Fixation probability in spatially changing environments.

The fixation probability of a mutant in a subdivided population with spatially varying environments is investigated using a finite island model. This probability is different from that in a panmictic population if selection is intermediate to strong and migration is weak. An approximation is used to compute the fixation probability when migration among subpopulations is very weak. By numerically solving the two-dimensional partial differential equation for the fixation probability in the two subpopulation case, the approximation was shown to give fairly accurate values. With this approximation, we show in the case of two subpopulations that the fixation probability in subdivided populations is greater than that in panmictic populations mostly. The increase is most pronounced when the mutant is selected for in one subpopulation and is selected against in the other subpopulation. Also it is shown that when there are two types of environments, further subdivision of subpopulations does not cause much change of the fixation probability in the no dominance case unless the product of the selection coefficient and the local population size is less than one. With dominance, the effect of subdivision becomes more complex.     

The propagation of a cultural or biological trait by neutral genetic drift in a subdivided population

We study fixation probabilities and times as a consequence of neutral genetic drift in subdivided populations, motivated by a model of the cultural evolutionary process of language change that is described by the same mathematics as the biological process. We focus on the growth of fixation times with the number of subpopulations, and variation of fixation probabilities and times with initial distributions of mutants. A general formula for the fixation probability for arbitrary initial condition is derived by extending a duality relation between forwards- and backwards-time properties of the model from a panmictic to a subdivided population. From this we obtain new formulae(formally exact in the limit of extremely weak migration) for the mean fixation time from an arbitrary initial condition for Wright's island model, presenting two cases as examples. For more general models of population subdivision, formulae are introduced for an arbitrary number of mutants that are randomly located, and a single mutant whose position is known. These formulae contain parameters that typically have to be obtained numerically, a procedure we follow for two contrasting clustered models. These data suggest that variation of fixation time with the initial condition is slight, but depends strongly on the nature of subdivision. In particular, we demonstrate conditions under which the fixation time remains finite even in the limit of an infinite number of demes. In many cases-except this last where fixation in a finite time is seen--the time to fixation is shown to be in precise agreement with predictions from formulae for the asymptotic effective population size.     

Effective size of the early-run sockeye salmon Oncorhynchus nerka population of Lake Azabach'e, Kamchatka Peninsula evaluation of the effect of interaction between subpopulations within a subdivided population

The effect of subdivision on the effective size (Ne) of the early-run sockeye salmon Oncorhynchus nerka population of Lake Azabach'e (Kamchatka Peninsula) has been studied. The mode of this effect is determined by the relative productivity of the subpopulations and its magnitude, by the rate of individual migration among subpopulations and genetic differentiation. If the contributions of subpopulations (offspring numbers) are different, genetic differentiation can reduce the Ne of the subdivided population. At equal subpopulation contributions, genetic differentiation always increases the Ne of the subdivided population in comparison with a panmictic population. We have found that all sockeye salmon subpopulations of Azabach'e Lake produce equal offspring numbers contributing to the next generation. The genetic differentiation between sockeye salmon subpopulations is low, and the subdivision increases the Ne of the early-run race with reference to the sum of the effective sizes of the subpopulations by as little as 2%.     

The evolution of reproductive isolation in spatially structured populations

Abstract.— Recent models of speciation have incorporated population structure and migration into the classic model of speciation in which reproductive isolation arises as a by-product of divergence. In this paper, we expanded these models to explore the joint effects of migration and population subdivision on speciation in a spatially explicit context. The results of our simulation support previous results concerning the influence of population subdivision on the accumulation of reproductive isolation. The simulation also shows that speciation in subdivided populations occurs most rapidly when subpopulations are not strictly allopatric. These results counter the widespread notion that speciation is most likely to occur in allopatric populations and suggest that there are useful insights to be gained by incorporating increasingly realistic types of population structure into models of speciation.     

Match probabilities in a finite, subdivided population

We generalize a recently introduced graphical framework to compute the probability that haplotypes or genotypes of two individuals drawn from a finite, subdivided population match. As in the previous work, we assume an infinite-alleles model. We focus on the case of a population divided into two subpopulations, but the underlying framework can be applied to a general model of population subdivision. We examine the effect of population subdivision on the match probabilities and the accuracy of the product rule which approximates multi-locus match probabilities as a product of one-locus match probabilities. We quantify the deviation from predictions of the product rule by R, the ratio of the multi-locus match probability to the product of the one-locus match probabilities. We carry out the computation for two loci and find that ignoring subdivision can lead to underestimation of the match probabilities if the population under consideration actually has subdivision structure and the individuals originate from the same subpopulation. On the other hand, under a given model of population subdivision, we find that the ratio R for two loci is only slightly greater than 1 for a large range of symmetric and asymmetric migration rates. Keeping in mind that the infinite-alleles model is not the appropriate mutation model for STR loci, we conclude that, for two loci and biologically reasonable parameter values, population subdivision may lead to results that disfavor innocent suspects because of an increase in identity-by-descent in finite populations. On the other hand, for the same range of parameters, population subdivision does not lead to a substantial increase in linkage disequilibrium between loci. Those results are consistent with established practice.     

