The stationary distribution of allele frequencies when selection acts at unlinked loci

We consider population genetics models where selection acts at a set of unlinked loci. It is known that if the fitness of an individual is multiplicative across loci, then these loci are independent. We consider general selection models, but assume parent-independent mutation at each locus. For such a model, the joint stationary distribution of allele frequencies is proportional to the stationary distribution under neutrality multiplied by a known function of the mean fitness of the population. We further show how knowledge of this stationary distribution enables direct simulation of the genealogy of a sample at a single-locus. For a specific selection model appropriate for complex disease genes, we use simulation to determine what features of the genealogy differ between our general selection model and a multiplicative model.     

Stabilizing Selection,Purifying Selection,and Mutational Bias in Finite Populations

Genomic traits such as codon usage and the lengths of noncoding sequences may be subject to stabilizing selection rather than purifying selection. Mutations affecting these traits are often biased in one direction. To investigate the potential role of stabilizing selection on genomic traits, the effects of mutational bias on the equilibrium value of a trait under stabilizing selection in a finite population were investigated, using two different mutational models. Numerical results were generated using a matrix method for calculating the probability distribution of variant frequencies at sites affecting the trait, as well as by Monte Carlo simulations. Analytical approximations were also derived, which provided useful insights into the numerical results. A novel conclusion is that the scaled intensity of selection acting on individual variants is nearly independent of the effective population size over a wide range of parameter space and is strongly determined by the logarithm of the mutational bias parameter. This is true even when there is a very small departure of the mean from the optimum, as is usually the case. This implies that studies of the frequency spectra of DNA sequence variants may be unable to distinguish between stabilizing and purifying selection. A similar investigation of purifying selection against deleterious mutations was also carried out. Contrary to previous suggestions, the scaled intensity of purifying selection with synergistic fitness effects is sensitive to population size, which is inconsistent with the general lack of sensitivity of codon usage to effective population size.     

Population Growth Inflates the Per-Individual Number of Deleterious Mutations and Reduces Their Mean Effect

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

The Structure of Genealogies and the Distribution of Fixed Differences between DNA Sequence Samples from Natural Populations

When two samples of DNA sequences are compared, one way in which they may differ is in the presence of fixed differences, which are defined as sites at which all of the sequences in one sample are different from all of the sequences in a second sample. The probability distribution of the number of fixed differences is developed. The theory employs Wright-Fisher genealogies and the infinite sites mutation model. For the case when both samples are drawn randomly from the same population it is found that genealogies permitting fixed differences are very unlikely. Thus the mere presence of fixed differences between samples is statistically significant, even for small samples. The theory is extended to samples from populations that have been separated for some time. The relationship between a simple Poisson distribution of mutations and the distribution of fixed differences is described as a function of the time since populations have been isolated. It is shown how these results may contribute to improved tests of recent balancing or directional selection.     

Estimating the distribution of selection coefficients from phylogenetic data with applications to mitochondrial and viral DNA

The distribution of selection coefficients of new mutations is of key interest in population genetics. In this paper we explore how codon-based likelihood models can be used to estimate the distribution of selection coefficients of new amino acid replacement mutations from phylogenetic data. To obtain such estimates we assume that all mutations at the same site have the same selection coefficient. We first estimate the distribution of selection coefficients from two large viral data sets under the assumption that the viral population size is the same along all lineages of the phylogeny and that the selection coefficients vary among sites. We then implement several new models in which the lineages of the phylogeny may have different population sizes. We apply the new models to a data set consisting of the coding regions from eight primate mitochondrial genomes. The results suggest that there might be little power to determine the exact shape of the distribution of selection coefficient but that the normal and gamma distributions fit the data significantly better than the exponential distribution.     

INBREEDING DEPRESSION UNDER JOINT SELFING,OUTCROSSING, AND ASEXUALITY

Partial asexual reproduction was introduced into a model of inbreeding depression due to nearly recessive lethal mutations in a partially selfing population. The frequencies of asexuality, selfing, and outcrossing were either constant or occurred in cycles of a single sexual generation followed by one or more asexual generations. We found that increasing the degree of asexuality generally increases the inbreeding depression maintained in an equilibrium population with a given selfing rate. This is due to the increase in the number of mutations relative to sexual generations during which selfing-induced purging of mutations may take place. For very high genomic mutation rates, sufficient to produce a threshold rate of self-fertilization for purging recessive lethal mutations, asexuality can have the opposite effect, decreasing equilibrium inbreeding depression, because of an increase in the efficiency of selection against mutations in heterozygotes with asexuality.     

