Effect of unsampled populations on the estimation of population sizes and migration rates between sampled populations

Current estimators of gene flow come in two methods; those that estimate parameters assuming that the populations investigated are a small random sample of a large number of populations and those that assume that all populations were sampled. Maximum likelihood or Bayesian approaches that estimate the migration rates and population sizes directly using coalescent theory can easily accommodate datasets that contain a population that has no data, a so-called 'ghost' population. This manipulation allows us to explore the effects of missing populations on the estimation of population sizes and migration rates between two specific populations. The biases of the inferred population parameters depend on the magnitude of the migration rate from the unknown populations. The effects on the population sizes are larger than the effects on the migration rates. The more immigrants from the unknown populations that are arriving in the sample populations the larger the estimated population sizes. Taking into account a ghost population improves or at least does not harm the estimation of population sizes. Estimates of the scaled migration rate M (migration rate per generation divided by the mutation rate per generation) are fairly robust as long as migration rates from the unknown populations are not huge. The inclusion of a ghost population does not improve the estimation of the migration rate M; when the migration rates are estimated as the number of immigrants Nm then a ghost population improves the estimates because of its effect on population size estimation. It seems that for 'real world' analyses one should carefully choose which populations to sample, but there is no need to sample every population in the neighbourhood of a population of interest.     

G'ST and D do not replace FST

The genetic differentiation among populations is affected by mutation as well as by migration, drift and selection. For loci with high mutation rates, such as microsatellites, the amount of mutation can influence the values of indices of differentiation such as G(ST) and F(ST). For many purposes, this effect is undesirable, and as a result, new indices such as G'(ST) and D have been proposed to measure population differentiation. This paper shows that these new indices are not effective measures of the causes or consequences of population structure. Both G'(ST) and D depend heavily on mutation rate, but both are insensitive to any population genetic process when the mutation rate is high relative to the migration rate. Furthermore, D is specific to the locus being measured, and so little can be inferred about the population demography from D. However, at equilibrium, D may provide an index of whether a particular marker is more strongly affected by mutation than by migration. I argue that F(ST) is a more important summary of the effects of population structure than D and that R(ST) or other measures that explicitly account for the mutation process are much better than G(ST), G'(ST), or D for highly mutable markers. Markers with lower mutation rates will often be easier to interpret.     

Evolutionary and statistical properties of three genetic distances

Many genetic distances have been developed to summarize allele frequency differences between populations. I review the evolutionary and statistical properties of three popular genetic distances: DS, DA, and theta;, using computer simulation of two simple evolutionary histories: an isolation model of population divergence and an equilibrium migration model. The effect of effective population size, mutation rate, and mutation mechanism upon the parametric value between pairs of populations in these models explored, and the unique properties of each distance are described. The effect of these evolutionary parameters on study design is also investigated and similar results are found for each genetic distance in each model of evolution: large sample sizes are warranted when populations are relatively genetically similar; and loci with more alleles produce better estimates of genetic distance.     

Effective population size of natural populations of Drosophila buzzatii, with a comparative evaluation of nine methods of estimation

Allozyme and microsatellite data from numerous populations of Drosophila buzzatii have been used (i) to determine to what degree N(e) varies among generations within populations, and among populations, and (ii) to evaluate the congruence of four temporal and five single-sample estimators of N(e) . Effective size of different populations varied over two orders of magnitude, most populations are not temporally stable in genetic composition, and N(e) showed large variation over generations in some populations. Short-term N(e) estimates from the temporal methods were highly correlated, but the smallest estimates were the most precise for all four methods, and the most consistent across methods. Except for one population, N(e) estimates were lower when assuming gene flow than when assuming populations that were closed. However, attempts to jointly estimate N(e) and immigration rate were of little value because the source of migrants was unknown. Correlations among the estimates from the single-sample methods generally were not significant although, as for the temporal methods, estimates were most consistent when they were small. These single-sample estimates of current N(e) are generally smaller than the short-term temporal estimates. Nevertheless, population genetic variation is not being depleted, presumably because of past or ongoing migration. A clearer picture of current and short-term effective population sizes will only follow with better knowledge of migration rates between populations. Different methods are not necessarily estimating the same N(e) , they are subject to different bias, and the biology, demography and history of the population(s) may affect different estimators differently.     

