FIXATION OF MUTATORS IN ASEXUAL POPULATIONS: THE ROLE OF GENETIC DRIFT AND EPISTASIS

We study the evolutionary dynamics of an asexual population of nonmutators and mutators on a class of epistatic fitness landscapes. We consider the situation in which all mutations are deleterious and mutators are produced from nonmutators continually at a constant rate. We find that in an infinitely large population, a minimum nonmutator‐to‐mutator conversion rate is required to fix the mutators but an arbitrarily small conversion rate results in the fixation of mutators in a finite population. We calculate analytical expressions for the mutator fraction at mutation‐selection balance and fixation time for mutators in a finite population when the difference between the mutation rate for mutator and nonmutator is smaller (regime I) and larger (regime II) than the selection coefficient. Our main result is that in regime I, the mutator fraction and the fixation time are independent of epistasis but in regime II, mutators are rarer and take longer to fix when the decrease in fitness with the number of deleterious mutations occurs at an accelerating rate (synergistic epistasis) than at a diminishing rate (antagonistic epistasis). Our analytical results are compared with numerics and their implications are discussed.     

Deleterious Passengers in Adapting Populations

Most new mutations are deleterious and are eventually eliminated by natural selection. But in an adapting population, the rapid amplification of beneficial mutations can hinder the removal of deleterious variants in nearby regions of the genome, altering the patterns of sequence evolution. Here, we analyze the interactions between beneficial “driver” mutations and linked deleterious “passengers” during the course of adaptation. We derive analytical expressions for the substitution rate of a deleterious mutation as a function of its fitness cost, as well as the reduction in the beneficial substitution rate due to the genetic load of the passengers. We find that the fate of each deleterious mutation varies dramatically with the rate and spectrum of beneficial mutations and the deleterious substitution rate depends nonmonotonically on the population size and the rate of adaptation. By quantifying this dependence, our results allow us to estimate which deleterious mutations will be likely to fix and how many of these mutations must arise before the progress of adaptation is significantly reduced.     

The rate of compensatory mutation in the DNA bacteriophage phiX174

A compensatory mutation occurs when the fitness loss caused by one mutation is remedied by its epistatic interaction with a second mutation at a different site in the genome. This poorly understood biological phenomenon has important implications, not only for the evolutionary consequences of mutation, but also for the genetic complexity of adaptation. We have carried out the first direct experimental measurement of the average rate of compensatory mutation. An arbitrary selection of 21 missense substitutions with deleterious effects on fitness was introduced by site-directed mutagenesis into the bacteriophage phiX174. For each deleterious mutation, we evolved 8-16 replicate populations to determine the frequency at which a compensatory mutation, instead of the back mutation, was acquired to recover fitness. The overall frequency of compensatory mutation was approximately 70%. Deleterious mutations that were more severe were significantly more likely to be compensated for. Furthermore, experimental reversion of deleterious mutations revealed that compensatory mutations have deleterious effects in a wild-type background. A large diversity of intragenic compensatory mutations was identified from sequencing fitness-recovering genotypes. Subsequent analyses of intragenic mutation diversity revealed a significant degree of clustering around the deleterious mutation in the linear sequence and also within folded protein structures. Moreover, a likelihood analysis of mutation diversity predicts that, on average, a deleterious mutation can be compensated by about nine different intragenic compensatory mutations. We estimate that about half of all compensatory mutations are located extragenically in this organism.     

On the accumulation of deleterious mutations during range expansions

We investigate the effect of spatial range expansions on the evolution of fitness when beneficial and deleterious mutations cosegregate. We perform individual‐based simulations of 1D and 2D range expansions and complement them with analytical approximations for the evolution of mean fitness at the edge of the expansion. We find that deleterious mutations accumulate steadily on the wave front during range expansions, thus creating an expansion load. Reduced fitness due to the expansion load is not restricted to the wave front, but occurs over a large proportion of newly colonized habitats. The expansion load can persist and represent a major fraction of the total mutation load for thousands of generations after the expansion. The phenomenon of expansion load may explain growing evidence that populations that have recently expanded, including humans, show an excess of deleterious mutations. To test the predictions of our model, we analyse functional genetic diversity in humans and find patterns that are consistent with our model.     

