## Abstract

Natural selection acts in three ways on heritable variation for mutation rates. A modifier allele that increases the mutation rate is (i) disfavored due to association with deleterious mutations, but is also favored due to (ii) association with beneficial mutations and (iii) the reduced costs of lower fidelity replication. When a unique beneficial mutation arises and sweeps to fixation, genetic hitchhiking may cause a substantial change in the frequency of a modifier of mutation rate. In previous studies of the evolution of mutation rates in sexual populations, this effect has been underestimated. This article models the long-term effect of a series of such hitchhiking events and determines the resulting strength of indirect selection on the modifier. This is compared to the indirect selection due to deleterious mutations, when both types of mutations are randomly scattered over a given genetic map. Relative to an asexual population, increased levels of recombination reduce the effects of beneficial mutations more rapidly than those of deleterious mutations. However, the role of beneficial mutations in determining the evolutionarily stable mutation rate may still be significant if the function describing the cost of high-fidelity replication has a shallow gradient.

THE evolution of the genetic system has been the subject of much theoretical research, ever since Fisher (1928) first studied the evolution of dominance. More recent studies have employed population genetic models that include modifier loci with alleles that modify the values of various genetic parameters. Examples include recombination rate (reviewed by Otto and Michalakis 1998), sex ratio (Charnov 1982), transposition rate (Charlesworth and Langley 1986), or deleterious mutation rate (Kondrashov 1995). A modifier allele may be subject to direct selection and also to indirect selection due to linkage disequilibrium with other loci that are under selection (see Ewens 1979, p. 195). Because there is heritable variation for mutation rates, they are subject to alteration through the action of natural selection (Sturtevant 1937). This article examines indirect selection acting on a modifier of mutation rates, through its association with both beneficial and deleterious mutations.

When a new beneficial mutation arises, it may be lost by genetic drift, or it may rise in frequency and become fixed. In either of these cases, the genetic background in which the beneficial mutation arose remains associated with it until separated by recombination. If the beneficial mutation is fixed, then other alleles initially associated with it will rise in frequency, and in an asexual population will also become fixed. This phenomenon was first observed in bacteria and termed *periodic selection* (Atwood*et al.* 1951; Dykhuizen 1990). In a continuous culture of bacteria, recurrent mutation causes rare neutral markers to increase linearly in frequency. Periodically, beneficial mutations sweeping to fixation cause clonal replacements: sudden decreases in the frequency of rare alleles not initially associated with the mutations. The more general term genetic hitchhiking (Maynard Smith and Haigh 1974) describes this process in both asexual and sexual populations. This is important in the evolution of mutation rates, because a modifier that increases the mutation rate is more likely to increase in frequency by hitchhiking on beneficial mutations. Linkage disequilibrium is generated when the beneficial mutation arises, and so the frequency of the modifier changes by indirect selection (Sniegowski*et al.* 1997; Taddei*et al.* 1997).

A second form of indirect selection acts on a modifier of the mutation rate, because a greater number of deleterious mutations arise in the higher mutation rate modifier background. In an asexual population, the net effect of these two forces is to move the mutation rate toward a stable equilibrium value that is also the value that maximizes the population mean fitness (Kimura 1967). This result is reproduced below. In this article, I study whether the genetic hitchhiking of a modifier allele affecting the mutation rate can be important in a sexually reproducing population, when both beneficial and deleterious mutations are modeled.

The indirect selection resulting from beneficial mutations on a modifier of mutation rate has been studied before in sexual populations (Leigh 1973; Gillespie 1981b; Ishii*et al.* 1989). Leigh (1973) concluded that the effect of beneficial mutations on the evolution of mutation rates was negligible in sexual populations. In contrast, both Gillespie (1981b) and Ishii *et al.* (1989) concluded that changing environments could favor increases in the mutation rate. These conclusions differ because only Leigh's (1973) model included a large class of unconditionally deleterious mutations.

