Selection on mutators is not frequency-dependent

The evolutionary fate of mutator mutations – genetic variants that raise the genome-wide mutation rate – in asexual populations is often described as being frequency (or number) dependent. Mutators can invade a population by hitchhiking with a sweeping beneficial mutation, but motivated by earlier experiments results, it has been repeatedly suggested that mutators must be sufficiently frequent to produce such a driver mutation before non-mutators do. Here, we use stochastic, agent-based simulations to show that neither the strength nor the sign of selection on mutators depend on their initial frequency, and while the overall probability of hitchhiking increases predictably with frequency, the per-capita probability of fixation remains unchanged.

Whether or not mutators can successfully hitchhike to fixation has often been described as depending on the initial prevalence of mutator alleles in a population -most commonly referred to as frequency or number dependence (Raynes and Sniegowski, 2014;Sniegowski et al., 2000). This view holds that to replace the resident non-mutators, mutators must generate a beneficial mutation that escapes genetic drift and sweeps to fixation before their non-mutator competitors do. Accordingly, it has been proposed that mutators may be expected to invade (i.e., are favored by selection) only when already present in sufficient numbers to produce the successful beneficial mutation first, and lose their advantage (i.e., are disfavored by selection) when too rare to do so (reviewed in Raynes and Sniegowski, 2014;Sniegowski et al., 2000).
This frequency-dependent interpretation of mutator success has been primarily motivated by mutator dynamics observed in experimental studies of laboratory microbial populations. Most famously, in a series of pioneering experiments, Lin Chao and colleagues showed that mutator strains of the bacterium E. coli could supplant otherwise isogenic non-mutator strains by hitchhiking with beneficial mutations when initialized above a critical threshold frequency but would decline when initialized below it (Chao and Cox, 1983: reproduced in Figure 1A; . Since then, a similar pattern has been recapitulated in several other studies in E. coli and S. cerevisiae (Thompson et al., 2006;Gentile et al., 2011;de Visser and Rozen, 2006;Le Chat et al., 2006). Critically, a frequency-dependent framing of indirect selection on mutators implies a change in the sign or the strength of indirect selection with frequency. Here, we use stochastic, agent-based computer simulations to demonstrate that on the contrary, indirect selection on mutators is independent of frequency.

