Equilibrium between Genetic Drift and Migration at Various Mutation Rates: Simulation Analysis

Rates of approach to equilibrium values of F _ST /R _ST at various mutation rates and using different mutation models (K-allele model KAM and stepwise model SMM) were analyzed numerically for the finite island model and the one-dimensional stepping stone models of migration, using simulation. In the island model of migration and the KAM mutation model, the rate of approach to the equilibrium F _ST value was appreciably higher and the equilibrium value was almost twofold lower at μ (mutation rate) = m (migration rate) than at μ ≪ m. In the one-dimensional stepping stone model of migration and the KAM model of mutation, the mutation rate significantly affected both the rate of approaching F _ST equilibrium and the equilibrium value. In both island and one-dimensional stepping stone models and SMM, R _ST was not influenced by various mutation rates. The rate of approach to the equilibrium values of both F _ST and R _ST was lower for the stepping stone model than to the island model. R _ST was rather resistant to deviations from the SMM mutation model. __________ Translated from Genetika, Vol. 41, No. 9, 2005, pp. 1283–1288. Original Russian Text Copyright ? 2005 by Efremov.     

The rate of approach to the equilibrium value of F(ST) in island and one-dimensional stepping-stone models of migration

The rate of approach to the equilibrium value of FST was analyzed numerically for the finite island and one-dimensional stepping-stone models using computer simulation. For both models, this rate was shown to decrease with decreasing migration rate among subpopulations but in the case of the stepping-stone model, it takes thousands rather than tens of generations to reach the equilibrium. Unlike the island structure of migration, in the stepping-stone model an increase in the subpopulation number reduces the rate of reaching the equilibrium state.     

The Rate of Approach to the Equilibrium Value of F ST in Island and One-Dimensional Stepping-Stone Models of Migration

The rate of approach to the equilibrium value of F _ST was analyzed numerically for the finite island and one-dimensional stepping-stone models using computer simulation. For both models, this rate was shown to decrease with decreasing migration rate among subpopulations but in the case of the stepping-stone model, it takes thousands rather than tens of generations to reach the equilibrium. Unlike the island structure of migration, in the stepping-stone model an increase in the subpopulation number reduces the rate of reaching the equilibrium state.     

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.     

Quasi equilibrium,variance effective size and fixation index for populations with substructure

In this paper, we develop a method for computing the variance effective size \(N_{eV}\) , the fixation index \(F_{ST}\) and the coefficient of gene differentiation \(G_{ST}\) of a structured population under equilibrium conditions. The subpopulation sizes are constant in time, with migration and reproduction schemes that can be chosen with great flexibility. Our quasi equilibrium approach is conditional on non-fixation of alleles. This is of relevance when migration rates are of a larger order of magnitude than the mutation rates, so that new mutations can be ignored before equilibrium balance between genetic drift and migration is obtained. The vector valued time series of subpopulation allele frequencies is divided into two parts; one corresponding to genetic drift of the whole population and one corresponding to differences in allele frequencies among subpopulations. We give conditions under which the first two moments of the latter, after a simple standardization, are well approximated by quantities that can be explicitly calculated. This enables us to compute approximations of the quasi equilibrium values of \(N_{eV}\) , \(F_{ST}\) and \(G_{ST}\) . Our findings are illustrated for several reproduction and migration scenarios, including the island model, stepping stone models and a model where one subpopulation acts as a demographic reservoir. We also make detailed comparisons with a backward approach based on coalescence probabilities.     

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.     

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.     

Microsatellite allele sizes: a simple test to assess their significance on genetic differentiation

The mutation process at microsatellite loci typically occurs at high rates and with stepwise changes in allele sizes, features that may introduce bias when using classical measures of population differentiation based on allele identity (e.g., F(ST), Nei's Ds genetic distance). Allele size-based measures of differentiation, assuming a stepwise mutation process [e.g., Slatkin's R(ST), Goldstein et al.'s (deltamu)(2)], may better reflect differentiation at microsatellite loci, but they suffer high sampling variance. The relative efficiency of allele size- vs. allele identity-based statistics depends on the relative contributions of mutations vs. drift to population differentiation. We present a simple test based on a randomization procedure of allele sizes to determine whether stepwise-like mutations contributed to genetic differentiation. This test can be applied to any microsatellite data set designed to assess population differentiation and can be interpreted as testing whether F(ST) = R(ST). Computer simulations show that the test efficiently identifies which of F(ST) or R(ST) estimates has the lowest mean square error. A significant test, implying that R(ST) performs better than F(ST), is obtained when the mutation rate, mu, for a stepwise mutation process is (a) >/= m in an island model (m being the migration rate among populations) or (b) >/= 1/t in the case of isolated populations (t being the number of generations since population divergence). The test also informs on the efficiency of other statistics used in phylogenetical reconstruction [e.g., Ds and (deltamu)(2)], a nonsignificant test meaning that allele identity-based statistics perform better than allele size-based ones. This test can also provide insights into the evolutionary history of populations, revealing, for example, phylogeographic patterns, as illustrated by applying it on three published data sets.     

