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.     

Likelihood and approximate likelihood analyses of genetic structure in a linear habitat: performance and robustness to model mis-specification

We evaluate the performance of maximum likelihood (ML) analysis of allele frequency data in a linear array of populations. The parameters are a mutation rate and either the dispersal rate in a stepping stone model or a dispersal rate and a scale parameter in a geometric dispersal model. An approximate procedure known as maximum product of approximate conditional (PAC) likelihood is found to perform as well as ML. Mis-specification biases may occur because the importance sampling algorithm is formally defined in term of mutation and migration rates scaled by the total size of the population, and this size may differ widely in the statistical model and in reality. As could be expected, ML generally performs well when the statistical model is correctly specified. Otherwise, mutation rate estimates are much closer to mutation probability scaled by number of demes in the statistical model than scaled by number of demes in reality when mutation probability is high and dispersal is most limited. This mis-specification bias actually has practical benefits. However, opposite results are found in opposite conditions. Migration rate estimates show roughly similar trends, but they may not always be easily interpreted as low-bias estimates of dispersal rate under any scaling. Estimation of the dispersal scale parameter is also affected by mis-specification of the number of demes, and the different biases compensate each other in such a way that good estimation of the so-called neighborhood size (or more precisely the product of population density and mean-squared parent-offspring dispersal distance) is achieved. Results congruent with these findings are found in an application to a damselfly data set.     

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.     

MAINTENANCE OF POLYGENIC VARIATION IN SPATIALLY STRUCTURED POPULATIONS: ROLES FOR LOCAL MATING AND GENETIC REDUNDANCY

Although heritable genetic variation is critical to the evolutionary process, we know little about how it is maintained. Obviously, mutation-selection balance must play a role, but there is considerable doubt over whether it can account for heritabilities as high as 0.5, which are commonly found in natural populations. Most models of mutation-selection balance assume panmictic populations. In this paper we use Monte Carlo simulations to examine the effect of isolation by distance on the variation maintained by mutation in a polygenic trait subject to optimizing selection. We show that isolation by distance can substantially increase the total variation maintained in continuous populations over a wide range of dispersal patterns, but only if more than one genotype produces the optimal phenotype (genetic redundancy). Isolation by distance alone has only a slight effect on the variation maintained in the total population for neutral alleles. The combined effect of isolation by distance and genetic redundancy, however, allows the maintenance of substantial variation despite strong stabilizing selection. The mechanism is straightforward. Isolation by distance allows mutation and drift to operate independently in different parts of the population. Because of their independent evolutionary histories, different parts of the population independently draw from the available set of redundant genotypes. Because the genotypes are redundant, selection does not discriminate among them, and they will persist until eliminated by drift. The population as a whole maintains many distinct genotypes. We show that this process allows mutation to maintain high levels of variation, even under strong stabilizing selection, and that over a moderate range of dispersal patterns the amount of variation maintained in the entire population is independent of both the strength of selection and the variance of the dispersal distance. Furthermore, we show that individual heterozygosity is increased in locally mating populations when selection is strong. Finally, our simulations provide a rough picture of how selection and the dispersal pattern influence the spatial distribution of genetic and phenotypic variation.     

Evolution of stepping-stone dispersal rates

We present a general model of the evolution of dispersal in a population with any distribution of dispersal distance. We use this model to analyse evolutionarily stable (ES) dispersal rates for the classical island model of dispersal and for three different stepping-stone models. Using general techniques to compute relatedness coefficients in the different dispersal models which we consider, we find that the distribution of dispersal distance may affect the ES dispersal rate when the cost of dispersal is low. In this case the ES dispersal rate increases with the number of demes that can be reached by one dispersal event. However, for increasing cost the ES dispersal rate converges to a value independent of the distribution of dispersal distance. These results are in contrast to previous analyses of similar models. The effects of the size (number of demes) and shape (ratio between the width and the length) of the population on the evolution of dispersal are also studied. We find that larger and more elongated populations lead generally to higher ES dispersal rates. However, both of these effects can only be observed for extreme parameter values (i.e. for very small and very elongated populations). The direct fitness method and the analytical techniques used here to compute relatedness coefficients provide an efficient way to analyse ES strategies in subdivided populations.     

