Fluctuations of fitness distributions and the rate of Muller's ratchet

Reversions mask the contribution of adaptive evolution in microbiomes

When examining bacterial genomes for evidence of past selection, the results depend heavily on the mutational distance between chosen genomes. Even within a bacterial species, genomes separated by larger mutational distances exhibit stronger evidence of purifying selection as assessed by dN/dS, the normalized ratio of nonsynonymous to synonymous mutations. Here, we show that the classical interpretation of this scale dependence, weak purifying selection, leads to problematic mutation accumulation when applied to available gut microbiome data. We propose an alternative, adaptive reversion model with opposite implications for dynamical intuition and applications of dN/dS. Reversions that occur and sweep within-host populations are nearly guaranteed in microbiomes due to large population sizes, short generation times, and variable environments. Using analytical and simulation approaches, we show that adaptive reversion can explain the dN/dS decay given only dozens of locally fluctuating selective pressures, which is realistic in the context of Bacteroides genomes. The success of the adaptive reversion model argues for interpreting low values of dN/dS obtained from long timescales with caution as they may emerge even when adaptive sweeps are frequent. Our work thus inverts the interpretation of an old observation in bacterial evolution, illustrates the potential of mutational reversions to shape genomic landscapes over time, and highlights the importance of studying bacterial genomic evolution on short timescales.

Muller's ratchet in a near-critical regime: Tournament versus fitness proportional selection

Muller's ratchet, in its prototype version, models a haploid, asexual population whose size N is constant over the generations. Slightly deleterious mutations are acquired along the lineages at a constant rate, and individuals carrying less mutations have a selective advantage. The classical variant considers fitness proportional selection, but other fitness schemes are conceivable as well. Inspired by the work of Etheridge et al. (2009) we propose a parameter scaling which fits well to the "near-critical" regime that was in the focus of Etheridge et al. (2009) (and in which the mutation-selection ratio diverges logarithmically as N→∞). Using a Moran model, we investigate the"rule of thumb" given in Etheridge et al. (2009) for the click rate of the "classical ratchet" by putting it into the context of new results on the long-time evolution of the size of the best class of the ratchet with (binary) tournament selection. This variant of Muller's ratchet was introduced in González Casanova et al. (2023), and was analysed there in a subcritical parameter regime. Other than that of the classical ratchet, the size of the best class of the tournament ratchet follows an autonomous dynamics up to the time of its extinction. It turns out that, under a suitable correspondence of the model parameters, this dynamics coincides with the so called Poisson profile approximation of the dynamics of the best class of the classical ratchet.

Reversions mask the contribution of adaptive evolution in microbiomes

When examining bacterial genomes for evidence of past selection, the results obtained depend heavily on the mutational distance between chosen genomes. Even within a bacterial species, genomes separated by larger mutational distances exhibit stronger evidence of purifying selection as assessed byd N / d S , the normalized ratio of nonsynonymous to synonymous mutations. Here, we show that the classical interpretation of this scale-dependence, weak purifying selection, leads to problematic mutation accumulation when applied to available gut microbiome data. We propose an alternative, adaptive reversion model with exactly opposite implications for dynamical intuition and applications ofd N / d S . Reversions that occur and sweep within-host populations are nearly guaranteed in microbiomes due to large population sizes, short generation times, and variable environments. Using analytical and simulation approaches, we show that adaptive reversion can explain thed N / d S decay given only dozens of locally-fluctuating selective pressures, which is realistic in the context of Bacteroides genomes. The success of the adaptive reversion model argues for interpreting low values ofd N / d S obtained from long-time scales with caution, as they may emerge even when adaptive sweeps are frequent. Our work thus inverts the interpretation of an old observation in bacterial evolution, illustrates the potential of mutational reversions to shape genomic landscapes over time, and highlights the importance of studying bacterial genomic evolution on short time scales.

A Path Integral Approach for Allele Frequency Dynamics Under Polygenic Selection

Many phenotypic traits have a polygenic genetic basis, making it challenging to learn their genetic architectures and predict individual phenotypes. One promising avenue to resolve the genetic basis of complex traits is through evolve-and-resequence experiments, in which laboratory populations are exposed to some selective pressure and trait-contributing loci are identified by extreme frequency changes over the course of the experiment. However, small laboratory populations will experience substantial random genetic drift, and it is difficult to determine whether selection played a roll in a given allele frequency change. Predicting how much allele frequencies change under drift and selection had remained an open problem well into the 21st century, even those contributing to simple, monogenic traits. Recently, there have been efforts to apply the path integral, a method borrowed from physics, to solve this problem. So far, this approach has been limited to genic selection, and is therefore inadequate to capture the complexity of quantitative, highly polygenic traits that are commonly studied. Here we extend one of these path integral methods, the perturbation approximation, to selection scenarios that are of interest to quantitative genetics. In particular, we derive analytic expressions for the transition probability (i.e., the probability that an allele will change in frequency from x , to y in time t ) of an allele contributing to a trait subject to stabilizing selection, as well as that of an allele contributing to a trait rapidly adapting to a new phenotypic optimum. We use these expressions to characterize the use of allele frequency change to test for selection, as well as explore optimal design choices for evolve-and-resequence experiments to uncover the genetic architecture of polygenic traits under selection.

A quantitative genetic model of background selection in humans

Across the human genome, there are large-scale fluctuations in genetic diversity caused by the indirect effects of selection. This "linked selection signal" reflects the impact of selection according to the physical placement of functional regions and recombination rates along chromosomes. Previous work has shown that purifying selection acting against the steady influx of new deleterious mutations at functional portions of the genome shapes patterns of genomic variation. To date, statistical efforts to estimate purifying selection parameters from linked selection models have relied on classic Background Selection theory, which is only applicable when new mutations are so deleterious that they cannot fix in the population. Here, we develop a statistical method based on a quantitative genetics view of linked selection, that models how polygenic additive fitness variance distributed along the genome increases the rate of stochastic allele frequency change. By jointly predicting the equilibrium fitness variance and substitution rate due to both strong and weakly deleterious mutations, we estimate the distribution of fitness effects (DFE) and mutation rate across three geographically distinct human samples. While our model can accommodate weaker selection, we find evidence of strong selection operating similarly across all human samples. Although our quantitative genetic model of linked selection fits better than previous models, substitution rates of the most constrained sites disagree with observed divergence levels. We find that a model incorporating selective interference better predicts observed divergence in conserved regions, but overall our results suggest uncertainty remains about the processes generating fitness variation in humans.

