + k ′ ) ε v v

(8.42)

H=

rp2  σ 2L 2DL 2ε v k ′ 2  rp = + +  2 v µ1 1 − ε 1 + k ′  3k f 15De 

( )

(8.43)

Thus, the ﬁrst moment is the same as that obtained with the LDF model and coincides with the local equilibrium value. Comparing Equations 8.35 and 8.43 shows that the HETP can also be matched to that obtained from the LDF model by deﬁning the LDF rate coefﬁcient as:

( ) (1 −εvε )D + 3rk

1 k′ = 1 + k′ kc

2

L

2

p

+ f

rp2 15De

(8.44)

This equation can be used to estimate kc given axial dispersion and mass transfer parameters. In turn, the estimated value can be used to predict the column response with the appropriate equation from Table 8.1. While some deviations

Example 8.3 Estimate the effective pore diffusivity from a pulse response experiment and compare results with a rigorous model Based on the results of Example 8.2, determine the effective pore diffusivity for lysozyme in SP-Sepharose-FF assuming that axial dispersion and ﬁlm mass transfer resistance are negligible. Solution – From Equation 8.44, with negligible axial dispersion and ﬁlm resistance, we have: kc =

15De rp2

(8.45)

using the value of kc determined in Example 8.2, we ﬁnd De = 2.78 × 10−7 cm2/s, based on rp = 50 µm . The corresponding calculated proﬁle based on the exact solution of the pore diffusion model reported in [11] is shown in Figure 8.11. The description of the experimental peak is improved considerably compared

255

8 Effects of Dispersion and Adsorption Kinetics on Column Performance

to the LDF model, although the simpler calculations shown in Figure 8.8 are probably adequate for most practical purposes. Further improvement of the ﬁt is possible by regressing De directly.

0.03 Experimental Pore diffusion model

0.02

C/CF

256

0.01

0 0

5

10

15

V (mL)

Figure 8.11 Comparison of experimental data and pore diffusion model prediction for a pulse injection of lysozyme onto a 1 × 10 cm SP-Sepharose-FF column eluted at a ﬂow rate of 4 ml/min according to Example 8.3.

between the proﬁles obtained in this fashion and those predicted with a rigorous model can be expected, the proﬁles will have the same ﬁrst moment and the same variance. In practice this will often sufﬁce for process calculations. Other transport mechanisms can also be considered and are described by Equation 8.44 with simple substitutions. If intra-particle transport occurs by solid phase diffusion instead of pore diffusion, then the quantity mDs replaces De. Conversely, if pore and solid phase diffusion occur in parallel De is replaced by the quantity (mDs + De). It is useful to consider a dimensionless form of Equation 8.43. In general, the axial dispersion coefﬁcient, DL, can be thought of as the result of two additive contributions: axial diffusion and hydrodynamic dispersion. Thus, DL = De ,b + ED =

εD0 + ED τb

(8.46)

where De,b is an effective diffusivity in the mobile phase, τb a tortuosity factor for the packed bed, and ED is the eddy diffusivity. The latter represents hydrodynamic dispersion. A number of correlations are available to predict ED in packed beds, as reviewed by Lange et al. [40], Gunn [41] and LeVan and Carta [10]. At the low Reynolds numbers normally encountered in liquid chromatography, for wellpacked columns with uniform ﬂow the following result is obtained:

8.3 Modeling and Prediction of Column Efﬁciency

ED ∼ γ d p v

(8.47)

where dp is the particle diameter and γ is a constant, typically in the range of 1 to 4. Combining Equation 8.43 with Equations 8.46 and 8.47 yields: H=

( )

2 rp 2  2εD0 1 2 k ′  rp + 2γ d p + + v  τb v φ 1 + k ′  3k f 15De 

(8.48)

where φ = (1 − ε)/ε. This result has the same form as the well-known empirical van Deemter equation given by the following expression [42]: H=

B + A + Cv v

(8.49)

where A, B, and C are constants. In terms of reduced quantities, Equation 8.48 becomes: h=

H b = + a + cv ′ dp v′

(8.50)

v′ =

vd p D0

(8.51)

where

b=

2ε τb

(8.52)

a = 2γ

(8.53)

τp  1 1  k ′   10 + 30 φ  1 + k ′   Sh ε pψ p  2

c=

(8.54)

The c term is usually dominant for protein chromatography applications and the relevant parameters, Sh, τp, and ψp, are estimated as a function of the molecular properties of the diffusing solute and the characteristics of the stationary phase as discussed in Chapter 6. For a stationary phase whose particles can be represented as an aggregate of microparticles with radius rs (cf. item F in Table 6.5), the c term is expressed by the following relationship [43]: 2

c=

2 τp m − ε p  rs  De  1 1  k ′   10 +  +    εm 2  rp  Ds  30 φ 1 + k ′  Sh ε pψ p

(8.55)

where Ds is the adsorbed phase diffusivity in the microparticles. Finally, if intraparticle convection occurs in the pores, the enhancement factor f(Pep) deﬁned in Section 6.2.4 is included in the c term to give the following result [44–46]: 2

c=

2 m − ε p  rs  De  1 1  k ′   10 τ p 1 +  +    εm 2  rp  Ds  30 φ 1 + k ′  Sh ε p f (Pe p )

(8.56)

where f (Pe p ) =

(Pe p 6)  3

1 1   tanh (Pe 6 ) − Pe 6  p p  

−1

(8.57)

257

258

8 Effects of Dispersion and Adsorption Kinetics on Column Performance

and Pe p =

Fpud p Fp ετ p = v′ De εp

(8.58)

Note that in this case we have neglected the hindrance factor, ψp, since intraparticle convection is usually signiﬁcant only in extremely large pores where ψp ∼ 1. 8.3.3 Rate Models with Non-Linear Isotherms

Analytical solutions of rate models with non-linear isotherms are available only for special cases and particular rate mechanisms. General cases, especially those involving competitive adsorption, are best handled by direct computer simulation [47–50]. However, a number of approximate analytical solutions are available to predict frontal analysis for conditions that are relevant to bioprocess columns. Of particular interest is the prediction of the dynamic binding capacity (DBC) which was deﬁned in Chapter 2 as the amount of protein held in the column before the efﬂuent concentration reaches the breakthrough value corresponding to a speciﬁed percentage of the feed concentration. A value of 10% of the feed concentration is typical, but lower values are often used, particularly if the adsorption process is designed to remove a toxic or otherwise objectionable impurity. Neglecting the leakage from the column before this value is reached, the DBC, normalized by the column volume, is approximated by the following equation: DBC =

CFVb CFQtb CF ε vtb = = Vc Vc L

(8.59)

where tb and Vb are the breakthrough time and corresponding load volume, respectively. In terms of dimensionless time, the ratio of dynamic and equilibrium binding capacities is given by the following equation: ˆ q 1 + φ F τ1 DBC CF = ˆ q EBC 1+φ F CF

(8.60)

where φ = (1 − ε)/ε and qˆF is the adsorbed concentration in equilibrium with the feed. When the ratio qˆF/CF is high, we have: DBC ∼ τ1 EBC

(8.61)

In the following we consider the prediction of the breakthrough curve in frontal chromatography assuming, for simplicity, that the adsorption capacity is high, so that qˆ ∼ q. In this case the following Thomas solution [51] is obtained when axial dispersion is neglected and the rate equation has the form

8.3 Modeling and Prediction of Column Efﬁciency

C (qref − q ) − Rq (Cref − C ) ∂q = ka (1 − R ) ∂t

(8.62)

where Cref is the concentration of a reference liquid phase and qref is the corresponding adsorbed equilibrium concentration. At equilibrium, Equation 8.62 reduces to the result: q C qref Cref q C 1− qref Cref

1− R=

(8.63)

which is the constant separation factor isotherm (cf. Equation 5.8). For frontal analysis, if equilibrium is described by the Langmuir isotherm and Cref is equal to the feed concentration, we obtain the relationship (cf. Equation 5.7): R=

1 1 + KCF

(8.64)

Equation 8.62 can be derived for an equivalent ion exchange process by assuming that the kinetics is limited by the rate of the exchange reaction [24, 51]. This is unrealistic for most protein chromatography processes, which are typically diffusion controlled. However, despite this limitation the Thomas solution still provides a valuable approximation, provided the rate constant ka is treated as an empirical rate coefﬁcient. The solution for the breakthrough curve is expressed in terms of the J function and is given by the following equation: J (Rn, nτ 1 ) C = CF J (Rn, nτ 1 ) + e − n (R −1)(1−τ1 )[1 − J (n , Rnτ 1 )]

(8.65)

where n=

(1 − ε ) ka qF L (1 − R ) ε v

(8.66)

and τ1 is as previously deﬁned. For the Langmuir isotherm (Equation 5.4) the expression for n becomes: n=

(1 − ε ) ka qm L εv

(8.67)

It should be noted that the solution is valid for both positive and negative concentration steps at the column inlet. For a negative step with a column initially saturated with concentration C0, the same equations apply with C0 and q0 = q*(C0) replacing Cref and qref, respectively. Figure 8.12 shows calculated breakthrough curves for R = 0.1, corresponding to a rather favorable isotherm, and different values of n. Compared to the results in Figure 8.9 for the linear isotherm case, it can be seen that the Thomas solution predicts much steeper breakthrough curves for the same number of transfer units. This occurs as a result of the favorable nature of the isotherm when R < 1. For

259

8 Effects of Dispersion and Adsorption Kinetics on Column Performance 1 Exact Constant pattern 0.8

C/CF

260

n=1 0.5 2 5

0.3

10 20 50 0 0

0.5

1

1.5

2

1

Figure 8.12 Breakthrough curves calculated using the Thomas model (Equation 8.65) and with the corresponding constant pattern solution (Equation 8.68) for R = 0.1 and different values of n. The two solutions are essentially coincident for n > 5.

local equilibrium conditions, this leads to a shock front. When the adsorption rate is ﬁnite, the result is a self-sharpening pattern whose spreading results from the competing effects of adsorption equilibrium, which tends to give a sharp front and dispersive factors, which tend to broaden the front. The opposite takes place when R > 1. This occurs for adsorption with an unfavorable (upwardly concave) isotherm or for desorption with a favorable isotherm. In either of these cases, both effects lead to broad proﬁles. As can be seen in Figure 8.12, for sufﬁciently large values of n the Thomas solution is very well approximated by a constant pattern solution shown by the dashed lines and given by the following equation [10]: C 1 = CF 1 + exp [(1 − R ) n (1 − τ 1 )]

(8.68)

For the conditions shown in Figure 8.12, the exact solution and the constant pattern approximation are essentially indistinguishable for n > 5. In general, however, the degree to which the constant pattern solution approaches the exact solution for a given n is also a function of R. As the name implies, the constant pattern solution is obtained by assuming that the concentration proﬁle in the column attains a constant shape that travels through the column at a uniform speed, which is equal to the shock velocity determined for local equilibrium conditions. The existence of a constant pattern as a limit of the Thomas solution has been demonstrated by Gilliland and Baddour [52]. However, it should be noted that the approach to a constant pattern is not limited to the Thomas model and is

8.3 Modeling and Prediction of Column Efﬁciency

the asymptotic behavior obtained when a positive concentration step is applied to a column with a favorable isotherm, that is, with d2q*/dC2 < 0 (see e.g. [53]). Physically, this limit is attained when the self-sharpening tendency caused by the favorable isotherm and the spreading due to dispersive factors are exactly balanced. When this occurs, the concentration proﬁle acquires a constant shape that travels through the column at a constant velocity. Thus, analytic solutions for the breakthrough curve can be found for any arbitrary favorable isotherm and various rate mechanisms in this asymptotic limit. For these conditions, the liquid phase concentration can be expressed as a function of a single variable, t′, deﬁned as follows: z vsh

t′ = t −

(8.69)

where vsh =

v 1+φ

qF CF

(8.70)

φ = (1 − ε)/ε. Since in the constant pattern limit we have C = C(t′) and ¯ q =¯ q (t′), Equation 7.5 can be written as:  1 − v  dC + φ dq = 0   vsh  dt ′ dt ′

(8.71)

which yields q qF = C CF

(8.72)

This relationship amounts to an ‘operating line’ for the constant pattern where the adsorbed and liquid phase concentrations are related through the constant ratio qF/CF. In turn, since this ratio is known from the isotherm, the constant pattern proﬁle can be found by direct integration of the rate equation. From Equation 7.2, in general we have: dq = f (C , q ) dt ′

(8.73)

where f(C, ¯ q ) is the rate equation describing the kinetics of the adsorption process. Combining this equation with Equation 8.72 and integrating we obtain the following general result: t=

qF CF

∫

dC +b q  f C, F C  CF 

(8.74)

where b is an integration constant. The latter can be obtained by material balance with the following procedure. With reference to Figure 8.13, assuming that the ﬂow rate is constant, it can be seen that the total shaded area is proportional to the difference between the amount of solute loaded and the amount that leaked

261

8 Effects of Dispersion and Adsorption Kinetics on Column Performance Area to left of t b tb

(

DBC

)

CF C dt

0

CF

Total area

C

262

(CF

EBC CF

C)dt = tdC

0

0

Cb tb Time, t

Figure 8.13 Diagram of a breakthrough curve for a feed with concentration CF. Cb and tb are the assumed breakthrough concentration and the corresponding breakthrough time, respectively. The total

shaded area is proportional to the column equilibrium binding capacity (EBC). The shaded area up to tb is proportional to the amount of solute retained in the column at breakthrough.

from the column. As t → ∞, this difference must equal the equilibrium capacity of the column. Thus, we obtain the following relationship: CF

L

∫ tdC = u [εC

F

+ (1 − ε ) qF ]

(8.75)

0

The ﬁnal result is obtained by combining Equations 8.74 and 8.75 which yields the following expression for the integration constant b: b=

L q q 1 + φ F  − F2 v CF  CF

CF



dC

∫  ∫ f (C , q C C 0

F



 F )

dC

(8.76)

Table 8.2 provides a summary of constant pattern breakthrough curves obtained with the constant separation factor isotherm and various rate mechanisms useful for the prediction of the DBC. These equations neglect axial dispersion. In dimensionless form, each rate mechanism gives rise to a different mathematical result for the breakthrough curve with a different expression for n. Only a numerical solution is available for the pore and solid diffusion mechanisms, except in the rectangular isotherm case (R = 0), for which exact solutions are available. Figure 8.14 illustrates the behavior of the solutions presented in Table 8.2 for a moderately favorable isotherm (R = 0.5), each plotted against the corresponding n. The differences among the breakthrough curves predicted for different mechanisms are obviously relatively small. The differences are even smaller for predicting the DBC at 10% of breakthrough. This important result suggests that even a fairly simple model, for example, the LDF approximation, will serve as a useful tool for design provided that the number of transfer units n is estimated according

Mechanism

External ﬁlm

LDF approximation

Pore diffusion

Solid diffusion

Item

A

B

C

D

Numerical solution for 0 < R < 1

15φDe L rp2 v

15φDsqF L rp2CF v

n=

n=

2

1

} π2 C 6 ∞ 1 = 1 − 2 ∑ 2 exp − j 2 0.64 − n (1 − τ 1 ) π i =0 j CF 15  

{

Analytical solution for R = 0

Numerical solution for 0 < R < 1

 2 15   C 3 C 3  15 C 3 1  5π 5 + − = n (1 − τ 1 ) ln 1 +  1 −  +  1 −   − tan −1   1 −  + 2   CF   CF   3 2 3 2 3 3  CF      

1

Analytical solution for R = 0

1  1 − C CF  ln 1 − R  (C CF )R 

n (1 − τ 1 ) = 1 +

φkqF L CF v

R 1  (1 − C CF )  ln   1 − R  C CF 

n=

n (1 − τ 1 ) = −1 +

3φk f L rp v

Constant pattern solution

n=

Number of transfer units

Constant pattern expressions for the breakthrough curve with the Langmuir or constant separation factor isotherm (Equation 5.8) with R < 1. Adapted from [10].

Table 8.2

8.3 Modeling and Prediction of Column Efﬁciency 263

8 Effects of Dispersion and Adsorption Kinetics on Column Performance 1.0

1.0

(b)

(a) 0.9

= DBC10% /EBC

0.8

0.6

C/CF 0.4

1,10%

264

External film LDF Pore diffusion Solid diffusion

0.2

0.0 -10

0.8 0.7 0.6 0.5 0.4

External film

0.3

LDF Pore diffusion Solid diffusion

0.2 -5

0

5

10

0

n( 1-1)

10

20

30

40

50

60

70

n

Figure 8.14 Constant pattern breakthrough curves (a) and dynamic binding capacity (b) predicted by the models shown in Table 8.2 with R = 0.5.

to Table 8.2 for the actual dominant rate mechanism. Cases where multiple resistances are important and the adsorption isotherm is not extremely favorable (e.g. R > 0.5), can also be handled with a LDF rate coefﬁcient deﬁned according to the following approximate relationships: Parallel pore and solid diffusion k=

15De CF 15Ds + 2 rp2 qF rp

(8.77)

Pore and solid diffusion in bi-disperse particles rp2 qF 1 r2 = + s k 15De CF 15Ds

(8.78)

External ﬁlm and intra-particle diffusion in series rp qF 1 1 = + k0 k 3φk f CF

(8.79)

Correction factors can be included in these approximate relationships for more favorable isotherms and can be found in [10]. Even the effects of intra-particle convection can be accounted for with these relationships by simply introducing ˜ e, deﬁned by Equation 6.20, in lieu of De. the convection-enhanced diffusivity D Breakthrough curves calculated in this manner have been shown to be in good agreement with those derived from the numerical solution of the rigorous mass

8.3 Modeling and Prediction of Column Efﬁciency

transfer model [54, 55]. Note, however, that these simple relationships based on additivity of resistances usually break down when the isotherm is highly favorable. The case of a rectangular or irreversible isotherm, where R ∼ 0, is practically relevant in many protein chromatography applications, since protein adsorption isotherms for capture conditions are often highly favorable (see Chapter 5). For example, in the case of IgG binding on Protein A adsorbents, the afﬁnity constant is of the order of 1 × 108 M−1 (see Table 6.4). Based on the IgG molecular mass of 150 kDa, this value translates into K ∼ 660 ml/mg. In turn for a typical protein feed concentration of 1 g/l, this corresponds to R ∼ 0.001. Even with ion exchangers, when the ionic strength is low, protein binding is often extremely favorable. As discussed in Chapter 6, when pore diffusion is dominant, this behavior results in a shell progressive uptake mechanism, which is represented by the shrinking core model. The corresponding constant pattern breakthrough curve for this case is item C in Table 8.2. The general solution for non-constant pattern conditions, which occur, for example, for short columns and high velocities, and including the external ﬁlm resistance is also available from Weber and Chackraborty [56]. The constant pattern expression is as follows: 1 1 2 1  15  15 2 ln 1 + (1 − X )3 + (1 − X ) 3  − tan −1  (1 − X )3 +     3  2 3 3 n pore 5π 5 − − [ ln ( X ) + 1] + n film 2 3 2 (8.80)

n pore (1 − τ 1 ) = +

where X=

3φk f L C 15φDe L , n pore = , n film = rp v CF rp 2 v

(8.81)

At X = 0.1 (10% of breakthrough), Equation 8.80 yields the following result: DBC10% τ 1,10% = ∼1− EBC

1.03 + 1.30 n pore

n pore n film

(8.82)

where n pore 5 10 De = = n film Bi Sh D0

(8.83)

In most practical situations for ﬂow in packed columns, as discussed in Chapter 6, Sh > 10 and De/D0 < 1, where pore diffusion is dominant. As a result the ratio npore/nﬁlm is typically much less than unity so that the intra-particle resistance is dominant. As a consequence we have:

τ 1,10% =

1.03 DBC10% ∼1− EBC n pore

(8.84)

265

8 Effects of Dispersion and Adsorption Kinetics on Column Performance 1

1

(a)

(b)

0.8

= DBC10%/EBC

0.8

npore = 1

0.6

C/CF 0.4

1,10%

266

2 npore/nfilm = 0.1

0.2

0.6

0.4

npore/nfilm = 0

0.2

General sol.

5

General solution

Const. pattern

Constant pattern 0

0 0

0.5

1

1.5 1

2

2.5

3

0

2

4

6

8

10

12

n=15 DeL/rp2v

Figure 8.15 Breakthrough curves (a) and dynamic binding capacity (b) calculated for a rectangular isotherm with R = 0. The general and constant pattern solutions merge at nporeτ1 = 2.5(1 + npore/nﬁlm).

The validity of Equations 8.80 and 8.84 is limited to constant pattern n pore  5 . The general conditions, which are established when n pore τ 1 >  1 +  2 n film  solution given in [56] is required for smaller values of the dimensionless time. The corresponding calculations require a trial and error procedure, which is cumbersome. However, the following approximation valid up to nporeτ1 = 2.5, derived from an empirical ﬁt of the numerical results with X = 0.1 and with negligible ﬁlm resistance (npore/nﬁlm = 0), can be used in lieu of the more complicated numerical calculation [18]:

τ 1,10% =

DBC10% ∼ 0.364n pore − 0.0612n 2pore + 0.00423n 3pore EBC

(8.85)

The corresponding expression for X = 0.01 (i.e. 1% breakthrough) can be found in Chapter 10. Examples of breakthrough curves and DBC calculated for R = 0 with the general and constant pattern solutions are shown in Figure 8.15a and 8.15b for npore/nﬁlm = 0.1 and 0, respectively. Note that in each case the general solution merges with the constant pattern solution when nporeτ1 = 2.5(1 + npore/nﬁlm). At npore = 5, the two solutions are practically indistinguishable. An interesting observation from Figure 8.15b is that the dynamic binding capacity is about 90% of the equilibrium binding capacity when npore ∼ 10. Hence, as a general rule when the isotherm is very favorable, as in the case of many protein capture processes, a small number of transfer units is sufﬁcient to obtain near ideal chromatography conditions.

