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 micro (mutation rate) = m (migration rate) than at micro < 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. RST was rather resistant to deviations from the SMM mutation model.     

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.     

Genealogy and subpopulation differentiation under various models of population structure

 The structured coalescent is used to calculate some quantities relating to the genealogy of a pair of homologous genes and to the degree of subpopulation differentiation, under a range of models of subdivided populations and assuming the infinite alleles model of neutral mutation. The classical island and stepping-stone models of population structure are considered, as well as two less symmetric models. For each model, we calculate the Laplace transform of the distribution of the coalescence time of a pair of genes from specified locations and the corresponding mean and variance. These results are then used to calculate the values of Wright’s coefficient F _ST , its limit as the mutation rate tends to zero and the limit of its derivative with respect to the mutation rate as the mutation rate tends to zero. From this derivative it is seen that F _ST can depend strongly on the mutation rate, for example in the case of an essentially one-dimensional habitat with many subpopulations where gene flow is restricted to neighbouring subpopulations. Received: 1 October 1997 / Revised version: 15 March 1998     

Moderately and Highly Polymorphic Microsatellites Provide Discordant Estimates of Population Divergence in Sockeye Salmon, Oncorhynchus nerka

Mutation rate can vary widely among microsatellite loci. This variation may cause discordant single-locus and multi-locus estimates of F_ST, the commonly used measure of population divergence. We use 16 microsatellite and five allozyme loci from 14 sockeye salmon populations to address two questions about the affect of mutation rate on estimates of F_ST: (1) does mutation rate influence F_ST estimates from all microsatellites to a similar degree relative to allozymes?; (2) does the influence of mutation rate on F_ST estimates from microsatellites vary with geographic scale in spatially structured populations? For question one we find that discordant estimates of F_ST among microsatellites as well as between the two marker classes are correlated with mean within-population heterozygosity (H_S) and thus are likely due to differences in mutation rate. Highly polymorphic microsatellites (H_S > 0.84) provide significantly lower estimates of F_ST than moderately polymorphic microsatellites and allozymes (H_S < 0.60). Estimates of F_ST from binned allele frequency data and R_ST provide more accurate measures of population divergence for highly polymorphic but not for moderately polymorphic microsatellites. We conclude it is more important to pool loci of like H_S rather than marker class when estimating F_ST. For question two we find the F_ST values for moderately and highly polymorphic loci, while significantly different, are positively correlated for geographically proximate but not geographically distant population pairs. These results are consistent with expectations from the equilibrium approximation of Wright's infinite island model and confirm that the influence of mutation on estimates of F_ST can vary in spatially structured populations presumably because the rate of migration varies inversely with geographic scale.     

A new method for the partition of allelic diversity within and between subpopulations

A method is proposed for the analysis of allelic diversity in the context of subdivided populations. The definition of an allelic distance between subpopulations allows for the partition of total allelic diversity into within- and between-subpopulation components, in a way analogous to the classical partition of gene diversity. A new definition of allelic differentiation, A _ST , between subpopulations results from this partition, and is contrasted with the concept of allelic richness differentiation. The partition of allelic diversity makes it possible to establish the relative contribution of each subpopulation to within and between-subpopulation components of diversity with implications in priorisation for conservation. A comparison between this partition and that corresponding to allelic richness is illustrated with an example. Computer simulations are used to investigate the behaviour of the new statistic A _ST in comparison with F _ST for a finite island model under a range of mutation and migration rates. A _ST has less dependence on migration rate than F _ST for large values of migration rate, but the opposite occurs for low migration rates. In addition, the variance in the estimates of A _ST is higher than that of F _ST for low mutation rates, but the opposite for high mutation rates.     

