Local extinction and recolonization, species effective population size, and modern human origins

A primary objection from a population genetics perspective to a multiregional model of modern human origins is that the model posits a large census size, whereas genetic data suggest a small effective population size. The relationship between census size and effective size is complex, but arguments based on an island model of migration show that if the effective population size reflects the number of breeding individuals and the effects of population subdivision, then an effective population size of 10,000 is inconsistent with the census size of 500,000 to 1,000,000 that has been suggested by archeological evidence. However, these models have ignored the effects of population extinction and recolonization, which increase the expected variance among demes and reduce the inbreeding effective population size. Using models developed for population extinction and recolonization, we show that a large census size consistent with the multiregional model can be reconciled with an effective population size of 10,000, but genetic variation among demes must be high, reflecting low interdeme migration rates and a colonization process that involves a small number of colonists or kin-structured colonization. Ethnographic and archeological evidence is insufficient to determine whether such demographic conditions existed among Pleistocene human populations, and further work needs to be done. More realistic models that incorporate isolation by distance and heterogeneity in extinction rates and effective deme sizes also need to be developed. However, if true, a process of population extinction and recolonization has interesting implications for human demographic history.     

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.     

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.     

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.     

Joint estimation of migration rate and effective population size using the island model

Using the island model of population demography, I report that the demographic parameters migration rate and effective population size can be jointly estimated with equilibrium probabilities of identity in state calculated using a sample of genotypes collected at a single point in time from a single generation. The method, which uses moment-type estimators, applies to dioecious populations in which females and males have identical demography and monoecious populations with no selfing and requires that offspring genotypes are sampled following reproduction and prior to migration. I illustrate the estimation procedure using the infinite-island model with no mutation and the finite-island model with three kinds of mutation models. In the infinite-island model with no mutation, the estimators can be expressed as simple functions of estimates of the F-statistic parameters F(IT) and F(ST). In the finite-island model with mutation among k alleles, mutation rate, migration rate, and effective population size can be simultaneously estimated. The estimates of migration rate and effective population size are somewhat robust to violations in assumptions that may arise in empirical applications such as different kinds of mutation models and deviations from temporal equilibrium.     

Population size and time since island isolation determine genetic diversity loss in insular frog populations

Understanding the factors that contribute to loss of genetic diversity in fragmented populations is crucial for conservation measurements. Land‐bridge archipelagoes offer ideal model systems for identifying the long‐term effects of these factors on genetic variations in wild populations. In this study, we used nine microsatellite markers to quantify genetic diversity and differentiation of 810 pond frogs (Pelophylax nigromaculatus) from 24 islands of the Zhoushan Archipelago and three sites on nearby mainland China and estimated the effects of the island area, population size, time since island isolation, distance to the mainland and distance to the nearest larger island on reduced genetic diversity of insular populations. The mainland populations displayed higher genetic diversity than insular populations. Genetic differentiations and no obvious gene flow were detected among the frog populations on the islands. Hierarchical partitioning analysis showed that only time since island isolation (square‐root‐transformed) and population size (log‐transformed) significantly contributed to insular genetic diversity. These results suggest that decreased genetic diversity and genetic differentiations among insular populations may have been caused by random genetic drift following isolation by rising sea levels during the Holocene. The results provide strong evidence for a relationship between retained genetic diversity and population size and time since island isolation for pond frogs on the islands, consistent with the prediction of the neutral theory for finite populations. Our study highlights the importance of the size and estimated isolation time of populations in understanding the mechanisms of genetic diversity loss and differentiation in fragmented wild populations.     

From inclusive fitness to fixation probability in homogeneous structured populations

The methods of inclusive fitness provide a powerful analysis of the action of selection on social behaviour. The key component of this analysis is the concept of relatedness R. In infinite populations, a standard method of calculating relatedness coefficients is through coefficients of consanguinity using the notion of genetic identity by descent. In this paper, we show that this approach can also be made to work in finite populations and we assume here that the population has a homogeneous structure, such as an island model. We demonstrate that, under the assumption that genetic effects are small and additive, the resulting formulation of inclusive fitness is equivalent to other significant measures of selection in finite populations, including the change in average allele frequency and fixation probability. The results are illustrated for a model of the evolution of cooperation in a finite island population.     