Maintenance of Genetic Variability under the Joint Effect of Mutation, Selection and Random Drift

Formulae are developed for the distribution of allele frequencies (the frequency spectrum), the mean number of alleles in a sample, and the mean and variance of heterozygosity under mutation pressure and under either genic or recessive selection. Numerical computations are carried out by using these formulae and Watterson's (1977) formula for the distribution of allele frequencies under overdominant selection. The following properties are observed: (1) The effect of selection on the distribution of allele frequencies is slight when 4Ns 相似文献   

Population Genetics Inference for Longitudinally-Sampled Mutants Under Strong Selection

Longitudinal allele frequency data are becoming increasingly prevalent. Such samples permit statistical inference of the population genetics parameters that influence the fate of mutant variants. To infer these parameters by maximum likelihood, the mutant frequency is often assumed to evolve according to the Wright–Fisher model. For computational reasons, this discrete model is commonly approximated by a diffusion process that requires the assumption that the forces of natural selection and mutation are weak. This assumption is not always appropriate. For example, mutations that impart drug resistance in pathogens may evolve under strong selective pressure. Here, we present an alternative approximation to the mutant-frequency distribution that does not make any assumptions about the magnitude of selection or mutation and is much more computationally efficient than the standard diffusion approximation. Simulation studies are used to compare the performance of our method to that of the Wright–Fisher and Gaussian diffusion approximations. For large populations, our method is found to provide a much better approximation to the mutant-frequency distribution when selection is strong, while all three methods perform comparably when selection is weak. Importantly, maximum-likelihood estimates of the selection coefficient are severely attenuated when selection is strong under the two diffusion models, but not when our method is used. This is further demonstrated with an application to mutant-frequency data from an experimental study of bacteriophage evolution. We therefore recommend our method for estimating the selection coefficient when the effective population size is too large to utilize the discrete Wright–Fisher model.     

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

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

Population genetics of polymorphism and divergence under fluctuating selection

Current methods for detecting fluctuating selection require time series data on genotype frequencies. Here, we propose an alternative approach that makes use of DNA polymorphism data from a sample of individuals collected at a single point in time. Our method uses classical diffusion approximations to model temporal fluctuations in the selection coefficients to find the expected distribution of mutation frequencies in the population. Using the Poisson random-field setting we derive the site-frequency spectrum (SFS) for three different models of fluctuating selection. We find that the general effect of fluctuating selection is to produce a more "U"-shaped site-frequency spectrum with an excess of high-frequency derived mutations at the expense of middle-frequency variants. We present likelihood-ratio tests, comparing the fluctuating selection models to the neutral model using SFS data, and use Monte Carlo simulations to assess their power. We find that we have sufficient power to reject a neutral hypothesis using samples on the order of a few hundred SNPs and a sample size of approximately 20 and power to distinguish between selection that varies in time and constant selection for a sample of size 20. We also find that fluctuating selection increases the probability of fixation of selected sites even if, on average, there is no difference in selection among a pair of alleles segregating at the locus. Fluctuating selection will, therefore, lead to an increase in the ratio of divergence to polymorphism similar to that observed under positive directional selection.     

Evolution of the human immunodeficiency virus envelope gene is dominated by purifying selection

The evolution of the human immunodeficiency virus (HIV-1) during chronic infection involves the rapid, continuous turnover of genetic diversity. However, the role of natural selection, relative to random genetic drift, in governing this process is unclear. We tested a stochastic model of genetic drift using partial envelope sequences sampled longitudinally in 28 infected children. In each case the Bayesian posterior (empirical) distribution of coalescent genealogies was estimated using Markov chain Monte Carlo methods. Posterior predictive simulation was then used to generate a null distribution of genealogies assuming neutrality, with the null and empirical distributions compared using four genealogy-based summary statistics sensitive to nonneutral evolution. Because both null and empirical distributions were generated within a coalescent framework, we were able to explicitly account for the confounding influence of demography. From the distribution of corrected P-values across patients, we conclude that empirical genealogies are more asymmetric than expected if evolution is driven by mutation and genetic drift only, with an excess of low-frequency polymorphisms in the population. This indicates that although drift may still play an important role, natural selection has a strong influence on the evolution of HIV-1 envelope. A negative relationship between effective population size and substitution rate indicates that as the efficacy of selection increases, a smaller proportion of mutations approach fixation in the population. This suggests the presence of deleterious mutations. We therefore conclude that intrahost HIV-1 evolution in envelope is dominated by purifying selection against low-frequency deleterious mutations that do not reach fixation.     

