The average number of sites separating DNA sequences drawn from a subdivided population

The "infinite sites" model in the absence of recombination is examined in a subdivided population in which there is arbitrary migration among demes. It is shown that, if the migration matrix is symmetric and irreducible, the average number of sites that differ in two alleles chosen from the same deme depends only on an effective size of the whole population and not on either the elements of the migration matrix or the size of each deme separately. If there are n demes all of size N, the average number of sites that differ in two alleles chosen from the same deme is 4nN mu, where mu is the average mutation rate per site. This is the same value as for two alleles drawn from a panmictic population of size nN. The average number of sites that differ in alleles drawn from the same and from different demes can provide some information about the degree of population subdivision, as is illustrated by using the data of Kreitman and Aquadé (1986, Proc. Nat. Acad. Sci. U.S.A., 83, 3562) on Drosophila melanogaster.     

Populational heritability: empirical studies of evolution in metapopulations

Using demes from experimental metapopulations of the flour beetle, Tribolium castaneum, we investigated phase 3 of Wright's shifting balance process. Using parent demes of high, intermediate, and low mean fitness, we experimentally modeled migration of varying amounts from demes of high mean fitness into demes of lower mean fitness (like phase 3) as well as the reciprocal (the opposite of phase 3). In natural populations, some migration among demes occurs independently of deme fitness. In this case, demes of high mean fitness are likely to receive migrants from demes of lower mean fitness; these effects might limit the effectiveness of phase 3 but have not been studied experimentally. We estimated the populational heritability of mean fitness by the regression of offspring deme means on the weighted parental means and found moderate levels of demic heritability one (0.641-0.690) and two (0.518-0.552) generations after migration. We discuss our findings in relation to the role of interdemic migration in "adaptive peak shifts" in metapopulations and the controversies over group selection and the units of inheritance.     

The coalescent in an island model of population subdivision with variation among demes

A simple genealogical structure is found for a general finite island model of population subdivision. The model allows for variation in the sizes of demes, in contributions to the migrant pool, and in the fraction of each deme that is replaced by migrants every generation. The ancestry of a sample of non-recombining DNA sequences has a simple structure when the sample size is much smaller than the total number of demes in the population. This allows an expression for the probability distribution of the number of segregating sites in the sample to be derived under the infinite-sites mutation model. It also yields easily computed estimators of the migration parameter for each deme in a multi-deme sample. The genealogical process is such that the lineages ancestral to the sample tend to accumulate in demes with low migration rates and/or which contribute disproportionately to the migrant pool. In addition, common ancestor or coalescent events tend to occur in demes of small size. This provides a framework for understanding the determinants of the effective size of the population, and leads to an expression for the probability that the root of a genealogy occurs in a particular geographic region, or among a particular set of demes.     

Theory of the effects of population structure and sampling on patterns of linkage disequilibrium applied to genomic data from humans

We develop predictions for the correlation of heterozygosity and for linkage disequilibrium between two loci using a simple model of population structure that includes migration among local populations, or demes. We compare the results for a sample of size two from the same deme (a single-deme sample) to those for a sample of size two from two different demes (a scattered sample). The correlation in heterozygosity for a scattered sample is surprisingly insensitive to both the migration rate and the number of demes. In contrast, the correlation in heterozygosity for a single-deme sample is sensitive to both, and the effect of an increase in the number of demes is qualitatively similar to that of a decrease in the migration rate: both increase the correlation in heterozygosity. These same conclusions hold for a commonly used measure of linkage disequilibrium (r(2)). We compare the predictions of the theory to genomic data from humans and show that subdivision might account for a substantial portion of the genetic associations observed within the human genome, even though migration rates among local populations of humans are relatively large. Because correlations due to subdivision rather than to physical linkage can be large even in a single-deme sample, then if long-term migration has been important in shaping patterns of human polymorphism, the common practice of disease mapping using linkage disequilibrium in "isolated" local populations may be subject to error.     