Dynamics of Fst for the island model

F(st) is a measure of genetic differentiation in a subdivided population. Sewall Wright observed that F(st)=1/1+2Nm in a haploid diallelic infinite island model, where N is the effective population size of each deme and m is the migration rate. In demonstrating this result, Wright relied on the infinite size of the population. Natural populations are not infinite and therefore they change over time due to genetic drift. In a finite population, F(st) becomes a random variable that evolves over time. In this work we ask, given an initial population state, what are the dynamics of the mean and variance of F(st) under the finite island model? In application both of these quantities are critical in the evaluation of F(st) data. We show that after a time of order N generations the mean of F(st) is slightly biased below 1/1+2Nm. Further we show that the variance of F(st) is of order 1/d where d is the number of demes in the population. We introduce several new mathematical techniques to analyze coalescent genealogies in a dynamic setting.     

A MORPHOLOGIC AND GENETIC STUDY OF THE ISLAND FOX,UROCYON LITTORALIS

The Island Fox, Urocyon littoralis, is a dwarf form found on six of the Channel Islands located 30–98 km off the coast of southern California. The island populations differ in two variables that affect genetic variation: effective population size and duration of isolation. We estimate that the effective population size of foxes on the islands varies from approximately 150 to 1,000 individuals. Archeological and geological evidence suggests that foxes likely arrived on the three northern islands minimally 10,400–16,000 years ago and dispersed to the three southern islands 2,200–4,300 years ago. We use morphometrics, allozyme electrophoresis, mitochondrial DNA (mtDNA) restriction-site analysis, and analysis of hypervariable minisatellite DNA to measure variability within and distances among island fox populations. The amount of within-population variation is lowest for the smallest island populations and highest for the mainland population. However, the larger populations are sometimes less variable, with respect to some genetic measures, than expected. No distinct trends of variability with founding time are observed. Genetic distances among the island populations, as estimated by the four techniques, are not well correlated. The apparent lack of correspondence among techniques may reflect the effects of mutation rate and colonization history on the values of each genetic measure.     

Exploring demographic, physical, and historical explanations for the genetic structure of two lineages of Greater Antillean bats

Observed patterns of genetic structure result from the interactions of demographic, physical, and historical influences on gene flow. The particular strength of various factors in governing gene flow, however, may differ between species in biologically relevant ways. We investigated the role of demographic factors (population size and sex-biased dispersal) and physical features (geographic distance, island size and climatological winds) on patterns of genetic structure and gene flow for two lineages of Greater Antillean bats. We used microsatellite genetic data to estimate demographic characteristics, infer population genetic structure, and estimate gene flow among island populations of Erophylla sezekorni/E. bombifrons and Macrotus waterhousii (Chiroptera: Phyllostomidae). Using a landscape genetics approach, we asked if geographic distance, island size, or climatological winds mediate historical gene flow in this system. Samples from 13 islands spanning Erophylla's range clustered into five genetically distinct populations. Samples of M. waterhousii from eight islands represented eight genetically distinct populations. While we found evidence that a majority of historical gene flow between genetic populations was asymmetric for both lineages, we were not able to entirely rule out incomplete lineage sorting in generating this pattern. We found no evidence of contemporary gene flow except between two genetic populations of Erophylla. Both lineages exhibited significant isolation by geographic distance. Patterns of genetic structure and gene flow, however, were not explained by differences in relative effective population sizes, island area, sex-biased dispersal (tested only for Erophylla), or surface-level climatological winds. Gene flow among islands appears to be highly restricted, particularly for M. waterhousii, and we suggest that this species deserves increased taxonomic attention and conservation concern.     