Surfing on the seascape: Adaptation in a changing environment

The environment changes constantly at various time scales and, in order to survive, species need to keep adapting. Whether these species succeed in avoiding extinction is a major evolutionary question. Using a multilocus evolutionary model of a mutation‐limited population adapting under strong selection, we investigate the effects of the frequency of environmental fluctuations on adaptation. Our results rely on an “adaptive‐walk” approximation and use mathematical methods from evolutionary computation theory to investigate the interplay between fluctuation frequency, the similarity of environments, and the number of loci contributing to adaptation. First, we assume a linear additive fitness function, but later generalize our results to include several types of epistasis. We show that frequent environmental changes prevent populations from reaching a fitness peak, but they may also prevent the large fitness loss that occurs after a single environmental change. Thus, the population can survive, although not thrive, in a wide range of conditions. Furthermore, we show that in a frequently changing environment, the similarity of threats that a population faces affects the level of adaptation that it is able to achieve. We check and supplement our analytical results with simulations.     

The evolution of mutation rate in finite asexual populations

In this article, we model analytically the evolution of mutation rate in asexual organisms. Three selective forces are present. First, everything else being equal, individuals with higher mutation rate have a larger fitness, thanks to the energy and time saved by not replicating DNA accurately. Second, as a flip side, the genome of these individuals is replicated with errors that may negatively affect fitness. Third, and conversely, replication errors have a potential benefit if beneficial mutations are to be generated. Our model describes the fate of modifiers of mutation rate under the three forces and allows us to predict the long-term evolutionary trajectory of mutation rate. We obtain three major results. First, in asexuals, the needs for both adaptation and genome preservation are not evolutionary forces that can stabilize mutation rate at an intermediate optimum. When adaptation has a significant role, it primarily destabilizes mutation rate and yields the emergence of strong-effect mutators. Second, in contrast to what is usually believed, the appearance of modifiers with large mutation rate is more likely when the fitness cost of each deleterious mutation is weak, because the cost of replication errors is then paid after a delay. Third, in small populations, and even if adaptations are needed, mutation rate is always blocked at the minimum attainable level, because the rate of adaptation is too slow to play a significant role. Only populations whose size is above a critical mass see their mutation rate affected by the need for adaptation.     

Mutator dynamics in fluctuating environments

Populations with high mutation rates (mutator clones) are being found in increasing numbers of species, and a clear link is being established between the presence of mutator clones and drug resistance. Mutator clones exist despite the fact that in a constant environment most mutations are deleterious, with the spontaneous mutation rate generally held at a low value. This implies that mutator clones have an important role in the adaptation of organisms to changing environments. Our study examines how mutator dynamics vary according to the frequency of environmental fluctuations. Although recent studies have considered a single environmental switch, here we investigate mutator dynamics in a regularly varying environment, seeking to mimic conditions present, for example, under certain drug or pesticide regimes. Our model provides four significant new insights. First, the results demonstrate that mutators are most prevalent under intermediate rates of environmental change. When the environment oscillates more rapidly, mutators are unable to provide sufficient adaptability to keep pace with the frequent changes in selection pressure and, instead, a population of 'environmental generalists' dominates. Second, our findings reveal that mutator dynamics may be complex, exhibiting limit cycles and chaos. Third, we demonstrate that when each beneficial mutation provides a greater gain in fitness, mutators achieve higher densities in more rapidly fluctuating environments. Fourth, we find that mutators of intermediate strength reach higher densities than very weak or strong mutators.     

The fitness effect of mutations across environments: Fisher's geometrical model with multiple optima

When are mutations beneficial in one environment and deleterious in another? More generally, what is the relationship between mutation effects across environments? These questions are crucial to predict adaptation in heterogeneous conditions in a broad sense. Empirical evidence documents various patterns of fitness effects across environments but we still lack a framework to analyze these multivariate data. In this article, we extend Fisher's geometrical model to multiple environments determining distinct peaks. We derive the fitness distribution, in one environment, among mutants with a given fitness in another and the bivariate distribution of random mutants’ fitnesses across two or more environments. The geometry of the phenotype‐fitness landscape is naturally interpreted in terms of fitness trade‐offs between environments. These results may be used to fit/predict empirical distributions or to predict the pattern of adaptation across heterogeneous conditions. As an example, we derive the genomic rate of substitution and of adaptation in a metapopulation divided into two distinct habitats in a high migration regime and show that they depend critically on the geometry of the phenotype‐fitness landscape.     