A dynamical limit to evolutionary adaptation

Natural selection makes evolutionary adaptation possible even if the overwhelming majority of new mutations are deleterious. However, in rapidly evolving populations where numerous linked mutations occur and segregate simultaneously, clonal interference and genetic hitchhiking can limit the efficiency of selection, allowing deleterious mutations to accumulate over time. This can in principle overwhelm the fitness increases provided by beneficial mutations, leading to an overall fitness decline. Here, we analyze the conditions under which evolution will tend to drive populations to higher versus lower fitness. Our analysis focuses on quantifying the boundary between these two regimes, as a function of parameters such as population size, mutation rates, and selection pressures. This boundary represents a state in which adaptation is precisely balanced by Muller's ratchet, and we show that it can be characterized by rapid molecular evolution without any net fitness change. Finally, we consider the implications of global fitness-mediated epistasis and find that under some circumstances, this can drive populations toward the boundary state, which can thus represent a long-term evolutionary attractor.

Mutational meltdown in asexual populations doomed to extinction

Asexual populations are expected to accumulate deleterious mutations through a process known as Muller's ratchet. Lynch and colleagues proposed that the ratchet eventually results in a vicious cycle of mutation accumulation and population decline that drives populations to extinction. They called this phenomenon mutational meltdown. Here, we analyze mutational meltdown using a multi-type branching process model where, in the presence of mutation, populations are doomed to extinction. We analyse the change in size and composition of the population and the time of extinction under this model.

Stem cell mutations, associated cancer risk, and consequences for regenerative medicine

Mutation accumulation in stem cells has been associated with cancer risk. However, the presence of numerous mutant clones in healthy tissues has raised the question of what limits cancer initiation. Here, we review recent developments in characterizing mutation accumulation in healthy tissues and compare mutation rates in stem cells during development and adult life with corresponding cancer risk. A certain level of mutagenesis within the stem cell pool might be beneficial to limit the size of malignant clones through competition. This knowledge impacts our understanding of carcinogenesis with potential consequences for the use of stem cells in regenerative medicine.

Universality of evolutionary trajectories under arbitrary forms of self-limitation and competition

The assumption of constant population size is central in population genetics. It led to a large body of results that is robust to modeling choices and that has proven successful to understand evolutionary dynamics. In reality, allele frequencies and population size are both determined by the interaction between a population and the environment. Relaxing the constant-population assumption has two big drawbacks. It increases the technical difficulty of the analysis, and it requires specifying a mechanism for the saturation of the population size, possibly making the results contingent on model details. Here we develop a framework that encompasses a great variety of systems with an arbitrary mechanism for population growth limitation. By using techniques based on scale separation for stochastic processes, we are able to calculate analytically properties of evolutionary trajectories, such as the fixation probability. Remarkably, these properties assume a universal form with respect to our framework, which depends on only three parameters related to the intergeneration timescale, the invasion fitness, and the carrying capacity of the strains. In other words, different systems, such as Lotka-Volterra or a chemostat model (contained in our framework), share the same evolutionary outcomes after a proper remapping of their parameters. An important and surprising consequence of our results is that the direction of selection can be inverted, with a population evolving to reach lower values of invasion fitness.

A dynamical limit to evolutionary adaptation

Natural selection makes evolutionary adaptation possible even if the overwhelming majority of new mutations are deleterious. However, in rapidly evolving populations where numerous linked mutations occur and segregate simultaneously, clonal interference and genetic hitchhiking can limit the efficiency of selection, allowing deleterious mutations to accumulate over time. This can in principle overwhelm the fitness increases provided by beneficial mutations, leading to an overall fitness decline. Here, we analyze the conditions under which evolution will tend to drive populations to higher versus lower fitness. Our analysis focuses on quantifying the boundary between these two regimes, as a function of parameters such as population size, mutation rates, and selection pressures. This boundary represents a state in which adaptation is precisely balanced by Muller's ratchet, and we show that it can be characterized by rapid molecular evolution without any net fitness change. Finally, we consider the implications of global fitness-mediated epistasis, and find that under some circumstances this can drive populations towards the boundary state, which can thus represent a long-term evolutionary attractor.

Statistical genetics in and out of quasi-linkage equilibrium

This review is about statistical genetics, an interdisciplinary topic between statistical physics and population biology. The focus is on the phase ofquasi-linkage equilibrium(QLE). Our goals here are to clarify under which conditions the QLE phase can be expected to hold in population biology and how the stability of the QLE phase is lost. The QLE state, which has many similarities to a thermal equilibrium state in statistical mechanics, was discovered by M Kimura for a two-locus two-allele model, and was extended and generalized to the global genome scale byNeher&Shraiman (2011). What we will refer to as the Kimura-Neher-Shraiman theory describes a population evolving due to the mutations, recombination, natural selection and possibly genetic drift. A QLE phase exists at sufficiently high recombination rate (r) and/or mutation ratesµwith respect to selection strength. We show how in QLE it is possible to infer the epistatic parameters of the fitness function from the knowledge of the (dynamical) distribution of genotypes in a population. We further consider the breakdown of the QLE regime for high enough selection strength. We review recent results for the selection-mutation and selection-recombination dynamics. Finally, we identify and characterize a new phase which we call the non-random coexistence where variability persists in the population without either fixating or disappearing.

A field theoretic approach to non-equilibrium population genetics in the strong selection regime

Natural populations are virtually never observed in equilibrium, yet equilibrium approximations comprise the majority of our understanding of population genetics. Using standard tools from statistical physics, a formalism is presented that re-expresses the stochastic equations describing allelic evolution as a partition functional over all possible allelic trajectories ('paths') governed by selection, mutation, and drift. A perturbative field theory is developed for strong additive selection, relevant to disease variation, that facilitates the straightforward computation of closed-form approximations for time-dependent moments of the allele frequency distribution across a wide range of non-equilibrium scenarios; examples are presented for constant population size, exponential growth, bottlenecks, and oscillatory size, all of which align well to simulations and break down just above the drift barrier. Equilibration times are computed and, even for static population size, generically extend beyond the order 1/s timescale associated with exponential frequency decay. Though the mutation load is largely robust to variable population size, perturbative drift-based corrections to the deterministic trajectory are readily computed. Under strong selection, the variance of a new mutation's frequency (related to homozygosity) is dominated by drift-driven dynamics and a transient increase in variance often occurs prior to equilibrating. The excess kurtosis over skew squared is roughly constant (i.e., independent of selection, provided 2Ns ≳ 5) for static population size, and thus potentially sensitive to deviation from equilibrium. These insights highlight the value of such closed-form approximations, naturally generated from Feynman diagrams in a perturbative field theory, which can simply and accurately capture the parameter dependences describing a variety of non-equilibrium population genetic phenomena of interest.

