The fixation probability of rare mutators in finite asexual populations

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 \textit{E. 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 $P_{fix}$ 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 $P_{fix}$ successfully captures effects (ii) and (iii) when selection is fast compared to mutation ($\mu/s \ll 1$). 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 semi-quantitatively with existing laboratory evolution experiments and suggest future experimental directions.

💡 Research Summary

This study addresses a fundamental question in microbial evolution: under what conditions does a mutator allele—an allele that dramatically raises the genome‑wide mutation rate—successfully fix in a finite asexual population when it first appears at a very low frequency? The authors identify three forces that can impede fixation: (i) competition from beneficial mutations that arise in wild‑type backgrounds, (ii) the additional deleterious mutational load carried by the mutator, and (iii) stochastic genetic drift in a finite population. To quantify the joint impact of these forces, they construct a multi‑locus Wright–Fisher model in which each locus can acquire a beneficial mutation (rate μ_b, selective advantage s) or a deleterious mutation (rate μ_d, selective disadvantage –s). A mutator increases both μ_b and μ_d by a factor κ (empirically observed to be on the order of 100 in E. coli). The initial frequency of the mutator is set to 1/N, where N ranges from 10² to 10⁶, allowing the authors to explore a broad spectrum of drift intensities.

Analytically, the authors employ a diffusion approximation under the biologically realistic regime μ/s ≪ 1 (mutation rates are small compared with the strength of selection). Solving the resulting backward Kolmogorov equation yields a compact expression for the fixation probability:

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