8.3 Modeling and Prediction of Column Efﬁciency

Example 8.4

Constant pattern breakthrough curve with SMA model

Obtain the constant pattern breakthrough curve based on the LDF approximation for the adsorption of a protein on an ion exchange column when equilibrium is described by the SMA model (Equation 5.22) with the following parameters: q0 = 567 mM, Ke = 0.0106, z = 6.0, σ = 53.6. The feed contains 0.1 mM protein and 100 mM Na+ and is fed at 300 cm/h. The column is 10 cm long with ε = 0.3. The effective pore diffusivity is De = 7.0 × 10−8 cm2/s and the particle diameter is 100 µm. Solution – Adsorption equilibrium is described by the following equation: q*  q − ( z + σ ) q*  = K e 0  C CNa+  

z

(8.86)

Solving this equation with C = CF = 0.1 mM and CNa + = 100 mM gives qF = 3.15 mM. The shock velocity is thus vsh = v/(1 + φqF/CF) = 0.278/(1 + 2.33 × 3.15/0.1) = 0.00372 cm/s. Thus, tsh = L/vsh = 2686 s. In order to satisfy a material balance, the constant pattern breakthrough curve will be centered around this time. From Equation 8.77, the LDF rate coefﬁcient for pore diffusion is given by the expression: k=

15De CF rp2 qF

(8.87)

Thus, we obtain k=

15 × 7 × 10 −8 0.1 = 1.33 × 10 −3 s−1 2 (50 × 10 −4 ) 3.15

0.1

C (mM)

0.08

0.06

0.04

0.02 LDF with SMA model Pore diff. model with R = 0 0 0

1000

2000

3000

4000

5000

6000

7000

8000

Time (s)

Figure 8.16 Breakthrough curves for Example 8.4 based on the SMA model with LDF approximation and pore diffusion model with R = 0.

267

8 Effects of Dispersion and Adsorption Kinetics on Column Performance

The operating line for the constant pattern is given by: C=

CF 0.1 q q= q= qF 3.15 31.5

The constant pattern is obtained by numerical integration of Equation 8.74, which, in this case, takes the following form: t=

1 qF dC +b ∫ k CF q* − q

(8.88)

q* is obtained at each increment solving Equation 8.86 by trial and error with C = q/31.5. Finally, the integration constant b is obtained from Equation 8.75, which gives b = 1151 s. A plot of the breakthrough curve is shown in Figure 8.16, where it is compared with the solution obtained from Equation 8.80 for R = 0. The discrepancy is signiﬁcant mainly because the isotherm, while favorable, is not rectangular in this example, which, in turn affects the shape of the breakthrough curve.

Example 8.5 Comparison of experimental and predicted breakthrough curves Compare the experimental data for breakthrough of 1 mg/ml lysozyme in a 2 cm long SP-Sepharose-FF column operated at 300 cm/h superﬁcial velocity. Experimental data from [18] are given in Figure 8.17 for lysozyme in a buffer

1

0.8

C (mg/mL)

268

0.6

0.4 Column data 0.2

Model fit with PSD Model fit with avg. particle size

0 0

100

200

300

400

500

Protein loaded (mg/mL)

Figure 8.17 Experimental and predicted breakthrough curves for 1 g/l lysozyme in a 2 cm long SP-Sepharose-FF laboratory column loaded at 300 cm/h. The dashed line

is predicted from Equation 8.80. The solid line is predicted taking into account the particle size distribution using the equations in [57]. Adapted from [18].

8.3 Modeling and Prediction of Column Efﬁciency

containing 20 mM Na2HPO4 adjusted to pH 6.5 with phosphoric acid at 298 K. The isotherm is described by the Langmuir model (Equation 5.4) with qm = 232 mg/ml and K = 128 ml/mg. The mean particle diameter is 100 µm. Solution – For these conditions, R = 1/(1 + KCF) = 0.00775. Since this value is much less than 0.1, the rectangular isotherm approximation is appropriate. For lysozyme, assuming a solution viscosity of 1 mPa s, we have D0 = 1.1 × 10−6 cm2/s. Following the methods in Chapter 6, we obtain Sh = 34. Based on Example 8.3, the intra-particle effective pore diffusivity is expected to be De ∼ 2.8 × 10−7 cm2/s. Thus, from Equation 8.83 we have: npore 10 2.7 × 10 −7 = = 0.072 n film 34 1.1 × 10 −6 Since this ratio is much less than unity, the external resistance is negligible. Thus, the number of transfer units is approximately: npore =

15φDe L 15 × 2.3 × 2.7 × 10 −7 2 = = 2.7 2 rp2 v 0.28 (50 × 10 −4 )

A comparison of the experimental data with Equation 8.80 is shown in Figure 8.17 for De = 2.5 × 10−7 cm2/s. The predicted curve overestimates the rate of approach to equilibrium for long separation times. As can be seen in the ﬁgure, the prediction can be improved by considering explicitly the particle size distribution (PSD) of the stationary phase as described in [57]. This approach recognizes that the largest particles in the PSD will be far from equilibrium long after the smallest ones have reached complete saturation. In batch adsorption this results in rates that are faster initially and slower for long periods, than would be predicted using the average particle size. A similar phenomenon affects the breakthrough curve resulting in a long tail as the feed concentration is approached. It is worth noting that the PSD also affects the column behavior when the isotherm is linear or when intra-particle transport is dominated by solid diffusion (see for example [13]). On the other hand, in these cases the effects are much less pronounced than for pore diffusion with a highly favorable isotherm so that using the volume-average particle size, calculated from Equation 3.16, is usually adequate.

Example 8.6 Correlation of dynamic binding capacity for IgG binding on a Protein A adsorbent The data shown in Figure 8.18 from Malmquist [58] are for IgG binding on a MabSelect column (GE Healthcare) and are presented as dynamic binding capacity at 10% of breakthrough as a function of the residence time, L/u, based on an empty column. Determine the effective pore diffusivity assuming that the average particle radius is 42.5 µm.

269

8 Effects of Dispersion and Adsorption Kinetics on Column Performance 1

= DBC10% /EBC

0.8

0.6

0.4

1,10%

270

Experimental 0.2

General solution Constant pattern

0 0

2

4

6

8

10

Residence time, L/u (min)

Figure 8.18 Comparison of experimental and predicted DBC data for IgG adsorption on a MabSelect (GE Healthcare) Protein A column. Experimental data are from [56]. The 8 lines were calculated from Equations 8.84 and 8.85 with De = 3.0 × 10−8 cm2/s.

Solution – Assuming that the protein adsorption isotherm is rectangular (R ∼ 0), the DBC curve is represented by Equations 8.84 and 8.85. The best ﬁt is obtained with De = 3.0 × 10−8 cm2/s. As seen in Figure 8.18, for these conditions the constant pattern solution is applicable for residence times greater than about 3 min while the general solution is needed at shorter times. Note that at a residence time of about 8 min, the DBC is about 90% of the equilibrium binding capacity (EBC). For these conditions, the number of transfer units is given by the following equation: npore =

15 × (1 − 0.3) × 3.0 × 10 −8

(42.5 × 10 −4 )

2

(8 × 60 ) = 8.4

8.3.4 Rate Models for Competitive Adsorption Systems

As previously noted, a general analytical treatment of dispersion effects for multicomponent competitive adsorption systems is not available. Thus, in the general case, direct numerical solutions of the conservation equations need to be used. Various methods are available and an extensive review has been provided by Guiochon et al. [47]. However, constant pattern solutions for shock transitions can be determined either numerically or, in some special cases, analytically. Rhee and Amundson [59] introduced the concept of the shock layer thickness, which is deﬁned as the region over which the actual concentration proﬁles are spread out around

8.3 Modeling and Prediction of Column Efﬁciency

Shock front

CiI

Ci

Ci+1I

Distance, z

Figure 8.19 Transition zone or ‘shock layer’ between adjacent bands in the isotachic train for displacement development. The dashed line indicates the position of the shock front.

a shock. They provide analytical solutions that include both axial dispersion and external ﬁlm resistance for the special case of a multi-component Langmuir isotherm. Cooney and Strusi [60] approached essentially the same problem, but neglected axial dispersion and used the LDF model to describe the rate of adsorption. These authors provided numerical solutions for frontal analysis with two components obeying the Langmuir isotherm. However, analytical solutions are also possible in the special case where the LDF rate coefﬁcients are the same for both components. This assumption is reasonable for the separation of closely related species whose transport processes are likely to be similar. The method of Cooney and Strusi can be readily extended to displacement development as shown below. The main assumption is that there has been enough time to develop the isotachic train and that the shock layer is not so thick that it erodes the proﬁles to the point that the isotachic concentrations are no longer achieved. With reference to Figure 8.19, we consider the shock wave that deﬁnes the transition between bands i and i + 1, where the components have been numbered again in order of decreasing afﬁnity for the stationary phase. The multi-component Langmuir isotherm (Equation 5.26) is assumed to describe the adsorption equilibrium. The shock velocity is common to all waves in the isotachic train and is the same as the velocity of the displacer shock front vsh = v (1 + φq1*/C1D ), while the concentration of the displaced components are given by Equation 7.86, which yields: CiI = C1D −

K 1  1− i  K i  K1 

(8.89)

The remaining problem is to determine the width of the mixed transition zone and the shapes of the concentration proﬁles. The starting point is Equation 8.71 which can be extended to each component in the transition zone yielding:

271

272

8 Effects of Dispersion and Adsorption Kinetics on Column Performance

 1 − v  dC + φdq = 0 i   i vsh 

(8.90)

This equation is integrated twice, ﬁrst from an arbitrary position in the transition zone upstream to CiI and then again from 0 to CiI . Dividing the results of these integrals gives the following result: qi qi* = Ci CiI

(8.91)

where qi* = qi*(CiI ). This equation is analogous to Equation 8.72 for the onecomponent case and serves as the operating line relating liquid and adsorbed phase concentrations along the constant pattern. Using the LDF approximation, the rate equation is given by the following: dqi = ki [qi*(Ci , Ci +1 ) − qi ] dt ′

(8.92)

Combining this result with Equation 8.91, we obtain the following relationships for each component:  qi* dCi q*  = ki qi*(Ci , Ci +1 ) − i I Ci  CiI dt ′ C i  

(8.93)

  qi*+1 dCi +1 q* = ki +1 qi*+1(Ci , Ci +1 ) − iI+1 Ci +1  CiI+1 dt ′ C i +1  

(8.94)

The relationship between Ci+1 and Ci can be found by dividing the last two equations by each other, which gives: dCi +1 ki +1 qi*(CiI ) CiI+1 = dCi ki CiI qi*+1(CiI+1 )

qi*+1 C i +1 CiI+1 q* qi*(Ci , Ci +1 ) − i I Ci Ci

qi*+1(Ci , Ci +1 ) −

(8.95)

In the general case, integration of Equation 8.95 requires a numerical algorithm and solutions for a range of values of ki+1/ki can be found in Cooney and Strusi [58]. However, as shown by these authors, when ki = ki+1, the following linear relationship is obtained: Ci +1 = CiI+1 −

CiI+1 Ci CiI

(8.96)

The ﬁnal result for the constant pattern Ci(t′) is thus obtained by substituting Equation 8.96 in Equation 8.93 and integrating with the integration constant determined by material balance. Note that when Equation 8.96 is substituted in the expression for qi*(Ci , Ci +1 ), we obtain the following results: qi* =

qmK iCi = 1 + K iCi + K i +1Ci +1

qmK iCi CI 1 + K iCi + K i +1 CiI+1 − i +I1 Ci    Ci

(8.97)

8.3 Modeling and Prediction of Column Efﬁciency

The latter can also be written as: qi* =

AiCi 1 + BiCi

(8.98)

where qm K i 1 + K i +1CiI+1 CI K i − i +I1 K i +1 Ci Bi = 1 + K i +1CiI+1 Ai =

(8.99)

Equation 8.98 has the form of a Langmuir isotherm but with parameters deﬁned by Equation 8.99. Thus, the ﬁnal result for Ci(t′) is the same as that obtained for the one-component case (cf. item B in Table 8.2) but with Ri = 1 (1 + BiCiI ). In dimensional form this is written as: t=−

 1 − Ci CiI  1 1 ln  +b ki 1 − Ri  (Ci CiI )Ri 

(8.100)

where b is an integration constant which is obtained by material balance.

Example 8.7 Shock layer thickness and the constant pattern concentration proﬁles in the mixed zone in displacement development Determine the concentration proﬁles in the mixed transition zone between components 2 and 3 in the isotachic train in Example 7.8 if the adsorbent particles are 30 µm in diameter, the column length is 10 cm, the superﬁcial velocity is 240 cm/h, and the effective pore diffusivity is 2.5 × 10−7 cm2/s. Solution – From Example 7.8 or Equation 8.89 we have C2I = 1.5 and C3I = 1.0 in consistent units. Thus, from Equation 8.99 1.0 0.667 1.5 = 0.333 B2 = 1+ 0.667 × 1.0 1−

and R2 =

1 1 = = 0.667 I 1 + B2C 2 1 + 0.333 × 1.5

The LDF rate coefﬁcient is estimated as k=

15De C1D 15 × 2.5 × 10 −7 2 = = 3.47 × 10 −2 s−1 2 rp2 qi* (15 × 10 −4 ) 96

273

8 Effects of Dispersion and Adsorption Kinetics on Column Performance 2 Local equilibrium LDF model

C2I

1.5

Ci

274

C3I

1

Displacer front

0.5

0 0

10

20

30

40

CV Figure 8.20 Prediction of transition between components 3 and 2 in displacement development according to Example 8.7. The thin lines are based on the local equilibrium model. The thick lines are based on the approach of Cooney and Strusi [60].

Equation 8.100 is plotted versus CV = ut/L in Figure 8.20 with b = 3360 s. C3 is obtained from Equation 8.96, which gives C 3 = C 3I −

C 3I 1.0 C 2 = 1.0 − C2 C 2I 1.5

The thickness of the shock layer is around 5 CV for this example. Note that component 2 will also overlap with the displacer by the thickness of the corresponding mixed transition zone determined in an analogous manner.

References 1 Yau, W.W., Kirkland, J.J., and Bly, D.D. (1979) Modern Size Exclusion Liquid Chromatography, John Wiley & Sons, Inc., New York. 2 Kirkland, J.J., Yau, W.W., Stoklosa, H.J., and Dilks, C.H. (1977) J. Chromatogr. Sci., 15, 303. 3 Bidlingmeyer, B.A., and Warren, F.V. (1984) Anal. Chem., 56, A583. 4 Foley, J.P., and Dorsey, J.G. (1983) Anal. Chem., 55, 730. 5 Yau, W.W., and Kirkland, J.J. (1992) J. Chromatogr., 556, 111.

6 Lightfoot, E.N., Coffman, J.L., Lode, F., Yuan, Q.S., Perkins, T.W., and Root, T.W. (1997) J. Chromatgr. A, 760, 139. 7 Teeters, M.A., Root, T.W., and Lightfoot, E.N. (2002) J. Chromatogr. A, 944, 129. 8 Zochling, A., Hahn, R., Ahrer, K., Urthaler, J., and Jungbauer, A. (2004) J. Sep. Sci., 27, 819. 9 Roper, D.K., and Lightfoot, E.N. (1995) J. Chromatogr. A, 702, 69. 10 LeVan, M.D., and Carta, G. (2007) Adsorption and Ion Exchange, Section 16, in Perry’s Chemical Engineers’

References

11 12 13 14

15 16 17 18 19 20

21 22 23 24

25 26 27 28 29 30 31 32 33

Handbook, 8th edn (ed. D.W. Green), McGraw-Hill. Carta, G. (1988) Chem. Eng. Sci., 43, 2877. Lenhoff, A.M. (1987) J. Chromatogr. A, 384, 285. Carta, G., and Bauer, J.S. (1990) AIChE J., 36, 147. Farkas, T., Sepaniak, M.J., and Guiochon, G. (1996) J. Chromatogr. A, 740, 169. Farkas, T., Sepaniak, M.J., and Guiochon, G. (1997) AIChE J., 43, 1964. Miyabe, K., and Guiochon, G. (1999) J. Chromatogr. A, 857, 69. Grushka, E. (1972) Anal. Chem., 44, 1733. Carta, G., Ubiera, A.R., and Pabst, T.M. (2005) Chem. Eng. Technol., 28, 1252. Martin, A.J.P., and Synge, R.L.M. (1941) Biochem. J., 35, 1359. Villermaux, J. (1981) Theory of linear chromatography, in Percolation Processes: Theory and Applications, NATO ASI, Series E, No. 33 (eds A.E. Rodrigues and D. Tondeur), Sijtoff & Noordoff, pp. 83–140. Van der Laan, E.Th. (1958) Chem. Eng. Sci., 7, 187. Czok, M., and Guiochon, G. (1990) Anal. Chem., 62, 189. Ma, Z., and Guiochon, G. (1991) Comput. Chem. Eng., 15, 415. Sherwood, T.K., Pigford, R.L., and Wilke, C.R. (1975) Mass Transfer, McGraw-Hill, New York, Chapter 10. Anzelius, A. (1926) Z. Angew. Math. Mech., 6, 291. Hiester, N.K., and Vermuelen, T. (1952) Chem. Eng. Progr., 48, 505. Tan, H. (1977) Chem. Eng., 84, 158. Klinkenberg, A. (1948) Ind. Eng. Chem., 40, 1970. Schneider, P., and Smith, J.M. (1969) AIChE J., 14, 762. Hsu, J.T., and Dranoff, J.S. (1987) Comput. Chem. Eng., 11, 101. Yiacoumi, S., and Tien, C. (1995) Comput. Chem. Eng., 19, 1041. Rosen, J.B. (1952) J. Chem. Phys., 20, 387. Rosen, J.B. (1954) Ind. Eng. Chem., 46, 1590.

34 Rasmuson, A., and Neretnieks, I. (1980) AIChE J., 26, 686. 35 Rasmuson, A. (1982) Chem. Eng. Sci., 37, 787. 36 Acrivos, A. (1956) Ind. Eng. Chem., 48, 703. 37 Carey, G.B., and Finlayson, B.A. (1975) Chem. Eng. Sci., 30, 587. 38 Costa, C., and Rodrigues, A.E. (1985) AIChE J., 31, 1645. 39 Saunders, M.S., Vierow, J.B., and Carta, G. (1989) AIChE J., 35, 53. 40 Langer, G., Roethe, A., Roethe, K.P., and Gelbin, D. (1978) Int. J. Heat Mass Transfer, 21, 751. 41 Gunn, D.J. (1987) Chem. Eng. Sci., 42, 363. 42 Van Deemter, J.J., Zuiderweg, F.J., and Klinkenberg, A. (1956) Chem. Eng. Sci., 5, 271. 43 Haynes, H.W., and Sarma, P.N. (1973) AIChE J., 19, 1043. 44 Carta, G., Gregory, M.E., Kirwan, D.J., and Massaldi, H.A. (1992) Sep. Technol., 2, 62. 45 Carta, G., and Rodrigues, A.E. (1993) Chem. Eng. Sci., 48, 3927. 46 Rodrigues, A.E., Lu, Z.P., Loureiro, J.M., and Carta, G. (1993) J. Chromatogr. A, 653, 189. 47 Guiochon, G., Felinger, A., Shirazi, D.G., and Katti, A.M. (2006) Fundamentals of Preparative and Nonlinear Chromatography, 2nd edn, Elsevier Acadmic Press, San Diego. 48 Finlayson, B.A. (1992) Numerical Methods for Problems with Moving Fronts, Ravenna Park,, Washingon. 49 Holland, C.D., and Liapis, A.I. (1982) Computer Methods for Solving Dynamic Separation Problems, McGraw-Hill, New York. 50 Villadsen, J., and Michelsen, M. (1978) Solution of Differential Equations Models by Polynomyal Approximations, Prentice Hall, Englewood Cliffs, NJ. 51 Thomas, H. (1944) J. Am. Chem. Soc., 66, 1664. 52 Gilliland, E.R., and Baddour, R.F. (1953) Ind. Eng. Chem., 45, 330. 53 Cooney, D.O., and Lightfoot, E.N. (1965) Ind. Eng. Chem. Fundam., 4, 233. 54 Carta, G. (1995) Chem. Eng. Sci., 50, 887.

275

276

8 Effects of Dispersion and Adsorption Kinetics on Column Performance 55 Liapis, A.I., and McCoy, M.A. (1992) J. Chromatogr. A, 599, 87. 56 Weber, T.W., and Chakraborti, R.K. (1974) AIChE J., 20, 228. 57 Carta, G., and Ubiera, A.R. (2003) AIChE J., 49, 3066. 58 Malmquist, G. (2000) Paper presented at the International Symposium of the

Separation and Analysis of Proteins, Peptides, and Polynucleotides, ISPPP2000, Ljubljana, Slovenia. 59 Rhee, H.K., and Amundson, N.R. (1972) Chem. Eng. Sci., 27, 199. 60 Cooney, D.O., and Strusi, A. (1972) Ind. Eng. Chem. Fundam., 11, 123.

