Beneficial mutation selection balance and the effect of linkage on positive selection

When beneficial mutations are rare, they accumulate by a series of selective sweeps. But when they are common, many beneficial mutations will occur before any can fix, so there will be many different mutant lineages in the population concurrently. In an asexual population, these different mutant lineages interfere and not all can fix simultaneously. In addition, further beneficial mutations can accumulate in mutant lineages while these are still a minority of the population. In this article, we analyze the dynamics of such multiple mutations and the interplay between multiple mutations and interference between clones. These result in substantial variation in fitness accumulating within a single asexual population. The amount of variation is determined by a balance between selection, which destroys variation, and beneficial mutations, which create more. The behavior depends in a subtle way on the population parameters: the population size, the beneficial mutation rate, and the distribution of the fitness increments of the potential beneficial mutations. The mutation-selection balance leads to a continually evolving population with a steady-state fitness variation. This variation increases logarithmically with both population size and mutation rate and sets the rate at which the population accumulates beneficial mutations, which thus also grows only logarithmically with population size and mutation rate. These results imply that mutator phenotypes are less effective in larger asexual populations. They also have consequences for the advantages (or disadvantages) of sex via the Fisher-Muller effect; these are discussed briefly.     

Fixation probability of rare nonmutator and evolution of mutation rates

Although mutations drive the evolutionary process, the rates at which the mutations occur are themselves subject to evolutionary forces. Our purpose here is to understand the role of selection and random genetic drift in the evolution of mutation rates, and we address this question in asexual populations at mutation‐selection equilibrium neglecting selective sweeps. Using a multitype branching process, we calculate the fixation probability of a rare nonmutator in a large asexual population of mutators and find that a nonmutator is more likely to fix when the deleterious mutation rate of the mutator population is high. Compensatory mutations in the mutator population are found to decrease the fixation probability of a nonmutator when the selection coefficient is large. But, surprisingly, the fixation probability changes nonmonotonically with increasing compensatory mutation rate when the selection is mild. Using these results for the fixation probability and a drift‐barrier argument, we find a novel relationship between the mutation rates and the population size. We also discuss the time to fix the nonmutator in an adapted population of asexual mutators, and compare our results with experiments.     

THE ADAPTATION RATE OF ASEXUALS: DELETERIOUS MUTATIONS,CLONAL INTERFERENCE AND POPULATION BOTTLENECKS

The rate at which a population adapts to its environment is a cornerstone of evolutionary theory, and recent experimental advances in microbial populations have renewed interest in predicting and testing this rate. Efforts to understand the adaptation rate theoretically are complicated by high mutation rates, to both beneficial and deleterious mutations, and by the fact that beneficial mutations compete with each other in asexual populations (clonal interference). Testable predictions must also include the effects of population bottlenecks, repeated reductions in population size imposed by the experimental protocol. In this contribution, we integrate previous work that addresses each of these issues, developing an overall prediction for the adaptation rate that includes: beneficial mutations with probabilistically distributed effects, deleterious mutations of arbitrary effect, population bottlenecks, and clonal interference.     

Estimation of the rate and effect of new beneficial mutations in asexual populations

The rate and effect of available beneficial mutations are key parameters in determining how a population adapts to a new environment. However, these parameters are poorly known, in large part because of the difficulty of designing and interpreting experiments to examine the rare and intrinsically stochastic process of mutation occurrence. We present a new approach to estimate the rate and selective advantage of beneficial mutations that underlie the adaptation of asexual populations. We base our approach on the analysis of experiments that track the effect of newly arising beneficial mutations on the dynamics of a neutral marker in evolving bacterial populations and develop efficient estimators of mutation rate and selective advantage. Using extensive simulations, we evaluate the accuracy of our estimators and conclude that they are quite robust to the use of relatively low experimental replication. To validate the predictions of our model, we compare theoretical and experimentally determined estimates of the selective advantage of the first beneficial mutation to fix in a series of ten replicate populations. We find that our theoretical predictions are not significantly different from experimentally determined selection coefficients. Application of our method to suitably designed experiments will allow estimation of how population evolvability depends on demographic and initial fitness parameters.     

The effect of population bottlenecks on mutation rate evolution in asexual populations

In the absence of recombination, a mutator allele can spread through a population by hitchhiking with beneficial mutations that appear in its genetic background. Theoretical studies over the past decade have shown that the survival and fixation probability of beneficial mutations can be severely reduced by population size bottlenecks. Here, we use computational modelling and evolution experiments with the yeast S. cerevisiae to examine whether population bottlenecks can affect mutator dynamics in adapting asexual populations. In simulation, we show that population bottlenecks can inhibit mutator hitchhiking with beneficial mutations and are most effective at lower beneficial mutation supply rates. We then subjected experimental populations of yeast propagated at the same effective population size to three different bottleneck regimes and observed that the speed of mutator hitchhiking was significantly slower at smaller bottlenecks, consistent with our theoretical expectations. Our results, thus, suggest that bottlenecks can be an important factor in mutation rate evolution and can in certain circumstances act to stabilize or, at least, delay the progressive elevation of mutation rates in asexual populations. Additionally, our findings provide the first experimental support for the theoretically postulated effect of population bottlenecks on beneficial mutations and demonstrate the usefulness of studying mutator frequency dynamics for understanding the underlying dynamics of fitness‐affecting mutations.     

