Table Of ContentLecture Notes in Biomathematics
Vol. 1 : P. Waltman, Deterministic Threshold Models in the Theory of Epidemics. V,
101 pages. 1974.
Vol. 2: Mathematical Problems in Biology, Victoria Conference 1973. Edited by P. van
den Driessche. VI, 280 pages. 1974.
Vol. 3: D. Ludwig, Stochastic Population Theories. VI, 108 pages. 1974.
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Vol. 7: M. C. Mackey, Ion Transport through Biological Membranes, An Integrated
Theoretical Approach. IX, 240 pages. 1975.
Lectu re Notes in
Biomathematics
Managing Editor: S. Levin
7
Michael C. Mackey
Ion Transport through
Biological Membranes
An Integrated Theoretical Approach
Springer-Verlag
Editorial Board
W. Bossert· H. J. Bremermann . J. D. Cowan . W. Hirsch
S. Karlin' J. B. Keller' M. Kimura' S. Levin (Managing Editor)
R. C. Lewontin . G. F. Oster' L. A. Segel
Author
Dr. Michael C. Mackey
McGill University
Department of Physiology
P.O.Box 6070
Station A
Montreal, Quebec
Canada H3C 3G1
Library of Congress Cataloging in Publication Data
Mackey, Michael Charles, 1942-
Ion transport through biological membranes.
B'
(Lecture notes in~hematics ; 7)
Bibliography: p.
Includes index.
1. Biological transport--Mathematical models.
2. Membranes (Biology)--Mathematical models.
I. Title. II. Series. [DNLM: 1. Biological
transport. 2. Ions--Metabolism. 3. Cell membrane
permeability. WI LE334 v. 7 / QH601 Ml57iJ
QH509.M3 574.8'75 75-35923
AMS Subject Classifications (1970): 92A05
ISBN-13: 978-3-540-07532-5 e-ISBN-13: 978-3-642-81008-4
001: 10.1007/978-3-642-81008-4
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© by Springer-Verlag Berlin' Heidelberg 1975
TO NANCY
PREFACE
This book illustrates some of the ways physics and mathematics
have been, and are being, used to elucidate the underlying mechan
isms of passive ion movement through biological membranes in general,
and the membranes of excltable cells in particular. I have made no
effort to be comprehensive in my introduction of biological material
and the reader interested in a brief account of single cell electro
physlology from a physically-oriented biologists viewpoint will find
the chapters by Woodbury (1965) an excellent introduction.
Part I is introductory in nature, exploring the basic electrical
properties of inexcitable and excitable cell plasma membranes. Cable
theory is utilized to illustrate the function of the non-decrementing
action potential as a signaling mechanism for the long range trans
mission of information in the nervous system, and to gain some in
sight into the gross behaviour of neurons. The detailed analysis of
Hodgkin and Huxley on the squid giant axon membrane ionic conductance
properties is reviewed briefly, and some facets of membrane behaviour
that have been revealed since the appearance of their work are dis
cussed.
Part II examines the foundations of electrodiffusion theory,
and the use of that theory in trying to develop quantitative expla
nationsof the observed membrane properties of excitable cells, in
particular the squid giant axon. In addition, an ad hoc formulation
of electrodiffusion theory including active transport is presented
to illustrate the qualitative nature of cellular homeostasis with
respect to intracellular ionic concentrations and membrane potential,
and cellular responses to prolonged stimUlation.
Part III consists entirely of personal efforts, and collab
orative work with M.L. McNeel, to examine the molecular foundations
VI
of electrodiffusion theory and arrive at a (hopefully) more realistic
treatment of transmembrane ion movement. The work described in this
last part is not complete, and I hope that others will be encouraged
to extend and/or modify the analysis in a number of obvious ways.
The material of Parts I and II forms the basis of a final year
undergraduate course that I have given for the past two years in the
Department of Physics at McGill, and which was offered in the
Department of physiology before that.
I would like to thank American Elsevier Publishing Company for
their permission to reproduce much of the material in Mathematical
Biosciences (1975, 22, 67-80) in Chapter 9; Robin Barnett and Eugene
Benjamin for computations on which some of the figures of Chapter 10
are based; and the Rockefeller university Press for permission to
reproduce material from the Biophysical Journal in Chapters 11
through 15. My special thanks go to Ms. Celia Lang who reduced a ver-
itable mountain of rough drafts to a final typescript, and Mr. Karl
Holeczek for his speedy and faithful execution of the figures and
photography work.
