Fundamentals of Mathematical Evolutionary Genetics

Mathematics and Its Applications (Soviet Series)

Managing Editor:

M. HAZEWINKEL Centre for Mathematics and Computer Science, Amsterdam, The Netherlands

Editorial Board:

A. A. KIRILLOV, MGU, Moscow, U.S.S.R. Yu. I. MANIN, Steklov Institute of Mathematics, Moscow, U.S.S.R. N. N. MOISEEV, Computing Centre, Academy of Sciences, Moscow, U.S.S.R. S. P. NOVIKOV, Landau Institute of Theoretical Physics, Moscow, U.S.S.R. M. C. POL YVANOV, Steklov Institute of Mathematics, Moscow, U.S.S.R. Yu. A. ROZANOV, Steklov Institute of Mathematics, Moscow, U.S.S.R.

Volume 22

Yuri M. Svirezhev and Vladimir P. Passekov Department of Mathematical Modeling in Ecology and Medicine, U .s.S.R. Academy of Sciences, Moscow, U.S.S.R.

Fundamentals of Mathematical Evolutionary Genetics

Kluwer Academic Publishers Dordrecht / Boston / London

Library of Congress Cataloging in Publication Data '" Svirezhev, IV. M.

22)

Fundamentals of mathematical evolutionar'y genetics.

(Mathematics and its applications. Soviet series

IT'anslation of: Osnovy matematicheskoi genet,iki. Includes index. 1. Population genetics--Mathematical models.

2. Genetics--Mathematical models. I. Pasekov, V. P. II. Title. III. Series: Mathematics and its appli­cations (D. Reidel Publishing Company). Soviet ser'ies ; 22. QH438.4.M3S9413 1989 575.] '5'0724 88-12798 ISBN-B: 978-94-010-7670-8 e-ISBN-B: 978-94-009-2589-2 om: 10.1007/978-94-009-2589-2

Published by Kluwer Academic Publishers, P.O. Box 17, 3300 AA Dordrecht, The Netherlands.

Kluwer Academic Publishers incorporates the publishing programmes of D. Reidel, Martinus Nijhoff, Dr W. Junk and MTP Press

Sold and distributed in the U.S.A. and Canada by Kluwer Academic Publishers, 101 Philip Drive, Norwell, MA 02061, U.S.A.

In all other countries, sold and distributed by Kluwer Academic Publishers Group, P.O. Box 322,3300 AH Dordrecht, The Netherlands.

Translated from Russian by Alexey A. Voinov and Dmitri1 O. Logofet

printed on acid-free paper

All Rights Reserved © 1990 by Kluwer Academic Publishers Softcover reprint of the hardcover 1st edition 1990 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopy­ing, recording or by any information storage and retrieval system, without written permission from the copyright owner.

N. V. TIMOFEEFF-RESSOVSKY

To the Teacher

Table of Contents

Editor's Preface Preface

Part 1. Deterministic Models in Mathematical Genetics

Chapter 1. Brief Outline of Microevolution Theory with Some Facts from

xiii xv

Genetics 3 1.1. History and Personalia 3 1.2. Conceptual Model of Microevolution 6 1.3. Elementary Evolutionary Structure and Elementary Evolutionary

Phenomenon 6 1.4. Elementary Evolutionary Material 7 1.5. Elementary Evolutionary Factors 8 1.6. An Introduction to Principles of Inheritance 9 1.7. Notes and Bibliography 13

Chapter 2. Basic Equations of Population Genetics 16 2.1. Description of a Population 16 2.2. Sexless Population 17 2.3. Equations for Populations ill Evolution 18 2.4. Evolution of Populations and Integral Renewal Equations 19 2.5. Panmixia and Other Systems of Mating 22 2.6. Principles of Inheritance 25 2.7. Multi-Allele Autosomal Gene: Equations of Evolution 26 2.8. Equations of Evolution with Specific Demographic Functions 29

2.8.1. Global Panmixia, Multiplicative Fccundity 29 2.8.2. Global Panmixia, Additive Fecundity 33 2.8.3. Local Panmixia 34

2.9. Equations of Evolution: Fecundity of a Couple is Determined by that of the Female 35

2.10. Equal Fecundity, Different Mortality: Another Form for Evolutionary Equations 37

2.11. Semelparity: Models with Discrete Time 39 2.12. More Realistic Assumptions About the Particular Form of Fecundity

and Mortality Functions 41 2.13. Some Generalizations of Classical Equations in Population Genetics.

