Fixation probabilities depend on life history: fecundity, generation time and survival in a burst-death model

The burst-death model has been developed to describe the life history of organisms with variable generation times and a burst of a fixed number of offspring. The model also includes an optional constant clearance rate, such as washout from a chemostat, and the possibility of sustained periods of population growth followed by severe bottlenecks, as in serial passaging. In this model, a beneficial mutation can either increase the burst rate or the burst size, or reduce the clearance rate, thus increasing survival. In this article we examine the effects of these three possible mechanisms on both the Malthusian fitness and the fixation probability of the lineage. We find that equivalent relative increases in the burst rate or burst size confer equivalent increases in the Malthusian fitness of a lineage, whereas increasing survival typically has a more moderate effect on Malthusian fitness. In contrast, for beneficial mutations that confer the same increase in fitness, mutations that increase survival are the most likely to fix, followed by mutations that increase the burst rate. Mutations that increase the burst size are the least likely to fix. These results imply that mutant lineages with the highest Malthusian fitness are not, in many cases, the most likely to escape extinction.     

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.     

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 of a beneficial mutation conferring decreased generation time in changing environments

Background

One central building block of population genetics is the fixation probability. It is a probabilistic understanding of the eventual fate of new mutations. Moreover, the fixation probability of new beneficial mutations plays an important effect on the adaptation of populations to environmental challenges. Great progress has been made in the study of the beneficial mutations that increases offspring number. However, the fixation probability of beneficial mutations with a shorter generation time under various genetic and ecological conditions has not been explored.

Results

Here we extend the classical result of the fixation probability of beneficial mutations obtained by Haldane, and estimate the fixation probability of a beneficial mutation with a reduced generation time in a changing environment. Assuming that the selective advantage is very small, we concentrate all the changing factors of environment on a single quantity: effective selective advantage. Using a time-dependent branching process, we get the analytic approximation for the fixation probability of beneficial mutations that decrease the generation time. Then, we apply this approximation to four interesting biological cases.

Conclusions

In these instances, we show the comparison of the approximation with the accurate values. We find that they are consistent, demonstrating the effectiveness of our result for the fixation probability of beneficial mutations conferring a reduced replication time.

     

Fixation probability favors increased fecundity over reduced generation time

The cornerstone of population genetics is a probabilistic understanding of the ultimate fate--survival or extinction--of rare mutations. If a mutation is beneficial, it enables its carrier to reproduce faster than native wild-type individuals. In classic derivations and in the considerable body of research that has followed, "faster" has been defined mathematically to mean "able to produce more surviving offspring per generation." Many organisms, however, may increase their reproductive rate by producing the same number of offspring in a shorter generation time: a mutant bacterium, for example, may complete the cell cycle and produce two offspring more quickly than the wild type. We find that the ultimate fixation probability of a mutation conferring a shorter generation time differs from that of a mutation conferring more offspring by a factor of 2 ln(2)-nearly 40%. This predicts a reduction in the overall substitution rate for any mutation that decreases the generation time: fixation probability is biased toward increased offspring number.     

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.     

Evaluating the impact of population bottlenecks in experimental evolution

Experimental evolution involves severe, periodic reductions in population size when fresh media are inoculated during serial transfer. These bottlenecks affect the dynamics of evolution, reducing the probability that a beneficial mutation will reach fixation. We quantify the impact of these bottlenecks on the evolutionary dynamics, for populations that grow exponentially between transfers and for populations in which growth is curbed by a resource-limited environment. We find that in both cases, mutations that survive bottlenecks are equally likely to occur, per unit time, at all times during the growth phase. We estimate the total fraction of beneficial mutations that are lost due to bottlenecks during experimental evolution protocols and derive the "optimal" dilution ratio, the ratio that maximizes the number of surviving beneficial mutations. Although more severe dilution ratios are often used in the literature, we find that a ratio of 0.1-0.2 minimizes the chances that rare beneficial mutations are lost. Finally, we provide a number of useful approximate results and illustrate our approach with applications to experimental evolution protocols in the literature.     