The Impact of Macroscopic Epistasis on Long-Term Evolutionary Dynamics

Genetic interactions can strongly influence the fitness effects of individual mutations, yet the impact of these epistatic interactions on evolutionary dynamics remains poorly understood. Here we investigate the evolutionary role of epistasis over 50,000 generations in a well-studied laboratory evolution experiment in Escherichia coli. The extensive duration of this experiment provides a unique window into the effects of epistasis during long-term adaptation to a constant environment. Guided by analytical results in the weak-mutation limit, we develop a computational framework to assess the compatibility of a given epistatic model with the observed patterns of fitness gain and mutation accumulation through time. We find that a decelerating fitness trajectory alone provides little power to distinguish between competing models, including those that lack any direct epistatic interactions between mutations. However, when combined with the mutation trajectory, these observables place strong constraints on the set of possible models of epistasis, ruling out many existing explanations of the data. Instead, we find that the data are consistent with a “two-epoch” model of adaptation, in which an initial burst of diminishing-returns epistasis is followed by a steady accumulation of mutations under a constant distribution of fitness effects. Our results highlight the need for additional DNA sequencing of these populations, as well as for more sophisticated models of epistasis that are compatible with all of the experimental data.     

The coupon collector and the suppressor mutation: estimating the number of compensatory mutations by maximum likelihood

Compensatory mutation occurs when a loss of fitness caused by a deleterious mutation is restored by its epistatic interaction with a second mutation at a different site in the genome. How many different compensatory mutations can act on a given deleterious mutation? Although this quantity is fundamentally important to understanding the evolutionary consequence of mutation and the genetic complexity of adaptation, it remains poorly understood. To determine the shape of the statistical distribution for the number of compensatory mutations per deleterious mutation, we have performed a maximum-likelihood analysis of experimental data collected from the suppressor mutation literature. Suppressor mutations are used widely to assess protein interactions and are under certain conditions equivalent to compensatory mutations. By comparing the maximum likelihood of a variety of candidate distribution functions, we established that an L-shaped gamma distribution (alpha=0.564, theta=21.01) is the most successful at explaining the collected data. This distribution predicts an average of 11.8 compensatory mutations per deleterious mutation. Furthermore, the success of the L-shaped gamma distribution is robust to variation in mutation rates among sites. We have detected significant differences among viral, prokaryotic, and eukaryotic data subsets in the number of compensatory mutations and also in the proportion of compensatory mutations that are intragenic. This is the first attempt to characterize the overall diversity of compensatory mutations, identifying a consistent and accurate prior distribution of compensatory mutation diversity for theoretical evolutionary models.     

Detecting epistasis from an ensemble of adapting populations

The role that epistasis plays during adaptation remains an outstanding problem, which has received considerable attention in recent years. Most of the recent empirical studies are based on ensembles of replicate populations that adapt in a fixed, laboratory controlled condition. Researchers often seek to infer the presence and form of epistasis in the fitness landscape from the time evolution of various statistics averaged across the ensemble of populations. Here, we provide a rigorous analysis of what quantities, drawn from time series of such ensembles, can be used to infer epistasis for populations evolving under weak mutation on finite‐site fitness landscapes. First, we analyze the mean fitness trajectory—that is, the time course of the ensemble average fitness. We show that for any epistatic fitness landscape and starting genotype, there always exists a non‐epistatic fitness landscape that produces the exact same mean fitness trajectory. Thus, the presence of epistasis is not identifiable from the mean fitness trajectory. By contrast, we show that two other ensemble statistics—the time evolution of the fitness variance across populations, and the time evolution of the mean number of substitutions—can detect certain forms of epistasis in the underlying fitness landscape.     

The influence of deleterious mutations on adaptation in asexual populations

We study the dynamics of adaptation in asexual populations that undergo both beneficial and deleterious mutations. In particular, how the deleterious mutations affect the fixation of beneficial mutations was investigated. Using extensive Monte Carlo simulations, we find that in the "strong-selection weak mutation (SSWM)" regime or in the "clonal interference (CI)" regime, deleterious mutations rarely influence the distribution of "selection coefficients of the fixed mutations (SCFM)"; while in the "multiple mutations" regime, the accumulation of deleterious mutations would lead to a decrease in fitness significantly. We conclude that the effects of deleterious mutations on adaptation depend largely on the supply of beneficial mutations. And interestingly, the lowest adaptation rate occurs for a moderate value of selection coefficient of deleterious mutations.     

