Fixation dynamics of beneficial alleles in prokaryotic polyploid chromosomes and plasmids

Theoretical population genetics has been mostly developed for sexually reproducing diploid and for monoploid (haploid) organisms, focusing on eukaryotes. The evolution of bacteria and archaea is often studied by models for the allele dynamics in monoploid populations. However, many prokaryotic organisms harbor multicopy replicons—chromosomes and plasmids—and theory for the allele dynamics in populations of polyploid prokaryotes remains lacking. Here, we present a population genetics model for replicons with multiple copies in the cell. Using this model, we characterize the fixation process of a dominant beneficial mutation at 2 levels: the phenotype and the genotype. Our results show that depending on the mode of replication and segregation, the fixation of the mutant phenotype may precede genotypic fixation by many generations; we term this time interval the heterozygosity window. We furthermore derive concise analytical expressions for the occurrence and length of the heterozygosity window, showing that it emerges if the copy number is high and selection strong. Within the heterozygosity window, the population is phenotypically adapted, while both alleles persist in the population. Replicon ploidy thus allows for the maintenance of genetic variation following phenotypic adaptation and consequently for reversibility in adaptation to fluctuating environmental conditions.     

Selective sweeps in multilocus models of quantitative traits

We study the trajectory of an allele that affects a polygenic trait selected toward a phenotypic optimum. Furthermore, conditioning on this trajectory we analyze the effect of the selected mutation on linked neutral variation. We examine the well-characterized two-locus two-allele model but we also provide results for diallelic models with up to eight loci. First, when the optimum phenotype is that of the double heterozygote in a two-locus model, and there is no dominance or epistasis of effects on the trait, the trajectories of selected mutations rarely reach fixation; instead, a polymorphic equilibrium at both loci is approached. Whether a polymorphic equilibrium is reached (rather than fixation at both loci) depends on the intensity of selection and the relative distances to the optimum of the homozygotes at each locus. Furthermore, if both loci have similar effects on the trait, fixation of an allele at a given locus is less likely when it starts at low frequency and the other locus is polymorphic (with alleles at intermediate frequencies). Weaker selection increases the probability of fixation of the studied allele, as the polymorphic equilibrium is less stable in this case. When we do not require the double heterozygote to be at the optimum we find that the polymorphic equilibrium is more difficult to reach, and fixation becomes more likely. Second, increasing the number of loci decreases the probability of fixation, because adaptation to the optimum is possible by various combinations of alleles. Summaries of the genealogy (height, total length, and imbalance) and of sequence polymorphism (number of polymorphisms, frequency spectrum, and haplotype structure) next to a selected locus depend on the frequency that the selected mutation approaches at equilibrium. We conclude that multilocus response to selection may in some cases prevent selective sweeps from being completed, as described in previous studies, but that conditions causing this to happen strongly depend on the genetic architecture of the trait, and that fixation of selected mutations is likely in many instances.     

Transition between Stochastic Evolution and Deterministic Evolution in the Presence of Selection: General Theory and Application to Virology

We present here a self-contained analytic review of the role of stochastic factors acting on a virus population. We develop a simple one-locus, two-allele model of a haploid population of constant size including the factors of random drift, purifying selection, and random mutation. We consider different virological experiments: accumulation and reversion of deleterious mutations, competition between mutant and wild-type viruses, gene fixation, mutation frequencies at the steady state, divergence of two populations split from one population, and genetic turnover within a single population. In the first part of the review, we present all principal results in qualitative terms and illustrate them with examples obtained by computer simulation. In the second part, we derive the results formally from a diffusion equation of the Wright-Fisher type and boundary conditions, all derived from the first principles for the virus population model. We show that the leading factors and observable behavior of evolution differ significantly in three broad intervals of population size, N. The “neutral limit” is reached when N is smaller than the inverse selection coefficient. When N is larger than the inverse mutation rate per base, selection dominates and evolution is “almost” deterministic. If the selection coefficient is much larger than the mutation rate, there exists a broad interval of population sizes, in which weakly diverse populations are almost neutral while highly diverse populations are controlled by selection pressure. We discuss in detail the application of our results to human immunodeficiency virus population in vivo, sampling effects, and limitations of the model.     

