Mathematics Colloquium: Linked selected and neutral loci in heterogeneous environments

2007-02-15

1:30 p.m. 255 Cleveland

Judith R. Miller

Abstract: We analyze a system of ordinary differential equations modeling a physically linked pair of loci (or genes), in a population consisting of two demes (or subpopulations). Each locus has two alleles; at the “selected” locus, natural selection favors different alleles in different demes, while at the “marker” locus both alleles are equally favorable in both demes. The system is singularly perturbed, with the migration rate m between the demes serving as a small parameter. We use geometric singular perturbation theory to show that when m is sufficiently small, almost every solution approaches one of a 1-dimensional continuum of equilibria; we also obtain asymptotic expansions of the solutions. Finally, we obtain formulas for the transient dynamics of FST (a measure of population structure) at both loci. as well as for the rate of genotyping error if the allelic state at the selected locus is inferred from that at the neutral marker locus. We conclude that in typical biological systems and with currently typical densities of markers in the genome, associations between selected loci and markers will decay too rapidly to allow reliable inference about a gene from a nearby marker.