Detecting Isolation by Distance Using Phylogenies of Genes

We introduce a method for analyzing phylogenies of genes sampled from a geographically structured population. A parsimony method can be used to compute s, the minimum number of migration events between pairs of populations sampled, and the value of s can be used to estimate the effective migration rate M, the value of Nm in an island model with local populations of size N and a migration rate m that would yield the same value of s. Extensive simulations show that there is a simple relationship between M and the geographic distance between pairs of samples in one- and two-dimensional models of isolation by distance. Both stepping-stone and lattice models were simulated. If two demes k steps apart are sampled, then, s, the average value of s, is a function only of k/(Nm) in a one-dimensional model and is a function only of k/(Nm)2 in a two-dimensional model. Furthermore, log(M) is approximately a linear function of log(k). In a one-dimensional model, the regression coefficient is approximately -1 and in a two-dimensional model the regression coefficient is approximately -0.5. Using data from several locations, the regression of log(M) on log(distance) may indicate whether there is isolation by distance in a population at equilibrium and may allow an estimate of the effective migration rate between adjacent sampling locations. Alternative methods for analyzing DNA sequence data from a geographically structured population are discussed. An application of our method to the data of R. L. Cann, M. Stoneking and A. C. Wilson on human mitochondrial DNA is presented.     

Evolution of learning in subdivided populations that occupy environmentally heterogeneous sites

We study the effects of natural selection and migration on the numbers of individual learners and social learners in subdivided populations that occupy environmentally heterogeneous sites. The island model and the circular stepping model each have four classes of globally stable equilibria (fixation of individual learners, polymorphism of individual and social learners, fixation of social learners, and extinction). The linear stepping stone model has an additional class of equilibria, which are characterized by the complete absence of phenotypes adapted to the interior sites. Low and high rates of migration favor social and individual learners, respectively, in all three models. In addition, we use the stepping stone models to study the range expansion of a species, initially confined to one environmentally homogeneous site, into the spatially heterogeneous world. The successive peaks of the transient spatial distributions of the number of individual learners occur at initially empty sites.     

Evolution of learning in subdivided populations that occupy environmentally heterogeneous sites

We study the effects of natural selection and migration on the numbers of individual learners and social learners in subdivided populations that occupy environmentally heterogeneous sites. The island model and the circular stepping model each have four classes of globally stable equilibria (fixation of individual learners, polymorphism of individual and social learners, fixation of social learners, and extinction). The linear stepping stone model has an additional class of equilibria, which are characterized by the complete absence of phenotypes adapted to the interior sites. Low and high rates of migration favor social and individual learners, respectively, in all three models. In addition, we use the stepping stone models to study the range expansion of a species, initially confined to one environmentally homogeneous site, into the spatially heterogeneous world. The successive peaks of the transient spatial distributions of the number of individual learners occur at initially empty sites.     

Multiscale analysis of Hymenocallis coronaria (Amaryllidaceae) genetic diversity, genetic structure, and gene movement under the influence of unidirectional stream flow

Understanding gene movement patterns in unidirectional flow environments and their effect on patterns of genetic diversity and genetic structure is necessary to manage these systems. Hypotheses and models to explain genetic patterns in streams are rare, and the results of macrophyte studies are inconsistent. This study addresses Ritland's (Canadian Journal of Botany 67: 2017-2024) unidirectional diversity hypothesis, the one-dimensional stepping stone model, and the metapopulation model within and among populations. Hymenocallis coronaria, an aquatic macrophyte of rocky river shoals of the SE USA, was sampled in four river basins. Within populations and among populations <16.2 km apart had significant isolation by distance. However, the rate of gene flow decay was not consistent with a one-dimensional stepping stone model, nor was evidence strong or consistent for Ritland's hypothesis. Some evidence indicates that localized metapopulation processes may be affecting genetic diversity and structure; however, gene flow patterns inconsistent with the assumptions of the linear and unidirectional models are also a possible influence. We discuss three variants on the one-dimensional stepping stone model. Future research in linear environments should examine the expectations of these models. This study is also one of the first efforts to calculate population genetic parameters using a new program, TETRASAT.     