Environmental Fluctuations and Level of Density-Compensation Strongly Affects the Probability of Fixation and Fixation Times

The probability of, and time to, fixation of a mutation in a population has traditionally been studied by the classic Wright–Fisher model where population size is constant. Recent theoretical expansions have covered fluctuating populations in various ways but have not incorporated models of how the environment fluctuates in combination with different levels of density-compensation affecting fecundity. We tested the hypothesis that the probability of, and time to, fixation of neutral, advantageous and deleterious mutations is dependent on how the environment fluctuates over time, and on the level of density-compensation. We found that fixation probabilities and times were dependent on the pattern of autocorrelation of carrying capacity over time and interacted with density-compensation. The pattern found was most pronounced at small population sizes. The patterns differed greatly depending on whether the mutation was neutral, advantageous, or disadvantageous. The results indicate that the degree of mismatch between carrying capacity and population size is a key factor, rather than population size per se, and that effective population sizes can be very low also when the census population size is far above the carrying capacity. This study highlights the need for explicit population dynamic models and models for environmental fluctuations for the understanding of the dynamics of genes in populations.     

The impact of host-cell dynamics on the fixation probability for lytic viruses

Lytic viruses are obligate parasites whose population dynamics are necessarily coupled to the dynamics of their host-cell population. The adaptation rate of these viruses has attracted considerable scientific interest, as they are a key model organism in experimental evolution. Nevertheless, to date mathematical models of experimental evolution have largely ignored the host-cell population. In this paper we incorporate two important features of host-cell dynamics—the possibility of clearance or death of an infected cell before lysis, and the possibility of changing host-cell density—into previous models for the fixation probability of lytic viruses. We compute the fixation probabilities of rare alleles that confer reproductive benefit through either an increase in attachment rate or burst size, or a reduction in lysis time. We find that host-cell clearance significantly reduces the fixation probabilities of all types of beneficial mutations, having the largest impact on mutations which reduce the lysis time, but has only modest effects on the pattern of fixation probabilities previously observed. We further predict that exponential growth of the host-cell population preferentially selects for mutations that affect burst size or lysis time, and exacerbates the sensitive dependence of fixation probabilities on the time between population bottlenecks. Even when burst size and lysis time are constrained to vary together, our results suggest that lytic viruses should readily adapt to optimize these traits to the timing between population bottlenecks.     

Note on the computation of critical effective population sizes.

This work extends the work of Whitlock in examining the critical effective population sizes from the fixation of both deleterious and beneficial mutations under drift and selection to prevent mutation breakdown of the population. The validity of approximations for the probability of fixation depends on the nature of the assumed distribution for the fitness effect of both types of mutations. Using no approximation for the probability of fixation and assuming a heavy tailed fitness effect distribution, the current model indicates that the coefficients of variation for the fitness effect distributions of both types of mutations and the fitness effect distribution mean for the beneficial mutations are important predictors of the critical effective population size. The current model further predicts that very small populations can be sustained if the fitness effect variances for both types of mutations and the mean for beneficial mutations are large.     

The Evolution of Recombination: Removing the Limits to Natural Selection

One of the oldest hypotheses for the advantage of recombination is that recombination allows beneficial mutations that arise in different individuals to be placed together on the same chromosome. Unless recombination occurs, one of the beneficial alleles is doomed to extinction, slowing the rate at which adaptive mutations are incorporated within a population. We model the effects of a modifier of recombination on the fixation probability of beneficial mutations when beneficial alleles are segregating at other loci. We find that modifier alleles that increase recombination do increase the fixation probability of beneficial mutants and subsequently hitchhike along as the mutants rise in frequency. The strength of selection favoring a modifier that increases recombination is proportional to λ(2)Sδr/r when linkage is tight and λ(2)S(3)δ r/N when linkage is loose, where λ is the beneficial mutation rate per genome per generation throughout a population of size N, S is the average mutant effect, r is the average recombination rate, and δr is the amount that recombination is modified. We conclude that selection for recombination will be substantial only if there is tight linkage within the genome or if many loci are subject to directional selection as during periods of rapid evolutionary change.     