Most cancers carry a substantial deleterious load due to Hill-Robertson interference

Cancer genomes exhibit surprisingly weak signatures of negative selection^1,2. This may be because selective pressures are relaxed or because genome-wide linkage prevents deleterious mutations from being removed (Hill-Robertson interference)³. By stratifying tumors by their genome-wide mutational burden, we observe negative selection (dN/dS ~ 0.56) in low mutational burden tumors, while remaining cancers exhibit dN/dS ratios ~1. This suggests that most tumors do not remove deleterious passengers. To buffer against deleterious passengers, tumors upregulate heat shock pathways as their mutational burden increases. Finally, evolutionary modeling finds that Hill-Robertson interference alone can reproduce patterns of attenuated selection and estimates the total fitness cost of passengers to be 46% per cell on average. Collectively, our findings suggest that the lack of observed negative selection in most tumors is not due to relaxed selective pressures, but rather the inability of selection to remove deleterious mutations in the presence of genome-wide linkage.

Diffusion approximations in population genetics and the rate of Muller's ratchet

The Wright-Fisher binomial model of allele frequency change is often approximated by a scaling limit in which selection, mutation and drift all decrease at the same 1/N rate. This construction restricts the applicability of the resulting 'Wright-Fisher diffusion equation' to the weak selection, weak mutation regime of evolution. We argue that diffusion approximations of the Wright-Fisher model can be used more generally, for instance in cases where genetic drift is much weaker than selection. One important example of this regime is Muller's ratchet phenomenon, whereby deleterious mutations slowly but irreversibly accumulate through rare stochastic fluctuations. Using a modified diffusion equation we derive improved analytical estimates for the mean click time of the ratchet.

Sheltering of deleterious mutations explains the stepwise extension of recombination suppression on sex chromosomes and other supergenes

Many organisms have sex chromosomes with large nonrecombining regions that have expanded stepwise, generating "evolutionary strata" of differentiation. The reasons for this remain poorly understood, but the principal hypotheses proposed to date are based on antagonistic selection due to differences between sexes. However, it has proved difficult to obtain empirical evidence of a role for sexually antagonistic selection in extending recombination suppression, and antagonistic selection has been shown to be unlikely to account for the evolutionary strata observed on fungal mating-type chromosomes. We show here, by mathematical modeling and stochastic simulation, that recombination suppression on sex chromosomes and around supergenes can expand under a wide range of parameter values simply because it shelters recessive deleterious mutations, which are ubiquitous in genomes. Permanently heterozygous alleles, such as the male-determining allele in XY systems, protect linked chromosomal inversions against the expression of their recessive mutation load, leading to the successive accumulation of inversions around these alleles without antagonistic selection. Similar results were obtained with models assuming recombination-suppressing mechanisms other than chromosomal inversions and for supergenes other than sex chromosomes, including those without XY-like asymmetry, such as fungal mating-type chromosomes. However, inversions capturing a permanently heterozygous allele were found to be less likely to spread when the mutation load segregating in populations was lower (e.g., under large effective population sizes or low mutation rates). This may explain why sex chromosomes remain homomorphic in some organisms but are highly divergent in others. Here, we model a simple and testable hypothesis explaining the stepwise extensions of recombination suppression on sex chromosomes, mating-type chromosomes, and supergenes in general.

Discussions of the "Not So Fit": How Ableism Limits Diverse Thought and Investigative Potential in Evolutionary Biology

AbstractEvolutionary biology and many of its foundational concepts are grounded in a history of ableism and eugenics. The field has not made a concerted effort to divest our concepts and investigative tools from this fraught history, and as a result, an ableist investigative lens has persisted in present-day evolutionary research, limiting the scope of research and harming the ability to communicate and synthesize knowledge about evolutionary processes. This failure to divest from our eugenicist and ableist history has harmed progress in evolutionary biology and allowed principles from evolutionary biology to continue to be weaponized against marginalized communities in the modern day. To rectify this problem, scholars in evolutionary research must come to terms with how the history of the field has influenced their investigations and work to establish a new framework for defining and investigating concepts such as selection and fitness.

Population genetics of polymorphism and divergence in rapidly evolving populations

In rapidly evolving populations, numerous beneficial and deleterious mutations can arise and segregate within a population at the same time. In this regime, evolutionary dynamics cannot be analyzed using traditional population genetic approaches that assume that sites evolve independently. Instead, the dynamics of many loci must be analyzed simultaneously. Recent work has made progress by first analyzing the fitness variation within a population, and then studying how individual lineages interact with this traveling fitness wave. However, these "traveling wave" models have previously been restricted to extreme cases where selection on individual mutations is either much faster or much slower than the typical coalescent timescale Tc. In this work, we show how the traveling wave framework can be extended to intermediate regimes in which the scaled fitness effects of mutations (Tcs) are neither large nor small compared to one. This enables us to describe the dynamics of populations subject to a wide range of fitness effects, and in particular, in cases where it is not immediately clear which mutations are most important in shaping the dynamics and statistics of genetic diversity. We use this approach to derive new expressions for the fixation probabilities and site frequency spectra of mutations as a function of their scaled fitness effects, along with related results for the coalescent timescale Tc and the rate of adaptation or Muller's ratchet. We find that competition between linked mutations can have a dramatic impact on the proportions of neutral and selected polymorphisms, which is not simply summarized by the scaled selection coefficient Tcs. We conclude by discussing the implications of these results for population genetic inferences.