Frequent beneficial mutations during single-colony serial transfer of Streptococcus pneumoniae

The appearance of new mutations within a population provides the raw material for evolution. The consistent decline in fitness observed in classical mutation accumulation studies has provided support for the long-held view that deleterious mutations are more common than beneficial mutations. Here we present results of a study using a mutation accumulation design with the bacterium Streptococcus pneumoniae in which the fitness of the derived populations increased. This rise in fitness was associated specifically with adaptation to survival during brief stationary phase periods between single-colony population bottlenecks. To understand better the population dynamics behind this unanticipated adaptation, we developed a maximum likelihood model describing the processes of mutation and stationary-phase selection in the context of frequent population bottlenecks. Using this model, we estimate that the rate of beneficial mutations may be as high as 4.8×10(-4) events per genome for each time interval corresponding to the pneumococcal generation time. This rate is several orders of magnitude higher than earlier estimates of beneficial mutation rates in bacteria but supports recent results obtained through the propagation of small populations of Escherichia coli. Our findings indicate that beneficial mutations may be relatively frequent in bacteria and suggest that in S. pneumoniae, which develops natural competence for transformation, a steady supply of such mutations may be available for sampling by recombination.     

NONADAPTIVE PROCESSES CAN CREATE THE APPEARANCE OF FACULTATIVE CHEATING IN MICROBES

Adaptations to social life may take the form of facultative cheating, in which organisms cooperate with genetically similar individuals but exploit others. Consistent with this possibility, many strains of social microbes like Myxococcus bacteria and Dictyostelium amoebae have equal fitness in single‐genotype social groups but outcompete other strains in mixed‐genotype groups. Here we show that these observations are also consistent with an alternative, nonadaptive scenario: kin selection‐mutation balance under local competition. Using simple mathematical models, we show that deleterious mutations that reduce competitiveness within social groups (growth rate, e.g.) without affecting group productivity can create fitness effects that are only expressed in the presence of other strains. In Myxococcus, mutations that delay sporulation may strongly reduce developmental competitiveness. Deleterious mutations are expected to accumulate when high levels of kin selection relatedness relax selection within groups. Interestingly, local resource competition can create nonzero “cost” and “benefit” terms in Hamilton's rule even in the absence of any cooperative trait. Our results show how deleterious mutations can play a significant role even in organisms with large populations and highlight the need to test evolutionary causes of social competition among microbes.     

Natural selection fails to optimize mutation rates for long-term adaptation on rugged fitness landscapes

The rate of mutation is central to evolution. Mutations are required for adaptation, yet most mutations with phenotypic effects are deleterious. As a consequence, the mutation rate that maximizes adaptation will be some intermediate value. Here, we used digital organisms to investigate the ability of natural selection to adjust and optimize mutation rates. We assessed the optimal mutation rate by empirically determining what mutation rate produced the highest rate of adaptation. Then, we allowed mutation rates to evolve, and we evaluated the proximity to the optimum. Although we chose conditions favorable for mutation rate optimization, the evolved rates were invariably far below the optimum across a wide range of experimental parameter settings. We hypothesized that the reason that mutation rates evolved to be suboptimal was the ruggedness of fitness landscapes. To test this hypothesis, we created a simplified landscape without any fitness valleys and found that, in such conditions, populations evolved near-optimal mutation rates. In contrast, when fitness valleys were added to this simple landscape, the ability of evolving populations to find the optimal mutation rate was lost. We conclude that rugged fitness landscapes can prevent the evolution of mutation rates that are optimal for long-term adaptation. This finding has important implications for applied evolutionary research in both biological and computational realms.     

