Linkage Disequilibrium in Subdivided Populations

The linkage disequilibrium in a subdivided populaton is shown to be equal to the sum of the average linkage disequilibrium for all subpopulations and the covariance between gene frequencies of the loci concerned. Thus, in a subdivided population the linkage disequilibrium may not be 0 even if the linkage disequilibrium in each subpopulation is 0. If a population is divided into two subpopulations between which migration occurs, the asymptotic rate of approach to linkage equilibrium is equal to either r or 2(m(1) + m(2)) - (m(1) + m(2))(2), whichever is smaller, where r is the recombination value and m(1) and m(2) are the proportions of immigrants in subpopulations 1 and 2, respectively. Thus, if migration rate is high compared with recombination value, the change of linkage disequilibrium in subdivided populations is similar to that of a single random mating population. On the other hand, if migration rate is low, the approach to lnkage equilibrium may be retarded in subdivided populations. If isolated populations begin to exchange genes by migration, linkage disequilibrium may increase temporarily even for neutral loci. If overdominant selection operates and the equilibrium gene frequencies are different in the two subpopulations, a permanent linkage disequilibrium may be produced without epistasis in each subpopulation.     

Stable linkage disequilibrium without epistasis in subdivided populations

In a large random mating population stable linkage disequilibrium occurs only when there is epistasis. However if a population is divided into a number of subpopulations among which migration occurs, stable linkage disequilibrium in each subpopulation may be produced without epistasis. In the case of two subpopulations a necessary condition for linkage equilibrium in the absence of epistasis is that at least at one of the two loci under consideration the gene frequency must be the same for the two populations. This condition is rather severe and any violation of this will lead to stable linkage disequilibrium. A similar conclusion can be made with more than two populations. In general the presence of linkage disequilibrium does not necessarily imply the existence of epistasis even in equilibrium populations.     

Genetic dynamics of population systems represented by small subpopulations: analytical modeling and numeric results

Genetic dynamics of population systems consisting of a finite number of small (Ne < 10(2)) semiisolated subpopulations was studied. A method of quantitative estimation of statistical parameters was developed for different types of population systems and different directions or intensities of selection. The following regularities were established: (1) optimal numbers of subpopulations, their effective size and rates of gene migration promoting continuous maintenance of genetic diversity can be chosen; (2) the genetic process in a population system is stationary only in the case of a specific structure of gene migrations corresponding to Wright's island model; (3) cyclic dynamics can stabilize the population system at high levels of gene diversity in a heterogeneous environment if gene migration and subpopulation size change in time. Similarities and differences between the concept of population system and the concept of metapopulation, which have been simultaneously proposed in Russia and abroad, are discussed in the final section.     

Cytonuclear disequilibria in a spatially structured hybrid zone

A hybrid zone model was developed that uses a stepping-stone model of population structure along with both nuclear and cytoplasmic genotypes to evaluate the effect that migration has on random mating organisms when no selection is present. Numerical simulations indicate that a number of different allele frequency and cytonuclear disequilibrium patterns can be found across the hybrid zone when it has reached equilibrium, and these results are discussed in the context of relative migration rates of the two sexes and the two source populations. The importance of numbers of subpopulations sampled, census time, and the persistence of pure species individuals into the hybrid zone are also discussed. Although this model does not consider selection or assortative mating in the hybrid zone, it should provide a useful baseline for evaluating joint cytonuclear genetic data from a structurally complex hybrid zone.     

Interactive use of computer models in teaching population genetics

A technique using a computer with a graphical display unit to teach students the effects of genetic drift, selection and migration is described. Both diallelic and triallelic loci are discussed. The Fokker-Planck equation is used as the mathematical model of the genetic system, and its validity as an approximation in this context is demonstrated by an investigation into selection at the ABO locus. An appendix contains a derivation from the Fokker-Planck equation of the formula used in the paper for the gene frequency distribution at a multiallelic locus in equilibrium under selection and migration.     

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.     

