The Age of a Neutral Mutant Persisting in a Finite Population

Formulae for the mean and the mean square age of a neutral allele which is segregating with frequency x in a population of effective size N(e) have been obtained using the diffusion equation method, for the case of 4N(e)v<1 where v is the mutation rate. It has been shown that the average ages of neutral alleles, even if their frequencies are relatively low, are quite old. For example, a neutral mutant whose current frequency is 10% has the expected age roughly equal to the effective population size N(e) and the standard deviation 1.4N(e) (in generations), assuming that this mutant has increased by random drift from a very low frequency. Also, formulae for the mean "first arrival time" of a neutral mutant to a certain frequency x have been presented. In addition, a new, approximate method has been developed which enables us to obtain the condition under which frequencies of "rare" polymorphic alleles among local populations are expected to be uniform if the alleles are selectively neutral.-It was concluded that exchange of only a few individuals on the average between adjacent colonies per generation is enough to bring about such a uniformity of frequencies.     

Analytical results on the neutral non-equilibrium allele frequency spectrum based on diffusion theory

The allele frequency spectrum has attracted considerable interest for the simultaneous inference of the demographic and adaptive history of populations. In a recent study, Evans et al. (2007) developed a forward diffusion equation describing the allele frequency spectrum, when the population is subject to size changes, selection and mutation. From the diffusion equation, the authors derived a system of ordinary differential equations (ODEs) for the moments in a Wright–Fisher diffusion with varying population size and constant selection. Here, we present an explicit solution for this system of ODEs with variable population size, but without selection, and apply this result to derive the expected spectrum of a sample for time-varying population size. We use this forward-in-time-solution of the allele frequency spectrum to obtain the backward-in-time-solution previously derived via coalescent theory by Griffiths and Tavaré (1998). Finally, we discuss the applicability of the theoretical results to the analysis of nucleotide polymorphism data.     

The effect of linkage and population size on inbreeding depression due to mutational load.

Using a stochastic model of a finite population in which there is mutation to partially recessive detrimental alleles at many loci, we study the effects of population size and linkage between the loci on the population mean fitness and inbreeding depression values. Although linkage between the selected loci decreases the amount of inbreeding depression, neither population size nor recombination rate have strong effects on these quantities, unless extremely small values are assumed. We also investigate how partial linkage between the loci that determine fitness affects the invasion of populations by alleles at a modifier locus that controls the selfing rate. In most of the cases studied, the direction of selection on modifiers was consistent with that found in our previous deterministic calculations. However, there was some evidence that linkage between the modifier locus and the selected loci makes outcrossing less likely to evolve; more losses of alleles promoting outcrossing occurred in runs with linkage than in runs with free recombination. We also studied the fate of neutral alleles introduced into populations carrying detrimental mutations. The times to loss of neutral alleles introduced at low frequency were shorter than those predicted for alleles in the absence of selected loci, taking into account the reduction of the effective population size due to inbreeding. Previous studies have been confined to outbreeding populations, and to alleles at frequencies close to one-half, and have found an effect in the opposite direction. It therefore appears that associations between neutral and selected loci may produce effects that differ according to the initial frequencies of the neutral alleles.     

Maintenance of Genetic Variability under the Pressure of Neutral and Deleterious Mutations in a Finite Population

In order to assess the effect of deleterious mutations on various measures of genic variation, approximate formulas have been developed for the frequency spectrum, the mean number of alleles in a sample, and the mean homozygosity; in some particular cases, exact formulas have been obtained. The assumptions made are that two classes of mutations exist, neutral and deleterious, and that selection is strong enough to keep deleterious alleles in low frequencies, the mode of selection being either genic or recessive. The main findings are: (1) If the expected value (q) of the sum of the frequencies of deleterious alleles is about 10% or less, then the presence of deleterious alleles causes only a minor reduction in the mean number of neutral alleles in a sample, as compared to the case of q = 0. Also, the low- and intermediate-frequency parts of the frequency spectrum of neutral alleles are little affected by the presence of deleterious alleles, though the high-frequency part may be changed drastically. (2) The contribution of deleterious mutations to the expected total number of alleles in a sample can be quite large even if q is only 1 or 2%. (3) The mean homozygosity is roughly equal to (1--2q)/(1 + theta 1), where theta 1 is twice the number of new neutral mutations occurring in each generation in the total population. Thus, deleterious mutations increase the mean heterozygosity by about 2q/(1 + theta 1). The present results have been applied to study the controversial problem of how deleterious mutations may affect the testing of the neutral mutation hypothesis.     