So many people have shaped my thinking in this area of biophysics
that it is difficult to acknowledge their individual contributions.
The one person I can single out is Professor J. Walter Woodbury, who
for twelve years has been a close friend, teacher, and colleague.
Much of the material presented bears the imprint of his deft touch.
TABLE OF CONTENTS
PART I. INTRODUCTION........................................... 1
CHAPTER 1. BAS IC MEMBRANE STRUCTURE AND ELECTRICAL
PROPERTIES. . . . . . . . . . . . . . . . . . . . • . . . • . . . . . . • . . . . 5
A. Some Simple Concepts About Membrane
Structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
B. The General Nature of Membrane
Electrical properties ..................... 7
CHAPTER 2. PASSIVE ELECTRICAL PROPERTIES OFAXONS ........ 12
A. Non-Myelinated Fibres ..................... 12
B. Myelinated Fibres ......................... 18
CHAPTER 3. OVERVIEW OF THE GROSS PROPERTIES OF
EXCITABLE CELLS............................... 27
A. The Membrane Theory of Excitation
and pr opaga t ion. . . . . . . • . . . . . . . . . . . . . . . . . .. 27
B. Strength-Duration Relation, Accomodation .. 30
C. Refractory States and Firing Frequency .... 34
D. Axonal Characteristics and Action
potential propagation .............•...... 35
CHAPTER 4. THE HODGKIN-HUXLEY AXON ....................... 42
A. Voltage Clamping.......................... 42
B. Ionic Current Flow as Revealed by the
voltage Clamp ............................. 44
C. Empirical Formulae for the Conductances ... 49
D. Reconstruction of the Action Potential .... 50
CHAPTER 5. CURRENT LEVELS OF KNOWLEDGE ABOUT THE
EARLY AND LATE CURRENT FLOW PATHWAyS .......... 51
A. The Action of Tetrodotoxin (TTX) .......... 51
B. The Action of Tetraethylammonium
Ion (TEA) ................................. 54
C. Ionic Strength Effects on the Kinetic
Parameters of Excitation .........•........ 54
D. The Effect of Calcium or! the Membrane..... 56
E. The Effects of pH Changes................. 57
F. cation Selective Properties of the Mem-
brane at Rest and During Excitation ....... 57
G. Kinetic Behaviour in the K Channel ........ 59
H. High Potassium Effects on the K Channel
Current-voltage Curve ..................... 59
I. The Nature of the Leakage pathway ......... 60
PART II. CLASSICAL ELECTRODIFFUSION THEORIES OF MEMBRANE
ELECTRICAL PROPERTIES................................ 63
CHAPTER 6. CONSERVATION AND FIELD EQUATIONS .............. 65
VIII
CHAPTER 7. THE STEADY STATE PROBLEM: APPROXIMATE
SOLUTIONS. • • • . • • • • . . . • . • . • • • • . • • • • • • • • • . . • . •• 72
A. Constant Field Approximation •••••••••.••• 73
B. The Microscopic Electroneutrality
Approximation ••••.••••..••••••••.••..•••• 78
CHAPTER 8. ACTIVE TRANSPORT AND THE MAINTENANCE
OF TRANSMEMBRANE IONIC DISTRIBUTIONS .•••••••. 84
A. General Characteristics of Active
Transport Systems... . • . . • • • • • • • • . • • . . . • •• 85
B. The Consequences of Including Active
Transport of Na+and K+in Steady State
Electrodiffusion •••.•.•.•••.•••••.•.•.•.• 90
C. Active Transport and the Recovery
From Excitation •••••••••.••••.•.••.....•. 96
CHAPTER 9. ADMITTANCE PROPERTIES OF THE ELECTRO-
DIFFUSION EQUATIONS •.••••.•.••••.......••.... 103
A. Analysis •.••.••..•••••••••••••.•.••..•••. 106
B. The Behaviour of the Transformed
Membrane Admittance ••••.••••••.••.•.....• 108
C. Quantitative Comparisons with Data ...•••• 112
CHAPTER 10. THE STEADY STATE AGAIN: APPROXIMATIONS
TO INVESTIGATE THE ROLE OF MEMBRANE
FIXED CHARGE •...•.••.••••...•.••...•..•...•.. 116
A. volume Charge Considerations .•..••...•••• 116
B. Surface Charge Effects .••••........•.•... 121
PART III. A MOLECUIAR TREATMENT OF TRANSMEMBRANE ION
MOVEMENT .•••••••.....•••••••.•••..•••.....•••••..•• 129
CHAPTER 11. MATHEMATICAL FORMUIATION OF THE MODEL ..••.••• 133
A. The Concept of the Distribution
Function •••••••..•••••.•••••••.••••...••• 133
B. The Relation of Some Macroscopic
Quantities to the Distribution Function •• 1~
C. The Derivation of an Equation of Change
for the Distribution Function ••••••..••.• 136
D. The Dynamics of a Binary Collision ....•.• 138
E. Evaluation of the Boltzmann Equation
collision Term ••.••••••.•••...•..•••••.•• 141
F. Simplification of the Boltzmann Equation. 143
G. Dependence of the Collision Frequency,
V , on v ..••..•••••••..•••••••.•••.•.••• 152
1 l.