Competition between high- and higher-mutating strains of Escherichia coli

Experimental studies have shown that a mutator allele can readily hitchhike to fixation with beneficial mutations in an asexual population having a low, wild-type mutation rate. Here, we show that a genotype bearing two mutator alleles can supplant a population already fixed for one mutator allele. Our results provide experimental support for recent theory predicting that mutator alleles will tend to accumulate in asexual populations by hitchhiking with beneficial mutations, causing an ever-higher genomic mutation rate.     

The Properties of Adaptive Walks in Evolving Populations of Fungus

The rarity of beneficial mutations has frustrated efforts to develop a quantitative theory of adaptation. Recent models of adaptive walks, the sequential substitution of beneficial mutations by selection, make two compelling predictions: adaptive walks should be short, and fitness increases should become exponentially smaller as successive mutations fix. We estimated the number and fitness effects of beneficial mutations in each of 118 replicate lineages of Aspergillus nidulans evolving for approximately 800 generations at two population sizes using a novel maximum likelihood framework, the results of which were confirmed experimentally using sexual crosses. We find that adaptive walks do indeed tend to be short, and fitness increases become smaller as successive mutations fix. Moreover, we show that these patterns are associated with a decreasing supply of beneficial mutations as the population adapts. We also provide empirical distributions of fitness effects among mutations fixed at each step. Our results provide a first glimpse into the properties of multiple steps in an adaptive walk in asexual populations and lend empirical support to models of adaptation involving selection towards a single optimum phenotype. In practical terms, our results suggest that the bulk of adaptation is likely to be accomplished within the first few steps.     

Contrasting dynamics of a mutator allele in asexual populations of differing size

Mutators have been shown to hitchhike in asexual populations when the anticipated beneficial mutation supply rate of the mutator subpopulation, NU(b) (for subpopulation of size N and beneficial mutation rate U(b)) exceeds that of the wild-type subpopulation. Here, we examine the effect of total population size on mutator dynamics in asexual experimental populations of Saccharomyces cerevisiae. Although mutators quickly hitchhike to fixation in smaller populations, mutator fixation requires more and more time as population size increases; this observed delay in mutator hitchhiking is consistent with the expected effect of clonal interference. Interestingly, despite their higher beneficial mutation supply rate, mutators are supplanted by the wild type in very large populations. We postulate that this striking reversal in mutator dynamics is caused by an interaction between clonal interference, the fitness cost of the mutator allele, and infrequent large-effect beneficial mutations in our experimental populations. Our work thus identifies a potential set of circumstances under which mutator hitchhiking can be inhibited in natural asexual populations, despite recent theoretical predictions that such populations should have a net tendency to evolve ever-higher genomic mutation rates.     

The two-mutant problem: clonal interference in evolutionary graph theory

In large asexual populations, clonal interference, whereby different beneficial mutations compete to fix in the population simultaneously, may be the norm. Results extrapolated from the spread of individual mutations in homogeneous backgrounds are found to be misleading in such situations: clonal interference severely inhibits the spread of beneficial mutations. In contrast with results gained in systems with just one mutation striving for fixation at any one time, the spatial structure of the population is found to be an important factor in determining the fixation probability when there are two beneficial mutations.     

Adaptive evolution in a spatially structured asexual population

We study the process of adaptation in a spatially structured asexual haploid population. The model assumes a local competition for replication, where each organism interacts only with its nearest neighbors. We observe that the substitution rate of beneficial mutations is smaller for a spatially structured population than that seen for populations without structure. The difference between structured and unstructured populations increases as the adaptive mutation rate increases. Furthermore, the substitution rate decreases as the number of neighbors for local competition is reduced. We have also studied the impact of structure on the distribution of adaptive mutations that fix during adaptation.     

