Theoretical Population Biology

Interpretations of values of the FST measure of genetic differentiation rely on an understanding of its mathematical constraints. Previously, it has been shown that FST values computed from a biallelic locus in a set of multiple populations and FST values computed from a multiallelic locus in a pair of populations are mathematically constrained as a function of the frequency of the allele that is most frequent across populations. We generalize from these cases to report here the mathematical constraint on FST given the frequency M of the most frequent allele at a multiallelic locus in a set of multiple populations. Using coalescent simulations of an island model of migration with an infinitely-many-alleles mutation model, we argue that the joint distribution of FST and M helps in disentangling the separate influences of mutation and migration on FST. Finally, we show that our results explain a puzzling pattern of microsatellite differentiation: the lower FST in an interspecific comparison between humans and chimpanzees than in the comparison of chimpanzee populations. We discuss the implications of our results for the use of FST.

The consequences of assortative model in the additive polygenic model have been extensively explored by several authors. In this note we extend their results by introducing a correlation between parental and offspring environments.

The Wright-Fisher diffusion and its dual, the coalescent process, are at the core of many results and methods in population genetics. Approaches have been developed to study the dynamics of its moments under genetic drift, mutation, and recombination using ordinary differential equations. The dynamics of these moments can be used to study population genetic processes and are key building blocks of efficient methods to infer population genetic parameters, like demographic histories or fine-scale recombination rates. However, the system of equations does not close under recombination; that is, computing moments of a certain order requires knowledge of moments of higher order. By applying a coordinate transformation to the diffusion generator, we show that the canonical moments in these alternative coordinates yield a closed system, enabling more accurate numerical computations. Compared to previous approaches in the literature, we believe that this approach can be more readily extended to general scenarios. Through simulations, we verify that the derived system of differential equations can accurately capture the dynamics of the moments, and can be used to efficiently compute expected diversity and linkage statistics in population genetic samples.

AO_SCPLOWBSTRACTC_SCPLOWThe site frequency spectrum (SFS) is a statistic that summarizes the distribution of derived allele frequencies in a sample of DNA sequences. The SFS provides useful information about genetic variation within and among populations and it can used to make population genetic inferences. Methods for computing the SFS based on the diffusion approximation are computationally efficient when computing all terms of the SFS simultaneously and they can handle complicated demographic scenarios. However, in practice it is sometimes only necessary to compute a subset of terms of the SFS, in which case coalescent-based methods can achieve greater computational efficiency. Here, we present simple and accurate approximate formulas for the expected joint SFS for multiple populations connected by migration. Compared with existing exact approaches, our approximate formulas greatly reduce the complexity of computing each entry of the SFS and have simple forms. The computational complexity of our method depends on the index of the entry to be computed, rather than on the sample size, and the accuracy of our approximation improves as the sample size increases.

Selection against deleterious mutations, called purifying selection, plays a central role in evolution and acts in all populations. It is known that the genetic patterns observed in genomic regions undergoing purifying selection differ from those resulting from neutral evolution. However, a comprehensive understanding of the underlying mechanisms shaping those patterns is still lacking. In the present work, we use simulations combined with a genealogical approach to identify the effect of purifying selection on the ancestry and thus on the genetic diversity. Our analysis relies on the postulate that the genealogy belongs to the universality class of Beta-coalescents. Under this assumption, we derive statistics measuring the distortion of the genealogy. This approach allows us to consider a wide range of regimes (i.e. arbitrary selection and mutation strengths) and uncover a rich phase diagram. We find that, for strong selection, the limiting genealogy is given by Kingmans coalescent on a polynomial timescale. As selection gets weaker, Mullers ratchet starts operating, setting off the emergence of multiple mergers in the genealogical structures. Our results show that while multiple-merger coalescents are often interpreted as the signature of selective sweeps in rapidly adapting populations, these structures can also appear in the context of Mullers ratchet.

Recent analyses of the effects that organismal genealogies or pedigrees of populations have on times to common ancestry for samples of genetic data are extended to cases of population subdivision and migration. Traditional coalescent models marginalize over pedigrees. A finding of a pedigree effect implies that data analysis and interpretation should not be based on the corresponding traditional coalescent model but rather on a coalescent model obtained by conditioning on the pedigree. We apply a straightforward test based on the distribution of pairwise coalescence times to four previously described scenarios of subdivision and migration. These scenarios are defined by the relative magnitudes of four parameters: the number of the local populations or demes, the deme size, the migration fraction, and the probability that migration can occur at all. We find pedigree effects in three scenarios. In two, the effect is weak if the deme size is large. The one scenario without a pedigree effect corresponds to the well known structured-coalescent model. The one scenario with a persistent pedigree effect even in the limit as the deme size tends to infinity involves long periods without gene flow interrupted by pulses of migration. We illustrate our results using simulations and numerical analysis.

