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OKS, OT, and LC contributed to the conception of the experiments. OKS and OT performed the experimental evolution and the mutation accumulation. OKS performed the maximum likelihood analysis of the mutation accumulation data and the analysis of the sequence data. OT performed the simulation analyses of substitution rates.

The authors have declared that no competing interests exist.

The most consistent result in more than two decades of experimental evolution is that the fitness of populations adapting to a constant environment does not increase indefinitely, but reaches a plateau. Using experimental evolution with bacteriophage, we show here that the converse is also true. In populations small enough such that drift overwhelms selection and causes fitness to decrease, fitness declines down to a plateau. We demonstrate theoretically that both of these phenomena must be due either to changes in the ratio of beneficial to deleterious mutations, the size of mutational effects, or both. We use mutation accumulation experiments and molecular data from experimental evolution to show that the most significant change in mutational effects is a drastic increase in the rate of beneficial mutation as fitness decreases. In contrast, the size of mutational effects changes little even as organismal fitness changes over several orders of magnitude. These findings have significant implications for the dynamics of adaptation.

The process of adaptation in finite populations is determined by the distribution of mutational effects on fitness and the size of the population. In large populations, beneficial mutations are frequently fixed, whereas deleterious mutations will only be fixed if they are of vanishingly small effect size (approximately the inverse of the effective population size [

An accurate description of the distribution of mutational effects is thus fundamental to understanding how adaptive changes occur in biological populations. Experimental manipulation of organisms in the laboratory (i.e., mutation accumulation [MA]) [

Importantly, the rate and effect size of beneficial and deleterious mutations are not constant. These quantities change if the fitness of the evolving organism changes. One manifestation of this effect is the fitness plateaus reached by populations adapting to constant environments [

There is an analogous process that occurs in populations that experience decreases in fitness. This process is equally important for evolutionary dynamics, but has received less attention. When populations are so small that drift overwhelms selection, deleterious mutations frequently fix [

The distribution of deleterious mutations appears in red, beneficial mutations in green. Arrows indicate the mean mutational effect; the width of the arrow reflects the rate of deleterious or beneficial mutations. (The total mutation rate remains constant.) Three mutational distributions are illustrated for each epistatic context: individuals of high, intermediate, and low fitness.

(A) No epistasis. Mutational distributions are constant regardless of fitness.

(B) Negative epistasis. The mean mutational effect increases as fitness decreases.

(C) Compensatory epistasis. The mean effect of each mutational class stays constant, whereas the fraction of mutations that are beneficial increases as fitness decreases.

(D) Positive epistasis. The mean mutational effect decreases as fitness decreases.

We used experimental evolution of a bacteriophage to investigate how the distribution of beneficial and deleterious mutations changes (i.e., epistasis) when fitness changes in the course of evolution, and how this impacts adaptation. We corroborate the finding that in large populations, fitness increases to a plateau. We also report the complementary effect that in small populations, fitness does not decrease indefinitely, but reaches a lower plateau. The equilibrium level of fitness is thus largely determined by the effective population size, and we show that this is a consequence of simple changes in mutational effects that cause qualitative changes in the processes of selection and drift.

To study the impact of population size on adaptation, we used experimental evolution in the bacteriophage ϕX174, an organism in which population sizes can be controlled by adjusting the bottleneck size at transfer. We used phage populations to investigate two related questions: First, do populations evolve to a fitness equilibrium? Second, do populations that differ in effective population size reach different fitness equilibria? As a measure for fitness, we used competition with a reference strain [

To identify fitness equilibria, we initiated experimental populations with either an ancestor of high fitness or an ancestor of low fitness. We then investigated whether the populations converged toward the same fitness value, irrespective of their starting condition. If populations that were initialized with high- or low-fitness ancestors converged in fitness during evolution, the evolved fitness levels of these populations were concluded to bracket a fitness equilibrium. The high-fitness ancestor was a clone derived from a phage line that had been passaged in our laboratory for 100 transfers at a large population size (≈10^{4}). Two lower fitness ancestors were derived by serially bottlenecking this clone and selecting small plaques. This procedure resulted in two strains that had low fitness in competition with the ancestral strain.