The effect of unequal migration rates on FST

Using the structured coalescent model, it is shown that unequal migration rates between different pairs of subpopulations can increase the value of Wright's coefficient F(ST) and its dependence on the mutation rate, and decrease the effective level of gene flow. Two specific models of population structure are considered: (i) an 'island model with barrier' where migration rates between subpopulations on the same side of the barrier are higher than migration rates between subpopulations on opposite sides of the barrier, and (ii) the two-dimensional stepping-stone model with unequal migration rates in the two dimensions of the model.     

Equilibrium Values of Measures of Population Subdivision for Stepwise Mutation Processes

Expected values of WRIGHT's F-statistics are functions of probabilities of identity in state. These values may be quite different under an infinite allele model and under stepwise mutation processes such as those occurring at microsatellite loci. However, a relationship between the probability of identity in state in stepwise mutation models and the distribution of coalescence times can be deduced from the relationship between probabilities of identity by descent and the distribution of coalescence times. The values of F(IS) and F(ST) can be computed using this property. Examination of the conditional probability of identity in state given some coalescence time and of the distribution of coalescence times are also useful for explaining the properties of F(IS) and F(ST) at high mutation rate loci, as shown here in an island model of population structure.     

The rule of "one migrant per generation" and genetic differentiation in subdivided populations

The genetic differentiation in population with migration according island and two-dimensional stepping stone models was studied with simulation methods. It is shown that migration of one or few individuals per generation is insufficient for leveling differences between subpopulations in allele frequencies. Even if migration is estimated as m = 0.5 (exchange of 50 and 500 individuals per generation) statistically significant differences remain at least in the half of populations with insular structure. Spatial heterogeneity disappears completely only if m = 0.7-0.8. In case of two-dimensional step model the level of genetic differentiation is higher and statistically significant heterogeneity remains at all levels of genetic exchange including that which was estimated as m = 1.     

Coalescent and biophysical models of stepping‐stone gene flow in neritid snails

Marine species in the Indo‐Pacific have ranges that can span thousands of kilometres, yet studies increasingly suggest that mean larval dispersal distances are less than historically assumed. Gene flow across these ranges must therefore rely to some extent on larval dispersal among intermediate ‘stepping‐stone’ populations in combination with long‐distance dispersal far beyond the mean of the dispersal kernel. We evaluate the strength of stepping‐stone dynamics by employing a spatially explicit biophysical model of larval dispersal in the tropical Pacific to construct hypotheses for dispersal pathways. We evaluate these hypotheses with coalescent models of gene flow among high‐island archipelagos in four neritid gastropod species. Two of the species live in the marine intertidal, while the other two are amphidromous, living in fresh water but retaining pelagic dispersal. Dispersal pathways predicted by the biophysical model were strongly favoured in 16 of 18 tests against alternate hypotheses. In regions where connectivity among high‐island archipelagos was predicted as direct, there was no difference in gene flow between marine and amphidromous species. In regions where connectivity was predicted through stepping‐stone atolls only accessible to marine species, gene flow estimates between high‐island archipelagos were significantly higher in marine species. Moreover, one of the marine species showed a significant pattern of isolation by distance consistent with stepping‐stone dynamics. While our results support stepping‐stone dynamics in Indo‐Pacific species, we also see evidence for nonequilibrium processes such as range expansions or rare long‐distance dispersal events. This study couples population genetic and biophysical models to help to shed light on larval dispersal pathways.     