277

9 Gradient Elution Chromatography 9.1 Introduction

This chapter is devoted to the theory and practice of gradient elution in protein chromatography applications. The general principles of gradient elution chromatography were presented in Chapter 2. The basic process involves a mobile phase modiﬁer (M), which is adsorbed less strongly than the components in the mixture that is to be separated. The modiﬁer modulates the strength of binding of the feed components thereby allowing control of the separation by varying its concentration at the column entrance. The role of the modiﬁer in liquid chromatography is thus similar to the role of temperature or pressure in gas chromatography. However, the effects of the modiﬁer are usually more complex than the effects of temperature and pressure since the modiﬁer itself can be adsorbed in competition with the feed components. Moreover, in general, changes in modiﬁer concentration can only be applied at the column entrance and propagate through the column at a ﬁnite rate, while changes in temperature and/or pressure can be more easily applied uniformly throughout the column. Gradient elution is commonly used in protein chromatography for various reasons. Firstly, the mixtures encountered in these applications are typically complex and often involve a broad range of adsorption strengths, with some components only weakly bound and others very strongly retained at a given mobile phase composition. For these mixtures, varying the mobile phase composition in such a way that adsorption becomes progressively weaker following the feed load, leads to the separation of weakly retained components early on in the gradient and of strongly retained species later when the eluting strength of the mobile phase is high. A second reason for the utility of gradient elution in protein chromatography is the extreme sensitivity of the retention of biopolymers to the exact composition of the mobile phase discussed in Chapter 3. Such extreme sensitivity is observed in nearly all branches of chromatography, but is especially pronounced in reversed phase chromatography (RPC), where minute changes in the volume fraction of the organic modiﬁer can sometimes shift the behavior of the protein from complete retention to no retention at all. Although usually not as extreme, the dependence of protein adsorption on modiﬁer conProtein Chromatography: Process Development and Scale-Up. Giorgio Carta and Alois Jungbauer © 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 978-3-527-31819-3

278

9 Gradient Elution Chromatography

centration is also very strong in ion exchange chromatography (IEC) and hydrophobic interaction chromatography (HIC) so that, in general, very precise control of the mobile phase composition is needed to operate isocratically. Gradient operation averts the need for such precise control allowing the chromatographic column to experience a range of mobile phase compositions where protein separation occurs. The result is a much more robust process where chromatographic elution can be monitored by following the changing composition of the mobile phase at the column outlet. The type of gradient implemented is generally dependent on the chemistry of interaction. Thus, gradients with increasing organic modiﬁer are used in RPC, gradients with increasing salt concentration are used in IEC, and gradients with decreasing salt concentration are used in HIC. pH gradients can also be used in IEC. Varying pH by traversing the protein pI will change the sign of the protein net charge, generally allowing the protein to elute faster at high pH and slower at low pH when using cation exchangers, with the opposite occurring with anion exchangers. While potentially powerful, pH gradients are less commonly used than salt gradients for various reasons. On one hand, pH gradients are usually more difﬁcult to control. On the other, crossing the protein pI will also cross the region where the protein solubility is usually minimal (see Chapter 1). For these conditions, the protein is often more prone to precipitation, aggregation and, potentially, unfolding. An advantage of pH elution is, however, that the separated components can emerge from the column at low ionic strengths, which may facilitate the next processing step. Approaches for implementing pH gradients by exploiting interactions with the stationary phase are discussed in Section 9.5. The shape of the applied gradient is also variable. Linear gradients, where the modiﬁer concentration at the column entrance is a linear function of time, are most commonly used. In RPC of hydrocarbons, such gradients tend to give equal resolution in the analysis of homologous series of components with increasing molecular mass. However, in the case of protein chromatography, non-linear gradients or even so-called step gradients can be advantageous. The latter are especially useful when the selectivity is high and the proteins can be separated with a simple bind–wash–elute sequence. On the other hand, difﬁcult separations, such as, for example, the resolution of protein charge variants, the separation of glycosylated isoforms, and the removal of deamidated species, often require shallow, smooth gradients. Predicting protein retention, band broadening, and resolution as a function of gradient slope and ﬂow rate then becomes critical. In general, gradient elution can be handled with the same approaches and equations as those outlined in Chapters 7 and 8, simply regarding the modiﬁer as an added components that is adsorbed in competition with the feed mixture. Thus, in the general case, a numerical solution of the relevant material balances, multicomponent equilibrium isotherm expressions, and rate equations is needed. In the case of protein chromatography, however, limiting relationships valid for the linear isotherm case have been developed and are useful in practice both in the analysis of laboratory data as well as for design. The applicability of the

9.2 General Theory for Gradient Elution with Linear Isotherms

linear theory to practical cases is made possible by the noted sensitivity of protein adsorption to mobile phase composition. While the proteins are strongly and competitively bound in the initial load step, the chromatographic separation of closely related species occurs mainly after the eluting strength of the mobile phase has reached a sufﬁciently high value for the protein bands to begin to migrate. As a result the adsorption isotherm reaches the linear limit quickly, thereby rendering the linear theory useful in practice.

9.2 General Theory for Gradient Elution with Linear Isotherms

Figure 9.1 illustrates the general set up of a gradient elution system. The feed mixture is ﬁrst loaded onto the column with a load pump or injector system. The mobile phase composition is then changed gradually with a slope β, which is deﬁned

A+B Modifier in Mobile phase from pumps

CMf

Feed from injector or feed pump CM 0

tG Time or volume

Chromatography column A CM

f

Modifier out B

CM,BR

Detector

CM,AR CM 0

To product collection

Time or volume

Figure 9.1 General scheme of linear gradient elution. Note that the separated components R R emerge at modiﬁer concentrations CM , A and C M ,B which are functions of the gradient slope β.

279

280

9 Gradient Elution Chromatography

in either time, volume, or CV units. Linear gradient elution (LGE) is obtained when the modiﬁer concentration increases linearly over time. Such linearity is not required but is typical for practical applications. In either case, the modiﬁer gradient is delayed in the column. If the modiﬁer adsorption isotherm is linear or the modiﬁer is not adsorbed, under ideal chromatography conditions the gradient at the outlet will be the same as that at the column entrance, merely shifted in time. Axial dispersion and ﬂow heterogeneities will of course broaden the gradient, although in many practical systems, since the modiﬁer is usually a small molecule, broadening of the gradient will be unaffected by mass transfer. Thus, for well-packed columns, the modiﬁer will often behave ideally. Note that in reality there are two gradients. One is the temporal gradient generated at the column entrance and the other is the axial gradient that exists at each time point between inlet and outlet. The axial gradient is responsible for band compression, which gives sharper peaks than those obtained in isocratic elution. However, if the modiﬁer is not adsorbed or the stationary phase can be considered to be completely saturated with the modiﬁer, the instantaneous axial gradient will be quite shallow unless the temporal gradient is very steep. For these conditions, which are typically needed for high-resolution applications, limited peak compression will take place. Note that in general, as shown in Figure 9.1, the separated R R components will emerge at modiﬁer concentrations CM , A and C M ,B that depend not only on the components’ interactions with the stationary phase but that are also functions of the gradient slope β. In the following we assume that retention of both the modiﬁer and the feed components is described by linear isotherms. Further, we assume that retention of the modiﬁer is not affected by the concentration of the feed components. This will occur when the modiﬁer is in excess and the adsorbent surface is essentially saturated with the modiﬁer. Such conditions are frequently met in protein chromatography. Based on these assumptions, we recognize that there are two types of characteristic velocities: one for the modiﬁer and the other for each of the feed components. Accordingly, we have the following expressions: vcM = vci =

v 1 + kM′

v , i = 1, 2,… M − 1 1 + ki′(CM )

(9.1) (9.2)

where kM′ and ki′(CM ) are the retention factors of the modiﬁer and of each feed components, respectively. These quantities are related to the slopes of the linear isotherms by the following equation: ki′ =

1−ε mi = φmi , i = 1, 2,… M ε

(9.3)

where φ = (1 − ε)/ε is the phase ratio and mi is the linear isotherm slope. If the modiﬁer is not adsorbed or the adsorbent surface is saturated, we have simply mM = εp. Finally, assuming a pulse injection of the feed followed by a linear

9.2 General Theory for Gradient Elution with Linear Isotherms

modiﬁer gradient at the column entrance we have the following boundary conditions: F CM = CM0 + βt , 0 < t ≤ tG

CiF =

Mi δ (t ) , i = 1, 2, … M − 1 Q

(9.4) (9.5)

where Mi is the amount injected, Q is the ﬂow rate, δ(t) is the delta function and

β=

CMf − CM0 tG

(9.6)

is the gradient slope where CM0 and CMf are the initial and ﬁnal modiﬁer concentrations, respectively, and tG is the duration of the gradient. A plot of the characteristics for linear gradient elution is shown in Figure 9.2. Based on Equation 9.1, the characteristics of the modiﬁer are parallel lines. However, the characteristics for the feed components are curves with increasing slope that corresponds to the increasing eluting strength of the mobile phase. As can be seen from Figure 9.2, the modiﬁer concentration at each point z and time t can be determined from the following equation: z CM ( z, t ) = CM0 + β t − (1 + kM′ )  v 

(9.7)

In turn, the characteristics of each component can be calculated by taking the differential of Equation 9.7, which yields the result: dCM dt β = β − (1 + kM′ ) dz dz v

(9.8)

Next, from Equation 9.2, we have: dt 1 + ki′(CM ) = dz v

(9.9)

Finally, by combining the last two equations we obtain: dCM β = [ki′(CM ) − kM′ ] dz v

(9.10)

The characteristics can be determined by integrating this equation deﬁning the path followed by each feed component on the z-t plane. The ﬁnal result is obtained by integrating from the column entrance to the outlet, which gives the following relationship:

βL γ = = v

R CM ,i

∫

0 CM

dCM , i = 1, 2,… M − 1 ki′(CM ) − kM′

(9.11)

R where γ = βL/v is a normalized gradient slope, and CM ,i is the modiﬁer concentration at which component i emerges from the column. Note that while γ is the same

281

Inlet Conc.

9 Gradient Elution Chromatography

CMf

CM0 tG vcM

vcB

Axial position, z Outlet Conc.

282

vcA

CM,BR

CMf CM,AR CM0 tR,M

tR,A

Figure 9.2 Characteristics of linear gradient elution of a two-component mixture. The thin solid lines are the characteristics of the mobile phase modiﬁer. The thick solid lines are the characteristics for feed components A

tR,B

Time

and B. The retention times tR,A and tR,B and the modiﬁer concentrations at which the two R R peaks elute, CM , A and C M ,B , are functions of the gradient slope and can be calculated using Equation 9.11.

for all components in a given run, the relationship between ki′ and CM is not. Thus, each component will follow a different path, eluting at a modiﬁer concentration that is a function of γ. Given the relationship between ki′ and CM, Equation 9.11 can be integrated to give either the CM value at which a protein will elute or the R value of β that will yield a desired CM ,i . In either case, Equation 9.7 can be used to calculate the time of elution. Accordingly, we have the following result: tR ,i =

0 R CM L ,i − C M + (1 + kM′ ) v β

(9.12)

Differentiate

A = L v

vs. C RM,i B

A

ki = kM +

vs. C RM,i

d dCRM,i

Plot LGE experiments varying gradient slope

CRM,i

Alternatively, as suggested by Yamamoto [1, 2], Equation 9.11 can be used to determine the relationship between ki′ and CM from a series of gradient elution experiments. This is especially useful for protein chromatography applications since isocratic elution experiments are often unreliable because of the difﬁculties encountered in maintaining an exactly constant mobile phase compositions. Moreover, gradient elution data are often available from early process development runs and can be used to deﬁne equilibrium parameters that are useful for process design. Taking the derivative of Equation 9.11 yields the following result: (9.13)

In turn, solving for ki′ gives the following relationship:  dγ  R ki′(CM ′ + R  ,i ) = kM  dCM ,i 

B

CM

Figure 9.3 General approach to the determination of retention factors as functions of the mobile phase composition from linear gradient elution experiments.

dγ 1 = R R dCM ,i ki′(CM ,i ) − kM′

283

1

9.2 General Theory for Gradient Elution with Linear Isotherms

−1

(9.14)

The procedure is shown schematically in Figure 9.3. Gradient elution experiments are ﬁrst conducted with a range of values of γ obtained by varying either β, L, R or v or any combinations thereof and the values of CM ,i are recorded for each R experimental condition. The derivative dγ dCM ,i is then evaluated graphically or numerically. Finally, ki′ for each component is obtained from Equation 9.14 R at CM = CM ,i . The theory presented above allows a prediction of the time of elution of the component peaks, tR,i, under ideal chromatography conditions. Provided the column efﬁciency is not too low, the actual peaks will be centered around these times, but, of course, spread by non-idealities. Even for systems where the isotherms are strictly linear, prediction of band broadening in gradient elution is complicated by the existence of the axial gradient, which leads to peak compression. The physical origin of this effect is illustrated in Figure 9.4 which shows a protein peak located at some point along the column length. Because of the axial gradient,

284

9 Gradient Elution Chromatography

Instantaneous axial gradient in modiﬁer concentration

vc

vc

Figure 9.4 Schematic of the instantaneous axial gradient in modiﬁer concentration established as a result of a temporal modiﬁer gradient imposed at the column entrance. vc′ and vc′′ are characteristic velocities downstream and upstream of the peak.

the characteristic velocity of the protein is higher at the rear of the peak where the eluting strength is greater, than it is at the front of the peak where the eluting strength is lower. As a result, the peak obtained in gradient elution is sharper than it would be if it were eluted isocratically. A peak compression factor, Cf,i, can thus be deﬁned as follows: C f ,i =

Bandwidth for gradient elution R Bandwidth for isocratic elution with CM = CM ,i

(9.15)

In this deﬁnition the actual bandwidth obtained in gradient elution is normalized by the bandwidth obtained if the protein peak were eluted isocratically at the modiﬁer concentration at which the peak actually emerges from the column. Thus, Cf,i varies between 0 and 1 and is a function of the gradient slope. In general, a steeper temporal gradient will be accompanied by a steeper axial gradient resulting in greater peak compression and lower values of Cf,i. Conversely, a shallower temporal gradient will result in a shallower axial gradient and values of Cf,i closer to unity. Based on Equation 9.15, if the peak eluted in gradient elution is reasonably symmetrical, band broadening can be related to column efﬁciency by the following equation obtained by analogy to Equation 8.10:

9.2 General Theory for Gradient Elution with Linear Isotherms 2

L  Wi  1   16  tRiso,i  C f ,i 2

H=

(9.16)

where Wi is the actual baseline width and tRiso,i is the retention time calculated R iso for isocratic elution at CM = CM ,i . Accordingly, tR ,i is given by the following equation: L [1 + ki′(CMR ,i )] v

tRiso,i =

(9.17)

In general, the column efﬁciency and, hence, H, will be different for different components. In practice, however, similar values can be expected when gradient elution is used to separate closely related species. By analogy to the case of isocratic elution, resolution between adjacent components A and B can also be deﬁned according to Equation 2.9, which gives: Rs =

tR ,B − tR , A 1 (WA + WB ) 2

(9.18)

The difference is that Rs is now also a function of the gradient slope. The latter can thus be adjusted to attain the desired degree of separation. Other than the direct numerical solution of the governing equations, there is no general theory to predict the peak proﬁles obtained in gradient elution even with linear isotherms. Useful empirical models are available for linear gradients with speciﬁc functional forms relating k′ to CM and are discussed in the following sections. Approximate solutions are also available for preparative injections with volume overload from Frey [3] and Carta et al. [4], which permit approximate predictions for more general cases. The solution of Frey, based on the linear driving force (LDF) approximation, is expressed in terms of the error function and is valid for feed injections of arbitrary duration and for arbitrary gradient shapes. An expression that is simpler to use is based on the solution of Carta et al. [4], which was derived based on the LDF model for periodic, rectangular feed injections, each followed by an arbitrary temporal gradient in mobile phase composition. This solution is given by the following equation:

( )

j 2n  jπ rF C k ′( t )  rF 2 ∞ 1 jrn     jϑ jπ rF + ∑ exp  − 2 2  sin = cos  − − 2 2   j +r   r CF k0′  2r π j =1 j 2r j +r  2r (9.19) where n=

(1 − ε ) kc L εv

(9.20)

φ kc dt k ′( t ) 0

(9.21)

t

ϑ=∫

285

286

9 Gradient Elution Chromatography

r= rF =

1 2π

tC

φ kc

∫ k ′(t ) dt

(9.22)

0

φkc tF π k0′

(9.23)

In these equations, n is the number of transfer units, kc is the LDF rate coefﬁcient based on a liquid phase driving force, k′(t) is the function that describes the temporal variation of the retention factor at the column entrance, k0′ is the value of k′ when the feed is loaded, and tF and tC = tF + tG are the durations of the feed injection and the total cycle time, respectively. As discussed in Chapter 8, when pore diffusion is the dominant rate mechanism, we have kc = 15De rp2 , where De is the effective pore diffusivity and rp the particle radius. For other rate mechanisms, kc can be approximated by Equations 8.77 to 8.79. It should be noted that Equation 9.19 is based on the assumption that changes in modiﬁer concentrations propagate instantly through the column. This is a good approximation only for shallow temporal gradients and when the characteristic velocity of the protein peaks is small compared to the characteristic velocity of the modiﬁer throughout the gradient. Accordingly, since the axial gradient is neglected, the peaks predicted by this equation will be somewhat broader than expected. A corrected value of n or kc accounting for peak compression can be used to account for this effect using the relationships in [3]. This correction is, however, generally small, especially when gradient elution is used to resolve components with similar retention. Integrated forms of Equation 9.19 can be found in [4] for the special case of linear gradients with either ion exchange or hydrophobic adsorption.

9.3 LGE Relationships for Ion Exchange Chromatography

Special relationships can be derived from Equation 9.11 for ion exchange (IEC) and hydrophobic adsorption (RPC or HIC). The critical assumption is that the adsorption isotherm may be approximated as linear while the protein migrates through the column. The ensuing discussion is adapted from the work of Yamamoto [1, 2, 5, 6] for ion exchange. In IEC the mobile phase modiﬁer is typically salt. As shown in Chapter 5, based on the stoichiometric displacement model [1, 7, 8], Equation 5.17, the protein retention factor in the linear limit of the isotherm is a function of the counter-ion concentration according to the following relationship: z

i q −z ki′ = ki′,∞ + φK e ,i  0  = ki′,∞ + Ai (CM ) i  CM 

(9.24)

where Ke,i is the equilibrium constant for the protein–counter-ion exchange, q0 is the total concentration of functional groups in the ion exchanger, zi is the effective

9.3 LGE Relationships for Ion Exchange Chromatography

charge of the protein, Ai = φK e ,i qozi , and ki,′ ∞ is the protein retention factor obtained when CM → ∞. Note that the units of Ai depend on zi and are, for instance, ( mM)zi if the modiﬁer concentration is in mM units. The limiting retention factor ki,′ ∞ is approximated by φεp, where εp is the macroporosity of the ion exchanger if the protein is not bound at high salt and, under these conditions, merely diffuses in and out of the intra-particle pore space without binding. If the pores in the stationary phase are sufﬁciently large and the counter-ion is present in excess, we have ki′,∞ ∼ kM′ , where kM′ is the retention factor for the salt modiﬁer. For these conditions, Equations 9.11 and 9.12 yield the following relationships: R 0 CM ,i =   Ai ( zi + 1) γ + (CM )

 Ai ( zi + 1) γ + (CM0 ) tR , i =  β

1 zi + 1 z + 1 i

zi + 1





1 zi + 1

− CM0

(9.25)

L + (1 + kM′ ) v

(9.26)

In many practical situations, where the feed components are strongly bound at z i +1 the initial modiﬁer concentration, the term (CM0 ) appearing in brackets in these equations is negligible. Thus, as a good approximation we have: 1

R CM ,i ∼ [ Ai ( zi + 1) γ ] zi +1

(9.27)

R In this case, a plot of logγ versus log (CM ,i ) yields a straight line with slope zi + 1 and intercept −log[Ai(zi + 1)], which can be conveniently used to determine Ai and zi from LGE experimental data. Equations 9.25 and 9.26 predict elution at earlier times and higher CM values as the normalized gradient slope increases. Thus, steeper gradients lead to earlier elution at higher salt concentrations. Yamamoto et al. [9] have derived an empirical expression for the peak compression factor based on extensive numerical calculations for conditions where Equation 9.27 is valid. A slightly modiﬁed form of their result is given by the following equations:

 Λi  C f ,i =  3.2Λ i  1 + 3.2 Λ

for Λ i < 0.25 for Λ i > 0.25

(9.28)

i

where Λi is a parameter that depends on the gradient slope and is given by: Λi =

R 1 1 + ki′ (CM ,i ) z i + 1 2 1 + kM′ zi

(9.29)

In many practical protein chromatography applications designed to separate closely related species with relatively shallow gradients we have, zi >> 1 and R ki′(CM ′ . For these conditions, Λi >> 1. In turn, Equation 9.28 predicts ,i ) >> kM that peak compression will be relatively unimportant, i.e. Cf,i ∼ 1, in these applications. An approximate prediction of peak proﬁles can be based on either the plate model, using an HETP or plate number corrected using Equation 9.28, or with

287

288

9 Gradient Elution Chromatography

the LDF model using either Equation 9.19 or the relationships of Frey [3]. For pulse injections, where the load volume is much smaller than the peak volume, the following relationship adapted from Yamamoto [2] can be used: Ci t = F CF ,i tRiso,i

N LGE N exp  − LGE (τ LGE − 1)2  3 2  τ LGE  2π (τ LGE )

(9.30)

where tRiso,i =

{

zi L L [1 + ki′ (CMR ,i )] = 1 + kM′ + Ai[ Ai ( zi + 1) γ ]zi +1 v v

τ LGE =

}

t − tR ,i + tRiso,i tRiso,i

(9.31) (9.32)

In Equation 9.30 NLGE is the effective plate number, which depends on peak compression and can be estimated as: N LGE =

L 1 H C 2f ,i

(9.33)

where H is the HETP determined for isocratic conditions. The corresponding baseline width, Wi, is given by: Wi = 4tRiso,iC f ,i

H L

(9.34)

Example 9.1 Estimate retention factor and mass action law parameters from LGE experiments with different gradient slopes The results in Figure 9.5 were reported in [10] for the LGE separation of mixtures of cytochrome c and lysozyme in a 1 × 10 cm SP-Sepharose-FF column with ε = 0.3 operated at 2 ml/min (150 cm/h). The injected volume was 0.1 ml. The mobile phase consisted of 10 mM Na2HPO4 buffer at pH 6.5 with a linear NaCl gradient from 100 to 500 mM using gradient slopes from 2.5 to 30 CV. Determine the parameters Ai and zi in the stoichiometric displacement model (Equation 9.24). Solution – From Figure 9.5 it can be seen that the resolution increases and the peaks emerge at lower conductivity values as the gradient duration increases, even though the ﬂow rate is the same. The retention factor of the modiﬁer (Na+ in this case) can be determined from the small negative peak visible in the conductivity trace. This peak was caused by the fact that the salt concentration of the protein sample was lower than that in the initial buffer. After correction for the extra-column volume, this peak emerges at 4.5 ml. Thus, kM′ = VR εVc − 1 = 4.5 (0.3 × 7.85) − 1 = 1.9 . This corresponds to εp ∼ 0.8, which is close to the actual intra-particle porosity. Gradient durations are given in column

9.3 LGE Relationships for Ion Exchange Chromatography 50 A 40

40 50

30

mS/cm

30

mS/cm

80

mAU

A

mAU

5 CV gradient

75

B

120

50

B

2.5 CV gradient

25

40

20

20 0

0 0

5

10

15

20

25

30

35

mL

0

10

20

30

10 CV gradient A

20 CV gradient 40

50

10 60 50

B

A

40

42

40

mL

50

50

B

56

10 40

40

mAU

30

14

30

20

mS/cm

28

mS/cm

mAU

30

10 20

20 0

0

-10

10 0

20

40

60

0

80

50

mL

100

10 150

mL

Figure 9.5 LGE separation of cytochrome c/lysozyme mixtures varying the gradient slope at constant ﬂow rate according to Example 9.1. Reproduced from [10].

volumes and it is possible to work the problem entirely in these units. However, in time units, the corresponding gradient durations are given by tG = CVG × Vc/Q. The CMR ,i values at which each peak emerges from the column are read on the conductivity trace assuming that the mS/cm values are proportional to Na+ concentrations. (If this is not the case, a calibration curve can be constructed to enable this conversion – see Figure 4.9). The ﬁrst major peak (identiﬁed as A) is cytochrome c and the second (identiﬁed as B) is lysozyme. Note that a few impurity peaks are also detected but are neglected in this analysis. The results are as follows:

Gradient

β (mM/min)

γ (mM)

R (mM) C M,A

R (mM) CM,B

2.5 CV 5.0 CV 10 CV 20 CV

40.7 20.4 10.2 5.09

48 24 12 6

365 332 293 264

430 394 355 334

289

9 Gradient Elution Chromatography 100

140

(a)

(b) 120 Cytochrome c A=5.59 x 1013 z=5.33

Lysozyme

100

10 Lysozyme A=7.09 x 1015 z=5.97

Reduced HETP, h

290

80 60

Cytochrome c

40 20 0

200

300

CM,iR (mM)

400

500

0

1000

2000

3000

Reduced velocity, v'

Figure 9.6 Results for Example 9.1 (a) and for Example 9.2 (b). Adapted from [10].

