Mathematical Theory of Electrophoresis PDF
Preview Mathematical Theory of Electrophoresis
Mathematical Theory of Electrophoresis Mathematical Theory of Electrophoresis v. G. 8abskii M. Yu. Zhukov V. I. Yudovich Institute of Molecular Biology and Genetics Academy of Sciences of the Ukrainian SSR Kiev, USSR Translated from Russian by Cathy Flick CONSULTANTS BUREAU NEW YORK AND LONDON Library of Congress Cataloging in Publication Data Babskil, V. G. (Vitalfi Genrikhovich) [Matematicheska(a teorila elektroforeza. English] Mathematical theory of electrophoresis f V. G. BabskiY, M. Yu. Zhukov, and V. I. Yudovich; translated from Russian by Cathy Flick. p. cm. Translation of: Matematicheskala teoriia elektroforeza. Bibliography: p. ISBN-13:978-1-4612-8225-9 e-ISBN-13:978-1-4613-0879-9 001: 10.1007/978-1-46\3-0879-9 1. Electrophoresis-Mathematical models. I. Zhukov, M. IU. (Mikhail t'iJr'evich) II. IUdovich, V. I. (Viktor losifovich) III. Title. QP519.9.E434B3313 1988 88·25282 574.87'072-dc19 This translation is published under an agreement with the Copyright Agency of the USSR (VAAP) (S) 1989 Consultants Bureau, New York Softcover reprint of the hardcover 1st edition 1989 A Division of Plenum Publishing Corporation 233 Spring Street, New York, N.Y. 10013 All rights reserved No part of this book may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording, or otherwise, without written permission from the Publisher PREFACE The development of contemporary molecular biology with its growing tendency toward in-depth study of the mechanisms of biological processes, structure, function, and identification of biopolymers requires application of accurate physicochemical methods. Electrophoresis occupies a key position among such methods. A wide range of phenomena fall un der the designation of electrophoresis in the literature at the present time. One common characteristic of all such phenomena is transport by an elec tric field of a substance whose particles take on a net charge as a result of interaction with the solution. The most important mechanisms for charge generation are dissociation of the substance into ions in solution and for mation of electrical double layers with uncompensated charges on particles of dispersed medium in the liquid. As applied to the problem of separation, purification, and analysis of cells, cell organelles, and biopolymers, there is a broad classification of electrophoretic methods primarily according to the methodological charac teristics of the process, the types of supporting media, etc. An extensive literature describes the use of these methods for the investigation of differ ent systems. A number of papers are theoretical in nature. Thus, the mi croscopic theory has been developed rather completely [13] by considering electrophoresis within the framework of electrokinetic phenomena based on the concept of the electrical double layer. Having begun to work with several problems in electrophoresis, we observed that most questions of practical importance which are of interest to experimenters are macroscopic in nature. Examples are the formation of zones with elevated amounts of individual components, the degree of separation, phenomena connected with diffusion and heat evolution, and the distribution of the electric field and charges. Such questions may be answered only by studying the physicomechanical properties of the v vi PREFACE medium in which electrophoresis is carried out, the interactions of individ ual components of the medium with each other and with the electric field, chemical reactions, and heat and mass transport. All these processes are closely interconnected and cannot be studied in isolation. We have become convinced that electrophoretic theory should be based on equations describing the hydrodynamics, electrodynamics, and thermodynamics of multicomponent, chemically reactive mixtures, i.e., on the most general equations of the physics of continuous media. In this case, we no longer need phenomenological verification of a specific electrophoresis model, and, in particular, we eliminate the possibility of omitting various cross-coupled phenomena. To our knowledge, there are no such general equations in the literature (see, for example, [11, 19,30, 32, 34, 43]). The derivation of these equations as presented in Chapter I is based, as usual, on application of the methods of nonequilibrium ther modynamics [3, 10, 11, 14,38, 54]. We hope that these equations will be useful far beyond the range of phenomena connected with electrophoresis. The extreme complexity of this system practically eliminates its direct use due to its generality. However, such a system may serve as a basis for constructing mathematical models describing different specific situations. Such simplifications decrease the dimensionality of the system and are based on comparative analysis of the importance of different effects. Secondary effects are discarded and the important ones taken into account using asymptotic methods. In the literature, the process usually begins with special models, from which we go to the more general ones. In our view, a method based on going from the most general model to a specific one has a number of ad vantages. There is no need to go over the entire route each time the balance equations for mass, energy, etc., are derived. Reliable control over the as sumptions made in the derivation of a specific model and the conditions for its applicability are achieved. The greater flexibility of the approach is ap parent in the possibilities for improving the model by more accurate ac counting of different effects. An unavoidable limitation of the physics of continuous media and, in particular, the thermodynamics of irreversible processes is the presence in the basic equations of transport coefficients (viscosity, thermal conduc tivity, diffusion, mobility, etc.) which must be determined from experi ment. The difficulties are aggravated if we need to know the dependence of these coefficients on the thermodynamic parameters (temperature, pres sure, concentration, electric and magnetic field, etc.). In principle, these dependencies should be described by a different (microscopic) theory, perhaps from statistical physics, but at the present time these dependencies are known only in isolated cases. The experimental difficulties arising in PREFACE vii this case are obvious: it is easy to