Effective population size,genetic diversity,and coalescence time in subdivided populations

A formula for the effective population size for the finite island model of subdivided populations is derived. The formula indicates that the effective size can be substantially greater than the actual number of individuals in the entire population when the migration rate among subpopulations is small. It is shown that the mean nucleotide diversity, coalescence time, and heterozygosity for genes sampled from the entire population can be predicted fairly well from the theory for randomly mating populations if the effective population size for the finite island model is used.     

High level of polymorphism and spatial structure in a selfing plant species,Medicago truncatula (Leguminosae), shown using RAPD markers

Using RAPD markers and one morphological marker, we studied the among- and within-population structure in a selfing annual plant species, Medicago truncatula GAERTN. About 200 individuals, sampled from four populations subdivided into three subpopulations each, were scored for 22 markers. It was found that the within-population variance component accounted for 55% of the total variance, while the among-population variance component accounted for 45%. Eighteen percent of the total variance was due to within-population structure (i.e., among subpopulations). Thus, 37% of the total variance was within subpopulations. Using a multilocus approach, it was found that no multilocus genotype was common to two populations. Two of the four studied populations were composed of few (≤6) multilocus genotypes, whereas the other two had many (≥15) multilocus genotypes. In the most polymorphic population (37 genotypes), only one genotype was found to be common to two subpopulations. Resampling experiments show that, depending on the population, three to 16 polymorphic loci were necessary and sufficient to score all multilocus genotypes in the population. When these data are compared to published results, it appears that on some occasions, the number of genotypes per population of selfing species might be larger than would be expected from the sole consideration of effective population size. The large within-subpopulation genetic variance observed in some populations could be explained by either small neighborhood sizes within subpopulations, or by outcrossing following migration through seed and/or pollen.     

Census vs. effective population size in chinook salmon: large- and small-scale environmental perturbation effects

Population viability has often been assessed by census of reproducing adults. Recently this method has been called into question and estimation of the effective population size (Ne) proposed as a complementary method to determine population health. We examined genetic diversity in five populations of chinook salmon (Oncorhynchus tshawytscha) from the upper Fraser River watershed (British Columbia, Canada) at 11 microsatellite loci over 20 years using DNA extracted from archived scale samples. We tested for changes in genetic diversity, calculated the ratio of the number of alleles to the range in allele size to give the statistic M, calculated Ne from the temporal change in allele frequency, used the maximum likelihood method to calculate effective population size (NeM), calculated the harmonic mean of population size, and compared these statistics to annual census estimates. Over the last two decades population size has increased in all five populations of chinook examined; however, Ne calculated for each population was low (81-691) and decreasing over the time interval measured. Values of NeM were low, but substantially higher than Ne calculated using the temporal method. The calculated values for M were generally low (M < 0.70), indicating recent population reductions for all five populations. Large-scale historic barriers to migration and development activities do not appear to account for the low values of Ne; however, available spawning area is positively correlated with Ne. Both Ne and M estimates indicate that these populations are potentially susceptible to inbreeding effects and may lack the ability to respond adaptively to stochastic events. Our findings question the practice of relying exclusively on census estimates for interpreting population health and show the importance of determining genetic diversity within populations.     

Samples from subdivided populations yield biased estimates of effective size that overestimate the rate of loss of genetic variation

Many empirical studies estimating effective population size apply the temporal method that provides an estimate of the variance effective size through the amount of temporal allele frequency change under the assumption that the study population is completely isolated. This assumption is frequently violated, and the magnitude of the resulting bias is generally unknown. We studied how gene flow affects estimates of effective size obtained by the temporal method when sampling from a population system and provide analytical expressions for the expected estimate under an island model of migration. We show that the temporal method tends to systematically underestimate both local and global effective size when populations are connected by gene flow, and the bias is sometimes dramatic. The problem is particularly likely to occur when sampling from a subdivided population where high levels of gene flow obscure identification of subpopulation boundaries. In such situations, sampling in a manner that prevents biased estimates can be difficult. This phenomenon might partially explain the frequently reported unexpectedly low effective population sizes of marine populations that have raised concern regarding the genetic vulnerability of even exceptionally large populations.     

