The fixation probability of rare mutators in finite asexual populations

Abstract

A mutator is an allele that increases the mutation rate throughout the genome by disrupting some aspect of DNA replication or repair. Mutators that increase the mutation rate by the order of 100-fold have been observed to spontaneously emerge and achieve high frequencies in natural populations and in long-term laboratory evolution experiments with Escherichia coli. In principle, the fixation of mutator alleles is limited by (i) competition with mutations in wild-type backgrounds, (ii) additional deleterious mutational load, and (iii) random genetic drift. Using a multiple-locus model and employing both simulation and analytic methods, we investigate the effects of these three factors on the fixation probability Pfix of an initially rare mutator as a function of population size N, beneficial and deleterious mutation rates, and the strength of mutations s. Our diffusion-based approximation for Pfix successfully captures effects ii and iii when selection is fast compared to mutation (micro/s<<1). This enables us to predict the conditions under which mutators will be evolutionarily favored. Surprisingly, our simulations show that effect i is typically small for strong-effect mutators. Our results agree semiquantitatively with existing laboratory evolution experiments and suggest future experimental directions.

Figure 1.—

Some sample runs from simulations where the wild-type mutation rate is zero. (Top) The number of mutators in the population vs. , where r is the birth probability per time step that is proportional to the (initial) mean population fitness. (Bottom) The average number of beneficial mutations in the mutator subpopulation is shown. The solid lines resulted in fixation of the mutator allele, whereas the shaded lines resulted in its loss. When the mutation rate of the mutators (μ₊) is not too large, the mutator hitchhikes to fixation with a single beneficial mutation (left). When μ₊ is larger, many beneficial mutations occur during the fixation process (middle and right). Our analytic approximation scheme assumes that the fixation process is triggered by merely the first beneficial mutation to survive drift. Note that in each case the population is always far from the fitness maximum when the mutator achieves fixation since there are 80 possible beneficial mutations. Parameters are N = 10⁵, x_o = 0.005, δ = 0, wild-type mutation rate μ₋ = 0, and μ₊ = 10⁻⁵ (left), μ₊ = 10⁻³ (middle), and μ₊ = 1 (right). α = 0.4, s = (initial values).

Figure 2.—

Averaged results of simulations, and the utility of S_μ as the measure of mutator success. When , P_fix increases linearly with x_o (data not shown). (Left) The (least-squares) slope of said linear increase when the population is well adapted (bottom) and poorly adapted (top) to its environment is shown. The data on the bottom row are quite noisy because of the small number of trials resulting in fixation. (Right) The same data are expressed, but in terms of the effective selection coefficient S_μ of the mutator allele obtained by inverting Equation 2. Whereas the values from the left obviously depend on N, the values on the right panels are independent of N when . This suggests that S_μ, which exposes an underlying simplicity to the simulation results, is a more natural measure of mutator success than P_fix. Note that when the mutator is favored, S_μ is always less than the selective advantage s of a single beneficial mutation; this is due both to deleterious mutations and to loss due to random drift. Parameters are s = , μ₋ = 0, δ = 0, and α = 0.4 (top) and 0.008 (bottom). See the supplemental information for details concerning averaging.

Figure 3.—

Dependence on the underlying selective advantage s. The data corresponding to two values of s, i.e., two values of L, approximately collapse onto a single curve when S_μ and μ₊ are each scaled by s. The scaling of the independent variable underscores the fact that mutator success for fixed α is largely controlled by the ratio of timescales for mutation (1/μ₊) and selection (1/s). In particular, the sharp decrease in S_μ at large μ₊ occurs when these timescales become comparable, i.e., when deleterious mutations accumulate in an expanding lineage before it has sufficient time to achieve fixation. Parameters are N = 5000, μ₋ = 0, α = 0.4, and δ = 0.

Figure 4.—

Behavior near the transition from favored to disfavored mutators. When α_e is greater than a critical value , the mutator allele is favored (S_μ > 0) for small enough μ₊. Our analytic approach (ISLA) predicts that the transition occurs at (Ns + 1) = 1, which agrees extremely well with simulation data. Parameters are N = 5000, s = , μ₋ = 0, and δ = 0. The numbers of available beneficial mutations are, in order of decreasing mutator success, 10, 5, 3, and 1.

Figure 5.—

Comparison of simulation, numerical solution of Equation 6, and the analytic approximation Equation 10. The exact numeric solutions to our ISLA Equation 6 for different N converge to the analytic approximation Equation 10 when (left). Solutions to Equation 6 show, in agreement with simulation, that S_μ/s depends on μ₊/s rather than μ₊ and s separately (right). Parameters are those used in Figures 2 and 3.

Figure 6.—

Small effect of mutations arising in wild-type backgrounds. ISLA predicts that these mutations will become important in the weak-effect mutator regime defined by , where R ≡ μ₊/μ₋. However, the simulation data show that mutations in wild-type backgrounds sometimes have a negligible impact even in the weak-effect mutator regime. (Right) has the values 18, 3.6, and 0.18, respectively, as N is increased. Accordingly, ISLA predicts a decrease in S_μ, but S_μ did not change in simulations. (Left) Beneficial mutations in wild-type backgrounds eventually decrease S_μ for large enough R, although the decrease here is smaller than what ISLA predicts. Parameters are α = 0.4, s = , δ = 0, and μ₊/μ₋ = 100 (right).

Figure 7.—

The role of nonlethal deleterious mutations. We “turned off” deleterious mutations, both in simulations and in ISLA, by setting the deleterious mutation rate to zero and leaving the beneficial mutation rate unchanged (left). The difference between these results and the corresponding ones with deleterious mutations is plotted on the vertical axis on the left. For μ₊/s ≲ 1, deleterious mutations have the same effects in ISLA Equation 6 as in simulations (left). ISLA essentially treats deleterious mutations as lethal (A3), instead of merely having a selective disadvantage −s. We tested this approximation directly in simulations by varying the parameters α and δ while holding the product α(1 − δ) ≡ α_e constant (right). Parameters are s = , N = 5000, μ₋ = 0, and α = 0.4 and δ = 0 (left only).

Figure 8.—

The scaling behaviors of Equation 1 and ISLA are qualitatively different. If the initial number of mutators Nx_o is kept constant while N is increased, then ISLA predicts that P_fix remains invariant, whereas the frequency-dependent Equation 1 predicts a large change. Simulations are in better accord with ISLA than with Equation 1. These scaling predictions could be experimentally tested by observing whether the “threshold” number of initial mutators changes with N. Here, we have defined the threshold as the number of mutators for which and depicted these values with vertical dotted lines. Parameters are α = 0.4, δ = 0, and .