Heterogeneous selection in subdivided populations

The dynamics of allele frequencies changing under migration and heterogeneous selection in a subdivided population are investigated. Using perturbation techniques, a stationary state is obtained when migration and selection are both small. Heterogeneous selection leads to a positive correlation between values of F-statistics and heterozygosities when these are compared among sets of subdivided populations. This contrasts with a negative value of the correlation obtained under Wright's classical model of homogeneous selection, and with the absence of correlation in the completely neutral situation.Research supported in part by NIH grants GM 28016 and GM 10452 and a grant from the John D. and Catherine T. MacArthur Foundation     

Fixation indices in subdivided populations.

Without restricting the evolutionary forces that may be present, the theory of fixation indices, or F-statistics, in an arbitrarily subdivided population is developed systematically in terms of allelic and genotypic frequencies. The fixation indices for each homozygous genotype are expressed in terms of the fixation indices for the heterozygous genotypes. Therefore, together with the allelic frequencies, the latter suffice to describe population structure. Possible random fluctuations in the allelic frequencies (which may be caused, e.g., by finiteness of the subpopulations) are incorporated so that the fixation indices are parameters, rather than random variables, and these parameters are expressed in terms of ratios of evolutionary expectations of heterozygosities. The interpretation of some measures of population differentiation is also discussed. In particular, F(ST) is an appropriate index of gene-frequency differentiation if and only if the genetic diversity is low.     

Some circumstances assuring monomorphism in subdivided populations

It is well known that in a subdivided population subject to soft selection with two alleles at one locus, instability of both fixation states (a “protected polymorphism”) entails at least one stable polymorphic equilibrium. Although stable polymorphic and monomorphic equilibria can coexist in general, a stable fixation state (monomorphic equilibrium) precludes the existence of any polymorphic equilibrium under the circumstances of haploid or submultiplicative diploid viabilities. This provides that a stable monomorphism is robust against random fluctuations in allele frequencies. It also increases the known circumstances where there is a unique globally attracting stable equilibrium, i.e., where allele frequencies are determined by the selection-migration structure independent of the history of the system.     

Fixation probability and time in subdivided populations

New alleles arising in a population by mutation ultimately are either fixed or lost. Either is possible, for both beneficial and deleterious alleles, because of stochastic changes in allele frequency due to genetic drift. Spatially structured populations differ from unstructured populations in the probability of fixation and the time that this fixation takes. Previous results have generally made many assumptions: that all demes contribute to the next generation in exact proportion to their current sizes, that new mutations are beneficial, and that new alleles have additive effects. In this article these assumptions are relaxed, allowing for an arbitrary distribution among demes of reproductive success, both beneficial and deleterious effects, and arbitrary dominance. The effects of population structure can be expressed with two summary statistics: the effective population size and a variant of Wright's F(ST). In general, the probability of fixation is strongly affected by population structure, as is the expected time to fixation or loss. Population structure changes the effective size of the species, often strongly downward; smaller effective size increases the probability of fixing deleterious alleles and decreases the probability of fixing beneficial alleles. On the other hand, population structure causes an increase in the homozygosity of alleles, which increases the probability of fixing beneficial alleles but somewhat decreases the probability of fixing deleterious alleles. The probability of fixing new beneficial alleles can be simply described by 2hs(1 - F(ST))N(e)/N(tot), where hs is the change in fitness of heterozygotes relative to the ancestral homozygote, F(ST) is a weighted version of Wright's measure of population subdivision, and N(e) and N(tot) are the effective and census sizes, respectively. These results are verified by simulation for a broad range of population structures, including the island model, the stepping-stone model, and a model with extinction and recolonization.     

Analysis of disruptive selection in subdivided populations

Background

Analytical methods have been proposed to determine whether there are evolutionarily stable strategies (ESS) for a trait of ecological significance, or whether there is disruptive selection in a population approaching a candidate ESS. These criteria do not take into account all consequences of small patch size in populations with limited dispersal.