Results and discussion
Our computer simulations (Raynes, 2019) model asexual populations that mimic microbial evolution experiments under generally-accepted parameter values (Raynes et al., 2018). Figure 1B shows mutator frequency dynamics in randomly chosen simulations initialized across four log-orders of starting frequency, x 0 , which recapitulate experimental observations of the critical frequency threshold for hitchhiking reproduced in Figure 1A. As in Figure 1A, single, randomly-chosen realizations (i.e., simulation replicates) started below a threshold frequency end in mutator loss, while randomlychosen realizations started above end in fixation ( Figure 1B   Critically, fixation of an allele in a finite population is a probabilistic process influenced both by selection and random genetic drift, and even beneficial mutations will frequently be lost by chance alone. As such, whether an allele is truly favored or disfavored by selection can only be ascertained by evaluating its expected behavior averaged across many replicate, independent realizations. Indeed, if we consider the expected mutator frequency averaged across many replicate simulations, the threshold-frequency effect disappears ( Figure 1C). Instead, the average mutator frequency ultimately rises above the starting frequency at all x 0 , suggesting that mutators are, in fact, favored by selection in these populations regardless of starting frequency. (For more on why mutators are favored in large populations such as these see Raynes et al., 2018). The transient decline in average frequency seen in Figure 1C reflects selection against the deleterious load inherent to mutators (Kimura, 1967), and will be explored in a forthcoming publication].
To confirm that selection on mutators is independent of starting frequency, we measured the fixation probability of a mutator allele, P fix x 0 ð Þ, at each initial frequency, x 0 simulated in Figure 1. Given the stochasticity of the fixation process (and following Good and Desai, 2016;Raynes et al., 2018;Wylie et al., 2009), we compare P fix x 0 ð Þ to the probability of fixation of a neutral allele, given simply by x 0 . If a mutator is favored, we expect it to fare better than neutral (i.e., P fix x 0 ð Þ>x 0 ), and worse than neutral (i.e., P fix x 0 ð Þ<x 0 ) if disfavored. As Figure 2 shows, P fix x 0 ð Þ exceeds the fixation probability of a neutral allele for all x 0 , as anticipated in Figure 1C and confirming that the sign of selection on mutators does not depend on starting frequency.
Furthermore, while P fix x 0 ð Þ of a mutator increases with x 0 , it does so exactly as expected for a frequency-independent mutation. Under frequency-independent selection P fix x 0 ð Þ is simply the probability that at least one of the x 0 N alleles reaches fixation (where N is the population size). By definition of frequency-independent selection, the per-capita fixation probability is a constant, written, P fix x 0 ¼ 1=N ð Þ. Correspondingly, P fix x 0 ð Þ for any x 0 can be calculated as As the orange line in Figure 2 shows, P fix x 0 ð Þ calculated with Equation 1 is indistinguishable from P fix x 0 ð Þ observed in simulations, confirming that the per-capita fixation probability is independent of x 0 and equal to P fix x 0 ¼ 1=N ð Þ at any x 0 . Thus, while the expected fixation probability of a mutator increases with x 0 , the per-capita fixation probability remains unchanged, confirming that individual mutators do not become more likely to hitchhike to fixation when present at higher frequencies in a population.
Why then do mutators in experimental populations appear destined to go extinct when initially rare (e.g. ? Given that this behavior has been documented across different systems and selective environments (as well as in our stochastic simulations in Figure 1B), it seems unlikely to depend on any shared biological property of the experimental systems. Consider, however, that the per-capita fixation probability of a mutator is relatively low -in our simulations, operating under realistic parameter values, P fix x 0 ¼ 1=N ð Þ = 5.6Â10 À4 . Thus even when mutators are favored, most experimental replicates with rare mutators are expected to end in mutator extinction, and only those started at frequencies higher than roughly 1= NP fix x 0 ¼ 1=N ð Þ Â Ã are expected to end mostly with mutator fixation. Considering only a single or even a few realizations at each starting frequency (as in Figure 1A or B) would, most likely, result in observing only the most expected outcome for each x 0 . Indeed, all experimental studies that have documented the frequency-based transition included only a few populations at each starting frequency. Such limited sampling across a broad range of starting frequencies in these experiments would explain the sharp transition between fixation at high frequencies and loss at lower ones even when selection is frequency-independent (see also Tanaka et al., 2003). We expect that observing the dynamics in Figure 1C would be possible with more experimental replication, which, however, may not always be experimentally feasible.
In fact, the critical frequency-dependent transition observed in Figure 1A and 1B is not unique to mutators. Recall that P fix x 0 ð Þ of any mutation not under frequency-dependent selection, nevertheless, increases with starting frequency, x 0 (Equation 1). For example, even for a directly beneficial mutation, the probability of fixation from low frequencies is relatively low (Figure 3A Inset), Accordingly, as Figure 3A illustrates, single realizations of the dynamics of a directly beneficial mutation also exhibit a threshold-like switch from fixation to loss. In contrast, expected frequency dynamics averaged across many independent realizations confirm that beneficial mutations are favored by selection independent of starting frequency ( Figure 3B). Indeed, only for mutations under truly frequency-dependent selection do both the individual realizations ( Figure 3C) and the expected dynamics averaged across many realizations ( Figure 3D) exhibit an actual frequency-dependent transition.
In summary, our results demonstrate that neither the strength nor the sign of selection on mutators depend on initial frequency or number. Instead, we show that in populations favoring higher mutation rates, mutators consistently fare better than the neutral expectation (Figure 1 and Figure 2) regardless of starting frequency. Most importantly, the per-capita probability of fixation remains unchanged with frequency. We conclude that the frequency threshold observed in earlier experiments is, therefore, an artifact of limited experimental sampling rather than a frequencydependent change in selective effect.  Figure 1. P fix x 0 ð Þ scales with but never crosses the fixation probability of a neutral mutation (x 0 ; black dashed line). Thus, mutators are favored at all starting frequencies. The expected fixation probability P fix x 0 ð Þ (solid orange line), calculated from the fixation probability of a single mutator, P fix x 0 ¼ 1=N ð Þ = 5.6Â10 À4 (orange point) using Equation 1 is indistinguishable from the P fix x 0 ð Þ observed in simulations, demonstrating that the per-capita fixation probability at every frequency is independent of x 0 and equal to P fix x 0 ¼ 1=N ð Þ. The online version of this article includes the following source data for figure 2: Source data 1. Numerical data represented in Figure 2.

Materials and methods
Individual-based, stochastic simulations employed here have been previously described (Raynes et al., 2018). In brief, we consider haploid asexual populations of constant size, N, evolving in discrete, non-overlapping generations according to the Wright-Fisher model (Ewens, 2004). Populations are composed of genetic lineages -subpopulations of individuals with the same genotype. A genotype is modeled as an array of 99 fitness-affecting loci and 1 mutation rate modifier locus, which in a mutator state raises the genomic mutation rate m-fold. For computational efficiency, simulations in Figure 1 assume constant fitness effects: beneficial mutations at the fitness loci increase fitness by a constant effect s ben , while deleterious mutations decrease fitness by a constant effect s del . We assume additive fitness effects and so calculate fitness of a lineage with x beneficial and y deleterious mutations as w xy ¼ 1 þ xs ben À ys del . In simulations in Figure 1-figure supplement 1, beneficial and deleterious fitness effects are randomly drawn from an exponential distribution with the mean s ben = 0.1 and s del = -0.1 respectively. Simulations start with the mutator allele at a frequency of x 0 and continue until it either fixes or is lost from a population. Every generation the size of each lineage i is randomly sampled from a multinomial distribution with expectation N Á f i Á w i =w À À Á , where f i is the frequency of the lineage in the previous generation, w i is the lineage's fitness, and w À is the average fitness of the population (w À ¼ P f i Á w i ). Upon reproduction, each lineage acquires a random number of fitness-affecting mutations M, drawn from a Poisson distribution with mean equal to the product of its size and its total per-individual mutation rate, U ben þ U del ð Þ, where U ben and U del are the deleterious and beneficial mutation rates respectively. The number of beneficial and deleterious mutations is then drawn from a binomial distribution with n=M and P = U ben = U ben þ U del ð Þ and new mutations are assigned to randomly chosen non-mutated fitness loci.