Genetic and phylogenetic consequences of island biogeography

Abstract.— Island biogeography theory predicts that the number of species on an island should increase with island size and decrease with island distance to the mainland. These predictions are generally well supported in comparative and experimental studies. These ecological, equilibrium predictions arise as a result of colonization and extinction processes. Because colonization and extinction are also important processes in evolution, we develop methods to test evolutionary predictions of island biogeography. We derive a population genetic model of island biogeography that incorporates island colonization, migration of individuals from the mainland, and extinction of island populations. The model provides a means of estimating the rates of migration and extinction from population genetic data. This model predicts that within an island population the distribution of genetic divergences with respect to the mainland source population should be bimodal, with much of the divergence dating to the colonization event. Across islands, this model predicts that populations on large islands should be on average more genetically divergent from mainland source populations than those on small islands. Likewise, populations on distant islands should be more divergent than those on close islands. Published observations of a larger proportion of endemic species on large and distant islands support these predictions.     

Hidden Genetic Variability within Electromorphs in Finite Populations

The amount of hidden genetic variability within electromorphs in finite populations is studied by using the infinite site model and stepwise mutation model simultaneously. A formula is developed for the bivariate probability generating function for the number of codon differences and the number of electromorph state differences between two randomly chosen cistrons. Using this formula, the distribution as well as the mean and variance of the number of codon differences between two identical or nonidentical electromorphs are studied. The distribution of the number of codon differences between two randomly chosen identical electromorphs is similar to the geometric distribution but more leptokurtic. Studies are also made on the number of codon differences between two electromorphs chosen at random one from each of two populations which have been separated for an arbitrary number of generations. It is shown that the amount of hidden genetic variability is very large if the product of effective population size and mutation rate is large.     

Genetic Differentiation in Populations with Different Classes of Individuals

I formulate and analyse a model of population structure with different classes of individuals. These different classes may be age classes, other demographic classes, or different types of habitats homogeneously distributed over a geographical area. The value of population differentiation under an island model of dispersal and the increase of differentiation with geographical distance in one- and two-dimensional "isolation by distance" models are then obtained for a generalization of the FST measure of population structure, as a function of "effective" mutation, migration, and population size parameters. The relevant effective subpopulation size is related to the "mutation effective population size" of a single isolated subpopulation and, in models of age-structured populations, to the inbreeding effective population size.     

Examination of the relationship between inbreeding and population size

This chapter presents a method for examining the relationship between effective population size and accumulated random inbreeding in human populations. Using a linear regression model on 9 Irish isolates, results show that this method is very useful in assessing differential influences on population structure. Inbreeding refers to the level expected at random due to finite population size, offset by migration into the population. The data used consist of effective population size estimates and kinship estimates derived from surnames for 9 isolates on, or near, the west coast of Ireland. Based on the non-parametric correlation results, there is no monotonic relationship between effective population size and the inverse of kinship. The demographic data available show that, with the exception of Garumna, Lettermullen, and the Aran Islands, the other populations changed little in population size during the latter part of the 19th century. The fact that observed kinship is higher than predicted kinship suggests an increse in population size. These analyses suggest that there is little, if any, relationship between population size and inbreeding among these populations, using 1891 effective population size estimates. Given the range of demogrphic, ecological, and cultural environments of human populations, perhaps it is unexpected to see a set of populations adhering strongly to a given theoretical model. The more important aspect of such model fitting is not whether or not a given model shows a significant fit, but rather the analysis of deviations from an expected relationship.     

Variance of Neutral Genetic Variances within and between Populations for a Quantitative Character

The variances of genetic variances within and between finite populations were systematically studied using a general multiple allele model with mutation in terms of identity by descent measures. We partitioned the genetic variances into components corresponding to genetic variances and covariances within and between loci. We also analyzed the sampling variance. Both transient and equilibrium results were derived exactly and the results can be used in diverse applications. For the genetic variance within populations, sigma 2 omega, the coefficient of variation can be very well approximated as [formula: see text] for a normal distribution of allelic effects, ignoring recurrent mutation in the absence of linkage, where m is the number of loci, N is the effective population size, theta 1(0) is the initial identity by descent measure of two genes within populations and t is the generation number. The first term is due to genic variance, the second due to linkage disequilibrium, and third due to sampling. In the short term, the variation is predominantly due to linkage disequilibrium and sampling; but in the long term it can be largely due to genic variance. At equilibrium with mutation [formula: see text] where u is the mutation rate. The genetic variance between populations is a parameter. Variance arises only among sample estimates due to finite sampling of populations and individuals. The coefficient of variation for sample gentic variance between populations, sigma 2b, can be generally approximated as [formula: see text] when the number of loci is large where S is the number of sampling populations.     