Multilocus selection in subdivided populations II. Maintenance of polymorphism under weak or strong migration

The potential of maintaining multilocus polymorphism by migration-selection balance is studied. A large population of diploid individuals is distributed over finitely many demes connected by migration. Generations are discrete and nonoverlapping, selection may vary across demes, and loci are multiallelic. It is shown that if migration and recombination are strong relative to selection, then with weak or no epistasis and intermediate dominance at every locus and in every deme, arbitrarily many alleles can be maintained at arbitrarily many loci at a stable equilibrium. If migration is weak relative to selection and recombination, then with weak or no epistasis and intermediate dominance at every locus and in every deme, as many alleles as there are demes can be maintained at arbitrarily many loci at equilibrium. In both cases open sets of such parameter combinations are constructed, thus the results are robust with respect to small, but arbitrary, perturbations in the parameters. For weak migration, the number of demes is, in fact, a generic upper bound to the number of alleles that can be maintained at any locus. Thus, several scenarios are identified under which multilocus polymorphism can be maintained by migration-selection balance when this is impossible in a panmictic population.      

The Role of Life Cycle and Migration in Selection for Variance in Offspring Number

For two genotypes that have the same mean number of offspring but differ in the variance in offspring number, naturalselection will favor the genotype with lower variance. In such cases, the average growth rate is not sufficient as a measure of fitness or as a predictor of fixation probability. However, the effect of variance in offspring number on the fixationprobability of mutant strategies has been calculated under several scenarios with the general conclusion that variance in offspring number reduces fitness in proportion to the inverse of the population size [Gillespie, J., Genetics 76:601–606, 1974; Proulx, S.R., Theor. Popul. Biol. 58:33–47, 2000]. This relationship becomes more complicated under a metapopulation scenario where the “effective” population size depends on migration rate, population structure, and lifecycle. It is shown that in a life cycle where reproduction and migration (the birth-migration-regulation life cycle, or BMR)occur prior to density regulation within every deme, the fitness of a strategy depends on migration rate. When migration rates are near zero, the fitness of the strategy is determined by the size of individual demes, so that the strategy favoredin small populations tends to be fixed. As migration rate increases and approaches panmixis between demes, the fitness ofa reproductive strategy approaches what its value would be in a single, panmictic deme with a population size correspondingtothe census size of the metapopulation. Interestingly, when the life cycle is characterized by having density regulation in each deme prior to migration (the BRM life cycle) the fixation probability of a strategy is independent of migration rate. These results are found to be qualitatively consistent with the individual-based simulation results in Shpak [Theor. Biosci.124:65–85, 2005]. An erratum to this article can be found at     

A SIMULATION OF WRIGHT'S SHIFTING-BALANCE PROCESS: MIGRATION AND THE THREE PHASES

Wright partitioned the shifting-balance process into three phases. Phase one is the shift of a deme within a population to the domain of a higher adaptive peak from that of the historical peak. Phase two is mass selection within a deme towards that higher peak. Phase three is the conversion of additional demes to the higher peak. The migration rate between demes is critical for the existence of phases one and three. Phase one requires small effective population sizes, hence low migration rates. Phase three is optimal under high migration rates that spread the most-fit genotype from deme to deme. Thus, a population-wide peak shift requires intermediate levels of migration. By altering the rates of phases one and three, migration affects the predominant direction of mass selection within a population. This study examines the degree to which migration, through its effects on phases one and three, determines the probability of a simulated population arriving at its genotypic optimum after 12,000 generations. These simulations reveal that there is a range of migration rates for which an entire population might be expected to shift to a higher peak. Below m = 0.001 peak shifts occur frequently (phases I and II) but are not successfully exported out of subpopulations (phase III), and above 0.01 peak shifts within demes (phase I and II), required to initiate phase III, become increasingly uncommon. Because it is unlikely that real populations will have uniform migration rates from generation to generation, the probable effects of varying migration rates on broadening the range of conditions producing peak shifts are discussed.     

The maintenance of sexual reproduction in a structured population

We present the results of a computer simulation model in which a sexual population produces an asexual mutant. We estimate the probability that the new asexual lineage will go extinct. We find that whenever the asexual lineage does not go extinct the sexual population is out-competed, and only asexual individuals remain after a sufficiently long period of time has elapsed. We call this type of outcome an asexual takeover. Our results suggest that, given repeated mutations to asexuality, asexual takeover is likely in an unstructured environment. However, if the environment is subdivided into demes that are connected by migration, then asexual takeover becomes less likely. The probability of asexual takeover declines towards zero as the number of demes increases and as the rate of migration decreases. The reason for this is that asexuality leads to a greater loss of fitness due to mutation and genetic drift, in comparison to what occurs under sexual reproduction. Population subdivision slows the spread of asexual lineages, which allows more time for the genetic degeneration caused by asexuality to take place.     

