theoretical evolutionary genetics

THEORETICAL EVOLUTIONARY GENETICS JOSEPH FELSENSTEIN Theoretical Evolutionary Genetics GENOME 562 Joseph Felsenstein Department of Genome Sciences University of Washington Box 357730 Seattle, Washington 98195-7730 April, 2003 Copyright (c) 1978, 1983, 1988, 1991, 1992, 1994, 1995, 1997, 1999, 2001, 2003 by Joseph Felsenstein. All rights reserved. Not to be reproduced without author’s permission. Contents PREFACE ix 1 RANDOM MATING POPULATIONS 1 I.1 Asexual inheritance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 I.2 Haploid inheritance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 I.3 Diploids with two alleles: Hardy-Weinberg laws. . . . . . . . . . . . . . . . . . . . . 4 I.4 Multiple alleles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 I.5 Overlapping generations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 I.6 Different Gene Frequencies in the Two Sexes . . . . . . . . . . . . . . . . . . . . . . 12 I.7 Sex linkage. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 I.8 Linkage. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 I.9 Estimating Gene Frequencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 I.10 Testing Hypotheses about Frequencies . . . . . . . . . . . . . . . . . . . . . . . . . . 25 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 Complements/Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2 NATURAL SELECTION 33 II.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 II.2 Selection in Asexuals - Discrete Generations . . . . . . . . . . . . . . . . . . . . . . 34 II.3 Selection in Asexuals - Continuous Reproduction . . . . . . . . . . . . . . . . . . . . 39 II.4 Selection in Diploids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 II.5 Rates of Change of Gene Frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 II.6 Overdominance and Underdominance . . . . . . . . . . . . . . . . . . . . . . . . . . 58 II.7 Selection and Fitness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 II.8 Selection and Fitness : Multiple Alleles . . . . . . . . . . . . . . . . . . . . . . . . . 73 II.9 Selection Dependent on Population Density . . . . . . . . . . . . . . . . . . . . . . . 77 II.10 Temporal Variation in Fitnesses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 II.11 Frequency-Dependent Fitnesses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 II.12 Kin selection: a specific case of frequency - dependence . . . . . . . . . . . . . . . . 91 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 Complements/Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 3 MUTATION 103 III.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 III.2 Effect of Mutation on Gene Frequencies . . . . . . . . . . . . . . . . . . . . . . . . . 104 v III.3 Mutation with Multiple Alelles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 III.4 Mutation versus Selection: Haploids . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 III.5 Mutation vs. Selection: Effects of Dominance . . . . . . . . . . . . . . . . . . . . . . 110 III.6 Mutational Load. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 III.7 Mutation and Linkage Disequilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . 121 III.8 History and References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 Complements/Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 4 MIGRATION 127 IV.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 IV.2 The Effect of Migration on Gene Frequencies . . . . . . . . . . . . . . . . . . . . . . 127 IV.3 Migration and Genotype Frequencies: Gene Pools . . . . . . . . . . . . . . . . . . . 128 IV.4 Estimating Admixture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 IV.5 Recurrent Migration: Models of Migration . . . . . . . . . . . . . . . . . . . . . . . 132 IV.6 Recurrent Migration: Effects on Gene Frequencies . . . . . . . . . . . . . . . . . . . 135 IV.7 History and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 IV.8 Migration vs. Selection: Patches of Adaptation . . . . . . . . . . . . . . . . . . . . . 137 IV.9 Two-Population Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 IV.10 The Levene Model: Large Amounts of Migration . . . . . . . . . . . . . . . . . . . . 145 IV.11 Selection-Migration Clines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 IV.12 The Wave of Advance of an Advantageous Allele . . . . . . . . . . . . . . . . . . . . 157 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 Complements/Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 5 INBREEDING 161 V.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 V.2 Inbreeding Coefficients and Genotype Frequencies . . . . . . . . . . . . . . . . . . . 162 V.3 The Loop Calculus: A Simple Example. . . . . . . . . . . . . . . . . . . . . . . . . . 164 V.4 The Loop Calculus: A Pedigree With Several Loops. . . . . . . . . . . . . . . . . . . 166 V.5 The Loop Calculus: Sex Linkage. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 V.6 The Method of Coefficients of Kinship. . . . . . . . . . . . . . . . . . . . . . . . . . 169 V.7 The Complication of Linkage. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 V.8 More Elaborate Probabilities of Identity. . . . . . . . . . . . . . . . . . . . . . . . . 172 V.9 Regular Systems of Inbreeding: Selfing. . . . . . . . . . . . . . . . . . . . . . . . . . 174 V.10 Regular Systems of Inbreeding: Full Sib Mating . . . . . . . . . . . . . . . . . . . . 175 V.11 Regular Systems of Inbreeding: Matrix Methods . . . . . . . . . . . . . . . . . . . . 179 V.12 Repeated double first cousin mating. . . . . . . . . . . . . . . . . . . . . . . . . . . 181 V.13 The Effects of Inbreeding. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 V.14 Some Comments About Pedigrees . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 Complements/Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 6 FINITE POPULATION SIZE 191 VI.1 Genetic Drift and Inbreeding: their relationship . . . . . . . . . . . . . . . . . . . . 191 VI.2 Inbreeding due to finite population size . . . . . . . . . . . . . . . . . . . . . . . . . 192 VI.3 Genetic drift: the Wright model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 VI.4 Inbreeding coefficients, variances, and fixation probabilities. . . . . . . . . . . . . . . 198 VI.5 Effective population number: avoidance of selfing, two sexes, monogamy. . . . . . . 201 VI.6 Varying population size, varying offspring number. . . . . . . . . . . . . . . . . . . . 205 VI.7 Other effects on effective population number. . . . . . . . . . . . . . . . . . . . . . . 209 VI.8 Hierarchical population structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 Complements/Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 7 GENETIC DRIFT AND OTHER EVOLUTIONARY FORCES 215 VII.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 VII.2 Drift Versus Mutation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 VII.3 Genetic distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 VII.4 Drift Versus Migration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 VII.5 Drift vs. Migration: the Island Model . . . . . . . . . . . . . . . . . . . . . . . . . . 229 VII.6 Drift vs. Migration: the stepping stone model. . . . . . . . . . . . . . . . . . . . . . 236 VII.7 Drift versus Selection: Probability of Fixation of a Mutant . . . . . . . . . . . . . . 242 VII.8 The Diffusion Approximation to Fixation Probabilities. . . . . . . . . . . . . . . . . 247 VII.9 Diffusion Approximation to Equilibrium Distributions. . . . . . . . . . . . . . . . . 258 VII.10 The Relative Strength of Evolutionary Forces . . . . . . . . . . . . . . . . . . . . . 272 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 Complements/Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 8 MULTIPLE LINKED LOCI 279 VIII.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 VIII.2 A Haploid 2-locus Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 VIII.2.1 Selection with no recombination . . . . . . . . . . . . . . . . . . . . . . . . . 280 VIII.2.2 Epistasis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281 VIII.2.3 Selection and recombination . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 VIII.2.4 Interaction and Linkage – An Example . . . . . . . . . . . . . . . . . . . . . 284 VIII.3 Linkage and Selection in Diploids . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286 VIII.4 Lewontin and Kojima’s symmetric model . . . . . . . . . . . . . . . . . . . . . . . . 289 VIII.4.1 Fitness and Disequilibrium: Moran’s Counterexample . . . . . . . . . . . . 294 VIII.4.2 Coadapted Gene Complexes and Recombination . . . . . . . . . . . . . . . 294 VIII.5 The General Symmetric Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296 9 QUANTITATIVE CHARACTERS 299 IX.1 What is a Quantitative Character? . . . . . . . . . . . . . . . . . . . . . . . . . . . 299 IX.2 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300 IX.3 Means . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304 IX.4 Additive and Dominance Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308 IX.5 Covariances Between Relatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318 IX.6 Regression of Offspring on Parents . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322 IX.7 Estimating variance components and heritability. . . . . . . . . . . . . . . . . . . . . 325 IX.8 History and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328 IX.9 Response to artificial selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329 IX.10 History and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337 Complements/Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339 10 MOLECULAR POPULATION GENETICS 341 X.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341 X.2 Mutation models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341 X.3 The Coalescent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343 Problems/Complements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346 11 POLYGENIC CHARACTERS IN NATURAL POPULATIONS 349 XI.1 Phenotypic Evolution Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349 XI.2 Kimura’s model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350 XI.3 Lande’s model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350 XI.4 Bulmer’s model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354 XI.5 Other models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355 Complements/Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356 REFERENCES 357 viii PREFACE These are chapters I-XI of a set of notes which when completed will serve as a text for Genome Sciences 562 (Population Genetics). The material omitted will complete chapter VIII on the in- teraction of linkage and selection and cover some additional topics in chapters X and XI, such as quantitative characters in natural populations and coalescents. Each chapter ends with two sets of problems. Those labeled Exercises are intended to be relatively straightforward application of principles given in the text. They usually involve numerical calculation or simple algebra. The set labeled Problems/Complements are more algebraic, and often involve extension or re-examination of the material in the text. The level of mathematics required to read this text is not high, although the volume of algebra is sometimes heavy. It is probably sufficient to know elementary Calculus, and parts of elementary