Simulating the Dynamic Intra-Tumor Heterogeneity and Therapeutic Responses

A tumor is a complex tissue comprised of heterogeneous cell subpopulations which exhibit substantial diversity at morphological, genetic and epigenetic levels. Under the selective pressure of cancer therapies, a minor treatment-resistant subpopulation could survive and repopulate. Therefore, the intra-tumor heterogeneity is recognized as a major obstacle to effective treatment. In this paper, we propose a stochastic clonal expansion model to simulate the dynamic evolution of tumor subpopulations and the therapeutic effect at different times during tumor progression. The model is incorporated in the CES webserver, for the convenience of simulation according to initial user input. Based on this model, we investigate the influence of various factors on tumor progression and treatment consequences and present conclusions drawn from observations, highlighting the importance of treatment timing. The model provides an intuitive illustration to deepen the understanding of temporal intra-tumor heterogeneity dynamics and treatment responses, thus helping the improvement of personalized diagnostic and therapeutic strategies.

Comparative Population Genetics in the Human Gut Microbiome

Genetic variation in the human gut microbiome is responsible for conferring a number of crucial phenotypes like the ability to digest food and metabolize drugs. Yet, our understanding of how this variation arises and is maintained remains relatively poor. Thus, the microbiome remains a largely untapped resource, as the large number of coexisting species in the microbiome presents a unique opportunity to compare and contrast evolutionary processes across species to identify universal trends and deviations. Here we outline features of the human gut microbiome that, while not unique in isolation, as an assemblage make it a system with unparalleled potential for comparative population genomics studies. We consciously take a broad view of comparative population genetics, emphasizing how sampling a large number of species allows researchers to identify universal evolutionary dynamics in addition to new genes, which can then be leveraged to identify exceptional species that deviate from general patterns. To highlight the potential power of comparative population genetics in the microbiome, we reanalyze patterns of purifying selection across ∼40 prevalent species in the human gut microbiome to identify intriguing trends which highlight functional categories in the microbiome that may be under more or less constraint.

Muller's ratchet of the Y chromosome with gene conversion

Muller's ratchet is a process in which deleterious mutations are fixed irreversibly in the absence of recombination. The degeneration of the Y chromosome, and the gradual loss of its genes, can be explained by Muller's ratchet. However, most theories consider single-copy genes, and may not be applicable to Y chromosomes, which have a number of duplicated genes in many species, which are probably undergoing concerted evolution by gene conversion. We developed a model of Muller's ratchet to explore the evolution of the Y chromosome. The model assumes a nonrecombining chromosome with both single-copy and duplicated genes. We used analytical and simulation approaches to obtain the rate of gene loss in this model, with special attention to the role of gene conversion. Homogenization by gene conversion makes both duplicated copies either mutated or intact. The former promotes the ratchet, and the latter retards, and we ask which of these counteracting forces dominates under which conditions. We found that the effect of gene conversion is complex, and depends upon the fitness effect of gene duplication. When duplication has no effect on fitness, gene conversion accelerates the ratchet of both single-copy and duplicated genes. If duplication has an additive fitness effect, the ratchet of single-copy genes is accelerated by gene duplication, regardless of the gene conversion rate, whereas gene conversion slows the degeneration of duplicated genes. Our results suggest that the evolution of the Y chromosome involves several parameters, including the fitness effect of gene duplication by increasing dosage and gene conversion rate.

Genetic and epigenetic Muller's ratchet as a mechanism of frailty and morbidity during aging: a demographic genetic model

Mutation accumulation has been proposed as a cause of senescence. During this process, age-related genetic and epigenetic mutations steadily accumulate. Cascading deleterious effects of mutations might initiate a steady "accumulation of deficits" in cells, despite the existence of repair mechanisms, leading to cellular senescence and functional decline of tissues and organs, which ultimately manifest as frailty and disease. Here, we investigate several of these aspects in differentiating cell populations through modeling and simulation using the Moran birth-death (demographic) process, under several scenarios of mutation accumulation. Deleterious mutations seem to rapidly accumulate particularly early in the course of life, during which the rate of cell division is high, thereby exerting a greater effect on subsequent cellular senescence. Our results are compatible with the principle of the Muller's ratchet taking place in asexually reproducing organisms. The ratchet speed in a given tissue depends on the size of the cell population, mutation rate and the impact of such mutations on cell phenotypes. It varies substantially among cells in different tissues and organs due to heterogeneity in relation to cell and organ-specific demographic features. Ratchet accelerates particularly after middle age, resulting in a synergistic fitness decay at the level of cell populations. We extend Fisher's average excess concept and rank order scale to interpret differential phenotypic effects of the increase of the mutation load among cell populations within a given tissue. We postulate that classical evolutionary genetic models can explain, at least in part, the origins of frailty, subclinical conditions, morbidity and the health consequences of senescence.

A little bit of sex prevents mutation accumulation even in apomictic polyploid plants

Background

In the absence of sex and recombination, genomes are expected to accumulate deleterious mutations via an irreversible process known as Muller’s ratchet, especially in the case of polyploidy. In contrast, no genome-wide mutation accumulation was detected in a transcriptome of facultative apomictic, hexaploid plants of the Ranunculus auricomus complex. We hypothesize that mutations cannot accumulate in flowering plants with facultative sexuality because sexual and asexual development concurrently occurs within the same generation. We assume a strong effect of purging selection on reduced gametophytes in the sexual developmental pathway because previously masked recessive deleterious mutations would be exposed to selection.

Results

We test this hypothesis by modeling mutation elimination using apomictic hexaploid plants of the R. auricomus complex. To estimate mean recombination rates, the mean number of recombinants per generation was calculated by genotyping three F1 progeny arrays with six microsatellite markers and character incompatibility analyses. We estimated the strength of purging selection in gametophytes by calculating abortion rates of sexual versus apomictic development at the female gametophyte, seed and offspring stage. Accordingly, we applied three selection coefficients by considering effects of purging selection against mutations on (1) male and female gametophytes in the sexual pathway (additive, s = 1.000), (2) female gametophytes only (s = 0.520), and (3) on adult plants only (sporophytes, s = 0.212). We implemented recombination rates into a mathematical model considering the three different selection coefficients, and a genomic mutation rate calculated from genome size of our plants and plant-specific mutation rates. We revealed a mean of 6.05% recombinants per generation. This recombination rate eliminates mutations after 138, 204 or 246 generations, depending on the respective selection coefficients (s = 1.000, 0.520, and 0.212).

Conclusions

Our results confirm that the empirically observed frequencies of facultative recombination suffice to prevent accumulation of deleterious mutations via Muller’s ratchet even in a polyploid genome. The efficiency of selection is in flowering plants strongly increased by acting on the haplontic (reduced) gametophyte stage.