Hitchhiking of Deleterious Alleles and the Cost of Adaptation in Partially Selfing Species

Self-fertilization is generally seen to be disadvantageous in the long term. It increases genetic drift, which subsequently reduces polymorphism and the efficiency of selection, which also challenges adaptation. However, high selfing rates can increase the fixation probability of recessive beneficial mutations, but existing theory has generally not accounted for the effect of linked sites. Here, we analyze a model for the fixation probability of deleterious mutants that hitchhike with selective sweeps in diploid, partially selfing populations. Approximate analytical solutions show that, conditional on the sweep not being lost by drift, higher inbreeding rates increase the fixation probability of the deleterious allele, due to the resulting reduction in polymorphism and effective recombination. When extending the analysis to consider a distribution of deleterious alleles, as well as the average fitness increase after a sweep, we find that beneficial alleles generally need to be more recessive than the previously assumed dominance threshold (h < 1/2) for selfing to be beneficial from one-locus theory. Our results highlight that recombination aiding the efficiency of selection on multiple loci amplifies the fitness benefits of outcrossing over selfing, compared to results obtained from one-locus theory. This effect additionally increases the parameter range under which obligate outcrossing is beneficial over partial selfing.     

Cumulant dynamics of a population under multiplicative selection, mutation, and drift

We revisit the classical population genetics model of a population evolving under multiplicative selection, mutation, and drift. The number of beneficial alleles in a multilocus system can be considered a trait under exponential selection. Equations of motion are derived for the cumulants of the trait distribution in the diffusion limit and under the assumption of linkage equilibrium. Because of the additive nature of cumulants, this reduces to the problem of determining equations of motion for the expected allele distribution cumulants at each locus. The cumulant equations form an infinite dimensional linear system and in an authored appendix Adam Prügel-Bennett provides a closed form expression for these equations. We derive approximate solutions which are shown to describe the dynamics well for a broad range of parameters. In particular, we introduce two approximate analytical solutions: (1) Perturbation theory is used to solve the dynamics for weak selection and arbitrary mutation rate. The resulting expansion for the system's eigenvalues reduces to the known diffusion theory results for the limiting cases with either mutation or selection absent. (2) For low mutation rates we observe a separation of time-scales between the slowest mode and the rest which allows us to develop an approximate analytical solution for the dominant slow mode. The solution is consistent with the perturbation theory result and provides a good approximation for much stronger selection intensities.     

Allele frequency distribution under recurrent selective sweeps

The allele frequency of a neutral variant in a population is pushed either upward or downward by directional selection on a linked beneficial mutation ("selective sweeps"). DNA sequences sampled after the fixation of the beneficial allele thus contain an excess of rare neutral alleles. This study investigates the allele frequency distribution under selective sweep models using analytic approximation and simulation. First, given a single selective sweep at a fixed time, I derive an expression for the sampling probabilities of neutral mutants. This solution can be used to estimate the time of the fixation of a beneficial allele from sequence data. Next, I obtain an approximation to mean allele frequencies under recurrent selective sweeps. Under recurrent sweeps, the frequency spectrum is skewed toward rare alleles. However, the excess of high-frequency derived alleles, previously shown to be a signature of single selective sweeps, disappears with recurrent sweeps. It is shown that, using this approximation and multilocus polymorphism data, genomewide parameters of directional selection can be estimated.     

Multilocus genetics and the coevolution of quantitative traits

We develop and analyze an explicit multilocus genetic model of coevolution. We assume that interactions between two species (mutualists, competitors, or victim and exploiter) are mediated by a pair of additive quantitative traits that are also subject to direct stabilizing selection toward intermediate optima. Using a weak-selection approximation, we derive analytical results for a symmetric case with equal locus effects and no mutation, and we complement these results by numerical simulations of more general cases. We show that mutualistic and competitive interactions always result in coevolution toward a stable equilibrium with no more than one polymorphic locus per species. Victim-exploiter interactions can lead to different dynamic regimes including evolution toward stable equilibria, cycles, and chaos. At equilibrium, the victim is often characterized by a very large genetic variance, whereas the exploiter is polymorphic in no more than one locus. Compared to related one-locus or quantitative genetic models, the multilocus model exhibits two major new properties. First, the equilibrium structure is considerably more complex. We derive detailed conditions for the existence and stability of various classes of equilibria and demonstrate the possibility of multiple simultaneously stable states. Second, the genetic variances change dynamically, which in turn significantly affects the dynamics of the mean trait values. In particular, the dynamics tend to be destabilized by an increase in the number of loci.     