determine the parameter, it is more com plicated to obtain the experimental curve, and it is practically impossible to determine how the transport coefficients depend on two or more thermodynamic parameters. The general approach developed in this monograph allows us to de termine the relevant relations for multicomponent mixtures from the general equations using an asymptotic method which separates fast and slow vari ables. Thus, we can determine the concentration dependence of the mobility and the electrical conductivity, as specified by the chemistry of a given multicomponent mixture. Finally, the form of the material relations is determined only for pseudocomponents, chemical complexes formed as a result of chemical reactions in solution. The determination of transport coefficients for the individual components included in the complex still remains unsolved. It is also important that with the deductive approach it is easy to discover diverse connections and parallels between different specific models. Thus, we perceive a profound internal commonality in the phenomena and models of isotachophoresis, zone electrophoresis, and isoelectric focusing. In constructing specific models, it was found that the extant classifi cation of types of electrophoresis is often inadequate. Expansion of the methodological nomenclature often follows explicit commercial goals. A meaningful classification scheme should obviously be derived within the framework of the theory of electromigration phenomena, and be based pri marily on the composition and properties of the buffer systems and mixtures to be separated. For example, when the concentrations of the substances to be separated are comparable with the buffer concentration, zone electrophoresis transforms into isotachophoresis. Also, if the iso electric point of a specific substance falls within the proper pH range, zone electrophoresis and isoelectric focusing may be carried out simultaneously in the chamber. Nevertheless, for convenience in comparison with known results, we have decided to adhere to generally accepted terminology. The current theory makes it possible to describe the time evolution of electrophoretic processes, including such phenomena as the rearrangement of zones and the establishment of a steady state in isotachophoresis, diffusion dispersion and the creation of meniscus-forming zones in zone electrophoresis, and the evolution of an artificial pH gradient and zones close to the isoelectric point in isoelectric focusing. In addition, known theoretical relationships, which usually pertain to the steady state, acquire a more .profound physical interpretation and prove to be special cases of the universal patterns for these phenomena. Thus, the Kohlrausch relation for a concentration jump in isotachophoresis proves to be no different than the Hugoniot condition across a shock wave. viii PREFACE The concept of an "infinite-component" mixture developed in this work is important. It allows us to describe phenomena, such as the cre ation of a natural pH gradient in a mixture with a very large number of ampholine-type carrier ampholytes, which are at first glance inaccessible to theory. These mixtures are not described by discrete sets of concentra tions, but instead by distribution functions in a space of continuous latent parameters. Work with the models presented in this book has disclosed that ap plications of the theory are not exhausted by phenomena known from ex periment. Within the framework of this theory, we have obtained a num ber of new qualitative results, e.g., the phenomena of electrolytic memory and electrolytic mixing observed in the study of isotachophoresis. The former is connected with the appearance of an absolute Riemann invariant, a function of concentration conserved over time for each point in the solu tion. The second phenomenon is described from the mathematical point of view as a progressive rarefaction wave. In constructing the theory, we have tried to consider all experimental data available to us. The subsequent fate of the theory depends on the existence of an inverse relationship with experiment. We need a thorough test of its qualitative and quantitative predictions, measurement of the parameters introduced, and estimates of the relative roles of various effects. In conclusion, we mention some peculiarities of this monograph connected with the fact that this book may arbitrarily be classified as physicochemical biology, and the methods described in it are the methods of the physics of continuous media. Therefore, our physicist readers may lack information from chemistry and biology, while our biologist readers will hardly recognize all the mathematical fine points. Nevertheless, we hope that the attraction of this frontier area, the unusual and unexpected nature of the physical effects, the novelty of the mathematical formulations, and the prospects for biologically interesting applications will reward read ers for their efforts. FOREWORD TO THE ENGLISH EDITION Electrophoresis of low-molecular-weight compounds and biopoly mers is a dynamic and rapidly developing field both from the standpoint of instrumentation and techniques and from the standpoint of the theory of the process and its mathematical model. Since the publication of the Russian edition of this book, progress has been made in several directions, and new results have been obtained by us and other researchers. We will point out some of these developments in this foreword. The method for calculating titration curves demonstrated in Chapter III for amino acids with several ionogenic groups has been extended to peptides and proteins with a known primary structure. By specifying the dissociation constants of the ionogenic groups in a given protein, we can determine the approximate pI value and then refine this value by varying the values of the constants and taking into account the secondary and ter tiary structure of the protein - where, of course, the inverse problem also arises: detennination of the dissociation constants from the titration curve [1]. The non-steady-state model of isotachophoresis presented in Chapter IV has been significantly developed. Explicit fonnulas that describe the motion of zones have been obtained for weak and strong electrolytes, thus eliminating the somewhat cumbersome reasoning used in constructing the tables and graphs [2-4], and algorithms