A unified treatment of the probability of fixation when population size and the strength of selection change over time

Developmental hematopoietic stem cell variation explains clonal hematopoiesis later in life

Clonal hematopoiesis becomes increasingly common with age, but its cause is enigmatic because driver mutations are often absent. Serial observations infer weak selection indicating variants are acquired much earlier in life with unexplained initial growth spurts. Here we use fluctuating CpG methylation as a lineage marker to track stem cell clonal dynamics of hematopoiesis. We show, via the shared prenatal circulation of monozygotic twins, that weak selection conferred by stem cell variation created before birth can reliably yield clonal hematopoiesis later in life. Theory indicates weak selection will lead to dominance given enough time and large enough population sizes. Human hematopoiesis satisfies both these conditions. Stochastic loss of weakly selected variants is naturally prevented by the expansion of stem cell lineages during development. The dominance of stem cell clones created before birth is supported by blood fluctuating CpG methylation patterns that exhibit low correlation between unrelated individuals but are highly correlated between many elderly monozygotic twins. Therefore, clonal hematopoiesis driven by weak selection in later life appears to reflect variation created before birth.

Information encoded in gene-frequency trajectories

In this work we present a systematic mathematical approximation scheme that exposes the way that information, about the evolutionary forces of selection and random genetic drift, is encoded within gene-frequency trajectories. We determine approximate, time-dependent, gene-frequency trajectory statistics, assuming additive selection. We use the probability of fixation to test and illustrate the approximation scheme introduced. For the case where the strength of selection and the effective population size have constant values, we show how a standard diffusion approximation result, for the probability of fixation, systematically emerges when increasing numbers of approximate trajectory statistics are taken into account. We then provide examples of how time-dependent parameters influence gene-frequency statistics.

On the hidden temporal dynamics of plant adaptation

Adaptation to a wide range of environments is a major driver of plant diversity. It is now possible to catalog millions of potential adaptive genomic differences segregating between environments within a plant species in a single experiment. Understanding which of these changes contributes to adaptive phenotypic divergence between plant populations is a major goal of evolutionary biologists and crop breeders. In this review, we briefly highlight the approaches frequently used to understand the genetic basis of adaptive phenotypes in plants, and we discuss some of the limitations of these methods. We propose that direct observation of the process of adaptation using multigenerational studies and whole genome sequencing is a crucial missing component of recent studies of plant adaptation because it complements several shortcomings of sampling-based techniques.

The influence of biological, epidemiological, and treatment factors on the establishment and spread of drug-resistant Plasmodium falciparum

The effectiveness of artemisinin-based combination therapies (ACTs) to treat Plasmodium falciparum malaria is threatened by resistance. The complex interplay between sources of selective pressure-treatment properties, biological factors, transmission intensity, and access to treatment-obscures understanding how, when, and why resistance establishes and spreads across different locations. We developed a disease modelling approach with emulator-based global sensitivity analysis to systematically quantify which of these factors drive establishment and spread of drug resistance. Drug resistance was more likely to evolve in low transmission settings due to the lower levels of (i) immunity and (ii) within-host competition between genotypes. Spread of parasites resistant to artemisinin partner drugs depended on the period of low drug concentration (known as the selection window). Spread of partial artemisinin resistance was slowed with prolonged parasite exposure to artemisinin derivatives and accelerated when the parasite was also resistant to the partner drug. Thus, to slow the spread of partial artemisinin resistance, molecular surveillance should be supported to detect resistance to partner drugs and to change ACTs accordingly. Furthermore, implementing more sustainable artemisinin-based therapies will require extending parasite exposure to artemisinin derivatives, and mitigating the selection windows of partner drugs, which could be achieved by including an additional long-acting drug.

Adapt or Perish: Evolutionary Rescue in a Gradually Deteriorating Environment

We investigate the evolutionary rescue of a microbial population in a gradually deteriorating environment, through a combination of analytical calculations and stochastic simulations. We consider a population destined for extinction in the absence of mutants, which can survive only if mutants sufficiently adapted to the new environment arise and fix. We show that mutants that appear later during the environment deterioration have a higher probability to fix. The rescue probability of the population increases with a sigmoidal shape when the product of the carrying capacity and of the mutation probability increases. Furthermore, we find that rescue becomes more likely for smaller population sizes and/or mutation probabilities if the environment degradation is slower, which illustrates the key impact of the rapidity of environment degradation on the fate of a population. We also show that our main conclusions are robust across various types of adaptive mutants, including specialist and generalist ones, as well as mutants modeling antimicrobial resistance evolution. We further express the average time of appearance of the mutants that do rescue the population and the average extinction time of those that do not. Our methods can be applied to other situations with continuously variable fitnesses and population sizes, and our analytical predictions are valid in the weak-to-moderate mutation regime.