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.     

Microgeographic population structure of green swordail fish: genetic differentiation despite abundant migration

Swordtails (Xiphophorus; Poeciliidae) have figured prominently in research on fish mating behaviours, sexual selection, and carcinogenesis, but their population structures and dispersal patterns have been relatively neglected. Using nine microsatellite loci, we estimated genetic differentiation in Xiphophorus helleri within and between adjacent streams in Belize. The genetic data were complemented by a tagging study of movement within one stream. In the absence of physical dispersal barriers (waterfalls), population structure followed an isolation by distance (IBD) pattern. Genetic differentiation (F_ST up to 0.07) was significant between and within creeks, despite high dispersal in the latter as judged by the tagging data. Such heterogeneity apparently was a result of genetic drift in local demes, due to small population sizes and highly skewed paternity. The IBD pattern was interrupted by waterfalls, boosting F_ST above 0.30 between adjacent samples across these barriers. Overall, our results are helpful in understanding the interplay of evolutionary forces and population dynamics in a small fish living in a changeable habitat.     

Polymorphism and divergence for island-model species

Estimates of the scaled selection coefficient, gamma of Sawyer and Hartl, are shown to be remarkably robust to population subdivision. Estimates of mutation parameters and divergence times, in contrast, are very sensitive to subdivision. These results follow from an analysis of natural selection and genetic drift in the island model of subdivision in the limit of a very large number of subpopulations, or demes. In particular, a diffusion process is shown to hold for the average allele frequency among demes in which the level of subdivision sets the timescale of drift and selection and determines the dynamic equilibrium of allele frequencies among demes. This provides a framework for inference about mutation, selection, divergence, and migration when data are available from a number of unlinked nucleotide sites. The effects of subdivision on parameter estimates depend on the distribution of samples among demes. If samples are taken singly from different demes, the only effect of subdivision is in the rescaling of mutation and divergence-time parameters. If multiple samples are taken from one or more demes, high levels of within-deme relatedness lead to low levels of intraspecies polymorphism and increase the number of fixed differences between samples from two species. If subdivision is ignored, mutation parameters are underestimated and the species divergence time is overestimated, sometimes quite drastically. Estimates of the strength of selection are much less strongly affected and always in a conservative direction.     

Consequences of population structure on genes under balancing selection

This paper describes a new approach to modeling population structure for genes under strong balancing selection of the type seen in plant self-incompatibility systems and the major histocompatibility complex (MHC) system of vertebrates. Simple analytic solutions for the number of alleles maintained at equilibrium and the expected proportion of alleles shared between demes at various levels are derived and checked against simulation results. The theory accurately captures the dynamics of allele number in a subdivided population and identifies important values of m (migration rate) at which allele number and distribution change qualitatively. Starting from a panmictic population, as migration among demes decreases a qualitative change in dynamics is seen at approximately m(crit) approximately equal to the square root of(s/4piNT) where NT is the total population size and s is a measure of the strength of selection. At this point, demes can no longer maintain their panmictic allele number, due to increasing isolation from the total population. Another qualitative change occurs at a migration rate on the same order of magnitude as the mutation rate, mu. At this point, the demes are highly differentiated for allele complement, and the total number of alleles in the population is increased. Because in general u < m<(crit) at intermediate migration rates slightly fewer alleles may be maintained in the total population than are maintained at panmixia. Within this range, total allele number may not be the best indicator of whether a population is effectively panmictic, and some caution should be used when interpreting samples from such populations. The theory presented here can help to analyze data from genes under balancing selection in subdivided populations.     

Invasion and fixation of sex-reversal genes

We simulated a meta-population with random dispersal among demes but local mating within demes to investigate conditions under which a dominant female-determining gene W, with no individual selection advantage, can invade and become fixed in females, changing the population from male to female heterogamety. Starting with one mutant W in a single deme, the interaction of sex ratio selection and random genetic drift causes W to be fixed among females more often than a comparable neutral mutation with no influence on sex determination, even when YY males have slightly reduced viability. Meta-population structure and interdeme selection can also favour the fixation of W. The reverse transition from female to male heterogamety can also occur with higher probability than for a comparable neutral mutation. These results help to explain the involvement of sex-determining genes in the evolution of sex chromosomes and in sexual selection and speciation.     

