Abstract
I study the population genetics of adaptation in asexuals. I show that the rate of adaptive substitution in an asexual species or nonrecombining chromosome region is a bellshaped function of the mutation rate: at some point, increasing the mutation rate decreases the rate of substitution. Curiously, the mutation rate that maximizes the rate of adaptation depends solely on the strength of selection against deleterious mutations. In particular, adaptation is fastest when the genomic rate of mutation, U, equals the harmonic mean of selection coefficients against deleterious mutations, where we assume that selection for favorable alleles is milder than that against deleterious ones. This simple result is independent of the shape of the distribution of effects among favorable and deleterious mutations, population size, and the action of clonal interference. In the course of this work, I derive an approximation to the probability of fixation of a favorable mutation in an asexual genome or nonrecombining chromosome region in which both favorable and deleterious mutations occur.
CONSIDER an asexual species that encounters a novel or changing environment. Under what conditions will it adapt fastest? In particular, what rate of mutation allows the fastest adaptation?
The problem of how adaptation rate depends on mutation rate in asexuals (or in chromosome regions that do not recombine) is subtle. Adaptive evolution obviously requires the production of beneficial mutants. Thus all else being equal, that clonal lineage having the highest mutation rate might seem best poised for longterm evolution. Such a clone enjoys what Leigh (1970, 1973) has called high “adaptability,” as opposed to present “adaptedness.” But all else is not equal, as there are at least two complications. The first is that as favorable mutations grow too common they begin to get in each other's way. The fixation of a first favorable mutation can, for instance, be blocked by the appearance of a more strongly favored one that arises during the first's transit to fixation. Gerrish and Lenski (1998) recently studied this phenomenon—which they call “clonal interference,” a variant of the HillRobertson effect (Hill and Robertson 1966). [In the Drosophila literature, clonal interference is often referred to as a “traffic problem” (Kirby and Stephan 1996; Stephan 1995).] Gerrish and Lenski showed that clonal interference increases the mean time between fixation events and so slows the rate of adaptive substitution. In particular, they showed that the rate of substitution does not increase without bound as the mutation rate increases. Instead it plateaus. They thus suggested that clonal interference may impose a “speed limit” on the rate of adaptation in asexuals. Miralles et al. (1999) recently attempted to detect this speed limit experimentally in asexual populations of vesicular stomatitis virus.
Second, an increase in the mutation rate increases the number of deleterious mutations. These deleterious alleles have several effects. For one, they cause asexuals to suffer an increased mutational load. The resulting tradeoff between longterm adaptability and shortterm adaptedness forms the foundation of a large body of work on the evolution of mutation rates in asexuals (Kimura 1967; Leigh 1970, 1973; Gillespie 1981; Dawson 1998, 1999; Johnson 1999a). But as we will see, deleterious mutation also plays an important role in determining the maximal adaptability, i.e., the rate of mutation that yields the fastest adaptation.
Here, following Fisher (1930), Peck (1994), and Barton (1995), I consider the fate of asexual lineages experiencing mutation to both favorable and deleterious alleles. In particular, I derive the rate of adaptive substitution when favorable mutations encounter traffic problems due to both other favorable mutations and to deleterious mutations. I find that the rate of adaptation does not plateau with increasing mutation rate, as claimed by Gerrish and Lenski (1998). Instead it peaks. Contrary to intuition, the rate of mutation that yields the fastest adaptation depends solely on the strength of selection against deleterious mutations.
THE MODEL AND RESULTS
A simple model: Consider an asexual haploid with large population size N (extensions to asexual diploids are straightforward but not pursued here). The total rate of mutation per genome (or nonrecombining chromosome region) is U. In the present environment, a fixed proportion p_{b} of all mutations are beneficial. Because our species is nonrecombining, adaptive evolution gets complicated by the presence of linked deleterious alleles: favorable mutations often arise in genomes bearing one or more deleterious mutations, as emphasized by Fisher (1930), Manning and Thompson (1984), and Peck (1994). I assume that selection coefficients for new beneficial mutations, s_{b}, are typically smaller than those for new deleterious mutations, s_{d}. Although surely not universally true, this is likely to be biologically realistic in many cases. [Indeed, analysis of Fisher's geometric model of adaptation (Hartl and Taubes 1996; Orr 1998) shows that the mean effect of new deleterious mutations must be greater than that for new favorable ones.] In any case, this assumption is relaxed somewhat in the simulations described below.