Results

We derive local stability conditions which account for the consequences of small and constant patch size. All results are derived from considering R_m, the overall production of successful emigrants from a patch initially colonized by a single mutant immigrant. Further, the results are interpreted in term of concepts of inclusive fitness theory. The condition for convergence to an evolutionarily stable strategy is proportional to some previous expressions for inclusive fitness. The condition for evolutionary stability stricto sensu takes into account effects of selection on relatedness, which cannot be neglected. It is function of the relatedness between pairs of genes in a neutral model and also of a three-genes relationship. Based on these results, I analyze basic models of dispersal and of competition for resources. In the latter scenario there are cases of global instability despite local stability. The results are developed for haploid island models with constant patch size, but the techniques demonstrated here would apply to more general scenarios with an island mode of dispersal.

Conclusions

The results allow to identity and to analyze the relative importance of the different selective pressures involved. They bridge the gap between the modelling frameworks that have led to the R_m concept and to inclusive fitness.

     

Stable linkage disequilibrium without epistasis in subdivided populations

In a large random mating population stable linkage disequilibrium occurs only when there is epistasis. However if a population is divided into a number of subpopulations among which migration occurs, stable linkage disequilibrium in each subpopulation may be produced without epistasis. In the case of two subpopulations a necessary condition for linkage equilibrium in the absence of epistasis is that at least at one of the two loci under consideration the gene frequency must be the same for the two populations. This condition is rather severe and any violation of this will lead to stable linkage disequilibrium. A similar conclusion can be made with more than two populations. In general the presence of linkage disequilibrium does not necessarily imply the existence of epistasis even in equilibrium populations.     

Three components of genetic drift in subdivided populations

Wright's metaphor of sampling is extended to consider three components of genetic drift: those occurring before, during, and after migration. To the extent that drift at each stage behaves like an independent random sample, the order of events does not matter. When sampling is not random, the order does matter, and the effect of population size is confounded with that of mobility. The widely cited result that genetic differentiation of local groups depends only on the product of group size and migration rate holds only when nonrandom sampling does not occur prior to migration in the life cycle.     

Stability properties of underdominance in finite subdivided populations

IN ISOLATED populations underdominance leads to bistable evolutionary dynamics: below a certain mutant allele frequency the wildtype succeeds. Above this point, the potentially underdominant mutant allele fixes. In subdivided populations with gene flow there can be stable states with coexistence of wildtypes and mutants: polymorphism can be maintained because of a migration-selection equilibrium, i.e., selection against rare recent immigrant alleles that tend to be heterozygous. We focus on the stochastic evolutionary dynamics of systems where demographic fluctuations in the coupled populations are the main source of internal noise. We discuss the influence of fitness, migration rate, and the relative sizes of two interacting populations on the mean extinction times of a group of potentially underdominant mutant alleles. We classify realistic initial conditions according to their impact on the stochastic extinction process. Even in small populations, where demographic fluctuations are large, stability properties predicted from deterministic dynamics show remarkable robustness. Fixation of the mutant allele becomes unlikely but the time to its extinction can be long.     

Quasi-species evolution in subdivided populations favours maximally deleterious mutations

Most models of quasi-species evolution predict that populations will evolve to occupy areas of sequence space with the greatest concentration of neutral sequences, thus minimizing the deleterious mutation rate and creating mutationally 'robust' genomes. In contrast, empirical studies of the principal model of quasi-species evolution, RNA viruses, suggest that the effects of deleterious mutations are more severe than in similar DNA-based microbes. We demonstrate that populations divided into discrete patches connected by dispersal may favour genotypes where the deleterious effect of non-neutral mutations is maximized. This effect is especially strong in the absence of back mutation and when the amount of time spent in hosts prior to dispersal is intermediate. Our results indicate that RNA viruses that produce acute infections initiated by a small number of virions are expected to evolve fragile genetic architectures when compared with other RNA viruses.     

Characterization and conservation of genetic diversity in subdivided populations

We review the available tools for analysing genetic diversity in conservation programmes of subdivided populations. Ways for establishing conservation priorities have been developed in the context of livestock populations, both from the classical population genetic analysis and from the more recent Weitzman's approach. We discuss different reasons to emphasize either within or between-population variation in conservation decisions and the methodology to establish some compromise. The comparison between neutral and quantitative variation is reviewed from both theoretical and empirical points of view, and the different procedures for the dynamic management of conserved subdivided populations are discussed.     