Microsatellite Analyses of Blacktip Reef Sharks (Carcharhinus melanopterus) in a Fragmented Environment Show Structured Clusters

The population dynamics of shark species are generally poorly described because highly mobile marine life is challenging to investigate. Here we investigate the genetic population structure of the blacktip reef shark (Carcharhinus melanopterus) in French Polynesia. Five demes were sampled from five islands with different inter-island distances (50–1500 km). Whether dispersal occurs between islands frequently enough to prevent moderate genetic structure is unknown. We used 11 microsatellites loci from 165 individuals and a strong genetic structure was found among demes with both F-statistics and Bayesian approaches. This differentiation is correlated with the geographic distance between islands. It is likely that the genetic structure seen is the result of all or some combination of the following: low gene flow, time since divergence, small effective population sizes, and the standard issues with the extent to which mutation models actually fit reality. We suggest low levels of gene flow as at least a partial explanation of the level of genetic structure seen among the sampled blacktip demes. This explanation is consistent with the ecological traits of blacktip reef sharks, and that the suitable habitat for blacktips in French Polynesia is highly fragmented. Evidence for spatial genetic structure of the blacktip demes we studied highlights that similar species may have populations with as yet undetected or underestimated structure. Shark biology and the market for their fins make them highly vulnerable and many species are in rapid decline. Our results add weight to the case that total bans on shark fishing are a better conservation approach for sharks than marine protected area networks.     

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.     

Between migration load and evolutionary rescue: dispersal,adaptation and the response of spatially structured populations to environmental change

The evolutionary potential of populations is mainly determined by population size and available genetic variance. However, the adaptability of spatially structured populations may also be affected by dispersal: positively by spreading beneficial mutations across sub-populations, but negatively by moving locally adapted alleles between demes. We develop an individual-based, two-patch, allelic model to investigate the balance between these opposing effects on a population''s evolutionary response to rapid climate change. Individual fitness is controlled by two polygenic traits coding for local adaptation either to the environment or to climate. Under conditions of selection that favour the evolution of a generalist phenotype (i.e. weak divergent selection between patches) dispersal has an overall positive effect on the persistence of the population. However, when selection favours locally adapted specialists, the beneficial effects of dispersal outweigh the associated increase in maladaptation for a narrow range of parameter space only (intermediate selection strength and low linkage among loci), where the spread of beneficial climate alleles is not strongly hampered by selection against non-specialists. Given that local selection across heterogeneous and fragmented landscapes is common, the complex effect of dispersal that we describe will play an important role in determining the evolutionary dynamics of many species under rapidly changing climate.     

PHASE III OF WRIGHT'S SHIFTING BALANCE PROCESS AND THE VARIANCE AMONG DEMES IN MIGRATION RATE

Interdemic selection by the differential migration of individuals out from demes of high fitness and into demes of low fitness (Phase III) is one of the most controversial aspects of Wright's Shifting Balance Theory. I derive a relationship between Phase III migration and the interdemic selection differential, S, and show its potential effect on F_ST. The relationship reveals a diversifying effect of interdemic selection by Phase III migration on the genetic structure of a metapopulation. Using experimental metapopulations, I explored the effect of Phase III migration on F_ST by comparing the genetic variance among demes for two different patterns of migration: (1) island model migration and (2) Wright's Phase III migration. Although mean migration rates were the same, I found that the variance among demes in migration rate was significantly higher with Phase III than with island model migration. As a result, F_ST for the frequency of a neutral marker locus was higher with Phase III than it was with island model migration. By increasing F_ST, Phase III enhanced the genetic differentiation among demes for traits not subject to interdemic selection. This feature makes Wright's process different from individual selection which, by reducing effective population size, decreases the genetic variance within demes for all other traits. I discussed this finding in relation to the efficacy of Phase III and random migration for effecting peak shifts, and the contribution of genes with indirect effects to among‐deme variation.     