Expected Shannon Entropy and Shannon Differentiation between Subpopulations for Neutral Genes under the Finite Island Model

Shannon entropy H and related measures are increasingly used in molecular ecology and population genetics because (1) unlike measures based on heterozygosity or allele number, these measures weigh alleles in proportion to their population fraction, thus capturing a previously-ignored aspect of allele frequency distributions that may be important in many applications; (2) these measures connect directly to the rich predictive mathematics of information theory; (3) Shannon entropy is completely additive and has an explicitly hierarchical nature; and (4) Shannon entropy-based differentiation measures obey strong monotonicity properties that heterozygosity-based measures lack. We derive simple new expressions for the expected values of the Shannon entropy of the equilibrium allele distribution at a neutral locus in a single isolated population under two models of mutation: the infinite allele model and the stepwise mutation model. Surprisingly, this complex stochastic system for each model has an entropy expressable as a simple combination of well-known mathematical functions. Moreover, entropy- and heterozygosity-based measures for each model are linked by simple relationships that are shown by simulations to be approximately valid even far from equilibrium. We also identify a bridge between the two models of mutation. We apply our approach to subdivided populations which follow the finite island model, obtaining the Shannon entropy of the equilibrium allele distributions of the subpopulations and of the total population. We also derive the expected mutual information and normalized mutual information (“Shannon differentiation”) between subpopulations at equilibrium, and identify the model parameters that determine them. We apply our measures to data from the common starling (Sturnus vulgaris) in Australia. Our measures provide a test for neutrality that is robust to violations of equilibrium assumptions, as verified on real world data from starlings.     

The Dynamics of Genetic Draft in Rapidly Adapting Populations

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

A two-locus mutation—Selection model and some of its evolutionary implications

The deterministic properties of a two-locus model with mutation and selection have been investigated. The mutation process is unidirectional, and the model is so constructed that the genetic variation at one locus is selectively neutral in the absence of a mutant allele at the other locus. All genotypes with three or four mutant alleles are deleterious, while the double heterozygotes may have the same fitness as the standard genotype. If one of the mutant alleles becomes fixed in the population, then the other locus will show a regular one-locus mutation-selection balance. Such a boundary equilibrium may be unstable or stable in the full two-locus setting. In the symmetric case, which is analyzed in details, the population will either go to one of the two boundary equilibria, or to a fully polymorphic equilibrium at which both the mutant alleles are rare. The origin of reproductive separation between two populations via the fixation of complementary deleterious mutants at different loci, and the fixation of nonfunctional alleles at duplicated loci, are two biological processes which both can be studied with the present model. In the last part of the paper we show how the results from the deterministic analysis can be used to predict how different factors will influence the rates of evolution in these systems.     

A Coalescent Model for a Sweep of a Unique Standing Variant

The use of genetic polymorphism data to understand the dynamics of adaptation and identify the loci that are involved has become a major pursuit of modern evolutionary genetics. In addition to the classical “hard sweep” hitchhiking model, recent research has drawn attention to the fact that the dynamics of adaptation can play out in a variety of different ways and that the specific signatures left behind in population genetic data may depend somewhat strongly on these dynamics. One particular model for which a large number of empirical examples are already known is that in which a single derived mutation arises and drifts to some low frequency before an environmental change causes the allele to become beneficial and sweeps to fixation. Here, we pursue an analytical investigation of this model, bolstered and extended via simulation study. We use coalescent theory to develop an analytical approximation for the effect of a sweep from standing variation on the genealogy at the locus of the selected allele and sites tightly linked to it. We show that the distribution of haplotypes that the selected allele is present on at the time of the environmental change can be approximated by considering recombinant haplotypes as alleles in the infinite-alleles model. We show that this approximation can be leveraged to make accurate predictions regarding patterns of genetic polymorphism following such a sweep. We then use simulations to highlight which sources of haplotypic information are likely to be most useful in distinguishing this model from neutrality, as well as from other sweep models, such as the classic hard sweep and multiple-mutation soft sweeps. We find that in general, adaptation from a unique standing variant will likely be difficult to detect on the basis of genetic polymorphism data from a single population time point alone, and when it can be detected, it will be difficult to distinguish from other varieties of selective sweeps. Samples from multiple populations and/or time points have the potential to ease this difficulty.     

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.     

