Geographical Variation in a Quantitative Character

A model for the evolution of the local averages of a quantitative character under migration, selection, and random genetic drift in a subdivided population is formulated and investigated. Generations are discrete and nonoverlapping; the monoecious, diploid population mates at random in each deme. All three evolutionary forces are weak, but the migration pattern and the local population numbers are otherwise arbitrary. The character is determined by purely additive gene action and a stochastically independent environment; its distribution is Gaussian with a constant variance; and it is under Gaussian stabilizing selection with the same parameters in every deme. Linkage disequilibrium is neglected. Most of the results concern the covariances of the local averages. For a finite number of demes, explicit formulas are derived for (i) the asymptotic rate and pattern of convergence to equilibrium, (ii) the variance of a suitably weighted average of the local averages, and (iii) the equilibrium covariances when selection and random drift are much weaker than migration. Essentially complete analyses of equilibrium and convergence are presented for random outbreeding and site homing, the Levene and island models, the circular habitat and the unbounded linear stepping-stone model in the diffusion approximation, and the exact unbounded stepping-stone model in one and two dimensions.     

THE EFFECT OF COLLECTIVE DISPERSAL ON THE GENETIC STRUCTURE OF A SUBDIVIDED POPULATION

Correlated dispersal paths between two or more individuals are widespread across many taxa. The population genetic implications of this collective dispersal have received relatively little attention. Here we develop two‐sample coalescent theory that incorporates collective dispersal in a finite island model to predict expected coalescence times, genetic diversities, and F‐statistics. We show that collective dispersal reduces mixing in the system, which decreases expected coalescence times and increases F_ST. The effects are strongest in systems with high migration rates. Collective dispersal breaks the invariance of within‐deme coalescence times to migration rate, whatever the deme size. It can also cause F_ST to increase with migration rate because the ratio of within‐ to between‐deme coalescence times can decrease as migration rate approaches unity. This effect is most biologically relevant when deme size is small. We find qualitatively similar results for diploid and gametic dispersal. We also demonstrate with simulations and analytical theory the strong similarity between the effects of collective dispersal and anisotropic dispersal. These findings have implications for our understanding of the balance between drift–migration–mutation in models of neutral evolution. This has applied consequences for the interpretation of genetic structure (e.g., chaotic genetic patchiness) and estimation of migration rates from genetic data.     

The two-locus ancestral graph in a subdivided population: convergence as the number of demes grows in the island model

We study the ancestral recombination graph for a pair of sites in a geographically structured population. In particular, we consider the limiting behavior of the graph, under Wrights island model, as the number of subpopulations, or demes, goes to infinity. After an instantaneous sample-size adjustment, the graph becomes identical to the two-locus graph in an unstructured population, but with a time scale that depends on the migration rate and the deme size. Interestingly, when migration is gametic, this rescaling of time increases the population mutation rate but does not affect the population recombination rate. We compare this to the case of a partially-selfing population, in which both mutation and recombination depend on the selfing rate. Our result for gametic migration holds both for finite-sized demes, and in the limit as the deme size goes to infinity. However, when migration occurs during the diploid phase of the life cycle and demes are finite in size, the population recombination rate does depend on the migration rate, in a way that is reminiscent of partial selfing. Simulations imply that convergence to a rescaled panmictic ancestral recombination graph occurs for any number of sites as the number of demes approaches infinity.Send offprint request to: Sabin LessardS. Lessard was supported by grants from the Natural Sciences and Research Council of Canada, the Fonds Québécois de la Recherche sur la Nature et les Technologies, and the Université de Montréal.J. Wakeley was supported by a Career Award (DEB-0133760) and by a grant (DEB-9815367) from the National Science Foundation.     

The influence of spatial inhomogeneities on neutral models of geographical variation IV. Discontinuities in the population density and migration rate

The equilibrium structure of the infinite, one-dimensional stepping-stone model with coincident discontinuities in the population density and migration rate is investigated in the diffusion approximation. The monoecious, diploid population is subdivided into an infinite linear array of equally large, panmictic colonies that exchange gametes isotropically. The population density and the migration rate have a discontinuity at the origin, but are elsewhere uniform. Generations are discrete and nonoverlapping; the analysis is restricted to a single locus without selection; every allele mutates to new alleles at the same rate. The three dimensionless parameters in the theory are alpha=(rho(+)/rho(-))(2) (V(+)/V(-))(3/2), and beta(+/-)=4rho(+/-) 2uV(+/-), where rho(+) (rho(-)) and V(+) (V(-)) designate the population density and variance of gametic dispersion per generation to the right (left) of the discontinuity, respectively, and u denotes the mutation rate. The characteristic length on the right (left) is V(+)/(2u) (V(-)/(2u)). The probability of identity is continuous at the origin, but its partial derivatives have a discontinuity unless migration is conservative (rho(-) V(-)=rho(+) V(+)). At least for nonconservative migration, the probability of identity (including the expected homozygosity) can be nonmonotonic even if the migration rate is uniform and the population density is monotonic. Thus, there can be a nonmonotonic genetic response in a neutral model to a monotonic environment.     

