Selection in a subdivided population with dominance or local frequency dependence

The interplay between population structure and natural selection is an area of great interest. It is known that certain types of population subdivision do not alter fixation probabilities of selected alleles under genic, frequency-independent selection. In the presence of dominance for fitness or frequency-dependent selection these same types of subdivision can have large effects on fixation probabilities. For example, the barrier to fixation of a fitter allele due to underdominance is reduced by subdivision. Analytic results presented here relate a subdivided population that conforms to a finite island model to an approximately equivalent panmictic population. The size of this equivalent population is different from (larger than) the actual size of the subdivided population. Selection parameters are also different in the hypothetical equivalent population. As expected, the degree of dominance is lower in the equivalent population. The results are not limited to dominance but cover any form of polynomial frequency dependence.     

Selection in a subdivided population with local extinction and recolonization

In a subdivided population, local extinction and subsequent recolonization affect the fate of alleles. Of particular interest is the interaction of this force with natural selection. The effect of selection can be weakened by this additional source of stochastic change in allele frequency. The behavior of a selected allele in such a population is shown to be equivalent to that of an allele with a different selection coefficient in an unstructured population with a different size. This equivalence allows use of established results for panmictic populations to predict such quantities as fixation probabilities and mean times to fixation. The magnitude of the quantity N(e)s(e), which determines fixation probability, is decreased by extinction and recolonization. Thus deleterious alleles are more likely to fix, and advantageous alleles less likely to do so, in the presence of extinction and recolonization. Computer simulations confirm that the theoretical predictions of both fixation probabilities and mean times to fixation are good approximations.     

The Effective Size of a Subdivided Population

This paper derives the long-term effective size, N(e), for a general model of population subdivision, allowing for differential deme fitness, variable emigration and immigration rates, extinction, colonization, and correlations across generations in these processes. We show that various long-term measures of N(e) are equivalent. The effective size of a metapopulation can be expressed in a variety of ways. At a demographic equilibrium, N(e) can be derived from the demography by combining information about the ultimate contribution of each deme to the future genetic make-up of the population and Wright's F(ST)'s. The effective size is given by N(e) = 1/(1 + var ( &))<(1 - f(STi))/N(i)n>, where n is the number of demes, &(i) is the eventual contribution of individuals in deme i to the whole population (scaled such that σ(i) &(i) = n), and < > denotes an average weighted by &(i)(2). This formula is applied to a catastrophic extinction model (where sites are either empty or at carrying capacity) and to a metapopulation model with explicit dynamics, where extinction is caused by demographic stochasticity and by chaos. Contrary to the expectation from the standard island model, the usual effect of population subdivision is to decrease the effective size relative to a panmictic population living on the same resource.     

Linkage disequilibrium and historical effective population size in the Thoroughbred horse

Many genomic methodologies rely on the presence and extent of linkage disequilibrium (LD) between markers and genetic variants underlying traits of interest, but the extent of LD in the horse has yet to be comprehensively characterized. In this study, we evaluate the extent and decay of LD in a sample of 817 Thoroughbreds. Horses were genotyped for over 50,000 single nucleotide polymorphism (SNP) markers across the genome, with 34,848 autosomal SNPs used in the final analysis. Linkage disequilibrium, as measured by the squared correlation coefficient (r(2)), was found to be relatively high between closely linked markers (>0.6 at 5 kb) and to extend over long distances, with average r(2) maintained above non-syntenic levels for single nucleotide polymorphisms (SNPs) up to 20 Mb apart. Using formulae which relate expected LD to effective population size (N(e)), and assuming a constant actual population size, N(e) was estimated to be 100 in our population. Values of historical N(e), calculated assuming linear population growth, suggested a decrease in N(e) since the distant past, reaching a minimum twenty generations ago, followed by a subsequent increase until the present time. The qualitative trends observed in N(e) can be rationalized by current knowledge of the history of the Thoroughbred breed, and inbreeding statistics obtained from published pedigree analyses are in agreement with observed values of N(e). Given the high LD observed and the small estimated N(e), genomic methodologies such as genomic selection could feasibly be applied to this population using the existing SNP marker set.     

Effective size and F-statistics of subdivided populations for sex-linked loci.