Group selection in plant populations

Summary Several mechanisms have been proposed for group selection, to account for the evolution of altruistic traits. One type, Neighbourhood models, suggests that individuals react with those immediately around them, but with no recognition mechanism. The organization of plant populations seems especially favorable for this type of selection. The possibility of Neighbourhood selection was investigated by simulating a plant population. It was possible for an altruistic trait to evolve, though only under restricted conditions. The main requirement was gene flow only by very restricted pollen dispersal, and a high benefit : cost ratio in the altruistic relationship. Under conditions favourable for such evolution, the starting frequency of the allele, the initial pattern, and the population size, had little effect. Inbreeding tended to prevent the increase of the altruism allele, though this depended on the mechanism of selfing. Known ecological features of plants are discussed that could be considered altruistic and hence require some form of group selection for their evolution, and whether the benefit : cost requirements are likely to be met. Neighbourhood models of group selection are a possibility in plant populations, and we therefore cannot exclude the possibility of altruism in plants. However, Neighbourhood selection is weak force, unlikely to be effective in the face of opposing individual selection. It may be more important as reinforcement of individual selection.     

IS THE POPULATION SIZE OF A SPECIES RELEVANT TO ITS EVOLUTION?

Abstract This paper examines aspects of genetic draft, the stochastic force induced by substitutions at one locus on the dynamics of a closely linked locus. Of particular interest is the role of population size on genetic draft. Remarkably, the rate of substitution of weakly selected advantageous mutations decreases with increasing population size, whereas that for deleterious mutations increases with population size. This dependency on population size is the opposite of that for genetic drift. Moreover, these rates are only weakly dependent on population size, again contrary to the strong dependency of drift‐based dynamics. Four models of the strongly selected loci responsible for genetic draft are examined. Three of these exhibit a very weak dependency on population size, which implies that their induced effects will also be weakly dependent on population size. Together, these results suggest that population size and binomial sampling may not be relevant to a species' evolution. If this is the case, then a number of evolutionary conundrums are resolved.     

Most recent common ancestor probability distributions in gene genealogies under selection

A computational study is made of the conditional probability distribution for the allelic type of the most recent common ancestor in genealogies of samples of n genes drawn from a population under selection, given the initial sample configuration. Comparisons with the corresponding unconditional cases are presented. Such unconditional distributions differ from samples drawn from the unique stationary distribution of population allelic frequencies, known as Wright's formula, and are quantified. Biallelic haploid and diploid models are considered. A simplified structure for the ancestral selection graph of S. M. Krone and C. Neuhauser (1997, Theor. Popul. Biol. 51, 210-237) is enhanced further, reducing the effective branching rate in the graph. This improves efficiency of such a nonneutral analogue of the coalescent for use with computational likelihood-inference techniques.     

Dynamics of inbreeding depression due to deleterious mutations in small populations: mutation parameters and inbreeding rate

A multilocus stochastic model is developed to simulate the dynamics of mutational load in small populations of various sizes. Old mutations sampled from a large ancestral population at mutation-selection balance and new mutations arising each generation are considered jointly, using biologically plausible lethal and deleterious mutation parameters. The results show that inbreeding depression and the number of lethal equivalents due to partially recessive mutations can be partly purged from the population by inbreeding, and that this purging mainly involves lethals or detrimentals of large effect. However, fitness decreases continuously with inbreeding, due to increased fixation and homozygosity of mildly deleterious mutants, resulting in extinctions of very small populations with low reproductive rates. No optimum inbreeding rate or population size exists for purging with respect to fitness (viability) changes, but there is an optimum inbreeding rate at a given final level of inbreeding for reducing inbreeding depression or the number of lethal equivalents. The interaction between selection against partially recessive mutations and genetic drift in small populations also influences the rate of decay of neutral variation. Weak selection against mutants relative to genetic drift results in apparent overdominance and thus an increase in effective size (Ne) at neutral loci, and strong selection relative to drift leads to a decrease in Ne due to the increased variance in family size. The simulation results and their implications are discussed in the context of biological conservation and tests for purging.     