Interfering waves of adaptation promote spatial mixing

A fundamental problem of asexual adaptation is that beneficial substitutions are not efficiently accumulated in large populations: Beneficial mutations often go extinct because they compete with one another in going to fixation. It has been argued that such clonal interference may have led to the evolution of sex and recombination in well-mixed populations. Here, we study clonal interference, and mechanisms of its mitigation, in an evolutionary model of spatially structured populations with uniform selection pressure. Clonal interference is much more prevalent with spatial structure than without, due to the slow wave-like spread of beneficial mutations through space. We find that the adaptation speed of asexuals saturates when the linear habitat size exceeds a characteristic interference length, which becomes shorter with smaller migration and larger mutation rate. The limiting speed is proportional to μ(1/2) and μ(1/3) in linear and planar habitats, respectively, where the mutational supply μ is the product of mutation rate and local population density. This scaling and the existence of a speed limit should be amenable to experimental tests as they fall far below predicted adaptation speeds for well-mixed populations (that scale as the logarithm of population size). Finally, we show that not only recombination, but also long-range migration is a highly efficient mechanism of relaxing clonal competition in structured populations. Our conservative estimates of the interference length predict prevalent clonal interference in microbial colonies and biofilms, so clonal competition should be a strong driver of both genetic and spatial mixing in those contexts.     

Stochastic interplay between mutation and recombination during the acquisition of drug resistance mutations in human immunodeficiency virus type 1

The emergence of drug resistance mutations in human immunodeficiency virus (HIV) has been a major setback in the treatment of infected patients. Besides the high mutation rate, recombination has been conjectured to have an important impact on the emergence of drug resistance. Population genetic theory suggests that in populations limited in size recombination may facilitate the acquisition of beneficial mutations. The viral population in an infected patient may indeed represent such a population limited in size, since current estimates of the effective population size range from 500 to 10(5). To address the effects of limited population size, we therefore expand a previously described deterministic population genetic model of HIV replication by incorporating the stochastic processes that occur in finite populations of infected cells. Using parameter estimates from the literature, we simulate the evolution of drug-resistant viral strains. The simulations show that recombination has only a minor effect on the rate of acquisition of drug resistance mutations in populations with effective population sizes as small as 1,000, since in these populations, viral strains typically fix beneficial mutations sequentially. However, for intermediate effective population sizes (10(4) to 10(5)), recombination can accelerate the evolution of drug resistance by up to 25%. Furthermore, a reduction in population size caused by drug therapy can be overcome by a higher viral mutation rate, leading to a faster evolution of drug resistance.     

The impact of population size on the evolution of asexual microbes on smooth versus rugged fitness landscapes

Background  

It is commonly thought that large asexual populations evolve more rapidly than smaller ones, due to their increased rate of beneficial mutations. Less clear is how population size influences the level of fitness an asexual population can attain. Here, we simulate the evolution of bacteria in repeated serial passage experiments to explore how features such as fitness landscape ruggedness, the size of the mutational target under selection, and the mutation supply rate, interact to affect the evolution of microbial populations of different sizes.     

Dynamic mutation-selection balance as an evolutionary attractor

The vast majority of mutations are deleterious and are eliminated by purifying selection. Yet in finite asexual populations, purifying selection cannot completely prevent the accumulation of deleterious mutations due to Muller's ratchet: once lost by stochastic drift, the most-fit class of genotypes is lost forever. If deleterious mutations are weakly selected, Muller's ratchet can lead to a rapid degradation of population fitness. Evidently, the long-term stability of an asexual population requires an influx of beneficial mutations that continuously compensate for the accumulation of the weakly deleterious ones. Hence any stable evolutionary state of a population in a static environment must involve a dynamic mutation-selection balance, where accumulation of deleterious mutations is on average offset by the influx of beneficial mutations. We argue that such a state can exist for any population size N and mutation rate U and calculate the fraction of beneficial mutations, ε, that maintains the balanced state. We find that a surprisingly low ε suffices to achieve stability, even in small populations in the face of high mutation rates and weak selection, maintaining a well-adapted population in spite of Muller's ratchet. This may explain the maintenance of mitochondria and other asexual genomes.     

Clonal interference is alleviated by high mutation rates in large populations

When a beneficial mutation is fixed in a population that lacks recombination, the genetic background linked to that mutation is fixed. As a result, beneficial mutations on different backgrounds experience competition, or "clonal interference," that can cause asexual populations to evolve more slowly than their sexual counterparts. Factors such as a large population size (N) and high mutation rates (mu) increase the number of competing beneficial mutations, and hence are expected to increase the intensity of clonal interference. However, recent theory suggests that, with very large values of Nmu, the severity of clonal interference may instead decline. The reason is that, with large Nmu, genomes including both beneficial mutations are rapidly created by recurrent mutation, obviating the need for recombination. Here, we analyze data from experimentally evolved asexual populations of a bacteriophage and find that, in these nonrecombining populations with very large Nmu, recurrent mutation does appear to ameliorate this cost of asexuality.     