For a population subdivided into an arbitrary number (s) of subpopulations, each consisting of different numbers of separate sexes, with arbitrary distributions of family size and variable migration rates by males (dm) and females (df), the recurrence equations for inbreeding coefficient and coancestry between individuals within and among subpopulations for a sex-linked locus are derived and the corresponding expressions for asymptotic effective size are obtained by solving the recurrence equations. The usual assumptions are made which are stable population size and structure, discrete generations, the island migration model, and without mutation and selection. The results show that population structure has an important effect on the inbreeding coefficients in any generation, asymptotic effective size, and F-statistics. Gene exchange among subpopulations inhibits inbreeding in initial generations but increases inbreeding in later generations. The larger the migration rate, the greater the final inbreeding coefficients and the smaller the effective size. Thus if the inbreeding coefficient is to be restricted to a specific value within a given number of generations, the appropriate population structure (the values of s, dm, and df) can be obtained by using the recurrence equations. It is shown that the greater the extent of subdivision (large s, small dm and df), the larger the effective size. For a given subdivided population, the effective size for a sex-linked locus may be larger or smaller than that for an autosomal locus, depending on the sex ratio, variance and covariance of family size, and the extend of subdivision. For the special case of a single unsubdivided population, our recurrence equations for inbreeding coefficient and coancestry and formulas for effective size reduce to the simple expressions derived by previous authors.     

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   

Fixation probability in spatially changing environments.

The fixation probability of a mutant in a subdivided population with spatially varying environments is investigated using a finite island model. This probability is different from that in a panmictic population if selection is intermediate to strong and migration is weak. An approximation is used to compute the fixation probability when migration among subpopulations is very weak. By numerically solving the two-dimensional partial differential equation for the fixation probability in the two subpopulation case, the approximation was shown to give fairly accurate values. With this approximation, we show in the case of two subpopulations that the fixation probability in subdivided populations is greater than that in panmictic populations mostly. The increase is most pronounced when the mutant is selected for in one subpopulation and is selected against in the other subpopulation. Also it is shown that when there are two types of environments, further subdivision of subpopulations does not cause much change of the fixation probability in the no dominance case unless the product of the selection coefficient and the local population size is less than one. With dominance, the effect of subdivision becomes more complex.     

Consequences of population structure on genes under balancing selection

This paper describes a new approach to modeling population structure for genes under strong balancing selection of the type seen in plant self-incompatibility systems and the major histocompatibility complex (MHC) system of vertebrates. Simple analytic solutions for the number of alleles maintained at equilibrium and the expected proportion of alleles shared between demes at various levels are derived and checked against simulation results. The theory accurately captures the dynamics of allele number in a subdivided population and identifies important values of m (migration rate) at which allele number and distribution change qualitatively. Starting from a panmictic population, as migration among demes decreases a qualitative change in dynamics is seen at approximately m(crit) approximately equal to the square root of(s/4piNT) where NT is the total population size and s is a measure of the strength of selection. At this point, demes can no longer maintain their panmictic allele number, due to increasing isolation from the total population. Another qualitative change occurs at a migration rate on the same order of magnitude as the mutation rate, mu. At this point, the demes are highly differentiated for allele complement, and the total number of alleles in the population is increased. Because in general u < m<(crit) at intermediate migration rates slightly fewer alleles may be maintained in the total population than are maintained at panmixia. Within this range, total allele number may not be the best indicator of whether a population is effectively panmictic, and some caution should be used when interpreting samples from such populations. The theory presented here can help to analyze data from genes under balancing selection in subdivided populations.     

Patterns of inbreeding depression and architecture of the load in subdivided populations

Inbreeding depression is a general phenomenon that is due mainly to recessive deleterious mutations, the so-called mutation load. It has been much studied theoretically. However, until very recently, population structure has not been taken into account, even though it can be an important factor in the evolution of populations. Population subdivision modifies the dynamics of deleterious mutations because the outcome of selection depends on processes both within populations (selection and drift) and between populations (migration). Here, we present a general model that permits us to gain insight into patterns of inbreeding depression, heterosis, and the load in subdivided populations. We show that they can be interpreted with reference to single-population theory, using an appropriate local effective population size that integrates the effects of drift, selection, and migration. We term this the "effective population size of selection" (NS(e)). For the infinite island model, for example, it is equal to NS(e) = N1 + m/hs, where N is the local population size, m the migration rate, and h and s the dominance and selection coefficients of deleterious mutation. Our results have implications for the estimation and interpretation of inbreeding depression in subdivided populations, especially regarding conservation issues. We also discuss the possible effects of migration and subdivision on the evolution of mating systems.     