Electronic supplementary material

The online version of this article (10.1186/s12862-019-1495-z) contains supplementary material, which is available to authorized users.

Background selection as null hypothesis in population genomics: insights and challenges from Drosophila studies

The consequences of selection at linked sites are multiple and widespread across the genomes of most species. Here, I first review the main concepts behind models of selection and linkage in recombining genomes, present the difficulty in parametrizing these models simply as a reduction in effective population size (N_e) and discuss the predicted impact of recombination rates on levels of diversity across genomes. Arguments are then put forward in favour of using a model of selection and linkage with neutral and deleterious mutations (i.e. the background selection model, BGS) as a sensible null hypothesis for investigating the presence of other forms of selection, such as balancing or positive. I also describe and compare two studies that have generated high-resolution landscapes of the predicted consequences of selection at linked sites in Drosophila melanogaster. Both studies show that BGS can explain a very large fraction of the observed variation in diversity across the whole genome, thus supporting its use as null model. Finally, I identify and discuss a number of caveats and challenges in studies of genetic hitchhiking that have been often overlooked, with several of them sharing a potential bias towards overestimating the evidence supporting recent selective sweeps to the detriment of a BGS explanation. One potential source of bias is the analysis of non-equilibrium populations: it is precisely because models of selection and linkage predict variation in N_e across chromosomes that demographic dynamics are not expected to be equivalent chromosome- or genome-wide. Other challenges include the use of incomplete genome annotations, the assumption of temporally stable recombination landscapes, the presence of genes under balancing selection and the consequences of ignoring non-crossover (gene conversion) recombination events.

This article is part of the themed issue ‘Evolutionary causes and consequences of recombination rate variation in sexual organisms’.

The Effect of Strong Purifying Selection on Genetic Diversity

Purifying selection reduces genetic diversity, both at sites under direct selection and at linked neutral sites. This process, known as background selection, is thought to play an important role in shaping genomic diversity in natural populations. Yet despite its importance, the effects of background selection are not fully understood. Previous theoretical analyses of this process have taken a backward-time approach based on the structured coalescent. While they provide some insight, these methods are either limited to very small samples or are computationally prohibitive. Here, we present a new forward-time analysis of the trajectories of both neutral and deleterious mutations at a nonrecombining locus. We find that strong purifying selection leads to remarkably rich dynamics: neutral mutations can exhibit sweep-like behavior, and deleterious mutations can reach substantial frequencies even when they are guaranteed to eventually go extinct. Our analysis of these dynamics allows us to calculate analytical expressions for the full site frequency spectrum. We find that whenever background selection is strong enough to lead to a reduction in genetic diversity, it also results in substantial distortions to the site frequency spectrum, which can mimic the effects of population expansions or positive selection. Because these distortions are most pronounced in the low and high frequency ends of the spectrum, they become particularly important in larger samples, but may have small effects in smaller samples. We also apply our forward-time framework to calculate other quantities, such as the ultimate fates of polymorphisms or the fitnesses of their ancestral backgrounds.

Limiting fitness distributions in evolutionary dynamics

Natural selection works on variation in fitness, but how should we measure "variation" to predict the rate of future evolution? Fisher's fundamental theorem of natural selection provides the short-run answer: the instantaneous rate of growth of a population's mean fitness is its variance in fitness. This identity captures an important feature of the evolutionary process, but, because it does not specify how the variance itself evolves in time, it cannot be used to predict evolutionary dynamics in the long run. In this paper we reconsider the problem of computing evolutionary trajectories from limited statistical information. We identify the feature of fitness distributions which controls their late-time evolution: their (suitably defined) tail indices. We show that the location, scale and shape of the fitness distribution can be predicted far into the future from the measurement of this tail index at some initial time. Unlike the "fitness waves" studied in the literature, this pattern encompasses both positive and negative selection and is not restricted to rapidly adapting populations. Our results are well supported by numerical simulations, both from the Wright-Fisher model and from a less structured genetic algorithm.

Joint Prediction of the Effective Population Size and the Rate of Fixation of Deleterious Mutations

Mutation, genetic drift, and selection are considered the main factors shaping genetic variation in nature. There is a lack, however, of general predictions accounting for the mutual interrelation between these factors. In the context of the background selection model, we provide a set of equations for the joint prediction of the effective population size and the rate of fixation of deleterious mutations, which are applicable both to sexual and asexual species. For a population of N haploid individuals and a model of deleterious mutations with effect s appearing with rate U in a genome L Morgans long, the asymptotic effective population size (Ne) and the average number of generations (T) between consecutive fixations can be approximated by [Formula: see text] and [Formula: see text] The solution is applicable to Muller's ratchet, providing satisfactory approximations to the rate of accumulation of mutations for a wide range of parameters. We also obtain predictions of the effective size accounting for the expected nucleotide diversity. Predictions for sexual populations allow for outlining the general conditions where mutational meltdown occurs. The equations can be extended to any distribution of mutational effects and the consideration of hotspots of recombination, showing that Ne is rather insensitive and not proportional to changes in N for many combinations of parameters. This could contribute to explain the observed small differences in levels of polymorphism between species with very different census sizes.

Inevitability of Genetic Parasites

Almost all cellular life forms are hosts to diverse genetic parasites with various levels of autonomy including plasmids, transposons and viruses. Theoretical modeling of the evolution of primordial replicators indicates that parasites (cheaters) necessarily evolve in such systems and can be kept at bay primarily via compartmentalization. Given the (near) ubiquity, abundance and diversity of genetic parasites, the question becomes pertinent: are such parasites intrinsic to life? At least in prokaryotes, the persistence of parasites is linked to the rate of horizontal gene transfer (HGT). We mathematically derive the threshold value of the minimal transfer rate required for selfish element persistence, depending on the element duplication and loss rates as well as the cost to the host. Estimation of the characteristic gene duplication, loss and transfer rates for transposons, plasmids and virus-related elements in multiple groups of diverse bacteria and archaea indicates that most of these rates are compatible with the long term persistence of parasites. Notably, a small but non-zero rate of HGT is also required for the persistence of non-parasitic genes. We hypothesize that cells cannot tune their horizontal transfer rates to be below the threshold required for parasite persistence without experiencing highly detrimental side-effects. As a lower boundary to the minimum DNA transfer rate that a cell can withstand, we consider the process of genome degradation and mutational meltdown of populations through Muller's ratchet. A numerical assessment of this hypothesis suggests that microbial populations cannot purge parasites while escaping Muller's ratchet. Thus, genetic parasites appear to be virtually inevitable in cellular organisms.