Effective Size and F-Statistics of Subdivided Populations. I. Monoecious Species with Partial Selfing

Assuming discrete generations and autosomal inheritance involving genes that do not affect viability or reproductive ability, we have derived recurrence equations for the inbreeding coefficient and coancestry between individuals within and among subpopulations for a subdivided monoecious population with arbitrary distributions of male and female gametes per family, variable pollen and seed migration rates, and partial selfing. From the equations, formulas for effective size and expressions for F-statistics are obtained. For the special case of a single unsubdivided population, our equations reduce to the simple expressions derived by previous authors. It is shown that population structure (subdivision and migration) is important in determining the inbreeding coefficient and effective size. Failure to recognize internal structures of populations may lead to considerable bias in predicting effective size. Inbreeding coefficient, coancestry between individuals within and among subpopulations accrue at different and variable rates over initial generations before they converge to the same asymptotic rate of increase. For a given population, the smaller the pollen and seed migration rates, the more generations are required to attain the asymptotic rate and the larger the asymptotic effective size. The equations presented herein can be used for the study of evolutionary biology and conservation genetics.     

Effective Size and F-Statistics of Subdivided Populations. II. Dioecious Species

Assuming discrete generations and autosomal inheritance involving genes that do not affect viability or reproductive ability, we have derived recurrence equations for the inbreeding coefficient and coancestry between individuals within and among subpopulations for a subdivided monoecious population with arbitrary distributions of male and female gametes per family, variable pollen and seed migration rates, and partial selfing. From the equations, formulas for effective size and expressions for F-statistics are obtained. For the special case of a single unsubdivided population, our equations reduce to the simple expressions derived by previous authors. It is shown that population structure (subdivision and migration) is important in determining the inbreeding coefficient and effective size. Failure to recognize internal structures of populations may lead to considerable bias in predicting effective size. Inbreeding coefficient, coancestry between individuals within and among subpopulations accrue at different and variable rates over initial generations before they converge to the same asymptotic rate of increase. For a given population, the smaller the pollen and seed migration rates, the more generations are required to attain the asymptotic rate and the larger the asymptotic effective size. The equations presented herein can be used for the study of evolutionary biology and conservation genetics.     

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.     

Molecular ecological approaches to studying the evolutionary impact of selective harvesting in wildlife

Harvesting of wildlife populations by humans is usually targeted by sex, age or phenotypic criteria, and is therefore selective. Selective harvesting has the potential to elicit a genetic response from the target populations in several ways. First, selective harvesting may affect population demographic structure (age structure, sex ratio), which in turn may have consequences for effective population size and hence genetic diversity. Second, wildlife-harvesting regimes that use selective criteria based on phenotypic characteristics (e.g. minimum body size, horn length or antler size) have the potential to impose artificial selection on harvested populations. If there is heritable genetic variation for the target characteristic and harvesting occurs before the age of maturity, then an evolutionary response over time may ensue. Molecular ecological techniques offer ways to predict and detect genetic change in harvested populations, and therefore have great utility for effective wildlife management. Molecular markers can be used to assess the genetic structure of wildlife populations, and thereby assist in the prediction of genetic impacts by delineating evolutionarily meaningful management units. Genetic markers can be used for monitoring genetic diversity and changes in effective population size and breeding systems. Tracking evolutionary change at the phenotypic level in the wild through quantitative genetic analysis can be made possible by genetically determined pedigrees. Finally, advances in genome sequencing and bioinformatics offer the opportunity to study the molecular basis of phenotypic variation through trait mapping and candidate gene approaches. With this understanding, it could be possible to monitor the selective impacts of harvesting at a molecular level in the future. Effective wildlife management practice needs to consider more than the direct impact of harvesting on population dynamics. Programs that utilize molecular genetic tools will be better positioned to assess the long-term evolutionary impact of artificial selection on the evolutionary trajectory and viability of harvested populations.     