We initiated three populations from the high-fitness ancestor at each of four population sizes: three, ten, 30, and 100. We also initiated three populations from one low-fitness ancestor (two at a population size of ten and one at a population size of three, as well as one population from a very low-fitness ancestor, at a population size of three (

Dotted lines indicate the fitness of ancestral clones: the high-fitness clone (black circles), lower fitness clones (bright red triangles and dark red diamonds), and the evolved highest fitness ancestor (green squares); this ancestor was derived from the population having the highest fitness after 90 transfers (population 100c). Each point indicates the average fitness of an evolved population, with error bars indicating one standard error. Some populations are slightly displaced on the _{10} scale, and all other fitness values are relative.

After 90 transfers, the fitness of populations bottlenecked at sizes of three, ten, 30, and 100 were measured to test for convergence (_{e} = 3), two maintained almost constant fitness (_{e} = 10), and one declined only slightly in fitness (_{e} = 3) (_{e} = 30 and _{e} = 100), five of the six evolved to a mean fitness above that of the high-fitness ancestor. It was thus difficult to ascertain whether these populations were evolving towards an equilibrium fitness. To establish whether these larger populations were evolving toward an equilibrium, we took the _{e} = 100 population that had evolved the highest fitness and propagated it at an expanded population size of _{e} = 250 for an additional 55 transfers (275 generations). Three _{e} = 250 population replicates were propagated, and none changed significantly in fitness (_{e} = 100 populations were at or near fitness equilibrium. Thus, as is illustrated in

Although care was taken to minimize the opportunity for cross-contamination between populations, it was important to directly rule out that contamination was responsible for the fitness convergence. To ascertain this, we sequenced the complete genome of one clone from each evolved population. Convergence of substitutions at a large fraction of sites would imply contamination. However, the number of sites at which convergence occurred was not significantly greater than expected by chance (see

The sequence data revealed that the fitness convergence was achieved despite little indication of genetic convergence. Hence our system reproduced one previously observed phenomenon in experimental evolution: fitness in a large, adapting population does reach a plateau with different starting genotypes. Interestingly, our results show that this effect can be extended to small population sizes: small populations converge to low-fitness equilibria without genetic convergence.

The presence of fitness equilibria has specific implications for how epistasis influences the rate and shape of the mutational distribution. In previous studies that have focused on the accumulation of deleterious mutations in small populations, two general hypotheses have been put forth that can prevent continuous fitness decline [

We focus on testing these two models here, by using an explicit model to describe how epistatic changes affect fitness convergence and equilibria. In this model, the deleterious and beneficial substitution rates (_{d} and _{b}, respectively) change as fitness changes. For simplicity, we illustrate the case in which the effect on fitness of all beneficial mutations is identical, as is the effect on fitness of all deleterious mutations (in all later analyses, we use simulations in which mutational effects are modeled as a distribution). The substitution rates of deleterious and beneficial mutations are thus:
_{e} is the population size, _{d} is the deleterious mutation rate, _{b} is the beneficial mutation rate, _{d}(_{e},_{d}) = (1 − e^{2s})/(1 − e^{2Ns}) [_{d}, and _{b}(_{e},_{b}) = (1 − e^{−2s})/(1 − e^{−2Ns}) is the probability of fixation of a beneficial mutation of effect size _{b}. If a population is reduced in size such that fitness begins to decline, by definition,

For the fitness decline to stop, Expression 4 must become an equality.

We can thus dichotomize the two epistatic hypotheses: The first hypothesis, negative epistasis, proposes that a decrease in the left side of Expression 4 occurs through an increase in the average selective coefficients, _{d} and _{b}. The second hypothesis, compensatory epistasis, states that decreases in fitness cause an increase in the ratio between the beneficial mutation rate and the deleterious mutation rate, such that the right side of Expression 4 increases. We focus here on testing these two forms of epistasis, because they are the most commonly modeled forms. If there is either no epistasis or positive epistasis, then the fitness decline will continue unchecked, and no fitness equilibrium occurs. Similar logic applies to the case of large populations adapting to a constant environment. In the absence of epistatic effects, or if positive epistasis operates, fitness would increase continuously without reaching an equilibrium.