The Effect of Linkage on Establishment and Survival of Locally Beneficial Mutations

We study invasion and survival of weakly beneficial mutations arising in linkage to an established migration–selection polymorphism. Our focus is on a continent–island model of migration, with selection at two biallelic loci for adaptation to the island environment. Combining branching and diffusion processes, we provide the theoretical basis for understanding the evolution of islands of divergence, the genetic architecture of locally adaptive traits, and the importance of so-called “divergence hitchhiking” relative to other mechanisms, such as “genomic hitchhiking”, chromosomal inversions, or translocations. We derive approximations to the invasion probability and the extinction time of a de novo mutation. Interestingly, the invasion probability is maximized at a nonzero recombination rate if the focal mutation is sufficiently beneficial. If a proportion of migrants carries a beneficial background allele, the mutation is less likely to become established. Linked selection may increase the survival time by several orders of magnitude. By altering the timescale of stochastic loss, it can therefore affect the dynamics at the focal site to an extent that is of evolutionary importance, especially in small populations. We derive an effective migration rate experienced by the weakly beneficial mutation, which accounts for the reduction in gene flow imposed by linked selection. Using the concept of the effective migration rate, we also quantify the long-term effects on neutral variation embedded in a genome with arbitrarily many sites under selection. Patterns of neutral diversity change qualitatively and quantitatively as the position of the neutral locus is moved along the chromosome. This will be useful for population-genomic inference. Our results strengthen the emerging view that physically linked selection is biologically relevant if linkage is tight or if selection at the background locus is strong.     

Genetic structuring in fluctuating populations

The effect of genetic drift in spatially distributed dispersal-linked and density-regulated populations is studied in a classical one-locus two-allele system. We analyse emergence of genetic differentiation assuming random drift only, where the noise-like variability is due to demographic stochasticity. We find emergence of clusters of sub-units with local allele fixation and persistence of both alleles in lengthy simulations. We demonstrate that local allele fixation (extending over a number of adjoining spatial sub-units) – without global loss of alleles – may occur when the carrying capacities of local patches are small, under a full range population dynamic regimes, when dispersal rate is small, and when redistribution (through dispersal) does not act as global mixer. These results are novel. The key to the observations is that drift is simultaneously influenced by distance-dependent dispersal, demographic stochasticity and autocorrelated population fluctuations due to delayed-density dependence. These are standard elements of contemporary population models in spatially structured context. With stable large populations, no stochasticity and dispersal limited to neighbours only, our model collapses to the stepping-stone model, while with dispersal being random and global, the model collapses to Wright's island model.     

Rate of Gene Silencing at Duplicate Loci: A Theoretical Study and Interpretation of Data from Tetraploid Fishes

A large-scale simulation has been conducted on the rate of gene loss at duplicate loci under irreversible mutation. It is found that tight linkage does not provide a strong sheltering effect, as thought by previous authors; indeed, the mean loss time for the case of tight linkage is of the same order of magnitude as that for no linkage, as long as Nu is not much larger than 1, where N is the effective population size and u the mutation rate. When Nu is 0.01 or less, the two loci behave almost as neutral loci, regardless of linkage, and the mean loss time is about only half the mean extinction time for a neutral allele under irreversible mutation. However, the former becomes two or more times larger than the latter when Nu ≥ 1.——In the simulation, the sojourn times in the frequency intervals (0, 0.01) and (0.99, 1) and the time for the frequency of the null allele to reach 0.99 at one of the two loci have also been recorded. The results show that the population is monomorphic for the normal allele most of the time if Nu ≤ 0.01, but polymorphic for the null and the normal alleles most of the time if Nu ≥ 0.1.——The distribution of the frequency of the null allele in an equilibrium tetraploid population has been studied analytically. The present results have been applied to interpret data from some fish groups that are of tetraploid origin, and a model for explaining the slow rate of gene loss in these fishes is proposed.     

An analytical model for genetic hitchhiking in the evolution of antimalarial drug resistance