Evolution of Mutation Rates in Rapidly Adapting Asexual Populations

Mutator and antimutator alleles often arise and spread in both natural microbial populations and laboratory evolution experiments. The evolutionary dynamics of these mutation rate modifiers are determined by indirect selection on linked beneficial and deleterious mutations. These indirect selection pressures have been the focus of much earlier theoretical and empirical work, but we still have a limited analytical understanding of how the interplay between hitchhiking and deleterious load influences the fates of modifier alleles. Our understanding is particularly limited when clonal interference is common, which is the regime of primary interest in laboratory microbial evolution experiments. Here, we calculate the fixation probability of a mutator or antimutator allele in a rapidly adapting asexual population, and we show how this quantity depends on the population size, the beneficial and deleterious mutation rates, and the strength of a typical driver mutation. In the absence of deleterious mutations, we find that clonal interference enhances the fixation probability of mutators, even as they provide a diminishing benefit to the overall rate of adaptation. When deleterious mutations are included, natural selection pushes the population toward a stable mutation rate that can be suboptimal for the adaptation of the population as a whole. The approach to this stable mutation rate is not necessarily monotonic: even in the absence of epistasis, selection can favor mutator and antimutator alleles that "overshoot" the stable mutation rate by substantial amounts.

Collective Fluctuations in the Dynamics of Adaptation and Other Traveling Waves

The dynamics of adaptation are difficult to predict because it is highly stochastic even in large populations. The uncertainty emerges from random genetic drift arising in a vanguard of particularly fit individuals of the population. Several approaches have been developed to analyze the crucial role of genetic drift on the expected dynamics of adaptation, including the mean fitness of the entire population, or the fate of newly arising beneficial deleterious mutations. However, little is known about how genetic drift causes fluctuations to emerge on the population level, where it becomes palpable as variations in the adaptation speed and the fitness distribution. Yet these phenomena control the decay of genetic diversity and variability in evolution experiments and are key to a truly predictive understanding of evolutionary processes. Here, we show that correlations induced by these emergent fluctuations can be computed at any arbitrary order by a suitable choice of a dynamical constraint. The resulting linear equations exhibit fluctuation-induced terms that amplify short-distance correlations and suppress long-distance ones. These terms, which are in general not small, control the decay of genetic diversity and, for wave-tip dominated ("pulled") waves, lead to anticorrelations between the tip of the wave and the lagging bulk of the population. While it is natural to consider the process of adaptation as a branching random walk in fitness space subject to a constraint (due to finite resources), we show that other traveling wave phenomena in ecology and evolution likewise fall into this class of constrained branching random walks. Our methods, therefore, provide a systematic approach toward analyzing fluctuations in a wide range of population biological processes, such as adaptation, genetic meltdown, species invasions, or epidemics.

Do cell-autonomous and non-cell-autonomous effects drive the structure of tumor ecosystems?

By definition, a driver mutation confers a growth advantage to the cancer cell in which it occurs, while a passenger mutation does not: the former is usually considered as the engine of cancer progression, while the latter is not. Actually, the effects of a given mutation depend on the genetic background of the cell in which it appears, thus can differ in the subclones that form a tumor. In addition to cell-autonomous effects generated by the mutations, non-cell-autonomous effects shape the phenotype of a cancer cell. Here, we review the evidence that a network of biological interactions between subclones drives cancer cell adaptation and amplifies intra-tumor heterogeneity. Integrating the role of mutations in tumor ecosystems generates innovative strategies targeting the tumor ecosystem's weaknesses to improve cancer treatment.

The Effects of Background and Interference Selection on Patterns of Genetic Variation in Subdivided Populations

It is well known that most new mutations that affect fitness exert deleterious effects and that natural populations are often composed of subpopulations (demes) connected by gene flow. To gain a better understanding of the joint effects of purifying selection and population structure, we focus on a scenario where an ancestral population splits into multiple demes and study neutral diversity patterns in regions linked to selected sites. In the background selection regime of strong selection, we first derive analytic equations for pairwise coalescent times and FST as a function of time after the ancestral population splits into two demes and then construct a flexible coalescent simulator that can generate samples under complex models such as those involving multiple demes or nonconservative migration. We have carried out extensive forward simulations to show that the new methods can accurately predict diversity patterns both in the nonequilibrium phase following the split of the ancestral population and in the equilibrium between mutation, migration, drift, and selection. In the interference selection regime of many tightly linked selected sites, forward simulations provide evidence that neutral diversity patterns obtained from both the nonequilibrium and equilibrium phases may be virtually indistinguishable for models that have identical variance in fitness, but are nonetheless different with respect to the number of selected sites and the strength of purifying selection. This equivalence in neutral diversity patterns suggests that data collected from subdivided populations may have limited power for differentiating among the selective pressures to which closely linked selected sites are subject.

Statistical properties and error threshold of quasispecies on single-peak Gaussian-distributed fitness landscapes

The stochastic Eigen model proposed by Feng et al. (2007) (Journal of Theoretical Biology, 246, 28) showed that error threshold is no longer a phase transition point but a crossover region whose width depends on the strength of the random fluctuation in an environment. The underlying cause of this phenomenon has not yet been well examined. In this article, we adopt a single peak Gaussian distributed fitness landscape instead of a constant one to investigate and analyze the change of the error threshold and the statistical property of the quasi-species population. We find a roughly linear relation between the width of the error threshold and the fitness fluctuation strength. For a given quasi-species, the fluctuation of the relative concentration has a minimum with a normal distribution of the relative concentration at the maximum of the averaged relative concentration, it has however a largest value with a bimodal distribution of the relative concentration near the error threshold. The above results deepen our understanding of the quasispecies and error threshold and are heuristic for exploring practicable antiviral strategies.