Effective size in management and conservation of subdivided populations

A numerical method for computing the eigenvalue variance effective size of a subdivided population connected by any fixed pattern of migration is described. Using specific examples it is shown that total effective size of a subdivided population can become less than the sum of the subpopulation sizes as a result of directionalities in the pattern of migration. For an extension of the model with threshold harvesting and local deterministic logistic population dynamic we consider the problem of maximizing the total harvesting yield with constraints on the total effective size. For some simple source-sink systems and more complicated population structures where subpopulations differ in their degree of isolation, it is shown to be optimal, for a given total effective size, to raise the harvesting thresholds relatively more in small and in isolated populations. Finally, we show how the method applies to populations which are supplemented, either intentionally or unintentionally. It is shown that the total effective size can be reduced by several orders of magnitude if the captive component of a population is much smaller than the wild component, even with symmetric backward migration.     

Geographical invariance and the strong-migration limit in subdivided populations

Invariance under population subdivision and the strong-migration limit are investigated for digenic samples in neutral models. The monoecious, diploid population is subdivided into a finite number of panmictic colonies that exchange gametes. The backward migration matrix is arbitrary, but time independent and ergodic (i.e., irreducible and aperiodic). Results are derived for the distribution of the place and time of coalescence, for the probability of identity in the model of infinitely many alleles, and for the distribution of the number of nucleotide differences in the model of infinitely many sites without recombination. Received: 5 October 1999 / Revised version: 1 February 2000 / Published online: 4 July 2000     

Management of genetic diversity of subdivided populations in conservation programmes

Population subdivision must be explicitly considered in the management of conservation programmes, as most populations of wild species at risk of extinction and those kept in captivity are spatially structured. The partition of gene and allelic diversity in within- and between-subpopulation components allows for the integral management of populations. We summarise the main aspects of this partition and some of its applications in terms of priorisation of populations for conservation and establishment of synthetic populations. The procedures for the maintenance of diversity in subdivided populations making use of molecular markers and its implementation by the software METAPOP are illustrated with empirical data.     

Polymorphic equilibria with assortative mating and selection in subdivided populations

Two modes of assortative mating, partial selfing and assorting by phenotypic classes, are studied in a subdivided population. Differential viability is allowed and the selection intensities and assorting tendencies are permitted to vary among the habitats. There exists a symmetric polymorphism; we delimit its level of heterozygosity and stability nature (dependent on the selection intensities and assorting propensities). This complements studies of the fixation states and thereby provides further insight into the global equilibrium structure in subdivided populations. Circumstances are given where the fixation states and symmetric polymorphism comprise the global equilibrium structure. Examples are also given where migration engenders stable polymorphic equilibria and stable polymorphic equilibrium cycles which are absent in single demes without migration.     

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.     

Genetic identity in subdivided populations. I. Two equal-sized subpopulations

Moments of the steady state frequency spectrum (probabilities of identity of samples of genes) are obtained for a subdivided population by using standard recursive identity by state calculations. These moments are used to obtain variances for some measures of genetic identity, including Nei's normalized genetic identity (I) and genetic distance (?log_eI). The results are compared with those obtained from the corresponding undivided population theory, including adjustments to the effective number to try to account for subdivision. Undivided population approximations based on effective number are surprisingly accurate, regardless of the migration rate, when sampling exclusively from one subpopulation.     

Evolution of learning in subdivided populations that occupy environmentally heterogeneous sites

We study the effects of natural selection and migration on the numbers of individual learners and social learners in subdivided populations that occupy environmentally heterogeneous sites. The island model and the circular stepping model each have four classes of globally stable equilibria (fixation of individual learners, polymorphism of individual and social learners, fixation of social learners, and extinction). The linear stepping stone model has an additional class of equilibria, which are characterized by the complete absence of phenotypes adapted to the interior sites. Low and high rates of migration favor social and individual learners, respectively, in all three models. In addition, we use the stepping stone models to study the range expansion of a species, initially confined to one environmentally homogeneous site, into the spatially heterogeneous world. The successive peaks of the transient spatial distributions of the number of individual learners occur at initially empty sites.     

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

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