All of these previous studies have used a model of a changing environment, in which there is a fixed set of alleles at a single locus. The selection coefficients change over time, in either a random (Gillespie 1981b) or a periodic manner (Leigh 1973; Ishii*et al.* 1989). All of the alleles are maintained at nonzero frequency by recurrent mutation. It has been suggested (Maynard Smith 1978, p. 192; see also Figure 1) that studies of such a model may underestimate the effect of beneficial mutations on the evolution of mutation rates, because when a selected allele starts to increase in frequency it will be in only weak linkage disequilibrium with the modifier. Therefore, this study models a succession of initially unique beneficial mutations arising in a stochastic manner, so that there is much stronger linkage disequilibrium between the new allele and the modifier background in which it arises.

The population genetic model that is used to study the fate of a modifier of mutation rate is described below. It is a multi-locus model, but the analysis is made tractable by treating only the simplest case of a single rare modifier of small effect. Linkage disequilibrium between sets of loci at which mutations occur can then be ignored, and only the two-way linkage disequilibrium between each mutable locus and the modifier needs to be considered. There are four main parts to the analysis, as follows: (i) the effect of many deleterious mutations scattered over a given genetic map is determined; (ii) the expectation of the change in allele frequency at the modifier locus is found for a single beneficial mutation sweeping through the population; (iii) this is used to find the long-term average fitness of the modifier allele for a series of beneficial mutations sweeping through the population. These results are presented in terms of a parameter that describes the average effect of hitchhiking events, and (iv) this parameter is estimated for a sexual population with beneficial mutations scattered over a given genetic map.

The main new results obtained in this article are expressions for the indirect selection coefficient acting at the modifier locus, caused by (i) deleterious mutations scattered over a genetic map and (ii) beneficial mutations sweeping through the population. The expressions are appropriate for a rare modifier, with a small effect on the mutation rate. The effect of beneficial mutations on the evolutionarily stable mutation rate toward which the population evolves is then discussed in the context of a “cost” function that describes the direct effect on fitness associated with a difference in mutation rate. Previously, such cost functions have only been included in models in which mutations are unconditionally deleterious (Kondrashov 1995; Dawson 1998).

## MODEL AND ANALYSIS

The notations used are summarized in Table 1.

**The modifier of mutation rate:** There is a randomly mating population of 2*N* haploid individuals. The population is polymorphic at a modifier locus that affects the genome-wide mutation rate. The deleterious mutation rate per genome, per generation, is *U* in genomes containing the Q allele and *U* + Δ*U* in genomes containing the P allele, which is rare. The mean mutation rate is *Ū* = *U* + *p*Δ*U*, where *q* and *p* are the frequencies of the two alleles. The beneficial mutation rate is proportional to the deleterious mutation rate. This haploid
model can be easily generalized to randomly mating diploids, because the P allele is rare and so PP homozygotes are vanishingly rare; it should be noted, however, that the definition of *U* remains as per haploid genome.

The fitness of genomes carrying the P allele, relative to genomes carrying the Q allele, is written *W.* For a modifier of small effect, *W* is close to unity, and so ln *W* is approximately the effective net selection coefficient favoring the P allele. The notation of fitness is used to avoid confusion with the selection coefficients for beneficial and deleterious mutations (*s*_{b} and *s*_{d}, see below). The term fitness is used to describe the effect of beneficial mutations, even though they will cause *p* to increase and decrease in a stochastic manner. I am considering a long-term limit expectation of the change in *p*, such that
*t*, is measured in generations, and *E*() stands for the expectation of a random variable. Because the main interest is determining the conditions under which P will spread (*i.e.*, when *W* > 1), rather than an exact description of the dynamics at the modifier locus, this definition of fitness is compatible with the restriction that *p* is small.

If evolutionary forces are weak then, to a good approximation, we have
*W*_{d}) and of beneficial mutations (*W*_{b}), and the direct effects on fitness (*W*_{c} or cost), act multiplicatively. This approximation holds only for a modifier of small effect because of second-order interactions between these effects. For example, the fixation probability of a beneficial mutation is reduced in the higher mutation rate background, because of its association with a greater number of deleterious mutations (Charlesworth 1994; Peck 1994; Barton 1995).

**Deleterious mutations:** The occurrence of deleterious mutations is assumed to be adequately described by a deterministic process. The net effect can then be represented as constant indirect selection at the modifier locus, which for a modifier of small effect will be proportional to Δ*U.* The precise relationship can be determined for any particular model of deleterious mutation.