The influence of spatial inhomogeneities on neutral models of geographical variation : I. Formulation

The influence of spatial variation in the carrying capacity and migration rate of a geographical barrier on the one-dimensional stepping-stone model is studied. The monoecious, diploid population is subdivided into an infinite linear array of panmictic colonies that exchange gametes. In each deme, the rate of self-fertilization is equal to the reciprocal of the number of individuals in that deme. Generations are discrete and nonoverlapping; the analysis is restricted to a single locus in the absence of selection; every allele mutates to new alleles at the same rate. In the diffusion approximation, a partial differential equation that incorporates spatial (and temporal) variation in the carrying capacity and migration rate is derived for the probability of identity. Transition conditions that simultaneously take into account discontinuities in the carrying capacity and migration rate are established: the probability of identity is continuous, but its partial derivatives are not, their ratio being a simple function of the carrying capacities and migrational variance on the two sides of the inhomogeneity. The partial derivatives of the probability of identity are continuous across a geographical barrier, whereas the probability of identity itself has a discontinuity proportional to the partial derivative at the barrier, the constant of proportionality being a measure of the difficulty of crossing the barrier.     

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.     

Genetic Variation and Random Drift in Autotetraploid Populations

The rate of decay of genetic variation is determined for randomly mating autotetraploid populations of finite size, and the equilibrium homozygosity under mutation and random drift is calculated. It is shown that heterozygosity is lost at a slower rate than in diploid populations, and that the equilibrium heterozygosity with mutation and random drift is higher than for diploids. Outcrossing populations as well as populations that randomly self are analyzed. A method of comparing genetic variation between autotetraploid and diploid populations is proposed. Our treatment suggests that the ``gametic homozygosity' provides a unified approach for comparing genotypes within a population as well as comparing genetic variation between populations with different levels of ploidy.     

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.     

Exact moment calculations for genetic models with migration,mutation, and drift

Using properties of moment stationarity we develop exact expressions for the mean and covariance of allele frequencies at a single locus for a set of populations subject to drift, mutation, and migration. Some general results can be obtained even for arbitrary mutation and migration matrices, for example: (1) Under quite general conditions, the mean vector depends only on mutation rates, not on migration rates or the number of populations. (2) Allele frequencies covary among all pairs of populations connected by migration. As a result, the drift, mutation, migration process is not ergodic when any finite number of populations is exchanging genes. In addition, we provide closed-form expressions for the mean and covariance of allele frequencies in Wright's finite-island model of migration under several simple models of mutation, and we show that the correlation in allele frequencies among populations can be very large for realistic rates of mutation unless an enormous number of populations are exchanging genes. As a result, the traditional diffusion approximation provides a poor approximation of the stationary distribution of allele frequencies among populations. Finally, we discuss some implications of our results for measures of population structure based on Wright's F-statistics.     

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.     

The Rate of Approach to the Equilibrium Value of F ST in Island and One-Dimensional Stepping-Stone Models of Migration

The rate of approach to the equilibrium value of F _ST was analyzed numerically for the finite island and one-dimensional stepping-stone models using computer simulation. For both models, this rate was shown to decrease with decreasing migration rate among subpopulations but in the case of the stepping-stone model, it takes thousands rather than tens of generations to reach the equilibrium. Unlike the island structure of migration, in the stepping-stone model an increase in the subpopulation number reduces the rate of reaching the equilibrium state.     

The evolution of recombination in permanent translocation heterozygotes

The evolution of a selectively neutral locus that controls the degree to which alleles at a single selected locus are linked with a particular set of chromosomes in a permanent translocation heterozygote is studied. With complete selfing and fitness overdominance a new allele at the modifying locus will increase in frequency if it increases the linkage of all alleles at the selected locus to a particular set of chromosomes. With random mating a new allele at the modifying locus will increase when rare if it increases the linkage of alleles at the selected locus to a particular set of chromosomes. In addition, a parameter analogous to the coefficient of linkage disequilibrium in usual two-locus models with random mating must be nonzero if a new allele at the modifying locus is to increase in frequency at a geometric rate when rare. With mixed selfing and random mating a new allele at the modifying locus will apparently increase when rare only if it increases the linkage of alleles at the selected locus to a particular set of chromosomes.     