The Impact of Dominance on Adaptation in Changing Environments

Natural environments are seldom static and therefore it is important to ask how a population adapts in a changing environment. We consider a finite, diploid population evolving in a periodically changing environment and study how the fixation probability of a rare mutant depends on its dominance coefficient and the rate of environmental change. We find that, in slowly changing environments, the effect of dominance is the same as in the static environment, that is, if a mutant is beneficial (deleterious) when it appears, it is more (less) likely to fix if it is dominant. But, in fast changing environments, the effect of dominance can be different from that in the static environment and is determined by the mutant's fitness at the time of appearance as well as that in the time-averaged environment. We find that, in a rapidly varying environment that is neutral on average, an initially beneficial (deleterious) mutant that arises while selection is decreasing (increasing) has a fixation probability lower (higher) than that for a neutral mutant as a result of which the recessive (dominant) mutant is favored. If the environment is beneficial (deleterious) on average but the mutant is deleterious (beneficial) when it appears in the population, the dominant (recessive) mutant is favored in a fast changing environment. We also find that, when recurrent mutations occur, dominance does not have a strong influence on evolutionary dynamics.

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.

A mathematical formalism for natural selection with arbitrary spatial and genetic structure

We define a general class of models representing natural selection between two alleles. The population size and spatial structure are arbitrary, but fixed. Genetics can be haploid, diploid, or otherwise; reproduction can be asexual or sexual. Biological events (e.g. births, deaths, mating, dispersal) depend in arbitrary fashion on the current population state. Our formalism is based on the idea of genetic sites. Each genetic site resides at a particular locus and houses a single allele. Each individual contains a number of sites equal to its ploidy (one for haploids, two for diploids, etc.). Selection occurs via replacement events, in which alleles in some sites are replaced by copies of others. Replacement events depend stochastically on the population state, leading to a Markov chain representation of natural selection. Within this formalism, we define reproductive value, fitness, neutral drift, and fixation probability, and prove relationships among them. We identify four criteria for evaluating which allele is selected and show that these become equivalent in the limit of low mutation. We then formalize the method of weak selection. The power of our formalism is illustrated with applications to evolutionary games on graphs and to selection in a haplodiploid population.

Effects of demographic stochasticity and life-history strategies on times and probabilities to fixation

How life-history strategies influence the evolution of populations is not well understood. Most existing models stem from the Wright-Fisher model which considers discrete generations and a fixed population size, thus not taking into account any potential consequences of overlapping generations and demographic stochasticity on allelic frequencies. We introduce an individual-based model in which both population size and genotypic frequencies at a single bi-allelic locus are emergent properties of the model. Demographic parameters can be defined so as to represent a large range of r and K life-history strategies in a stable environment, and appropriate fixed effective population sizes are calculated so as to compare our model to the Wright-Fisher diffusion. Our results indicate that models with fixed population size that stem from the Wright-Fisher diffusion cannot fully capture the consequences of demographic stochasticity on allele fixation in long-lived species with low reproductive rates. This discrepancy is accentuated in the presence of demo-genetic feedback. Furthermore, we predict that populations with K life-histories should maintain lower genetic diversity than those with r life-histories.

Fixation probabilities in populations under demographic fluctuations

We study the fixation probability of a mutant type when introduced into a resident population. We implement a stochastic competitive Lotka-Volterra model with two types and intra- and interspecific competition. The model further allows for stochastically varying population sizes. The competition coefficients are interpreted in terms of inverse payoffs emerging from an evolutionary game. Since our study focuses on the impact of the competition values, we assume the same net growth rate for both types. In this general framework, we derive a formula for the fixation probability [Formula: see text] of the mutant type under weak selection. We find that the most important parameter deciding over the invasion success of the mutant is its death rate due to competition with the resident. Furthermore, we compare our approximation to results obtained by implementing population size changes deterministically in order to explore the parameter regime of validity of our method. Finally, we put our formula in the context of classical evolutionary game theory and observe similarities and differences to the results obtained in that constant population size setting.