When s_{b} < s_{d}, adaptive evolution is essentially constrained to those favorable mutations that appear in deleteriousmutationfree genomes (Fisher 1930; Peck 1994; Barton 1995). Such mutations enjoy normal probabilities of fixation of about 2s_{b}. In all other cases, genomes carrying a new favorable mutation suffer a negative net selection coefficient and cannot get fixed. Peck (1994) has called this the “ruby in the rubbish” effect.
If adaptation reflects the substitution of new alleles and favorable mutations have independent fates, the rate of adaptive substitution approximately equals the number of favorable mutations appearing per generation multiplied by each mutation's probability of fixation, or
We now ask: What rate of mutation maximizes the rate of adaptation? Differentiating,
Three interesting results emerge from Equation 4. The first is that the rate of mutation yielding the fastest adaptation takes an intermediate (nonzero) value even though we have focused solely on longterm adaptability and ignored shortterm genetic load. The intuitive reason is straightforward. The optimal mutation rate walks a line at which the product of the number of favorable mutations and the size of the zero class—a quantity that might be viewed as the effective number of favorable mutations—is maximized. When U > U_{opt}, there are too many deleterious mutations, mutationfree genomes are too rare, and too many favorable mutations are thrown away. When U < U_{opt}, the zero class is larger, but the population produces too few favorable mutations to take advantage of the number of mutationfree genomes, and adaptation slows. Note that, to the order of our approximations, the optimal mutation rate in asexuals is independent of population size, the proportion of mutations that are favorable, and the selective advantage enjoyed by favorable alleles—all poorly known quantities.
The second point emerging from (4) is that those lineages that adapt fastest have their mutation rate set equal to a quantity that might be roughly constant across lineages, the selection coefficient against deleterious mutations. We pursue this point in the discussion. Third, because an average of U/s_{d} deleterious mutations exist per genome at mutationselection balance, an asexual population that enjoys a maximal rate of adaptation carries a mean of U_{opt}/s_{d} = 1 deleterious alleles per genome.
The fact that adaptation proceeds fastest when U = s_{d} is not intuitive. It also depends on a number of assumptions and approximations. One is that we assume the population always resides at mutationselection balance, which is unlikely to be true, especially following the fixation of a favorable mutation (Johnson 1999a). (Another assumption is discussed below.) Although it seems unlikely that our approximations would qualitatively affect our conclusions, it seemed worth checking Equation 4 against exact computer simulations.
These simulations were brute force, following a population composed of N haploid asexual genomes, each of which may experience mutation to deleterious and/or favorable alleles. In particular, the number of deleterious mutations per genome per generation was Poisson distributed with mean U(1 − p_{b}), while the number of favorable mutations per genome per generation was Poisson distributed with (much smaller) mean Up_{b} (see Figure 1 legend for parameter values). The order of events was mutation followed by selection, and fitness was multiplicative. Generations were discrete and the program recorded the number of generations between successive adaptive substitutions. Preliminary simulations showed that, when favorable mutations were introduced singly (i.e., no other beneficial mutations were segregating) into a population at mutationselection balance, probabilities of fixation were nearly perfectly predicted by 2s_{b}P_{0} (not shown). More important, Figure 1 shows that Equation 4 remains quite accurate over long stretches of time in which the simultaneous segregation of several favorable mutations as well as departures from mutationselection balance are allowed.
Distribution of fitness effects: We have restricted our attention to the case in which all deleterious mutations have the same fitness effect. This is not necessary. The above theory remains reasonably accurate if we replace s_{d} with the mean effect of deleterious mutations that segregate at mutationselection balance, a quantity that equals the harmonic mean,
Similarly, because Equations 1 and 2 are linear in s_{b}, the relevant selection coefficient is the arithmetic mean of effects among beneficial mutations,
This result was again tested against exact computer simulations. The simulations were identical to those above except that exponential distributions of both deleterious and favorable effects were allowed (see Figure 2 legend for details and note that in a small fraction of cases s_{b} > s_{d}). Once again, the simulations showed that our analytic solution is reasonably accurate. Although the predicted k tends to overestimate the rate of adaptation, the error is fairly small.
