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.      

Unequal allelic frequencies at the self‐incompatibility locus within local populations of Prunus avium L.: an effect of population structure?

In this paper, we investigated the genetic structure and distribution of allelic frequencies at the gametophytic self-incompatibility locus in three populations of Prunus avium L. In line with theoretical predictions under balancing selection, genetic structure at the self-incompatibility locus was almost three times lower than at seven unlinked microsatellites. Furthermore, we found that S-allele frequencies in wild cherry populations departed significantly from the expected isoplethic distribution towards which balancing selection is expected to drive allelic frequencies (i.e. identical frequency equal to the inverse of the number of alleles in the population). To assess whether this departure could be caused either by drift alone or by population structure, we used numerical simulations to compare our observations with allelic frequency distributions expected : (1) within a single deme from a subdivided population with various levels of differentiation; and (2) within a finite panmictic population with identical allelic diversity. We also investigated the effects of sample size and degree of population structure on tests of departure from isoplethic equilibrium. Overall, our results showed that the observed allele frequency distributions were consistent with a model of subdivided population with demes linked by moderate migration rate.     

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.     

Environmental variation, fluctuating selection and genetic drift in subdivided populations

Although there have many studies of the population genetical consequences of environmental variation, little is known about the combined effects of genetic drift and fluctuating selection in structured populations. Here we use diffusion theory to investigate the effects of temporally and spatially varying selection on a population of haploid individuals subdivided into a large number of demes. Using a perturbation method for processes with multiple time scales, we show that as the number of demes tends to infinity, the overall frequency converges to a diffusion process that is also the diffusion approximation for a finite, panmictic population subject to temporally fluctuating selection. We find that the coefficients of this process have a complicated dependence on deme size and migration rate, and that changes in these demographic parameters can determine both the balance between the dispersive and stabilizing effects of environmental variation and whether selection favors alleles with lower or higher fitness variance.     

The effect of subdivision on variation at multi-allelic loci under balancing selection

Simulations are used to investigate the expected pattern of variation at loci under different forms of multi-allelic balancing selection in a finite island model of a subdivided population. The objective is to evaluate the effect of restricted migration among demes on the distribution of polymorphism at the selected loci at equilibrium, and to compare the results with those expected for a neutral locus. The results show that the expected number of alleles maintained, and numbers of nucleotide differences between alleles, are relatively insensitive to the migration rate, and differentiation remains low even under very restricted migration. However, nucleotide divergence between copies of functionally identical alleles increases sharply when migration decreases. These results are discussed in relation to published surveys of allelic diversity in MHC and plant self-incompatibility systems, and to the possibility of inferring ancient population genetic events and processes. In addition, it is shown that, for sporophytic self-incompatibility systems, it is not necessarily true in a subdivided population that recessive alleles are more frequent than dominant ones.     

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.     

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.     

Geographical invariance in population genetics

Various quantities of evolutionary interest are shown to be independent of geographical structure. A diploid, monoecious population is subdivided into a finite number of panmictic colonies that exchange migrants. The migration pattern is fixed and ergodic, but otherwise arbitrary. Generations are discrete and non-overlapping; the analysis is restricted to a single locus. Previous results are generalized in the neutral multiallelic case. With selection, it is assumed that there are only two alleles, dominance is absent, selection has the same intensity in all demes, migration does not change the subpopulation numbers, and all evolutionary forces are weak. A diffusion approximation is established for the gene frequencies, and the invariance of the fixation probability and of the moments of the conditional and unconditional total heterozygosities before absorption is demonstrated by a martingale argument.     

Genealogy of Neutral Genes and Spreading of Selected Mutations in a Geographically Structured Population

In a geographically structured population, the interplay among gene migration, genetic drift and natural selection raises intriguing evolutionary problems, but the rigorous mathematical treatment is often very difficult. Therefore several approximate formulas were developed concerning the coalescence process of neutral genes and the fixation process of selected mutations in an island model, and their accuracy was examined by computer simulation. When migration is limited, the coalescence (or divergence) time for sampled neutral genes can be described by the convolution of exponential functions, as in a panmictic population, but it is determined mainly by migration rate and the number of demes from which the sample is taken. This time can be much longer than that in a panmictic population with the same number of breeding individuals. For a selected mutation, the spreading over the entire population was formulated as a birth and death process, in which the fixation probability within a deme plays a key role. With limited amounts of migration, even advantageous mutations take a large number of generations to spread. Furthermore, it is likely that these mutations which are temporarily fixed in some demes may be swamped out again by non-mutant immigrants from other demes unless selection is strong enough. These results are potentially useful for testing quantitatively various hypotheses that have been proposed for the origin of modern human populations.     