Evolutionary justifications for human reproductive limitations

Common human reproductive inefficiencies have multiple etiologies. Going against chance, many effects, such as polycystic ovaries, endometriosis, and folate metabolic issues, have genetic components, while aneuploid losses arise from diverse mitotic and meiotic errors at different stages, some transitory. This can be advantageous, since greater overall survival with fewer offspring can increase reproductive success. Benefits primarily accrue to mothers, who bear most child related costs, and for whom early losses are less costly than late. Different adaptations to different situations reflect human evolutionary history. For early speciation, periodic climate extremes repeatedly reduced resources, favoring limitations while contracted populations helped fix relevant genes. Later, under better conditions, evolving social cooperation could increase fecundity faster than it added resources, further supporting reproductive suppression through mitotic aneuploidy, with very early losses minimizing maternal costs. The grandmother hypothesis suggests benefits in limiting reproduction as maternal age increased pregnancy risks in order to support grandchildren as they arrived, selecting for maternal age-related meiotic aneuploidy. Finally, with variable short-term agricultural shortages, acute reproductive responses arose through chromatin "nutrient sensor"-regulated epigenetic effects that also shifted some lethal effects earlier, reducing both maternal and mutation load costs. Overall, despite suggestions to the contrary, it is likely that human selective pressures have not decreased with civilization, but that many of the costs have been shifted to early reproduction.

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.

Fluctuating Selection in the Moran

Contrary to classical population genetics theory, experiments demonstrate that fluctuating selection can protect a haploid polymorphism in the absence of frequency dependent effects on fitness. Using forward simulations with the Moran model, we confirm our analytical results showing that a fluctuating selection regime, with a mean selection coefficient of zero, promotes polymorphism. We find that increases in heterozygosity over neutral expectations are especially pronounced when fluctuations are rapid, mutation is weak, the population size is large, and the variance in selection is big. Lowering the frequency of fluctuations makes selection more directional, and so heterozygosity declines. We also show that fluctuating selection raises dn /ds ratios for polymorphism, not only by sweeping selected alleles into the population, but also by purging the neutral variants of selected alleles as they undergo repeated bottlenecks. Our analysis shows that randomly fluctuating selection increases the rate of evolution by increasing the probability of fixation. The impact is especially noticeable when the selection is strong and mutation is weak. Simulations show the increase in the rate of evolution declines as the rate of new mutations entering the population increases, an effect attributable to clonal interference. Intriguingly, fluctuating selection increases the dn /ds ratios for divergence more than for polymorphism, a pattern commonly seen in comparative genomics. Our model, which extends the classical neutral model of molecular evolution by incorporating random fluctuations in selection, accommodates a wide variety of observations, both neutral and selected, with economy.

Darwinism for the Genomic Age: Connecting Mutation to Diversification

A growing body of evidence suggests that rates of diversification of biological lineages are correlated with differences in genome-wide mutation rate. Given that most research into differential patterns of diversification rate have focused on species traits or ecological parameters, a connection to the biochemical processes of genome change is an unexpected observation. While the empirical evidence for a significant association between mutation rate and diversification rate is mounting, there has been less effort in explaining the factors that mediate this connection between genetic change and species richness. Here we draw together empirical studies and theoretical concepts that may help to build links in the explanatory chain that connects mutation to diversification. First we consider the way that mutation rates vary between species. We then explore how differences in mutation rates have flow-through effects to the rate at which populations acquire substitutions, which in turn influences the speed at which populations become reproductively isolated from each other due to the acquisition of genomic incompatibilities. Since diversification rate is commonly measured from phylogenetic analyses, we propose a conceptual approach for relating events of reproductive isolation to bifurcations on molecular phylogenies. As we examine each of these relationships, we consider theoretical models that might shine a light on the observed association between rate of molecular evolution and diversification rate, and critically evaluate the empirical evidence for these links, focusing on phylogenetic comparative studies. Finally, we ask whether we are getting closer to a real understanding of the way that the processes of molecular evolution connect to the observable patterns of diversification.

How Life History Can Sway the Fixation Probability of Mutants

In this work, we study the effects of demographic structure on evolutionary dynamics when selection acts on reproduction, survival, or both. In contrast to the previously discovered pattern that the fixation probability of a neutral mutant decreases while the population becomes younger, we show that a mutant with a constant selective advantage may have a maximum or a minimum of the fixation probability in populations with an intermediate fraction of young individuals. This highlights the importance of life history and demographic structure in studying evolutionary dynamics. We also illustrate the fundamental differences between selection on reproduction and selection on survival when age structure is present. In addition, we evaluate the relative importance of size and structure of the population in determining the fixation probability of the mutant. Our work lays the foundation for also studying density- and frequency-dependent effects in populations when demographic structures cannot be neglected.

