Evolution of Mutation Rates in Rapidly Adapting Asexual Populations

Abstract

Mutator and antimutator alleles often arise and spread in both natural microbial populations and laboratory evolution experiments. The evolutionary dynamics of these mutation rate modifiers are determined by indirect selection on linked beneficial and deleterious mutations. These indirect selection pressures have been the focus of much earlier theoretical and empirical work, but we still have a limited analytical understanding of how the interplay between hitchhiking and deleterious load influences the fates of modifier alleles. Our understanding is particularly limited when clonal interference is common, which is the regime of primary interest in laboratory microbial evolution experiments. Here, we calculate the fixation probability of a mutator or antimutator allele in a rapidly adapting asexual population, and we show how this quantity depends on the population size, the beneficial and deleterious mutation rates, and the strength of a typical driver mutation. In the absence of deleterious mutations, we find that clonal interference enhances the fixation probability of mutators, even as they provide a diminishing benefit to the overall rate of adaptation. When deleterious mutations are included, natural selection pushes the population toward a stable mutation rate that can be suboptimal for the adaptation of the population as a whole. The approach to this stable mutation rate is not necessarily monotonic: even in the absence of epistasis, selection can favor mutator and antimutator alleles that "overshoot" the stable mutation rate by substantial amounts.

Keywords: antimutator; deleterious load; hitchhiking; mutator.

Figure 1

The fixation probability of a mutation rate modifier in the successive mutations regime. Solid lines depict Equation (11) for four sets of parameters, which illustrate the four characteristic shapes of $N{p}_{\text{fix}}\left(r\right)$. In all four cases, the base parameters are $N={10}^7$, $s_b={10}^{-2}$, $U={10}^{-4}$, and $\mathit{ϵ}={10}^{-5}$, with modifications listed in the inset.

Figure 2

(Top) The fixation probability of a mutation rate modifier in the clonal interference regime when $U_d=0.$ Symbols denote the results of forward-time solutions (described in Appendix C) for $s_b={10}^{-2}$, $U_b={10}^{-5}$, and $N\in \left\{{10}^5,{10}^7,{10}^9\right\}$. Solid lines denote the theoretical predictions in Equation (38). For comparison, the successive mutations prediction ($N{p}_{\mathrm{fix}}\approx r$) and neutrality ($N{p}_{\mathrm{fix}}\approx 1$) are shown by the dashed lines. (Bottom) The rate of adaptation of a successful modifier lineage, relative to that of the wild type, for the same set of populations above. The solid black line denotes the asymptotic prediction, $v\left(r\right)/v\approx {\log }^2\left(s_b/U_b\right)/{\log }^2\left(s_b/r{U}_b\right)$, which is independent of N. For comparison, the successive mutations prediction ($v\left(r\right)/v\approx r$), and no change ($v\left(r\right)/v\approx 1$), are shown by the dashed lines.

Figure 3

The fixation probability of a mutation rate modifier in the clonal interference regime when $U_d>0.$ Symbols denote the results of forward-time simulations with the base parameters $N={10}^7$, $s_b={10}^{-2}$, $U_b={10}^{-5}$, $U_d={10}^{-4}$, and $s_d={10}^{-1}$, with modifications listed in the figure. Solid lines denote the theoretical predictions in Equation (38).

Figure 4

The predicted fixation probability of a modifier allele when $N{s}_b={10}^6$ and $\mathit{ϵ}={10}^{-6}$. Grid points are colored according to the value of $N{p}_{\mathrm{fix}}$ from Equation (38), and are capped at a maximum value of $\left|{\log }_{2}N{p}_{\mathrm{fix}}\right|=2$ to maintain contrast. For comparison, the solid lines denote 20 log-spaced contours that range from ${10}^{-1}$ to ${10}^4$.

Figure 5

(Top) A vertical “slice” of Figure 4 for three different values of U. Symbols denote the results of forward time simulations with $N{U}_0=5780$, and $s_d=5s_b;$ other parameters are the same as Figure 4. Solid lines denote theoretical predictions from Equation (38). (Bottom) The scaled rate of adaptation as a function of the mutation rate for the same set of parameters. Symbols denote the results of forward-time simulations, the solid lines show the theoretical predictions obtained by solving Equations (31) and (32) numerically, and the dashed line shows the asymptotic formula, $v/s_b^2\sim 2\log \left(N{s}_b\right)/\log {\left(s_b/\mathit{ϵ}{U}_m\right)}^2$.