Individual recruitment success, or the offspring number distribution of a given population, is a fundamental element in ecology and evolution. Sweepstakes reproduction refers to a highly skewed individual recruitment success without involving natural selection and may apply to individuals in broadcast spawning populations characterised by Type III survivorship. We consider an extension of the Schweinsberg (2003) model of sweepstakes reproduction for a haploid panmictic population of constant size N; the extension also works as an alternative to the Wright-Fisher model. Our model incorporates an upper bound on the random number of potential offspring (juveniles) produced by a given individual. Depending on how the bound behaves relative to the total population size, we obtain the Kingman (1982a,c,b) coalescent, an incomplete Beta-coalescent, or the (complete) Beta-coalescent of Schweinsberg (2003). We argue that applying such an upper bound is biologically reasonable. Moreover, we estimate the error of the coalescent approximation. The error estimates reveal that convergence can be slow, and small sample size can be sufficient to invalidate convergence, for example if the stated bound is of the form N/ log N. We use simulations to investigate the effect of increasing sample size on the site-frequency spectrum. When the limit is a Beta-coalescent, the site frequency spectrum will be as predicted by the limiting tree even though the full coalescent tree may deviate from the limiting one. When in the domain of attraction of the Kingman coalescent the effect of increasing sample size depends on the effective population size as has been noted in the case of the Wright-Fisher model. Conditioning on the population ancestry (the random ancestral relations of the entire population at all times) may have little effect on the site-frequency spectrum for the models considered here (as evidenced by simulation results).

1Neutral theory assumes that in a population of size N, diversity results from an equilibrium between new mutations arising at rate {micro} and genetic drift that purge them at rate 1/N, predicting an equilibrium value proportional to N{micro}. The difference between this expectation and the much lower observed molecular diversity is known as Lewontins paradox of variation. Here, we investigate the effect of genetic draft, a regime of evolution where recurrent sparse selective sweeps entirely drive the diversity of surrounding loci. More specifically, we focus on the neglected distant effect of selective sweeps on remote neutral loci, where the effect of a single sweep is almost negligible. We derived novel mathematical approximations of this underexplored regime and show that under weak genetic draft, diversity at neutral loci is a power law of the population size: [Formula], for A < 0.5, where A is the ratio between recombination rate and coefficient of selection (A = c/s). Interestingly the Site Frequency Spectrum at neutral loci is identical to the one produced by genetic drift, as the underlying coalescent tree is an n-Kingman coalescent. In brief, weak genetic draft produces patterns of diversity that look entirely neutral, while being drastically reduced in magnitude. Ultimately, our study points to the need to explore evolutionary models for which diversity looks neutral but does not scale linearly with population size.

Shuffling ones genetic material with another individual seems a risky endeavor more likely to decrease than to increase offspring fitness. This intuitive argument is commonly employed to explain why the ubiquity of sex and recombination in nature is enigmatic. It is predicated on the notion that natural selection assembles selectively well-matched combinations of genes that recombination would break up resulting in low-fitness offspring - a notion often stated in the literature as a self-evident premise. We show however that, upon closer examination, this premise is flawed: we find to the contrary that natural selection in fact has an encompassing tendency to assemble selectively mismatched gene combinations; recombination breaks up these selectively mismatched combinations (on average), assembles selectively matched combinations, and should thus be favored. The new perspective our findings offer suggests that sex and recombination are not so enigmatic but are instead unavoidable byproducts of natural selection.