Migration-selection polymorphism in dioecious populations

Summary Conditions are derived for a protected polymorphism in a dioecious population subdivided into an arbitrary number of demes which exchange migrants. Generations are discrete and nonoverlapping; mutation and random drift are neglected. The analysis is restricted to a diallelic autosomal locus. In contrast to the monoecious case, the protection criteria depend on the order of migration and selection; they become identical for adult and juvenile migration if both the male and female backward migration matrices are symmetric, or the migration or selection patterns in the two sexes are the same. The protection conditions are presented explicitly for the Levene model. A recessive allele is protected in a panmictic dioecious population if the unweighted average of the recessive-to-dominant fitness ratios in the two sexes exceeds unity.Supported by the National Science Foundation (Grant No. DEB77-21494)     

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.     

Dominance and the maintenance of polymorphism in multiallelic migration-selection models with two demes

The maintenance of genetic variation in a spatially heterogeneous environment has been one of the main research themes in theoretical population genetics. Despite considerable progress in understanding the consequences of spatially structured environments on genetic variation, many problems remain unsolved. One of them concerns the relationship between the number of demes, the degree of dominance, and the maximum number of alleles that can be maintained by selection in a subdivided population. In this work, we study the potential of maintaining genetic variation in a two-deme model with deme-independent degree of intermediate dominance, which includes absence of G×E interaction as a special case. We present a thorough numerical analysis of a two-deme three-allele model, which allows us to identify dominance and selection patterns that harbor the potential for stable triallelic equilibria. The information gained by this approach is then used to construct an example in which existence and asymptotic stability of a fully polymorphic equilibrium can be proved analytically. Noteworthy, in this example the parameter range in which three alleles can coexist is maximized for intermediate migration rates. Our results can be interpreted in a specialist-generalist context and (among others) show when two specialists can coexist with a generalist in two demes if the degree of dominance is deme independent and intermediate. The dominance relation between the generalist allele and the specialist alleles play a decisive role. We also discuss linear selection on a quantitative trait and show that G×E interaction is not necessary for the maintenance of more than two alleles in two demes.     

Clines with partial panmixia

In spatially distributed populations, global panmixia can be regarded as the limiting case of long-distance migration. The effect of incorporating partial panmixia into single-locus clines maintained by migration and selection is investigated. In a diallelic, two-deme model without dominance, partial panmixia can increase or decrease both the polymorphic area in the plane of the migration rates and the equilibrium gene-frequency difference between the two demes. For multiple alleles, under the assumptions that the number of demes is large and both migration and selection are arbitrary but weak, a system of integro-partial differential equations is derived. For two alleles with conservative migration, (i) a Lyapunov functional is found, suggesting generic global convergence of the gene frequency; (ii) conditions for the stability or instability of the fixation states, and hence for a protected polymorphism, are obtained; and (iii) a variational representation of the minimal selection-migration ratio λ₀ (the principal eigenvalue of the linearized system) for protection from loss is used to prove that λ₀ is an increasing function of the panmictic rate and to deduce the effect on λ₀ of changes in selection and migration. The unidimensional step-environment with uniform population density, homogeneous, isotropic migration, and no dominance is examined in detail: An explicit characteristic equation is derived for λ₀; bounds on λ₀ are established; and λ₀ is approximated in four limiting cases. An explicit formula is also deduced for the globally asymptotically stable cline in an unbounded habitat with a symmetric environment; partial panmixia maintains some polymorphism even as the distance from the center of the cline tends to infinity.     

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.     

Separation of time scales, fixation probabilities and convergence to evolutionary stable states under isolation by distance

To a first order of approximation, selection is frequency independent in a wide range of family structured models and in populations following an island model of dispersal, provided the number of families or demes is large and the population is haploid or diploid but allelic effects on phenotype are semidominant. This result underlies the way the evolutionary stability of traits is computed in games with continuous strategy sets. In this paper similar results are derived under isolation by distance. The first-order effect on expected change in allele frequency is given in terms of a measure of local genetic diversity, and of measures of genetic structure which are almost independent of allele frequency in the total population when the number of demes is large. Hence, when the number of demes increases the response to selection becomes of constant sign. This result holds because the relevant neutral measures of population structure converge to equilibrium at a rate faster than the rate of allele frequency changes in the total population. In the same conditions and in the absence of demographic fluctuations, the results also provide a simple way to compute the fixation probability of mutants affecting various ecological traits, such as sex ratio, dispersal, life-history, or cooperation, under isolation by distance. This result is illustrated and tested against simulations for mutants affecting the dispersal probability under a stepping-stone model.     

The robustness of neutral models of geographical variation

The approximation of diploid migration by gametic dispersion is studied. The monoecious, diploid population is subdivided into panmictic colonies that exchange migrants. Generations are discrete and nonoverlapping; the analysis is restricted to a single locus in the absence of selection; every allele mutates to a new allele at the same rate u. Diploid-migration models without self-fertilization and with selfing at the “random” rate (equal to the reciprocal of the deme size in each deme) are investigated; in the gametic-dispersion models, selfing occurs at the random rate. It is shown for the unbounded stepping-stone model in one and two dimensions, the circular stepping-stone model, and the island model that the probabilitities of identity in state at equilibrium for diploid migration are close to those for gametic dispersion if the mutation rate is small or the deme size is large. Explicit error bounds are presented in all the above cases. It is also proved that if the number of demes is finite and the migration matrix is arbitrary but time independent and ergodic, then in the strong-migration approximation the equilibrium and the ultimate rate and pattern of convergence of both diploid-dispersion models are close to the corresponding gametic-dispersion formulae. For the strong-migration approximation at equilibrium, migration must dominate both mutation and random drift; for the convergence results, it suffices that migration dominate random drift. All the results apply to a dioecious population if the migration pattern and mutation rate are sex independent.     