The rate of approach to the equilibrium value of F(ST) in island and one-dimensional stepping-stone models of migration

The rate of approach to the equilibrium value of FST was analyzed numerically for the finite island and one-dimensional stepping-stone models using computer simulation. For both models, this rate was shown to decrease with decreasing migration rate among subpopulations but in the case of the stepping-stone model, it takes thousands rather than tens of generations to reach the equilibrium. Unlike the island structure of migration, in the stepping-stone model an increase in the subpopulation number reduces the rate of reaching the equilibrium state.     

The diffusion model for migration and selection in a plant population

 The diffusion approximation is derived for migration and selection at a multiallelic locus in a partially selfing plant population subdivided into a lattice of colonies. Generations are discrete and nonoverlapping; both pollen and seeds disperse. In the diffusion limit, the genotypic frequencies at each point are those determined at equilibrium by the local rate of selfing and allelic frequencies. If the drift and diffusion coefficients are taken as the appropriate linear combination of the corresponding coefficients for pollen and seeds, then the migration terms in the partial differential equation for the allelic frequencies have the standard form for a monoecious animal population. The selection term describes selection on the local genotypic frequencies. The boundary conditions and the unidimensional transition conditions for a geographical barrier and for coincident discontinuities in the carrying capacity and migration rate have the standard form. In the diallelic case, reparametrization renders the entire theory of clines and of the wave of advance of favorable alleles directly applicable to plant populations. Received 30 August 1995; received in revised form 23 February 1996     

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.      

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.     

Gene Conversion, Linkage, and the Evolution of Multigene Families

The evolution of the probabilities of genetic identity within and between the loci of a multigene family is investigated. Unbiased gene conversion, equal crossing over, random genetic drift, and mutation to new alleles are incorporated. Generations are discrete and nonoverlapping; the diploid, monoecious population mates at random. The linkage map is arbitrary, and the location dependence of the probabilities of identity is formulated exactly. The greatest of the rates of gene conversion, random drift, and mutation is epsilon much less than 1. For interchromosomal conversion, the equilibrium probabilities of identity are within order epsilon [i.e., O(epsilon)] of those in a simple model that has no location dependence and, at equilibrium, no linkage disequilibrium. At equilibrium, the linkage disequilibria are of O(epsilon); they are evaluated explicitly with an error of O(epsilon 2); they may be negative if symmetric heteroduplexes occur. The ultimate rate and pattern of convergence to equilibrium are within O(epsilon 2) and O(epsilon), respectively, of that of the same simple model. If linkage is loose (i.e., all the crossover rates greatly exceed epsilon, though they may still be much less than 1/2), the linkage disequilibria are reduced to O(epsilon) in a time of O(-ln epsilon). If intrachromosomal conversion is incorporated, the same results hold for loose linkage, except that, if the crossover rates are much less than 1/2, then the linkage disequilibria generally exceed those for pure interchromosomal conversion.     

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.     

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.     

Kin groups and trait groups: population structure and epidemic disease selection

A Monte Carlo simulation based on the population structure of a small-scale human population, the Semai Senoi of Malaysia, has been developed to study the combined effects of group, kin, and individual selection. The population structure resembles D.S. Wilson's structured deme model in that local breeding populations (Semai settlements) are subdivided into trait groups (hamlets) that may be kin-structured and are not themselves demes. Additionally, settlement breeding populations are connected by two-dimensional stepping-stone migration approaching 30% per generation. Group and kin-structured group selection occur among hamlets the survivors of which then disperse to breed within the settlement population. Genetic drift is modeled by the process of hamlet formation; individual selection as a deterministic process, and stepping-stone migration as either random or kin-structured migrant groups. The mechanism for group selection is epidemics of infectious disease that can wipe out small hamlets particularly if most adults become sick and social life collapses. Genetic resistance to a disease is an individual attribute; however, hamlet groups with several resistant adults are less likely to disintegrate and experience high social mortality. A specific human gene, hemoglobin E, which confers resistance to malaria, is studied as an example of the process. The results of the simulations show that high genetic variance among hamlet groups may be generated by moderate degrees of kin-structuring. This strong microdifferentiation provides the potential for group selection. The effect of group selection in this case is rapid increase in gene frequencies among the total set of populations. In fact, group selection in concert with individual selection produced a faster rate of gene frequency increase among a set of 25 populations than the rate within a single unstructured population subject to deterministic individual selection. Such rapid evolution with plausible rates of extinction, individual selection, and migration and a population structure realistic in its general form, has implications for specific human polymorphisms such as hemoglobin variants and for the more general problem of the tempo of evolution as well.