Effect of balancing selection on spatial genetic structure within populations: theoretical investigations on the self-incompatibility locus and empirical studies in Arabidopsis halleri

The effect of selection on patterns of genetic structure within and between populations may be studied by contrasting observed patterns at the genes targeted by selection with those of unlinked neutral marker loci. Local directional selection on target genes will produce stronger population genetic structure than at neutral loci, whereas the reverse is expected for balancing selection. However, theoretical predictions on the intensity of this signal under precise models of balancing selection are still lacking. Using negative frequency-dependent selection acting on self-incompatibility systems in plants as a model of balancing selection, we investigated the effect of such selection on patterns of spatial genetic structure within a continuous population. Using numerical simulations, we tested the effect of the type of self-incompatibility system, the number of alleles at the self-incompatibility locus and the dominance interactions among them, the extent of gene dispersal, and the immigration rate on spatial genetic structure at the selected locus and at unlinked neutral loci. We confirm that frequency-dependent selection is expected to reduce the extent of spatial genetic structure as compared to neutral loci, particularly in situations with low number of alleles at the self-incompatibility locus, high frequency of codominant interactions among alleles, restricted gene dispersal and restricted immigration from outside populations. Hence the signature of selection on spatial genetic structure is expected to vary across species and populations, and we show that empirical data from the literature as well as data reported here on three natural populations of the herb Arabidopsis halleri confirm these theoretical results.     

The effects of multilocus balancing selection on neutral variability

We studied the effect of multilocus balancing selection on neutral nucleotide variability at linked sites by simulating a model where diallelic polymorphisms are maintained at an arbitrary number of selected loci by means of symmetric overdominance. Different combinations of alleles define different genetic backgrounds that subdivide the population and strongly affect variability. Several multilocus fitness regimes with different degrees of epistasis and gametic disequilibrium are allowed. Analytical results based on a multilocus extension of the structured coalescent predict that the expected linked neutral diversity increases exponentially with the number of selected loci and can become extremely large. Our simulation results show that although variability increases with the number of genetic backgrounds that are maintained in the population, it is reduced by random fluctuations in the frequencies of those backgrounds and does not reach high levels even in very large populations. We also show that previous results on balancing selection in single-locus systems do not extend to the multilocus scenario in a straightforward way. Different patterns of linkage disequilibrium and of the frequency spectrum of neutral mutations are expected under different degrees of epistasis. Interestingly, the power to detect balancing selection using deviations from a neutral distribution of allele frequencies seems to be diminished under the fitness regime that leads to the largest increase of variability over the neutral case. This and other results are discussed in the light of data from the Mhc.     

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   

The Coalescent Process in Models with Selection

Statistical properties of the process describing the genealogical history of a random sample of genes are obtained for a class of population genetics models with selection. For models with selection, in contrast to models without selection, the distribution of this process, the coalescent process, depends on the distribution of the frequencies of alleles in the ancestral generations. If the ancestral frequency process can be approximated by a diffusion, then the mean and the variance of the number of segregating sites due to selectively neutral mutations in random samples can be numerically calculated. The calculations are greatly simplified if the frequencies of the alleles are tightly regulated. If the mutation rates between alleles maintained by balancing selection are low, then the number of selectively neutral segregating sites in a random sample of genes is expected to substantially exceed the number predicted under a neutral model.     

Experimental analysis of molecular events during mutational periodic selections in bacterial evolution

A fundamental feature of bacterial evolution is a succession of adaptive mutational sweeps when fitter mutants take over a population. To understand the processes involved in mutational successions, Escherichia coli continuous cultures were analyzed for changes at two loci where mutations provide strong transport advantages to fitness under steady-state glucose limitation. Three separate sweeps, observed as classic periodic selection events causing a change in the frequency of neutral mutations (in fhuA causing phage T5 resistance), were identified with changes at particular loci. Two of the sweeps were associated with a reduction in the frequency of neutral mutations and the concurrent appearance of at least 13 alleles at the mgl or mlc loci, respectively. These mgl and mlc polymorphisms were of many mutational types, so were not the result of a mutator or directed mutation event. The third sweep observed was altogether distinct and involved hitchhiking between T5 resistance and advantageous mgl mutations. Moreover, the hitchhiking event coincided with an increase in mutation rates, due to the transient appearance of a strong mutator in the population. The spectrum of mgl mutations among mutator isolates was distinct and due to mutS. The mutator-associated periodic selection also resulted in mgl and fhuA polymorphism in the sweeping population. These examples of periodic selections maintained significant genotypic diversity even in a rapidly evolving culture, with no individual "winner clone" or genotype purging the population.     

