Stochastic Problems in Population Genetics

These are" notes based on courses in Theoretical Population Genetics given at the University of Texas at Houston during the winter quarter, 1974, and at the University of Wisconsin during the fall semester, 1976. These notes explore problems of population genetics and evolution involving stochastic processes. Biological models and various mathematical techniques are discussed. Special emphasis is given to the diffusion method and an attempt is made to emphasize the underlying unity of various problems based on the Kolmogorov backward equation. A particular effort was made to make the subject accessible to biology students who are not familiar with stochastic processes. The references are not exhaustive but were chosen to provide a starting point for the reader interested in pursuing the subject further. Acknowledgement I would like to use this opportunity to express my thanks to Drs. J. F. Crow, M. Nei and W. J. Schull for their hospitality during my stays at their universities. I am indebted to Dr. M. Kimura for his continuous encouragement. My thanks also go to the small but resolute groups of.students, visitors and colleagues whose enthusiasm was a great source of encouragement. I am especially obliged to Dr. Martin Curie-Cohen and Dr. Crow for reading a large part eX the manuscript and making many valuable comments. Special gratitude is expressed to Miss Sumiko Imamiya for her patience and endurance and for her efficient preparation of the manuscript.

1 Orientation.- 1.1 Discrete space, continuous time, random walk.- 1.2 Discrete space, discrete time.- 1.3 Circular space, continuous time.- 1.4 Continuous space, continuous time.- 1.5 Markov process.- 1.6 Dynkin’s formula.- 2 Population Genetics Models.- 2.1 Wright’s model.- 2.2 Feller’s model.- 2.3 Moran’s model.- 2.4 Variable population size.- 2.5 Wright’s model with mutation.- 2.6 A model of irreversible mutation or a model of infinite alleles.- 2.7 A selection model.- 2.8 A model of dominance.- 2.9 Birth-and-death process.- 2.10 Density or frequency dependent process.- 2.11 Time inhomogeneous process.- 2.12 A model of random environment.- 3 Classification of Boundaries.- 3.1 Regular boundary.- 3.2 Exit boundary.- 3.3 Entrance boundary.- 3.4 Natural boundary.- 3.5 Nature of boundary.- 3.6 Examples.- 4 Expectation of Integration Along Sample Paths.- 4.1 Integration along sample paths.- 4.2 The boundary conditions.- 4.3 An example.- 4.4 Green’s function for a pure random process.- 4.5 Computer simulation.- 4.6 Sum of heterozygotes.- 4.7 Process with reflecting boundary.- 4.8 Irreversible mutation model (or infinite alleles).- 4.9 General form of Green’s function.- 4.10 Probability of fixation.- 4.11 Behavior of sample paths near the origin.- 4.12 Higher moments.- 4.13 Nagylaki’s formula.- 5 Modification of Processes.- 5.1 Killing and creating paths.- 5.2 Selection of paths.- 5.3 Random drift and fixation time.- 5.4 A case of genic selection.- 5.5 A symmetric property of sample paths.- 5.6 A general formula.- 5.7 Age of sample paths.- 5.8 Number of affected individuals and genetic load.- 5.9 Sojourn time of conditional sample paths on the present-frequency.- 5.10 A conditional age.- 5.11 Computer simulation.- 5.12 Random time change.- 6 Numerical Integration of the Kolmogorov Backward Equation.- 6.1 Integration method.- 6.2 Examples.- 7 Eigenvalues and Eigenvectors of the KBE.- 7.1 Eigenvalues and eigenvectors.- 7.2 Pure random drift case.- 7.3 Mutation and random drift case.- 7.4 Irreversible mutation case.- 7.5 Hypergeometric differential equation.- 7.6 Orthogonality of eigenfunctions.- 7.7 Expansion by eigenfunctions.- 7.8 The steady-state distribution of gene frequencies.- 8 Approximation Methods.- 8.1 Perturbation.- 8.2 Examples.- 8.3 Numerical method.- 8.4 Singular perturbation.- 9 Geographical Structure of Populations.- 9.1 One-dimensional populations, discrete colonies.- 9.2 Continuous space.- 9.3 Two-dimensional populations.- 9.4 Two-dimensional continuous space.- 9.5 Higher order moments.- 9.6 Numerical analysis at equilibrium.- 9.7 A differential equation and asymptotic formulae.- 9.8 Random drift.- 9.9 Time to fixation.- 10 Geographically Invariant Properties.- 10.1 Discrete time model.- 10.2 Continuous time model.- 10.3 Markov process.- 10.4 Diffusion method.- 10.5 Computer simulation of gene frequency change.- 10.6 Invariance based on diffusion method.- 10.7 Computer simulation of heterozygote distribution and other invariant properties.- 11 Gene Frequency Distributions and Random Drift in Geographically Structured Populations.- 11.1 Gene frequency distribution (global) in a structured population.- 11.2 Distribution of local gene frequencies.- 11.3 Random drift in a structured population.- 12 Some Special Problems.- 12.1 Variance of homozygote probability for the infinite neutral allele model.- 12.2 Variance of homozygote probability in a geographically structured population.- 12.3 Number of alleles.- 12.4 Some properties of the stepwise mutation model.- A1.1 Mutation model I.- A1.2 Mutation model II.- A1.3 Derivation of Kolmogorov backward equations (KBE).- Appendix II A Supplementary Note on the Existence and Uniqueness of the Solution for the Recurrence Equation (9.7).- Appendix III Distribution of Stochastic Integrals.- References.