have been developed for computer calculations. A model for isotachophoresis has been constructed that is free from the restrictions on the composition of the mixture given in Sec tion,1 of Chapter IV. For reactions of the type H+B ~ B + H+, HAj ~ AC + H+ (i = 1, ... , n), where H+B is a base and HAj are the acids to be separated, algebraic algorithms written for the computer have been constructed which allow one to calculate the motion of the zones, the position of their boundaries at any instant of time, the time for complete ix x FOREWORD separation of the mixture, the concentrations of the substances, and the pH values in the zones. The mixture components HAj (i = 1, ... , n) may be strong or weak acids, or amphoteric substances displaying acidic properties (the pH of the zone in this case should differ considerably from the pI of the substance). The indicated reaction scheme is most typical for experiments when a strong acid (usually HCI) is chosen as the leader, and the mixture consists of amino acids. A mathematical model for isotachophoresis at constant potential has been constructed [5]. This isotachophoresis method was recommended in [6] for determining the electrophoretic mobility of substances; however, the theory of the process given in [6] describes only a special case that is not realizable experimentally. Coulophoretic titration has been investigated. This is a process for determining the electrolyte concentration when the zones move in opposing directions [7]. Analysis of the corresponding mathematical model has made it possible to describe [4] the electrosynthesis process: the develop ment in the electrophoretic chamber of moving and fixed regions of elec trolytes that did not exist in the solution at the initial instant of time. Models have also been constructed that take into account diffusion processes in isotachophoresis (a calculation of the width of the boundary in the steady-state case is given in [8]). Certain advances have been made in describing the motion of indi vidual zones in zone electrophoresis, extending further the results of Chapter V. The effect of the buffer on the mobility of an amphoteric sub stance present in low concentration has been clarified [9]. It is shown that the ratio of their conductivities plays an important role in the description of the interaction between the buffer and the amphoteric substance. An am photeric substance found in some local volume together with the buffer displaces buffer ions from this volume. Here, for the case where the con ductivity of the amphoteric substance is less than the conductivity of the buffer, a decrease in conductivity of the given volume occurs; conse quently, an increase in the electric field occurs within this volume accord ing to the law E = Eo(1 + ac), where Eo is the electric field in a region free of the amphoteric substance, c « 1 is the concentration of the amphoteric substance, and a is the weighting coefficient (in this case, a > 0; when the conductivity of the amphoteric substance is greater than the conductivity of the buffer, the opposite effect occurs, and a < 0). This means that the de pendence v = vo(1 + ac) is satisfied for the electrophoretic transport rate. For the case a> 0 (usually for high-molecular-weight compounds), non linear effects lead to broadening of the trailing edge of the zone: develop- FOREWORD xi ment of a "tail" of the substance behind the moving zone. Conversely, when a < 0, the leading edge of the zone is broadened. Calculations show that electromigrational broadening of the zones in most cases occurs sig nificantly earlier than their diffusional broadening, especially when the concentration of the substance in the zone is high [9]. Progress has also been made in constructing models for creating pH gradients in finite-component mixtures, as demonstrated in Chapter VI for borate-polyol systems. It has been shown (see [10]) that in order to carry out accurate calculations we need to detail the reaction scheme (1.1)-(1.4) in Section 1 of Chapter VI, taking into account the effect of the i<;mic strength and other factors on the dissociation constants. Such additional information is especially important for the traditionally used tris-borate buffer. Furthermore, the theory has been refined for borate-polyol sys tems by taking into account diffusion processes affecting the quality of the pH gradients [11]. We should also note one promising route for creating a pH gradient: formation of a pH gradient moving at a low constant velocity, using isota chophoresis, based in this case on the rather convincing argument that it is better to have a controlled, constant motion of the pH gradient than to have an uncontrolled cathode or anode drift, or a "plateau" for the pH gradient created by the carrier ampholytes. For example, for a set of weak acids RAi (i = 1, ... , n) and one weak base H+B used as the counterion, it is not difficult to obtain relations specifying the required pH gradient (see [5]). With regard to the theory of the creation of pH gradients in infinite component systems and isoelectric focusing in such pH gradients (Chapter VII), we emphasize again that the theory is based on the fine difference between the pH values at which the mobility (pIi) and charge (pIe) of the amphoteric substance go to zero. The amphoteric substance in solution is represented by an ionic complex, the anions and cations of which in the general case have different mobilities. The number, or more precisely the concentration, of the anions and cations in the complex depends on the pH of the medium. When pH = pIe, this does not at all mean that the mobility of the complex in an electric field is equal to zero. The point is that the electric field does not act on the complex as a whole, but rather on its cations and anions individually; as a result, the complex may have nonzero mobility even when pH = pIe. When pH = ph the concentration of anions and cations is such that the mobility of the complex is equal to zero. The difference between pIe and pIi is significant for low-molecular-weight compounds, in particular for low-molecular-weight carrier ampholytes, and the difference in practice is insignificant for high-molecular-weight compounds such as proteins since, in such proteins, the mobilities of the cations and anions are, as a rule, practically equal.
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