Mathematical Sciences: Stochastic Models in Population Genetics

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9505129 Donnelly Abstract The proposed research falls in three broad areas. The first of these concerns measure-valued population models. A particle representation is used to study properties of a general class (including the Dawson-Watanabe and Fleming-Viot) of "neutral" processes and their induced genealogical structure. A major aim of the proposal is to extend this construction, most notably to incorporate selection (equivalently location dependent branching rates in the spatial setting), but also (in the genetics context) to allow for recombination and population substructure. Motivated by problems in the interpretation of human population genetic data, the investigator and his colleagues study particular subclasses of the general models with a view to understanding the effects on genealogy, and hence on genetic data, of various assumptions about population dynamics, and possible demographic correlations. A second area of the proposal concerns properties of geographically structured population genetics models. Recent results relating to coalescence times in these models allow explicit calculation of moments of identity measures for mutation structures (beyond infinite alleles) relevant to molecular DNA data. Conversely, an understanding of the models motivates a study of measures of population differentiation (and associated estimators) for molecular genetic data which generalize Wright's F_st coefficient. The third area of the proposal concerns the application of existing and anticipated theoretical results to statistical questions in the analysis of genetic data. These include exact likelihood ratios for a wide class of selective models against neutrality (and hence efficient inference procedures), the potential for use of information on the age order of alleles in testing neutrality, the correlation structure of estimates of evolutionary parameters from samples of linked genes, and the (non-)existence of consistent estimators of certain evolutionary parameters from singl e-locus data. The project is concerned with advancing our understanding of evolutionary processes. The development of mathematical models allows an examination of the consequences of particular evolutionary mechanisms: one may ask "if evolution acted in the following way, what sort of patterns would we expect in genetic data?". Mathematical modeling is necessary because many evolutionary forces operate over extremely long time scales for which direct observation, or experimentation, is impossible. Comparisons of model predictions with actual genetic data then provides insight into the extent to which the posited mechanisms are acting. The investigator studies, at a general level, the consequences of particular assumptions about the structure and demography of populations of interest. Particular aspects of the project relate to models for the evolution of human populations. Various statistical questions which arise in the interpretation of genetic data are also included.