Selection for recombination in structured populations

In finite populations, linkage disequilibria generated by the interaction of drift and directional selection (Hill-Robertson effect) can select for sex and recombination, even in the absence of epistasis. Previous models of this process predict very little advantage to recombination in large panmictic populations. In this article we demonstrate that substantial levels of linkage disequilibria can accumulate by drift in the presence of selection in populations of any size, provided that the population is subdivided. We quantify (i) the linkage disequilibrium produced by the interaction of drift and selection during the selective sweep of beneficial alleles at two loci in a subdivided population and (ii) the selection for recombination generated by these disequilibria. We show that, in a population subdivided into n demes of large size N, both the disequilibrium and the selection for recombination are equivalent to that expected in a single population of a size intermediate between the size of each deme (N) and the total size (nN), depending on the rate of migration among demes, m. We also show by simulations that, with small demes, the selection for recombination is stronger than both that expected in an unstructured population (m = 1 - 1/n) and that expected in a set of isolated demes (m = 0). Indeed, migration maintains polymorphisms that would otherwise be lost rapidly from small demes, while population structure maintains enough local stochasticity to generate linkage disequilibria. These effects are also strong enough to overcome the twofold cost of sex under strong selection when sex is initially rare. Overall, our results show that the stochastic theories of the evolution of sex apply to a much broader range of conditions than previously expected.     

Effect of population structure on the amount of polymorphism and the fixation probability under overdominant selection

Under overdominant selection, mutants substantially contribute to increase the amount of polymorphism. It is also known that under neutrality as the migration rates among demes decrease in a subdivided population, the amount of polymorphism increases along with the increase of the effective population size, N(e). In this study, under overdominant selection the effect of population subdivision on the amount of polymorphism was investigated using the diffusion approximation and the low migration approximation. It was shown that if selection is medium or strong (e.g., N(T)s > 1, where N(T) is the population size and s is the selective advantage of heterozygotes), the nucleotide diversity, pi, decreases along with the decrease of Nm against the increase of N(e), where N is the size of demes and m is the migration rate per deme. In addition, the ratio of the nucleotide diversity to the evolutionary rate also decreases along with the decrease of Nm. In some cases the ratio becomes smaller than that expected under neutrality as Nm decreases.     

Gene identity and genetic differentiation of populations in the finite island model

A formula for the variance of gene identity (homozygosity) was derived for the case of neutral mutations using diffusion approximations for the changes of gene frequencies in a subdivided population. It is shown that when gene flow is extremely small, the variance of gene identity for the entire population at equilibrium is smaller than that of the panmictic population with the same mean gene identity. On the other hand, although a large amount of gene flow makes a subdivided population equivalent to a panmictic population, there is an intermediate range of gene flow in which population subdivision can increase the variance. This increase results from the increased variance between colonies. In such a case, each colony has a predominant allele, but the predominant type may differ from colony to colony. The formula for obtaining the variance allows us to study such statistics as the coefficient of gene differentiation and the correlation of heterozygosity. Computer simulations were conducted to study the distribution of gene identity as well as to check the validity of the analytical formulas. Effects of selection were also studied by simulations.     

The effective size of mixed sexually and asexually reproducing populations

Using the transition matrix of inbreeding and coancestry coefficients, the inbreeding (N(eI)), variance (N(eV)), and asymptotic (N(e lambda)) effective sizes of mixed sexual and asexual populations are formulated in terms of asexuality rate (delta), variance of asexual (C) and sexual (K) reproductive contributions of individuals, correlation between asexual and sexual contributions (rho(ck)), selfing rate (beta), and census population size (N). The trajectory of N(eI) toward N(e lambda) changes crucially depending on delta, N, and beta, whereas that of N(eV) is rather consistent. With increasing asexuality, N(e lambda) either increases or decreases depending on C, K, and rho(ck). The parameter space in which a partially asexual population has a larger N(e lambda) than a fully sexual population is delineated. This structure is destroyed when N(1 - delta) < 1 or delta > 1 - 1/N. With such a high asexuality, tremendously many generations are required for the asymptotic size N(e lambda) to be established, and N(e lambda) is extremely large with any value of C, K, and rho(ck) because the population is dominated eventually by individuals of the same genotype and the allelic diversity within the individuals decays quite slowly. In reality, the asymptotic state would occur only occasionally, and instantaneous rather than asymptotic effective sizes should be practical when predicting evolutionary dynamics of highly asexual populations.     

Selection, load and inbreeding depression in a large metapopulation

The subdivision of a species into local populations causes its response to selection to change, even if selection is uniform across space. Population structure increases the frequency of homozygotes and therefore makes selection on homozygous effects more effective. However, population subdivision can increase the probability of competition among relatives, which may reduce the efficacy of selection. As a result, the response to selection can be either increased or decreased in a subdivided population relative to an undivided one, depending on the dominance coefficient F(ST) and whether selection is hard or soft. Realistic levels of population structure tend to reduce the mean frequency of deleterious alleles. The mutation load tends to be decreased in a subdivided population for recessive alleles, as does the expected inbreeding depression. The magnitude of the effects of population subdivision tends to be greatest in species with hard selection rather than soft selection. Population structure can play an important role in determining the mean fitness of populations at equilibrium between mutation and selection.     