PHASE III OF WRIGHT'S SHIFTING BALANCE PROCESS AND THE VARIANCE AMONG DEMES IN MIGRATION RATE

Interdemic selection by the differential migration of individuals out from demes of high fitness and into demes of low fitness (Phase III) is one of the most controversial aspects of Wright's Shifting Balance Theory. I derive a relationship between Phase III migration and the interdemic selection differential, S, and show its potential effect on F_ST. The relationship reveals a diversifying effect of interdemic selection by Phase III migration on the genetic structure of a metapopulation. Using experimental metapopulations, I explored the effect of Phase III migration on F_ST by comparing the genetic variance among demes for two different patterns of migration: (1) island model migration and (2) Wright's Phase III migration. Although mean migration rates were the same, I found that the variance among demes in migration rate was significantly higher with Phase III than with island model migration. As a result, F_ST for the frequency of a neutral marker locus was higher with Phase III than it was with island model migration. By increasing F_ST, Phase III enhanced the genetic differentiation among demes for traits not subject to interdemic selection. This feature makes Wright's process different from individual selection which, by reducing effective population size, decreases the genetic variance within demes for all other traits. I discussed this finding in relation to the efficacy of Phase III and random migration for effecting peak shifts, and the contribution of genes with indirect effects to among‐deme variation.     

THE EFFECT OF COLLECTIVE DISPERSAL ON THE GENETIC STRUCTURE OF A SUBDIVIDED POPULATION

Correlated dispersal paths between two or more individuals are widespread across many taxa. The population genetic implications of this collective dispersal have received relatively little attention. Here we develop two‐sample coalescent theory that incorporates collective dispersal in a finite island model to predict expected coalescence times, genetic diversities, and F‐statistics. We show that collective dispersal reduces mixing in the system, which decreases expected coalescence times and increases F_ST. The effects are strongest in systems with high migration rates. Collective dispersal breaks the invariance of within‐deme coalescence times to migration rate, whatever the deme size. It can also cause F_ST to increase with migration rate because the ratio of within‐ to between‐deme coalescence times can decrease as migration rate approaches unity. This effect is most biologically relevant when deme size is small. We find qualitatively similar results for diploid and gametic dispersal. We also demonstrate with simulations and analytical theory the strong similarity between the effects of collective dispersal and anisotropic dispersal. These findings have implications for our understanding of the balance between drift–migration–mutation in models of neutral evolution. This has applied consequences for the interpretation of genetic structure (e.g., chaotic genetic patchiness) and estimation of migration rates from genetic data.     

The distribution of <Emphasis Type="Italic">F</Emphasis><Subscript><Emphasis Type="Italic">st</Emphasis></Subscript> and other genetic statistics for a class of population structure models

We examine genetic statistics used in the study of structured populations. In a 1999 paper, Wakeley observed that the coalescent process associated with the finite island model can be decomposed into a scattering phase and a collecting phase. This decomposition becomes exact in the large population limit with the coalescent at the end of the scattering phase converging to the Ewens sampling formula and the coalescent during the collecting phase converging to the Kingman coalescent. In this paper we introduce a class of limiting models, which we refer to as G/KC models, that generalize Wakeley’s decomposition. G in G/KC represents a completely general limit for the scattering phase, while KC represents a Kingman coalescent limit for the collecting phase. We show that both the island and two-dimensional stepping stone models converge to G/KC models in the large population limit. We then derive the distribution of the statistic F _st for all G/KC models under a large sample limit for the cases of strong or weak mutation, thereby deriving the large population, large sample limiting distribution of F _st for the island and two-dimensional stepping stone models as a special case of a general formula. Our methods allow us to take the large population and large sample limits simultaneously. In the context of large population, large sample limits, we show that the variance of F _st in the presence of weak mutation collapses as O(\frac1logd){O(\frac{1}{\log d})} where d is the number of demes sampled. Further, we show that this O(\frac1logd){O(\frac{1}{\log d})} is caused by a heavy tail in the distribution of F _st . Our analysis of F _st can be extended to an entire class of genetic statistics, and we use our approach to examine homozygosity measures. Our analysis uses coalescent based methods.     

A clever solution to a vexing problem

F_ST (as well as related measures such as G_ST) has long been used both as a measure of the relative amount of genetic variation between populations and as an indicator of the amount of gene flow among populations. Unfortunately, F_ST and its clones are also sensitive to mutation, particularly when the mutation rate per locus is greater than the migration rate among populations. Relatively high mutation rates cause estimates of F_ST and G_ST to be much lower than researchers sometimes expect, when migration rates are low in the studied species. Several recent suggestions for dealing with this problem have been unsatisfactory for one reason or another, and no general solution exists (if we are not to abandon these otherwise useful measures of differentiation). In an important article in this issue, Jinliang Wang (2015) shows that it is possible to identify whether the genetic markers in a given study are likely to give estimates of F_ST that are strongly affected by mutation. The proposed test is simple and elegant, and with it, molecular ecologists can determine whether the F_ST from their makers can be depended on for further inference about their species’ genome and the demographic forces which shaped its patterns.     