The fitness effects of a point mutation in Escherichia coli change with founding population density

Although intraspecific competition plays a seminal role in organismal evolution, little is known about the fitness effects of mutations at different population densities. We identified a point mutation in the cyclic AMP receptor protein (CRP) gene in Escherichia coli that confers significantly higher fitness than the wildtype at low founding population density, but significantly lower fitness at high founding density. Because CRP is a transcription factor that regulates the expression of nearly 500 genes, we compared global gene expression profiles of the mutant and wildtype strains. This mutation (S63F) does not affect expression of crp itself, but it does significantly affect expression of 170 and 157 genes at high and low founding density, respectively. Interestingly, acid resistance genes, some of which are known to exhibit density-dependent effects in E. coli, were consistently differentially expressed at high but not low density. As such, these genes may play a key role in reducing the crp mutant's fitness at high density, although other differentially expressed genes almost certainly also contribute to the fluctuating fitness differences we observed. Whatever the causes, we suspect that many mutations may exhibit density-dependent fitness effects in natural populations, so the fate of new mutations may frequently depend on the effective population size when they originate.

Eco-evolutionary dynamics of social dilemmas

Social dilemmas are an integral part of social interactions. Cooperative actions, ranging from secreting extra-cellular products in microbial populations to donating blood in humans, are costly to the actor and hence create an incentive to shirk and avoid the costs. Nevertheless, cooperation is ubiquitous in nature. Both costs and benefits often depend non-linearly on the number and types of individuals involved-as captured by idioms such as 'too many cooks spoil the broth' where additional contributions are discounted, or 'two heads are better than one' where cooperators synergistically enhance the group benefit. Interaction group sizes may depend on the size of the population and hence on ecological processes. This results in feedback mechanisms between ecological and evolutionary processes, which jointly affect and determine the evolutionary trajectory. Only recently combined eco-evolutionary processes became experimentally tractable in microbial social dilemmas. Here we analyse the evolutionary dynamics of non-linear social dilemmas in settings where the population fluctuates in size and the environment changes over time. In particular, cooperation is often supported and maintained at high densities through ecological fluctuations. Moreover, we find that the combination of the two processes routinely reveals highly complex dynamics, which suggests common occurrence in nature.

SpectralTDF: transition densities of diffusion processes with time-varying selection parameters, mutation rates and effective population sizes

MOTIVATION

In the Wright-Fisher diffusion, the transition density function describes the time evolution of the population-wide frequency of an allele. This function has several practical applications in population genetics and computing it for biologically realistic scenarios with selection and demography is an important problem.

RESULTS

We develop an efficient method for finding a spectral representation of the transition density function for a general model where the effective population size, selection coefficients and mutation parameters vary over time in a piecewise constant manner.

AVAILABILITY AND IMPLEMENTATION

The method, called SpectralTDF, is available at https://sourceforge.net/projects/spectraltdf/

CONTACT

yss@berkeley.edu

SUPPLEMENTARY INFORMATION

Supplementary data are available at Bioinformatics online.

Connecting within-host dynamics to the rate of viral molecular evolution

Viruses evolve rapidly, providing a unique system for understanding the processes that influence rates of molecular evolution. Neutral theory posits that the evolutionary rate increases linearly with the mutation rate. The occurrence of deleterious mutations causes this relationship to break down at high mutation rates. Previous studies have identified this as an important phenomenon, particularly for RNA viruses which can mutate at rates near the extinction threshold. We propose that in addition to mutation dynamics, viral within-host dynamics can also affect the between-host evolutionary rate. We present an analytical model that predicts the neutral evolution rate for viruses as a function of both within-host parameters and deleterious mutations. To examine the effect of more detailed aspects of the virus life cycle, we also present a computational model that simulates acute virus evolution using target cell-limited dynamics. Using influenza A virus as a case study, we find that our simulation model can predict empirical rates of evolution better than a model lacking within-host details. The analytical model does not perform as well as the simulation model but shows how the within-host basic reproductive number influences evolutionary rates. These findings lend support to the idea that the mutation rate alone is not sufficient to predict the evolutionary rate in viruses, instead calling for improved models that account for viral within-host dynamics.