Population Size Affects Adaptation in Complex Ways: Simulations on Empirical Adaptive Landscapes

Do large populations always outcompete smaller ones? Does increasing the mutation rate have a similar effect to increasing the population size, with respect to the adaptation of a population? How important are substitutions in determining the adaptation rate? In this study, we ask how population size and mutation rate interact to affect adaptation on empirical adaptive landscapes. Using such landscapes, we do not need to make many ad hoc assumption about landscape topography, such as about epistatic interactions among mutations or about the distribution of fitness effects. Moreover, we have a better understanding of all the mutations that occur in a population and their effects on the average fitness of the population than we can know in experimental studies. Our results show that the evolutionary dynamics of a population cannot be fully explained by the population mutation rate \(N\mu\); even at constant \(N\mu\), there can be dramatic differences in the adaptation of populations of different sizes. Moreover, the substitution rate of mutations is not always equivalent to the adaptation rate, because we observed populations adapting to high adaptive peaks without fixing any mutations. Finally, in contrast to some theoretical predictions, even on the most rugged landscapes we study, small population size is never an advantage over larger population size. These result show that complex interactions among multiple factors can affect the evolutionary dynamics of populations, and simple models should be taken with caution.     

Parasites and mutational load: an experimental test of a pluralistic theory for the evolution of sex

Ecological and mutational explanations for the evolution of sexual reproduction have usually been considered independently. Although many of these explanations have yielded promising theoretical results,experimental support for their ability to overcome a twofold cost of sex has been limited. For this reason, it has recently been argued that a pluralistic approach, combining effects from multiple models, may be necessary to explain the apparent advantage of sex. One such pluralistic model proposes that parasite load and synergistic epistasis between deleterious mutations might interact to create an advantage for recombination.Here, we test this proposal by comparing the fitness functions of parasitized and parasite-free genotypes of Escherichia coli bearing known numbers of transposon-insertion mutations. In both classes, we failed to detect any evidence for synergistic epistasis. However, the average effect of deleterious mutations was greater in parasitized than parasite-free genotypes. This effect might broaden the conditions under which another proposed model combining parasite-host coevolutionary dynamics and mutation accumulation can explain the maintenance of sex. These results suggest that, on average, deleterious mutations act multiplicatively with each other but in synergy with infection in determining fitness.     

EVOLUTIONARILY STABLE SEX RATIOS AND MUTATION LOAD

Frequency‐dependent selection should drive dioecious populations toward a 1:1 sex ratio, but biased sex ratios are widespread, especially among plants with sex chromosomes. Here, we develop population genetic models to investigate the relationships between evolutionarily stable sex ratios, haploid selection, and deleterious mutation load. We confirm that when haploid selection acts only on the relative fitness of X‐ and Y‐bearing pollen and the sex ratio is controlled by the maternal genotype, seed sex ratios evolve toward 1:1. When we also consider haploid selection acting on deleterious mutations, however, we find that biased sex ratios can be stably maintained, reflecting a balance between the advantages of purging deleterious mutations via haploid selection, and the disadvantages of haploid selection on the sex ratio. Our results provide a plausible evolutionary explanation for biased sex ratios in dioecious plants, given the extensive gene expression that occurs across plant genomes at the haploid stage.     

Functional origins of fitness effect-sizes of compensatory mutations in the DNA bacteriophage phiX174

Abstract Epistasis is an important and poorly understood aspect of mutations and strongly influences the evolutionary impact of genetic variation on adaptation and fitness. Although recent studies have begun to characterize the distribution of epistatic effects between mutations affecting fitness, there is currently a lack of empirical information on the underlying biological causes of these epistatic interactions. What are the functional constraints that determine the effectiveness of a compensatory mutation at restoring fitness? We have measured the effect‐sizes of 52 compensatory mutations affecting nine different deleterious mutations in the major capsid and spike proteins of the DNA bacteriophage X174. On average, an experimentally detectable compensatory mutation recovers about two‐thirds of the fitness cost of the preceding deleterious mutation. Variation in fitness effect‐sizes is only weakly associated with measures of the distance separating the deleterious and compensatory mutations in the amino acid sequence or the folded protein structure. However, there is a strong association of fitness effect‐size with the correlation in the effects of the mutations on the biochemical properties of amino acids. A compensatory mutation has the largest effect‐size, on average, when both the compensatory and deleterious mutations have radical effects on the overall biochemical make‐up of the amino acids. By examining the relative contributions of specific biochemical properties to variation in fitness effect‐size, we find that the area and charge of amino acids have a major influence, which suggests that the complexity of the amino acid phenotype is simplified by selection into a reduced number of phenotypic components.