Genetic population structure of central Oregon Coast coho salmon (<Emphasis Type="Italic">Oncorhynchus kisutch</Emphasis>)

We surveyed microsatellite variation from 22 spawning populations of coho salmon (Oncorhynchus kisutch) from the Oregon Coast to help identify populations for conservation planning. All of our samples were temporally replicated, with most samples obtained in 2000 and 2001. We had three goals: (1) to confirm the status of populations identified on the basis of spawning location and life history; (2) to estimate effective population sizes and migration rates in order to determine demographic independence at different spatial scales; and (3) to determine if releases of Washington hatchery coho salmon in the 1980's into Oregon Coast streams resulted in measurable introgression into nearby wild Oregon Coast coho populations. For the last question, our study included a hatchery broodstock sample from 1985, after the Puget Sound introduction, and a 1975 sample taken from the same area prior to the introduction. Our results generally supported previously hypothesized population structure. Most importantly, we found unique lake-rearing groups identified on the basis of a common life-history type were genetically related. Estimates of immigrant fraction using several different methods also generally supported previously identified populations. Estimates of effective population size were highly correlated with estimates of spawning abundance. The 1985 hatchery sample was genetically similar to contemporary Washington samples, and the contemporary Oregon Coast samples were similar to the 1975 Oregon Coast sample, suggesting that introductions of Washington coho salmon did not result in large scale introgression into Oregon populations.     

Effective Sizes for Subdivided Populations

Many derivations of effective population sizes have been suggested in the literature; however, few account for the breeding structure and none can readily be expanded to subdivided populations. Breeding structures influence gene correlations through their effects on the number of breeding individuals of each sex, the mean number of progeny per female, and the variance in the number of progeny produced by males and females. Additionally, hierarchical structuring in a population is determined by the number of breeding groups and the migration rates of males and females among such groups. This study derives analytical solutions for effective sizes that can be applied to subdivided populations. Parameters that encapsulate breeding structure and subdivision are utilized to derive the traditional inbreeding and variance effective sizes. Also, it is shown that effective sizes can be determined for any hierarchical level of population structure for which gene correlations can accrue. Derivations of effective sizes for the accumulation of gene correlations within breeding groups (coancestral effective size) and among breeding groups (intergroup effective size) are given. The results converge to traditional, single population measures when similar assumptions are applied. In particular, inbreeding and intergroup effective sizes are shown to be special cases of the coancestral effective size, and intergroup and variance effective sizes will be equal if the population census remains constant. Instantaneous solutions for effective sizes, at any time after gene correlation begins to accrue, are given in terms of traditional F statistics or transition equations. All effective sizes are shown to converge upon a common asymptotic value when breeding tactics and migration rates are constant. The asymptotic effective size can be expressed in terms of the fixation indices and the number of breeding groups; however, the rate of approach to the asymptote is dependent upon dispersal rates. For accurate assessment of effective sizes, initial, instantaneous or asymptotic, the expressions must be applied at the lowest levels at which migration among breeding groups is nonrandom. Thus, the expressions may be applicable to lineages within socially structured populations, fragmented populations (if random exchange of genes prevails within each population), or combinations of intra- and interpopulation discontinuities of gene flow. Failure to recognize internal structures of populations may lead to considerable overestimates of inbreeding effective size, while usually underestimating variance effective size.     

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.     

Sex-linked effective population size in control populations,with particular reference to honeybees (Apis mellifera L.)

Summary Sex-linked effective population size (Ne) is derived for a variety of control population structures relevant to normal diploid and/or, more importantly, to haplo-diploid species. For equal sex ratio, it is shown that the control population structure which doubles autosomal effective population size trebles sex-linked effective size. For haplo-diploid species where the number of males exceeds the number of reproductive females, several different control structures are described, which tend to increase effective population size by about 1/3. These would be suitable for stock maintenance of honeybees. Directional selection programmes employing within-family selection would maintain most of the minimum drift/inbreeding properties of these control populations.     