The many-demes limit for selection and drift in a subdivided population

A diffusion approximation is obtained for the frequency of a selected allele in a population comprised of many subpopulations or demes. The form of the diffusion is equivalent to that for an unstructured population, except that it occurs on a longer time scale when migration among demes is restricted. This many-demes diffusion limit relies on the collection of demes always being in statistical equilibrium with respect to migration and drift for a given allele frequency in the total population. Selection is assumed to be weak, in inverse proportion to the number of demes, and the results hold for any deme sizes and migration rates greater than zero. The distribution of allele frequencies among demes is also described.     

An evolutionary analysis of the relationship between spite and altruism

We investigate the selective pressures on a social trait when evolution occurs in a population of constant size. We show that any social trait that is spiteful simultaneously qualifies as altruistic. In other words, any trait that reduces the fitness of less related individuals necessarily increases that of related ones. Our analysis demonstrates that the distinction between "Hamiltonian spite" and "Wilsonian spite" is not justified on the basis of fitness effects. We illustrate this general result with an explicit model for the evolution of a social act that reduces the recipient's survival ("harming trait"). This model shows that the evolution of harming is favoured if local demes are of small size and migration is low (philopatry). Further, deme size and migration rate determine whether harming evolves as a selfish strategy by increasing the fitness of the actor, or as a spiteful/altruistic strategy through its positive effect on the fitness of close kin.     

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.     

The effects of subdivision on the genetic divergence of populations and species

Abstract. An island model of migration is used to study the effects of subdivision within populations and species on sample genealogies and on between-population or between-species measures of genetic variation. The model assumes that the number of demes within each population or species is large. When populations (or species), connected either by gene flow or historical association, are themselves subdivided into demes, changes in the migration rate among demes alter both the structure of genealogies and the time scale of the coalescent process. The time scale of the coalescent is related to the effective size of the population, which depends on the migration rate among demes. When the migration rate among demes within populations is low, isolation (or speciation) events seem more recent and migration rates among populations seem higher because the effective size of each population is increased. This affects the probability of reciprocal monophyly of two samples, the chance that a gene tree of a sample matches the species tree, and relative likelihoods of different types of polymorphic sites. It can also have a profound effect on the estimation of divergence times.     

The effect of hitch-hiking on genes linked to a balanced polymorphism in a subdivided population

The effect of multi-allelic balancing selection on nucleotide diversity at linked neutral sites was investigated by simulations of subdivided populations. The motivation is to understand the behaviour of self-recognition systems such as the MHC and plant self-incompatibility. For neutral sites, two types of subdivision are present: (1) into demes (connected by migration), and (2) into classes defined by different functional alleles at the selected locus (connected by recombination). Previous theoretical studies of each type of subdivision separately have shown that each increases diversity, and decreases the relative frequencies of low-frequency variants, at neutral sites or loci. We show here that the two types of subdivision act non-additively when sampling is at the whole population level, and that subdivision produces some non-intuitive results. For instance, in highly subdivided populations, genetic diversity at neutral sites may decrease with tighter linkage to a selected locus or site. Another conclusion is that, if there is population subdivision, balancing selection leads to decreased expected FST values for neutral sites linked to the selected locus. Finally, we show that the ability to detect balancing selection by its effects on linked variation, using tests such as Tajima's D, is reduced when genes in a subdivided population are sampled from the total population, rather than within demes.     

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.     

Extranuclear differentiation and gene flow in the finite island model

Use of sequence information from extranuclear genomes to examine deme structure in natural populations has been hampered by lack of clear linkage between sequence relatedness and rates of mutation and migration among demes. Here, we approach this problem in two complementary ways. First, we develop a model of extranuclear genomes in a population divided into a finite number of demes. Sex-dependent migration, neutral mutation, unequal genetic contribution of separate sexes and random genetic drift in each deme are incorporated for generality. From this model, we derive the relationship between gene identity probabilities (between and within demes) and migration rate, mutation rate and effective deme size. Second, we show how within- and between-deme identity probabilities may be calculated from restriction maps of mitochondrial (mt) DNA. These results, when coupled with our results on gene flow and genetic differentiation, allow estimation of relative interdeme gene flow when deme sizes are constant and genetic variants are selectively neutral. We illustrate use of our results by reanalyzing published data on mtDNA in mouse populations from around the world and show that their geographic differentiation is consistent with an island model of deme structure.     