A more exact model: The above theory depends on an important simplification: we assume that favorable mutations enjoy independent fates. That is, we ignore clonal interference (Gerrish and Lenski 1998). Although favorable mutations may be sufficiently rare that clonal interference is unimportant, we cannot be sure of this, particularly in taxa having large populations. Fortunately we can incorporate clonal interference into the above model, at least approximately. The required calculations, a straightforward combination of the above and those of Gerrish and Lenski (1998), are presented in the appendix.
It is shown there that, when both ruby in the rubbish and clonal interference effects are allowed, the probability of fixation of a beneficial mutation is
The expected rate of substitution is therefore
Equation 8 again shows that the plot of adaptive substitution rate vs. mutation rate peaks (Figure 2). To find the U that maximizes the rate of adaptation, we must solve ∂E[k]/∂U = 0. The Appendix shows that this occurs when
DISCUSSION
Our calculations, although mathematically trivial, lead to a counterintuitive result. The rate of mutation in asexuals that maximizes the rate of adaptation depends solely on the strength of selection against deleterious mutations. In particular, asexuals adapt fastest when the genomic mutation rate equals the harmonic mean of deleterious effects among new mutations. We assume only that selection for new favorable alleles is typically milder than that against new deleterious ones. (We do not assume that the favorable mutations that actually get fixed have such small effects.)
The reason for this dependence on deleterious mutation is clear. As U grows too large, too many genomes carry deleterious alleles and, consequently, too many favorable mutations arise in deleterious “loaded” genomes, thus suffering zero probabilities of fixation. But as U gets too small, there are too few favorable mutations to take advantage of the existing deleteriousmutationfree genomes and adaptation slows. At
Consequently, adaptation is fastest when U assumes an intermediate value. This contradicts traditional intuition, which held that longterm adaptability increases as the production of favorable mutations grows. Leigh (1973, pp. 15–17), for instance, asked “What mutation rate is best for evolutionary progress?” and concluded “[t]he larger u, the larger the eventual fitness.” But real mutation rates are obviously not infinite and thus shortterm costs were invoked to rein in such absurdly high mutation rates (Kimura 1967; Leigh 1970, 1973; Dawson 1998, 1999; Johnson 1999b; and see below). The present work shows that intermediate mutation rates are favored in asexuals even when ignoring shortterm costs.
Our finding was (as usual) anticipated by Fisher (1930, pp. 120–122). After noting that asexuals can use all of the favorable mutations that escape accidental loss only if the mutation rate is so low that the species adapts at a glacial pace, he points out that:
[I]f on the contrary the mutation rates, both of beneficial and of deleterious mutations, are high enough to maintain any considerable genetic diversity, it will be only the best adapted genotypes which can become the ancestors of future generations, and the beneficial mutations which occur will have only the minutest chance of not appearing in types of organisms so inferior to some of their competitors, that their offspring will certainly be supplanted by those of the latter. Between these two extremes there will doubtless be an optimum degree of mutability … .
Fisher (1930, pp. 120–122)
Fisher did not, however, find this optimum. [Indeed, the required distribution of number of deleterious mutations at multilocus mutationselection balance was determined fairly late in the history of population genetics, by Kimura and Maruyama (1966) and Haigh (1978).] Our results are likely also related to those of Woodcock and Higgs (1996), who found in computer simulations that when both deleterious and favorable mutations occur, fitness in asexuals increases fastest when U assumes small, but intermediate, values. (Woodcock and Higgs assumed, however, that s_{b} = s_{d}; their results are not, therefore, directly comparable to the present ones.)
