The effects of genetic drift in experimental evolution

Experimental evolution is characterized by exponential or logistic growth and periodic population bottlenecks. The fate of a rare beneficial mutation in these systems is influenced both by the bottleneck effect and by genetic drift. This paper explores the effects of incorporating genetic drift into models of fixation probability in populations with periodic bottlenecks. To model the inherent stochasticity during the growth phase in these populations, we assume a Poisson distribution of offspring. An analytical solution is developed to calculate the fixation probability and a computer simulation is used to verify the results. We find that the fixation rate of a favourable mutant is significantly lower when genetic drift is considered; fixation probability is reduced by over 25% for realistic experimental protocols. Our method is valid for both weak and strong selection; since very large selection coefficients have been reported in the experimental literature, this is an important improvement over previous results.     

THE EFFECTS OF POPULATION BOTTLENECKS ON CLONAL INTERFERENCE, AND THE ADAPTATION EFFECTIVE POPULATION SIZE

Clonal interference refers to the competition that arises in asexual populations when multiple beneficial mutations segregate simultaneously. A large body of theoretical and experimental work now addresses this issue. Although much of the experimental work is performed in populations that grow exponentially between periodic population bottlenecks, the theoretical work to date has addressed only populations of a constant size. We derive an analytical approximation for the rate of adaptation in the presence of both clonal interference and bottlenecks, and compare this prediction to the results of an individual-based simulation, showing excellent agreement in the parameter regime in which clonal interference prevails. We also derive an appropriate definition for the effective population size for adaptive evolution experiments in the presence of population bottlenecks. This "adaptation effective population size" allows for a good approximation of the expected rate of adaptation, either in the strong-selection weak-mutation regime, or when clonal interference comes into play. In the multiple mutation regime, when the product of the population size and mutation rate is extremely large, these results no longer hold.     

Estimation of 2Nes from temporal allele frequency data

We develop a new method for estimating effective population sizes, Ne, and selection coefficients, s, from time-series data of allele frequencies sampled from a single diallelic locus. The method is based on calculating transition probabilities, using a numerical solution of the diffusion process, and assuming independent binomial sampling from this diffusion process at each time point. We apply the method in two example applications. First, we estimate selection coefficients acting on the CCR5-delta 32 mutation on the basis of published samples of contemporary and ancient human DNA. We show that the data are compatible with the assumption of s = 0, although moderate amounts of selection acting on this mutation cannot be excluded. In our second example, we estimate the selection coefficient acting on a mutation segregating in an experimental phage population. We show that the selection coefficient acting on this mutation is approximately 0.43.     

Estimating the optimal bottleneck ratio for experimental evolution: the burst-death model

Population bottlenecks are ubiquitous in nature, and are an inherent feature of the experimental protocol for many laboratory selection experiments. These bottlenecks can have profound effects on the rate and trajectory of evolution. In particular, there is a trade-off between sampling the population too frequently and imposing infrequent, but more severe, bottlenecks. In this paper we consider the effects of population bottlenecks, assuming a burst-death model for the life history of the organism under study. This model assumes that generation times are exponentially distributed and that at each generation, individuals in the population have a fixed number of offspring. The model also allows for a constant death rate between bottlenecks. We use this model to estimate the optimal bottleneck ratio, that is, the fraction of the population that should be sampled at each bottleneck in order to maximize the probability that beneficial mutations occur and are not lost. We find that the optimal ratio is roughly constant with respect to many of the model parameters, and that sampling about 20% of the population will maximize the rate of adaptation.     

Fixation probability for lytic viruses: the attachment-lysis model

The fixation probability of a beneficial mutation is extremely sensitive to assumptions regarding the organism's life history. In this article we compute the fixation probability using a life-history model for lytic viruses, a key model organism in experimental studies of adaptation. The model assumes that attachment times are exponentially distributed, but that the lysis time, the time between attachment and host cell lysis, is constant. We assume that the growth of the wild-type viral population is controlled by periodic sampling (population bottlenecks) and also include the possibility that clearance may occur at a constant rate, for example, through washout in a chemostat. We then compute the fixation probability for mutations that increase the attachment rate, decrease the lysis time, increase the burst size, or reduce the probability of clearance. The fixation probability of these four types of beneficial mutations can be vastly different and depends critically on the time between population bottlenecks. We also explore mutations that affect lysis time, assuming that the burst size is constrained by the lysis time, for experimental protocols that sample either free phage or free phage and artificially lysed infected cells. In all cases we predict that the fixation probability of beneficial alleles is remarkably sensitive to the time between population bottlenecks.     

