Genetic demixing and evolution in linear stepping stone models

Abstract

Results for mutation, selection, genetic drift, and migration in a one-dimensional continuous population are reviewed and extended. 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 micro-organisms undergoing a range expansion on a Petri dish. 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 "zerodimensional") model occurs exponentially in time, it occurs only algebraically fast in the onedimensional model. An unusual sublinear increase is also found in the variance of the spatially averaged allele frequency with time. If selection is weak, selective sweeps occur exponentially fast in both well-mixed and one-dimensional populations, but the time constants are different. The relatively unexplored problem of evolutionary dynamics at the edge of an expanding circular colony is studied as well. Also reviewed are how the observed patterns of genetic diversity can be used for statistical inference and the differences are highlighted between the well-mixed and one-dimensional models. Although the focus is on two alleles or variants, q-allele Potts-like models of gene segregation are considered as well. Most of the analytical results are checked with simulations and could be tested against recent spatial experiments on range expansions of inoculations of Escherichia coli and Saccharomyces cerevisiae.

Stochasticity enhances the gaining of bet-hedging strategies in contact-process-like dynamics

“…Given that these conditions are often met by biological populations-as for instance, in bacterial colonies competing at the front of a range expansion in noisy environments [34][35][36]-our results support the importance of bet-hedging in nature. This being said, of course, more realistic models-including some realistic ingredients such as, for example, the possibility of "dormant" states and not just birth and death processes-would be required to approach viral or bacterial communities and their bet-hedging more closely.…”

Spreading of nonmotile bacteria on a hard agar plate: Comparison between agent-based and stochastic simulations

“…Our approach is different than earlier studies [7,8,33,41,42] where the bacterial morphological patterns were attributed to substrate properties such as irregularities on the agar substrate [41], substrate hardness that depends on the agar concentration and local lubrication created by bacteria [42], and nutrient concentration. However, these models ignore the role of population fluctuations that have been shown [10,43,44] to play a crucial rule in determining the growth, competition and cooperation in bacterial colonies under nutrient rich conditions.…”

“…Bacteria in a Petri dish environment exhibit a large variety of complex spatial patterns ranging from compact circular growth, concentric rings to long branched patterns [4][5][6][7][8][9][10][11]. The colony morphology depends upon various factors such as nutrient concentration, cell motility, growth-proliferation and death dynamics, and other chemical and physical variables [12][13][14][15][16][17][18][19][20].…”

“…Several studies have incorporated the effect of the nutrient concentration and bacteria motility by coupling Eq. (1.1) with an additional equation for each of these variables to obtain different morphological patterns that were discussed above [4][5][6][7][8][9][10][11][33][34][35]. Studies designed to investigate the role of demographic noise use the stochastic variants of Eq.…”

“…Studies designed to investigate the role of demographic noise use the stochastic variants of Eq. (1.1) [10,[36][37][38][39][40].…”