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.     

EVOLUTION OF ALTRUISM IN KIN-STRUCTURED AND RANDOM SUBDIVIDED POPULATIONS

A. population structure favorable to the evolution of an altruistic trait is studied by Monte Carlo simulation. The model is based on a small-scale nonindustrial human society but seems generalizable to other highly social mammals. Three hierarchical levels are recognized: 1) the ecologically isolated local group (hamlet) which may be composed of kin and/or unrelated individuals; 2) the deme (settlement) comprising several such groups which interbreed; and 3) the set of demes (metapopulation) among which gene flow occurs. The first two levels of the model are based on D. S. Wilson's structured deme concept; the third allows for gene flow among demes in the metapopulation and for the structured diffusion of alleles across a wider area than might be included within the scope of a single deme. The simulation models genetic drift by a process of hamlet formation which may be random, or variously kin-structured. Hamlets may then become extinct based on a probability function of their gene frequencies. Individual selection within settlements is modeled deterministically, and gene flow among settlements is modeled as two-dimensional steppingstone migration of random or kin-structured groups. Results of the simulations show that, with realistic values for group sizes, moderate extinction rate, and high rates of migration (m > 27%), disadvantageous alleles (s = 10% and 25%) may increase markedly due to differential hamlet extinction over the course of 50 generations. The greater the degree of kin-structuring of founder groups, the higher the variance among hamlets and the faster the rate of increase of the allele for altruism. Nonetheless, even in some randomly founded groups, a clear increase in the altruism gene frequency occurred. It is also notable that kin-structured group selection by hamlet extinction may be effective when the initial frequency of altruism genes is very low (average of one per deme) and among a relatively small number of demes (25). Thus the process of group extinction in a hierarchically structured population allows rapid increase of an allele for altruism under plausible demographic conditions.     

Spatial environmental variation can select for evolvability

Previous studies have shown that temporally fluctuating environments can create indirect selection for modifiers of evolvability. Here, we use a simple computational model to investigate whether spatially varying environments (multiple demes with limited migration among them, and a different, static selective optimum in each) can also create indirect selection for increased evolvability. The answer is surprisingly complicated. Spatial variation in the environment can sharply reduce the survival rate of migrants, because migrants may be maladapted to their new deme, relative to incumbents. The incumbent advantage can be removed by occasional extinctions in single demes. After all incumbents in a particular deme die, incoming migrants from other demes will, on average, be similarly maladapted to the new environment. This sets off a race to adapt rapidly. Over many extinction events, and the subsequent invasions by maladapted immigrants into a new environment, indirect selection for the ability to adapt rapidly, also known as high evolvability, may result.     

Genetic load,inbreeding depression and heterosis in an age‐structured metapopulation

In a metapopulation, the process of recurrent local extinction and recolonization gives rise to an age structure among demes. Recently established demes will tend to differ from older demes in terms of the levels of genetic diversity found within them and the way this diversity is distributed among demes in the same and different ages. The effects of population turnover on average levels of genetic diversity among demes in a metapopulation have been the focus of much attention, both for neutral and nonneutral loci, but much less is known about the distribution of nonneutral genetic diversity among demes of different ages. In this paper, we used computer simulations to study the distribution of genetic load, inbreeding depression and heterosis in an age‐structured metapopulation. We found that, for mildly deleterious mutations, within‐deme inbreeding depression increased, whereas heterosis and genetic load decreased with deme age following severe colonization bottlenecks. In contrast, recessive lethal alleles tended to be purged during colonization, with older populations showing higher genetic load and higher within‐deme inbreeding depression. Heterosis caused by recessive lethal alleles and resulting from gene flow among different demes tended to be greatest for young demes, because the mutations responsible tended to be purged in the first few generations after colonization, but its effects increased again as populations grow older as a result of immigration. Our results point to a need for estimates of genetic diversity, genetic load, within‐deme inbreeding depression and heterosis in demes of different age classes separately.     

Selection and drift in subdivided populations: a straightforward method for deriving diffusion approximations and applications involving dominance, selfing and local extinctions

Population structure affects the relative influence of selection and drift on the change in allele frequencies. Several models have been proposed recently, using diffusion approximations to calculate fixation probabilities, fixation times, and equilibrium properties of subdivided populations. We propose here a simple method to construct diffusion approximations in structured populations; it relies on general expressions for the expectation and variance in allele frequency change over one generation, in terms of partial derivatives of a "fitness function" and probabilities of genetic identity evaluated in a neutral model. In the limit of a very large number of demes, these probabilities can be expressed as functions of average allele frequencies in the metapopulation, provided that coalescence occurs on two different timescales, which is the case in the island model. We then use the method to derive expressions for the probability of fixation of new mutations, as a function of their dominance coefficient, the rate of partial selfing, and the rate of deme extinction. We obtain more precise approximations than those derived by recent work, in particular (but not only) when deme sizes are small. Comparisons with simulations show that the method gives good results as long as migration is stronger than selection.     