An experimental evaluation with Drosophila melanogaster of a novel dynamic system for the management of subdivided populations in conservation programs

A dynamic method (DM) recently proposed for the management of captive subdivided populations was evaluated using the pilot species Drosophila melanogaster. By accounting for the particular genetic population structure, the DM determines the optimal mating pairs, their contributions to progeny and the migration pattern that minimize the overall coancestry in the population with a control of inbreeding levels. After a pre-management period such that one of the four subpopulations had higher inbreeding and differentiation than the others, three management methods were compared for 10 generations over three replicates: (1) isolated subpopulations (IS), (2) one-migrant-per-generation rule (OMPG), (3) DM aimed to produce the same or lower inbreeding coefficient than OMPG. The DM produced the lowest coancestry and equal or lower inbreeding than the OMPG method throughout the experiment. The initially lower fitness and lower variation for nine microsatellite loci of the highly inbred subpopulation were restored more quickly with the DM than with the OMPG method. We provide, therefore, an empirical illustration of the usefulness of the DM as a conservation protocol for captive subdivided populations when pedigree information is available (or can be deduced) and manipulation of breeding pairs is possible.     

THE EFFECT OF DELAYED POPULATION GROWTH ON THE GENETIC DIFFERENTIATION OF LOCAL POPULATIONS SUBJECT TO FREQUENT EXTINCTIONS AND RECOLONIZATIONS

I investigated the effects of delayed population growth on the genetic differentiation among populations subjected to local extinction and recolonization, for two different migration functions; (1) a constant migration rate, and (2) a constant number of migrants. A delayed period of population growth reduces the size of the newly founded populations for one or several generations. Whether this increases differentiation among local populations depends on the actual pattern of migration. With a constant migration rate, fewer migrants move into small populations than into large, thus providing ample opportunity for drift to act within a population. A prolonged period of population growth thus makes the conditions for enhanced differentiation between local populations less restrictive and also inflates the actual levels of differentiation. The effect depends on the relative magnitudes of k_e, the effective number of colonizers and k, the actual number of colonizers. When there is a constant number of migrants into a population per generation, migration into small populations is increased. This increase of migration in small populations counteracts the effects of genetic drift due to small population size. It increases the rate by which populations approach equilibrium, as small populations are swamped by migrants from larger populations closer to genetic equilibrium, and overall levels of differentiation are thus reduced. I also discuss situations for which the results of this paper are relevant.     

Craniometric variation,genetic theory,and modern human origins

Recent controversies surrounding models of modern human origins have focused on among-group variation, particularly the reconstruction of phylogenetic trees from mitochondrial DNA (mtDNA) and, the dating of population divergence. Problems in tree estimation have been seen as weakening the case for a replacement model and favoring a multiregional evolution model. There has been less discussion of patterns of within-group variation, although the mtDNA evidence has consistently shown the greatest diversity within African populations. Problems of interpretation abound given the numerous factors that can influence within-group variation, including the possibility of earlier divergence, differences in population size, patterns of population expansion, and variation in migration rates. We present a model of within-group phenotypic variation and apply it to a large set of craniometric data representing major Old World geographic regions (57 measurements for 1,159 cases in four regions: Europe, Sub-Saharan Africa, Australasia, and the Far East). The model predicts a linear relationship between variation within populations (the average within-group variance) and variation between populations (the genetic distance of populations to pooled phenotypic means). On a global level this relationship should hold if the long-term effective population sizes of each region are correctly specified. Other potential effects on withingroup variation are accounted for by the model. Comparison of observed and expected variances under the assumption of equal effective sizes for four regions indicates significantly greater within-group variation in Africa and significantly less within-group variation in Europe. These results suggest that the long-term effective population size was greatest in Africa. Closer examination of the model suggests that the long-term African effective size was roughly three times that of any other geographic region. Using these estimates of relative population size, we present a method for analyzing ancient population structure, which provides estimates of ancient migration. This method allows us to reconstruct migration history between geographic regions after adjustment for the effect of genetic drift on interpopulational distances. Our results show a clear isolation of Africa from other regions. We then present a method that allows direct estimation of the ancient migration matrix, thus providing us with information on the actual extent of interregional migration. These methods also provide estimates of time frames necessary to reach genetic equilibrium. The ultimate goal is extracting as much information from present-day patterns of human variation relevannt to issues of human origins. Our results are in agreement with mismatch distribution analysis of mtDNA, and they support a “weak Garden o Eden” model. In this model, modern-day variation can be explained by divergence from an initial source (perhaps Africa) into a number o small isolated populations, followed by later population expansion throughout our species. The major populationn expansions of Homo sapiens during and after the late Pleistocene have had the effect of “freezing” ancient patterns of population structure. While this is not the only possible scenario, we do note the close agreement with ecent analyses of mtDNA mismatch distibutions. © 1994 Wiley-Liss, Inc.     