Spatial-temporal stratifications in natural populations and how they affect understanding and estimation of effective population size

The concept of effective population size (N(e) ) is based on an elegantly simple idea which, however, rapidly becomes very complex when applied to most real-world situations. In natural populations, spatial and temporal stratifications create different classes of individuals with different vital rates, and this in turn affects (generally reduces) N(e) in complex ways. I consider how these natural stratifications influence our understanding of effective size and how to estimate it, and what the consequences are for conservation and management of natural populations. Important points that emerge include the following: 1 The relative influences of local vs metapopulation N(e) depend on a variety of factors, including the time frame of interest. 2 Levels of diversity in local populations are strongly influenced by even low levels of migration, so these measures are not reliable indicators of local N(e) . 3 For long-term effective size, obtaining a reliable estimate of mutation rate is the most important consideration; unless this is accomplished, estimates can be biased by orders of magnitude. 4 At least some estimators of contemporary N(e) appear to be robust to relatively high (approximately 10%) equilibrium levels of migration, so under many realistic scenarios they might yield reliable estimates of local N(e) . 5 Age structure probably has little effect on long-term estimators of N(e) but can strongly influence contemporary estimates. 6 More research is needed in several key areas: (i) to disentangle effects of selection and drift in metapopulations connected by intermediate levels of migration; (ii) to elucidate the relationship between N(b) (effective number of breeders per year) and N(e) per generation in age-structured populations; (iii) to perform rigorous sensitivity analyses of new likelihood and coalescent-based methods for estimating demographic and evolutionary histories.     

Statistical properties of population differentiation estimators under stepwise mutation in a finite island model

Microsatellite loci mutate at an extremely high rate and are generally thought to evolve through a stepwise mutation model. Several differentiation statistics taking into account the particular mutation scheme of the microsatellite have been proposed. The most commonly used is R(ST) which is independent of the mutation rate under a generalized stepwise mutation model. F(ST) and R(ST) are commonly reported in the literature, but often differ widely. Here we compare their statistical performances using individual-based simulations of a finite island model. The simulations were run under different levels of gene flow, mutation rates, population number and sizes. In addition to the per locus statistical properties, we compare two ways of combining R(ST) over loci. Our simulations show that even under a strict stepwise mutation model, no statistic is best overall. All estimators suffer to different extents from large bias and variance. While R(ST) better reflects population differentiation in populations characterized by very low gene-exchange, F(ST) gives better estimates in cases of high levels of gene flow. The number of loci sampled (12, 24, or 96) has only a minor effect on the relative performance of the estimators under study. For all estimators there is a striking effect of the number of samples, with the differentiation estimates showing very odd distributions for two samples.     

Genetic effective size of a wild primate population: influence of current and historical demography

A comprehensive assessment of the determinants of effective population size (N(e)) requires estimates of variance in lifetime reproductive success and past changes in census numbers. For natural populations, such information can be best obtained by combining longitudinal data on individual life histories and genetic marker-based inferences of demographic history. Independent estimates of the variance effective size (N(ev), obtained from life-history data) and the inbreeding effective size (N((eI), obtained from genetic data) provide a means of disentangling the effects of current and historical demography. The purpose of this study was to assess the demographic determinants of N(e) in one of the most intensively studied natural populations of a vertebrate species: the population of savannah baboons (Papio cynocephalus) in the Amboseli Basin, southern Kenya. We tested the hypotheses that N(eV) < N < N(eI) (where N = population census number) due to a recent demographic bottleneck. N(eV) was estimated using a stochastic demographic model based on detailed life-history data spanning a 28-year period. Using empirical estimates of age-specific rates of survival and fertility for both sexes, individual-based simulations were used to estimate the variance in lifetime reproductive success. The resultant values translated into an N(eV)/N estimate of 0.329 (SD = 0.116, 95% CI = 0.172-0.537). Historical N(eI), was estimated from 14-locus microsatellite genotypes using a coalescent-based simulation model. Estimates of N(eI) were 2.2 to 7.2 times higher than the contemporary census number of the Amboseli baboon population. In addition to the effects of immigration, the disparity between historical N(eI) and contemporary N is likely attributable to the time lag between the recent drop in census numbers and the rate of increase in the average probability of allelic identity-by-descent. Thus, observed levels of genetic diversity may primarily reflect the population's prebottleneck history rather than its current demography.     