Local extinction and recolonization, species effective population size, and modern human origins

A primary objection from a population genetics perspective to a multiregional model of modern human origins is that the model posits a large census size, whereas genetic data suggest a small effective population size. The relationship between census size and effective size is complex, but arguments based on an island model of migration show that if the effective population size reflects the number of breeding individuals and the effects of population subdivision, then an effective population size of 10,000 is inconsistent with the census size of 500,000 to 1,000,000 that has been suggested by archeological evidence. However, these models have ignored the effects of population extinction and recolonization, which increase the expected variance among demes and reduce the inbreeding effective population size. Using models developed for population extinction and recolonization, we show that a large census size consistent with the multiregional model can be reconciled with an effective population size of 10,000, but genetic variation among demes must be high, reflecting low interdeme migration rates and a colonization process that involves a small number of colonists or kin-structured colonization. Ethnographic and archeological evidence is insufficient to determine whether such demographic conditions existed among Pleistocene human populations, and further work needs to be done. More realistic models that incorporate isolation by distance and heterogeneity in extinction rates and effective deme sizes also need to be developed. However, if true, a process of population extinction and recolonization has interesting implications for human demographic history.     

Polymorphism in multiallelic migration–selection models with dominance

The evolution of the gene frequencies at a single multiallelic locus under the joint action of migration and viability selection with dominance is investigated. The monoecious, diploid population is subdivided into finitely many panmictic colonies that exchange adult migrants independently of genotype. Underdominance and overdominance are excluded. If the degree of dominance is deme independent for every pair of alleles, then under the Levene model, the qualitative evolution of the gene frequencies (i.e., the existence and stability of the equilibria) is the same as without dominance. In particular: (i) the number of demes is a generic upper bound on the number of alleles present at equilibrium; (ii) there exists exactly one stable equilibrium, and it is globally attracting; and (iii) if there exists an internal equilibrium, it is globally asymptotically stable. Analytic examples demonstrate that if either the Levene model does not apply or the degree of dominance is deme dependent, then the above results can fail. A complete global analysis of weak migration and weak selection on a recessive allele in two demes is presented.     

Population Dynamic and Genetic Consequences of Spatial Density-Dependent Dispersal in Patchy Populations

Predictions about sex-specific, spatial density-dependent dispersal and their demographic and genetic consequences were tested in experimental populations of root voles (Microtus oeconomus). Each population consisted of two demes inhabiting equal-sized habitat patches imbedded in a barren matrix area. We used a neutral two-allele allozyme marker to monitor gene flow. Initially, the two demes were genetically distinct and had different densities so that the size of a high-density deme (genotype bb) was four times larger than that of a low-density deme (genotype aa). The sex-specific dispersal pattern was in accordance with our prediction. Male dispersal was unconditional on deme-specific densities, and the majority of the first-generation males became dispersed from both demes, whereas female dispersal was strongly density dependent, so that dispersal took place exclusively from the high-density to the low-density deme. The demographic implication of this dispersal pattern was that the initial density difference between the demes was quickly canceled out. We built a mathematical model that predicted that the initially rare allele (a) would increase in frequency given the dispersal pattern, and this was supported by our experimental data. This result relies mostly on the density-independent male-dispersal strategy, which presumably stems from inbreeding avoidance. Our study highlights the importance of incorporating sex-specific dispersal strategies in population genetic models. Sex-biased dispersal may act as a deterministic force counteracting the tendency for stochastic loss of alleles in small and fragmented populations.     

Evolution of altruism under group selection in large and small populations in fluctuating environments

A continuous, graded form of group selection which does not involve extinction of demes can effectively oppose selection on the individual level against an altruistic allele under fluctuating environments in infinitely large demes among which uniform mixing occurs every generation. Although group selection cannot alter the conditions necessary for the initial increase of altruistic alleles, group selection can significantly influence the stationary distribution of gene frequency which is attained once stochastic forces have allowed theirintroduction. Drift is a more effective source of variation than fluctuations in selection when the variance in selection is moderate to small. High numbers of demes promote polymorphism under both graded group selection and extinction group selection.     

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.     

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.     

Analysis of linkage disequilibrium in an island model

Linkage disequilibria for two loci in a finite island model were parameterized. The total linkage disequilibrium was decomposed into three components, gametic, demic, and population, for which corresponding unbiased estimators were established. Other statistics encountered provided measures of differentiation corresponding to the hierarchical structure of the ecological model. Under the assumption of linkage equilibrium, the variances and covariances of these estimators and statistics were formulated in terms of descent measures, functions of gene frequencies, and the numbers of individuals, demes, and populations sampled. The functions of gene frequencies fall into two classes, one representing the differentiation of genes at each locus, and the other representing the association of genes between the loci. For a neutral model with extinction, migration, and linkage, transition equations were derived for the descent measures which also take into account deme size and numbers of dems within the population. With the addition of unequal mutation rates for a finite number of alleles at each locus, the transition equations were solved for the descent measures in the equilibrium state. This permitted the exact numerical evaluation of the effects of the sampling and ecological dimensions and of extinction, migration, and mutation rates in any parameter range. Some numerical results were presented for the effects of linkage, extinction, migration, and sampling on the variances of various measures of linkage disequilibrium and genetic differentiation. Also, some results were compared with the approximate numerical results of Ohta which agreed fairly well in the parameter ranges she considered, but not so well in other ranges.     