The distribution of fitness effects in an uncertain world

The distribution of fitness effects (DFE) among new mutations plays a critical role in adaptive evolution and the maintenance of genetic variation. Although fitness landscape models predict several key features of the DFE, most theory to date focuses on predictable environmental conditions, while ignoring stochastic environmental fluctuations that feature prominently in the ecology of many organisms. Here, we derive an extension of Fisher's geometric model that incorporates two common effects of environmental variation: (1) nonadaptive genotype-by-environment interactions (G × E), in which the phenotype of a given genotype varies across environmental contexts; and (2) random fluctuation of the fitness optimum, which generates fluctuating selection. We show that both factors cause a mismatch between the DFE within single generations and the distribution of geometric mean fitness effects (averaged over multiple generations) that governs long-term evolutionary change. Such mismatches permit strong evolutionary constraints-despite an abundance of beneficial fitness variation within single environmental contexts-and to conflicting DFE estimates from direct versus indirect inference methods. Finally, our results suggest an intriguing parallel between the genetics and ecology of evolutionary constraints, with environmental fluctuations and pleiotropy placing qualitatively similar limits on the availability of adaptive genetic variation.

Survival probability of beneficial mutations in bacterial batch culture

The survival of rare beneficial mutations can be extremely sensitive to the organism's life history and the trait affected by the mutation. Given the tremendous impact of bacteria in batch culture as a model system for the study of adaptation, it is important to understand the survival probability of beneficial mutations in these populations. Here we develop a life-history model for bacterial populations in batch culture and predict the survival of mutations that increase fitness through their effects on specific traits: lag time, fission time, viability, and the timing of stationary phase. We find that if beneficial mutations are present in the founding population at the beginning of culture growth, mutations that reduce the mortality of daughter cells are the most likely to survive drift. In contrast, of mutations that occur de novo during growth, those that delay the onset of stationary phase are the most likely to survive. Our model predicts that approximately fivefold population growth between bottlenecks will optimize the occurrence and survival of beneficial mutations of all four types. This prediction is relatively insensitive to other model parameters, such as the lag time, fission time, or mortality rate of the population. We further estimate that bottlenecks that are more severe than this optimal prediction substantially reduce the occurrence and survival of adaptive mutations.

Formal properties of the probability of fixation: identities, inequalities and approximations

The formula for the probability of fixation of a new mutation is widely used in theoretical population genetics and molecular evolution. Here we derive a series of identities, inequalities and approximations for the exact probability of fixation of a new mutation under the Moran process (equivalent results hold for the approximate probability of fixation under the Wright-Fisher process, after an appropriate change of variables). We show that the logarithm of the fixation probability has particularly simple behavior when the selection coefficient is measured as a difference of Malthusian fitnesses, and we exploit this simplicity to derive inequalities and approximations. We also present a comprehensive comparison of both existing and new approximations for the fixation probability, highlighting those approximations that induce a reversible Markov chain when used to describe the dynamics of evolution under weak mutation. To demonstrate the power of these results, we consider the classical problem of determining the total substitution rate across an ensemble of biallelic loci and prove that, at equilibrium, a strict majority of substitutions are due to drift rather than selection.

Characteristics of neutral and deleterious protein-coding variation among individuals and populations

Whole-genome and exome data sets continue to be produced at a frenetic pace, resulting in massively large catalogs of human genomic variation. However, a clear picture of the characteristics and patterns of neutral and deleterious variation within and between populations has yet to emerge, given that recent large-scale sequencing studies have often emphasized different aspects of the data and sometimes appear to have conflicting conclusions. Here, we comprehensively studied characteristics of protein-coding variation in high-coverage exome sequence data from 6,515 European American (EA) and African American (AA) individuals. We developed an unbiased approach to identify putatively deleterious variants and investigated patterns of neutral and deleterious single-nucleotide variants and alleles between individuals and populations. We show that there are substantial differences in the composition of genotypes between EA and AA populations and that small but statistically significant differences exist in the average number of deleterious alleles carried by EA and AA individuals. Furthermore, we performed extensive simulations to delineate the temporal dynamics of deleterious alleles for a broad range of demographic models and use these data to inform the interpretation of empirical patterns of deleterious variation. Finally, we illustrate that the effects of demographic perturbations, such as bottlenecks and expansions, often manifest in opposing patterns of neutral and deleterious variation depending on whether the focus is on populations or individuals. Our results clarify seemingly disparate empirical characteristics of protein-coding variation and provide substantial insights into how natural selection and demographic history have patterned neutral and deleterious variation within and between populations.