A log–log plot of the results (γ versus CMR ,i ) according to Equation 9.27 is linear as seen in Figure 9.6a. The corresponding values of Ai and zi are 5.59 × 1013 and 5.33, respectively, for cytochrome c and 7.09 × 1015 and 5.97, respectively, for lysozyme. In both cases, the Ai values are for use with CM in mM units. Note that neither retention times nor the value of kM′ are needed for this analysis. However, the latter is needed to predict ki′ and retention in conjunction with the calculated values of Ai and zi.

Example 9.2 ﬂow rates

Estimate the HETP from LGE experiments at different

The results shown in Figure 9.7 were reported in [10] for the LGE separation of mixtures of cytochrome c and lysozyme for the same conditions as those in Example 9.1 but at different ﬂow rates with a constant 20 CV gradient. Determine the HETP as a function of ﬂow rate. Solution – Since the same gradient duration was maintained in CV units, the normalized gradient slope γ = 6 is the same for all these runs. However, it can be seen that the peaks broaden and the resolution decreases as the ﬂow rate is increased. Thus, the column efﬁciency is lower at higher ﬂow rates mainly as a result of mass transfer limitations. Since the peaks are reasonably symmetrical, the HETP can be estimated from the baseline width, W, according to Equation 9.16. Note that tRiso,i and Cf,i are evaluated for each component

9.3 LGE Relationships for Ion Exchange Chromatography 50

60 50

1 mL/min 40 40

40

30 20

30

20

mS/cm

30

mAU

30

mS/cm

mAU

40

10

10

20

20 0

0 -10 0

50

100

10 150

-10

50

30

mL

40

0

2 mL/min 40

mAU

10

50

4 mL/min 40

15 30 10 5

20

10 150

mS/cm

30

100

20

mS/cm

20

50

mL

25

30

mAU

50

50

0.5 mL/min

20

0

0 0

50

10 150

100

-5 0

50

mL

100

10 150

mL

Figure 9.7 LGE separation of cytochrome c/lysozyme mixtures using a variable ﬂow rate with a constant normalized gradient slope according to Example 9.2. Reproduced from [10].

from Equations 9.28–9.31 using the kM′ , Ai, and zi values determined in Example 9.1. For these conditions CMR ,A = 264 and C MR ,B = 320mM , with corresponding values of k A′ = 6.98 and kB′ = 7.78 . We also have ΛA = 0.5(1 + 6.98) (5.33+1)/5.33(1+1.0) = 1.63 and Cf,A = 0.84 for cytochrome c, and ΛB = 0.5(1 + 7.78) (5.97 + 1)/5.97(1 + 1.0) = 1.77 and Cf,B = 0.85 for lysozyme. These results show that band compression was relatively unimportant for these experiments. The ﬁnal results are as follows:

Q (ml/min)

WA (min)

WB (min)

iso (min) tR,A

iso (min) tR,B

HA (cm)

HB (cm)

0.5 1 2 4

18 11 7.0 4.8

48 24 12 6

38 19 9.4 4.7

41 21 10 5.2

0.20 0.30 0.49 0.91

0.20 0.34 0.65 1.21

291

9 Gradient Elution Chromatography

In Example 9.2, the HETP increases linearly with ﬂow rate as can be seen from the plot in Figure 9.6b showing the reduced HETP, h = H/dp, versus the reduced velocity, v′ = vdp/D0, where D0 ∼ 1 × 10−6 cm2/s is the free solution diffusivity. The slopes of these lines, 0.0355 for cytochrome c and 0.0499 for lysozyme, correspond to the c term in the generalized van Deemter equation (Equation 8.54). Neglecting the external mass transfer resistance, these slopes give values of τp/εpψp = 3.24 and 4.44 for the two proteins. The corresponding effective pore diffusivities are De = 1 × 10−6/3.24 = 3.1 × 10−7 cm2/s for cytochrome c and De = 1 × 10−6/4.40 = 2.3 × 10−7 cm2/s for lysozyme.

Example 9.3 Compare the experimental data of Examples 9.1 and 9.2 with model predictions All the parameters needed for the predictions are available from Examples 9.1 and 9.2 for use in Equations 9.28 and 9.30 to 9.32. Time units can be converted to volume by multiplying with the ﬂow rate. Predicted concentration proﬁles are compared to the experimental UV detector trace taking into account each protein’s extinction coefﬁcient. The results are shown in Figure 9.8. The agreement is obviously satisfactory, indicating that the model can be used to predict the effects of gradient slope and ﬂow rate on resolution. Equation 9.19 gives nearly identical results for these conditions.

0.05 2.5 CV gradient Data Model

20 CV gradient Data Model

0.016

Normalized Conc.

0.04

Normalized Conc.

292

0.03

0.02

0.012

0.008

0.004 0.01

0

0 0

10

20

V, mL

30

40

0

50

100

150

V, mL

Figure 9.8 Comparison of experimental and predicted LGE separations according to Example 9.3. The model lines are calculated from Equations 9.28 and 9.30 through 9.32 using parameters from Examples 9.1 and 9.2.

9.3 LGE Relationships for Ion Exchange Chromatography 2.5

500 Eq. 9.19, LDF model Eq. 9.30, plate model

2 400

300

CM (mM)

C/CF

1.5

1

200 0.5

0

100 0

50

100

150

Volume (mL)

Figure 9.9 Comparison of LGE peak predictions for LDF model (Equation 9.19) and plate model (Equation 9.30) for the conditions shown in Figure 9.8b in Example

9.2 but with a 20-ml feed volume. Only the ﬁrst peak is shown. The volume axis begins when the feed load is started.

Somewhat different peak shapes are predicted, however, for volume-overloaded conditions where the peak acquires a fronting shape, which cannot be predicted by Equation 9.30 (see Frey [3], Carta, et al. [4], and Guiochon et al. [11]). However, when the initial k′ is very high, the difference is negligible and the two solutions provide similar results. This is illustrated in Figure 9.9 for conditions identical to those in Figure 9.8 (right), but with a 20-ml feed load with the volume axis shown from the time the feed load is started. For simplicity calculations are shown only for the cytochrome c peak. Note that the peak is now concentrated above the feed value. Since the initial k′ is k0′ = 465 for these conditions, the difference between the two solutions is very small. For these calculations, n = 38 for the LDF model, Equation 9.19, and N = 19 for the plate model, Equation 9.30. The explicit expressions for ϑ, r, and rF for use in Equation 9.19 in the case of IEC can be found in [4]. The resolution in LGE ion exchange chromatography was deﬁned by Equation 9.18, which is appropriate for small feed pulses. For these conditions, Equation 9.30 is applicable and yields the following result:

293

9 Gradient Elution Chromatography

294

(a)

Comp. A

A

Comp. B

5

z 100

8.0x10

6 11

2.4x1014

(b)

Comp. A

k'

B A

100

10

A

8.0x1011

Comp. B 5 1.2x1012

A 10

(a) 1 100

200

300

400

500

Salt concentration, CM

1.5

(b) 1 100

200

1

300

400

500

(c)

Comp. A

0.5

Comp. B

5

z

100

7

8.0x10

A

11

2.0x1012

B A

k'

Resolution, Rs

5

B

k'

2

z

0

10

-0.5

-1

CM0=100, k'M=1.5, N=100 0

10

(c) 1 100

20

30

40

50

200

300

400

500

Salt concentration, CM

Normalized gradient slope,

Figure 9.10

Effect of normalized gradient slope on resolution for three different cases.

1

Rs =

1

[ AB ( zB + 1) γ ]zB +1 − [ AA ( z A + 1) γ ]zA +1

{

z z 1 − A − B 1 + kM′ + AB[ AB ( zB + 1) γ ] zB +1 + AA[ AA ( z A + 1) γ ] z A +1 2 R R 1 L CM 1 ,B − CM , A = × 1 z z − − B A 4 1 + k ′ + A CR R  γ C f H + A A (CM B ( M ,B ) ,A ) M 2

}

×

1 γCf

L H

(9.35)

where we have assumed that the HETP is the same and have introduced an average − peak compression factor Cf for the two adjacent components. These are reasonable assumptions for the separation of closely related species. The trends of resolution with normalized gradient slope, γ, are shown in Figure 9.10 for three different hypothetical cases. The ﬁrst, described by inset (a), corresponds to the separation of two proteins that have different effective charge and, thus, converging plots of log k′ versus log CM. In this case, the resolution increases substantially as γ is decreased as was seen in Example 9.1. The second case, described by inset (b),

9.4 LGE Relationships for RPC and HIC

corresponds to a case where the two proteins have the same charge but have a different afﬁnity for the stationary phase resulting in different values of A. Accordingly, the log k′ versus log CM curves are parallel, until the limit where k′ approaches kM′ . In this care Rs also increases as γ is reduced, but the effect is much less pronounced than in case (a). The ﬁnal case, described by inset (c) corresponds to a situation where the log k′ versus log CM curves cross. This occurs when the z values are different but the afﬁnity is greater for the protein of smaller z. As seen in Figure 9.10, this behavior results in the possibility of selectivity reversal. The actual behavior of a particular protein separation will of course depend on the speciﬁc bimolecular properties and the characteristics of the stationary phase and is difﬁcult to predict a priori. In most cases, however, a limited number of LGE experiments will provide the requisite parameters to predict the effects of a gradient slope on the separation.

9.4 LGE Relationships for RPC and HIC

The theory describing LGE in reversed phase chromatography (RPC) has been developed by Snyder [12] and tested extensively for small molecule applications where the retention factor is found to depend on the volume fraction of the organic modiﬁer, ϕ, according to the following relationship (see Equation 3.8): ki′ = k∞′ ,i + Ai e −Siϕ

(9.36)

k∞′ ,i is the retention factor at high values of ϕ, and Ai and Si are equilibrium parameters. The latter is sometimes called the sensitivity coefﬁcient. Si generally increases with the molecular mass of the adsorbed solute and is typically very large for macromolecules [13]. As in the case of IEC, if the organic modiﬁer is present in large excess so that we have k∞′ ,i ∼ kM′ . Thus, ki′ = kM′ + Ai e −Siϕ

(9.37)

With a linear gradient, substituting Equation 9.37 in Equations 9.11 and 9.12 yields:

ϕ iR = ϕ 0 + tR ,i =

1 ln (1 + γ AiSi e −Siϕ0 ) Si

ϕ iR − ϕ 0 L + (1 + kM′ ) v β

(9.38) (9.39)

which are analogous to Equations 9.25 and 9.26. In these equations, ϕ0 is the initial value of ϕ and ϕ iR is the value at which the peak emerges from the column. Snyder [12] has also provided an expression for the peak compression factor derived from the numerical simulations of LGE using Equation 9.37. His result can be expressed as follows:

295

9 Gradient Elution Chromatography

1.2

1

0.8

Cf,i

296

0.6

0.4

IEC

0.2

RPC 0 0

5

10

15

20

i

Figure 9.11 Prediction of the peak compression factor for IEC based on Equation 9.28 and RPC based on Equation 9.40.

1

1 + 1 + 1  2  Λ i 3Λ 2i  C f ,i = 1 1+ Λi

(9.40)

where Λi =

1 + kM′ + Ai e −Siϕ0 (1 + kM′ ) γ AiSi e −Siϕ0

(9.41)

Snyder’s result for RPC is analogous to that of Yamamoto et al. [9] for IEC. If the gradient is shallow, Λi is large and Equation 9.40 gives Cf,i ∼ 1, while lower values, indicating substantial peak compression, are obtained for steep gradients with high values of γ. The two results are also quantitatively very similar as shown in Figure 9.11, where both equations have been plotted versus Λi. The practical range of values for Λi that is useful in resolving closely related species is typically above 2 so that both models predict relatively little peak compression. Application of these relationships to the RPC of proteins and other biopolymers is discussed in [13]. These applications are complicated by the fact that such molecules are often partially or even completely unfolded on RPC surfaces. Moreover, the sensitivity coefﬁcient, S, for proteins is typically extremely large resulting in an adsorptive behavior that is almost on–off. Gradient elution is thus

9.4 LGE Relationships for RPC and HIC

critical since even extremely small changes in ϕ result in large variations in k′ and this effect is more pronounced when the protein molecular mass is high [13]. In these extreme cases, each protein will elute at a value of ϕ that is practically independent of the gradient slope. When this occurs, only extremely shallow and often impractical gradients will enable the effects of the gradient slope of ϕR to be discerned. The relationship between protein binding on HIC surfaces and salt concentration is generally complex for highly overloaded conditions (e.g. see Section 3.2.2 and [14–17]). However, in the linear isotherm limit, the relationship between retention and salt concentration is described approximately by the solvophobic theory [18, 19] discussed in Chapter 3. At sufﬁciently high salt concentration, the solvophobic theory yields a linear relationship between the logarithm of the retention factor and the salt molality, where the slope is equivalent to a ‘salting-out’ constant (cf. Equation 3.6). Thus, Snyder’s RPC treatment can be formally extended to treat HIC since the relationship between retention and modiﬁer concentration is also described by Equation 9.37 or 9.38, where the molality or, approximately, the concentration of a salt like ammonium sulfate replaces the organic modiﬁer. In the case of HIC, however, retention generally increases with salt concentration so that gradient elution is obtained with a negative modiﬁer gradient. Accordingly, we have: ki′ = kM′ + Ai eSiCM 0 R CM ,i = C M −

(9.42)

(

0 1 ln 1 − γ AiSi eSiCM Si

)

(9.43)

with γ < 0. An example of peaks obtained for different proteins in LGE with a Butyl Sepharose 4FF column is given in Figure 9.12 using ammonium sulfate gradients from 1.5 to 0 M at pH 6.9. For the smaller proteins, lysozyme and αchymotrypsinogen A, symmetrical peaks are obtained and the behavior of the peak in response to the varying gradient slopes is consistent with Equation 9.42 (see Example 9.4). On the other hand, the behavior of BSA is quite different. In this case, two peaks are obtained from each injection: one elutes early in the gradient and the other very late, with the two peaks connected by a broad proﬁle. A possible interpretation of this behavior attributes the phenomenon to partial unfolding of the protein on the HIC surface [20]. Accordingly, the early peak corresponds to folded protein and the late eluting peak to unfolded protein. In between there is a range of mixtures of folded and unfolded protein that elutes at intermediate ammonium sulfate concentrations. Ovalbumin appears to exhibit an intermediate behavior with some visible peak tailing probably associated with partial unfolding on the surface. Fernandez and co-workers [21–24] have recently provided direct evidence of protein unfolding on HIC surfaces. Although these surfaces are designed to minimize damage to the proteins, frequently unfolding will still occur, especially in high ammonium sulfate concentrations causing substantial deviations from Snyder’s theory, which, of course neglects any such complications.

297

9 Gradient Elution Chromatography 400

600

(a) Lyo

1 CV

300

mAU

mAU

250 5

300

7.5 10

200

5

200

7.5

150 12.5 15

10

100

100 0 0

2.5

2.5

400

12.5

15

50 1

2

3

4

5

6

7

0 0

8

2

4

CV

6

8

10

CV

140

35

(c) Ova

1 CV 120

30

100

(d) BSA

2.5 CV

25

60

mAU

2.5

80

5

20

5

15 7.5

7.5 40

10

2

4

10

10 12.5

20 0 0

(b) -Chy

1 CV

350

500

mAU

298

6

15

12.5 15

5 8

10

0 0

CV

Figure 9.12 Experimental LGE data for different proteins using a 1 × 10 cm Butyl Sepharose 4FF column and an elution ﬂow rate of 1 ml/min with ammonium sulfate

5

10

15

20

CV

gradients ranging from 1.5 to 0 M in 10 mM phosphate buffer at pH 6.9. Gradient durations are shown in column volumes. Data from [25].

Example 9.4 Obtain k′ versus salt from LGE data for proteins in HIC Obtain the relationship between k′ and ammonium sulfate concentration from the data given in Figure 9.12 for lysozyme and α-chymotrypsinogen A. Solution – The ammonium sulfate concentration at which each protein elutes can be derived from the conductivity trace. The latter is not shown explicitly, but can be found in [25]. Note that the relationship between conductivity and ammonium sulfate concentration is not linear over the range 0 to 1.5 M. Thus, a caliR bration curve is needed. The corresponding values of CM at which the peaks emerge from the column are shown in Figure 9.13. The normalized gradient slope is γ = βL/v = ε∆CM/CVG where ∆CM = 0 − 1.5 M and CVG is the duration of the gradient in column volume. For this column, ε = 0.37 and kM′ = 1.56 based on the data in [25]. If the protein is strongly retained under the initial

9.5 Separations with pH Gradients 0.5

20

(a)

-0.5

-Chy

-1.5

k'

ln (- )

15

Lysozyme A = 0.0131 S = 5.45

-1

(b)

LGE data Isocratic data Model

0

10 Lyo

-2 -2.5

-Chy A = 0.00388 S = 7.82

5

-3 -3.5 0.4

0.6

0.8

CMR

(M)

1

1.2

0

0

0.2

0.4

0.6

0.8

1

1.2

1.4

CM (M)

R Figure 9.13 Plot of ln(−γ) versus CM ,i for the LGE data shown in Figure 9.12a and 9.12b (a) and derived values of ki′(b). The isocratic data are from [25].

conditions, the second term in parenthesis in Equation 9.43 is dominant. Thus, we have: CMR ,i ∼ −

ln ( −γ Ai Si ) Si

(9.44)

In this case, a plot of ln(−γ) versus CMR ,i will give a straight line with slope −Si and intercept−ln(AiSi). Figure 9.13a shows that this is indeed the case for our system. The corresponding values of Ai and Si are 0.0131 and 5.45, respectively, for lysozyme, and 0.00388 and 7.82, respectively, for α-chymotrypsinogen A. The relationship between k′ and the ammonium sulfate concentration obtained from the LGE data is shown in Figure 9.13b. Note that the LGE results are in good agreement with the k′ values determined by Siebenmann-Lucas [25] from isocratic elution experiments. The same approach also works in an approximate manner for the ovalbumin data in Figure 9.12c (which give A = 0.000366 and S = 8.77), but not for BSA, which shows split peak behavior for the reasons discussed above.

9.5 Separations with pH Gradients

Use of pH as a mobile phase modiﬁer in IEC can be an effective alternative to using salt gradients. The concept is based on the dependence of the net protein

299

300

9 Gradient Elution Chromatography

charge on pH. Using a cation exchanger, for example, a mixture of acidic proteins can be loaded at a moderately low pH, where the proteins bear a positive charge and are thus strongly bound. The bound proteins can then be eluted with a mobile phase having a pH that increases over time. As the net positive protein charge is reduced, the binding strength decreases and elution takes place. An analogous pH gradient process can be implemented with anion exchangers, with a mixture of basic proteins loaded at moderately high pH and eluted at gradually decreasing pH values. Difﬁculties associated with pH elution are complexities in establishing and controlling pH gradients at the column entrance and the possibility of reaching pH values where the protein associates, precipitates, or is degraded. Advantages include the possibility of greater resolving power in cases when salt gradients are ineffective and the fact that elution may be conducted at low ionic strengths, thereby facilitating subsequent processing steps. In the case of strong cation and anion exchange resins, using unadsorbed buffering species (i.e. negatively charged with cation exchangers, positively charged with anion exchangers) allows pH gradient elution to be conducted in a manner analogous to elution with salt gradients (see e.g. [26]). In such cases the temporal pH gradient is essentially unretained and travels quickly through the column. Thus, there is only a shallow instantaneous axial gradient, as in the case of salt gradients, unless the temporal gradient is very steep. In this case, the analysis follows the same equations as presented in Section 9.2, except that the relationship between protein retention and pH is usually complex. Over narrow pH ranges, some simpliﬁcations are possible where the net charge of the protein is modeled taking into account only a small number of amino acid residues. Even in these cases, however, modeling and predicting protein retention as a function of pH is not easy, since binding on ion exchangers is typically affected by partial charges on the protein surface. It is not uncommon, for example, to encounter proteins that are strongly retained by cation exchangers even if the pH is well above the protein pI. Empirical determination of the relationship between protein retention and pH is thus usually required. pH gradients can also be generated within the column itself resulting in the separation of proteins along a traveling pH wave. The process is generally known as chromatofocusing [27, 28]. One approach is to use strong cation or anion exchangers using retained buffer systems that generate pH gradients by applying a step change at the column entrance. Special buffer combinations containing so-called ampholytes which are designed to yield smooth gradients are commercially available [29]. However, as discussed, for example, by Strong and Frey [30], these buffers have certain disadvantages in terms of cost and reproducibility. Moreover, these species are adsorbed, obviously in competition with the protein, making the column behavior difﬁcult to predict. Regeneration of the column is also required after each separation and removal of the adsorbed ampholytes is not always easy. As an alternative, using either weak cation or weak anion exchangers allows for the on-column generation of induced pH gradients with simple mixtures of nonadsorbed buffering species using step changes at the column entrance. The advan-

9.5 Separations with pH Gradients

tages are that the process is more robust and predictable and the costs are generally lower since inexpensive buffers such as acetate and phosphate can be used with cation exchangers and TRIS or ethanolamine with anion exchangers. Moreover, since the gradient is generated as a result of a step change there is no need for variable ﬂow pumps or external mixing devices. Frey and co-workers have studied various applications focusing primarily on weak anion exchange columns with analytical-size particles and basic buffer species [31–33]. More recent work by Kang and Frey [34] has also examined applications of weak cation exchangers using both adsorbed and non-adsorbed buffers. Although prior work has focused mainly on stationary phases for high performance analytical applications (5–10 µm particles) and very low protein loadings (∼ 0.01 mg/ml column), the use of preparative-sized ion exchangers to achieve separations with induced pH gradients has also recently been shown using larger particles (∼ 100 µm), high ﬂow rates (100–600 cm/h), and relatively high protein loads (1–10 mg/ml) on both weak cation exchangers [35] and weak anion exchangers [36]. A practical requirement for implementing successful separations with such induced pH gradients is the ability to predict the gradient shapes generated by different buffers and resin combinations. Unlike temporal gradients, the generation of an induced pH gradient with a desired duration and degree of linearity is dependent on being able to predict the exchange of the buffer counter-ions for hydrogen ions in the case of cation exchangers and hydroxyl ions in the case of anion exchangers. Since the exchange of these species is rapid compared to the time scales of protein binding, these predictions can usually be made on the basis of the local equilibrium model. The starting points for these predictions are the resin’s potentiometric titration curve and the dissociation behavior of the buffering species in the mobile phase. A shown in Example 7.7, this information enables prediction of the pH gradients that are generated in response to step changes in salt concentration at the column entrance. The same approach can be used to predict the pH gradients that are generated in response to step changes in pH at the column entrance. Such steps result in two different waves traveling through the column: (i) a fast pH wave, accompanying the unretained buffering species, and (ii) a slow wave associated with exchange of counter-ions between the mobile phase and the stationary phase. For certain conditions such as those given in [35, 36], appropriate buffers or buffer mixtures result in smooth pH transitions between the fast and the slow wave. For example, mixtures of acetate and phosphate buffers can be used to generate nearly linear pH gradients between pH 5 and 7 with cation exchangers, while mixtures of TRIS and BTP can be used to generate nearly linear pH gradients between pH 7 and 5 with anion exchangers. It should be noted that such gradients are intrinsically different from those obtained with strong cation and anion exchange resins by applying temporal pH gradients at the column entrance. The main difference is that the induced pH gradients extend across the column length so that there is a strong instantaneous axial gradient. Conversely, unretained pH gradients generated by temporal gradients at the column entrance result only in very shallow axial gradients unless the temporal gradients are very steep. As a result, induced pH gradients cause a focusing effect where the protein