For example, consider a model (Kimura and Maruyama 1966) that takes the limiting case of an infinite number of unlinked loci segregating for infinitesimally rare alleles. Selection occurs before mutation, both in the haploid phase of the life cycle. In the case where each deleterious mutation has an equal, multiplicative, effect on fitness of (1 – *s*_{d}), an exact expression for the reduction in log fitness experienced by a rare neutral modifier was derived by Dawson (1999),
*U*, was obtained by Kondrashov (1995).

In a large population (*i.e.*,2*Ns*_{d} > 1) with no recombination, any individual carrying more than the minimum number of deleterious mutations ultimately leaves no descendants (Fisher 1930, p. 136), and so

Here, I use a result derived by Leigh (1973) for a two-locus model with arbitrary linkage to estimate ln *W*_{d} for deleterious mutations randomly scattered over a genetic map of *n* chromosomes, each of length *M* morgans. By analyzing a model in which both mutation and selection are deterministic processes, Leigh (1973) obtained an equation for the strength of indirect selection on a modifier, which increases the mutation rate at a single linked locus by Δμ. His analysis of a continuous-time model assumes that the linkage disequilibrium between the modifier and the selected locus changes rapidly relative to the allele frequency of the modifier. This quasi-linkage equilibrium approach is appropriate for a modifier of small effect and yields

A similar result has been derived by Kimura (1967). A more general result for a deterministic multi-locus model has been derived by K. J. Dawson (unpublished results). Dawson's analysis further demonstrates that, if there is no epistasis in log fitness between deleterious mutations, then linkage disequilibrium between them is only generated because a modifier segregates in the population. The linkage disequilibrium is of order (Δ*U*)^{2}; when Δ*U* is small, the individual effects on the modifier therefore combine multiplicatively, to a good approximation.

Now consider deleterious mutations scattered randomly over a genome of *n* chromosomes, each of length *M* morgans. A deleterious mutation is unlinked to the modifier with probability (*n* – 1)/*n*, and otherwise the map distance, *z*, between it, and a modifier in the middle of a chromosome is a random variable with a uniform distribution on [0, *M*/2]. This gives
*r*(*z*) is the recombination probability obtained from *z* by using Haldane's (1919) mapping function, *r*(*z*) = ½ (1 – *e*^{–2}* ^{z}*). The quantity contained in braces in Equation 1b describes the increase over the free linkage (

*nM*→ ∞) case. Equation 1b is obtained from Equation 1a in the limiting case where

*s*

_{d}⪡ 1 and

*M*⪢ 1 and is surprisingly accurate for almost all plausible values of these parameters. The approximation is least accurate when

*n*= 1, but as long as

*s*

_{d}< 0.1, the error is <2% for

*M*> 2, and <11% for

*M*> 1. The error is reduced for larger

*n*; it is roughly halved for

*n*= 4. Note that, in the case of free recombination, this result differs by a factor of two from Dawson's (1999) analysis of the infinitesimally rare alleles model, where mutation occurs after selection, and hence each deleterious mutation has a 50% chance of being separated from the modifier by recombination before selection acts on it.

**Beneficial mutations:** In this model, I consider only a single beneficial mutation to be segregating at any one time. However, as is seen below, in sexual populations only beneficial mutations that are tightly linked to the modifier locus and that are destined to be fixed have any role to play in the evolution of mutation rates, and so this is only a weak restriction on the total rate of beneficial mutations. Because the effect at the modifier locus depends on whether the beneficial mutation arises in the Q or the P background, which is a single random event, it is necessary to study the long-term dynamics over the course of many beneficial mutations, each sweeping through the population in turn. The approach is to calculate the expectation of the effect of a single beneficial mutation, and then to combine the individual effects to estimate the net effect.