Effect of drift, selection and recombination on the equilibrium frequency of deleterious mutations

We study the stationary state of a population evolving under the action of random genetic drift, selection and recombination in which both deleterious and reverse beneficial mutations can occur. We find that the equilibrium fraction of deleterious mutations decreases as the population size is increased. We calculate exactly the steady state frequency in a nonrecombining population when population size is infinite and for a neutral finite population, and obtain bounds on the fraction of deleterious mutations. We also find that for small and very large populations, the number of deleterious mutations depends weakly on recombination, but for moderately large populations, recombination alleviates the effect of deleterious mutations. An analytical argument shows that recombination decreases disadvantageous mutations appreciably when beneficial mutations are rare as is the case in adapting microbial populations, whereas it has a moderate effect on codon bias where the mutation rates between the preferred and unpreferred codons are comparable.

Tug-of-war between driver and passenger mutations in cancer and other adaptive processes

Cancer progression is an example of a rapid adaptive process where evolving new traits is essential for survival and requires a high mutation rate. Precancerous cells acquire a few key mutations that drive rapid population growth and carcinogenesis. Cancer genomics demonstrates that these few driver mutations occur alongside thousands of random passenger mutations--a natural consequence of cancer's elevated mutation rate. Some passengers are deleterious to cancer cells, yet have been largely ignored in cancer research. In population genetics, however, the accumulation of mildly deleterious mutations has been shown to cause population meltdown. Here we develop a stochastic population model where beneficial drivers engage in a tug-of-war with frequent mildly deleterious passengers. These passengers present a barrier to cancer progression describable by a critical population size, below which most lesions fail to progress, and a critical mutation rate, above which cancers melt down. We find support for this model in cancer age-incidence and cancer genomics data that also allow us to estimate the fitness advantage of drivers and fitness costs of passengers. We identify two regimes of adaptive evolutionary dynamics and use these regimes to understand successes and failures of different treatment strategies. A tumor's load of deleterious passengers can explain previously paradoxical treatment outcomes and suggest that it could potentially serve as a biomarker of response to mutagenic therapies. The collective deleterious effect of passengers is currently an unexploited therapeutic target. We discuss how their effects might be exacerbated by current and future therapies.

Clonal interference and Muller's ratchet in spatial habitats

Competition between independently arising beneficial mutations is enhanced in spatial populations due to the linear rather than exponential growth of clones. Recent theoretical studies have pointed out that the resulting fitness dynamics is analogous to a surface growth process, where new layers nucleate and spread stochastically, leading to the build up of scale-invariant roughness. This scenario differs qualitatively from the standard view of adaptation in that the speed of adaptation becomes independent of population size while the fitness variance does not. Here we exploit recent progress in the understanding of surface growth processes to obtain precise predictions for the universal, non-Gaussian shape of the fitness distribution for one-dimensional habitats, which are verified by simulations. When the mutations are deleterious rather than beneficial the problem becomes a spatial version of Muller's ratchet. In contrast to the case of well-mixed populations, the rate of fitness decline remains finite even in the limit of an infinite habitat, provided the ratio [Formula: see text] between the deleterious mutation rate and the square of the (negative) selection coefficient is sufficiently large. Using, again, an analogy to surface growth models we show that the transition between the stationary and the moving state of the ratchet is governed by directed percolation.

Genetic diversity in the interference selection limit

Pervasive natural selection can strongly influence observed patterns of genetic variation, but these effects remain poorly understood when multiple selected variants segregate in nearby regions of the genome. Classical population genetics fails to account for interference between linked mutations, which grows increasingly severe as the density of selected polymorphisms increases. Here, we describe a simple limit that emerges when interference is common, in which the fitness effects of individual mutations play a relatively minor role. Instead, similar to models of quantitative genetics, molecular evolution is determined by the variance in fitness within the population, defined over an effectively asexual segment of the genome (a "linkage block"). We exploit this insensitivity in a new "coarse-grained" coalescent framework, which approximates the effects of many weakly selected mutations with a smaller number of strongly selected mutations that create the same variance in fitness. This approximation generates accurate and efficient predictions for silent site variability when interference is common. However, these results suggest that there is reduced power to resolve individual selection pressures when interference is sufficiently widespread, since a broad range of parameters possess nearly identical patterns of silent site variability.

A path integral formulation of the Wright-Fisher process with genic selection

The Wright-Fisher process with selection is an important tool in population genetics theory. Traditional analysis of this process relies on the diffusion approximation. The diffusion approximation is usually studied in a partial differential equations framework. In this paper, I introduce a path integral formalism to study the Wright-Fisher process with selection and use that formalism to obtain a simple perturbation series to approximate the transition density. The perturbation series can be understood in terms of Feynman diagrams, which have a simple probabilistic interpretation in terms of selective events. The perturbation series proves to be an accurate approximation of the transition density for weak selection and is shown to be arbitrarily accurate for any selection coefficient.

Horizontal gene transfer can rescue prokaryotes from Muller's ratchet: benefit of DNA from dead cells and population subdivision

Horizontal gene transfer (HGT) is a major factor in the evolution of prokaryotes. An intriguing question is whether HGT is maintained during evolution of prokaryotes owing to its adaptive value or is a byproduct of selection driven by other factors such as consumption of extracellular DNA (eDNA) as a nutrient. One hypothesis posits that HGT can restore genes inactivated by mutations and thereby prevent stochastic, irreversible deterioration of genomes in finite populations known as Muller’s ratchet. To examine this hypothesis, we developed a population genetic model of prokaryotes undergoing HGT via homologous recombination. Analysis of this model indicates that HGT can prevent the operation of Muller’s ratchet even when the source of transferred genes is eDNA that comes from dead cells and on average carries more deleterious mutations than the DNA of recipient live cells. Moreover, if HGT is sufficiently frequent and eDNA diffusion sufficiently rapid, a subdivided population is shown to be more resistant to Muller’s ratchet than an undivided population of an equal overall size. Thus, to maintain genomic information in the face of Muller’s ratchet, it is more advantageous to partition individuals into multiple subpopulations and let them “cross-reference” each other’s genetic information through HGT than to collect all individuals in one population and thereby maximize the efficacy of natural selection. Taken together, the results suggest that HGT could be an important condition for the long-term maintenance of genomic information in prokaryotes through the prevention of Muller’s ratchet.