Another Way to Derive these Equations 45 2.14. Discrete-time Equations of Evolution 48 2.15. On the Relationship Between Continuous and Discrete Models 51 2.16. Notes and Bibliography 52

viii Table of Contents

Chapter 3. Simplest Population Models 54 3.1. Introduction 54 3.2. Equations of Evolution 54 3.3. Existence Conditions for Polymorphism 55 3.4. Sufficient Conditions for Stability of Limiting States of a Population 58 3.5. Population Without Age Structure. Continuous Model 63 3.6. Population Without Age Structure. Discrete Model 65 3.7. Polymorphism. Experiments and Theory. What Are the Malthusian

Parameters or Genotype Fitnesses? 68 3.8. Genetico-Ecological Models 72 3.9. Special Cases of Genetico-Ecological Models 75 3.10. Passage from Genetico-Ecological Models to Models in Frequency

Form 79 3.11. NotesandBibliography 81

Chapter 4. Multiple Alleles 84 4.1. Introduction 84 4.2. State of Genetic EqUilibrium. Polymorphism 84 4.3. Mean Population Fitness. Fisher's Fundamental Theorem 86 4.4. Mean Fitness as a Lyapunov Function 88 4.5. Adaptive Topography of a Population 88 4.6. The Case of Three Alleles. Search for Domains of Asymptotic

Stability 90 4.7. Necessary and Sufficient Existence Conditions for Polymorphism 97 4.8. Theorem About Limited Variations and Another Form of Existence

Conditions for Polymorphism 97 4.9. Elimination of Alleles and Theorem About Dominance 100 4.10 Simple Necessary Conditions of Existence for Polymorphic and Pure

Equilibria 104 4.11. Population Trajectory as a Trajectory of Steepest Ascent 105

4.11.1. Introduction of a New Metric Space 105 4.11.2. Equations of Evolution and Local Extremal Principle 106

4.12. Another Form for Equations of Evolution 109 4.13. Notes and Bibliography 111

Chapter 5. Sex-Limited and Sex-Linked Characters. Models Taking Account of Sex Distinctions 114

5.1. Introduction 114 5.2. Model Taking Account of Sex Distinctions 114

5.2.1. Autosomal Gene. Continuous Model 114 5.3. New Types of Polymorphism and Their Stability 117 5.4. Model Taking Account of Sex Distinctions 122

5.4.1. Sex-Linked Gene. Continuos Model 122 5.5. Sex-Linked Gene. Discrete Model 126

Table of Contents ix

5.6. Sex-Linked Gene. Multiple Alleles 129 5.7. Minimax Properties of the Mean Fitness Function in a Model of an

Age-Structured Population 130 5.8. Notes and Bibliography 132

Chapter 6. Populations With Deviations from Panmixia 134 6.1. Introduction 134 6.2. Preference in Crossing and Preference Matrix 134 6.3. Model of Population with Deviations from Panmixia Caused by

Preference in Crossing 135 6.4. Evolution and Stability of Deviations from Hardian Equilibrium.

Inbreeding 138 6.5. Preference in Crossing. Discrete Model 141 6.6. Evolution of Genetic Structure of Population Under Inbreeding.

Discrete Model 143 6.7. Isolation by Distance and Deviations from Panmixia 144 6.8. Models with Particular Functions of Deviations from Panmixia 147 6.9. Notes and Bibliography 149

Chapter 7. Systems of Linked Populations. Migration 152 7.1. Introduction 152 7.2. Migration Between Populations of Different Sizes 152 7.3. Migration Between Population Occupying Two Similar Ecological

Niches 155 7.4. On 'Past' and 'Slow' Variables in Systems of Linked Populations 160 7.5. Genetic Interpretation. Why Stable Divergence is Important in

Systems of Linked Populations 162 7.6. Systems of Weakly Linked Populations 163 7.7. Populations with Continuous Area (Spatially Distributed Populations) 166 7.8. Genic Waves in Spatially Distributed Populations 171 7.9. Notes and Bibliography 174

Chapter 8. Population Dynamics in Changing Environment 177 8.1. Introduction 177 8.2. Seasonal Oscillations in Coefficients of Relative Viability. Discrete