Nonequilibrium migration in human history

A nonequilibrium migration model is proposed and applied to genetic data from humans. The model assumes symmetric migration among all possible pairs of demes and that the number of demes is large. With these assumptions it is straightforward to allow for changes in demography, and here a single abrupt change is considered. Under the model this change is identical to a change in the ancestral effective population size and might be caused by changes in deme size, in the number of demes, or in the migration rate. Expressions for the expected numbers of sites segregating at particular frequencies in a multideme sample are derived. A maximum-likelihood analysis of independent polymorphic restriction sites in humans reveals a decrease in effective size. This is consistent with a change in the rates of migration among human subpopulations from ancient low levels to present high ones.     

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.     

Polymorphism in the two-locus Levene model with nonepistatic directional selection

For the Levene model with soft selection in two demes, the maintenance of polymorphism at two diallelic loci is studied. Selection is nonepistatic and dominance is intermediate. Thus, there is directional selection in every deme and at every locus. We assume that selection is in opposite directions in the two demes because otherwise no polymorphism is possible. If at one locus there is no dominance, then a complete analysis of the dynamical and equilibrium properties is performed. In particular, a simple necessary and sufficient condition for the existence of an internal equilibrium and sufficient conditions for global asymptotic stability are obtained. These results are extended to deme-independent degree of dominance at one locus. A perturbation analysis establishes structural stability within the full parameter space. In the absence of genotype-environment interaction, which requires deme-independent dominance at both loci, nongeneric equilibrium behavior occurs, and the introduction of arbitrarily small genotype-environment interaction changes the equilibrium structure and may destroy stable polymorphism. The volume of the parameter space for which a (stable) two-locus polymorphism is maintained is computed numerically. It is investigated how this volume depends on the strength of selection and on the dominance relations. If the favorable allele is (partially) dominant in its deme, more than 20% of all parameter combinations lead to a globally asymptotically stable, fully polymorphic equilibrium.     

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.     

Impact of Migration and Fitness on the Stability of Lethal T-Haplotype Polymorphism in Mus Musculus: A Computer Study

The t-haplotype is a chromosomal region in Mus musculus characterized by meiotic drive such that heterozygous males transmit t-bearing chromosomes to roughly 90% of their offspring. Most naturally occurring t-haplotypes express a recessive embryonic lethality, preventing fixation of the t-haplotype. Surprisingly, the t-haplotype occurs in nature as a persistent, low-frequency polymorphism. Early modeling studies led LEWONTIN to hypothesize that this low level polymorphism results from a balance between genetic drift in small demes and interdemic migration. Here, we show that while combinations of deme size and migration rate that predict natural t-haplotype frequencies exist, the range of such values is too narrow to be biologically plausible, suggesting that small deme size and interdemic migration alone do not explain the observed t-haplotype frequencies. In response, we tested other factors that might explain the observed t-polymorphism. Two led to biologically plausible models: substantially reduced heterozygous fitness and reduced meiotic drive. This raises the question whether these phenomena occur in nature. Our data suggest an alternative explanation: there is no stable, low-level t-polymorphism. Rather wild populations are in one of two stable states characterized by extinction of the t-haplotype and a high t-haplotype frequency, respectively, or in transition between the two.     

Ethological isolation between local populations of house mice (Mus musculus) based on olfaction

Some rodent populations are composed of demes which remain reproductively isolated within the major population. Territoriality has been advanced as a possible isolating mechanism. This research investigated the possible role of olfaction in the isolation of feral house mouse (Mus musculus) demes. When given a choice, male and female mice spent more time with the air-transported odours of members of their own deme than with those of neighbouring deme members. Tests with a fresh-air control showed that mice avoided the odours of members of an adjacent deme. In addition, more time was spent with urine and faeces odours of members of a mouse's own group than with those of the members of an adjacent deme. The production and detection of odorous substances may contribute to the autonomy and relative reproductive isolation of some local populations of mice.     

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.     

Fixation probability for a beneficial allele and a mutant strategy in a linear game under weak selection in a finite island model

The effect of population structure on the probability of fixation of a newly introduced mutant under weak selection is studied using a coalescent approach. Wright's island model in a framework of a finite number of demes is assumed and two selection regimes are considered: a beneficial allele model and a linear game among offspring. A first-order approximation of the fixation probability for a single mutant with respect to the intensity of selection is deduced. The approximation requires the calculation of expected coalescence times, under neutrality, for lineages starting from two or three sampled individuals. The results are obtained in a general setting without assumptions on the number of demes, the deme size or the migration rate, which allows for simultaneous coalescence or migration events in the genealogy of the sampled individuals. Comparisons are made with limit cases as the deme size or the number of demes goes to infinity or the migration rate goes to zero for which a diffusion approximation approach is possible. Conditions for selection to favor a mutant strategy replacing a resident strategy in the context of a linear game in a finite island population are addressed.