Fixation probability of rare nonmutator and evolution of mutation rates

Abstract

Although mutations drive the evolutionary process, the rates at which the mutations occur are themselves subject to evolutionary forces. Our purpose here is to understand the role of selection and random genetic drift in the evolution of mutation rates, and we address this question in asexual populations at mutation‐selection equilibrium neglecting selective sweeps. Using a multitype branching process, we calculate the fixation probability of a rare nonmutator in a large asexual population of mutators and find that a nonmutator is more likely to fix when the deleterious mutation rate of the mutator population is high. Compensatory mutations in the mutator population are found to decrease the fixation probability of a nonmutator when the selection coefficient is large. But, surprisingly, the fixation probability changes nonmonotonically with increasing compensatory mutation rate when the selection is mild. Using these results for the fixation probability and a drift‐barrier argument, we find a novel relationship between the mutation rates and the population size. We also discuss the time to fix the nonmutator in an adapted population of asexual mutators, and compare our results with experiments.

Fixation probability of a nonmutator in a large population of asexual mutators

“…where p(k) is given by (A4). In [19], the frequency p(k) was obtained by solving (A2) numerically; however, an exact expression for p(k) that was obtained recently [16] allows us to extend our previous results (see (27) below). When U d /s ≪ 1, the average number of deleterious mutations is smaller than one (⌊k⌋ = 0) and we have Π ≈ 2sk which, on using (A7), yields the relative fixation probability Π/Π d ≈ 1 − (U b /s), in agreement with (11) of [19].…”

“…When U d /s ≫ 1, we need to consider the parameter regimes U b /s ≪ 1 and ≫ 1 separately. In the former case, the fixation probability Π is larger than Π d as supported by a perturbation theory in U b /s which yields Π/Π d = 1+(U b /s) [19]. But as Fig.…”

“…In this case, a nonmutator is favored only if it arises in mutator backgrounds with few deleterious mutations. But such mutator subpopulations are rare when U d is large thereby rendering the nonmutator fixation unlikely (also, see the discussion below (19)).…”

“…For u d = u b = 0, as the surprising effect of beneficial mutations described above survives [19] and the recursion equations for the probability π(k) are amenable to analysis (see below), we now study this case using the multitype branching process.…”

Evolution of mutation rates in rapidly adapting asexual populations

“…In the absence of beneficial mutations, natural selection will therefore act to decrease the mutation rate, until it is eventually balanced by genetic drift, mutation, or other physiological costs. A large body of previous theoretical work has explored these dynamics (Dawson, 1998(Dawson, , 1999Desai and Fisher, 2011;James and Jain, 2015;Johnson, 1999a;Kimura, 1967;Liberman and Feldman, 1986;Lynch, 2008Lynch, , 2011Soderberg and Berg, 2011).…”