ISOLATION BY DISTANCE IN EQUILIBRIUM AND NON-EQUILIBRIUM POPULATIONS

It is shown that for allele frequency data a useful measure of the extent of gene flow between a pair of populations is , which is the estimated level of gene flow in an island model at equilibrium. For DNA sequence data, the same formula can be used if F_ST is replaced by N_ST. In a population with restricted dispersal, analytic theory shows that there is a simple relationship between M? and geographic distance in both equilibrium and non-equilibrium populations and that this relationship is approximately independent of mutation rate when the mutation rate is small. Simulation results show that with reasonable sample sizes, isolation by distance can indeed be detected and that, at least in some cases, non-equilibrium patterns can be distinguished. This approach to analyzing isolation by distance is used for two allozyme data sets, one from gulls and one from pocket gophers.     

Generalization of the QST framework in hierarchically structured populations: Impacts of inbreeding and dominance

Q_ST is a differentiation parameter based on the decomposition of the genetic variance of a trait. In the case of additive inheritance and absence of selection, it is analogous to the genic differentiation measured on individual loci, F_ST. Thus, Q_ST?F_ST comparison is used to infer selection: selective divergence when Q_ST > F_ST, or convergence when Q_ST < F_ST. The definition of Q‐statistics was extended to two‐level hierarchical population structures with Hardy–Weinberg equilibrium. Here, we generalize the Q‐statistics framework to any hierarchical population structure. First, we developed the analytical definition of hierarchical Q‐statistics for populations not at Hardy–Weinberg equilibrium. We show that the Q‐statistics values obtained with the Hardy–Weinberg definition are lower than their corresponding F‐statistics when F_IS > 0 (higher when F_IS < 0). Then, we used an island model simulation approach to investigate the impact of inbreeding and dominance on the Q_ST?F_ST framework in a hierarchical population structure. We show that, while differentiation at the lower hierarchical level (Q_SR) is a monotonic function of migration, differentiation at the upper level (Q_RT) is not. In the case of additive inheritance, we show that inbreeding inflates the variance of Q_RT, which can increase the frequency of Q_RT > F_RT cases. We also show that dominance drastically reduces Q‐statistics below F‐statistics for any level of the hierarchy. Therefore, high values of Q‐statistics are good indicators of selection, but low values are not in the case of dominance.     

The relative power of genome scans to detect local adaptation depends on sampling design and statistical method

Although genome scans have become a popular approach towards understanding the genetic basis of local adaptation, the field still does not have a firm grasp on how sampling design and demographic history affect the performance of genome scans on complex landscapes. To explore these issues, we compared 20 different sampling designs in equilibrium (i.e. island model and isolation by distance) and nonequilibrium (i.e. range expansion from one or two refugia) demographic histories in spatially heterogeneous environments. We simulated spatially complex landscapes, which allowed us to exploit local maxima and minima in the environment in ‘pair’ and ‘transect’ sampling strategies. We compared F_ST outlier and genetic–environment association (GEA) methods for each of two approaches that control for population structure: with a covariance matrix or with latent factors. We show that while the relative power of two methods in the same category (F_ST or GEA) depended largely on the number of individuals sampled, overall GEA tests had higher power in the island model and F_ST had higher power under isolation by distance. In the refugia models, however, these methods varied in their power to detect local adaptation at weakly selected loci. At weakly selected loci, paired sampling designs had equal or higher power than transect or random designs to detect local adaptation. Our results can inform sampling designs for studies of local adaptation and have important implications for the interpretation of genome scans based on landscape data.     