In order to understand which mechanism was responsible for the fitness equilibria we observed in our experimental populations, we needed to discriminate between the two main epistatic mechanisms (negative or compensatory epistasis). This required determining the ratio of beneficial and deleterious mutations, and the distribution of the effects of these mutations. To estimate these quantities, we used MA. In an MA experiment, mutations are allowed to accumulate by randomly collecting the progeny of a parental phage with no bias against low-fitness progeny. By measuring the fitness of a number of parents and a number of randomly chosen progeny, it is possible to estimate the fitness effects of mutations that occur in a single generation. We determined the fitness of 50 parents and 50 progeny for three high-fitness populations and three low-fitness populations (

(A) Fitness measurements of the high-fitness MA lines. The fitness values of the parents are shown in gray, the offspring in black. Note that all fitness values for each MA experiment were scaled such that the mean of the parental log_{10} fitness values was zero; this was done so that the error in fitness measurements was approximately normal, and so all three separate experiments in the low- and high-fitness lines could be combined on the same axis.

(B) Fitness measurements of the low-fitness MA lines. Shading and scaling are the same as in (A).

In both (A) and (B), a considerable shift to the left (lower fitness) can be observed after a single round of mutagenesis. In the low-fitness lines, several offspring clones with fitness greater than the parental clones can be observed. In the high-fitness lines, no such offspring clones are observed.

We used maximum-likelihood (ML) analysis [_{b}/(_{b} + _{d}). Importantly, this analysis assumes that the distribution of mutational effects can be modeled as a reflected gamma distribution. The gamma distribution is used because it has great flexibility in terms of the shapes it can assume, with shapes ranging from highly leptokurtic to exponential to log-normal to Gaussian (

Estimating _{e} = 3 and _{e} = 10 to estimate the neutral mutation rate. (We expect that in these populations synonymous substitutions behaved neutrally, i.e., the selective coefficient is less than the inverse of the population size.) The average rate of synonymous substitution in these lineages was 0.18 per transfer. We extrapolated this rate to a genomic rate by multiplying by the ratio of total sites to synonymous sites in the ancestral genome while accounting for a proportion of lethal mutations (see

The ML estimate of the mean mutational effect over all populations had an expected value of 0.35 (0.082 per generation) (^{2} 4 ^{2} 4 ^{2} 2 ^{2} 1 ^{2} 2

(A) Shape of the joint ML gamma distribution of mutational effects for all MA datasets. The distribution is extremely leptokurtic (L-shaped), with the majority of mutations having very small selection coefficients, although a significant proportion have large selection coefficients.

(B) Relationship between fitness and rate of compensatory mutation. Filled circles indicate the inferred proportion of compensatory mutations using the ML gamma distribution (illustrated in [A]) and the experimental fitness equilibria (see

We tested the reliability of our results by using a different method to estimate _{e} and a specified distribution of _{e} = 3 populations. We used additional simulations to derive the value of _{e} (

The results of the MA experiment and the above analysis suggest that an increase in the ratio between the rates of beneficial and deleterious mutations _{b}, the rate of beneficial mutations, or through a decrease in _{d}, the rate of deleterious mutations. The two alternatives have different biological meanings. An increase in _{b} with decreasing fitness would correspond to the hypothesis of compensatory epistasis. A decrease in _{d}, the rate of deleterious mutations, could result if a large fraction of deleterious mutations become lethal in less fit genotypes. In our experimental regimes, the number of lethal mutations was not directly measured, and it was thus important to test for such an increase directly.

A change in the number of lethal mutations with changing fitness would manifest as a change in the genomic mutation rate, _{b} and _{d}. We derived an independent measure of ^{2} 15 _{e} = 3 and _{e} = 10 lines. Again, this suggests that our results are robust and internally consistent, and that increases in _{b}, and not a change in the genomic mutation rate, accounted for the observed changes in

In the present paper, using experimental evolution of viruses, we have shown that populations tend to converge to a fitness equilibrium that depends on their population size. Large population–size lines converged to high fitness, and small population–size lines to low fitness (