Soft selective sweeps in complex demographic scenarios

Adaptation from de novo mutation can produce so-called soft selective sweeps, where adaptive alleles of independent mutational origin sweep through the population at the same time. Population genetic theory predicts that such soft sweeps should be likely if the product of the population size and the mutation rate toward the adaptive allele is sufficiently large, such that multiple adaptive mutations can establish before one has reached fixation; however, it remains unclear how demographic processes affect the probability of observing soft sweeps. Here we extend the theory of soft selective sweeps to realistic demographic scenarios that allow for changes in population size over time. We first show that population bottlenecks can lead to the removal of all but one adaptive lineage from an initially soft selective sweep. The parameter regime under which such "hardening" of soft selective sweeps is likely is determined by a simple heuristic condition. We further develop a generalized analytical framework, based on an extension of the coalescent process, for calculating the probability of soft sweeps under arbitrary demographic scenarios. Two important limits emerge within this analytical framework: In the limit where population-size fluctuations are fast compared to the duration of the sweep, the likelihood of soft sweeps is determined by the harmonic mean of the variance effective population size estimated over the duration of the sweep; in the opposing slow fluctuation limit, the likelihood of soft sweeps is determined by the instantaneous variance effective population size at the onset of the sweep. We show that as a consequence of this finding the probability of observing soft sweeps becomes a function of the strength of selection. Specifically, in species with sharply fluctuating population size, strong selection is more likely to produce soft sweeps than weak selection. Our results highlight the importance of accurate demographic estimates over short evolutionary timescales for understanding the population genetics of adaptation from de novo mutation.

Variation in the fitness effects of mutations with population density and size in Escherichia coli

The fitness effects of mutations are context specific and depend on both external (e.g., environment) and internal (e.g., cellular stress, genetic background) factors. The influence of population size and density on fitness effects are unknown, despite the central role population size plays in the supply and fixation of mutations. We addressed this issue by comparing the fitness of 92 Keio strains (Escherichia coli K12 single gene knockouts) at comparatively high (1.2×10(7) CFUs/mL) and low (2.5×10(2) CFUs/mL) densities, which also differed in population size (high: 1.2×10(8); low: 1.25×10(3)). Twenty-eight gene deletions (30%) exhibited a fitness difference, ranging from 5 to 174% (median: 35%), between the high and low densities. Our analyses suggest this variation among gene deletions in fitness responses reflected in part both gene orientation and function, of the gene properties we examined (genomic position, length, orientation, and function). Although we could not determine the relative effects of population density and size, our results suggest fitness effects of mutations vary with these two factors, and this variation is gene-specific. Besides being a mechanism for density-dependent selection (r-K selection), the dependence of fitness effects on population density and size has implications for any population that varies in size over time, including populations undergoing evolutionary rescue, species invasions into novel habitats, and cancer progression and metastasis. Further, combined with recent advances in understanding the roles of other context-specific factors in the fitness effects of mutations, our results will help address theoretical and applied biological questions more realistically.

An experimental test on the probability of extinction of new genetic variants

In 1927, J.B.S. Haldane reasoned that the probability of fixation of new beneficial alleles is twice their fitness effect. This result, later generalized by M. Kimura, has since become the cornerstone of modern population genetics. There is no experimental test of Haldane's insight that new beneficial alleles are lost with high probability. Here we demonstrate that extinction rates decrease with increasing initial numbers of beneficial alleles, as expected, by performing invasion experiments with inbred lines of the nematode Caenorhabditis elegans. We further show that the extinction rates of deleterious alleles are higher than those of beneficial alleles, also as expected. Interestingly, we also find that for these inbred lines, when at intermediate frequencies, the fate of invaders might not result in their ultimate fixation or loss but on their maintenance. Our study confirms the key results from classical population genetics and highlights that the nature of adaptation can be complex.