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.     

Gene Flow and Genetic Differentiation

A brief analysis is presented for the effects of gene flow upon genetic differentiation within and between populations generated by mutation and drift. Previous results obtained with the "island" model are developed into a form that lends itself to biological interpretation. Attention is focused upon the effective local population size and the ratio of the genetic identity of two genes in different populations to that of two genes in the same population. The biological significance of this ratio, which is independent of population size, is discussed. Similarities between the results of this model and those of the "stepping-stone" model are noted.     

The Effective Size of a Subdivided Population

This paper derives the long-term effective size, N(e), for a general model of population subdivision, allowing for differential deme fitness, variable emigration and immigration rates, extinction, colonization, and correlations across generations in these processes. We show that various long-term measures of N(e) are equivalent. The effective size of a metapopulation can be expressed in a variety of ways. At a demographic equilibrium, N(e) can be derived from the demography by combining information about the ultimate contribution of each deme to the future genetic make-up of the population and Wright's F(ST)'s. The effective size is given by N(e) = 1/(1 + var ( &))<(1 - f(STi))/N(i)n>, where n is the number of demes, &(i) is the eventual contribution of individuals in deme i to the whole population (scaled such that σ(i) &(i) = n), and < > denotes an average weighted by &(i)(2). This formula is applied to a catastrophic extinction model (where sites are either empty or at carrying capacity) and to a metapopulation model with explicit dynamics, where extinction is caused by demographic stochasticity and by chaos. Contrary to the expectation from the standard island model, the usual effect of population subdivision is to decrease the effective size relative to a panmictic population living on the same resource.     

Influence of Gene Flow and Breeding Tactics on Gene Diversity within Populations

Expressions describing the accumulation of gene correlations within and among lineages and individuals of a population are derived. The model permits different migration rates by males and females and accounts for various breeding tactics within lineages. The resultant equations enable calculation of the probabilistic quantities for the fixation indices, rates of loss of genetic variation, accumulation of inbreeding, and coefficients of relationship for the population at any generation. All fixation indices were found to attain asymptotic values rapidly despite the consistent loss of genetic variation and accumulation of inbreeding within the population. The time required to attain asymptotic values, however, was prolonged when gene flow among lineages was relatively low (less than 20%). The degree of genetic differentiation among breeding groups, inbreeding coefficients, and gene correlations within lineages were found to be primarily functions of breeding tactics within groups rather than gene flow among groups. Thus, the asymptotic value of S. Wright's island model is not appropriate for describing genetic differences among groups within populations. An alternative solution is provided that under limited conditions will reduce to the original island model. The evolution of polygynous breeding tactics appears to be more favorable for promoting intragroup gene correlations than modification of migration rates. Inbreeding and variance effective sizes are derived for populations that are structured by different migration and breeding tactics. Processes that reduce the inbreeding effective population size result in a concomitant increase in variance effective population size.     

Differentiation with drift: a spatio-temporal genetic analysis of Galápagos mockingbird populations (Mimus spp.)

Small and isolated island populations provide ideal systems to study the effects of limited population size, genetic drift and gene flow on genetic diversity. We assessed genetic diversity within and differentiation among 19 mockingbird populations on 15 Galápagos islands, covering all four endemic species, using 16 microsatellite loci. We tested for signs of drift and gene flow, and used historic specimens to assess genetic change over the last century and to estimate effective population sizes. Within-population genetic diversity and effective population sizes varied substantially among island populations and correlated strongly with island size, suggesting that island size serves as a good predictor for effective population size. Genetic differentiation among populations was pronounced and increased with geographical distance. A century of genetic drift did not change genetic diversity on an archipelago-wide scale, but genetic drift led to loss of genetic diversity in small populations, especially in one of the two remaining populations of the endangered Floreana mockingbird. Unlike in other Galápagos bird species such as the Darwin''s finches, gene flow among mockingbird populations was low. The clear pattern of genetically distinct populations reflects the effects of genetic drift and suggests that Galápagos mockingbirds are evolving in relative isolation.