Separation of time scales, fixation probabilities and convergence to evolutionarily stable states under isolation by distance

To a first order of approximation, selection is frequency independent in a wide range of family structured models and in populations following an island model of dispersal, provided the number of families or demes is large and the population is haploid or diploid but allelic effects on phenotype are semidominant. This result underlies the way the evolutionary stability of traits is computed in games with continuous strategy sets. In this paper similar results are derived under isolation by distance. The first-order effect on expected change in allele frequency is given in terms of a measure of local genetic diversity, and of measures of genetic structure which are almost independent of allele frequency in the total population when the number of demes is large. Hence, when the number of demes increases the response to selection becomes of constant sign. This result holds because the relevant neutral measures of population structure converge to equilibrium at a rate faster than the rate of allele frequency changes in the total population. In the same conditions and in the absence of demographic fluctuations, the results also provide a simple way to compute the fixation probability of mutants affecting various ecological traits, such as sex ratio, dispersal, life-history, or cooperation, under isolation by distance. This result is illustrated and tested against simulations for mutants affecting the dispersal probability under a stepping-stone model.     

Multilocus selection in subdivided populations I. Convergence properties for weak or strong migration

The dynamics and equilibrium structure of a deterministic population-genetic model of migration and selection acting on multiple multiallelic loci is studied. A large population of diploid individuals is distributed over finitely many demes connected by migration. Generations are discrete and nonoverlapping, migration is irreducible and aperiodic, all pairwise recombination rates are positive, and selection may vary across demes. It is proved that, in the absence of selection, all trajectories converge at a geometric rate to a manifold on which global linkage equilibrium holds and allele frequencies are identical across demes. Various limiting cases are derived in which one or more of the three evolutionary forces, selection, migration, and recombination, are weak relative to the others. Two are particularly interesting. If migration and recombination are strong relative to selection, the dynamics can be conceived as a perturbation of the so-called weak-selection limit, a simple dynamical system for suitably averaged allele frequencies. Under nondegeneracy assumptions on this weak-selection limit which are generic, every equilibrium of the full dynamics is a perturbation of an equilibrium of the weak-selection limit and has the same stability properties. The number of equilibria is the same in both systems, equilibria in the full (perturbed) system are in quasi-linkage equilibrium, and differences among allele frequencies across demes are small. If migration is weak relative to recombination and epistasis is also weak, then every equilibrium is a perturbation of an equilibrium of the corresponding system without migration, has the same stability properties, and is in quasi-linkage equilibrium. In both cases, every trajectory converges to an equilibrium, thus no cycling or complicated dynamics can occur.      

The Distribution of Mutant Alleles in a Subdivided Population

The results are presented from a simulation study of the spatial distribution of mutant alleles in a subdivided population. Statistical measures of the spatial pattern are defined in such a way that the same quantities could be measured in a geographic survey of allele frequencies in natural populations. Two types of quantities are discussed in this paper: (1) the occupancy distribution provides information on the presence or absence of the mutant in different numbers of demes; and (2) the conditional frequency distribution provides information about the extent of local differentiation when the mutant is present in different numbers of demes. Properties of these distributions are found for different types of natural selection acting on the mutant. Some results are presented for the same statistical measures based on samples of individuals from a fraction of the total number of demes. The simulation results for intermediate levels of the migration rates are compared with analytic results obtained on the limits of high and low migration rates. The main conclusion is that these measures of the spatial distribution of mutants in a subdivided population have simple properties that could provide a new perspective on data from natural populations.     

The structured ancestral selection graph and the many-demes limit

We show that the unstructured ancestral selection graph applies to part of the history of a sample from a population structured by restricted migration among subpopulations, or demes. The result holds in the limit as the number of demes tends to infinity with proportionately weak selection, and we have also made the assumptions of island-type migration and that demes are equivalent in size. After an instantaneous sample-size adjustment, this structured ancestral selection graph converges to an unstructured ancestral selection graph with a mutation parameter that depends inversely on the migration rate. In contrast, the selection parameter for the population is independent of the migration rate and is identical to the selection parameter in an unstructured population. We show analytically that estimators of the migration rate, based on pairwise sequence differences, derived under the assumption of neutrality should perform equally well in the presence of weak selection. We also modify an algorithm for simulating genealogies conditional on the frequencies of two selected alleles in a sample. This permits efficient simulation of stronger selection than was previously possible. Using this new algorithm, we simulate gene genealogies under the many-demes ancestral selection graph and identify some situations in which migration has a strong effect on the time to the most recent common ancestor of the sample. We find that a similar effect also increases the sensitivity of the genealogy to selection.