We analytically study a deterministic model for the spread of drug resistance among human malaria parasites. The model incorporates all major characteristics of the complex malaria transmission cycle and accounts for the fact that only a fraction α of infected hosts receive drug treatment. Furthermore, the model incorporates that hosts can be co-infected. The number m of parasites co-infecting a host is either a constant or, more generally, follows a given frequency distribution.Although the model is formulated in a multilocus setup, for our results we assume that drug resistance is caused by a single locus with two alleles — a sensitive one and a resistant one. We assume that the resistant allele has a selective advantage only in treated hosts and pays metabolic costs, which causes this allele to be deleterious in untreated hosts. We provide necessary and sufficient conditions for the fixation of the resistant allele. Moreover, provided the resistant allele will sweep through the population, we derive a formula for the time until it reaches a given frequency and in particular for the time until quasi-fixation.Furthermore, we establish an analytical solution for allele frequency changes at a linked neutral biallelic locus due to the rapid increase in frequency of the resistant allele. Our solution describes a local reduction in heterozygosity among parasite chromosomes around the resistant allele, the effect commonly referred to as the hitchhiking effect, as a function of α and m. The result therefore allows the investigation of selective sweep patterns under specific demographic settings. We find that the hitchhiking effect is similar but different from the standard model of genetic hitchhiking that assumes random mating and homogeneous selection. In particular, the process of recombination and selection cannot be decoupled. We further explain why standard hitchhiking theory cannot be applied to drug resistance in malaria. Furthermore, we will show that a genome-wide reduction in relative heterozygosity can occur provided a fraction of hosts is infected by a single parasite haplotype.Finally, we show how to incorporate host heterogeneity, and generalize our results to this biologically more realistic case.     

An inclusive fitness analysis of synergistic interactions in structured populations

We study the evolution of a pair of competing behavioural alleles in a structured population when there are non-additive or ‘synergistic’ fitness effects. Under a form of weak selection and with a simple symmetry condition between a pair of competing alleles, Tarnita et al. provide a surprisingly simple condition for one allele to dominate the other. Their condition can be obtained from an analysis of a corresponding simpler model in which fitness effects are additive. Their result uses an average measure of selective advantage where the average is taken over the long-term—that is, over all possible allele frequencies—and this precludes consideration of any frequency dependence the allelic fitness might exhibit. However, in a considerable body of work with non-additive fitness effects—for example, hawk–dove and prisoner''s dilemma games—frequency dependence plays an essential role in the establishment of conditions for a stable allele-frequency equilibrium. Here, we present a frequency-dependent generalization of their result that provides an expression for allelic fitness at any given allele frequency p. We use an inclusive fitness approach and provide two examples for an infinite structured population. We illustrate our results with an analysis of the hawk–dove game.     

Competition Between the Sperm of a Single Male Can Increase the Evolutionary Rate of Haploid Expressed Genes

The population genetic behavior of mutations in sperm genes is theoretically investigated. We modeled the processes at two levels. One is the standard population genetic process, in which the population allele frequencies change generation by generation, depending on the difference in selective advantages. The other is the sperm competition during each genetic transmission from one generation to the next generation. For the sperm competition process, we formulate the situation where a huge number of sperm with alleles A and B, produced by a single heterozygous male, compete to fertilize a single egg. This “minimal model” demonstrates that a very slight difference in sperm performance amounts to quite a large difference between the alleles’ winning probabilities. By incorporating this effect of paternity-sharing sperm competition into the standard population genetic process, we show that fierce sperm competition can enhance the fixation probability of a mutation with a very small phenotypic effect at the single-sperm level, suggesting a contribution of sperm competition to rapid amino acid substitutions in haploid-expressed sperm genes. Considering recent genome-wide demonstrations that a substantial fraction of the mammalian sperm genes are haploid expressed, our model could provide a potential explanation of rapid evolution of sperm genes with a wide variety of functions (as long as they are expressed in the haploid phase). Another advantage of our model is that it is applicable to a wide range of species, irrespective of whether the species is externally fertilizing, polygamous, or monogamous. The theoretical result was applied to mammalian data to estimate the selection intensity on nonsynonymous mutations in sperm genes.     

Perturbation expansions of multilocus fixation probabilities for frequency-dependent selection with applications to the Hill-Robertson effect and to the joint evolution of helping and punishment

Natural populations are of finite size and organisms carry multilocus genotypes. There are, nevertheless, few results on multilocus models when both random genetic drift and natural selection affect the evolutionary dynamics. In this paper we describe a formalism to calculate systematic perturbation expansions of moments of allelic states around neutrality in populations of constant size. This allows us to evaluate multilocus fixation probabilities (long-term limits of the moments) under arbitrary strength of selection and gene action. We show that such fixation probabilities can be expressed in terms of selection coefficients weighted by mean first passages times of ancestral gene lineages within a single ancestor. These passage times extend the coalescence times that weight selection coefficients in one-locus perturbation formulas for fixation probabilities. We then apply these results to investigate the Hill-Robertson effect and the coevolution of helping and punishment. Finally, we discuss limitations and strengths of the perturbation approach. In particular, it provides accurate approximations for fixation probabilities for weak selection regimes only (Ns?1), but it provides generally good prediction for the direction of selection under frequency-dependent selection.