1In population genetics, accurately inferring the selection coefficient and the time of onset of advantageous mutations from genetic data is fundamental for understanding evolutionary processes. Here, we investigate how mismatches between the true evolutionary process and the inference model--specifically in the reproductive variance ({sigma}2) and the number of generations (L)--affect the posterior distributions of the selection coefficient and the time of onset. Using the Kolmogorov forward and backward equations, we model the stochastic dynamics of gene frequencies under selection and drift. We show that while the posterior distribution of the selection coefficient remains unaffected by changes in{sigma} 2 and L, this invariance does not apply to the time of onset. By framing the problem as a first passage time issue, we derive explicit expressions for the offsets in the posterior mean and variance of the time of onset that result from incorrect assumptions about{sigma} 2 and L. Our analysis reveals that these offsets are related to the mean and variance of the first passage time required for the allele frequency to reach a certain threshold, starting from an initial frequency determined by the model parameters. Under the assumption of a uniform prior for the time of onset, we find that the offset in the inferred mean is given by the difference in the effective generation duration ({Delta} = 1/{sigma}2) between the true process and the inference model. We validate our theoretical findings through simulations, demonstrating that the empirical offsets closely match our predictions. Furthermore, we generalize our results to accommodate non-uniform prior distributions, such as exponential priors, and provide numerical methods for calculating offsets under arbitrary priors. Stochastic fluctuations due to genetic drift, which are influenced by the reproductive variance and generational structure, can introduce significant biases in the posterior distribution of time of onset of advantageous mutations. By quantifying these biases, our framework enables more accurate adjustments to inferences drawn from genetic data, thereby enhancing our understanding of evolutionary dynamics and improving the reliability of population genetic analyses.

We model the introgression of a genome with many weakly selected linked loci into a large homogeneous population, under the simple assumption that a block of introduced genome has a selective effect proportional to its map length. Using a diffusion approximation, we compute the probability that some part of the initial genome survives the initial phase of the introgression and give the typical length of the surviving blocks. Our results quantify the effect of recombination on selection and drift during an introgression and indicate that the fate of the genome depends on the strength of selection relative to recombination. When selection is positive some parts of the genome are able to survive at large times, but large blocks can only persist if selection is stronger than recombination. Surprisingly, the probability of such a successful introgression is independent of the strength of recombination and is the same as that for a single beneficial allele. Conversely, a deleterious or neutral genome is eventually lost, but at a much slower rate than a single allele with the same selective effect. In this case, surviving blocks are very small. We also consider the introgression of a genome made of a single beneficial allele linked to a deleterious background and compute the amount of deleterious material that hitchhikes during fixation.

Recruitment dynamics, or the distribution of the number of offspring among individuals, is central for understanding ecology and evolution. Sweepstakes reproduction (heavy right-tailed offspring number distribution) is central for understanding the ecology and evolution of highly fecund natural populations. Sweepstakes reproduction can induce jumps in type frequencies and multiple mergers in gene genealogies of sampled gene copies. We take sweepstakes reproduction to be skewed offspring number distribution due to mechanisms not involving natural selection, such as in chance matching of broadcast spawning with favourable environmental conditions. Here, we consider population genetic models of sweepstakes reproduction in a diploid panmictic populations absent selfing and evolving in a random environment. Our main results are (i) continuous-time Beta and Poisson-Dirichlet coalescents, when combining the results the skewness parameter of the Beta-coalescent ranges from 0 to 2, and the Beta-coalescents may be incomplete due to an upper bound on the number of potential offspring produced by any pair of parents; (ii) in large populations time is measured in units proportional to either N/ log N or N generations (where 2N is the population size when constant); (iii) it follows that incorporating population size changes leads to time-changed coalescents with the time-change independent of ; (iv) using simulations we show that the ancestral process is not well approximated by the corresponding coalescent (as measured through certain functionals of the processes); (v) whenever the skewness of the offspring number distribution is increased the conditional (conditioned on the population ancestry) and the unconditional ancestral processes are not in good agreement.

Over the past decade there has been an increasing focus on the application of the Wright-Fisher diffusion to the inference of natural selection from genetic time series. A key ingredient for modelling the trajectory of gene frequencies through the Wright-Fisher diffusion is its transition probability density function. Recent advances in DNA sequencing techniques have made it possible to monitor genomes in great detail over time, which presents opportunities for investigating natural selection while accounting for genetic recombination and local linkage. However, most existing methods for computing the transition probability density function of the Wright-Fisher diffusion are only applicable to one-locus problems. To address two-locus problems, in this work we propose a novel numerical scheme for the Wright-Fisher stochastic differential equation of population dynamics under natural selection at two linked loci. Our key innovation is that we reformulate the stochastic differential equation in a closed form that is amenable to simulation, which enables us to avoid boundary issues and reduce computational costs. We also propose an adaptive importance sampling approach based on the proposal introduced by Fearnhead (2008) for computing the transition probability density of the Wright-Fisher diffusion between any two observed states. We show through extensive simulation studies that our approach can achieve comparable performance to the method of Fearnhead (2008) but can avoid manually tuning the parameter{rho} to deliver superior performance for different observed states.