301

9 Gradient Elution Chromatography

pHin

a

b

c

7.5 6.5 5.5 4.5 5

10

15

20

Distance from entrance

0

25

30

35

40

30

35

40

pH 7.0 6.5 6.0

6.25

5.5 5.75

pHout

302

7.5 6.5 5.5 4.5 0

5

a, b, c

c

a,b 10

15

20

25

CV Figure 9.14 Separation of a two-component protein mixture with a weak cation exchange column using an induced pH gradient. A gradual pH wave is established as a result of a step change in buffer composition form pH 4.5 to 7 at the column entrance. Thin lines

show the pH wave characteristics. The thick lines show the characteristics for two proteins that are focused at pH 6 and 6.5. Pulse injections are shown at 0, 5, and 15 CV. Reproduced from [35].

peaks are compressed and generally sharper than they would be in ordinary gradient elution. The separation principle of pH elution with induced gradients is shown in Figure 9.14, using a weak cation exchanger as an example. In this case, a buffer step from pH 4.5 to 7 is applied at the column entrance. With a proper buffer combination, a gradual pH wave is obtained which corresponds to the fan of characteristics that is shown starting at 0 CV. pH and counter-ion concentrations are constant on each characteristic line. A temporal pH gradient emerges at the column outlet and is accompanied by a substantial instantaneous axial pH gradi-

9.5 Separations with pH Gradients

ent. The injection of a mixture of two proteins is also shown in Figure 9.14 at three different times corresponding to 0, 5, and 15 CV. Thick dashed lines represent the paths followed by the two proteins. With the injection at 0 CV, both proteins are immediately trapped or focused on characteristics where the combination of pH and counter-ion concentration is such that the protein characteristic velocity equals the characteristic velocity of the pH wave. Note that any protein molecule moving faster than the pH wave at this point ends up in a region of lower pH where it acquires a higher positive charge and thus binds more strongly. In turn, this would allow the pH wave to catch up and virtually trap that molecule along the wave. Conversely, any protein molecule that lags behind will end up in a region of higher pH where it acquires a smaller positive charge thereby binding less strongly and thus catching up with the pH wave. The net result is a focusing effect that traps each protein at particular pH values. The latter values depend on speciﬁc interactions between the protein and the stationary phase and on the counter-ion concentration. The latter can be adjusted to control the time of elution and the resolution as shown in [35]. Note that for a given counter-ion concentration in the qualitative example shown in Figure 9.14, both proteins are still trapped at the same pH values even when the feed is injected at 5 CV. However, only the second, more strongly retained protein is trapped when the feed is injected at 15 CV. In this case, the more weakly bound species travels through the column quickly eluting in about 1 CV. Obviously, as shown in Figure 9.15, a continuous feed between 0 and 5 CV will also be trapped, resulting in a high degree of concentration as each protein is focused along the pH wave. The experimental validation of these results is shown in Figure 9.16 for mixtures of albumin and transferrin separated with an induced pH gradient using a UNOsphere S column. Note that highly reproducible and smooth pH gradients are obtained allowing resolution of these acid proteins at very low ionic strengths. Applications of induced pH gradient separations to the resolution of mAb variants with weak anion exchangers have been reported in [36]. An example using a ANX-Sepharose column with a pH gradient from 9.5 to 8 generated with a 10 mM ethanolamine buffer step is shown in Figure 9.17. The accompanying analytical isoelectric focusing (IEF) results show a substantial resolution of the charge variants in the mAb sample. The approach is obviously effective and can be scaled up easily, since only a pH step at the column entrance is required. There is a ﬁnal consideration regarding the pH at which the proteins are trapped. Chromatofocusing is, in some respects, analogous to isoelectric focusing (IEF) and is used for analytical separations. In IEF, however, where, in principle, there is no interaction with the matrix, the proteins are focused at their pI. In contrast, induced pH gradients in chromatography columns focus the proteins along the pH wave at pH values that are higher than the pI when using cation exchangers and lower than the pI when using anion exchangers. This occurs, in part, as a result of competition between protein and counterion for binding to the stationary phase. Elution at a pH different from the pI is obviously advantageous to reduce the possibility of precipitation or aggregation.

303

9 Gradient Elution Chromatography

pHin

7.5 6.5 5.5 4.5 5

10

15

20

Distance from entrance

0

25

30

35

40

30

35

40

pH 7.0 6.5 6.25 6.0 5.5

pHout

304

5.75

7.5 6.5 5.5 4.5 0

5

10

15

20

25

CV Figure 9.15 Separation of a two-component protein mixture with a weak cation exchange column as in Figure 9.14 but with a continuous protein feed load from 0 to 5CV using an induced pH gradient. Reproduced from [35].

9.6 Modeling Gradient Elution with Non-linear Isotherms

The prediction of gradient elution when the isotherms are non-linear and competitive requires a description of the multi-component adsorption isotherm. Numerical simulations are required for the general case where both equilibrium and dispersive factors are considered. A general framework for the numerical simulation of non-linear preparative gradient elution chromatography has been presented by Antia and Horvath [37] using the so-called equilibrium dispersive model and an extended form of the multi-component Langmuir isotherm, were each adsorption constant, K, is expressed as a function of the modiﬁer concentration. Antia and Horvath later extended the approach to more general single component

9.6 Modeling Gradient Elution with Non-linear Isotherms 7.5

100 Inject at 0 CV Inject at 5 CV Inject at 15 CV

80

7.0 6.5

60

pH

mAU

6.0

40

5.5

T A

5.0 20 4.5 0

4.0 0

5

10

15

20

25

30

35

40

CV 7.5

150 1 mL pulse at 0 CV (20 mg tot. protein) 5 CV load (21 mg tot. protein) 5 CV load (85 mg tot. protein)

7.0 6.5

100

pH

mAU

6.0 5.5

T 50

5.0

A

4.5 0

4.0 0

5

10

15

20

25

30

35

40

CV Figure 9.16 Separation of albumin/ transferrin mixtures on a 1 × 10 cm UNOsphere S column eluted at 4 ml/min (300 cm/h) using steps in pH from 4.5 to 7 and buffers containing 10/1 acetate/ phosphate and a 0.03 M Na+ concentration. The top ﬁgure shows the results when 1 ml

of a 10-mg/ml solution of each protein is injected at 0, 5, and 15 CV. The bottom ﬁgure shows the results for a 1-ml injection load and 5 CV (∼40 ml) protein feed loads using 0.25 and 1 mg/ml of each protein. Reproduced from [35].

isotherm shapes, using the ideal adsorbed solution (IAS) theory to describe multicomponent adsorption [38]. The approach Antia and Horvath used was originally intended for small molecule RPC, but could also be considered as an approximate representation of protein chromatography when mass transfer resistances are not too substantial. The main difﬁculty is expressing the multi-component, competitive binding behavior, which is often more complex than can be predicted by the

305

9 Gradient Elution Chromatography 300

10 2

250

9.5

200 9.0

pH

mAU

306

150 1

8.5

100 8.0

3

50 0

7.5 0

10

20

30

40

50

CV Cathode

Anode L

1

2

3

Figure 9.17 Chromatographic separation of mAb charge variants using steps in the pH of the buffer from 9.5 to 8 and 0.01 M ethanolamine in a 1.1 × 20 cm ANX-Sepharose column eluted at 1.6 ml/min (100 cm/h) and a

L

5-mg/ml protein load. The dashed line shows the experimentally-induced pH gradient. The lower graph shows IEF analyses of the peak fractions indicated. Lanes labeled ‘L’ are the crude protein load. Reproduced from [36].

empirical Langmuir model or the thermodynamics-based IAS theory. Several authors, including Gu et al. [39], Bellot and Condoret [40], and Whitley et al. [41], have presented general rate models incorporating axial dispersion and intra-particle diffusion for the description of non-linear gradient elution. Commercial simulators capable of such predictions have also been made available [42, 43]. Use of these more complex models affords, in general, greater precision and more accurate predictions, especially at very high protein loads, where competitive binding results in displacement effects. Such models, however, often require many parameters whose empirical determination is not always easy or even possible. In many cases, semi-empirical approaches, such as the LDF approximation will sufﬁce for practical engineering design calculations (e.g. see [44–46]). Finally, as pointed out, for example, by Frey [3] even the limiting descriptions developed for linear isotherms and presented in Sections 9.2 to 9.5 are often sufﬁcient to describe protein gradient elution for these purposes and can be used for actual process calculations especially in IEC for the reasons already discussed (e.g. see refs. [2, 6, 47, 48]).

References

References 1 Yamamoto, S., Nakanishi, K., and Matsuno, R. (1988) Ion Exchange Chromatography of Proteins, Marcel Dekker, New York. 2 Yamamoto, S. (1995) Biotechnol. Bioeng., 48, 444. 3 Frey, D.D. (1990) Biotechnol. Bioeng., 35, 1055. 4 Carta, G., and Stringﬁeld, W.B. (1992) J. Chromatogr. A., 605, 151. 5 Yamamoto, S., Nomura, M., and Sano, Y. (1987) J. Chromatogr. A, 409, 101. 6 Ishihara, T., Kadoya, T., and Yamamoto, S. (2007) J. Chromatogr. A, 1162, 34. 7 Regnier, F.E., and Mazsaroff, I. (1987) Biotechnol. Progr., 3, 22. 8 Brooks, C.A., and Cramer, S.M. (1992) AIChE J., 38, 1969. 9 Yamamoto, S., Suehisa, T., and Sano, Y. (1993) Chem. Eng. Commun., 119, 221. 10 Carta, G., Ubiera, A.R., and Pabst, T.M. (2005) Chem. Eng. Technol., 28, 1252. 11 Guiochon, G., Felinger, A., Shirazi, D.G., and Katti, A.M. (2006) Fundamentals of Preparative and Nonlinear Chromatography, 2nd edn, Elsevier Academic Press, New York, Chapter 15. 12 Snyder, L.R. (1985) Gradient elution, in High Performance Liquid Chromatography: Advances and Perspectives, vol. 1 (ed. Cs. Horvath), Academic Press, New York, p. 208. 13 Snyder, L.R., and Stadalius, M.A. (1986) HPLC separations of large molecules: a general model, in High Performance Chromatography of Biopolymers: Advances and Perspectives, vol. 4 (ed. Cs. Horvath), Academic Press, New York, p. 195. 14 Antia, F.D., Fellegvari, I., and Horvath, Cs. (1995) Ind. Eng. Chem. Res., 34, 2796. 15 Xia, F., Nagrath, D., and Cramer, S.M. (2003) J. Chromatogr. A, 989, 47. 16 Chen, J., and Cramer, S.A. (2007) J. Chromatogr. A, 1165, 67. 17 To, B.C.S., and Lenhoff, A.M. (2007) J. Chromatogr. A, 1141, 235. 18 Melander, W.R., and Horvath, Cs. (1977) Arch. Biochem. Biophys., 183, 393.

19 Melander, W.R., Corradini, D., and Horvath, Cs. (1984) J. Chromatogr. A, 317, 67. 20 Jungbauer, A., Machold, C., and Hahn, R. (2005) J. Chromatogr. A, 1079, 211. 21 Xiao, Y.Z., Freed, A.S., Jones, T.T., Makrodimitris, K., O’Connell, J.P., and Fernandez, E.J. (2006) Biotechnol. Bioeng., 93, 1177. 22 Fogle, J.L., O’Connell, J.P., and Fernandez, E.J. (2006) J. Chromatogr. A, 1121, 209. 23 Xiao, Y.Z., Rathore, A., O’Connell, J.P., and Fernandez, E.J. (2007) J. Chromatogr. A, 1157, 197. 24 Xiao, Y.Z., Jones, T.T., Laurent, A.H., O’Connell, J.P., Przybycien, T.M., and Fernandez, E.J. (2007) Biotechnol. Bioeng., 96, 80. 25 Siebenmann-Lucas, J.N. (2006) Modeling hydrophobic interaction chromatography under linear gradient elution using Butyl Sepharose 4FF. Undergraduate Thesis. University of Virginia, Charlottesville, Virginia, USA. 26 Anderson, D.J., and Shan, L. (2002) Anal. Chem., 74, 5641. 27 Sluyterman, L.A.A.E., and Elgersma, O. (1978) J. Chromatogr., 150, 17. 28 Sluyterman, L.A.A.E., and Wijdenes, J. (1978) J. Chromatogr., 150, 31. 29 Hutchens, T.W. (1989) Chromatofocusing, in Protein Puriﬁcation (eds J. Janson and L. Ryden), John Wiley & Sons, New York. 30 Strong, J.C., and Frey, D.D. (1997) J. Chromatogr. A, 769, 129. 31 Bates, R.C., and Frey, D.D. (1998) J. Chromatogr. A, 814, 43. 32 Bates, R.C., Kang, X., and Frey, D.D. (2000) J. Chromatogr. A, 890, 25. 33 Kang, X., Bates, R.C., and Frey, D.D. (2000) J. Chromatogr. A, 890, 37. 34 Kang, X., and Frey, D.D. (2003) J. Chromatogr. A, 991, 117. 35 Pabst, T.M., Antos, D., Ramasubramanyan, N., Hunter, A.K., and Carta, G. (2008) J. Chromatogr. A, 1181, 83. 36 Pabst, T.M., Carta, G., Ramasubramanyan, N., Hunter, A.K.,

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37 38

39

40 41

Mensah, P., and Gustafson, M.E. (2008) Biotechnol. Progr., 24, 1096. Antia, F.D., and Horvath, Cs. (1989) J. Chromatogr., 484, 1. Antia, F.D., and Horvath, Cs. (1991) Gradient elution chromatography, in Chromatographic and Membrane Processes in Biotechnology (eds C.A. Costa and J. Cabral), Kluwer Academic Publishers, The Netherlands, pp. 115–136. Gu, T., Truei, Y.H., Tsai, G.J., and Tsao, G.T. (1992) Chem. Eng. Sci., 47, 253. Bellot, J.C., and Condoret, J.S. (1993) J. Chromatogr. A, 635, 1. Whitley, R.D., Zhang, X., and Wang, N.H.L. (1994) AIChE J., 40, 1067.

42 Purdue Research Foundation, Purdue University, West Lafayette, IN, USA. 43 Chromatography®, Aspen Technology, Inc., Burlington, MA, USA. 44 Gallant, S.R., Kundu, A., and Cramer, S.M. (1995) Biotechnol. Bioeng., 47, 355. 45 Gallant, S.R., Vunnum, S., and Cramer, S.M. (1996) J. Chromatogr. A, 725, 295. 46 Melter, L., Strohlein, G., Butte, A., and Morbidelli, M. (2007) J. Chromatogr. A, 1154, 121. 47 Yamamoto, S., Nakanishi, R., Matsuno, R., and Maikubo, T. (1983) Biotechnol. Bioeng., 25, 1373. 48 Yamamoto, S., Nomura, M., and Sano, Y. (1987) AIChE J., 33, 1426.

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10 Design of Chromatographic Processes 10.1 Introduction

The optimum design of chromatographic processes is generally subject to a broad range of constraints that include product quality, process robustness, and overall economic viability. Product quality requirements were discussed in general terms in Chapter 1. Quality attributes vary widely depending on the type of product and the intended use, but generally purities in excess of 99% with less than 0.5% of any individual contaminant are required for biopharmaceuticals. As discussed in Chapter 1, special requirements apply to genetic contaminants, especially for products derived from animal cells, and to endotoxin for products of microbial origin. Adventitious agents such as virus, prions, bioburden, and process chemicals, as well as aggregates also require special consideration. Process robustness and reproducibility are of course critical to product safety and must be taken into account particularly in the manufacture of biopharmaceuticals. A special concern is the microheterogeneity of the puriﬁed product. Many biopharmaceutical products are glycosylated, resulting in the possibility of substantial microheterogeneity, which can be affected by process conditions. Since, in turn, glycosylation can affect bioavailabilty, pharmakokinetics, and, potentially, toxicity, the impact of downstream processing on the glycosylation patterns of the puriﬁed product must be considered. In some cases, downstream processing can be designed to attain the desired glycosylation proﬁles by separating undesirable isoforms. Finally, economic viability is dependent on product yield, process productivity, the associated consumption of solvents and process chemicals, labor, and depreciation. The well-known relationship between the overall yield and the yield of an individual step for a multi-step process is shown in Figure 10.1. Most biopharmaceutical downstream processes require several steps in order to meet product quality and robustness requirements. As a result, the yields of individual steps need to be high in order to attain acceptable overall yields. Thus, as a general rule, economically viable chromatographic process designs should aim for greater than 90% step yields.

Protein Chromatography: Process Development and Scale-Up. Giorgio Carta and Alois Jungbauer © 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 978-3-527-31819-3

10 Design of Chromatographic Processes 1

97.5%

0.8

95% Overall yield

310

0.6

90% 0.4

85% 80%

0.2

Step yield 0

1

2

3

4

5

6

7

8

9

10

11

Number of steps Figure 10.1

Overall yield as a function of the number of steps and the individual step yield.

The productivity of a chromatographic process is deﬁned as the amount of product processed per unit time and per unit volume of stationary phase. In some applications, the cost of the stationary phase is a dominant factor. Thus, improving productivity can have a dramatic impact on the overall process costs. In general productivity will depend on the properties of the stationary phase and the choice of operating conditions. Thus, these choices are critical for engineering biochromatography processes. The consumption of solvents and process chemicals and the labor associated with the process are also inﬂuenced by these choices. The preparation of buffers and, in some instances, their ultimate disposal can be signiﬁcant cost components thereby requiring careful consideration. Column packing and unpacking and validation also have signiﬁcant impact on labor requirements and are dependent on both stationary phase properties and on the column hardware design (see Chapter 4). Finally, depreciation takes into account all ﬁxed equipment costs with the appropriate amortization schedules. Equipment costs can be minimized with efﬁcient designs although, in general, a compromise will be needed between ﬁxed and operating costs. For example, countercurrent or simulated moving bed systems will in most cases provide greater productivity and lower consumption of process chemicals, but usually at the expense of greater equipment costs. A comprehensively optimized chromatographic process will require consideration of all the technical, regulatory, and economic factors described above. In the following however, we will limit the discussion to technical factors with the primary concern of designing for maximum productivity. The resulting designs will approach a global economic optimum only when the cost of the stationary phase is dominant, which is often the case in many protein chromatography processes.

10.2 Chromatographic Process Steps and Constraints D (finish) B

Separation

Purity (%)

Polishing

A (start)

Concentration

C

Product Concentration (g/L) Figure 10.2 Downstream processing paths from a dilute, impure feedstock (A) to a pure, concentrated product (D). Path ACD (concentration ﬁrst followed by separation and polishing) is typically preferred. Based on [1].

10.2 Chromatographic Process Steps and Constraints

Figure 10.2, based on the work of Lightfoot and Cockrem [1], provides a qualitative description of product concentration and purity in an overall downstream process. Ordinarily, the starting point is a dilute and impure solution, while the goal is a pure and concentrated product. Two extreme paths could be considered: ABD, where the product is ﬁrst puriﬁed and then concentrated, and ACD, where the product is ﬁrst concentrated and then puriﬁed. As noted by Lightfoot, the processing costs for dilute solutions are often directly related to the volume of solution that needs to be processed, which, in turn, is higher for more dilute solutions. As a result, path ACD, where concentration and, hence volume reduction occurs ﬁrst, is generally preferred. In addition to volume reduction, concentration on a highcapacity selective adsorbent may lead to reduced product loss if, for example, proteases are separated from the product during the capture step. Based on these considerations, a typical downstream process for biopharmaceutical manufacturing will normally involve the following major steps: 1) Capture; 2) Separation; and 3) Polishing. The main goals of each step are different. The principal goal of capture is generally the concentration of the product on the stationary phase with separation from unrelated impurities such a small molecules, DNA, endotoxin, and contaminant proteins whose properties are vastly different from those of the target product. High yield is desirable. Resolution of related impurities is also desirable but is

311

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10 Design of Chromatographic Processes

usually a secondary goal. Thus, capture steps will often use stationary phases with high binding capacities and larger particles and columns that are shorter and operated at higher ﬂow rates. Use of selective afﬁnity adsorbents such as Protein A for IgG capture will of course permit substantial puriﬁcation during capture. Capture steps will often use step elution or relatively steep gradients for product recovery. The goal of subsequent chromatographic separation steps will generally be to resolve impurities that are more closely related to the target product. This includes proteins with similar pI and hydrophobicity as well as misfolded, deamidated, or otherwise altered isoforms. Since resolution requires greater plate numbers or, equivalently, a larger number of transfer units, chromatographic separations processes will typically use smaller particles with greater emphasis on selectivity rather than capacity. Lower ﬂow rates and longer column will also typically be used along with shallow gradient operations. The ﬁnal process chromatography steps will be dedicated to polishing, including the removal of process chemicals and the separation of impurities generated in the process itself, such as dimers or aggregates. Size exclusion chromatography (SEC) will sometimes be effective for this step. Although SEC has limited capacity, it is mild and can be used simultaneously to remove aggregates and place the product in a desired buffer. Since the product concentration will generally be high at this point in the process, the limited capacity of SEC may not greatly impact process economics. Intermediate steps also have to be considered in the overall design of a downstream processing train. Frequently, downstream processing will make use of orthogonal stationary phase chemistries, such as, for example, cation exchange followed by anion exchange. In such cases, a buffer exchange step may be needed and can be achieved by SEC or, preferably, diaﬁltration. Alternatively, at times it is possible to arrange orthogonal steps in such as way as to avoid buffer exchange steps. For example, the product eluted from an IEC column at high salt concentration can sometimes be loaded directly onto an HIC column, thereby avoiding the need for buffer exchange. In other cases, the high-salt eluate emerging from an ion exchange column can be simply diluted and directly loaded onto a second IEC column. Dilution will reduce the protein concentration and increase the feed volume. However, since protein binding is often very strongly dependent on the concentration of the mobile phase modiﬁer (e.g. salt), high binding capacities will still be realized with the diluted feedstock even if the protein concentration is low. Finally, hold points, such as frozen storage, may also have to be included in order to provide emergency shutdown options without loss of a batch as well as the scheduling of equipment to maximize productivity. In the following sections we provide guidelines and design equations for three different cases: capture with a selective stationary phase, chromatographic resolution under linear isotherm conditions, and SMB separation of a binary mixture. The ﬁrst two are directly relevant to current biopharmaceutical manufacturing practice. SMB shows promise for the future and is likely to be implemented as advances are made in process control and analytical technologies.