Fluctuating selection models and McDonald-Kreitman type analyses

It is likely that the strength of selection acting upon a mutation varies through time due to changes in the environment. However, most population genetic theory assumes that the strength of selection remains constant. Here we investigate the consequences of fluctuating selection pressures on the quantification of adaptive evolution using McDonald-Kreitman (MK) style approaches. In agreement with previous work, we show that fluctuating selection can generate evidence of adaptive evolution even when the expected strength of selection on a mutation is zero. However, we also find that the mutations, which contribute to both polymorphism and divergence tend, on average, to be positively selected during their lifetime, under fluctuating selection models. This is because mutations that fluctuate, by chance, to positive selected values, tend to reach higher frequencies in the population than those that fluctuate towards negative values. Hence the evidence of positive adaptive evolution detected under a fluctuating selection model by MK type approaches is genuine since fixed mutations tend to be advantageous on average during their lifetime. Never-the-less we show that methods tend to underestimate the rate of adaptive evolution when selection fluctuates.

The characteristic trajectory of a fixing allele: a consequence of fictitious selection that arises from conditioning

This work is concerned with the historical progression, to fixation, of an allele in a finite population. This progression is characterized by the average frequency trajectory of alleles that achieve fixation before a given time, T. Under a diffusion analysis, the average trajectory, conditional on fixation by time T, is shown to be equivalent to the average trajectory in an unconditioned problem involving additional selection. We call this additional selection "fictitious selection"; it plays the role of a selective force in the unconditioned problem but does not exist in reality. It is a consequence of conditioning on fixation. The fictitious selection is frequency dependent and can be very large compared with any real selection that is acting. We derive an approximation for the characteristic trajectory of a fixing allele, when subject to real additive selection, from an unconditioned problem, where the total selection is a combination of real and fictitious selection. Trying to reproduce the characteristic trajectory from the action of additive selection, in an infinite population, can lead to estimates of the strength of the selection that deviate from the real selection by >1000% or have the opposite sign. Strong evolutionary forces may be invoked in problems where conditioning has been carried out, but these forces may largely be an outcome of the conditioning and hence may not have a real existence. The work presented here clarifies these issues and provides two useful tools for future analyses: the characteristic trajectory of a fixing allele and the force that primarily drives this, namely fictitious selection. These should prove useful in a number of areas of interest including coalescence with selection, experimental evolution, time series analyses of ancient DNA, game theory in finite populations, and the historical dynamics of selected alleles in wild populations.

Population size and the rate of evolution

Does evolution proceed faster in larger or smaller populations? The relationship between effective population size (Ne) and the rate of evolution has consequences for our ability to understand and interpret genomic variation, and is central to many aspects of evolution and ecology. Many factors affect the relationship between Ne and the rate of evolution, and recent theoretical and empirical studies have shown some surprising and sometimes counterintuitive results. Some mechanisms tend to make the relationship positive, others negative, and they can act simultaneously. The relationship also depends on whether one is interested in the rate of neutral, adaptive, or deleterious evolution. Here, we synthesize theoretical and empirical approaches to understanding the relationship and highlight areas that remain poorly understood.

Population growth inflates the per-individual number of deleterious mutations and reduces their mean effect

This study addresses the question of how purifying selection operates during recent rapid population growth such as has been experienced by human populations. This is not a straightforward problem because the human population is not at equilibrium: population genetics predicts that, on the one hand, the efficacy of natural selection increases as population size increases, eliminating ever more weakly deleterious variants; on the other hand, a larger number of deleterious mutations will be introduced into the population and will be more likely to increase in their number of copies as the population grows. To understand how patterns of human genetic variation have been shaped by the interaction of natural selection and population growth, we examined the trajectories of mutations with varying selection coefficients, using computer simulations. We observed that while population growth dramatically increases the number of deleterious segregating sites in the population, it only mildly increases the number carried by each individual. Our simulations also show an increased efficacy of natural selection, reflected in a higher fraction of deleterious mutations eliminated at each generation and a more efficient elimination of the most deleterious ones. As a consequence, while each individual carries a larger number of deleterious alleles than expected in the absence of growth, the average selection coefficient of each segregating allele is less deleterious. Combined, our results suggest that the genetic risk of complex diseases in growing populations might be distributed across a larger number of more weakly deleterious rare variants.