The Effect of Deleterious Mutations on Neutral Molecular Variation

Selection against deleterious alleles maintained by mutation may cause a reduction in the amount of genetic variability at linked neutral sites. This is because a new neutral variant can only remain in a large population for a long period of time if it is maintained in gametes that are free of deleterious alleles, and hence are not destined for rapid elimination from the population by selection. Approximate formulas are derived for the reduction below classical neutral values resulting from such background selection against deleterious mutations, for the mean times to fixation and loss of new mutations, nucleotide site diversity, and number of segregating sites. These formulas apply to random-mating populations with no genetic recombination, and to populations reproducing exclusively asexually or by self-fertilization. For a given selection regime and mating system, the reduction is an exponential function of the total mutation rate to deleterious mutations for the section of the genome involved. Simulations show that the effect decreases rapidly with increasing recombination frequency or rate of outcrossing. The mean time to loss of new neutral mutations and the total number of segregating neutral sites are less sensitive to background selection than the other statistics, unless the population size is of the order of a hundred thousand or more. The stationary distribution of allele frequencies at the neutral sites is correspondingly skewed in favor of rare alleles, compared with the classical neutral result. Observed reductions in molecular variation in low recombination genomic regions of sufficiently large size, for instance in the centromere-proximal regions of Drosophila autosomes or in highly selfing plant populations, may be partly due to background selection against deleterious mutations.     

The effects of deleterious mutations on linked, neutral variation in small populations.

The effects of recessive, deleterious mutations on genetic variation at linked neutral loci can be heterozygosity-decreasing because of reduced effective population sizes or heterozygosity-increasing because of associative overdominance. Here we examine the balance between these effects by simulating individual diploid genotypes in small panmictic populations. The haploid genome consists of one linkage group with 1000 loci that can have deleterious mutations and a neutral marker. Combinations of the following parameters are studied: gametic mutation rate to harmful alleles (U), population size (N), recombination rate (r), selection coefficient (s), and dominance (h). Tight linkage (r 相似文献   

The signature of positive selection at randomly chosen loci

In Drosophila and humans, there are accumulating examples of loci with a significant excess of high-frequency-derived alleles or high levels of linkage disequilibrium, relative to a neutral model of a random-mating population of constant size. These are features expected after a recent selective sweep. Their prevalence suggests that positive directional selection may be widespread in both species. However, as I show here, these features do not persist long after the sweep ends: The high-frequency alleles drift to fixation and no longer contribute to polymorphism, while linkage disequilibrium is broken down by recombination. As a result, loci chosen without independent evidence of recent selection are not expected to exhibit either of these features, even if they have been affected by numerous sweeps in their genealogical history. How then can we explain the patterns in the data? One possibility is population structure, with unequal sampling from different subpopulations. Alternatively, positive selection may not operate as is commonly modeled. In particular, the rate of fixation of advantageous mutations may have increased in the recent past.     

The neutral coalescent process for recent gene duplications and copy-number variants

I describe a method for simulating samples from gene families of size two under a neutral coalescent process, for the case where the duplicate gene either has fixed recently in the population or is still segregating. When a duplicate locus has recently fixed by genetic drift, diversity in the new gene is expected to be reduced, and an excess of rare alleles is expected, relative to the predictions of the standard coalescent model. The expected patterns of polymorphism in segregating duplicates ("copy-number variants") depend both on the frequency of the duplicate in the sample and on the rate of crossing over between the two loci. When the crossover rate between the ancestral gene and the copy-number variant is low, the expected pattern of variability in the ancestral gene will be similar to the predictions of models of either balancing or positive selection, if the frequency of the duplicate in the sample is intermediate or high, respectively. Simulations are used to investigate the effect of crossing over between loci, and gene conversion between the duplicate loci, on levels of variability and the site-frequency spectrum.     

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.     

Statistical inference of selection and divergence from a time-dependent Poisson random field model

We apply a recently developed time-dependent Poisson random field model to aligned DNA sequences from two related biological species to estimate selection coefficients and divergence time. We use Markov chain Monte Carlo methods to estimate species divergence time and selection coefficients for each locus. The model assumes that the selective effects of non-synonymous mutations are normally distributed across genetic loci but constant within loci, and synonymous mutations are selectively neutral. In contrast with previous models, we do not assume that the individual species are at population equilibrium after divergence. Using a data set of 91 genes in two Drosophila species, D. melanogaster and D. simulans, we estimate the species divergence time t(div) = 2.16 N(e) (or 1.68 million years, assuming the haploid effective population size N(e) = 6.45 x 10(5) years) and a mean selection coefficient per generation μ(γ) = 1.98/N(e). Although the average selection coefficient is positive, the magnitude of the selection is quite small. Results from numerical simulations are also presented as an accuracy check for the time-dependent model.     