Selection in a subdivided population with dominance or local frequency dependence

The interplay between population structure and natural selection is an area of great interest. It is known that certain types of population subdivision do not alter fixation probabilities of selected alleles under genic, frequency-independent selection. In the presence of dominance for fitness or frequency-dependent selection these same types of subdivision can have large effects on fixation probabilities. For example, the barrier to fixation of a fitter allele due to underdominance is reduced by subdivision. Analytic results presented here relate a subdivided population that conforms to a finite island model to an approximately equivalent panmictic population. The size of this equivalent population is different from (larger than) the actual size of the subdivided population. Selection parameters are also different in the hypothetical equivalent population. As expected, the degree of dominance is lower in the equivalent population. The results are not limited to dominance but cover any form of polynomial frequency dependence.     

Demographic and genetic consequences of population subdivision in Formosa land-locked salmon <Emphasis Type="Italic">Oncorhynchus masou formosanus</Emphasis>, the southernmost subspecies of the salmonids

A few studies have evaluated demographic and genetic consequences of population subdivision by damming on the population as a whole. Formosa landlocked salmon Oncorhynchus masou formosanus have persisted as a relatively large population that has recently been subdivided into seven subpopulations by erosion-control dams and a natural waterfall. The present study simulated the demographic and genetic dynamics of this subdivided salmon population using VORTEX, an individual-based stochastic model. Although the population as a whole did not experience extinction over 200 simulation years, the loss of genetic variation was often detrimental (>10% loss). Isolated headwater subpopulations frequently reached extinction. Even when the subpopulations did persist through the 200 years, they experienced dramatic loss of genetic variation, suggesting short-term genetic threats. Due to the unidirectional dispersal of the fish, the damming puts headwater subpopulations at higher risk of extinction and loss of genetic variation, which affects persistence of the population as a whole, particularly from a genetic perspective.     

Effective Size of the Early-Run Sockeye Salmon <Emphasis Type="Italic">Oncorhynchus nerka</Emphasis> Population of Lake Azabach’e,Kamchatka Peninsula Evaluation of the Effect of Interaction between Subpopulations within a Subdivided Population

The effect of subdivision on the effective size (N _e) of the early-run sockeye salmon Oncorhynchus nerka population of Lake Azabach’e (Kamchatka Peninsula) has been studied. The mode of this effect is determined by the relative productivity of the subpopulations and its magnitude, by the rate of individual migration among subpopulations and genetic differentiation. If the contributions of subpopulations (offspring numbers) are different, genetic differentiation can reduce the N _e of the subdivided population. At equal subpopulation contributions, genetic differentiation always increases the N _e of the subdivided population in comparison with a panmictic population. We have found that all sockeye salmon subpopulations of Azabach’e Lake produce equal offspring numbers contributing to the next generation. The genetic differentiation between sockeye salmon subpopulations is low, and the subdivision increases the N _e of the early-run race with reference to the sum of the effective sizes of the subpopulations by as little as 2%.__________Translated from Genetika, Vol. 41, No. 5, 2005, pp. 680–685.Original Russian Text Copyright © 2005 by Efremov.     

Genetic load in subdivided populations: interactions between the migration rate, the size and the number of subpopulations

We assess the relative importance of migration rate, size and number of subpopulations on the genetic load of subdivided populations. Using diffusion approximations, we show that in most cases subdivision has detrimental effects on fitness. Moreover, our results suggest that fitness increases with subpopulation size, so that for the same total population size, genetic load is relatively lower when there are a small number of large subpopulations. Using elasticity analysis, we show that the size of the subpopulations appears to be the parameter that most strongly determines genetic load. interconnecting subpopulations via migration would also be of importance for population fitness when subpopulations are small and gene flow is low. Interestingly, the number of subpopulations has minor influence on genetic load except for the case of both very slightly deleterious mutations and small subpopulations. Elasticities decrease as the magnitude of deleterious effects increases. In other words, population structure does not matter for very deleterious alleles, but strongly affects fitness for slightly deleterious alleles.     

Effective size and F-statistics of subdivided populations for sex-linked loci.