CORRELATION OF PAIRWISE GENETIC AND GEOGRAPHIC DISTANCE MEASURES: INFERRING THE RELATIVE INFLUENCES OF GENE FLOW AND DRIFT ON THE DISTRIBUTION OF GENETIC VARIABILITY

Attempts to relate estimates of regional F_ST to gene flow and drift via Wright's (1931) equation F_ST ≈ 1/ (4Nm + 1) are often inappropriate because most natural sets of populations probably are not at equilibrium (McCauley 1993), as assumed by the island model upon which the equation is based, or ineffective because the influences of gene flow and drift are confounded in the product Nm. Evaluations of the association between genetic (F_ST) and geographic distances separating all pairwise populations combinations in a region allows one to test for regional equilibrium, to evaluate the relative influences of gene flow and drift on population structure both within and between regions, and to visualize the behavior of the association across all degrees of geographic separation. Tests of the model using microsatellite data from 51 populations of eastern collared lizards (Crotaphytus collaris collaris) collected from four distinct geographical regions gave results highly consistent with predicted patterns of association based on regional differences in various historical and ecological factors that affect the amount of drift and gene flow. The model provides a prerequisite for and an alternative to regional F_ST analyses, which often simply assume regional equilibrium, thus potentially leading to erroneous and misleading inferences regarding regional population structure.     

Emergent patterns of population genetic structure for a coral reef community

What shapes variation in genetic structure within a community of codistributed species is a central but difficult question for the field of population genetics. With a focus on the isolated coral reef ecosystem of the Hawaiian Archipelago, we assessed how life history traits influence population genetic structure for 35 reef animals. Despite the archipelago's stepping stone configuration, isolation by distance was the least common type of genetic structure, detected in four species. Regional structuring (i.e. division of sites into genetically and spatially distinct regions) was most common, detected in 20 species and nearly in all endemics and habitat specialists. Seven species displayed chaotic (spatially unordered) structuring, and all were nonendemic generalist species. Chaotic structure also associated with relatively high global F_ST. Pelagic larval duration (PLD) was not a strong predictor of variation in population structure (R² = 0.22), but accounting for higher F_ST values of chaotic and invertebrate species, compared to regionally structured and fish species, doubled the power of PLD to explain variation in global F_ST (adjusted R² = 0.50). Multivariate correlation of eight species traits to six genetic traits highlighted dispersal ability, taxonomy (i.e. fish vs. invertebrate) and habitat specialization as strongest influences on genetics, but otherwise left much variation in genetic traits unexplained. Considering that the study design controlled for many sampling and geographical factors, the extreme interspecific variation in spatial genetic patterns observed for Hawaìi marine species may be generated by demographic variability due to species‐specific abundance and migration patterns and/or seascape and historical factors.     

Microsatellite variation in grey seals (Halichoerus grypus) shows evidence of genetic differentiation between two British breeding colonies

Eight highly variable microsatellite loci were used to examine the genetic variability and differentiation of grey seals (Halichoerus grypus) at two widely spaced British breeding colonies. Samples were collected from adults and pups on the island of North Rona, off the north-west coast of Scotland, and on the Isle of May, situated at the mouth of the Firth of Forth on the east coast Highly significant differences in allele frequencies between these two sites were found for all eight loci, indicating considerable genetic differentiation. Thus, although grey seals are known to range over very large areas outside the breeding season, site fidelity of adults and philopatry of pups for these breeding colonies must be sufficiently common to have effects, through genetic drift, at the sub-population level. Migration rate was estimated using Wrighf's fixation index (F_ST), Slatkin's private alleles model and the new statistic, R_ST, which is analogous to (F_ST) but which takes into account the process of microsatellite mutation. An almost 8-fold discrepancy between the values we obtained provides cautionary evidence that microsatellite loci may contravene one or more of the assumptions on which these methods are based.     

Estimation in an Island Model Using Simulation

Estimation for an island model where mutation maintains ak-allele neutral polymorphism at a single locus on each island is considered. The likelihood of an observed sample type configuration is obtained by applying a computational algorithm analogous to Griffiths and Tavaré (Theor. Popul. Biol.46(1994), 131–159). This allows the computation of sampling distributions in an island model and investigation of their properties. Given a sample type configuration, the maximum likelihood estimate of the migration parameter is obtained by simulating independently the likelihood at a grid of points and, also, using a surface simulation method. The latter method generates the whole likelihood trajectory in a single application of the simulation program. An estimate of variance of the estimate of the migration parameter is obtained using the likelihood trajectory. A comparison of the maximum likelihood estimates of the gene flow between subpopulations is made with those obtained by using Wright'sF_STstatistic.     

