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Genetic Demixing and Evolutionary Forces in the One-Dimensional Stepping Stone Model

We review and extend results for mutation, selection, genetic drift, and migration in a one-dimensional continuous population. The population is described by a continuous limit of the stepping stone model, which leads to the stochastic Fisher-Kolmogorov-Petrovsky-Piscounov equation with additional terms describing mutations. Although the stepping stone model was first proposed for population genetics, it is closely related to "voter models" of interest in nonequilibrium statistical mechanics. The stepping stone model can also be regarded as an approximation to the dynamics of a thin layer of actively growing pioneers at the frontier of a colony of microorganisms undergoing a range expansion on a Petri dish. We find that the population tends to segregate into monoallelic domains. This segregation slows down genetic drift and selection because these two evolutionary forces can only act at the boundaries between the domains; the effects of mutation, however, are not significantly affected by the segregation. Although fixation in the neutral well-mixed (or "zero dimensional") model occurs exponentially in time, it occurs only algebraically fast in the one-dimensional model. If selection is weak, selective sweeps occur exponentially fast in both well-mixed and one-dimensional populations, but the time constants are different. We also find an unusual sublinear increase in the variance of the spatially averaged allele frequency with time. Although we focus on two alleles or variants, q-allele Potts-like models of gene segregation are considered as well. We also investigate the effects of geometry at the frontier by considering growth of circular colonies. Our analytical results are checked with simulations, and could be tested against recent spatial experiments on range expansions off linear inoculations of Escherichia coli and Saccharomyces cerevisiae.