Population structure analysis using polygenic traits: estimation of migration matrices

Many studies of subdivided populations have attempted to determine the underlying migration rates that generate observed patterns of genetic differentiation. Most previous analyses have yielded only qualitative inferences about migration. In this paper I present a new method for estimating the full migration matrix from information on polygenic trait variation. The method employs multivariate quantitative genetic theory to provide a matrix formulation of the expected covariance structure in multigenerational subdivided populations for which information is available at different points in the life cycle. I develop a restricted maximum likelihood technique (REML) to take account of this additional life-cycle information and to estimate both the migration matrix and the ratio of effective population size to census size. To make the problem computationally tractable, the migration matrix is modeled as a log-linear function of a few covariates, such as subdivision size and geographic distance. I apply the technique to data on dermatoglyphic ridge counts for 1015 individuals of the Jirel population of east Nepal, considering two different age cohorts. In the adult cohort (individuals over 21 years of age) I examine data by both birthplace and residence and for the subadult cohort (under 21 years of age), by birthplace. Results from the REML technique reveal that the best-fitting migration model is a finite island model with an estimated endemicity of 0.730 +/- 0.105 and an estimated ratio of effective size to census size of 0.287 +/- 0.095. Both estimates are reasonable given known demographic data. In addition, Fst values predicted by the migration model are concordant with REML estimates obtained directly from the dermatoglyphic variation.     

Using molecular markers with high mutation rates to obtain estimates of relative population size and to distinguish the effects of gene flow and mutation: a demonstration using data from endemic Mauritian skinks

We propose a method of analysing genetic data to obtain separate estimates of the size (N(p)) and migration rate (m(p)) for the sampled populations, without precise prior knowledge of mutation rates at each locus ( micro(L)). The effects of migration and mutation can be distinguished because high migration has the effect of reducing genetic differentiation across all loci, whereas a high mutation rate will only affect the locus in question. The method also takes account of any differences between the spectra of immigrant alleles and of new mutant alleles. If the genetic data come from a range of population sizes, and the loci have a range of mutation rates, it is possible to estimate the relative sizes of the different N(p) values, and likewise the m(p) and the micro(L). Microsatellite loci may also be particularly appropriate because loci with a high mutation rate can reach mutation-drift-migration equilibrium more quickly, and because the spectra of mutants arriving in a population can be particularly distinct from the immigrants. We demonstrate this principle using a microsatellite data set from Mauritian skinks. The method identifies low gene flow between a putative new species and populations of its sister species, whereas the differentiation of two other populations is attributed to small population size. These distinct interpretations were not readily apparent from conventional measures of genetic differentiation and gene diversity. When the method is evaluated using simulated data sets, it correctly distinguishes low gene flow from small population size. Loci that are not at mutation-migration-drift equilibrium can distort the parameter estimates slightly. We discuss strategies for detecting and overcoming this effect.     