10.3 Design for Capture Unbound impurities Bound product

= =

C/CF ~ 1 CV 1

10% BT 0

tLoad

tWash

tElute

tCIP

tEquil

tCycle Figure 10.3 Schematic of efﬂuent proﬁles tracking target protein and unretained contaminants in a capture step.

10.3 Design for Capture

We consider a batch capture process designed to adsorb a target protein with a selective stationary phase and separate it from unbound or weakly retained contaminants. The process consists of at least ﬁve steps: 1) 2) 3) 4) 5)

Feed load with contaminants in ﬂow through; Wash to remove unbound and weakly retained contaminants; Elution to desorb the bound protein; Clean-in-place to remove irreversibly bound species; and Equilibration to remove CIP liquids and restore the starting conditions.

A pre-equilibration step may also be needed to remove storage chemicals and prepare the column for the ﬁrst feed load step. A diagram of the efﬂuent proﬁles tracking the target protein and the contaminants is shown in Figure 10.3. Note that unbound impurities emerge from the column in about a column volume. Such contaminants are subsequently quickly eluted during the wash step. Other adsorbed contaminants, however, will require increasing intermediate modiﬁer concentrations for elution or may emerge together with the captured product in the elution step. In the following we assume that elution is carried our with a mobile phase modiﬁer that completely desorbs the protein. This is typical in many actual protein adsorption systems where the protein binding isotherm is extremely sensitive to the mobile phase composition. With reference to Figure 10.3, the productivity, P, is deﬁned as [2, 3]: P=

Amount of Protein Recovered Column Volume × Total Cycle Time

(10.1)

313

314

10 Design of Chromatographic Processes

where tCycle = tLoad + tWash + tElute + tCIP + tEquil is the total cycle time. With a suitably small breakthrough concentration, we can neglect leakage of the protein during the load step and obtain the following relationships: Amt. protein recovered ∼ ηE × DBC10% × Vc tLoad =

VF DBC10%Vc DBC10% L ~ = Q Load CFQ Load CF uLoad

(10.2) (10.3)

In these equations, ηE is the fraction of protein recovered on elution, QLoad is the volumetric ﬂow rate, VF and CF are the load volume and protein feed concentration, respectively, L/uLoad is the residence time during the load step, and DBC10% is the dynamic binding capacity at 10% of breakthrough. Other percentages can of course be used, and lower values may be preferable for robustness. Combining Equations 10.2 and 10.3 with Equation 10.1 yields the following expression for P: P=

ηE × DBC10% DBC10% L + tWash + tElute + tEquil + tCIP CF uLoad

(10.4)

All of the terms in this equation can be estimated or determined experimentally. In general, the DBC increases with residence time during the load step. Thus, since the denominator in Equation 10.4 contains the product of DBC and residence time, there will be a residence time that maximizes productivity. The speciﬁc relationship between DBC and residence time depends on the isotherm shape and the controlling band-broadening mechanism. For example, if the isotherm is very favorable and pore diffusion is controlling, from Equations 8.84 and 8.85 we have the following expressions1):

(

DBC10% = EBC × 1 −

1.03 n

)

for n > 2.5

(10.5)

DBC10% = EBC × (0.364n − 0.0612n 2 + 0.00423n 3 ) for n < 2.5 where n=

15 (1 − ε )De L rp2 uLoad

(10.6)

is the number of transfer units for pore diffusion, De is the protein effective pore diffusivity, and rp is the particle radius. Expressions for other rate mechanisms under constant pattern conditions can be found in Table 8.2 and more general situations can be handled by numerical simulation. In other cases, an empiricallyderived relationship between DBC and residence time may be used. The times required for each subsequent step can be estimated as follows.

1) Analogous expressions for the DBC at 1% breakthrough are as follows: DBC1% = EBC × (1 − 1.19 n ) for n > 2.5

DBC1% = EBC × (0.288n − 0.0307n 2 + 0.00083n 3 ) for n < 2.5

10.3 Design for Capture

10.3.1 Wash Step

The time required for the wash step can be estimated from the following relationship if we assume that the contaminants are unbound or retained with a linear isotherm. If the column is saturated with contaminants at the end of the load step, from item B in Table 8.1 and Equation 8.39 we have: 1 Cc = 1 − J (nc , nc τ 1,c ) ∼ [1 − erf ( nc τ 1,c − nc )] 2 CF ,c

(10.7)

where Cc and CF,c are the contaminant concentrations in the efﬂuent and in the protein feed, respectively. nc and τ1,c are deﬁned as follows: nc =

15 (1 − ε )De,c L rp2 uWash

(10.8)

tWashuWash εL − 1 kc′

(10.9)

τ 1,c =

where kc′ is the retention factor of the contaminant and De,c is its effective pore diffusivity. A common case is where the contaminant is not adsorbed. In this case, kc′ ∼ φε p , where φ = (1 − ε)/ε is the phase ratio and εp the intra-particle porosity. If we require the wash step to remove 99% of the unbound contaminants, we have erf ( nc τ 1,c − nc ) = 1 − 2

Cc = 0.980 CF ,c

(10.10)

Based on the properties of the error function, Equation 10.10 yields the result nc τ 1,c − nc = 1.645

(10.11)

Finally, solving this equation for τ1,c we obtain the following result:

(

τ 1,c = 1 +

1.645 nc

)

2

(10.12)

or tWash =

εL uWash

(

)

1.645 2  L  1 + k ′ 1 + n  = CVWash × u c Wash  

(10.13)

Note that different removal percentages will result in different numerical coefﬁcients in Equation 10.13, which can be obtained from a table of the error function. 10.3.2 Elution Step

In general, the time required for elution of the captured protein will depend on the isotherm parameters that prevail in this step as well as on mass transfer

315

316

10 Design of Chromatographic Processes

limitations. In many practical cases, however, conditions are often chosen so that there is virtually no protein binding with the mobile phase used for elution at equilibrium. This is possible because of the previously noted high sensitivity of protein binding to the mobile phase composition. For these conditions, when pore diffusion is dominant, mass transfer limitations typically play a small role during elution since the driving force for diffusion is high. As a result, the protein concentration in the eluted product will also be high. In many cases, however, limits will be imposed on the maximum protein concentration allowable in the eluted product pool, CE,max. Such limits may arise from the need to prevent aggregation or precipitation or to avoid viscous ﬁngering caused by the potentially high viscosity of the eluted band (see Chapters 1 and 2). If CE,max is used as a design criterion, we obtain the following relationship: tElute ∼

L DBC10%Vc DBC10% L = = CVElute × CE ,maxQ Elute CE ,max uElute uElute

(10.14)

10.3.3 CIP Step

Clean-in-place (CIP) protocols are often needed to remove strongly bound impurities or irreversibly adsorbed product, which may come from aggregates, unfolded proteins, or clipped fractions. The time required for CIP depends on the ability of the cleaning agent to hydrolyze or otherwise remove these materials from the column and is generally independent of residence time. Thus, only the actual time of exposure to the CIP agent is relevant. Validation studies are needed to determine the appropriate concentration of the cleaning agent and the duration of the cleaning step. Low ﬂow rates are often used for this step. 10.3.4 Equilibration Step

The time required to remove the CIP liquid and restore the column to its original state will vary depending on the chemical nature of the stationary phase. With strong acid and strong base ion exchange resins as well as with HIC and afﬁnity media, equilibration times are generally short and are in the order of a few column volumes. However, much longer times may be needed with weak acid and weak base resins, as illustrated, for instance, in Example 7.7. Frequently, media manufacturers’ recommendations or simple laboratory tests are often sufﬁcient to determine the number of column volumes needed. Thus we have: tEquil = CVEquil ×

L uEquil

(10.15)

The ﬁnal expression for the productivity is obtained by combining Equations 10.3–10.15 with Equation 10.1, yielding the following result:

10.3 Design for Capture

P=

ηE × DBC10% L DBC10% L L L + tCIP + CVWash + CVELute + CVEquil uEquil CF uLoad uWash uElute

(10.16)

In this equation all of the parameters except for ηE and tCIP can be estimated from adsorption and transport considerations. The remaining two can be determined experimentally from laboratory results. Column design is different depending on whether the adsorbent particles are rigid or compressible. With rigid particles, based on Darcy’s law, the column pressure is proportional to the product u × L and independent of column diameter (see Equation 3.18). On the other hand, with compressible stationary phases, the column pressure will typically exhibit a non-linear dependence on ﬂow velocity and will vary with column diameter (see Section 3.4.4). Examples 10.1 and 10.2 illustrate applications of the relevant design relationships for these two cases.

Example 10.1

Capture design with rigid particles

Design a capture column that maximizes productivity for the following conditions: Feed volume, VF = 1000 l Feed protein concentration, CF = 1 mg/ml Feed viscosity, η = 0.002 Pa · s (= 2 cp) Equilibrium binding capacity, EBC = 50 mg/ml (based on packed bed volume) Particle radius, rp = 42.5 µm Effective pore diffusivity, De = 3.0 × 10−8 cm2/s Extra-particle porosity, ε = 0.30 Intra-particle porosity, εp = 0.8 Maximum protein concentration in elution pool, CE,max = 5 mg/ml Maximum design pressure, ∆Pmax = 2 bar Design based on 10% breakthrough with 99% removal of unbound contaminants. Use 4 CV for equilibration assuming the same residence time L/u for all steps. Assume that mass transfer is controlled by pore diffusion and that the adsorption isotherm is rectangular. Solution – For the conditions in this problem we have φ = (1 − 0.3)/0.3 = 2.33 and k c′ = φε p = 2.33 × 0.8 = 1.87 . The column pressure can be estimated from Darcy’s law, Equation 3.18. The hydraulic permeability, B0 can be determined empirically or estimated from the Karman–Cozeny equation (Equation 3.19), which gives the following result:

B0 =

1 ε3 1 0.33 2 2 r = ( ) (42.5 × 10 −6 ) = 2.65 × 10 −12 m2 p 37.5 (1− ε )2 37.5 (1− 0.3)2

317

10 Design of Chromatographic Processes

The number of transfer units is given by the following equation (cf. item C in Table 8.2): n=

15(1− ε )De L 15 × (1− 0.3) × 3.0 × 10 −8 L L = = 0.0174 × 2 2 − 4 rp u u u (42.5 × 10 )

with L/u expressed in seconds. Assuming that the contaminants have the same effective pore diffusivity, De, as the target protein, the same expression also applies to nc. Lastly, for simplicity we assume ηE = 1. A plot of Equation 10.16 for these conditions is given in Figure 10.4. Note that P is at a maximum of 0.286 mg/ml/min when L/u = 1.11 min. This corresponds to tLoad = 19.2 min and tCycle = 60.5 min.

1

0.25

0.8

0.2 0.6 0.15 0.4 0.1 0.2

0.05 0

DBC10% /EBC

Productivity, P (mg/mL min)

0.3

0

1

2

3

4

5

6

7

8

0

Residence time L/u (min) 1000

Step and cycle times (min)

318

Total Cycle Load

100

10

Elute

CIP

Equil. Wash

1

0.1 0

1

2

3

4

5

6

7

8

Residence time, L/u (min) Figure 10.4 Productivity and step times for a chromatographic capture step according to Example 10.1.

10.3 Design for Capture

The pressure constraint of 2 bar (= 0.203 MPa) is met by designing the column geometry according to Equation 3.18 so that: uL =

∆PmaxB0 0.203 × 106 × 2.65 × 10 −12 = = 2.69 × 10 −4 m2 s η 0.002

or uL = 161 cm2/min. Thus, for this example we have u = 161 1.11 = 12.0 cm min and L = 13.4 cm. The ﬂow rate is calculated based on the load volume and is QLoad = VF/tLoad = 1000/19.2 = 52.1 l/min. Finally the column diameter can be determined from the equation dc = 4QLoad π u = 74.2 cm, which gives a column volume Vc = 57.8 l. Note that for this design, the DBC10% is 17.5 mg/ml which is about 35% of the EBC. This maximum productivity design tends to give high ﬂow rates and shallow columns. If the residence time is increased to 4 min, P is reduced to 0.161 mg/ml·min. In turn, we obtain tLoad = 149 min, tCycle = 233 min, u = 6.34 cm/min, L = 25.4 cm, QLoad = 6.71 L/min, dc = 36.7 cm, and Vc = 26.9 L. The smaller column volume is, of course, to be balanced against the longer processing time. Thus, a ﬁnal choice can only be made by taking into account ﬁxed and operating cost factors as well as technical factors regarding the availability of large diameter columns that can be packed reliably with short lengths. Note that it may also be possible to split the feed volume into multiple batches so that the system can be operated at maximum productivity with a smaller column. This possibility is dependent on the impact of quality control and the ability to use the stationary phase for multiple cycles.

Example 10.2

Capture design with compressible media

Design a capture column that maximizes productivity for the conditions of Example 10.1, but assuming that the stationary phase is compressible based on the model of Stickel and Fotopoulos [4] introduced in Chapter 3. The parameters describing bed compression are assumed to be ε0 = 0.30, λcri = 0.21, m = 1058 cm2/h, and b = 5055 cm2/h. Solution – The model of Stickel and Fotopoulos [4] assumes that bed compression is attained without an actual change in the particle volume. As ﬂow velocity is increased, the bed is compressed from its initial length L0 to a length L with an accompanying decrease in the extra-particle porosity from ε0 to ε and of the hydraulic permeability from B0 to B. As a result of the lower permeability, the pressure increases in a non-linear manner up to a maximum or critical velocity, ucri, where the bed length reaches a minimum value, Lcri, and the pressure becomes inﬁnite. Stickel and Fotopoulos have derived an empirical description of these effects based on experimental data obtained in columns of up to 100 cm in diameter. Their description can be expressed by the following equations: ucriL0 = m

L0 +b dc

(10.17)

319

320

10 Design of Chromatographic Processes

λ L = L0  1− cri u  ucri 

(10.18)

λcri u ucri ε= λ 1− cri u ucri

(10.19)

ε0 −

where ε0 is the extra-particle porosity of the uncompressed bed, and λcri = (L0 − Lcri)/L0 is the critical bed compression ratio. In general, the model parameters m, b, ε0, and λcri need to be determined empirically for each stationary phase following, for example, the approach of Stickel and Fotopoulos. Once these parameters have been determined, the hydraulic permeability of the compressed bed can be determined using the Karman–Cozeny equation: B=

1 ε3 rp2 37.5 (1− ε )2

(10.20)

where ε is the extra-particle porosity of the compressed bed obtained from Equation 10.19. Finally, the column backpressure can be calculated from Darcy’s law (Equation 3.18) which gives the following result: ∆P =

ηuL B

(10.21)

Note that since ucri depends on the initial bed length, L0, and on the column diameter, dc, the column pressure will vary with scale even if the product u × L is kept constant. A ﬁnal consideration is the number of transfer units and the EBC. The model of Stickel and Fotopoulos assumes that the particle size remains the same, so that (1 − ε0)L0 ∼ (1 − ε)L. Thus, to a ﬁrst approximation, n, and nc for the compressed bed are the same as the values for the uncompressed case at the same superﬁcial velocity. As a result, the DBC and the breakthrough behavior are also the same provided that the uncompressed column volume is used as a reference and protein retention in the inter-particle space is negligible. Since, in practice, the variations in ε are small, this is often a satisfactory approximation. With these assumptions, the relationship between productivity and residence time, L0/u, is the same as that in Example 10.1, provide both quantities are based on the uncompressed column volume. In general, an iterative procedure is needed for column sizing that will meet the pressure constraints of compressible media. For a chosen L0/u and corresponding tLoad the load ﬂow rate is calculated as QLoad = VF/tLoad, as in Example 10.1. A column diameter is then assumed as an initial guess and the superﬁcial velocity calculated as u = Q Load (π dc2 4 ). The quantities ucri, ε, L, B, and ∆P are then calculated from Equations 10.17 to 10.21. If the calculated column pressure is lower than the design value, a smaller column diameter can be used as a guess and the calculation repeated. For our case, with L0/u = 1.11 min and a 2-bar design pressure, the procedure above yields dc = 100.8 cm, L0 = 7.25 cm,

10.4 Design for Chromatographic Resolution

L = 6.41 cm, u = 6.53 cm/min, and ε = 0.207 with ucri = 11.8 cm/min. On the other hand, when L0/u = 4 min and the same design pressure, we obtain dc = 49.5 cm, L0 = 13.9 cm, L = 12.4 cm, u = 3.49 cm/min, and ε = 0.209 with ucri = 6.40 cm/min.

10.4 Design for Chromatographic Resolution

In Chapter 2 we introduced the chromatographic resolution as a simple measure of the separation attained between two components in adjacent bands (see Figure 2.11). For pulse injections, where the volume of feed injected is a small fraction of the volume of the eluted peak, the resolution was deﬁned as follows [5]: Rs =

tR ,B − tR ,A 1 (WA + WB ) 2

(10.22)

where tR,i and Wi are the retention time and baseline width, respectively, for each peak. As discussed in Chapter 8, when Rs = 1 the two peaks are almost resolved, while when Rs = 1.5 the separation is complete. If the isotherm is linear and the separated peaks are approximately symmetrical, tR,i and Wi are given by the following relationships based on the properties of Gaussian curves (see Section 8.2): L tR ,i = (1 + ki′) v

(10.23)

Hi L

(10.24)

Wi = 4tR ,i

where ki′ is the retention factor and Hi is the HETP. Combining Equations 10.23 and 10.24 with Equation 10.22 yields the result: Rs =

L kB′ − k A′ 2 (1 + k A′ ) H A + (1 + kB′ ) HB

(10.25)

For the separation of two similar components, we have H = HA ∼ HB. Accordingly, Equation 10.25 can be written as follows: Rs =

1 α − 1 k′ L 1 α − 1 k′ = N 2 α + 1 1 + k′ H 2 α + 1 1 + k′

(10.26)

which has already been given in Chapter 2 (Equation 2.11). In this equation,

α=

kB′ mB = k A′ m A

(10.27)

is the selectivity (which is equal to the ratio of the linear isotherm slopes, mA and mB) and

321

322

10 Design of Chromatographic Processes

k′ =

kA′ + kB′ 2

(10.28)

is the average retention factor. Equations 10.25 or 10.26 can be used as the basis for design for pulse injections. From Equation 10.26, the plate requirement to obtain a given resolution with a pulse injection, N0, is thus given by: 2

( )

 1 + k′  α + 1 N 0 = 4Rs2   k ′  α − 1

2

(10.29)

In turn, the required column length can be calculated if the HETP is known. The latter can be measured experimentally as illustrated in Section 8.2 or estimated as discussed in Section 8.3.2. Volume overloaded conditions, where the feed volume is a large fraction of the peak volume are however often encountered in practice and require the following modiﬁed deﬁnition of resolution [6, 7]: tR ,B − tR ,A − tF (10.30) 1 0 (WA + WB0 ) 2 where tF is the duration of the feed injection and WA0 and WB0 are the baseline widths obtained for each component with inﬁnitesimal pulse injections. The actual baseline widths obtained with a ﬁnite feed volume can be approximated as follows: Rs =

Wi ∼ Wi0 + tF

(10.31)

As in the case of inﬁnitesimal injection, Equation 10.30 predicts that the two components are fully separated when Rs = 1.5. Combining Equations 8.27 and 8.25 with Equation 10.30, we obtain the following result: Rs =

L kB′ − k A′ − tF t0 2 (1 + k A′ ) H A + (1 + kB′ ) HB

(10.32)

where t0 = L/v. For the separation of similar species, this can also be written as follows:  α − 1 − tF  tF L  1 k′ tR ,A − t0  1 k ′ α − 1 L   Rs =  = 2 1 + k α + 1 H  1 − t − t  α +1 2 1 + k ′ H  ′ R ,B R ,A    (10.33)

Accordingly, the plate requirement for a given resolution with a ﬁnite feed volume is given by: tF   N = N0 1 −  tR ,B − tR ,A 

−2

(10.34)

Since the term in parenthesis is smaller than unity, it can be see that if the same resolution is to be maintained with a preparative injection, a number

10.4 Design for Chromatographic Resolution

Feed

A+B

tF

Outlet concentration

A

A B

tR,A+0.5tF

B

tR,B+0.5tF Cycle time = 2(t R,B-tR,A)

Figure 10.5 Cyclic operation of a chromatographic resolution process for the separation of two components by isocratic elution.

of plates larger than the number required for an inﬁnitesimal pulse will be required. We can now derive a productivity expression for isocratic elution with a linear isotherm assuming that repeated injections are carried out in such a way that the resolution between bands of successive injections is equal to the resolution within an injection. With reference to Figure 10.5, we recognize that the cycle time is given by: tCycle = 2 (tR ,B − tR ,A ) = 2tF + Rs (WA0 + WB0 )

(10.35)

The fraction of the cycle over which feed is injected and number of plates required are thus given by the following equations:

θF =

tF 1 = tCycle 2 + R s (WA0 + WB0 ) tF

(10.36)

N = N 0 (1 − 2θ F )−2

(10.37)

Finally the productivity, P, deﬁned as the amount separated per unit time and unit column volume, is given by the following expression: 2

P=

( )

θF uCF uCFθF (1 − 2θF ) CF  k ′  α − 1 2 u = = θF (1 − 2θF )2 L HN 0 4Rs2  1 + k ′  α + 1 H

(10.38)

In order to maximize the productivity, the function θF(1 − 2θF)2 in Equation 10.38 should be maximized. This function is at a maximum when θF,opt = 1/6.