The joint effects of selection and dominance on the QST - FST contrast

Q_ST measures the differentiation of quantitative traits between populations. It is often compared to F_ST, which measures population differentiation at neutral marker loci due to drift, migration, and mutation. When Q_ST is different from F_ST, it is usually taken as evidence that selection has either restrained or accelerated the differentiation of the quantitative trait relative to neutral markers. However, a number of other factors such as inbreeding, dominance, and epistasis may also affect the Q_ST − F_ST contrast. In this study, we examine the effects of dominance, selection, and inbreeding on Q_ST − F_ST. We compare Q_ST with F_ST at selected and neutral loci for populations at equilibrium between selection, drift, mutation, and migration using both analytic and simulation approaches. Interestingly, when divergent selection is acting on a locus, inbreeding and dominance generally inflate Q_ST relative to F_ST when they are both measured at the quantitative locus at equilibrium. As a consequence, dominance is unlikely to hide the signature of divergent selection on the Q_ST − F_ST contrast. However, although in theory dominance and inbreeding affect the expectation for Q_ST − F_ST, of most concern is the very large variance in both Q_ST and F_ST, suggesting that we should be cautious in attributing small differences between Q_ST and F_ST to selection.     

A COMPARISON OF THREE INDIRECT METHODS FOR ESTIMATING AVERAGE LEVELS OF GENE FLOW

Three methods for estimating the average level of gene flow in natural population are discussed and compared. The three methods are F_ST, rare alleles, and maximum likelihood. All three methods yield estimates of the combination of parameters (the number of migrants [Nm] in a demic model or the neighborhood size [4πDσ²] in a continuum model) that determines the relative importance of gene flow and genetic drift. We review the theory underlying these methods and derive new analytic results for the expectation of F_ST in stepping-stone and continuum models when small sets of samples are taken. We also compare the effectiveness of the different methods using a variety of simulated data. We found that the F_ST and rare-alleles methods yield comparable estimates under a wide variety of conditions when the population being sampled is demographically stable. They are roughly equally sensitive to selection and to variation in population structure, and they approach their equilibrium values at approximately the same rate. We found that two different maximum-likelihood methods tend to yield biased estimates when relatively small numbers of locations are sampled but more accurate estimates when larger numbers are sampled. Our conclusion is that, although F_ST and rare-alleles methods are expected to be equally effective in analyzing ideal data, practical problems in estimating the frequencies of rare alleles in electrophoretic studies suggest that F_ST is likely to be more useful under realistic conditions.     

Mean and variance of FST in a finite number of incompletely isolated populations

In the presence of migration F_ST in a finite number of incompletely isolated populations first increases, but after reaching a certain maximum value, it starts to decline and eventually becomes 0. The mean and variance of F_ST in this process are studied by using the recurrence formulas for the moments of gene frequencies in the island model of finite size as well as by using Monte Carlo simulation. The mean and variance in the early generations can be predicted by the approximate formulas developed. On the other hand, if we exclude the cases of an allele being fixed in all subpopulations, the mean of F_ST eventually reaches a steady-state value. This value is given by 1 − 2N_T(1 − λ) approximately, where N_T is the total population size and λ is the rate of decay of heterozygosity at steady state. It is shown that the mean and variance of F_ST depend on the initial gene frequency and when this is close to 0 or 1, Lewontin and Krakauer's test of the neutrality of polymorphic genes is not valid.     

TEMPORAL FLUCTUATIONS IN DEMOGRAPHIC PARAMETERS AND THE GENETIC VARIANCE AMONG POPULATIONS

Contrary to assumptions commonly made in the study of population genetics, the demographic properties of many populations are not always constant. Important characteristics of populations such as migration rate and population size may vary in time and space. Moreover, local populations often come and go; the rate of extinction and the properties of colonization may also vary. In this paper, the approach to equilibrium following a disturbance in the genetic variance among populations is described. The rate of migration is shown to be critical in determining the extent to which extinction and recolonization affects genetic differentiation. Perturbations and variations through time and space in demographic parameters such as population size and migration rate are shown to be important in determining the partitioning of genetic variance. Equations are given to predict the average through time of genetic differentiation among populations in the event of a single disturbance or in constant fluctuations in the pertinent demographic parameters. In general, these fluctuations increase the F_ST of a species. Spatial demographic variation affects F_STmuch more than temporal variation. These demographic properties make some species unsuitable for the empirical analysis of migration with indirect genetic measures. Demographic instability may play a large role in the evolution of genetic variation.