Model 177 8.3. Polymorphism in Populations of Adalia Bipunctata 180 8.4. Environments Changing with Time. Continuous Model 182 8.5. How Variations in the Total Size of a Population Affect its Genetic

Dynamics 184 8.6. How Periodic Variations in Coefficients of Relative Viability Affect

the Total Population Size 186 8.7. Changing Environment. Adaption and Adaptability 187 8.8. Notes and Bibliography 189

x Table of Contents

Chapter 9. Multi-Locus Models 191 9.1. Discrete Two-Locus Model of Segregation-Recombination and its

Continuous Approximation 191 9.2. Continuous One- and Two-Locus Models with no Selection. Equa-

tions for Numbers and Frequencies, Fast and Slow Variables 196 9.3. Formalization of Recombination-Segregation in a Discrete-Time

Multi-Locus System. Equations of Dynamics, Equilibria 200 9.4. Recombination-Segregation Model in a Multi-Locus Continuous-

Time System 205 9.5. Additivity of Interaction Betwecn Selection and Recombination­

Segregation in Multi-Locus Models Presented by Differential Equations 208

9.6. Selection of Zygotes and Gametes in a Discrete-Time Model and its Continuous Approximation 210

9.7. Equations of Dynamics Under the Combined Action of Selection and Recombination-Segregation in Discrete- and Continuous-Time Models 216

9.8.

9.9. 9.10 9.11.

Comparing the Dynamics in One-Locus and Multi-Locus Systems in the presence of Selection Model of Additive Selection in a Multi-Locus System Models of Multiplicative and Additively Multiplicative Selection Notes and Bibliography

220 223 226 232

Part 2. Stochastic Models of Mathematical Genetics 237

Chapter 10. Diffusion Models of Population Genetics 239 10.1. Types of Random Processes Relevant to Models of Population

Genetics 239 10.2. Fundamental Problems in the Analysis of Stochastic Models 241 10.3. Forward and Backward Kolmogorov Equations 242 lOA. Diffusion Approximation of the Fisher-Wright and the Moran Models 247 10.5. Classification of Boundaries in Diffusion Models 249 10.6. Multidimensional Diffusion Models 251 10.7. Solutions to the Kolmogorov Equations by the Fourier Method.

Transformations of Diffusion Processes. The Steady-State Density 253 10.8. Search for Moments of Some Functionals on Diffusion Processes 259 10.9. An Approach to Search for the Mean of a Function Defined on States

of a Process 262 10.10. Notes and Bibliography 266

Chapter 11. Random Genetic Drift in the Narrow Sense 269 11.1. The Kolmogorov Equations for a Single-Locus Model of Random

Genetic Drift 269

Table of Contents xi

11.2. Approximating the Random Genetic Drift Process within Small Intervals of Time 270

11.3. Asymptotics of the Fundamental Solution for the Random Genetic Drift Process When t ~ 00 276

11.4. Boundary Attainment Probabilities 278 11.5. Characteristics of the Boundary Attainment Time 281 11.6. Probability Density Function for the Sojourn Time and the Age of an

Allele 288 11.7. Moments of the Random Genetic Drift Process 291 11.8. Fundamental Solution to the Kolmogorov Equations 294 11.9. A Random Genetic Drift Model with Two Loci 297 11.10. Notes and Bibliography 300

Chapter 12. Properties of Single-Locus Models under Several Microevolutionary Pressures 302

12.1. Ko1mogorov Equations in Case of Several Microevolutionary Conditions 302

12.2. Probabilities of Allele Fixation 304 12.3. Characteristics of the Homozygosity Attainment Time 311 12.4. Steady-State Probability Density Function for the Case of a Single

Diallelic Locus 313 12.5. Investigation of the Steady-State Probability Density Function for a

Single Diallelic Locus 316 12.6. Steady-State Density Function and the Adaptive Landscape in the

Two-Allele Case 319 12.7. Derivation of a Steady-State Density Function in the Case of Multiple

Alleles 321 12.8. Contribution to the Steady-State Density Caused by Selection 323 12.9. Contribution Caused by Migrations and Mutations. The General Form

of a Steady-State Density Function 326 12.10. Investigation of the Steady-State Probability Density Function for

Concentrations of Multiple Alleles. A Multi-Locus Case 329 12.11. Steady-state Density and Objective Functions in Case of Multiple

Alleles 332 12.12. Relation of Objective Functions to the Sphere Motion Potential.