323

324

10 Design of Chromatographic Processes

Accordingly, for maximum productivity the feed mixture should be injected for one-sixth of the cycle time. A more rigorous analysis, using the full time-domain solution for cyclic injections with the LDF approximation (item C in Table 8.1), applicable also when the peaks are non-symmetrical, shows that the optimum θF varies somewhat with solute retention and resolution. However, even in these cases, this result provides a good approximation of the optimum. The mobile velocity is the other variable in Equation 10.38 that can be optimized since it affects the ratio u/H. In order to optimize this variable, the dependence of H on u must be known. For liquid chromatography of biopolymers, the B term in the van Deemter equation (Equation 8.49) is generally negligible so that we have: H ∼ A + Cv

(10.39)

In this case, P is maximized when u or v → ∞. This, of course, must be balanced against the cost associated with the increased pressure drop. The particles should also be as small as possible (compatible with the admissible pressure drop) in order to minimize the C term in the van Deemter equation. Example 10.3 provides an illustrative numerical example.

Example 10.3 Design for chromatographic resolution with a linear isotherm and isocratic elution In this example, we consider the separation of a mixture of two components with a linear isotherm using isocratic elution. The parameters chosen are representative of the resolution of a binary mixture by SEC. The feed mixture contains a smaller protein (B), with hydrodynamic radius rm = 5 nm and a larger one (A) with hydrodynamic radius rm = 10 nm. The pore radius, rpore, and intra-particle porosity, εp, of the stationary phase are 20 nm and 0.8, respectively. Particles with diameters of 30, 50, and 100 µm are available. Effective intra-particle diffusivities are 3 × 10−8 cm2/s and 1 × 10−8 cm2/s for B and A, respectively. Assume that the particles are rigid, that ε = 0.30, and that the design column pressure is 10 bar. A resolution Rs = 1 is desired between adjacent bands. The total feed volume is 100 l, the concentration of the protein is 10 mg/ml and its viscosity is 2 mPa s. The total processing time is 8 h. Solution – Retention factors can be estimated from Equation 3.15, which gives the following result:  r  mi = K D = ε p  1− m,i   rpore 

2

(10.40)

For our case, we obtain mA = 0.20 and mB = 0.45. The corresponding retention factors are kA′ = 0.467 and kB′ = 1.05. The HETP is estimated from Equation 8.48. For well-packed columns, we take a = 4 in the van Deemter equation (Equations 8.48 or 8.50). Neglecting the ﬁlm resistance, we have the following result:

10.4 Design for Chromatographic Resolution 5

5

200

120 Productivity

50 µm 150

3 100 2 30 µm

50

Productivity, P (mg/mL h)

30 µm

1

100

4

Column length, L (cm)

Productivity, P (mg/mL h)

4

80 3 Col. length 60 2 40 Pressure

1

50 µm

20

100 µm

30 µm

0 0

0.1

0.2

0.3

Column length (cm) and Pressure (bar)

100 µm

0 0.5

0.4

Feed time/cycle time,

F

0 0

100

200

0 300

Mobile phase velocity, u (cm/h)

Figure 10.6 Isocratic resolution design according to Example 10.3. The left panel shows the effect of the ratio of feed time and cycle time for particles of different sizes at u = 50 cm/h. The right panel shows the effect of mobile phase velocity for 30 µm particles using θF = 1/6.

Hi = 4d p +

1 1  ki′  2 d p2 u 30 (1− ε )  1+ ki′  De,i

(10.41)

Hydraulic permeability and column pressure are estimated from Equation 10.20 and 10.21, respectively. Plots of productivity and column length are easily obtained from Equation 10.38 as a function of θF and mobile phase velocity and are shown in Figure 10.6, using the average of the estimated HETPs for the two components at each value of u. As seen in Example 10.3, the optimum value of θF = 1/6 that maximizes P is independent of particle size. The column length increases very rapidly at values of θF larger than the optimum and low productivity and very long columns are obtained with 50 and 100 µm particles. Thus, we focus on the 30 µm size. As can be seen in Figure 10.6 increasing the mobile phase velocity while keeping θF = 1/6 increases the productivity. However, the incremental beneﬁts are small when u is larger than about 100 cm/h. The column pressure, on the other hand, increases rapidly at higher ﬂow rates. The 10-bar design pressure is achieved at 125 cm/h with a column length of 49 cm and P = 4.2 mg/ml/h. The corresponding feed and cycle times are 83 and 500 s, respectively. The column volume required to process the 100-l feed in 8 h is thus Vc = (100 L × 10 g/L)/(4.2 g/L h × 8 h) = 29.8 l. In turn, the column diameter is dc = ( 4 × 29.8 × 103 ) (π × 49) = 27.8 cm while the ﬂow rate is Q = 1.26 l/min. Finally, the feed volume is found to be VF = (π × 27.82 × 125 × 82.7)/ (4 × 3600 × 1000) = 1.75 l. Thus, 100/1.75 = 57 cycles will be needed to process the 100-l batch. The total eluent consumption is VE = 5 6 × 1.26 × (8 × 60 ) = 504.

325

10 Design of Chromatographic Processes A

B

B-cut

A-cut

8

6

C (mg/mL)

326

4

2

0 0

500

1000

1500

Time (s) Figure 10.7 Peak proﬁles predicted for Example 10.3 for the periodic state at 125 cm/h with 30 µm particles and maximum productivity conditions.

Note that a smaller number of batches could be used if a larger column is designed, but only at the expense of a lower productivity. Peak proﬁles calculated from item C in Table 8.1 for the conditions of Example 10.3 are shown in Figure 10.7. For these calculations we have assumed that there is no change in stationary phase properties from cycle to cycle and the peaks are only shown for the periodic state. The product cuts shown in Figure 10.7 will have the following concentrations and purities: CA (mg/ml)

CB (mg/ml)

Purity

A-cut

3.65

0.0282

99.2%

B-cut

2.99

0.0229

99.2%

It is instructive to compare the results of Examples 10.1 or 10.2 with those of Example 10.3. In the ﬁrst two cases, optimum designs call for shallow bed depths using high ﬂow rates with relatively large particles and low operating pressures. This is possible because of the highly favorable nature of protein binding. For these conditions, a small number of transfer units is sufﬁcient to approach ideal chromatography behavior (see Chapter 8). On the other hand, optimum design in Example 10.3 calls for substantially longer columns, lower ﬂow rates, smaller particles, and relatively high operating pressures. This occurs because of the larger plate requirement associated with resolution for the relatively small alpha values often encountered in SEC. Note that with compressible stationary

10.5 SMB Design

phases it may not be possible to achieve the bed heights needed to attain resolution. In this case, it may become necessary to subdivide the column length into a number of shorter column sections each of a small length. This will, of course, increase the complexity of the equipment and the cost, thereby providing even greater incentives to optimize productivity. Optimum design for resolution when the adsorption isotherms are non-linear and competitive must take into account simultaneously the equilibrium and band-broadening factors. Analytical solutions have been developed under ideal chromatography conditions (local equilibrium) for the multi-component Langmuir isotherm and have been reviewed by Guiochon et al. [8]. A common practice in small-molecule HPLC is to optimize loading under so-called ‘touching band conditions’, were both recovery and purity of the separated components are essentially 100% [9, 10]. Extensions of the approach to overlapping bands including band-broadening effects have also been developed [11], which are suitable for conditions where equilibrium effects are dominant and the column efﬁciency is high. More complicated cases involving multi-component systems and strong mass transfer limitations generally require optimization via the numerical solution of the governing equations. A number of illustrative examples can be found in ref. [8]. A common difﬁculty encountered in applying these approaches to protein chromatography is associated with the lack of precise models to describe multi-component protein adsorption equilibrium. Nagrath et al. [12] and Natarajan et al. [13] provide examples of applications to proteins with idealized equilibrium models that provide approximate descriptions. In most cases involving proteins, adsorption equilibrium is highly sensitive to the mobile phase composition making operation with a modiﬁer gradient desirable and sometimes necessary to assure robustness. In these cases, even at high initial protein loads, however, use of linear isotherm models to predict resolution is often sufﬁcient as discussed in Chapter 9.

10.5 SMB Design

The basic principle of an SMB separator was discussed in Chapter 2. The layout is illustrated in Figure 10.8 with reference to an equivalent true countercurrent moving bed (TMB) system comprising four idealized moving bed columns or zones. The feed containing components A and B is supplied between Zones II and III. The least strongly retained species, A, is recovered between Zones III and IV, while the most strongly retained species, B, is recovered between Zones I and II. The adsorbent is re-circulated from the bottom of Zone I to the top of Zone IV. An eluent or desorbent make-up stream is added to the ﬂuid recycled from Zone IV and the combined stream is fed to the bottom of Zone I. As discussed in Chapter 2, the main purpose of each zone is as follows. Zone III adsorbs B while letting A pass through. Zone II desorbs A while adsorbing B. Zone I desorbs B allowing the adsorbent to be recycled. Zone IV adsorbs A allowing the mobile

327

10 Design of Chromatographic Processes

IV

C1,i

q

IV

1,i

=q

I

=q

IV

0,i

Zone IV Raffinate, C ,u R,i

IV

C

,u

0,i

IV

q

III

1,i

III

C1,i

R

0,i

C

Zone III III

Feed, C ,u F,i

C0,i , u C

F

III

q

II

1,i

II

=q

III 0,i

1,i

Extract, C ,u E,i

E

C

S

Zone II

Adsorbent recycle, u

Mobile phase recycle, u

328

II 0,i

, uII

C

I

I

II

q1,i = q0,i

1,i

Zone I I

I

C ,u 0,i

q

I

0,i

Desorbent, u

D

Figure 10.8 True Moving Bed (TMB) equivalent of a SMB separator. All mobile phase and adsorbent velocities are based on the same column cross-section.

phase to be recycled to Zone I. Proper selection of operating conditions is needed to obtain the desired separation. This can be achieved by considering ﬁrst the true moving bed process analog and then extending the results to the SMB process. The ensuing analysis is based on ideal chromatography conditions and assumes linear isotherms with a constant modiﬁer concentration. In the following, uj represents the ﬂuid phase velocity in Zone j and uS the adsorbent velocity, with both

10.5 SMB Design

velocities deﬁned on the basis of the column cross-sectional area. Net upward transport of each component is determined by the component velocity uij = u j − miu s

(10.42)

where mi is the linear isotherm slope. The following inequalities must be satisﬁed in order to obtain the desired separation [14]: uBI > 0

(10.43a)

u IIA > 0, uBII < 0

(10.43b)

> 0, u

(10.43c)

u

III A

III B

<0

u IV A <0

(10.43d)

When these constraints are met, component A will move downward in Zone IV and upward in Zone III and will be recovered between these two zones. Similarly, component B will move upward in Zone I and downward in Zone II and will be recovered between these two zones. At steady state, with a continuous supply of the feed mixture, the separated products will emerge continuously as the A-rich rafﬁnate and the B-rich extract. Combining Equations 10.42 and 10.43 yields the following relationships [15]: M I > mB

(10.44a)

mB > M > m A

(10.44b)

mB > M III > m A

(10.44c)

< mA

(10.44d)

uj uS

(10.45)

II

M

IV

where Mj =

is the ratio of the liquid phase and adsorbent velocities in zone j. The inequalities shown in Equations 10.44b and 10.44c determine whether separation will occur under ideal conditions and can be represented on the so-called MIII − MII plane shown in Figure 10.9 [15, 16]. Because of material balance constraints MIII > MII and only the region above the 45° line is meaningful. Values of MII and MIII below mA or above mB result in incomplete separation while operation within the shaded triangular region results in complete separation. The vertex of this triangle, where the difference between MIII and MII is greatest represents the point of minimum desorbent consumption under ideal conditions. In practice, mass transfer resistances and deviations from plug ﬂow will result in imperfect separation even within the shaded region. Moreover, operation at or close to the vertex is not robust and is greatly affected by small changes in operating conditions. This can be an especially signiﬁcant problem in protein chromatography applications where fouling or other forms of degradation of the stationary phase will lead to varying equilibrium constants. To avert such problems,

329

10 Design of Chromatographic Processes

Pure B extract only B

Complete separation region

S

m

III

M = u /u

Pure A raffinate only

III

330

m

A

mA

mB II

II

M = u /u

S

Figure 10.9 Triangle diagram deﬁning the operating conditions in Zones II and III of TMB and SMB separators. The vertex point with coordinates (mA, mB) corresponds to minimum eluent or desorbent consumption. Adapted from [16].

operating away from the vertex and closer to the 45° line is usually needed to gain robustness albeit at the expense of a greater desorbent consumption. A simple criterion useful for an approximate design is obtained by introducing a safety margin, β ≥ 1 [17−19]. This parameter deﬁnes the distance from the vertex of the complete separation region in the triangular area of the diagram. Accordingly, the inequality constraints are converted into the following equalities: M I = mB β

(10.46a)

M = m Aβ

(10.46b)

M III = mB β

(10.46c)

= mA β

(10.46d)

II

M

IV

These equations enable the mobile phase velocities within each zone to be deﬁned. β = 1 yields minimum desorbent consumption under ideal conditions, while larger values provide a more robust design up to the maximum allowable value of β = mB m A . For a given β, external and internal ﬂow velocities, deﬁned in Figure 10.8, are calculated from the following relationships:

10.5 SMB Design

uS = uF (mB β − m A β )

(10.47a)

uE = uS (mB − m A ) β

(10.47b)

uR = uS (mB − m A ) β

(10.47c)

uD = uE + uR − u F

(10.47d)

uI = uS mB β

(10.47e)

uII = uI − uE

(10.47f)

uIII = uII + uF

(10.47g)

uIV = uIII − uR

(10.47h)

uC = u I − uD

(10.47i)

Note that all velocities, both external and internal, given by these equations are deﬁned as ﬂow rates divided by the column cross-section. Analogous relationships are derived for SMB systems where each zone comprises a number of ﬁxed beds operated in a merry-go-round sequence as shown in Figure 2.5. External ﬂow velocities are calculated from Equations 10.47a to 10.47d replacing uS with uS,SMB =

(1 − ε ) p

(10.48)

where is the length of a single bed and p is the switching period. Internal ﬂow velocities equivalent to the TMB operation are increased from the values calculated from Equations 10.47e to 10.47i to compensate for the extra-particle ﬂuid carried along in each bed at each switch according to the following equation: j uSMB =uj +

ε p

(10.49)

In practice, a small number of beds in series in each zone closely approaches the performance of ideal, true countercurrent system. Industrial SMB systems normally use two to three beds per zone, but operation with even one bed per zone may be effective. Note, that if a single recirculation pump is used, as is typically the case, the maximum column pressure is the sum of the pressure drops in each of the four zones. In addition to the backpressure due to ﬂow through the packed columns in an SMB system, pressure drops through valves, connecting pipes, monitors, and control elements may also be signiﬁcant. The relationships presented thus far enable the selection of operating conditions for systems that approach ideal behavior. However, complete design of SMB systems also requires consideration of band-broadening factors. For an existing SMB unit, initial stream ﬂow rates and switching period can be selected based on Equations 10.47 to 10.49 so that the operating point lies within the desired separation region of the triangle diagram. A precise determination of the column length requires, in general, a dynamic model including a description of mass transfer

331

332

10 Design of Chromatographic Processes

rates, adsorption kinetics, and axial dispersion in addition to consideration of the start up and approach to steady state as well as dead volumes and mixing between columns. In practice, however, a simpler analysis based on the steady-state TMB analogy is often sufﬁcient for preliminary design and provides information about the limits of performance that can be expected from actual SBM operations. In the ensuing analysis we assume plug ﬂow and describe mass transfer based on the linear driving force model. More general relationships considering axial dispersion-limited band broadening and cases were both axial dispersion and mass transfer effects are important are illustrated in [7] and [8]. For protein chromatography applications, however, mass transfer is usually limiting so that the equations that follow are usually adequate. Steady-state material balances for each solute i in zone j yield the following equations [20]: uj

dqˆ dCi − us i = 0 dz dz

(10.50)

(1 − ε ) ki (1 − ε ) ki dqˆi (qˆ* − qˆi ) = − (miCi − qˆi ) =− dz us us

(10.51)

where ki is the LDF rate coefﬁcient deﬁned by Equations 8.77 to 8.79. Integrating these equations from z = 0 to z yields the following results: j

j ni ζ   Cζj,i qˆ0j,i  j (1−γ i ) γ ij γ ij ˆ q0j,i 1  − 1 γie = j 1− +   j j j  C 0,i γ i − 1  miC 0,i  miC 0,i  

(10.52)

miC 0j,i − γ ij q0j,i − miCζj ,i + γ ij qζj ,i = 0

(10.53)

where C 0,j i and ˆ q0j,i are the liquid phase and adsorbent concentrations at the j j entrance of each zone j and Cζ ,i and qˆζ ,i are the corresponding values at a dimensionless distance ζ from the entrance. The variables in these equations are deﬁned as follows:

γ ij = ζ=

miu s uj

z L

nij =

(1 − ε ) ki miL uj

(10.54) (10.55) (10.56)

Note that nij is the number of transfer units of component i in Zone j and is given in Table 8.2 for different rate mechanisms. Its value changes from zone to zone with the varying mobile phase velocities. However, the ratio Sti = nij γ ij = (1 − ε ) kiL u s , known as the Stanton number, is the same for all zones if ε and L are the same since us is obviously constant through all zones. The following additional relationships are easily derived from material balances and continuity at the interfaces or nodes between zones:

10.5 SMB Design

qˆ0I ,i = qˆ1IV,i , ˆ q0j,i = ˆ q1j,−i 1 C 0I ,i =

for

j = II , III , IV

(10.57)

u IV IV C1,i uI

(10.58)

C 0II,i = C1I,i = CE ,i C 0III,i =

(10.59)

uFCF ,i + u IIC1II,i u III

(10.60)

C 0IV,i = C1III,i = CR ,i

(10.61)

Setting ζ = 1, Equations 10.52 through 10.61 form the following system of linear equations, solvable, for example, by matrix inversion. I  γi I I γ (1 − δ iI ) [ i δ i − 1]   mi   I mi −γ i    0 0   0 0   0 0    0 0    uI IV 0  IV (1 − γ i ) u  uI γ iIV  − IV mi u

(1 − γ iI )

0

0

−m i

γ iI

0

[γ iIIδ iII − 1]

γ II   mi (1 − δ i ) −γ iII

(1 − γ iII )

0

0

0

0

0

0

γ iII

0

γ i (1 − δ iIII )  mi 

(1 − γ iIII )

−γ iIII

mi

0

[γ iIV δ iIV − 1]

0

mi

III

mi 0

II i

−m i II u  III III  u III (γ i δ i − 1) uII mi uIII

0

0

0

0

0

0

0

0

0

0

    0   0  CI    0 ,i  0     I   qˆ0,i 0     0  I   0   C1,i    I   ˆ q1,i    uFCF ,i    III III  = ⋅ 0  −  III (γ i δ i − 1)  II   C  u   1,i    ˆ uFCF ,i  q1II,i     − III mi 0   u  C1III,i       III 0   ˆ  γ iIII  q   1 i ,    0 IV   γ i IV   mi (1 − δ i )    −γ iIV 0

(10.62)

333

334

10 Design of Chromatographic Processes

where: nj   δ ij = exp (1 − γ ij ) ij  γi  

(10.63)

The steady state productivity is then calculated as P = uFCF/(4 × L) while the speciﬁc desorbent consumption is uD/(uFCF). Pressure drops in each zone are calculated using the SMB internal velocities (uI, uII, uIII, and uIV) and the hydraulic permeability. The following example provides a numerical illustration of the use of these relationships.

Example 10.4 Design an SMB separator for the resolution of a binary mixture with a linear isotherm In this example, we consider the separation of a mixture of two components with a linear isotherm as described in Example 10.3. Assume again that the particles are rigid, that ε = 0.30, and that the design column pressure is 10 bar. Minimum product purities of 99.2% are desired for both extract (B-rich) and rafﬁnate (A-rich) products. Design for 30 µm particles with a safety factor β = 1.05. Solution – From the solution of Example 10.3, the linear isotherm slopes are mA = 0.20 and mB = 0.45. Neglecting axial dispersion and assuming mass transfer is controlled by pore diffusion, the LDF rate coefﬁcient is given by the following equation: ki =

15De,i rp2mi

For 30 µm diameter particles, this equation gives kA = 20.0 min−1 and kB = 26.7 min−1. The design requires a trial and error calculation to determine the value of the quantity L/uF that yields the desired minimum product purities. For this purpose, for an assumed L, external and internal velocities can be calculated from Equations 10.47a through 10.47i for each uF value. Rafﬁnate and extract product concentrations and purities are then calculated by solving Equation 10.62 and taking CE ,i = C1I,i and CR,i = C1III,i . If the purities are insufﬁcient, uF is reduced (or L/uF increased) and the calculations repeated. Once the purity requirements are met, the productivity is calculated as P = uFCF/4L. The total SMB column volume is then obtained by dividing the required processing throughput in mg/h by the productivity, P. Column diameter and length can then be adjusted to meet the pressure drop constraint. The backpressure is calculated for each zone from the following equation based on Darcy’s law: ∆P j =

j ηLuSMB B0

(10.64)

10.5 SMB Design

where η is the viscosity (0.002 Pa·s in this example problem), B0 = 3.31 × 10−13 m2 j is the hydraulic permeability (estimated from Equation 10.20), and uSMB is the equivalent velocity in the actual SMB system. The latter is calculated from Equation 10.49. Finally, the switching period, p, is calculated from Equation 10.48 once the number of column sections per zone is chosen. The latter choice will require detail simulations of the transient behavior. In practice, however, SMB systems with two columns per zone will usually provide a reasonably close approximation of the ideal TMB performance. For the conditions at hand, we obtain the results shown in Table 10.1, with the switching period calculated assuming two columns per zone. Calculations were carried out in Microsoft Excel using the function {MMULT(MINVERSE(array1),(array2))} where array1 and array2 are the coefﬁcient matrices in Equation 10.62. Table 10.1 Summary of SMB design parameters according to Example 10.4 with a safety margin β = 1.05. External and internal volumetric ﬂow rates are obtained multiplying the superﬁcial velocities given by the column cross-sectional area.