The Probability of Fixation in Populations of Changing Size

The rate of adaptive evolution of a population ultimately depends on the rate of incorporation of beneficial mutations. Even beneficial mutations may, however, be lost from a population since mutant individuals may, by chance, fail to reproduce. In this paper, we calculate the probability of fixation of beneficial mutations that occur in populations of changing size. We examine a number of demographic models, including a population whose size changes once, a population experiencing exponential growth or decline, one that is experiencing logistic growth or decline, and a population that fluctuates in size. The results are based on a branching process model but are shown to be approximate solutions to the diffusion equation describing changes in the probability of fixation over time. Using the diffusion equation, the probability of fixation of deleterious alleles can also be determined for populations that are changing in size. The results developed in this paper can be used to estimate the fixation flux, defined as the rate at which beneficial alleles fix within a population. The fixation flux measures the rate of adaptive evolution of a population and, as we shall see, depends strongly on changes that occur in population size.     

Limits to adaptation in asexual populations

In asexual populations, the rate of adaptation is basically limited by the frequency and properties of spontaneous beneficial mutations. Hence, knowledge of these mutational properties and how they are affected by particular evolutionary conditions is a precondition for understanding the process of adaptation. Here, we address how the rate of adaptation of asexual populations is limited by its population size and mutation rate, as well as by two factors affecting the fraction of mutations that confer a benefit, i.e. the initial adaptedness of the population and the variability of the environment. These factors both influence which mutations are likely to occur, as well as the probability that they will ultimately contribute to adaptation. We attempt to separate the consequences of these basic population features in terms of their effect on the rate of adaptation by using results from evolution experiments with microorganisms.     

The evolution of mutation rates: separating causes from consequences

Natural selection can adjust the rate of mutation in a population by acting on allelic variation affecting processes of DNA replication and repair. Because mutation is the ultimate source of the genetic variation required for adaptation, it can be appealing to suppose that the genomic mutation rate is adjusted to a level that best promotes adaptation. Most mutations with phenotypic effects are harmful, however, and thus there is relentless selection within populations for lower genomic mutation rates. Selection on beneficial mutations can counter this effect by favoring alleles that raise the mutation rate, but the effect of beneficial mutations on the genomic mutation rate is extremely sensitive to recombination and is unlikely to be important in sexual populations. In contrast, high genomic mutation rates can evolve in asexual populations under the influence of beneficial mutations, but this phenomenon is probably of limited adaptive significance and represents, at best, a temporary reprieve from the continual selection pressure to reduce mutation. The physiological cost of reducing mutation below the low level observed in most populations may be the most important factor in setting the genomic mutation rate in sexual and asexual systems, regardless of the benefits of mutation in producing new adaptive variation. Maintenance of mutation rates higher than the minimum set by this "cost of fidelity" is likely only under special circumstances.     

The evolution of stress-induced hypermutation in asexual populations

Numerous empirical studies show that stress of various kinds induces a state of hypermutation in bacteria via multiple mechanisms, but theoretical treatment of this intriguing phenomenon is lacking. We used deterministic and stochastic models to study the evolution of stress-induced hypermutation in infinite and finite-size populations of bacteria undergoing selection, mutation, and random genetic drift in constant environments and in changing ones. Our results suggest that if beneficial mutations occur, even rarely, then stress-induced hypermutation is advantageous for bacteria at both the individual and the population levels and that it is likely to evolve in populations of bacteria in a wide range of conditions because it is favored by selection. These results imply that mutations are not, as the current view holds, uniformly distributed in populations, but rather that mutations are more common in stressed individuals and populations. Because mutation is the raw material of evolution, these results have a profound impact on broad aspects of evolution and biology.     

Sex and the evolution of intrahost competition in RNA virus phi6.

Sex allows beneficial mutations that occur in separate lineages to be fixed in the same genome. For this reason, the Fisher-Muller model predicts that adaptation to the environment is more rapid in a large sexual population than in an equally large asexual population. Sexual reproduction occurs in populations of the RNA virus phi6 when multiple bacteriophages coinfect the same host cell. Here, we tested the model''s predictions by determining whether sex favors more rapid adaptation of phi6 to a bacterial host, Pseudomonas phaseolicola. Replicate populations of phi6 were allowed to evolve in either the presence or absence of sex for 250 generations. All experimental populations showed a significant increase in fitness relative to the ancestor, but sex did not increase the rate of adaptation. Rather, we found that the sexual and asexual treatments also differ because intense intrahost competition between viruses occurs during coinfection. Results showed that the derived sexual viruses were selectively favored only when coinfection is common, indicating that within-host competition detracts from the ability of viruses to exploit the host. Thus, sex was not advantageous because the cost created by intrahost competition was too strong. Our findings indicate that high levels of coinfection exceed an optimum where sex may be beneficial to populations of phi6, and suggest that genetic conflicts can evolve in RNA viruses.