Evidence of spatial genetic structure in a California bunchgrass population

We investigated the scale of genetic variation of purple needlegrass (Nassella pulchra), a species commonly used in California for grassland restoration. Common garden and field data revealed evidence of genetic differentiation between two intermixed microhabitats characterized by differences in soil depth and community composition. We assessed the genetic variation within a single population using randomly amplified polymorphic DNA (RAPD) data collected from clusters of five individuals in 40 locations. We found no evidence for genetic structure at the whole population level. At smaller spatial scales, however, we found strong evidence that genetic subdivision of the population occurs at the level of the maternal neighborhood. We suggest that the interaction between widespread pollen dispersal and restricted seed dispersal may be the primary factor generating these results; panmictic pollen dispersal will make detection of genetic patterning difficult at larger spatial scales while limited seed dispersal will generate local genetic structure. As a result, the detection of population genetic structure will depend on the spatial scale of analysis. Local selection gradients related to topography and soil depth are also likely to play a role in structuring local genetic variation. Since N. pulchra is widely used in California in grassland and woodland habitat restoration, we suggest that, as a general rule, care should be exercised in transferring germplasm for the purposes of conservation when little is known about the within-population genetic subdivision of a plant species. Received: 23 December 1996 / Accepted: 20 May 1997     

Distinguishing between population bottleneck and population subdivision by a Bayesian model choice procedure

Although most natural populations are genetically subdivided, they are often analysed as if they were panmictic units. In particular, signals of past demographic size changes are often inferred from genetic data by assuming that the analysed sample is drawn from a population without any internal subdivision. However, it has been shown that a bottleneck signal can result from the presence of some recent immigrants in a population. It thus appears important to contrast these two alternative scenarios in a model choice procedure to prevent wrong conclusions to be made. We use here an Approximate Bayesian Computation (ABC) approach to infer whether observed patterns of genetic diversity in a given sample are more compatible with it being drawn from a panmictic population having gone through some size change, or from one or several demes belonging to a recent finite island model. Simulations show that we can correctly identify samples drawn from a subdivided population in up to 95% of the cases for a wide range of parameters. We apply our model choice procedure to the case of the chimpanzee (Pan troglodytes) and find conclusive evidence that Western and Eastern chimpanzee samples are drawn from a spatially subdivided population.     

When did the human population size start increasing?

We analyze the frequency spectra of all available human nuclear sequence data sets by using a model of constant population size followed by exponential growth. Parameters of growth (more extreme than or) comparable to what has been suggested from mtDNA data can be rejected for 6 out of the 10 largest data sets. When the data are separated into African and non-African samples, a constant size no-growth model can be rejected for 4 out of 8 non-African samples. Long-term growth (i.e., starting 50-100 kya) can be rejected for 2 out of 8 African samples and 5 out of 8 non-African ones. Under more complex demographic models, including a bottleneck or population subdivision, more of the data are compatible with long-term growth. One problem with the data used here is that a subset of loci may reflect the action of natural selection as well as of demography. It remains possible that the correct demographic model is one of constant population size followed by long-term growth but that at several loci the demographic signature has been obscured by balancing or diversifying selection. However, it is not clear that the data at these loci are consistent with a simple model of balancing selection; more complicated selective alternatives cannot be tested unless they are made explicit. An alternative explanation is that population size growth is more recent (e.g., upper Paleolithic) and that some of the loci have experienced recent directional selection. Given the available data, the latter hypothesis seems more likely.     

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.     

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.     

Navigating the unknown: model selection in phylogeography. Models of population structure: tools for thinkers

Despite the widespread use and obvious strengths of model-based methods for phylogeographic study, a persistent concern for such analyses is related to the definition of the model itself. The study by Peter et al. (2010) in this issue of Molecular Ecology demonstrates an approach for overcoming such hurdles. The authors were motivated by a deceptively simple goal; they sought to infer whether a population has remained at a low and stable size or has undergone a decline, and certainly there is no shortage of software packages for such a task (e.g., see list of programs in Excoffier & Heckel 2006). However, each of these software packages makes basic assumptions about the underling population (e.g., is the population subdivided or panmictic); these assumptions are explicit to any model-based approach but can bias parameter estimates and produce misleading inferences if the model does not approximate the actual demographic history in a reasonable manner. Rather than guessing which model might be best for analyzing the data (microsatellite data from samples of chimpanzees), Peter et al. (2010) quantify the relative fit of competing models for estimating the population genetic parameters of interest. Complemented by a revealing simulation study, the authors highlight the peril inherent to model-based inferences that lack a statistical evaluation of the fit of a model to the data, while also demonstrating an approach for model selection with broad applicability to phylogeographic analysis.