ADAPTATION RATES OF LYTIC VIRUSES DEPEND CRITICALLY ON WHETHER HOST CELLS SURVIVE THE BOTTLENECK

We use a branching process approach to estimate the substitution rate, the rate at which beneficial mutations occur and fix, in populations of lytic viruses whose growth is controlled by periodic population bottlenecks. Our model predicts that substitution rates, and by extension adaptation rates, are profoundly affected by the survival of infected host cells at the bottleneck. In particular, we find that direct transfer (or environmental) bottlenecks, in which some fraction of both free virus and host cells are preserved, are associated with relatively slow adaptation rates for the virus. In contrast, viruses can adapt much more quickly when only free virus is transferred to a new host population, as is typical in an epidemiological setting. Finally, when premature lysis of the host‐cell population is induced at the bottleneck, we predict that adaptation rates for the virus will, in general, be faster still. These results hold irrespective of the life‐history trait affected by the beneficial mutation. The substitution rates in the presence of environmental bottlenecks are predicted to be as much as an order of magnitude lower than in the other two cases.     

A method to infer positive selection from marker dynamics in an asexual population

MOTIVATION: The observation of positive selection acting on a mutant indicates that the corresponding mutation has some form of functional relevance. Determining the fitness effects of mutations thus has relevance to many interesting biological questions. One means of identifying beneficial mutations in an asexual population is to observe changes in the frequency of marked subsets of the population. We here describe a method to estimate the establishment times and fitnesses of beneficial mutations from neutral marker frequency data. RESULTS: The method accurately reproduces complex marker frequency trajectories. In simulations for which positive selection is close to 5% per generation, we obtain correlations upwards of 0.91 between correct and inferred haplotype establishment times. Where mutation selection coefficients are exponentially distributed, the inferred distribution of haplotype fitnesses is close to being correct. Applied to data from a bacterial evolution experiment, our method reproduces an observed correlation between evolvability and initial fitness defect.     

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.     

Effect of growth under selection on appearance of chromosomal mutations in Salmonella enterica

Populations adapt physiologically using regulatory mechanisms and genetically by means of mutations that improve growth. During growth under selection, genetic adaptation can be rapid. In several genetic systems, the speed of adaptation has been attributed to cellular mechanisms that increase mutation rates in response to growth limitation. An alternative possibility is that growth limitation serves only as a selective agent but acts on small-effect mutations that are common under all growth conditions. The genetic systems that initially suggested stress-induced mutagenesis have been analyzed without regard for multistep adaptation and some include features that make such analysis difficult. To test the selection-only model, a simpler system is examined, whose behavior was originally attributed to stress-induced mutagenesis (Yang et al. 2001, 2006). A population with a silent chromosomal lac operon gives rise to Lac+ revertant colonies that accumulate over 6 days under selection. Each colony contains a mixture of singly and doubly mutant cells. Evidence is provided that the colonies are initiated by pre-existing single mutants with a weak Lac+ phenotype. Under selection, these cells initiate slow-growing clones, in which a second mutation arises and improves growth of the resulting double mutant. The system shows no evidence of general mutagenesis during selection. Selection alone may explain rapid adaptation in this and other systems that give the appearance of mutagenesis.     

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.     

Analysis of a stochastic predator–prey model with applications to intrahost HIV genetic diversity

During an infection, HIV experiences strong selection by immune system T cells. Recent experimental work has shown that MHC escape mutations form an important pathway for HIV to avoid such selection. In this paper, we study a model of MHC escape mutation. The model is a predator–prey model with two prey, composed of two HIV variants, and one predator, the immune system CD8 cells. We assume that one HIV variant is visible to CD8 cells and one is not. The model takes the form of a system of stochastic differential equations. Motivated by well-known results concerning the short life-cycle of HIV intrahost, we assume that HIV population dynamics occur on a faster time scale then CD8 population dynamics. This separation of time scales allows us to analyze our model using an asymptotic approach. Using this model we study the impact of an MHC escape mutation on the population dynamics and genetic evolution of the intrahost HIV population. From the perspective of population dynamics, we show that the competition between the visible and invisible HIV variants can reach steady states in which either a single variant exists or in which coexistence occurs depending on the parameter regime. We show that in some parameter regimes the end state of the system is stochastic. From a genetics perspective, we study the impact of the population dynamics on the lineages of an HIV sample taken after an escape mutation occurs. We show that the lineages go through severe bottlenecks and that in certain parameter regimes the lineage distribution can be characterized by a Kingman coalescent. Our results depend on methods from diffusion theory and coalescent theory.     

The frequency of the perfect genotype in a population subject to pleiotropic mutation

We consider a large population of asexual organisms characterised by a number of quantitative traits that are subject to stabilising selection. Mutation is taken to act pleiotropically, with every mutation generally changing all of the traits under selection. We focus on the equilibrium distribution of the population, where mutation and selection are in balance. It has been previously established that the equilibrium distribution of genotypic effects may be anomalous, as it may contain a singular spike--a Dirac delta function--corresponding to a non-zero proportion of the population having exactly optimal genotypic values. In the present work, we present exact results for the case where three traits are under selection. These results give the equilibrium genetic variance of the population, and the proportion of the population that have the optimal genotype. This is achieved for two different spherically symmetric distributions of mutant effects. Additionally, a simple and robust numerical approach is also presented that allows the treatment of some other mutation distributions, where there are an arbitrary number of selected traits.     