L/uF = 15.6 min, P = 9.62 mg/ml h, Vc,total = 13.0 l L = 10.7 cm, uF = 0.686 cm/min, dc = 19.7 cm us = 3.14 cm/min, p = 1.19 min, uD = 0.885 cm/min, uC = 0.598 cm/min Products

Extract Rafﬁnate

u (cm/min)

CA (mg/ml)

CB (mg/ml)

Purity

0.824 0.747

0.067 9.11

8.26 0.069

99.2% 99.2%

j (cm min) uSMB

∆Pj (bar)

Internal ﬂows Zone

uj (cm/min)

I II III IV

1.48 0.659 1.34 0.598

2.83 3.13 2.86 2.07

3.01 2.13 2.86 2.07

The equivalent steady-state TMB concentration proﬁles in each zone for Example 10.4 can be obtained from Equation 10.52 and are shown in Figure 10.10. As shown by several authors [21–24], the actual SMB periodic steady proﬁle, averaged over the switching time can be expected to be close to those shown in this ﬁgure. Thus, the TMB approach is a useful approximation for preliminary design and optimization. It usually forms the basis for more detailed calculations that can be performed via the numerical solution of the transient balances. Such calculations are computationally intensive but have the advantage that various non-idealities,

335

10 Design of Chromatographic Processes 10 A B 8

C (mg/mL)

336

6

4

2

0

0

1

2

Desorbent uD = 0.885

Extract uE = 0.824

Feed uF = 0.686

3

4

Raffinate uE = 0.747

= z/L Figure 10.10 Concentration proﬁles for SMB separation according to Example 10.4 calculated with the TMB model for the conditions shown in Table 10.1.

including dead volumes in connecting lines and packing heterogeneities can be simulated along with the effects of periodic column switching. It should be noted that the conditions in Table 10.1 are based on a relatively small safety margin. Accordingly, the design is not very robust in the sense that small changes in the system will result in potentially large variations in product purity. Increasing β moves us away from the vertex of the complete separation region of the triangle diagram improving robustness but increasing desorbent consumption. The effects of increasing β predicted by the TMB model for the conditions in Example 10.4 are shown in Figure 10.11. The predictions were made assuming that the zone length L is kept constant at 10.7 cm. As can be seen in Figure 10.7, as β is increased, the residence time L/uF must be increased (by reducing uF) in order to meet the imposed 10-bar pressure constraint. As a result, the productivity decreases and consumption of the speciﬁc desorbent increases slowly at ﬁrst and then rapidly as the maximum value of β = 1.5 is approached. Obviously, optimum design will require careful consideration of economic and safety factors in order to arrive at conditions that are both stable and efﬁcient. It is instructive to compare the results of Example 10.4 to those obtained in Example 10.3 for batch elution chromatography. Production rates and pressure constraints are identical for both. However, with a safety margin of 1.05, the SMB operation reduces the total column volume from 29.8 to 13.0 l, increases

10.5 SMB Design 60 L/uF

L/uF (min)

8

8

6

uD/uF

Productivity (mg/mL x h)

40

10

6

4

30

4

20

2

2

0

0

10 1

1.1

1.2

1.3

Spec. desorbent usage (u D/uF)

Productivity 50

10

1.4

Safety margin, Figure 10.11 Effect of safety margin on productivity and speciﬁc desorbent consumption for the conditions in Example 10.4 with L = 10.7 cm.

the productivity from 4.20 to 9.62 mg/ml/h, and reduces the eluent or desorbent consumption from 504 to 129 l. Accordingly, the product concentrations are increased from 3.65 to 9.11 mg/ml for the A-rich product and from 2.99 to 8.26 mg/ ml for the B-rich product. Such improvements in productivity and eluent consumption are typical of what can be expected for these conditions and are even greater for more difﬁcult separations where the selectivity is lower. The improvements are, of course, less impressive if a more robust SMB design is implemented with a larger safety margin. In either case, however, the price of such improvements is, of course, the complexity of the equipment and the associated validation requirements. Several practical issues can reduce the efﬁciency of SMBs. One is the fact that, in practice, it is difﬁcult to pack identical columns and column-tocolumn variability can lead to signiﬁcant reduction in performance [24]. A second issue is associated with the possibility of degradation of the stationary phase. If the isotherm parameters change over time, product purities can change dramatically. Such changes can be accommodated by manually adjusting the SMB operating parameters. Sophisticated automatic control systems for SMB separators have been developed in recent years and have been applied to the separation of enantiomers and other small-molecule ﬁne chemicals [25–28]. However, applications to protein manufacturing on an industrial scale have not yet been reported. In general, SMB operation will be advantageous for difﬁcult separations (low alpha values), when the stationary phase is stable and fouling is not signiﬁcant, and when the stationary and/or mobile phases are expensive. In these cases, the efﬁciency gained with SMB operation will typically compensate for the increased cost and complexity of the equipment. Operation with a non-linear isotherm can

337

338

10 Design of Chromatographic Processes

be analyzed in a manner similar to that described above. In this case, the right triangle deﬁning the complete separation region in Figure 10.8 is distorted, acquiring one or more curved sides and further restricting the range of conditions leading to complete separation. As is evident from the analysis above, only binary separations are achievable with a four-zone system. Multi-component separations require multiple SMB units or integrated units comprising more than four zones. Useful references covering SMB design for linear and non-linear isotherms are [15, 17, 18, 29–31]. Examples of laboratory scale proteins separations by SMB can be found in [32–36], including design approaches.

References 1 Lightfoot, E.N., and Cockrem, M.C.M. (1987) Sep. Sci. Technol., 22, 165. 2 Fahrner, R.L., Iyer, H.V., and Blank, G.S. (1999) Bioprocess Eng., 21, 287. 3 Ghose, S., Nagrath, D., Hubbard, B., Brooks, C., and Cramer, S.M. (2004) Biotechnol. Progr., 20, 830. 4 Stickel, J.J., and Fotopoulos, A. (2001) Biotechnol. Progr., 17, 744. 5 Snyder, L.R., and Kirkland, J.J. (1979) Introduction to Modern Liquid Chromatography, John Wiley & Sons, Inc., New York. 6 Le Goff, P., and Midoux, N. (1981) Energetics and cost optimization of preparative chromatography columns, in Percolation Processes: Theory and Applications (eds A.E. Rodrigues and D. Tondeur), Sijthoff & Noordhoff, Alphen aan den Rijn, The Netherlands, pp. 197–247. 7 Ruthven, D.M. (1984) Principles of Adsorption and Adsorption Processes, John Wiley & Sons, Inc., New York. 8 Guiochon, G., Felinger, A., Shirazi, D.G., and Katti, A. (2006) Fundamentals of Preparative and Nonlinear Chromatography, Elsevier Academic Press, Amsterdam. 9 Knox, J.H., and Pyper, H.M. (1986) J. Chromatogr., 363, 1. 10 Golsham-Shirazi, S., and Guiochon, G. (1990) J. Chromatogr. A, 517, 229. 11 Golsham-Shirazi, S., and Guiochon, G. (1991) J. Chromatogr. A., 536, 57. 12 Nagrath, D., Bequette, B.W., Cramer, S.M., and Messac, A. (2005) AIChE J., 51, 511.

13 Natarajan, V., and Cramer, S.M. (2000) J. Chromatogr. A, 876, 63. 14 Ching, C.B., and Ruthven, D.M. (1984) Can. J. Chem. Eng., 62, 398. 15 Mazzotti, M., Storti, G., and Morbidelli, M. (1997) J. Chromatogr. A, 769, 3. 16 LeVan, M.D., and Carta, G. (2007) Adsorption and ion exchange, in Perry’s Chemical Engineers’ Handbook, Section 16, 8th edn (ed. D.W. Green), McGrawHill, New York, pp. 16-1–16-69. 17 Ruthven, D.M., and Ching, C.B. (1989) Chem. Eng. Sci., 44, 1011. 18 Zhong, G., and Guiochon, G. (1996) Chem. Eng. Sci., 51, 4307. 19 Seader, J.D., and Henley, E.J. (2006) Separation Process Principles, 2nd edn, John Wiley & Sons, Inc., New York, pp. 598–606. 20 Dunnebier, G., Fricke, J., and Klatt, K.U. (2000) Ind. Eng. Chem. Res., 39, 2290. 21 Ching, C.B., Hidajat, K., Ho, C., and Ruthven, D.M. (1987) React. Polym., 6, 15. 22 Ching, C.B., Ho, C., and Ruthvne, D.M. (1988) Chem. Eng. Sci., 43, 703. 23 Beste, Y.A., Lisso, M., Wozny, G., and Artl, W. (2000) J. Chromatogr. A, 868, 169. 24 Mihlbachler, K., Jupke, A., SeidelMorgenstern, A., Schmidt-Traub, H., and Guiochon, G. (2002) J. Chromatogr. A, 944, 3. 25 Klatt, K.U., Hanisch, F., and Dunnebier, G. (2002) J. Proc. Control, 12, 203. 26 Abel, S., Erdem, G., Mazzotti, M., Morari, M., and Morbidelli, M. (2004) J. Chromatogr. A, 1033, 229.

References 27 Erdem, G., Abel, S., Morari, M., Mazzotti, M., Morbidelli, M., and Lee, J.H. (2004) Ind. Eng. Chem. Res., 43, 405. 28 Grossmann, C., Erdem, G., Morari, M., Amanullah, M., Mazzotti, M., and Morbidelli, M. (2008) AIChE J., 54, 194. 29 Storti, G., Masi, M., Carra, S., and Morbidelli, M. (1989) Chem. Eng. Sci., 44, 1329. 30 Khattabi, S., Cherrak, D.E., Mihlbachler, K., and Guiochon, G. (2000) J. Chromatogr. A, 893, 307. 31 Minceva, M., and Rodrigues, A.E. (2002) Ind. Eng. Chem. Res., 41, 3454.

32 Gottschlich, N., and Kasche, V. (1997) J. Chromatogr. A, 765, 201. 33 Xie, Y., Mun, S.Y., Kim, J.H., and Wang, N.H.L. (2002) Biotechnol. Progr., 18, 1332. 34 Houwing, J., Billiet, H.A.H., and van der Wielen, L.A.M. (2003) AIChE J., 49, 1158. 35 Li, P., Xiu, G., and Rodrigues, A.E. (2007) AIChE J., 53, 2419. 36 Li, P., Yu, J.G., Xiu, G.H., and Rodrigues, A.E. (2008) Sep. Sci. Technol., 43, 11.

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Index a accessible intra-particle porosity 70 adsorbed phase diffusion 162ff. adsorption azeotropes 228 adsorption equilibria 145ff. adsorption isotherm 147f. adsorption kinetics 161ff., 237ff. adventitious agent 44 afﬁnity chromatography 60, 100 air sensor 141 amino acid 4f. ampholyte 300 anion exchanger 95 antibody 1 – puriﬁcation 50, 51, 60, 94, 100–103, 161, 168, 179, 269, 306, 317 asymmetry factor 242 attenuated total reﬂectance Fourier transform infrared (ATR FT-IR) spectroscopy 9 axial diffusion 256 axial dispersion coefﬁcient 202

b backpressure 334 backwards ﬁnite differences 247 bacterial protein – immunoglobulin-binding 102 band broadening 283 band compression 280 base matrices 109 baseline width 240 batch adsorption 183ff. – kinetics 179ff. bed compression ratio – critical 320 BET (Brunauer, Emmett, and Teller) isotherm 152 binding capacity – maximum 158

binding charge 153 binding kinetics 146 biomolecules 2 – chemistry and structure 2ff. – physiochemical property 19 biopharmaceuticals 1 – quality 45 – regulatory aspect 45 biospeciﬁc interaction chromatography (BIC) 60, 99 Biot number 191 biotechnology product – downstream processing 1ff. – role of chromatography in downstream processing 49 breakthrough curve 245ff., 258ff. bubble trap 135, 141 buffer 105ff.

c capture 311ff. capture design 313ff. – compressible media 319 – rigid particles 317 carbohydrate polymer – natural 109 cation exchanger 95 ceramic hydroxyapatite (CHT) 103 chaotropic salt 32 characteristic velocity 207, 225, 286 chromatofocusing 300ff. chromatographic process – design 309ff. – productivity 310 – step 311 chromatographic resolution 78, 321 – design 321ff. chromatographic velocity 205 chromatography column – dynamics 201ff.

Protein Chromatography: Process Development and Scale-Up. Giorgio Carta and Alois Jungbauer © 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 978-3-527-31819-3

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Index chromatography media 85ff. circular dichroism (CD) 7 clean-in-place (CIP) 316 coherent paths 218 column efﬁciency 74, 237ff. – empirical characterization 238 – modeling and prediction 246 column packing 141 column pressure 325 combined mass transfer resistances 188 competitive adsorption 270 complexation 97 compression factor 142 concentration overload 214 conductivity monitor 132 constant pattern breakthrough curve 267 – expressions 263ff. constant-separation factor isotherm 151, 217, 259ff. consumption of solvents and process chemicals 310 continuous annular chromatography (CAC) 58f. continuous stationary phase 70, 108 controlled pore glass (CPG) 112 cosmotropic salt 32 critical bed compression ratio 320 critical operational parameter 46f. critical velocity 121 culture broth – component 43 culture media 41

d Darcy’s law 81, 320 deamidation 14f. Debye length 96, 157 Debye–Hückel theory 29, 89 deoxyribonucleic acid (DNA) 16 detector 127 dextran-based media 109 differential migration 59 diffusion – adsorbed phase 170 – axial 256 diffusional hindrance coefﬁcient 167 diffusivity 36, 166 – eddy 256 – effective 189, 256 dispersion 237ff. – axial 72 – hydrodynamic 256 displacement development 65, 203, 217, 227, 273

displacement pump 128, 140 distribution coefﬁcient 24, 79, 86 disulﬁde bond 5 disulﬁde bridge – UV absorbance 20 dynamic binding capacity (DBC) 80, 258, 269, 314 dynamic light scattering (DLS) 23

e eddy diffusivity 256 effective charge 153 effective diffusivity 189, 256 effective pore diffusivity 166ff., 255 electrostatic interaction 94 ellipticity 7 elution chromatography 63, 203 endotoxin 16f. – chemical structure 18 – units (EU) 17 environmental scanning microscopy (ESM) technique 118 equilibration step 316 equilibrium binding capacity (EBC) 80, 258 exponentially modiﬁed Gaussian (EMG) – function 243 – peak 243ff. expression system 37 extract 329 extra-column effects 134 external ﬁlm and intra-particle diffusion in series 189, 264 external mass transfer coefﬁcient 166 external mass transfer control 184 extra-column effect 134

f fast protein liquid chromatography (FPLC) 129 ﬁlm mass transfer coefﬁcient 163 ﬂow velocity – external and internal 330 free surface model 174 Freundlich isotherm 151 frontal analysis 203, 259 frontal chromatography 64 frontal loading – Langmuir isotherm 209 – Langmuir–Freundlich isotherm 215 – multi-component Langmuir isotherm 219 fronting 215

Index fusion partner – protein puriﬁcation 102

g gel ﬁltration 86 generalized van Deemter equation 75, 77 gradient elution chromatography 63, 277ff. – linear isotherm 279 – modeling with non-linear isotherms

h h-transformation 217, 234 Hamaker constant 157 height equivalent to a theoretical plate (HETP) 74, 142, 238ff. – calculation from experimental pulse response peak 244 – estimation 290 Henry constant 150 high performance liquid chromatography (HPLC) 75, 129 His-tag 98 hodograph 218ff. Hofmeister series 30ff., 92 hold point 312 homogeneous diffusion 171 host cell composition 40 hydraulic permeability 174, 325 hydrodynamic dispersion 256 hydrodynamic radius 86 hydrophobic adsorption 286 hydrophobic interaction chromatography (HIC) 28, 54, 60, 87ff., 278 – LGE relationships 295 hydrophobicity 27 hydroxyapatite 103ff., 112

i ideal model of chromatography 71, 201, 205 iminodiacetic acid (IDA) 97 immobilized metal afﬁnity chromatography (IMAC) 54 immunoglobulin-binding bacterial protein 102 International Conference on Harmonization (ICH) 46 interstitial velocity 70, 125 intra-particle convection 71, 173ff. intra-particle Peclet number 176f. intra-particle permeability 175 intra-particle porosity 70, 167 inverse size exclusion chromatography (iSEC) 114f.

ion-exchange chromatography (IEC) 278 ionic strength 96, 146 irreversible isotherm 265 isocratic elution 63, 80 – design for chromatographic resolution 324 – preparative injection 212 isoelectric focusing (IEF) 303 – gel separation 13 isopropyl-β-D-thiogalactopyranoside (IPTG) 42 isotachic train 65, 228ff. – prediction 228ff.

60, 94,

j J function 252 Jones–Doyle B-coefﬁcient 32

k Karman–Cozeny equation 120, 174, 320 kinetic resistance to binding 162, 178 kinetic stability 32 Kohlrausch’s law 133 kosmotropic salt 32

l Langmuir isotherm 149ff., 259ff., 273 – elution of a saturated column 211 – frontal loading of a column 209 – prediction of isotachic train 230 Langmuir–Freundlich isotherm 151f. – frontal loading of a column 215 light scattering 23 linear driving force (LDF) – approximation 182, 285 – model 249ff. – rate coefﬁcient 250, 267, 286 – solution 187 linear gradient elution (LGE) 278, 280, 295 – relationships for ion exchange chromatography 286 linear isotherm 249f., 279 – design for chromatographic resolution 324 limulus amoebocyte lysate (LAL) test 17 loading factor 250 local equilibrium theory 205 log-virus reduction (LVR) factor 44

m macropore diffusion 165 macroporosity 287

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Index Mark–Houwink equation 35 mass transfer ﬂux 171, 183 mass transfer rate 182 mercury intrusion 114f. metal chelate afﬁnity chromatography (MIC) 97 metal ion interaction chromatography (MIC) 60 microheterogeneity 48, 309 micropore diffusion 171 mixed mode 103 molal surface tension increment 90 molar extinction coefﬁcient 20 molecular diffusion coefﬁcient 36 monolith 70, 76, 108, 109, 113ff., 118ff., 139, 178ff., 240 multi-component isotherm 202 multi-component Langmuir isotherm 219 – frontal loading of a binary mixture 219 multi-component protein adsorption kinetics 195 multicolumn countercurrent solvent gradient puriﬁcation process (MSGP) 69 multimodal interaction 103

n Navier–Stokes equation 174 nitrogen adsorption 114 normalized gradient slope 281 number of transfer units 250 – pore diffusion 314

o oligonucleotide 15 on-column generation of induced pH gradients 300

p parallel pore and solid diffusion 189, 264 particle size distribution (PSD) 119, 189, 269 peak asymmetry factor 142, 243 peak compression 283, 284 peak fronting 243 peak skew 242 peak width measured at mid height 240 Peclet number 176, 204 pellicular stationary phase 70 peptide bond 3 perfusion 71, 173 permeability – extra-particle 174

– hydraulic 174, 325 – intra-particle 175 pH elution 302 pH gradient 299ff. – on-column generation of induced pH gradients 300 pH monitor 133 piston pump 128, 129 plate model 246 plate number 74ff., 238 polynucleotide 15 pore and solid diffusion in bi-disperse particles 189, 264 pore diffusion 162ff., 267 – number of transfer units 314 pore diffusion control 186 pore size 113 porosity 70, 113 – extra-particle 70 – intra-particle 70, 167 post-translational modiﬁcation 12f. primary recovery 49 process-scale column 135 process-scale system 140 product quality requirement 43 productivity 313ff. – chromatographic process 310 protein 2f. – α-helix 6f. – β-sheet 6ff. – charge 27 – diffusivity 36 – expression system 37 – folding 11 – post-translational modiﬁcation 12f. – primary structure 3 – puriﬁcation 50, 102 – quaternary structure 10 – radius 167 – secondary structure 6 – size 21f. – solubility 29 – speciﬁc absorbance 20 – structural hierarchy 11 – tertiary structure 9 protein A 101ff. protein A adsorbent 269 pulse damper 129 pulse injection 203 pulse response 240ff. pumps 128ff. purity 43 – requirement 47 pyrogen 16

Index

r radial ﬂow chromatography 139 radius of gyration 86 rafﬁnate 329 rafﬁnate stream 67 rate coefﬁcient 272 rate equation 181, 202, 258, 272 rate model – competitive adsorption system 270 – linear isotherm 249 – non-linear isotherm 258 recombinant human erythropoietin (rhEPO) 13 rectangular isotherm 265 reduced HETP 74 reduced velocity 74 refolding 51 refractive index 23, 132 relative permittivity 96 resolution 285ff., 322 – binary mixture with a linear isotherm 334 retention factor 79, 92, 204, 286ff., 321 – estimation 288 retention time 285 reversed phase chromatography (RPC) 60, 88ff., 277 – LGE relationships 295 Reynolds number 163, 256 ribonucleic acid (RNA) 16

shock wave 217 shrinking core model 187, 265 simulated moving bed (SMB) 54, 60 – design 327ff. – separation 312 – system 334ff., 335 Sips isotherm 151 size exclusion chromatography (SEC) 22f., 86, 312 small angle X-ray scattering (SAXS) 24 solid diffusion 171, 189, 264 solid diffusion control 184 solid diffusion in bi-disperse particles 189 solubility 29, 89 solvophobic theory 297 stability 32 standard operating protocol (SOP) 46 Stanton number 332 Staphylococcal protein A (SpA) 51, 101, 103, 113, 179 steric hindrance 155 steric mass action (SMA) model 97, 153ff., 231ff., 267 stoichiometric displacement (SD) model 97, 153, 286 Stokes–Einstein equation 23, 36 superﬁcial velocity 70, 125, 173 surface area 113 surface diffusion 171 surface tension 89 system volume 134

s salting-in coefﬁcient 89 salting-out coefﬁcient 89 scaling relationships 81ff. scanning electron microscopy (SEM) 118 Scatchard plot 150 Schmidt number 163 selectivity 158ff. selectivity coefﬁcient 321 sensitivity coefﬁcient 90ff., 295 – protein 296 separation factor 150, 184 separation performance metrics 74 Sephadex 110, 172 Sepharose 66, 110, 117, 118, 120, 122, 189ff., 193, 244, 252, 255ff., 268, 288, 297ff., 306 Sherwood number 163f shielding factor 155 shock front 209 shock layer thickness 270ff. shock velocity 220ff.

t Temkin isotherm 151 theory of coherence 217 Thomas solution 258ff. throughput parameter 204 tortuosity factor 167, 256 total column porosity 70 Toth isotherm 151 transfer function 254 transient development 234 transmission electron microscopy (TEM) 118 transport mechanism – experimental veriﬁcation 190 tris(carboxymethyl)ethylenediamine (TED) 97 true countercurrent moving bed (TMB) system 67, 327ff.

u urea 32 ultraviolet/visible absorbance (UV/VIS)

132

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346

Index UV absorbance 19 – aromatic amino acid – disulﬁde bridge 20

20

v validation 45 van Deemter equation 257, 324 – generalized 77 velocity 121, 139 – characteristic 207, 225 – critical 121

– interstitial 70, 125 – superﬁcial 70, 125 viscosity 33 – intrinsic 34 – relative 34 viscous ﬁngering 72f. volume overload 254

w wash step

315