Gene genealogies in a metapopulation

A simple genealogical process is found for samples from a metapopulation, which is a population that is subdivided into a large number of demes, each of which is subject to extinction and recolonization and receives migrants from other demes. As in the migration-only models studied previously, the genealogy of any sample includes two phases: a brief sample-size adjustment followed by a coalescent process that dominates the history. This result will hold for metapopulations that are composed of a large number of demes. It is robust to the details of population structure, as long as the number of possible source demes of migrants and colonists for each deme is large. Analytic predictions about levels of genetic variation are possible, and results for average numbers of pairwise differences within and between demes are given. Further analysis of the expected number of segregating sites in a sample from a single deme illustrates some previously known differences between migration and extinction/recolonization. The ancestral process is also amenable to computer simulation. Simulation results show that migration and extinction/recolonization have very different effects on the site-frequency distribution in a sample from a single deme. Migration can cause a U-shaped site-frequency distribution, which is qualitatively similar to the pattern reported recently for positive selection. Extinction and recolonization, in contrast, can produce a mode in the site-frequency distribution at intermediate frequencies, even in a sample from a single deme.     

Testing neutrality in subdivided populations

Statistical tests of the neutrality hypothesis that are based on the sampling theory of Ewens (1972) require the assumption of panmixia. It is proposed that for a population comprising numerous local populations with weak gene flow among them, tests based on Ewens' theory can be applied separately to samples from each local population. At low levels of gene flow, migration acts primarily like mutation, introducing new alleles to each local population. It is shown with simulation results that, at low levels of migration, correlations in allele frequencies among demes are sufficiently small that the results from the application of Ewens' theory to each deme are statistically independent. It is also shown that, by combining the results of the tests in different demes, some statistical power to detect deviations from neutrality is gained. The method is illustrated with the application to data on a salamander species. At low levels of gene flow, population subdivision must be taken account of in testing neutrality and the proposed test provides one way to do so.     

Influence of finite-sites mutation, population subdivision and sampling schemes on patterns of nucleotide polymorphism for species with molecular hyperdiversity

Molecular hyperdiversity has been documented in viruses, prokaryotes and eukaryotes. Such organisms undermine the assumptions of the infinite-sites mutational model, because multiple mutational events at a site comprise a non-negligible portion of polymorphisms. Moreover, different sampling schemes of individuals from species with subdivided populations can profoundly influence resulting patterns and interpretations of molecular variation. Inspired by molecular hyperdiversity in the nematode Caenorhabditis sp. 5, which exhibits average pairwise differences among synonymous sites of >5% as well as modest population structure, we investigated via coalescent simulation the joint effects of a finite-sites mutation (FSM) process and population subdivision on the variant frequency spectrum. From many demes interconnected through a stepping-stone migration model, we constructed local samples from a single deme, pooled samples from several demes and scattered samples of a single individual from numerous demes. Compared with a single panmictic population at equilibrium, we find that high population mutation rates induce a deficit of rare variants (positive Tajima's D) under a FSM model. Population structure also induces such a skew for local samples when migration is high and for pooled samples when migration is low. Contrasts of sampling schemes for C. sp. 5 imply high mutational input coupled with high migration. We propose that joint analysis of local, pooled and scattered samples for species with subdivided populations provides a means of improving inference of demographic history, by virtue of the partially distinct patterns of polymorphism that manifest when sequences are analyzed according to differing sampling schemes.     

A framework for estimating the fixation time of an advantageous allele in stepping-stone models

Determining how population subdivision increases the fixation time of an advantageous allele is an important problem in evolutionary genetics as this influences many processes. Here, I lay out a framework for calculating the fixation time of a positively selected allele in a subdivided population, as a function of the number of demes present, the migration rate between them and the manner in which they are connected. Using this framework, it becomes clear that a beneficial allele's fixation time is significantly reduced through migration continuously introducing copies of the allele into a newly colonized subpopulation, increasing its frequency within these demes. The effect that migration has on allele frequency needs to be explicitly taken into account to produce a realistic estimate of fixation time. This behaviour is most prominent when demes are arranged on a two-dimensional torus, in comparison with populations where demes are arranged in a circle. This is because each subpopulation is connected to several neighbours over a torus, so that there are multiple paths that an allele can take in order to fix. As a consequence, some demes experience a greater influx and efflux of migrants than others. Analytical results are found to be very accurate when compared to stochastic simulations, and are generally robust if there are a large number of demes, or if the allele is weakly selected for.