Complete numerical solution of the diffusion equation of random genetic drift

A numerical method is presented to solve the diffusion equation for the random genetic drift that occurs at a single unlinked locus with two alleles. The method was designed to conserve probability, and the resulting numerical solution represents a probability distribution whose total probability is unity. We describe solutions of the diffusion equation whose total probability is unity as complete. Thus the numerical method introduced in this work produces complete solutions, and such solutions have the property that whenever fixation and loss can occur, they are automatically included within the solution. This feature demonstrates that the diffusion approximation can describe not only internal allele frequencies, but also the boundary frequencies zero and one. The numerical approach presented here constitutes a single inclusive framework from which to perform calculations for random genetic drift. It has a straightforward implementation, allowing it to be applied to a wide variety of problems, including those with time-dependent parameters, such as changing population sizes. As tests and illustrations of the numerical method, it is used to determine: (i) the probability density and time-dependent probability of fixation for a neutral locus in a population of constant size; (ii) the probability of fixation in the presence of selection; and (iii) the probability of fixation in the presence of selection and demographic change, the latter in the form of a changing population size.

Evolutionary rescue by beneficial mutations in environments that change in space and time

A factor that may limit the ability of many populations to adapt to changing conditions is the rate at which beneficial mutations can become established. We study the probability that mutations become established in changing environments by extending the classic theory for branching processes. When environments change in time, under quite general conditions, the establishment probability is approximately twice the 'effective selection coefficient', whose value is an average that gives most weight to a mutant's fitness in the generations immediately after it appears. When fitness varies along a gradient in a continuous habitat, increased dispersal generally decreases the chance a mutation establishes because mutations move out of areas where they are most adapted. When there is a patch of favourable habitat that moves in time, there is a maximum speed of movement above which mutations cannot become established, regardless of when and where they first appear. This critical speed limit, which is proportional to the mutation's maximum selective advantage, represents an absolute constraint on the potential of locally adapted mutations to contribute to evolutionary rescue.

Establishment of new mutations in changing environments

Understanding adaptation in changing environments is an important topic in evolutionary genetics, especially in the light of climatic and environmental change. In this work, we study one of the most fundamental aspects of the genetics of adaptation in changing environments: the establishment of new beneficial mutations. We use the framework of time-dependent branching processes to derive simple approximations for the establishment probability of new mutations assuming that temporal changes in the offspring distribution are small. This approach allows us to generalize Haldane's classic result for the fixation probability in a constant environment to arbitrary patterns of temporal change in selection coefficients. Under weak selection, the only aspect of temporal variation that enters the probability of establishment is a weighted average of selection coefficients. These weights quantify how much earlier generations contribute to determining the establishment probability compared to later generations. We apply our results to several biologically interesting cases such as selection coefficients that change in consistent, periodic, and random ways and to changing population sizes. Comparison with exact results shows that the approximation is very accurate.

Population growth enhances the mean fixation time of neutral mutations and the persistence of neutral variation

A fundamental result of population genetics states that a new mutation, at an unlinked neutral locus in a randomly mating diploid population, has a mean time of fixation of ∼4N(e) generations, where N(e) is the effective population size. This result is based on an assumption of fixed population size, which does not universally hold in natural populations. Here, we analyze such neutral fixations in populations of changing size within the framework of the diffusion approximation. General expressions are derived for the mean and variance of the fixation time in changing populations. Some explicit results are given for two cases: (i) the effective population size undergoes a sudden change, representing a sudden population expansion or a sudden bottleneck; (ii) the effective population changes linearly for a limited period of time and then remains constant. Additionally, a lower bound for the mean time of fixation is obtained for an effective population size that increases with time, and this is applied to exponentially growing populations. The results obtained in this work show, among other things, that for populations that increase in size, the mean time of fixation can be enhanced, sometimes substantially so, over 4N(e,0) generations, where N(e,0) is the effective population size at the time the mutation arises. Such an enhancement is associated with (i) an increased probability of neutral polymorphism in a population and (ii) an enhanced persistence of high-frequency neutral variation, which is the variation most likely to be observed.

On the fixation process of a beneficial mutation in a variable environment

A population that adapts to gradual environmental change will typically experience temporal variation in its population size and the selection pressure. On the basis of the mathematical theory of inhomogeneous branching processes, we present a framework to describe the fixation process of a single beneficial allele under these conditions. The approach allows for arbitrary time-dependence of the selection coefficient s(t) and the population size N(t), as may result from an underlying ecological model. We derive compact analytical approximations for the fixation probability and the distribution of passage times for the beneficial allele to reach a given intermediate frequency. We apply the formalism to several biologically relevant scenarios, such as linear or cyclic changes in the selection coefficient, and logistic population growth. Comparison with computer simulations shows that the analytical results are accurate for a large parameter range, as long as selection is not very weak.