Coalescence Times and FST Under a Skewed Offspring Distribution Among Individuals in a Population

Estimates of gene flow between subpopulations based on F_ST (or N_ST) are shown to be confounded by the reproduction parameters of a model of skewed offspring distribution. Genetic evidence of population subdivision can be observed even when gene flow is very high, if the offspring distribution is skewed. A skewed offspring distribution arises when individuals can have very many offspring with some probability. This leads to high probability of identity by descent within subpopulations and results in genetic heterogeneity between subpopulations even when Nm is very large. Thus, we consider a limiting model in which the rates of coalescence and migration can be much higher than for a Wright–Fisher population. We derive the densities of pairwise coalescence times and expressions for F_ST and other statistics under both the finite island model and a many-demes limit model. The results can explain the observed genetic heterogeneity among subpopulations of certain marine organisms despite substantial gene flow.     

F-statistics under alternation of sexual and asexual reproduction: a model and data from schistosomes (platyhelminth parasites)

Accurate inferences on population genetics data require a sound underlying theoretical null model. Nearly nothing is known about the gene dynamics of organisms with complex life cycles precluding any biological interpretation of population genetics parameters. In this article, we used an infinite island model to derive the expectations of those parameters for the life cycle of a dioecious organism obligatorily alternating sexual and asexual reproductions as it is the case for schistosomes (plathyhelminth parasites). This model allowed us to investigate the effects of the degree of mixing among individuals coming from different subpopulations at each new generation (represented in the model by the migration rates before and after clonal reproductions) and the variance in the reproductive success of individuals during the clonal phase. We also consider the effects of different migration rates and degrees of clonal reproductive skew between male and female individuals. Results show that the variance in the reproductive success of clones is very important in shaping the distribution of the genetic variability both within and among subpopulations. Thus, higher variance in the reproductive success of clones generates heterozygous excesses within subpopulations and also increases genetic differentiation between them. Migration occurring before and after asexual reproduction has different effects on the patterns of F(IS) and F(ST). When males and females display different degrees of reproductive skew or migration rates, we observe differences in their respective population genetic structure. While results of the model apply to any organism alternating sexual and clonal reproductions (e.g. all parasitic trematodes, many plants, and all aphididae), we finally confront some of these theoretical expectations to empirical data from Schistosoma mansoni infecting Rattus rattus in Guadeloupe.     

Polymorphism and divergence for island-model species

Estimates of the scaled selection coefficient, gamma of Sawyer and Hartl, are shown to be remarkably robust to population subdivision. Estimates of mutation parameters and divergence times, in contrast, are very sensitive to subdivision. These results follow from an analysis of natural selection and genetic drift in the island model of subdivision in the limit of a very large number of subpopulations, or demes. In particular, a diffusion process is shown to hold for the average allele frequency among demes in which the level of subdivision sets the timescale of drift and selection and determines the dynamic equilibrium of allele frequencies among demes. This provides a framework for inference about mutation, selection, divergence, and migration when data are available from a number of unlinked nucleotide sites. The effects of subdivision on parameter estimates depend on the distribution of samples among demes. If samples are taken singly from different demes, the only effect of subdivision is in the rescaling of mutation and divergence-time parameters. If multiple samples are taken from one or more demes, high levels of within-deme relatedness lead to low levels of intraspecies polymorphism and increase the number of fixed differences between samples from two species. If subdivision is ignored, mutation parameters are underestimated and the species divergence time is overestimated, sometimes quite drastically. Estimates of the strength of selection are much less strongly affected and always in a conservative direction.     

Expected Shannon Entropy and Shannon Differentiation between Subpopulations for Neutral Genes under the Finite Island Model