Fixation probability and time in subdivided populations

New alleles arising in a population by mutation ultimately are either fixed or lost. Either is possible, for both beneficial and deleterious alleles, because of stochastic changes in allele frequency due to genetic drift. Spatially structured populations differ from unstructured populations in the probability of fixation and the time that this fixation takes. Previous results have generally made many assumptions: that all demes contribute to the next generation in exact proportion to their current sizes, that new mutations are beneficial, and that new alleles have additive effects. In this article these assumptions are relaxed, allowing for an arbitrary distribution among demes of reproductive success, both beneficial and deleterious effects, and arbitrary dominance. The effects of population structure can be expressed with two summary statistics: the effective population size and a variant of Wright's F(ST). In general, the probability of fixation is strongly affected by population structure, as is the expected time to fixation or loss. Population structure changes the effective size of the species, often strongly downward; smaller effective size increases the probability of fixing deleterious alleles and decreases the probability of fixing beneficial alleles. On the other hand, population structure causes an increase in the homozygosity of alleles, which increases the probability of fixing beneficial alleles but somewhat decreases the probability of fixing deleterious alleles. The probability of fixing new beneficial alleles can be simply described by 2hs(1 - F(ST))N(e)/N(tot), where hs is the change in fitness of heterozygotes relative to the ancestral homozygote, F(ST) is a weighted version of Wright's measure of population subdivision, and N(e) and N(tot) are the effective and census sizes, respectively. These results are verified by simulation for a broad range of population structures, including the island model, the stepping-stone model, and a model with extinction and recolonization.     

The impact of male-killing bacteria on host evolutionary processes

Male-killing bacteria are maternally inherited endosymbionts that selectively kill male offspring of their arthropod hosts. Using both analytical techniques and computer simulations, we studied the impact of these bacteria on the population genetics of their hosts. In particular, we derived and corroborated formulas for the fixation probability of mutant alleles, mean times to fixation and fixation or extinction, and heterozygosity for varying male-killer prevalence. Our results demonstrate that infections with male-killing bacteria impede the spread of beneficial alleles, facilitate the spread of deleterious alleles, and reduce genetic variation. The reason for this lies in the strongly reduced fitness of infected females combined with no or very limited gene flow from infected females to uninfected individuals. These two properties of male-killer-infected populations reduce the population size relevant for the initial emergence and spread of mutations. In contrast, use of Wright's equation relating sex ratio to effective population size produces misleading predictions. We discuss the relationship to the similar effect of background selection, the impact of other sex-ratio-distorting endosymbionts, and how our results affect the interpretation of empirical data on genetic variation in male-killer-infected populations.     

Fixation of new alleles and the extinction of small populations: drift load, beneficial alleles, and sexual selection

With a small effective population size, random genetic drift is more important than selection in determining the fate of new alleles. Small populations therefore accumulate deleterious mutations. Left unchecked, the effect of these fixed alleles is to reduce the reproductive capacity of a species, eventually to the point of extinction. New beneficial mutations, if fixed by selection, can restore some of this lost fitness. This paper derives the overall change in fitness due to fixation of new deleterious and beneficial alleles, as a function of the distribution of effects of new mutations and the effective population size. There is a critical effective size below which a population will on average decline in fitness, but above which beneficial mutations allow the population to persist. With reasonable estimates of the relevant parameters, this critical effective size is likely to be a few hundred. Furthermore, sexual selection can act to reduce the fixation probability of deleterious new mutations and increase the probability of fixing new beneficial mutations. Sexual selection can therefore reduce the risk of extinction of small populations.     

The accumulation of mutations in asexual populations and the structure of genealogical trees in the presence of selection

We study the accumulation of unfavourable mutations in asexual populations by the process of Muller's ratchet, and the consequent inevitable decrease in fitness of the population. Simulations show that it is mutations with only moderate unfavourable effect that lead to the most rapid decrease in fitness. We measure the number of fixations as a function of time and show that the fixation rate must be equal to the ratchet rate once a steady state is reached. Large bursts of fixations are observed to occur simultaneously. We relate this to the structure of the genealogical tree. We derive equations relating the rate of the ratchet to the moments of the distribution of the number of mutations k per individual. These equations interpolate between the deterministic limit (an infinite population with selection present) and the neutral limit (a finite size population with no selection). Both these limits are exactly soluble. In the neutral case, the distribution of k is shown to be non-self-averaging, i.e. the fluctuations remain very large even for very large populations. We also consider the strong-selection limit in which only individuals in the fittest surviving class have offspring. This limit is again exactly soluble. We investigate the structure of the genealogical tree relating individuals in the same population, and consider the probability (T) that two individuals had their latest common ancestor T generations in the past. The function (T) is exactly calculable in the neutral limit and the strong-selection limit, and we obtain an empirical solution for intermediate selection strengths.     