Each beneficial mutation that is destined to be fixed is assumed to arise at a point in time such that it does not interfere with other beneficial mutations sweeping through the population. This allele, b, confers a selective advantage *s*_{b} compared with the alternative allele B. It is assumed that stochastic effects are important only when b is rare (*i.e.*,2*Ns*_{b} ⪢ 1). The probability of recombination between this locus and the modifier locus is *r.* For each beneficial mutation that arises, *r* is a random variable, and so the effect of many beneficial mutations can be found by taking the expectation of the effect of a single beneficial mutation over a distribution of values of *r.*

The rate of occurrence, in the whole population, of beneficial mutations that are destined to be fixed, is *K* per generation. *K* may implicitly be a function of 2*N* and *Ū* and may vary through time, depending on the model of adaptive evolution. If, for example, adaptation is limited by the rate of environmental change (as assumed by Kaplan*et al.* 1989), then *K* would be independent of both 2*N* and *Ū*. Note that even if the delay between an environmental change and the ensuing beneficial mutations arising is a function of 2*N* and *Ū*, the overall rate of beneficial mutations remains independent of these parameters. The opposite extreme is a model of adaptation where there are very many loci at which beneficial mutations could potentially arise, so *K* would be proportional to both 2*N* and *Ū*. A model intermediate between these two extremes seems most likely to be realistic.

The hitchhiking effect is simply represented by the parameter *h*, which is the fraction by which the frequency of the allele *not* initially associated with the beneficial mutation is multiplied, as a net effect of the entire selective sweep. If, for example, b arises in the P background, then
*h* = 0. For sexual populations, the hitchhiking effect was first studied in by Maynard Smith and Haigh (1974), who derived an approximate expression for *h.* However, their analysis ignored stochastic fluctuations in the frequency of the b allele while it is rare. Taking this into account and conditioning on the ultimate fixation of b, Barton (1998) has found an exact expression for *h* in terms of gamma functions,
*r*/*s*_{b} < 1 and 4*N*_{e}*s*_{b} > 1. The dependence on *N*_{e}, the effective population size, arises because this is conditional on the fixation of b, which has probability 2*s*_{b} (*N*_{e}/*N*). In sexual populations, the hitchhiking effect decreases with increasing population size, because of the greater number of generations (and hence recombination events) between a beneficial mutation arising and sweeping to fixation.

In the model studied here, the modifier allele is not neutral. However, the direct selection (ln *W*_{c}) and indirect selection due to deleterious mutations (ln *W*_{d}) are assumed to be weak relative to the selection acting on the beneficial mutation (*s*_{b}), and so the result for a neutral modifier should be a sufficiently accurate approximation.

*Effect of a single beneficial mutation:* In this part of the analysis, *q* and *p* denote the modifier allele frequencies at the moment the beneficial allele b arises. I derive an expression for the expectation of *p*′, the frequency of the P allele after the b allele has swept to high frequency. Because the rate of beneficial mutation in each modifier background is proportional to the deleterious mutation rate, the probability of b arising in the Q background is *qU*/*Ū*, and in the P background is *p*(*U* + Δ*U*)/*Ū*. In the former case, *p*′= *hp*, and in the latter case *p*′= (1 – *q*′) = (1 – *hq*). Because *h* is a random variable, independent of which background the mutation arises on,

*Net effect of a succession of beneficial mutations:* Consider a series of *x* beneficial mutations arising at rate *K* over a total time *t.* I make use of the fact that the expectation of the product of independent random variables is the product of the expectations. While *p* is small, *E*(*p*′*/p*) is independent of *p*, and hence of the outcome of previous events. In this case
*x* → *Kt* as *t* → ∞, using (3) we obtain

For asexual populations (*h* = 0), Equation 4 is identical to a result derived by Leigh (1973). Although the linkage disequilibrium is much stronger in the model analyzed here, when the consequently larger effects are averaged over the different genetic backgrounds, the net effect is the same as in Leigh's model.

For sexual populations, Leigh (1973) tabulated values of (*p*′–*p*) for a range of *r*/*s*_{b} found by approximate solution of similar equations to those used to study hitchhiking (Maynard Smith and Haigh 1974), but assuming deterministic mutation and hence weaker linkage disequilibrium. The result obtained here is much simpler and clearly shows the relationship between the indirect selection at the mutator locus and the mean magnitude of hitchhiking events in the population in question.

*Expectation of the hitchhiking effect:* The results obtained above depend on the expectation of (1 – *h*). For no recombination, this is equal to one, and hitchhiking events have maximum effect on the frequency of the modifier. For a sexual population, *E*(1 – *h*) can be estimated by assuming that the beneficial mutations that arise are randomly scattered over *n* chromosomes, each *M* morgans long. Only a small fraction of these mutations are likely to have any effect, because (1 – *h*) is insignificant unless *r* < *s*_{b}. Unless the selective advantage of the b allele is very large, *r* is small enough for it to be reasonable to directly equate *r* with map distance rather than use Haldane's (1919) mapping function (see Nordborg*et al.* 1996).