Relative influences of historical and contemporary forces shaping the distribution of genetic variation in the Atlantic killifish, Fundulus heteroclitus

A major goal of population genetics research is to identify the relative influences of historical and contemporary processes that serve to structure genetic variation. Most population genetic models assume that populations exist in a state of migration-drift equilibrium. However, in the past this assumption has rarely been verified, and is likely rarely achieved in natural populations. We assessed the equilibrium status at both local and regional scales of the Atlantic killifish, Fundulus heteroclitus . This species is a model organism for the study of adaptive clinal variation, but has also experienced a complicated history of range expansion and secondary contact following allopatric divergence, potentially obscuring the influence of contemporary evolutionary processes. Presumptively neutral genetic markers (microsatellites) demonstrated zones of secondary intergradation among coastal populations centred around northern New Jersey and the Chesapeake Bay region. Analysis of genetic variation indicated isolation by distance among some populations and provided supporting evidence that the Delaware Bay, but not the Chesapeake Bay, has acted as a barrier to dispersal among coastal populations. Bayesian estimates indicated large effective population sizes and low migration rates, and were in good agreement with empirically derived estimates of population and neighbourhood size from mark–recapture studies. These data indicate that populations are not in migration-drift equilibrium at a regional scale, and suggest that contributing factors include large population size combined with relatively low migration rates. These conditions should be considered when interpreting the evolutionary significance of the distribution of genetic variation among F. heteroclitus populations.     

Effect of changes in population size on genetic microdifferentiation

Changes in local population size are expected to have an effect on the degree of genetic microdifferentiation. A decrease in population size is expected to lead to an increase in microdifferentiation, and an increase in population size to a decrease in microdifferentiation. These expectations are routinely used with historical and/or demographic data to evaluate changes in estimates of microdifferentiation obtained over time for human populations. Here I look more closely at these expectations by using simple mathematical models that relate a change in average effective population size to the degree of microdifferentiation. The direction of change in microdifferentiation is influenced by the migration structure of the populations and the proximity of the region to an equilibrium state. A change in population size always leads to a new equilibrium, but the speed at which this new equilibrium is reached depends on migration and time depth. A decline in population size in one generation always leads to an immediate increase in the degree of microdifferentiation. An increase in population size in one generation could lead to an initial decrease or increase in the degree of microdifferentiation or to no change at all. Consideration of the parameters of the models shows under what conditions such changes occur. The relevance of these models is explored using summary data from a number of human populations.     

Maximum-likelihood estimation of migration rates and effective population numbers in two populations using a coalescent approach.

A new method for the estimation of migration rates and effective population sizes is described. It uses a maximum-likelihood framework based on coalescence theory. The parameters are estimated by Metropolis-Hastings importance sampling. In a two-population model this method estimates four parameters: the effective population size and the immigration rate for each population relative to the mutation rate. Summarizing over loci can be done by assuming either that the mutation rate is the same for all loci or that the mutation rates are gamma distributed among loci but the same for all sites of a locus. The estimates are as good as or better than those from an optimized FST-based measure. The program is available on the World Wide Web at http://evolution.genetics. washington.edu/lamarc.html/.     

Exact moment calculations for genetic models with migration,mutation, and drift

Using properties of moment stationarity we develop exact expressions for the mean and covariance of allele frequencies at a single locus for a set of populations subject to drift, mutation, and migration. Some general results can be obtained even for arbitrary mutation and migration matrices, for example: (1) Under quite general conditions, the mean vector depends only on mutation rates, not on migration rates or the number of populations. (2) Allele frequencies covary among all pairs of populations connected by migration. As a result, the drift, mutation, migration process is not ergodic when any finite number of populations is exchanging genes. In addition, we provide closed-form expressions for the mean and covariance of allele frequencies in Wright's finite-island model of migration under several simple models of mutation, and we show that the correlation in allele frequencies among populations can be very large for realistic rates of mutation unless an enormous number of populations are exchanging genes. As a result, the traditional diffusion approximation provides a poor approximation of the stationary distribution of allele frequencies among populations. Finally, we discuss some implications of our results for measures of population structure based on Wright's F-statistics.     