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.     

The Dynamics of Genetic Draft in Rapidly Adapting Populations

The accumulation of beneficial mutations on competing genetic backgrounds in rapidly adapting populations has a striking impact on evolutionary dynamics. This effect, known as clonal interference, causes erratic fluctuations in the frequencies of observed mutations, randomizes the fixation times of successful mutations, and leaves distinct signatures on patterns of genetic variation. Here, we show how this form of “genetic draft” affects the forward-time dynamics of site frequencies in rapidly adapting asexual populations. We calculate the probability that mutations at individual sites shift in frequency over a characteristic timescale, extending Gillespie’s original model of draft to the case where many strongly selected beneficial mutations segregate simultaneously. We then derive the sojourn time of mutant alleles, the expected fixation time of successful mutants, and the site frequency spectrum of beneficial and neutral mutations. Finally, we show how this form of draft affects inferences in the McDonald–Kreitman test and how it relates to recent observations that some aspects of genetic diversity are described by the Bolthausen–Sznitman coalescent in the limit of very rapid adaptation.     

FIXATION OF SLIGHTLY BENEFICIAL MUTATIONS: EFFECTS OF LIFE HISTORY

Recent studies of rates of evolution have revealed large systematic differences among organisms with different life histories, both within and among taxa. Here, we consider how life history may affect the rate of evolution via its influence on the fixation probability of slightly beneficial mutations. Our approach is based on diffusion modeling for a finite, stage‐structured population with stochastic population dynamics. The results, which are verified by computer simulations, demonstrate that even with complex population structure just two demographic parameters are sufficient to give an accurate approximation of the fixation probability of a slightly beneficial mutation. These are the reproductive value of the stage in which the mutation first occurs and the demographic variance of the population. The demographic variance also determines what influence population size has on the fixation probability. This model represents a substantial generalization of earlier models, covering a large range of life histories.     

Clonal interference and the periodic selection of new beneficial mutations in Escherichia coli

The conventional model of adaptation in asexual populations implies sequential fixation of new beneficial mutations via rare selective sweeps that purge all variation and preserve the clonal genotype. However, in large populations multiple beneficial mutations may co-occur, causing competition among them, a phenomenon called "clonal interference." Clonal interference is thus expected to lead to longer fixation times and larger fitness effects of mutations that ultimately become fixed, as well as to a genetically more diverse population. Here, we study the significance of clonal interference in populations consisting of mixtures of differently marked wild-type and mutator strains of Escherichia coli that adapt to a minimal-glucose environment for 400 generations. We monitored marker frequencies during evolution and measured the competitive fitness of random clones from each marker state after evolution. The results demonstrate the presence of multiple beneficial mutations in these populations and slower and more erratic invasion of mutants than expected by the conventional model, showing the signature of clonal interference. We found that a consequence of clonal interference is that fitness estimates derived from invasion trajectories were less than half the magnitude of direct estimates from competition experiments, thus revealing fundamental problems with this fitness measure. These results force a reevaluation of the conventional model of periodic selection for asexual microbes.     

Genetic variation and the fate of beneficial mutations in asexual populations

The fate of a newly arising beneficial mutation depends on many factors, such as the population size and the availability and fitness effects of other mutations that accumulate in the population. It has proved difficult to understand how these factors influence the trajectories of particular mutations, since experiments have primarily focused on characterizing successful clones emerging from a small number of evolving populations. Here, we present the results of a massively parallel experiment designed to measure the full spectrum of possible fates of new beneficial mutations in hundreds of experimental yeast populations, whether these mutations are ultimately successful or not. Using strains in which a particular class of beneficial mutation is detectable by fluorescence, we followed the trajectories of these beneficial mutations across 592 independent populations for 1000 generations. We find that the fitness advantage provided by individual mutations plays a surprisingly small role. Rather, underlying "background" genetic variation is quickly generated in our initially clonal populations and plays a crucial role in determining the fate of each individual beneficial mutation in the evolving population.