This distribution of mutational effects has significant implications for adaptation, because the direction and rate of fitness change in any population is governed by four quantities intrinsic to the organism: the rate and mutational distribution of beneficial mutations and the rate and mutational distribution of deleterious mutations. As shown above, that populations at a specific effective population size converge toward a single level of fitness necessarily constrains the values and dynamics of these quantities. In a population in which mean individual fitness is increasing, the rate of fixation of beneficial mutations multiplied by the mean effect size must outweigh this analogous quantity for deleterious mutations. In order for the rate of adaptation to decrease as fitness increases, one or more of the following must occur: the mean effect size of beneficial mutations must decrease; the rate of beneficial mutations must decrease (and consequently the rate of deleterious mutations must increase); or the effect size of deleterious mutations must decrease (although the consequence of this does depend on population size. When 1/_{e} is less than the average mutational effect, a decrease in effect size will make deleterious mutations less visible to the purifying action of selection, and this effect will outweigh the fact that each deleterious fixation event has a lesser effect on fitness). Conversely, if a population is decreasing in fitness, to stop this decrease, one or more of the following must occur: the mean effect size of beneficial mutations must increase; the rate of beneficial mutations must increase (and consequently the rate of deleterious mutations must decrease); or the effect size of deleterious mutations must increase.

To untangle the different possibilities, we performed MA experiments on lines differing by approximately 300-fold in fitness to characterize the distribution of deleterious mutation effects on fitness (_{e} = 3). Although the distribution differs slightly from others that have been recently proposed, it concurs well with others [

The deleterious mutation distributions found do not provide significant support for the hypothesis that the effect size of deleterious mutations changes with fitness (i.e., positive or negative epistasis between deleterious mutations) (

The absence of any change in the effect size of deleterious mutations suggests that a significant change must occur in either the rate or effect size of beneficial mutations. We provided strong support for compensatory epistasis using a ML analysis of MA data. We also showed that the increase in

The epistasis we have observed in our data is a result of a change in the probability that a given mutation is beneficial or deleterious and is dependent on the fitness of the organism in which it appears. The shape of the distribution of beneficial and deleterious effects remains approximately the same. This type of epistasis has been studied less often than other forms, but is consistent with several experimental estimations of epistasis that have focused only on deleterious mutations. The positive epistasis found to be operating between deleterious mutations in several recent studies [

Additionally, it is important to note that the model of compensatory epistasis suggested here is not subsumed by any of the simpler models of epistasis (

The nonlinear relationship that we observed between ^{i}

This study is the first to propose and test an explicit model of adaptation in which organismal fitness specifies both the rate and distribution of deleterious and beneficial mutations. Although the model necessarily relies upon an idealized form of the genotypic landscape, it presents specific and testable predictions of the circumstances under which populations will increase or decrease in fitness.

Additionally, the model offers a simple mechanistic explanation for the diminishing rate of fitness increase observed within large populations adapting in a constant environment [

The original ϕX174 bacteriophage used in this study was derived from a clone kindly provided by B. Fane. This clone was passaged for 100 transfers at 32 °C to allow general adaptation to laboratory conditions. The deleterious mutants were obtained through serial bottlenecks with mutagenesis (see below), during which small plaques were purposefully chosen. The host was

During passaging, phage were mutagenized in 250 mM hydroxylamine (HA) [

Competition experiments were designed and performed as described previously in Burch and Chao [

In an MA design, mutations are allowed to accumulate by randomly collecting the progeny of a parental phage with no bias against low-fitness progeny. The high-fitness parents were derived from two of the _{e} = 100 populations (_{e} = 3 populations (

The ML model used models the distribution of selective coefficients ^{−s}). The maximum likelihood shape and size parameters for the mutational distribution values were 5.3 and 1.0, respectively.

The likelihood surfaces were generated by calculating the likelihood values over a grid of parameter values while maintaining the other parameters at their ML-estimated values (for

Some sequencing reactions were performed using the BigDye Terminator v3.1 kit and analyzed using an ABI sequencer; others were performed by the University of California San Diego Cancer Center or at INSERM. Alignments were done using Sequencher. Because ϕX174 contains multiple overlapping reading frames, sequence analysis was done using software written in C or Perl.