Distribution of the fittest individuals and the rate of Muller's ratchet in a model with overlapping generations

Muller's ratchet is a paradigmatic model for the accumulation of deleterious mutations in a population of finite size. A click of the ratchet occurs when all individuals with the least number of deleterious mutations are lost irreversibly due to a stochastic fluctuation. In spite of the simplicity of the model, a quantitative understanding of the process remains an open challenge. In contrast to previous works, we here study a Moran model of the ratchet with overlapping generations. Employing an approximation which describes the fittest individuals as one class and the rest as a second class, we obtain closed analytical expressions of the ratchet rate in the rare clicking regime. As a click in this regime is caused by a rare, large fluctuation from a metastable state, we do not resort to a diffusion approximation but apply an approximation scheme which is especially well suited to describe extinction events from metastable states. This method also allows for a derivation of expressions for the quasi-stationary distribution of the fittest class. Additionally, we confirm numerically that the formulation with overlapping generations leads to the same results as the diffusion approximation and the corresponding Wright-Fisher model with non-overlapping generations.

Muller's ratchet is a paradigmatic model in population genetics which describes the fixation of a deleterious mutation in a population of finite size due to an unfortunate stochastic fluctuation. Obtaining quantitative predictions of the ratchet rate, i.e. the frequency with which such a mutation fixes, is believed to be important for understanding a broad range of effects ranging from the degeneration of the Y-chromosome to the evolution of sex as a means of avoiding the fixation of deleterious mutations. To obtain a better understanding of how Muller's ratchet operates, we have considered a model with overlapping generations, which allows for the application of methods specifically tailored for the analysis of rare stochastic fluctuations which drive the ratchet. We obtain concise and accurate results for the rate of Muller's ratchet. Additionally, we are able to predict the full distribution of the frequency of the fittest individuals, a quantity of central interest in understanding the ratchet rate and possibly experimentally much more accessible than the rate, in particular when the ratchet rate is very large.

Normal mutation rate variants arise in a Mutator (Mut S) Escherichia coli population

The rate at which mutations are generated is central to the pace of evolution. Although this rate is remarkably similar amongst all cellular organisms, bacterial strains with mutation rates 100 fold greater than the modal rates of their species are commonly isolated from natural sources and emerge in experimental populations. Theoretical studies postulate and empirical studies teort the hypotheses that these “mutator” strains evolved in response to selection for elevated rates of generation of inherited variation that enable bacteria to adapt to novel and/or rapidly changing environments. Less clear are the conditions under which selection will favor reductions in mutation rates. Declines in rates of mutation for established populations of mutator bacteria are not anticipated if such changes are attributed to the costs of augmented rates of generation of deleterious mutations. Here we report experimental evidence of evolution towards reduced mutation rates in a clinical isolate of Escherichia coli with an hyper-mutable phenotype due a deletion in a mismatch repair gene, (ΔmutS). The emergence in a ΔmutS background of variants with mutation rates approaching those of the normal rates of strains carrying wild-type MutS was associated with increase in fitness with respect to ancestral strain. We postulate that such an increase in fitness could be attributed to the emergence of mechanisms driving a permanent “aerobic style of life”, the negative consequence of this behavior being regulated by the evolution of mechanisms protecting the cell against increased endogenous oxidative radicals involved in DNA damage, and thus reducing mutation rate. Gene expression assays and full sequencing of evolved mutator and normo-mutable variants supports the hypothesis. In conclusion, we postulate that the observed reductions in mutation rate are coincidental to, rather than, the selective force responsible for this evolution.

Fixation of mutators in asexual populations: the role of genetic drift and epistasis

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

Impact of deleterious passenger mutations on cancer progression

Cancer progression is driven by the accumulation of a small number of genetic alterations. However, these few driver alterations reside in a cancer genome alongside tens of thousands of additional mutations termed passengers. Passengers are widely believed to have no role in cancer, yet many passengers fall within protein-coding genes and other functional elements that can have potentially deleterious effects on cancer cells. Here we investigate the potential of moderately deleterious passengers to accumulate and alter the course of neoplastic progression. Our approach combines evolutionary simulations of cancer progression with an analysis of cancer sequencing data. From simulations, we find that passengers accumulate and largely evade natural selection during progression. Although individually weak, the collective burden of passengers alters the course of progression, leading to several oncological phenomena that are hard to explain with a traditional driver-centric view. We then tested the predictions of our model using cancer genomics data and confirmed that many passengers are likely damaging and have largely evaded negative selection. Finally, we use our model to explore cancer treatments that exploit the load of passengers by either (i) increasing the mutation rate or (ii) exacerbating their deleterious effects. Though both approaches lead to cancer regression, the latter is a more effective therapy. Our results suggest a unique framework for understanding cancer progression as a balance of driver and passenger mutations.

Fluctuations in fitness distributions and the effects of weak linked selection on sequence evolution

Evolutionary dynamics and patterns of molecular evolution are strongly influenced by selection on linked regions of the genome, but our quantitative understanding of these effects remains incomplete. Recent work has focused on predicting the distribution of fitness within an evolving population, and this forms the basis for several methods that leverage the fitness distribution to predict the patterns of genetic diversity when selection is strong. However, in weakly selected populations random fluctuations due to genetic drift are more severe, and neither the distribution of fitness nor the sequence diversity within the population are well understood. Here, we briefly review the motivations behind the fitness-distribution picture, and summarize the general approaches that have been used to analyze this distribution in the strong-selection regime. We then extend these approaches to the case of weak selection, by outlining a perturbative treatment of selection at a large number of linked sites. This allows us to quantify the stochastic behavior of the fitness distribution and yields exact analytical predictions for the sequence diversity and substitution rate in the limit that selection is weak.

Genealogies of rapidly adapting populations

The genetic diversity of a species is shaped by its recent evolutionary history and can be used to infer demographic events or selective sweeps. Most inference methods are based on the null hypothesis that natural selection is a weak or infrequent evolutionary force. However, many species, particularly pathogens, are under continuous pressure to adapt in response to changing environments. A statistical framework for inference from diversity data of such populations is currently lacking. Towards this goal, we explore the properties of genealogies in a model of continual adaptation in asexual populations. We show that lineages trace back to a small pool of highly fit ancestors, in which almost simultaneous coalescence of more than two lineages frequently occurs. Whereas such multiple mergers are unlikely under the neutral coalescent, they create a unique genetic footprint in adapting populations. The site frequency spectrum of derived neutral alleles, for example, is nonmonotonic and has a peak at high frequencies, whereas Tajima's D becomes more and more negative with increasing sample size. Because multiple merger coalescents emerge in many models of rapid adaptation, we argue that they should be considered as a null model for adapting populations.

Dynamic mutation-selection balance as an evolutionary attractor

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