Landscape genetic analyses reveal cryptic population structure and putative selection gradients in a large-scale estuarine environment

Disentangling the relative contributions of selective and neutral processes underlying phenotypic and genetic variation under natural, environmental conditions remains a central challenge in evolutionary ecology. However, much of the variation that could be informative in this area of research is likely to be cryptic in nature; thus, the identification of wild populations suitable for study may be problematic. We use a landscape genetics approach to identify such populations of three-spined stickleback inhabiting the Saint Lawrence River estuary. We sampled 1865 adult fish over multiple years. Individuals were genotyped for nine microsatellite loci, and georeferenced multilocus data were used to infer population groupings, as well as locations of genetic discontinuities, under a Bayesian model framework ( geneland ). We modelled environmental data using nonparametric multiple regression to explain genetic differentiation as a function of spatio-ecological effects. Additionally, we used genotype data to estimate dispersal and gene flow to parameterize a simple model predicting adaptive vs. plastic divergence between demes. We demonstrate a bipartite division of the genetic landscape into freshwater and maritime zones, independent of geographical distance. Moreover, we show that the greatest proportion of genetic variation (31.5%) is explained by environmental differences. However, the potential for either adaptive or plastic divergence between demes is highly dependent upon the strength of migration and selection. Consequently, we highlight the utility of landscape genetics as a tool for hypothesis generation and experimental design, to identify focal populations and putative selection gradients, in order to distinguish between phenotypic plasticity and local adaptation.     

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.     

Germline bottlenecks, biparental inheritance and selection on mitochondrial variants: a two-level selection model

Selection on mitochondrial mutations potentially occurs at different levels: at the mitochondria, cell, and organism levels. Several factors affect the strength of selection at these different levels; in particular, mitochondrial bottlenecks during germline development and reduced paternal transmission decrease the genetic variance within cells, while they increase the variance between cells and between organisms, thus decreasing the strength of selection within cells and increasing the strength of selection between cells and organisms. However, bottlenecks and paternal transmission also affect the effective mitochondrial population size, thus affecting genetic drift. In this article, we use a simple model of a unicellular life cycle to investigate the effects of bottlenecks and paternal transmission on the probability of fixation of mitochondrial mutants and their frequency at mutation-selection equilibrium. We find that bottlenecks and reduced paternal transmission decrease the mean frequency of alleles with sm>sc (approximately), where sm and sc are the strengths of selection for an allele within and between cells, respectively, and increase the frequency of alleles with sm0 (unless sm is very small relative to sc) and increase the fixation probability of mutants with sm<0.     

An exact sampling formula for the Wright-Fisher model and a solution to a conjecture about the finite-island model

An exact sampling formula for a Wright-Fisher population of fixed size N under the infinitely many neutral alleles model is deduced. This extends the Ewens formula for the configuration of a random sample to the case where the sample is drawn from a population of small size, that is, without the usual large-N and small-mutation-rate assumption. The formula is used to prove a conjecture ascertaining the validity of a diffusion approximation for the frequency of a mutant-type allele under weak selection in segregation with a wild-type allele in the limit finite-island model, namely, a population that is subdivided into a finite number of demes of size N and that receives an expected fraction m of migrants from a common migrant pool each generation, as the number of demes goes to infinity. This is done by applying the formula to the migrant ancestors of a single deme and sampling their types at random. The proof of the conjecture confirms an analogy between the island model and a random-mating population, but with a different timescale that has implications for estimation procedures.     

Prediction of informativeness for microsatellite markers among progeny of sires used for detection of economic trait loci in dairy cattle

Individual loci affecting economic traits can be located using genetic linkage. Application of either daughter or granddaughter designs requires determination of allele origin in the progeny. If only the sires and their progeny are genotyped, the paternal allele origin of progeny having the same genotype as the sire cannot be determined. The expected frequency of informative sons can be predicted for each sire and genetic marker from the allele frequencies in the population. The accuracy of a predictor of the frequency of informative progeny was tested on 103 grandsire x microsatellite combinations. Number of sons per grandsire varied from 24 to 129. Allele frequencies in the population were estimated by genotyping seven sires. The regression of the frequency of informative sons on the predicted frequency was 1.04 with a zero intercept model. Thus, considering the large number of genetic markers available for analysis, predicted informative frequency is a useful criterion for selection of genetic markers.     

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.