The simplest method of estimating the mutation rate is to estimate the rate of mutation in a theoretically neutral class of sites, and then to extrapolate this rate to the entire genome. However, two problems arise in using such an analysis. Firstly, the mutagenesis resulted in a significant bias toward cytosine → thymine transitions: of all observed substitutions, 89% were of this nature. Additionally, a significant number of mutations are either lethal or highly deleterious, and such mutations will never occur in the population during evolution. It is very likely that the majority of such mutations are missense or nonsense substitutions. Because the ML model of the mutational distribution can only account for a unimodal distribution of mutations, directly extrapolating the genome-wide mutation rate from the rate at synonymous sites would result in an overestimation of the mutation rate, because this rate would include all missense and nonsense mutations that lead to nonviable phenotypes. Therefore we used the rate of synonymous substitution, as well as information about the specific nature of mutations (missense or nonsense) and the amount of over-dispersion of substitutions to infer an

We base the estimate of the mutation rate on the substitutions observed in the populations propagated at the smallest effective population sizes, because mutations appearing in these lineages should be least subject to selection. Within these lineages, a total of 162 synonymous or intergenic cytosine → thymine substitutions occurred at 117 different sites. These substitutions are slightly over-dispersed when compared to the random expectation, although not significantly. If we assume that some proportion, λ_{S}, of the synonymous cytosine → thymine transitions result in a nonviable phenotype, the over-dispersion is decreased. We fit the λ_{S} parameter by minimizing the χ^{2} goodness-of-fit statistic; the value of this statistic is approximately equal to the probability of observing the data. We estimate λ_{S} = 0.14, with 95% confidence limits ranging from λ_{S} = 0 to λ_{S} = 0.40. A similar analysis may be done with non-synonymous missense sites. During experimental evolution, a total of 141 missense cytosine → thymine substitutions at 120 different sites occurred. We removed one of these sites from the analysis, because the substitution occurred in more than four independent lines, and was thus likely subject to strong positive selection. The estimate of λ_{M} was 0.37, with 95% confidence limits ranging from λ_{M} = 0.01 to λ_{M} = 0.61. Finally, we examine nonsense mutations. Of the 560 substitutions observed over all lines during the period of evolution, only one was a nonsense mutation, although the expected frequency is 9%. We thus estimate that λ_{N} is essentially 1. All of these estimates are similar to previously estimates values in other viruses, with λ_{S} estimated at 0.11 and λ_{M} at 0.40 [

We calculated the effective mutation rate using these parameters. The genome of the ancestral ϕX174 clone contains 1,151 cytosines; at 309 (27%) of these residues, a transition from cytosine to thymine is synonymous or intergenic; at 736 (64%), a transition is non-synonymous; and at 106 (9%), a transition results in a change to or from a stop codon. The observed rate of synonymous cytosine → thymine substitution was 0.18 per transfer. With a total of 309 cytosines at which this transition is synonymous or intergenic, and an estimated λ_{S} = 0.14_{N} = 0

If cross-contamination occurred, one would expect that two lines would share substitutions at a large number of sites (thus significantly increasing over-dispersion). This is clearly not the case for synonymous substitutions. (There is no significant over-dispersion; the 95% confidence interval for λ_{S} includes zero.)

We modified an individual-based model of adaptation [^{−2}, so almost 99% of the phage within each plaque are genetically identical). Thus the bottleneck population sizes are an accurate characterization of the effective population size of the phage populations. Secondly, even if individual phage were tracked, this would give rise to similar rates of fixation (see ^{2} > 0.98), and the following equation was numerically solved for each population size:
_{fix}(^{−2s})/(1 − e^{−2Ns}) [

The relationship between log fitness and

(A) Confidence limits for the ML parameter values of the gamma distribution in the low-fitness clones. The proportion of beneficial mutations,

(B) Confidence limits for the ML parameter values of the gamma distribution in the high-fitness lines. The proportion of beneficial mutations,

(C) Confidence limits for the ML parameter values of the gamma distribution in all MA lines. The proportion of beneficial mutations,

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(A) Confidence limits for the ML parameter values of the gamma distribution in the low-fitness clones. The value of beta was kept constant, at one.

(B) Confidence limits for the maximum likelihood parameter values of the gamma distribution in the high-fitness lines. Again, the value of beta was kept constant, at one. The

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The GenBank (

Thanks to K. Wright for assisting in the sequence analysis, E. Salgado for technical help, and M. Ackermann, T. Berngruber, A. Poon, and D. Weinreich for helpful discussions.

likelihood ratio test

mutation accumulation

maximum likelihood