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 "zero-dimensional") model occurs exponentially in time, it occurs only algebraically fast in the one-dimensional 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.
Gene regulatory networks (GRNs) that make reliable decisions should have design features to cope with random fluctuations in the levels or activities of biological molecules. The phage ? GRN makes a lysis-lysogeny decision informed by the number of phages infecting the cell. To analyse the design of decision making GRNs, we generated random in silico GRNs comprised of two or three transcriptional regulators and selected those able to perform a ?-like decision in the presence of noise. Various two-protein networks analogous to the ? CI-Cro GRN worked in noise-less conditions but failed when noise was introduced. Adding a ? CII-like protein significantly improved robustness to noise. CII relieves the CI-like protein of its decider function, allowing CI to be optimized as a decision maintainer. CIIs lysogenic decider function was improved by its instability and rapid removal once the decision was taken, preventing its interference with maintenance. A more reliable decision also resulted from simulated co-transcription of the genes for CII and the Cro-like protein, which correlates fluctuations in these opposing decider functions and makes their ratio less noisy. Thus, the ? decision network contains design features for reducing and resisting noise.
Phage lambda is among the simplest organisms that make a developmental decision. An infected bacterium goes either into the lytic state, where the phage particles rapidly replicate and eventually lyse the cell, or into a lysogenic state, where the phage goes dormant and replicates along with the cell. Experimental observations by P. Kourilsky are consistent with a single phage infection deterministically choosing lysis and double infection resulting in a stochastic choice. We argue that the phage are playing a "game" of minimizing the chance of extinction and that the shift from determinism to stochasticity is due to a shift from a single-player to a multiplayer game. Crucial to the argument is the clonal identity of the phage.
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