The ubiquity of sex and recombination in nature has eluded unified explanation since the time of Darwin. Conditions that promote the evolution of recombination, broadly defined as any form of genetic mixing, are fairly well understood: it is favored when genomes tend to contain more selectively mismatched combinations of alleles than can be explained by chance alone. Yet, while a variety of theoretical approaches have been put forth to explain why such conditions would prevail in natural populations, each has turned out to be of limited scope and applicability. Here, we show, simply and surprisingly, that natural selection acting on standing heritable variation always creates conditions favoring the evolution of recombination, in expectation. Specifically, we find that, in expectation: 1) the mean selective advantage of recombinants is non-negative, 2) the mean selective advantage of a recombination-competent modifier is non-negative, and 3) the asymptotic frequency of a recombination-competent modifier is close to one and is independent of the strength of selection. Remarkably, these findings are independent of the distribution of genic fitnesses in the standing heritable variation upon which natural selection acts, implying that the source of this variation is immaterial. Taken together, our findings indicate that: 1) the evolution of recombination should be promoted in expectation wherever natural selection is operating, and 2) sex and recombination may have evolved more as a byproduct than as a catalyst of natural selection.

Recruitment dynamics, or the distribution of the number of offspring among individuals, is fundamental to ecology and evolution. We take sweepstakes reproduction to mean a skewed (heavy right-tailed) offspring number distribution without natural selection being involved. Sweepstakes may be generated by chance matching of reproduction with favorable environmental conditions. Gene genealogies generated by sweepstakes reproduction are in the domain of attraction of multiple-merger coalescents where a random number of lineages merges at such times. We consider population genetic models of sweepstakes reproduction for haploid panmictic populations of both constant (N), and varying population size, and evolving in a random environment. We construct our models so that we can recover the observed number of new mutations in a given sample without requiring strong assumptions regarding the population size or the mutation rate. Our main results are (i) continuous-time coalescents that are either the Kingman coalescent or specific families of Beta- or Poisson-Dirichlet coalescents; when combining the results the parameter of the Beta-coalescent ranges from 0 to 2, and the Beta-coalescents may be incomplete due to an upper bound on the number of potential offspring an arbitrary individual may produce; (ii) in large populations we measure time in units proportional to either N/log N or N generations; (iii) incorporating fluctuations in population size leads to time-changed multiple-merger coalescents where the time-change does not depend on ; (iv) using simulations we show that in some cases approximations of functionals of a given coalescent do not match the ones of the ancestral process in the domain of attraction of the given coalescent; (v) approximations of functionals obtained by conditioning on the population ancestry (the ancestral relations of all gene copies at all times) are broadly similar (for the models considered here) to the approximations obtained without conditioning on the population ancestry.

We study the behaviour of the F2-statistic and Fst-statistic, respectively, over time in a Wright-Fisher model with mutation and migration. We give precise conditions for when the F2-statistic is non-monotonic, that is, increases over time until a certain point and then starts decreasing. We show that even for small population sizes, the two statistics are well approximated by population size scaled expressions.

Our recent work on molecular evolution and population genetics postulated that individuals with a specific mutation exhibit a fluctuation in fitness, short for FSI (fluctuating selection among individuals), whereas the fitness effect of wildtype remains a constant. An intriguing phenomenon called selection-duality emerges, that is, a slightly beneficial mutation could be a negative selection (the substitution rate less than the mutation rate). It appears that selection-duality is bounded by two bounds: the generic neutrality where the mutation is neutral by the means of fitness on average, and the substitution neutrality where the substitution rate equals to the mutation rate. In addition, the middle point of generic neutrality and substitution neutrality is called the FSI-neutrality. An important problem is about the age profile of allele frequency, i.e., the arising timing of a mutation whose frequency in the current population is given (the allele-age problem for short). Solving this problem under selection duality would help extend the standard coalescent theory that based on strict neutrality to a more general form under selection duality. In this paper, we studied the allele-age problem under selection-duality by the first arrival time approach and the mean age approach, respectively. Since the general solution of allele-age problem under selection duality is not available, we focused on solving the problem at the substitution neutrality (the up-bound of selection duality), the FSI-neutrality (the middle-point) and the generic neutrality (the low-bound), respectively. Our analysis results in an overall picture that the mean first-arrival age of a mutation at the substitution neutrality is theoretically identical to that at the FSI-neutrality, which is numerically close to that at the generic neutrality. For illustration, we calculated the mean age of nonsynonymous mutations in the human population and demonstrated that the estimated allele-age could be overestimated considerably when the effect of FSI was neglected.