Given the robustness of our findings—our main conclusion is independent of
The finding that adaptation is fastest when U =
The present calculations also show that Gerrish and Lenski's (1998) and Miralles et al.'s (1999) attempts to use experimental evolution data from Escherichia coli and vesicular stomatitis virus (VSV) to estimate (i) the proportion of mutations that are favorable, and (ii) the mean selection coefficient among new favorable mutations, were compromised, at least quantitatively. (Gerrish and Lenski estimated that 1 in 10^{6} mutations are favorable in E. coli, while Miralles et al. arrived at 1 in 10^{8} in VSV. Similar analysis suggested that
Given our results, it may be tempting to conclude that asexuals will, over vast stretches of evolutionary time, evolve to the optimal mutation rate of U =
The difficulty is that we have assumed that—when our optimal clone competes with others having lower U—the optimal clone survives the intervals between adaptive fixation events (Leigh 1973, p. 16). But we have no such guarantee. During these intervals, any mutant clones having smaller U enjoy a shortterm advantage and so invade, causing the frequency of the “optimal” clone to fall.
Thus we can give no simple answer to the question of whether asexuals will converge on genomic mutation rates in the neighborhood of
It is, however, worth noting that if and when clones move to the optimal rate identified here, they will not suffer absurdly high mutation loads. Indeed, the resulting mutation load will be much closer to the smaller than to the larger values of s_{d}, a wellknown property of harmonic means. Assuming that selection coefficients on the order of 10^{−3} are realistic, a load of
In closing, it should be noted that this analysis required us to address a problem of perhaps wider interest. We have found a simple approximation to the probability of fixation of a favorable mutation in a nonrecombining genome or chromosome region. Such a mutation faces two kinds of traffic problems. First, it must escape stochastic loss due to linked deleterious mutations and, second, it must avoid being displaced by a later favorable mutation of greater advantage. When both forces act, the probability of fixation is
Acknowledgments
I thank Brian Charlesworth, Phil Gerrish, Peter Keightley, Yuseob Kim, Alex Kondrashov, Sally Otto, Daven Presgraves, Wolfgang Stephan, and especially Toby Johnson for very helpful comments. This work was supported by National Institutes of Health grant GM51932 and by the David and Lucile Packard Foundation.
APPENDIX
Combined effects of deleterious mutation and clonal interference: I derive the rate of adaptive substitution when both deleterious mutation and clonal interference are allowed. The derivation proceeds in two main steps. First, I find the number of interfering mutations that arise during a first favorable mutation's transit to fixation; second, I derive a rate of adaptation given this number of interfering mutations. These calculations are a straightforward combination of those of Gerrish and Lenski (1998) and those from the first half of the text. Unlike Gerrish and Lenski, however, I allow for deleterious mutation as well as for any arbitrary distribution of favorable selection coefficients.
Deleterious mutations have a harmonic mean effect of
A total of NUp_{b}t favorable mutations appear during time t, where t is the transit time to fixation of our favorable mutation. Because allele frequency under selection is logistic, t = (2/s_{b})ln N, where the mutation starts at p = 1/N and goes to pseudofixation at p = 1 − 1/N. But by symmetry over the logistic curve, only half of these mutations appear on ancestral (wild) chromosomes. Of these NUp_{b}t/2 relevant mutations, some fraction both has an effect greater than s_{b} and escapes stochastic loss (where we include loss due to mutations arising in deleterious loaded genomes). This fraction is
The expected number of interfering favorable mutations is thus ~I = NUp_{b}tF/2, yielding Equation 7 of the text. Our approach to calculating the number of interfering mutations is clearly approximate: we ignore the effect on N_{e} of subsequent mutations whose effects are less than s_{b}, as well as the fact that favorable mutations that are destined to fixation increase in frequency in the first few generations somewhat faster than expected under our logistic argument (see Otto and Barton 1997, Appendix C). Similarly, P. Gerrish (personal communication) has shown that estimates of I can be improved by taking into account favorable mutations that appear before the one whose fate we follow. (This quantitative improvement does not, however, affect our results, i.e., the value of the optimal U.) Despite these approximations, simulations show that our analytic estimate of I is fairly accurate.
We can now calculate the probability that our first favorable mutation is neither lost (e.g., by appearing on a chromosome bearing a deleterious mutation) nor displaced by an interfering mutation. This probability is
The maximum rate of adaptive substitution: We find the value of U that maximizes E[k]. This requires solving
Footnotes

Communicating editor: P. D. Keightley
 Received November 9, 1999.
 Accepted February 18, 2000.
 Copyright © 2000 by the Genetics Society of America