H. Expressions for Macroscopic Quantities
from the Expanded Distribution Function •• 156
I. Summary ...••.•..•...•...•••••••.•••...•.• 157
IX
CHAPTER 12. RELATIONSHIP BETWEEN THE MICROSCOPIC AND
MACROSCOPIC FORMULATIONS OF ELECTRO-
DIFFUSION THEORy ••••••••••••••••••••••••••••• 161
A. Conservation Equations ••.•••••••••••••••• 161
B. Relation to E1ectrodiffusion •.••••••••••• 164
CHAPTER 13. THE MICROSCOPIC MODEL IN A STEADY STATE:
NO CONCENTRATION GRADIENTS .••.•••••••...••.•. 168
A. Spatial Gradients in a Steady State •••••• 168
B. Solutions of the Kinetic Equations
For a DC Field ••.•••.•••••••••••••••••••• 169
C. Analytic Behaviour of the Current
Density as a Function of Electric
Field Strength ••.••.••...•••••••.•••••••• 171
D. Computed Behaviour of l' (E") and Gc (E) ••••• 174
E. Estimation of Some Membrane Related
Quantities •.••..........••••••••••••..•.• 193
CHAPTER 14. THE STEADY STATE MICROSCOPIC MODEL WITH
SOLUTION ASYMMETRY ....•.••••..••..•••.•...... 199
A. Solution of the Kinetic Equations ••••.•.. 199
B. A Generalized Goldman Equation ••••••••••• 201
C. Computed properties of the Model •••••••.• 204
D. One Way Fluxes and the Independence
principle •••••..•••..••.•.•••••••••••.... 209
CHAPTER 15. STEADY STATE AND DYNAMIC PROPERTIES OF
THE MACROSCOPIC MODEL ••.•.•••.•.•••••..•.••.. 217
REFERENCES .•........••.••...••.•........•.••.•..•••••••••.•..• 229
APPENDIX 1 •.••.••.•....•.•••.••.•...........•.•.••.•.•••••.... 239
APPENDIX 2 ........••••••••••••••••••.•.••••••••••••••••••••••• 240
PART I. INTRODUCTION
It is believed by most investigators (see Ling,1962, for the
opposite opinion) that all biological cells are surrounded by a
boundary layer that is chemically quite distinct from the extra
cellular fluid (ECF) and the intra-cellular fluid (ICF). This
boundary layer which is called the cell, or plasma, membrane seems
to perform many varied and specialized tasks necessary to cellular
integrity and thus the integrity of whole multicellular organisms.
I am going to deal with only two of these specialized membrane
functions, not because they are the most important (an impossible
judgment to make) or the simplest to understand (unclear at this
time), but rather because they afford an excellent opportunity to
examine the way in which the biological sciences have been able to
combine forces with the physical sciences and mathematics in
gaining a deeper understanding of fundamental biological processes.
One broad area of membrane function that I examine theoreti
cally is the way in which membranes can control the ionic composi
tion of the ECF and ICF (see Table 1.1). This is a basic area of
interest for all types of cells, because it is known that many of
the necessary intracellular chemical reactions in a viable cell are
strongly dependent on the ionic medium in which they proceed. Fur
ther, the extracellular environment in which a cell exists is
crucial for its continued survival and proper functioning. This
basic problem will lead to an examination of the way in which ions
move across biological membranes in response to a number of exter
nal forces, and the way in which they interact with the molecular
