Interdisciplinary Applied Mathematics

Contents Preface Introduction 1 Historical Background 1.1 Biometricians, Saltationists and Mendelians 1.2 The Hardy--Weinberg Law 1.3 The Correlation Between Relatives 1.4 Evolution 1.4.1 The Deterministic Theory 1.4.2 Non-Random-Mating Populations 1.4.3 The Stochastic Theory 1.5 Evolved Genetic Phenomena 1.6 Modelling 1.7 Overall Evolutionary Theories 2 Technicalities and Generalizations 2.1 Introduction 2.2 Random Union of Gametes 2.3 Dioecious Populations 2.4 Multiple Alleles 2.5 Frequency-Dependent Selection 2.6 Fertility Selection 2.7 Continuous-Time Models 2.8 Non-Random-Mating Populations 2.9 The Fundamental Theorem of Natural Selection 2.10 Two Loci 2.11 Genetic Loads 2.12 Finite Markov Chains 3 Discrete Stochastic Models 3.1 Introduction 3.2 Wright--Fisher Model: Two Alleles 3.3 The Cannings (Exchangeable) Model: Two Alleles 3.4 Moran Models: Two Alleles 3.5 K-Allele Wright--Fisher Models 3.6 Infinitely Many Alleles Models 3.6.1 Introduction 3.6.2 The Wright--Fisher In.nitely Many Alleles Model 3.6.3 The Cannings In.nitely Many Alleles Model 3.6.4 The Moran In.nitely Many Alleles Model 3.7 The Effective Population Size 3.8 Frequency-Dependent Selection 3.9 Two Loci 4 Diffusion Theory 4.1 Introduction 4.2 The Forward and Backward Kolmogorov Equations 4.3 Fixation Probabilities 4.4 Absorption Time Properties 4.5 The Stationary Distribution 4.6 Conditional Processes 4.7 Diffusion Theory 4.8 Multi-dimensional Processes 4.9 Time Reversibility 4.10 Expectations of Functions of Di.usion Variables 5 Applications of Diffusion Theory 5.1 Introduction 5.2 No Selection or Mutation 5.3 Selection 5.4 Selection: Absorption Time Properties 5.5 One-Way Mutation 5.6 Two-Way Mutation 5.7 Diffusion Approximations and Boundary Conditions 5.8 Random Environments 5.9 Time-Reversal and Age Properties 5.10 Multi-Allele Diffusion Processes 6 Two Loci 6.1 Introduction 6.2 Evolutionary Properties of Mean Fitness 6.3 Equilibrium Points 6.4 Special Models 6.5 Modifier Theory 6.6 Two-Locus Diffusion Processes 6.7 Associative Overdominance and Hitchhiking 6.8 The Evolutionary Advantage of Recombination 6.9 Summary 7 Many Loci 7.1 Introduction 7.2 Notation 7.3 The Random Mating Case 7.3.1 Linkage Disequilibrium, Means and Variances 7.3.2 Recurrence Relations for Gametic Frequencies 7.3.3 Components of Variance 7.3.4 Particular Models 7.4 Non-Random Mating 7.4.1 Introduction 7.4.2 Notation and Theory 7.4.3 Marginal Fitnesses and Average Effects 7.4.4 Implications 7.4.5 The Fundamental Theorem of Natural Selection 7.4.6 Optimality Principles 7.5 The Correlation Between Relatives 7.6 Summary 8 Further Considerations 8.1 Introduction 8.2 What is Fitness? 8.3 Sex Ratio 8.4 Geographical Structure 8.5 Age Structure 8.6 Ecological Considerations 8.7 Sociobiology 9 Molecular Population Genetics: Introduction 9.1 Introduction 9.2 Technical Comments 9.3 In.nitely Many Alleles Models: Population Properties 9.3.1 The Wright--Fisher Model 9.3.2 The Moran Model 9.4 In.nitely Many Sites Models: Population Properties 9.4.1 Introduction 9.4.2 The Wright--Fisher Model 9.4.3 The Moran Model 9.5 Sample Properties of In.nitely Many Alleles Models 9.5.1 Introduction 9.5.2 The Wright--Fisher Model 9.5.3 The Moran Model 9.6 Sample Properties of In.nitely Many Sites Models 9.6.1 Introduction 9.6.2 The Wright--Fisher Model 9.6.3 The Moran Model 9.7 Relation Between In.nitely Many Alleles and Infinitely Many Sites Models 9.8 Genetic Variation Within and Between Populations 9.9 Age-Ordered Alleles: Frequencies and Ages 10 Looking Backward in Time: The Coalescent 10.1 Introduction 10.2 Competing Poisson and Geometric Processes 10.3 The Coalescent Process 10.4 The Coalescent and Its Relation to Evolutionary Genetic Models 10.5 Coalescent Calculations: Wright--Fisher Models 10.6 Coalescent Calculations: Exact Moran Model Results 10.7 General Comments 10.8 The Coalescent and Human Genetics 11 Looking Backward: Testing the Neutral Theory 11.1 Introduction 11.2 Testing in the Infinitely Many Alleles Models 11.2.1 Introduction 11.2.2 The Ewens and the Watterson Tests 11.2.3 Procedures Based on the Conditional Sample Frequency Spectrum 11.2.4 Age-Dependent Tests 11.3 Testing in the Infinitely Many Sites Models 11.3.1 Introduction 11.3.2 Estimators of e 11.3.3 The Tajima Test 11.3.4 Other "Tajima-like" Testing Procedures 11.3.5 Testing for the Signature of a Selective Sweep 11.3.6 Combining Infinitely Many Alleles and In.nitely Many Sites Approaches 11.3.7 Data from Several Unlinked Loci 11.3.8 Data from Unlinked Sites 11.3.9 Tests Based on Historical Features 12 Looking Backward in Time: Population and Species Comparisons 12.1 Introduction 12.1.1 The Reversibility Criterion 12.2 Various Evolutionary Models 12.2.1 The Jukes--Cantor Model 12.2.2 The Kimura Model and Its Generalizations 12.2.3 The Felsenstein Models 12.3 Some Implications 12.3.1 Introduction 12.3.2 The Jukes--Cantor Model 12.3.3 The Kimura Model 12.4 Statistical Procedures Appendix A: Eigenvalue Calculations Appendix B: Significance Levels for F Appendix C: Means and Variances of F References Author Index Subject Index