Effective population size associated with self-fertilization: lessons from temporal changes in allele frequencies in the selfing annual Medicago truncatula

Despite its significance in evolutionary and conservation biology, few estimates of effective population size (N(e)) are available in plant species. Self-fertilization is expected to affect N(e), through both its effect on homozygosity and population dynamics. Here, we estimated N(e) using temporal variation in allele frequencies for two contrasted populations of the selfing annual Medicago truncatula: a large and continuous population and a subdivided population. Estimated N(e) values were around 5-10% of the population census size suggesting that other factors than selfing must contribute to variation in allele frequencies. Further comparisons between monolocus allelic variation and changes in the multilocus genotypic composition of the populations show that the local dynamics of inbred lines can play an important role in the fluctuations of allele frequencies. Finally, comparing N(e) estimates and levels of genetic variation suggest that H(e) is a poor estimator of the contemporaneous variance effective population size.     

POPULATION STRUCTURE OF DIPODOMYS INGENS (HETEROMYIDAE): THE ROLE OF SPATIAL HETEROGENEITY IN MAINTAINING GENETIC DIVERSITY

The giant kangaroo rat, Dipodomys ingens (Heteromyidae), is an endangered rodent that inhabits approximately 3% of its estimated historic range. Its current distribution is centered in two geographic areas, situated about 150 km apart, in south-central California. We sequenced a 293 base-pair fragment at the 5' end of the control region in 95 giant kangaroo rats from nine localities to examine the genetic structure of extant populations. We determine that mutations in this section of the control region follow a negative binominal distribution, rather than a Poisson. However, the distance between haplotypes is small enough that the difference between a tree that corrects for the non-Poisson distribution of mutations and one that does not, is minimal. This implies that the use of methods that assume a Poisson distribution of mutations, such as those based on coalescent theory, are justified. We find that the correlation between levels of genetic diversity and estimated census size is poor. This suggests that population sizes have fluctuated over time or that populations have not been isolated from one another, or both. We also examine the hierarchical structure of populations and find that the southern populations are not genetically subdivided but that there is significant subdivision between northern and southern populations and between some northern subpopulations. The phylogeographic relationship between northern and southern populations can primarily be attributed to isolation by distance, although the time since divergence between them appears to be less than the age of either. To examine the phylogeographic relationships in more detail we construct a minimum spanning tree based on Tamura-Nei gamma-corrected distances and superimpose on it the geographic position of haplotypes. This reveals that there is more genetic distance between some northern haplotypes than between any northern and southern haplotypes, despite the geographic distance separating north from south and the larger size of the southern population. It also reveals that one northern population, in the Panoche Valley, contains old allelic lineages and shares ancestral polymorphism with several other populations. It also shows that two, small, geographically remote populations contain a surprising amount of genetic diversity, but that different population/geographic processes have affected the structure of that diversity. We estimate the average migration rate among all populations to be 7.5 per generation, and conclude that a disproportionate number of migration events involve gene flow with one northern population, the Panoche Valley. We find evidence for the hypothesis that there has been an increase in population size in the remaining populations in the north and suggest that the Panoche Valley could play a role in these expansions. Finally we discuss the probabilitiy that the genetic structure of the southern populations has been affected by fluctuations in size. These results are briefly compared to other studies on the genetic structure of rodent populations.     

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.     

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.