For a population subdivided into an arbitrary number (s) of subpopulations, each consisting of different numbers of separate sexes, with arbitrary distributions of family size and variable migration rates by males (dm) and females (df), the recurrence equations for inbreeding coefficient and coancestry between individuals within and among subpopulations for a sex-linked locus are derived and the corresponding expressions for asymptotic effective size are obtained by solving the recurrence equations. The usual assumptions are made which are stable population size and structure, discrete generations, the island migration model, and without mutation and selection. The results show that population structure has an important effect on the inbreeding coefficients in any generation, asymptotic effective size, and F-statistics. Gene exchange among subpopulations inhibits inbreeding in initial generations but increases inbreeding in later generations. The larger the migration rate, the greater the final inbreeding coefficients and the smaller the effective size. Thus if the inbreeding coefficient is to be restricted to a specific value within a given number of generations, the appropriate population structure (the values of s, dm, and df) can be obtained by using the recurrence equations. It is shown that the greater the extent of subdivision (large s, small dm and df), the larger the effective size. For a given subdivided population, the effective size for a sex-linked locus may be larger or smaller than that for an autosomal locus, depending on the sex ratio, variance and covariance of family size, and the extend of subdivision. For the special case of a single unsubdivided population, our recurrence equations for inbreeding coefficient and coancestry and formulas for effective size reduce to the simple expressions derived by previous authors.     

The distribution of the ancestral haplotype in finite stepping-stone models with population expansion

Recent extensive analyses of human DNA polymorphism reveal that the ancestral haplotype at various genetic loci occurs almost exclusively in African samples. We develop a coalescence-based simulation method in stepping-stone models with population expansion and examine the probability (P(A)) that the ancestral haplotype is found in African samples and the probability (Q(A)) that the most recent common ancestor of sampled genes occurs in Africa. These probabilities and other summary statistics are used to infer the human demographic history. It is shown that the high observed P(A) value cannot be explained simply by sampling bias. Rather, it suggests that the African population has been more strongly subdivided and isolated from each other than the non-African population and that there must have been some African populations which were not directly involved in the Out-of-Africa expansion in the late Pleistocene.     

A model for the length of tracts of identity by descent in finite random mating populations

Linkage disequilibrium (LD) reflects coinheritance of an ancestral segment by chromosomes in a population. To begin to understand the effects of population history on the extent of LD, we model the length of a tract of identity-by-descent (IBD) between two chromosomes in a finite, random mating population. The variance of an IBD tract is large: a model described by (Genet. Res. Cambridge 35 (1980) 131) underestimates this variance. Using Fisher's concept of junctions, we predict the mean length of an IBD tract, given the age of the population and the population sizes over time. We derive results also for subdivided populations, given times of subdivision events and sizes of the resulting subpopulations. The model demonstrates that population growth and subdivision strongly affect the expected length of an IBD tract in small populations. These effects are less dramatic in large populations.     

THE DISTINCTIVE FOOTPRINTS OF LOCAL HITCHHIKING IN A VARIED ENVIRONMENT AND GLOBAL HITCHHIKING IN A SUBDIVIDED POPULATION

Loci with higher levels of population differentiation than the neutral expectation are traditionally interpreted as evidence of ongoing selection that varies in space. This article emphasizes an alternative explanation that has been largely overlooked to date: in species subdivided into large subpopulations, enhanced differentiation can also be the signature left by the fixation of an unconditionally favorable mutation on its chromosomal neighborhood. This is because the hitchhiking effect is expected to diminish as the favorable mutation spreads from the deme in which it originated to other demes. To discriminate among the two alternative scenarios one needs to investigate how genetic structure varies along the chromosomal region of the locus. Local hitchhiking is shown to generate a single sharp peak of differentiation centered on the adaptive polymorphism and the standard signature of a selective sweep only in those subpopulations in which the allele is favored. Global hitchhiking produces two domes of differentiation on either side of the fixed advantageous mutation and signatures of a selective sweep in every subpopulation, albeit of different magnitude. Investigating population differentiation around a locus that strongly differentiates two otherwise genetically similar populations of the marine mussel Mytilus edulis, plausible evidence for the global hitchhiking hypothesis has been obtained. Global hitchhiking is a neglected phenomenon that might prove to be important in species with large population sizes such as many marine invertebrates.     

Population substructure in Alkanna orientalis (Boraginaceae) in the Sinai Desert, in relation to its pollinator behaviour

Subpopulation genetic structure was studied in a population of the short-lived perennial plant Alkanna orientalis from the Sinai Desert, Egypt. The population investigated was subdivided for sampling into four subpopulations, which were located within three steep-sided wadis and a central plain area. Results from previous studies suggested that bee pollinator behaviour was likely to cause limited gene dispersal within the population and that subpopulations might have diverged from each other genetically. Seven RAPD primers were used to detect polymorphisms in the population. Differences between sub-populations in fragment frequency were found for several of the 45 polymorphic RAPD fragments scored. Population subdivision was evident from cluster analysis, and an analysis of genetic distances showed that there was significant genetic differentiation between all subpopulations. Nevertheless, more extensive gene flow appears to take place within the population than was expected, as demonstrated by a higher level of genetic similarity between subpopulations from two of the narrow wadis and the interconnecting plain. It is suggested that seed transport mediated by periodic flash floods is responsible for this pattern.     