Patterns of DNA sequence diversity and genetic structure after a range expansion: lessons from the infinite-island model

It has been long recognized that population demographic expansions lead to distinctive features in the molecular diversity of populations. However, recent simulation results have suggested that a distinction could be made between a pure demographic expansion in an unsubdivided population, and a range expansion in a subdivided population, both leading to a large increase in the total number of the individuals. In order to better characterize the effect of a range expansion, I introduce a simple model of instantaneous expansion under an infinite-island model, under which I derive the distribution of the number of mutation differences between pairs of genes (the mismatch distribution), the heterozygosity, the average number of pairwise difference, and the fixation index F(ST). These derivations are checked against simulations, and are shown to lead to results qualitatively similar to those one would obtain after a range expansion in a 2-dimensional stepping-stone model. I then apply these results to estimate immigration rates in hunter-gather and post-Neolithic human populations from patterns of mitochondrial (mtDNA) diversity. Potential problems with this estimation procedure are also discussed.     

Utility of DNA viruses for studying human host history: case study of JC virus

Microbial pathogens, and viruses in particular, can serve as important complements to traditional genetic markers when investigating the population histories of their human host. The range of mutation rates for DNA viruses suggests that DNA viruses can be useful markers of both recent and ancient events in their host histories. Here, we assess the utility of a well known DNA virus, JC virus (JCV), for investigating human history and demography. Using complete coding viral genomes, we confirm the phylogeographic structure of JCV in populations worldwide and provide coalescent estimates of its evolutionary rate under two alternative models of its history. Using these rate estimates, we compare Bayesian skyline plots of population size changes for JCV to those of its human host as estimated with coding mitochondrial genomes of the latter. These comparisons, when combined with other evidence including a log Bayes Factor model test, show that JCV is evolving rapidly and is therefore tracking the recent history of its human host. These results support the hypothesis that post-World War II societal changes are most likely responsible for the recent demographic patterns observed among different regional JCV populations. In sum, fast evolving DNA viruses, such as JCV, can complement RNA viruses to provide novel insights about the recent history and demography of their human host.     

A fast and reliable computational method for estimating population genetic parameters

The estimation of ancestral and current effective population sizes in expanding populations is a fundamental problem in population genetics. Recently it has become possible to scan entire genomes of several individuals within a population. These genomic data sets can be used to estimate basic population parameters such as the effective population size and population growth rate. Full-data-likelihood methods potentially offer a powerful statistical framework for inferring population genetic parameters. However, for large data sets, computationally intensive methods based upon full-likelihood estimates may encounter difficulties. First, the computational method may be prohibitively slow or difficult to implement for large data. Second, estimation bias may markedly affect the accuracy and reliability of parameter estimates, as suggested from past work on coalescent methods. To address these problems, a fast and computationally efficient least-squares method for estimating population parameters from genomic data is presented here. Instead of modeling genomic data using a full likelihood, this new approach uses an analogous function, in which the full data are replaced with a vector of summary statistics. Furthermore, these least-squares estimators may show significantly less estimation bias for growth rate and genetic diversity than a corresponding maximum-likelihood estimator for the same coalescent process. The least-squares statistics also scale up to genome-sized data sets with many nucleotides and loci. These results demonstrate that least-squares statistics will likely prove useful for nonlinear parameter estimation when the underlying population genomic processes have complex evolutionary dynamics involving interactions between mutation, selection, demography, and recombination.     

Demographic history and genetic diversity in West Indian Coereba flaveola populations

The bananaquit (Coereba flaveola) has been well studied throughout the Caribbean region from a phylogenetic perspective. However, data concerning the population genetics and long-term demography of this bird species are lacking. In this study, we focused on three populations within the Lesser Antilles and one on Puerto Rico and assessed genetic and demographic processes, using five nuclear and two mitochondrial markers. We found that genetic diversity of bananaquits on Puerto Rico exceeds that on the smaller islands (Dominica, Guadeloupe and Grenada); this might reflect either successive founder events from Puerto Rico to Grenada, or more rapid drift in smaller populations subsequent to colonization. Population growth rate estimates showed no evidence of rapid expansion and migration was indicated only between populations from the closest islands of Dominica and Guadeloupe. Overall, our results suggest that a "demographic fission" model, considering only mutation and drift, but without migration, can be applied to these bananaquit populations in the West Indies.     

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.     

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.