Mechanical Interpretation of Single-Locus Genetic Processes in Terms of Motion in a Force Field 335

12.13. Notes and Bibliography 341

Chapter 13. Random Genetic Drift in Subdivided Populations 343 13.1. Generating Operator for the Random Genetic Drift Process in a

Subdivided Finite-Sized Population with Migrations of the "Island" Type 343

13.2. Dynamics of Expected Allele Frequencies in a Subdivided Population 347 13.3. Behavior of Expected Heterozygosity Indices 349

xii Table of Contents

13.4. Dynamics of Expected Indices of Linkage Disequilibrium in Case of Two Loci 355

13.5. Model of a Hierarchically Subdivided Population 360 13.6. Investigation of the Asymptotic Rate of Decrease in Heterozygosity in

the Hierarchical Model 363 13.7. Model of Isolation by Distance 366 13.8. Properties of the Random Genetic Drift Process in a Subdivided

Population with Migrations of the General Type 372 13.9. Notes and Bibliography 380

Conclusion 382

Short Glossary of Genetic Terms 387

Subject Index 391

SERlES EDITOR'S PREFACE

~Et moi, ... , si j'avait su comment en revenir, je riy serais point aile.'

Jules Verne

The series is divergent; therefore we may be able to do something with it.

O. Heaviside

One service mathematics has rendered the human race. It has put common sense back where it belongs, on the topmost shelf next to the dusty canister labelled 'discarded non­sense'.

Eric T. Bell

Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and non­linearities abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for other sciences.

Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One service topology has rendered mathematical physics .. o'; 'One service logic has rendered com­puter science .. o'; 'One service category theory has rendered mathematics .. .'. All arguably true. And all statements obtainable this way form part of the raison d'etre of this series.

This series, Mathematics and Its Applications, started in 1977. Now that over one hundred volumes have appeared it seems opportune to reexamine its scope. At the time I wrote

"Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the 'tree' of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be completely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathematics applied in various sciences has changed drastically in recent years: measure theory is used (non-trivially) in regional and theoretical economics; algebraic geometry interacts with physics; the Minkowsky lemma, coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces. And in addition to this there are such new emerging subdisciplines as 'experimental mathematics', 'CFD', 'completely integrable systems', 'chaos, synergetics and large-scale order', which are almost impossible to fit into the existing classification schemes. They draw upon widely different sections of mathematics."

By and large, all this still applies today. It is still true that at first sight mathematics seems rather fragmented and that to find, see, and exploit the deeper underlying interrelations more effort is needed and so are books that can help mathematicians and scientists do so. Accordingly MIA will continue to try to make such books available.

If anything, the description I gave in 1977 is now an understatement. To the examples of interaction areas one should add string theory where Riemann surfaces, algebraic geometry, modu­lar functions, knots, quantum field theory, Kac-Moody algebras, monstrous moonshine (and more) all come together. And to the examples of things which can be usefully applied let me add the topic 'finite geometry'; a combination of words which sounds like it might not even exist, let alone be applicable. And yet it is being applied: to statistics via designs, to radar/sonar detection arrays (via finite projective planes), and to bus connections of VLSI chips (via difference sets). There seems to be no part of (so-called pure) mathematics that is not in immediate danger of being applied. And, accordingly, the applied mathematician needs to be aware of much more. Besides analysis and numerics, the traditional workhorses, he may need all kinds of combinatorics, algebra, probability, and so on.

In addition, the applied scientist needs to cope increasingly with the nonlinear world and the

xiv

extra mathematical sophistication that this requires. For that is where the rewards are. Linear models are honest and a bit sad and depressing: proportional efforts and results. It is in the non­linear world that infinitesimal inputs may result in macroscopic outputs (or vice versa). To appreci­ate what I am hinting at: if electronics were linear we would have no fun with transistors and com­puters; we would have no TV; in fact you would not be reading these lines.

There is also no safety in ignoring such outlandish things as nonstandard analysis, superspace and anticommuting integration, p-adic and ultrametric space. All three have applications in both electrical engineering and physics. Once, complex numbers were equally outlandish, but they fre­quently proved the shortest path between 'real' results. Similarly, the first two topics named have already provided a number of 'wormhole' paths. There is no telling where all this is leading -fortunately.