Estimation of the amount of inbreeding due to subdivision of a population

In a large population which is subdivided into isolated or partially isolated subpopulations polymorphic for a gene locus, there is an excess of homozygotes due to the subdivision. This excess increases with the variance of the gene frequency. The excess can be measured by the “coefficient of inbreeding.” The aim of this paper is to estimate this coefficient, which is a function of various population parameters. We suggest several different estimates, which are the same functional form of unbiased estimates of the population parameters. These estimates are shown to be consistent. They have been compared by numerical methods among themselves and with two other estimates suggested previously.     

Migration promotes mutator alleles in subdivided populations

Mutator alleles that elevate the genomic mutation rate may invade nonrecombining populations by hitchhiking with beneficial mutations. Mutators have been repeatedly observed to take over adapting laboratory populations and have been found at high frequencies in both microbial pathogen and cancer populations in nature. Recently, we have shown that mutators are only favored by selection in sufficiently large populations and transition to being disfavored as population size decreases. This population size‐dependent sign inversion in selective effect suggests that population structure may also be an important determinant of mutation rate evolution. Although large populations may favor mutators, subdividing such populations into sufficiently small subpopulations (demes) might effectively inhibit them. On the other hand, migration between small demes that otherwise inhibit hitchhiking may promote mutator fixation in the whole metapopulation. Here, we use stochastic, agent‐based simulations and evolution experiments with the yeast Saccharomyces cerevisiae to show that mutators can, indeed, be favored by selection in subdivided metapopulations composed of small demes connected by sufficient migration. In fact, we show that population structure plays a previously unsuspected role in promoting mutator success in subdivided metapopulations when migration is rare.     

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.     

Variation after a selective sweep in a subdivided population

The effect of genetic hitchhiking on neutral variation is analyzed in subdivided populations with differentiated demes. After fixation of a favorable mutation, the consequences on particular subpopulations can be radically different. In the subpopulation where the mutation first appeared by mutation, variation at linked neutral loci is expected to be reduced, as predicted by the classical theory. However, the effect in the other subpopulations, where the mutation is introduced by migration, can be the opposite. This effect depends on the level of genetic differentiation of the subpopulations, the selective advantage of the mutation, the recombination frequency, and the population size, as stated by analytical derivations and computer simulations. The characteristic outcomes of the effect are three. First, the genomic region of reduced variation around the selected locus is smaller than that predicted in a panmictic population. Second, for more distant neutral loci, the amount of variation increases over the level they had before the hitchhiking event. Third, for these loci, the spectrum of gene frequencies is dominated by an excess of alleles at intermediate frequencies when compared with the neutral theory. At these loci, hitchhiking works like a system that takes variation from the between-subpopulation component and introduces it into the subpopulations. The mechanism can also operate in other systems in which the genetic variation is distributed in clusters with limited exchange of variation, such as chromosome arrangements or genomic regions closely linked to targets of balancing selection.     

Hitchhiking effect of a beneficial mutation spreading in a subdivided population

A central problem in population genetics is to detect and analyze positive natural selection by which beneficial mutations are driven to fixation. The hitchhiking effect of a rapidly spreading beneficial mutation, which results in local removal of standing genetic variation, allows such an analysis using DNA sequence polymorphism. However, the current mathematical theory that predicts the pattern of genetic hitchhiking relies on the assumption that a beneficial mutation increases to a high frequency in a single random-mating population, which is certainly violated in reality. Individuals in natural populations are distributed over a geographic space. The spread of a beneficial allele can be delayed by limited migration of individuals over the space and its hitchhiking effect can also be affected. To study this effect of geographic structure on genetic hitchhiking, we analyze a simple model of directional selection in a subdivided population. In contrast to previous studies on hitchhiking in subdivided populations, we mainly investigate the range of sufficiently high migration rates that would homogenize genetic variation at neutral loci. We provide a heuristic mathematical analysis that describes how the genealogical structure at a neutral locus linked to the locus under selection is expected to change in a population divided into two demes. Our results indicate that the overall strength of genetic hitchhiking--the degree to which expected heterozygosity decreases--is diminished by population subdivision, mainly because opportunity for the breakdown of hitchhiking by recombination increases as the spread of the beneficial mutation across demes is delayed when migration rate is much smaller than the strength of selection. Furthermore, the amount of genetic variation after a selective sweep is expected to be unequal over demes: a greater reduction in expected heterozygosity occurs in the subpopulation from which the beneficial mutation originates than in its neighboring subpopulations. This raises a possibility of detecting a "hidden" geographic structure of population by carefully analyzing the pattern of a selective sweep.