Mathematical genetics and evolution

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This research is directed towards a wide variety of population genetic models and the concomitant evolutionary dynamics and statistical inferences. The long-term objectives are: a) to describe and explain the level of variability in biological populations including human populations; b) to develop mathematical, statistical and computational methods to analyse genetic polymorphisms and make inferences about related parameters or assumptions; c) to deduce general evolutionary principles that result from selection, mutation and drift; and d) to incorporate the effects of complex structures and individual interactions into the models. In the next five years, the efforts will be directed more precisely towards: (1) deducing exact conditions for the replacement of one allele by another under the combined effects of selection and drift in structured populations and applying these conditions to the evolution of individual behaviours or genetic parameters; (2) showing that non random mating creates hidden interactions between individuals in the presence of selection but the coefficients of relatedness necessary to describe their effects are generally more complicated than usually thought; (3) extending separation-of-timescales arguments to the study of genetic diversity and linkage disequilibrium in metapopulations; (4) finding exact and approximate formulas for the sample configurations of DNA sequences and improving composite-likelihood methods for ancestral inference and fine gene mapping. A combination of mathematical, statistical and computational tools will be used: local stability analyses, linear or quadratic, for deterministic selection models; diffusion approximations or coalescent approaches for finite populations; combinatorics for sampling formulas; separation of timescales for structured populations; simulations and algorithms for the estimation of genetic parameters. I expect that this research will contribute to a better understanding of the effects of genetic mechanisms underlying evolutionary processes in natural populations and the multiple factors responsible for genetic polymorphisms and, eventually, human diseases.