Persistence time of loss-of-function mutations at nonessential loci affecting eye color in Drosophila melanogaster

Persistence time of a mutant allele, the expected number of generations before its elimination from the population, can be estimated as the ratio of the number of segregating mutations per individual over the mutation rate per generation. We screened two natural populations of Drosophila melanogaster for mutations causing clear-cut eye phenotypes and detected 25 mutant alleles, falling into 19 complementation groups, in 1164 haploid genomes, which implies 0.021 eye mutations/genome. The de novo haploid mutation rate for the same set of loci was estimated as 2 x 10(-4) in a 10-generation mutation-accumulation experiment. Thus, the average persistence time of all mutations causing clear-cut eye phenotypes is approximately 100 generations (95% confidence interval: 61-219). This estimate shows that the strength of selection against phenotypically drastic alleles of nonessential loci is close to that against recessive lethals. In both cases, deleterious alleles are apparently eliminated by selection against heterozygous individuals, which show no visible phenotypic differences from wild type.     

Persistence of Common Alleles in Two Related Populations or Species

Mathematical studies are conducted on three problems that arise in molecular population genetics. (1) The time required for a particular allele to become extinct in a population under the effects of mutation, selection, and random genetic drift is studied. In the absence of selection, the mean extinction time of an allele with an initial frequency close to 1 is of the order of the reciprocal of the mutation rate when 4Nv << 1, where N is the effective population size and v is the mutation rate per generation. Advantageous mutations reduce the extinction time considerably, whereas deleterious mutations increase it tremendously even if the effect on fitness is very slight. (2) Mathematical formulae are derived for the distribution and the moments of extinction time of a particular allele from one or both of two related populations or species under the assumption of no selection. When 4Nv << 1, the mean extinction time is about half that for a single population, if the two populations are descended from a common original stock. (3) The expected number as well as the proportion of common neutral alleles shared by two related species at the tth generation after their separation are studied. It is shown that if 4Nv is small, the two species are expected to share a high proportion of common alleles even 4N generations after separation. In addition to the above mathematical studies, the implications of our results for the common alleles at protein loci in related Drosophila species and for the degeneration of unused characters in cave animals are discussed.     

Sojourn times and substitution rate at overdominant and linked neutral loci

The sojourn times until fixation of an overdominant allele were investigated based on the diffusion equation. Furthermore, the rate of accumulation of mutations, or the substitution rate, was predicted from the mean extinction time of a common overdominant allele. The substitution rate calculated theoretically agreed well with that determined by computer simulation. Overdominant selection enhances the polymorphism at linked loci, while its effect on the sojourn times and the substitution rate at linked loci has not been studied yet. To solve these problems, a model that assumed two linked loci, each with infinite alleles, was examined by computer simulation. A decrease in the recombination rate between two loci markedly changed the distribution of sojourn times of a neutral allele. Although overdominant selection obviously increased the sojourn times and the polymorphism at a linked locus, the rate of nucleotide substitution at the neutral locus was not influenced significantly even if complete linkage was assumed. These results suggest that, in regions containing overdominant genes, linked neutral loci will exhibit elevated levels of polymorphism, but their rate of molecular evolution remains that predicted by neutral theory.     

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.     

Variation after a selective sweep in a subdivided population

The effect of genetic hitchhiking on neutral variation is analyzed in subdivided populations with differentiated demes. After fixation of a favorable mutation, the consequences on particular subpopulations can be radically different. In the subpopulation where the mutation first appeared by mutation, variation at linked neutral loci is expected to be reduced, as predicted by the classical theory. However, the effect in the other subpopulations, where the mutation is introduced by migration, can be the opposite. This effect depends on the level of genetic differentiation of the subpopulations, the selective advantage of the mutation, the recombination frequency, and the population size, as stated by analytical derivations and computer simulations. The characteristic outcomes of the effect are three. First, the genomic region of reduced variation around the selected locus is smaller than that predicted in a panmictic population. Second, for more distant neutral loci, the amount of variation increases over the level they had before the hitchhiking event. Third, for these loci, the spectrum of gene frequencies is dominated by an excess of alleles at intermediate frequencies when compared with the neutral theory. At these loci, hitchhiking works like a system that takes variation from the between-subpopulation component and introduces it into the subpopulations. The mechanism can also operate in other systems in which the genetic variation is distributed in clusters with limited exchange of variation, such as chromosome arrangements or genomic regions closely linked to targets of balancing selection.