Thus the original scope of the series, which for various (sound) reasons now comprises five sub­series: white (Japan), yellow (China), red (USSR), blue (Eastern Europe), and green (everything else), still applies. It has been enlarged a bit to include books treating of the tools from one subdis­cipline which are used in others. Thus the series still aims at books dealing with:

- a central concept which plays an important role in several different mathematical and/or scientific specialization areas;

- new applications of the results and ideas from one area of scientific endeavour into another; - influences which the results, problems and concepts of one field of enquiry have, and have had,

on the development of another.

The present volume concerns mathematical biology: quite possibly the fastest growing area of applied mathematics. For this last fact several reasons can, with some hindsight, be discerned. For one thing, the potential applications are great and important. Think, for instance, of the harvesting of microbial (and other) biological populations. For another, the subject is good at generating new kinds of mathematical problems. Thus, after a somewhat slow and tentative start, mathematization of (parts of) biology is in full swing.

This book is on what is perhaps the most mathematical (for the moment) part of mathematical biology: the subject of mathematical genetics and population dynamics including the modern topic of age-structured populations. Both deterministic and stochastic models are considered.

As one referee replied, now several years ago, this volume is a nice solid, thorough, complete book in the justly famous USSR tradition of high level workmanlike monographs and textbooks. It is also by two top experts in the field and thus very welcome in this series.

Perusing the present volume is not guaranteed to turn you into an instant expert, but it will help, though perhaps only in the sense of the last quote on the right below.

The shortest path between two truths in the real domain passes through the complex domain.

J. Hadamard

La physique ne nous donne pas seulement I' occasion de resoudre des problemes ... elle nous fait pressentir la solution.

H. Poincare

Bussum, March 1989

Never lend books, for no one ever returns them; the only books I have in my library are books that other folk have lent me.

Anatole France

The function of an expert is not to be more

right than other people. but to be wrong for more sophisticated reasons.

David Butler

Michiel Hazewinkel

Preface

Mathematical genetics is one of the most formalized fields of biology. It is con­cerned both with speculations aimed at a better understanding of the evolutionary process and with purely practical aspects used, say, in animal and plant breeding plans, in artificial selection, etc. It would be difficult to cover all possible applica­tions of mathematical methods to problems of genetics within the limits of one book, therefore this book is mainly devoted to the evolutionary aspect of genetical problems. The first part is concerned with deterministic models of mathematical genetics and the second part - with the stochastic ones. Some basic facts from genetics are included, sufficient to understand the problems considered. It is assumed that the reader is familiar with fundamentals of integral and differential calculus, the qualitative theory of differential equations, and probability theory. The authors have tried to make the chapters independent. To understand Chapters 3-9 of Part I, which ignore the age-structure of populations, the reader may res­trict him- or herself to Sections 15 and 16 of Chapter 2. The Chapters of Part 2 can be read independently if the results of Chapter 10 are known; if the reader is familiar with diffusion equations, he may read only Sections 4-6 and 9 of this Chapter.

Every chapter is concluded with references and comments, given in such a way that the reader finds information not only about sources of the presented results and methods, but also about the directions of their further research and generaliza­tion.

A word about the numbering of formulas and sections in the book is in order. Numbers of formulas within one chapter are made up of two figures - the section number and the number of a formula itself. If we refer to a formula from another chapter, the first number presents the number of the chapter; the second one, of the section, the third one, of the formula. Similarly, references to sections within one chapter contain only its number, references to sections from other chapters also have the chapter number.

We have failed to give only a unique meaning to symbols, since a lot of variables had to be introduced. However, within sections, and to a great extend within chapters, each symbol represents one variable.

Chapters 1-8 have been written by Yu. M. Svirezhev. Chapters 9-13 - by V. P. Passekov.

xv

xvi Preface

In conclusion we would like to say some words about how the book has been written. In the mid-1960's our teacher, the outstanding biologist of today Nikolai Vladimirovich Timofeeff-Ressovsky formulated a grand program for the mathemat­ization of population genetics and evolution theory. Realizing this program we prepared chapters of the book. Each of them was discussed in much detail. Timofeeff-Ressovsky suggested a conceptual scheme, and we tried to translate it into the language of mathematics. Next came the stage of interpretation of the results; here again his role was also great.

Unfortunately, Timofeeff-Ressovsky did not live to see the appearance of this book. We wish to dedicate the book to his memory.

Yu. M. SvIREZHEV

V. P. PASSEKOV