Motivated originally by the increasing number of examples where a lone member of a once thriving population or even species now survives, we investigate what genomes of that population the survivor may represent. More generally it is of interest to consider what genomes of our ancestry each of us may represent. We consider only diploid dioecious organisms, and consider primarily the ancestry of a haploid genome, for example the maternal autosomes of the focal individual. Our ancestors are many and, in an unbounded population, increase exponentially in number. Our genetic ancestors are few, bounded by the number of ancestral genome segments which increases linearly over past generations. First we show that the major loss of potential ancestral lineages is at 8-11 generations, and that thereafter the number of genetic ancestors increases approximately linearly, but does not approach the upper bound. Over many generations, there remain tightly linked but not contiguous segments that result from the same ancestral lineage. Second we analyze the process of these "repeated" ancestral segments that continue to be formed in distant ancestry, even as others are lost by recombination. Thirdly, we consider the effect of a finite population, with one model of a geographically structured population. Ancestors are many, and soon fill the entire species range even with low migration rates. Genetic ancestors are not only few, but remain geographically local.

Despite increasing data from population-wide sequencing studies, the risk for recessive disorders in consanguineous partnerships is still heavily debated. An important aspect that has not sufficiently been investigated theoretically, is the influence of inbreeding on mutation load and incidence rates when the population sizes change. We therefore developed a model to study these dynamics for a wide range of growth and mating conditions. In the phase of population expansion and shortly afterwards, our simulations show that there is a drop of diseased individuals at the expense of an increasing mutation load for random mating, while both parameters remain almost constant in highly consanguineous partnerships. This explains the empirical observation in present times that a high degree of consanguinity is associated with an increased risk of autosomal recessive disorders. However, it also states that the higher frequency of severe recessive disorders with developmental delay in inbred populations is a transient phenomenon before a mutation-selection balance is reached again. Author summaryWhat determines the recessive disease burden? The empirical observation that the proportion of intellectual disability of autosomal recessive cause is usually higher in offspring of consanguineous partnerships may lead to a misunderstanding about the mechanisms at work. In any population, selection removes pathogenic alleles from the gene pool while the de novo mutation rate adds novel pathogenic alleles. For comparable mutation rates, the incidence of severe recessive disease should be comparable in mutation-selection balance, regardless of the mating scheme. Different incidences can therefore only be explained by population dynamics that are not in equilibrium. We studied in simulated populations the time scales in which mutation-selection balance is reached after a growth phase and found that the mating scheme has a big impact on this lag time. When cousins mate preferentially with cousins, a few generations after a ten-fold increase in size the new equilibrium is established. In contrast, for random-mating, the transient advantage of a lower incidence may last for hundreds of generations, while the mutation load increases. By this means, our findings also highlight the importance of better carrier screens in the future for genetic consultations.

The evolutionary mechanism that drove the establishment of self-incompatibility in early sexual eukaryotes is still a debated topic. While a number of competing hypotheses have been proposed, many have not received detailed theoretical attention. In particular, the hypothesis that self-incompatibility increases the benefits of genetic recombination in sexual haploids has been comparatively understudied. In this paper we address this topic by mathematically deriving how the probability of mating with a genetically distinct individual changes as a function of the presence or absence of self-incompatible mating type classes. We find that although populations with mating types successfully engage in sexual reproduction less frequently than their self-compatible competitors, they can nevertheless engage in useful sex with genetically distinct partners more frequently. This conclusion holds when the number of sexual reproductive events per generation is low (i.e. in small populations with low rates of facultative sexual reproduction). Finally we demonstrate the potential for frequency-dependent selection in competitive dynamics between self-compatible and self-incompatible types. These analytic results provide a baseline for studying the sex advantage enhancer model for the evolutionary origin of mating types within each specific hypothesis for the evolution of recombination. PACS87.23.-n Ecology and evolution, 87.23.Kg Dynamics of evolution, 02.50.Ey Stochastic processes 2000 MSC37N25: Dynamical systems in biology, 60J70: Applications of diffusion theory (population genetics, absorption problems, etc.)