Haplotype-based inference of the distribution of fitness effects

Recent genome sequencing studies with large sample sizes in humans have discovered a vast quantity of low-frequency variants, providing an important source of information to analyze how selection is acting on human genetic variation. In order to estimate the strength of natural selection acting on low-frequency variants, we have developed a likelihood-based method that uses the lengths of pairwise identity-by-state between haplotypes carrying low-frequency variants. We show that in some nonequilibrium populations (such as those that have had recent population expansions) it is possible to distinguish between positive or negative selection acting on a set of variants. With our new framework, one can infer a fixed selection intensity acting on a set of variants at a particular frequency, or a distribution of selection coefficients for standing variants and new mutations. We show an application of our method to the UK10K phased haplotype dataset of individuals.     

Mutation-selection balance in mixed mating populations

An approximation to the average number of deleterious mutations per gamete, Q, is derived from a model allowing selection on both zygotes and male gametes. Progeny are produced by either outcrossing or self-fertilization with fixed probabilities. The genetic model is a standard in evolutionary biology: mutations occur at unlinked loci, have equivalent effects, and combine multiplicatively to determine fitness. The approximation developed here treats individual mutation counts with a generalized Poisson model conditioned on the distribution of selfing histories in the population. The approximation is accurate across the range of parameter sets considered and provides both analytical insights and greatly increased computational speed. Model predictions are discussed in relation to several outstanding problems, including the estimation of the genomic deleterious mutation rates (U), the generality of "selective interference" among loci, and the consequences of gametic selection for the joint distribution of inbreeding depression and mating system across species. Finally, conflicting results from previous analytical treatments of mutation-selection balance are resolved to assumptions about the life-cycle and the initial fate of mutations.     

Beneficial mutation selection balance and the effect of linkage on positive selection

When beneficial mutations are rare, they accumulate by a series of selective sweeps. But when they are common, many beneficial mutations will occur before any can fix, so there will be many different mutant lineages in the population concurrently. In an asexual population, these different mutant lineages interfere and not all can fix simultaneously. In addition, further beneficial mutations can accumulate in mutant lineages while these are still a minority of the population. In this article, we analyze the dynamics of such multiple mutations and the interplay between multiple mutations and interference between clones. These result in substantial variation in fitness accumulating within a single asexual population. The amount of variation is determined by a balance between selection, which destroys variation, and beneficial mutations, which create more. The behavior depends in a subtle way on the population parameters: the population size, the beneficial mutation rate, and the distribution of the fitness increments of the potential beneficial mutations. The mutation-selection balance leads to a continually evolving population with a steady-state fitness variation. This variation increases logarithmically with both population size and mutation rate and sets the rate at which the population accumulates beneficial mutations, which thus also grows only logarithmically with population size and mutation rate. These results imply that mutator phenotypes are less effective in larger asexual populations. They also have consequences for the advantages (or disadvantages) of sex via the Fisher-Muller effect; these are discussed briefly.     

Transition Densities and Sample Frequency Spectra of Diffusion Processes with Selection and Variable Population Size

Advances in empirical population genetics have made apparent the need for models that simultaneously account for selection and demography. To address this need, we here study the Wright–Fisher diffusion under selection and variable effective population size. In the case of genic selection and piecewise-constant effective population sizes, we obtain the transition density by extending a recently developed method for computing an accurate spectral representation for a constant population size. Utilizing this extension, we show how to compute the sample frequency spectrum in the presence of genic selection and an arbitrary number of instantaneous changes in the effective population size. We also develop an alternate, efficient algorithm for computing the sample frequency spectrum using a moment-based approach. We apply these methods to answer the following questions: If neutrality is incorrectly assumed when there is selection, what effects does it have on demographic parameter estimation? Can the impact of negative selection be observed in populations that undergo strong exponential growth?     

Summary statistics of neutral mutations in longitudinal DNA samples

Longitudinal samples of DNA sequences are the DNA sequences sampled from the same population at different time points. For fast evolving organisms, e.g. RNA virus, these kind of samples have increasingly been used to study the evolutionary process in action. Longitudinal samples provide some interesting new summary statistics of genetic variation, such as the frequency of mutation of size i in one sample and size j in another, the average number of mutations accumulated since the common ancestor of two sequences each from a different sample, and number of private, shared and fixed mutations within samples. To make the results more applicable, we used in this study a general two-sample model, which assumes two longitudinal samples were taken from the same measurably evolving population. Inspired by the HIV study, we also studied a two-sample-two-stage model, which is a special case of two-sample model and assumes a treatment after the first sampling instantaneously changes the population size. We derived the formulas for calculating statistical properties, e.g. expectations, variances and covariances, of these new summary statistics under the two models. Potential applications of these results were discussed.