The robustness of neutral models of geographical variation

The approximation of diploid migration by gametic dispersion is studied. The monoecious, diploid population is subdivided into panmictic colonies that exchange migrants. Generations are discrete and nonoverlapping; the analysis is restricted to a single locus in the absence of selection; every allele mutates to a new allele at the same rate u. Diploid-migration models without self-fertilization and with selfing at the “random” rate (equal to the reciprocal of the deme size in each deme) are investigated; in the gametic-dispersion models, selfing occurs at the random rate. It is shown for the unbounded stepping-stone model in one and two dimensions, the circular stepping-stone model, and the island model that the probabilitities of identity in state at equilibrium for diploid migration are close to those for gametic dispersion if the mutation rate is small or the deme size is large. Explicit error bounds are presented in all the above cases. It is also proved that if the number of demes is finite and the migration matrix is arbitrary but time independent and ergodic, then in the strong-migration approximation the equilibrium and the ultimate rate and pattern of convergence of both diploid-dispersion models are close to the corresponding gametic-dispersion formulae. For the strong-migration approximation at equilibrium, migration must dominate both mutation and random drift; for the convergence results, it suffices that migration dominate random drift. All the results apply to a dioecious population if the migration pattern and mutation rate are sex independent.     

Diversity and geneflow in a migratory frugivorous fish: implications for Amazonian habitat connectivity

Colossoma macropomum is an ecologically and economically important fish distributed throughout the major tributaries of the Amazon River. C. macropomum require a suite of habitat types for different life stages making them potentially susceptible to the impacts of habitat fragmentation and alteration. As a means of better understanding the potential impacts of development, baseline data on connectivity and patterns of gene flow in species from relatively undisturbed habitat will be of value to monitor potential ecosystem impacts of anthropogenic habitat alteration on native fish communities. We used 13 single sequence repeat markers to determine if fine-scale structuring could be detected at the landscape scale at the Pacaya Samiria National Reserve, Perú. We also applied a model testing approach to evaluate the strength of different migration models, including panmixia, stepping stone and isolation models. Bayesian clustering detected a single genetic grouping across 131 fish. However, a comparison of marginal likelihoods for alternative migration models across PSNR supported a stepping stone model, rather than panmixia (Probability ~1.0). These results demonstrate that even in highly migratory fish with limited genetic structure, the effects of anthropogenic aquatic habitat alterations can be explored using genetic data.     

Continuous selective models with mutation and migration

The continuous selective model formulated previously for a single locus with multiple alleles in a monoecious population is extended to include mutation and migration. Somatic and germ line genotypic frequencies are distinguished, and the alternative hypotheses of constant mutation rates and age-independent mutation frequencies are analyzed in detail for arbitrary selection and mating schemes. With any mating pattern, if there is no selection, the equilibrium allelic frequencies are shown to be unaffected by the generalizations introduced in this paper. If, in addition, mating is at random, the equilibrium genotypic frequencies are proved to be in Hardy-Weinberg proportions. For both models, the nature of the approach to equilibrium is discussed. Migration is treated in the island model.     

Effects of metapopulation processes on measures of genetic diversity

Many species persist as a metapopulation under a balance between the local extinction of subpopulations or demes and their recolonization through dispersal from occupied patches. Here we review the growing body of literature dealing with the genetic consequences of such population turnover. We focus our attention principally on theoretical studies of a classical metapopulation with a 'finite-island' model of population structure, rather than on 'continent-island' models or 'source-sink' models. In particular, we concern ourselves with the subset of geographically subdivided population models in which it is assumed that all demes are liable to extinction from time to time and that all demes receive immigrants. Early studies of the genetic effects of population turnover focused on population differentiation, such as measured by F(ST). A key advantage of F(ST) over absolute measures of diversity is its relative independence of the mutation process, so that different genes in the same species may be compared. Another advantage is that F(ST) will usually equilibrate more quickly following perturbations than will absolute levels of diversity. However, because F(ST) is a ratio of between-population differentiation to total diversity, the genetic effects of metapopulation processes may be difficult to interpret in terms of F(ST) on its own, so that the analysis of absolute measures of diversity in addition is likely to be informative. While population turnover may either increase or decrease F(ST), depending on the mode of colonization, recurrent extinction and recolonization is expected always to reduce levels of both within-population and species-wide diversity (piS and piT, respectively). One corollary of this is that piS cannot be used as an unbiased estimate of the scaled mutation rate, theta, as it can, with some assumptions about the migration process, in species whose demes do not fluctuate in size. The reduction of piT in response to population turnover reflects shortened mean coalescent times, although the distribution of coalescence times under extinction colonization equilibrium is not yet known. Finally, we review current understanding of the effect of metapopulation dynamics on the effective population size.