Shannon entropy H and related measures are increasingly used in molecular ecology and population genetics because (1) unlike measures based on heterozygosity or allele number, these measures weigh alleles in proportion to their population fraction, thus capturing a previously-ignored aspect of allele frequency distributions that may be important in many applications; (2) these measures connect directly to the rich predictive mathematics of information theory; (3) Shannon entropy is completely additive and has an explicitly hierarchical nature; and (4) Shannon entropy-based differentiation measures obey strong monotonicity properties that heterozygosity-based measures lack. We derive simple new expressions for the expected values of the Shannon entropy of the equilibrium allele distribution at a neutral locus in a single isolated population under two models of mutation: the infinite allele model and the stepwise mutation model. Surprisingly, this complex stochastic system for each model has an entropy expressable as a simple combination of well-known mathematical functions. Moreover, entropy- and heterozygosity-based measures for each model are linked by simple relationships that are shown by simulations to be approximately valid even far from equilibrium. We also identify a bridge between the two models of mutation. We apply our approach to subdivided populations which follow the finite island model, obtaining the Shannon entropy of the equilibrium allele distributions of the subpopulations and of the total population. We also derive the expected mutual information and normalized mutual information (“Shannon differentiation”) between subpopulations at equilibrium, and identify the model parameters that determine them. We apply our measures to data from the common starling (Sturnus vulgaris) in Australia. Our measures provide a test for neutrality that is robust to violations of equilibrium assumptions, as verified on real world data from starlings.     

The many-demes limit for selection and drift in a subdivided population

A diffusion approximation is obtained for the frequency of a selected allele in a population comprised of many subpopulations or demes. The form of the diffusion is equivalent to that for an unstructured population, except that it occurs on a longer time scale when migration among demes is restricted. This many-demes diffusion limit relies on the collection of demes always being in statistical equilibrium with respect to migration and drift for a given allele frequency in the total population. Selection is assumed to be weak, in inverse proportion to the number of demes, and the results hold for any deme sizes and migration rates greater than zero. The distribution of allele frequencies among demes is also described.     

Seeing ghosts: the effect of unsampled populations on migration rates estimated for sampled populations

In 2004, the term 'ghost population' was introduced to summarize the effect of unsampled subpopulations that exchange migrants with other subpopulations that have been sampled. Estimated long-term migration rates among populations sampled will be affected by ghost populations. Although it would be convenient to be able to define an apparent migration matrix among sampled populations that incorporate the exchange of migrants with ghost populations, no such matrix can be defined in a way that predicts all features of the coalescent process for the true migration matrix. This paper shows that if the underlying migration matrix is symmetric, it is possible to define an apparent migration matrix among sampled subpopulations that predicts the same within-population and between-population homozygosities among sampled populations as is predicted by the true migration matrix. Application of this method shows that there is no simple relationship between true and apparent migration rates, nor is there a way to place an upper bound on the effect of ghost populations. In general, ghost populations can create the appearance of migration between subpopulations that do not actually exchange migrants. Comparison with published results from the application of the program, MIGRATE, shows that the apparent migration rates inferred with that program in a three-subpopulation model differ from those based on pairwise homozygosities. The apparent migration matrix determined by the method described in this paper probably represents the upper bound on the effect of ghost populations.     

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.     

Immigration can destabilize tri-trophic interactions: implications for conservation of top predators

Top predators often have large home ranges and thus are especially vulnerable to habitat loss and fragmentation. Increasing connectance among habitat patches is therefore a common conservation strategy, based in part on models showing that increased migration between subpopulations can reduce vulnerability arising from population isolation. Although three-dimensional models are appropriate for exploring consequences to top predators, the effects of immigration on tri-trophic interactions have rarely been considered. To explore the effects of immigration on the equilibrium abundances of top predators, we studied the effects of immigration in the three-dimensional Rosenzweig-MacArthur model. To investigate the stability of the top predator equilibrium, we used MATCONT to perform a bifurcation analysis. For some combinations of model parameters with low rates of top predator immigration, population trajectories spiral towards a stable focus. Holding other parameters constant, as immigration rate is increased, a supercritical Hopf bifurcation results in a stable limit cycle and thus top predator populations that cycle between high and low abundances. Furthermore, bistability arises as immigration of the intermediate predator is increased. In this case, top predators may exist at relatively low abundances while prey become extinct, or for other initial conditions, the relatively higher top predator abundance controls intermediate predators allowing for non-zero prey population abundance and increased diversity. Thus, our results reveal one of two outcomes when immigration is added to the model. First, over some range of top predator immigration rates, population abundance cycles between high and low values, making extinction from the trough of such cycles more likely than otherwise. Second, for relatively higher intermediate predator migration rates, top predators may exist at low values in a truncated system with impoverished diversity, again with extinction more likely.     