When Equation 2 is averaged over a distribution of *r*, the gamma functions in Equation 2 can be ignored to a good approximation if *N*_{e}*s*_{b} is large. This is because when *r*/*s*_{b} ⪡ 1, the gamma functions are all approximately one, and when *r*/*s*_{b} is larger, (4*N*_{e}*s*_{b})^{–}^{r}^{/}^{s}^{b} becomes very small. In the calculation that follows, the error in making this approximation is <3% when *N*_{e}*s*_{b} > 10^{3}, and <15% when *N*_{e}*s*_{b} > 10^{2}.

In the same way as for deleterious mutations, the probability that the modifier and a beneficial mutation are on the same chromosome is 1/*n.* When the map distance between the two is chosen from a uniform distribution on [0, *M*/2], the probability that *r* < *s*_{b} is simply 2*s*_{b}/*M.* In this case, *r* is uniformly distributed on the interval [0, *s*_{b}], and the expectation of (1 – *h*) according to Equation 2 without the gamma functions is given by
*N*_{e}*s*_{b}

**Direct selection on the modifier:** The log-fitness of the P allele relative to the Q allele is a function of both *U* and Δ*U.* The component of this, due to differences in the direct fitness effects of the Q and P alleles, is ln *W*_{c}, which is also a function of both *U* and Δ*U.* Let *w*(*U*) be the fitness of an individual with mutation rate *U*, carrying the B allele and no deleterious mutations. Assume that there is no epistasis between the modifier alleles and any fitness-affecting mutations. Then, for a modifier of small effect, ln *W*_{c} is linear in Δ*U*, as follows:

Although it is widely believed that increasing the fidelity of DNA replication is costly (Sturtevant 1937; Leigh 1973; Kirkwood*et al.* 1986; Kondrashov 1995), very little is known about the nature of such a cost. Here I assume that the direct selection results only from increasing costs of higher-fidelity replication or mutation repair. This cost approaches infinity for perfect fidelity (Kirkwood*et al.* 1986, p. 5), and therefore fitness *w* is zero for *U* = 0. If the general form of the cost is as shown in Figure 2, then it would be reasonable to assume that *w*(*U*) asymptotically approaches some maximum as *U* increases. In this case, the derivative of the fitness function, *d* ln *w*(*U*)/*dU*, is a strictly positive, monotonically decreasing function of *U.* This is important in determining the existence and uniqueness of an evolutionarily stable mutation rate (ESS; see below and Figure 3). It appears that it is not possible to make such a statement if the effect of the modifier is considered in relative (Δ*U*/*U*) rather than absolute (Δ*U*) terms.

**Asexual populations:** Although the model described here is a reasonable one with which to study the evolution of mutation rates in sexual populations, it is inappropriate for asexual populations. In a totally asexual population each beneficial mutation will cause a complete clonal replacement, and hence the restriction that *p* should remain small would be violated. Hypermutators (modifiers) increasing the rate of certain mutations by factors of up to a thousand have been found at low frequency in natural populations of the bacteria *Escherichia coli* and *Salmonella enterica* (LeClerc*et al.* 1996). The rate of mutation at modifier loci themselves would be increased in a mutator phenotype, and hence a mutator allele coupled to a beneficial mutation stands an appreciable chance of back-mutation once at high frequency. This can result in ultimate fixation of a genotype combining the low mutation rate modifier with the beneficial mutation (Taddei*et al.* 1997). In other words, clonal replacement need not occur, and the modifier that “caused” the beneficial mutation is not fixed, so *h* ≠ 0. Microorganisms maintained in continuous culture show population turnovers that are too rapid to be explained by sequential fixation of unique beneficial mutations (Dykhuizen 1990). A fundamentally different model such as the one studied by Taddei *et al.* (1997) is clearly more appropriate. However, this would not allow easy comparison with results from the model used here for sexual populations. Therefore the treatment of asexual populations in this article is better regarded as a limiting case for sexual populations, as recombination rates approach zero.