Experimental adaptation of Salmonella typhimurium to mice

Experimental evolution is a powerful approach to study the dynamics and mechanisms of bacterial niche specialization. By serial passage in mice, we evolved 18 independent lineages of Salmonella typhimurium LT2 and examined the rate and extent of adaptation to a mainly reticuloendothelial host environment. Bacterial mutation rates and population sizes were varied by using wild-type and DNA repair-defective mutator (mutS) strains with normal and high mutation rates, respectively, and by varying the number of bacteria intraperitoneally injected into mice. After <200 generations of adaptation all lineages showed an increased fitness as measured by a faster growth rate in mice (selection coefficients 0.11-0.58). Using a generally applicable mathematical model we calculated the adaptive mutation rate for the wild-type bacterium to be >10(-6)/cell/generation, suggesting that the majority of adaptive mutations are not simple point mutations. For the mutator lineages, adaptation to mice was associated with a loss of fitness in secondary environments as seen by a reduced metabolic capability. During adaptation there was no indication that a high mutation rate was counterselected. These data show that S. typhimurium can rapidly and extensively increase its fitness in mice but this niche specialization is, at least in mutators, associated with a cost.     

Quantifying the effects of migration and mutation on adaptation and demography in spatially heterogeneous environments

How do mutation and gene flow influence population persistence, niche expansion and local adaptation in spatially heterogeneous environments? In this article, we analyse a demographic and evolutionary model of adaptation to an environment containing two habitats in equal frequencies, and we bridge the gap between different theoretical frameworks. Qualitatively, our model yields four qualitative types of outcomes: (i) global extinction of the population, (ii) adaptation to one habitat only, but also adaptation to both habitats with, (iii) specialized phenotypes or (iv) with generalized phenotypes, and we determine the conditions under which each equilibrium is reached. We derive new analytical approximations for the local densities and the distributions of traits in each habitat under a migration–selection–mutation balance, compute the equilibrium values of the means, variances and asymmetries of the local distributions of phenotypes, and contrast the effects of migration and mutation on the evolutionary outcome. We then check our analytical results by solving our model numerically, and also assess their robustness in the presence of demographic stochasticity. Although increased migration results in a decrease in local adaptation, mutation in our model does not influence the values of the local mean traits. Yet, both migration and mutation can have dramatic effects on population size and even lead to metapopulation extinction when selection is strong. Niche expansion, the ability for the population to adapt to both habitats, can also be prevented by small migration rates and a reduced evolutionary potential characterized by rare mutation events of small effects; however, niche expansion is otherwise the most likely outcome. Although our results are derived under the assumption of clonal reproduction, we finally show and discuss the links between our model and previous quantitative genetics models.     

Microsatellite variation and evolution in the Mimulus guttatus species complex with contrasting mating systems

Mutational variability at microsatellite loci is shaped by both population history and the mating system. In turn, alternate mating systems in flowering plants can resolve aspects of microsatellite loci evolution. Five species of yellow monkeyflowers (Mimulus sect. Simiolis) differing for historical rates of inbreeding were surveyed for variation at six microsatellite loci. High levels of diversity at these loci were found in both outcrossing and selfing taxa. In line with allozyme studies, inbreeders showed more partitioning of diversity among populations, and diversity in selfing taxa was lower than expected from reductions in effective population size due to selfing alone, suggesting the presence of either population bottlenecks or background selection in selfers. Evaluation of the stepwise mutation model (a model of DNA replication slippage) suggests that these loci evolve in a stepwise fashion. Inferred coalescent times of microsatellite alleles indicate that past bottlenecks of population size or colonization events were important in reducing diversity in the inbreeding taxon.      

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.     

Mutators, population size, adaptive landscape and the adaptation of asexual populations of bacteria.

Selection of mutator alleles, increasing the mutation rate up to 10, 000-fold, has been observed during in vitro experimental evolution. This spread is ascribed to the hitchhiking of mutator alleles with favorable mutations, as demonstrated by a theoretical model using selective parameters corresponding to such experiments. Observations of unexpectedly high frequencies of mutators in natural isolates suggest that the same phenomenon could occur in the wild. But it remains questionable whether realistic in natura parameter values could also result in selection of mutators. In particular, the main parameters of adaptation, the size of the adapting population and the height and steepness of the adaptive peak characterizing adaptation, are very variable in nature. By simulation approach, we studied the effect of these parameters on the selection of mutators in asexual populations, assuming additive fitness. We show that the larger the population size, the more likely the fixation of mutator alleles. At a large population size, at least four adaptive mutations are needed for mutator fixation; moreover, under stronger selection stronger mutators are selected. We propose a model based on multiple mutations to illustrate how second-order selection can optimize population fitness when few favorable mutations are required for adaptation.