The effects of admixture and population subdivision on cytonuclear disequilibria

We examine the generation of cytonuclear disequilibria by admixture and continued gene flow. General formulas analogous to the nuclear case are first derived showing that the allelic and genotypic disequilibria from admixture or population subdivision equal their expected value across the contributing (sub) populations plus the covariance across these sources between the cytoplasmic gene frequency and the relevant nuclear frequency. A detailed study is then presented of the cytonuclear dynamics, in a random-mating population under two different migration scenarios. In both cases closed-form solutions are given for all variables as a function of the initial conditions and relevant migration parameters. The dynamics of the gene frequencies and allelic disequilibria, which dominate each system, are the same as those involving two unlinked nuclear loci, while the dynamics of the genotypic disequilibria and cytonuclear frequencies have no nuclear counterpart. The continent-island formulation focuses on a population receiving continued immigration from a large source of constant composition. A major discovery is that cytonuclear disequilibria can transiently build up on the "island" to levels far exceeding those found at equilibrium. In contrast, the admixture formulation focuses on the dynamics within two populations undergoing continued intermigration. Although in this case all cytonuclear associations must ultimately decay to zero, long-term transient disequilibria can develop which are many times their initial admixture values. For both migration scenarios it is shown that the time of population censusing relative to migration and reproduction dramatically affects both the amount and pattern of the nonrandom associations produced. The empirical relevance of these models is discussed in light of nuclear-mitochondrial data from a hybrid zone between European and North American eels and from a zone of racial admixture in humans.     

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.     

The Island Model of Population Differentiation: A General Solution

The island model deals with a species which is subdivided into a number of discrete finite populations, races or subspecies, between which some migration occurs. If the number of populations is small, an assumption of equal rates of migration between each pair of populations may be reasonable approximation. Mutation at a constant rate to novel alleles may also be assumed.-A general solution is given for the process of population divergence under this model following subdivision of a single parental population, expressed in terms of the observed average frequency of heterozygotes within and between subpopulations at a randomly chosen set of independently segregating loci. No restriction is imposed on the magnitude of the migration or mutation rates involved, nor on the number of populations exchanging migrants.-The properties of two fundamental measures of genetic divergence are deduced from the theory. One is a parameter related to varphi, the coefficient of kinship, and the other, gamma, measures the rate of mutational divergence between the sub-populations.     

Accounting for unmeasured population substructure in case-control studies of genetic association using a novel latent-class model

We propose a novel latent-class approach to detect and account for population stratification in a case-control study of association between a candidate gene and a disease. In our approach, population substructure is detected and accounted for using data on additional loci that are in linkage equilibrium within subpopulations but have alleles that vary in frequency between subpopulations. We have tested our approach using simulated data based on allele frequencies in 12 short tandem repeat (STR) loci in four populations in Argentina.     

Comparing relative rates of pollen and seed gene flow in the island model using nuclear and organelle measures of population structure

We describe a method for comparing nuclear and organelle population differentiation (F(ST)) in seed plants to test the hypothesis that pollen and seed gene flow rates are equal. Wright's infinite island model is used, with arbitrary levels of self-fertilization and biparental organelle inheritance. The comparison can also be applied to gene flow in animals. Since effective population sizes are smaller for organelle genomes than for nuclear genomes and organelles are often uniparentally inherited, organelle F(ST) is expected to be higher at equilibrium than nuclear F(ST) even if pollen and seed gene flow rates are equal. To reject the null hypothesis of equal seed and pollen gene flow rates, nuclear and organelle F(ST)'s must differ significantly from their expected values under this hypothesis. Finite island model simulations indicate that infinite island model expectations are not greatly biased by finite numbers of populations (>/=100 subpopulations). The power to distinguish dissimilar rates of pollen and seed gene flow depends on confidence intervals for fixation index estimates, which shrink as more subpopulations and loci are sampled. Using data from the tropical tree Corythophora alta, we rejected the null hypothesis that seed and pollen gene flow rates are equal but cannot reject the alternative hypothesis that pollen gene flow is 200 times greater than seed gene flow.