statistics and probability. Matrix algebra is used in several places, but these can be skipped without much loss. The most relevant mathematical technique for population genetics is probably factorization of simple polynomial expressions, which most people are taught in high school. Many people have contributed to the production of these notes, particularly students in earlier years of the course who caught many errors in earlier versions. The presentations were heavily influenced by lecture notes and courses on this subject by J. F. Crow and R. C. Lewontin. The cover illustration is adapted from an original by Helen Leung. Sean Lamont wrote the plotting program that produced the majority of the figures. I am indebted to many students for suggestions and corrections, particularly to Eric Anderson and Max Robinson. But most of all, I must thank Nancy Gamble and Martha Katz for doing the enormous job of typing out these notes, and Nancy Gamble for drawing some of the figures for earlier editions. I am still hoping to complete this set of notes one day. Joe Felsenstein Department of Genome Sciences University of Washington Seattle joe@gs.washington.edu ix x [...]...Chapter 1 RANDOM MATING POPULATIONS Theoretical population genetics is arguably the area of biology in which mathematics has been most successfully applied Other areas such as theoretical ecology model phenomena which are intrinsically more important to human welfare, and which have a much larger base of observations... successful The major reason why theory is more readily applied to population genetics is that there is a framework – Mendelian segregation – on which to hang it The Mendelian mechanism is a highly regular process with strong geometric and algebraic overtones The other reason why Mendelian segregation is particularly important to population genetics is that it occurs whether or not natural selection is present,... frequencies are the same among both a and α haploids, so that we need not take mating types into account We will shortly see the consequences of relaxing this assumption Many of the phenomena of population genetics can be seen most clearly in haploid cases, and we will return to the haploid case more frequently than its biological importance alone warrants I.3 Diploids with two alleles: Hardy-Weinberg laws... 1/2PAa )(1/2PAa + Paa ) aa : 1/4(PAa )2 + PAa Paa + (Paa )2 (I-9) = (1/2PAa + Paa )2 The two principles given above are often known as the Hardy-Weinberg Law They have two important impacts on population genetics The first implies that genotype frequencies can (under appropriate conditions) be predicted from gene frequencies Together with the second, it implies that we can carry through an analysis in terms... of the environment to a greater importance than he had hitherto assigned it, in order to provide the continuous torrent of new variation necessary to keep evolution operating With the rise of Mendelian genetics, and the realization of its consequences, the problem vanished The Hardy-Weinberg law was discovered by the famous English mathematician G H Hardy (1908), and simultaneously and independently... that Castle worked in terms of genotypes rather than gene frequencies The Hardy-Weinberg Law is as close to being trivially obvious as it can be, but it had a major impact on the practice of population genetics Before it, calculations of the effect of natural selection required one to keep track of three variables, the genotype frequencies, and the algebra required to do even simple cases was quite complicated... forces, the HardyWeinberg Law greatly simplified calculations The advances of the next two decades would come much more slowly and tortuously if it had not been true For a more detailed history of population genetics during the decade of the 1900s, the reader should consult the book by Provine (1968) The Hardy-Weinberg Law is sometimes referred to as the Hardy-Weinberg Equilibrium It is an equilibrium in only... assumptions We will not be able to cover all possibilities, even superficially, but we should be able to arrive at some intuitive understanding of the effects, singly and in combination, of these various evolutionary forces I.4 Multiple alleles If, instead of 2 alleles, a population contains n alleles, the principles stated in the previous section either apply or generalize naturally In a haploid population,... tractable than discrete-generations models This is mostly because Hardy-Weinberg proportions cannot be assumed As we have seen, they are approached only asymptotically even with random mating If there is any evolutionary force, such as natural selection, making the population continually depart from Hardy-Weinberg proportions, we will have to follow genotype frequencies rather than gene frequencies, which... excess of heterozygotes Biologically, the main implication of the results of this section is that for autosomal loci, we would not expect to see gene frequency differences between the sexes unless some evolutionary force 13 SA SA SA SA sa sa sa sa Figure 1.2: Segregation of an allele completely linked to a sex-determining locus in a haploid organism continually created such differences This has an interesting . THEORETICAL EVOLUTIONARY GENETICS JOSEPH FELSENSTEIN Theoretical Evolutionary Genetics GENOME 562 Joseph Felsenstein Department of Genome. 1 RANDOM MATING POPULATIONS Theoretical population genetics is arguably the area of biology in which mathematics has been most successfully applied. Other areas such as theoretical ecology model. Equilibrium Distributions. . . . . . . . . . . . . . . . . 258 VII.10 The Relative Strength of Evolutionary Forces . . . . . . . . . . . . . . . . . . . . . 272 Exercises . . . . . . . . . .