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.     

Coexistence of cytoplasmic incompatibility and male-killing-inducing endosymbionts, and their impact on host gene flow

Male-killing (MK) and cytoplasmic incompatibility (CI) inducing bacteria are among the most common endosymbionts of arthropods. Previous theoretical research has demonstrated that these two types of endosymbionts cannot stably coexist within a single unstructured host population if no doubly infected host individuals occur. Here, we analyse a model of two host subpopulations connected by migration. We demonstrate that coexistence of MK- and CI-inducing endosymbionts is possible if migration rates are sufficiently low. In particular, our results suggest that for coexistence to be possible, migration rates into the subpopulation infected predominantly with MK-inducing endosymbionts must be considerably low, while migration rates from the MK- to the CI-infected subpopulation can be very high. We also analyse how the presence of MK- and CI-inducing endosymbionts affects host gene flow between the two subpopulations. Employing the concept of the 'effective migration rate', we demonstrate that compared with an uninfected subdivided population, gene flow is increased towards the MK-infected island, but decreased towards the CI-infected island. We discuss our results with respect to the butterfly Hypolimnas bolina, in which infection polymorphism of CI- and MK-inducing Wolbachia has been reported across South-Pacific island populations.     

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.     

Allee effects can both conserve and create spatial heterogeneity in population densities

In order to determine conditions which allow the Allee effect (caused by biparental reproduction) to conserve and create spatial heterogeneity in population densities, we studied a deterministic model of a symmetric two-patch metapopulation. We proved that under certain conditions there exist stable equilibria with unequal population densities in the two patches, a situation which can be interpreted as conserved heterogeneity. Furthermore, the Allee effect can lead to instability of the equilibrium with equal population densities if some degree of competition is assumed to occur between the subpopulations (non-local competition). This indicates the potential of the Allee effect to create spatial heterogeneity. Neither of these effects appear under biologically realistic parameter values in a model where uniparental reproduction is assumed. We proved that both the between-patch migration intensity and the degree of non-local competition are decisive in determining boundaries between these types of behaviour of the spatial system with Allee effect. Therefore, we propose that the Allee effect, migration intensity, and non-local competition should be considered jointly in studies focusing on problems like pattern formation in space and invasions of spreading species.     

Extranuclear differentiation and gene flow in the finite island model

Use of sequence information from extranuclear genomes to examine deme structure in natural populations has been hampered by lack of clear linkage between sequence relatedness and rates of mutation and migration among demes. Here, we approach this problem in two complementary ways. First, we develop a model of extranuclear genomes in a population divided into a finite number of demes. Sex-dependent migration, neutral mutation, unequal genetic contribution of separate sexes and random genetic drift in each deme are incorporated for generality. From this model, we derive the relationship between gene identity probabilities (between and within demes) and migration rate, mutation rate and effective deme size. Second, we show how within- and between-deme identity probabilities may be calculated from restriction maps of mitochondrial (mt) DNA. These results, when coupled with our results on gene flow and genetic differentiation, allow estimation of relative interdeme gene flow when deme sizes are constant and genetic variants are selectively neutral. We illustrate use of our results by reanalyzing published data on mtDNA in mouse populations from around the world and show that their geographic differentiation is consistent with an island model of deme structure.     

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.     

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.     

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.     

Streams over mountains: influence of riparian connectivity on gene flow in the Pacific jumping mouse (Zapus trinotatus)

In species affiliated with heterogeneous habitat, we expect gene flow to be restricted due to constraints placed on individual movement by habitat boundaries. This is likely to impact both individual dispersal and connectivity between populations. In this study, a GIS-based landscape genetics approach was used, in combination with fine-scale spatial autocorrelation analysis and the estimation of recent intersubpopulation migration rates, to infer patterns of dispersal and migration in the riparian-affiliated Pacific jumping mouse (Zapus trinotatus). A total of 228 individuals were sampled from nine subpopulations across a system of three rivers and genotyped at eight microsatellite loci. Significant spatial autocorrelation among individuals revealed a pattern of fine-scale spatial genetic structure indicative of limited dispersal. Geographical distances between pairwise subpopulations were defined following four criteria: (i) Euclidean distance, and three landscape-specific distances, (ii) river distance (distance travelled along the river only), (iii) overland distance (similar to Euclidean, but includes elevation), and (iv) habitat-path distance (a least-cost path distance that models movement along habitat pathways). Pairwise Mantel tests were used to test for a correlation between genetic distance and each of the geographical distances. Significant correlations were found between genetic distance and both the overland and habitat-path distances; however, the correlation with habitat-path distance was stronger. Lastly, estimates of recent migration rates revealed that migration occurs not only within drainages but also across large topographic barriers. These results suggest that patterns of dispersal and migration in Pacific jumping mice are largely determined by habitat connectivity.     

Predicted impact of barriers to migration on the Serengeti wildebeest population

The Serengeti wildebeest migration is a rare and spectacular example of a once-common biological phenomenon. A proposed road project threatens to bisect the Serengeti ecosystem and its integrity. The precautionary principle dictates that we consider the possible consequences of a road completely disrupting the migration. We used an existing spatially-explicit simulation model of wildebeest movement and population dynamics to explore how placing a barrier to migration across the proposed route (thus creating two disjoint but mobile subpopulations) might affect the long-term size of the wildebeest population. Our simulation results suggest that a barrier to migration--even without causing habitat loss--could cause the wildebeest population to decline by about a third. The driver of this decline is the effect of habitat fragmentation (even without habitat loss) on the ability of wildebeest to effectively track temporal shifts in high-quality forage resources across the landscape. Given the important role of the wildebeest migration for a number of key ecological processes, these findings have potentially important ramifications for ecosystem biodiversity, structure, and function in the Serengeti.     

Variance and covariance of homozygosity in a structured population

The variance of homozygosity for a K-allele model with n partially isolated subpopulations is derived numerically using identity coefficients. The variance of homozygosity within a subpopulation is shown to depend strongly upon the migration rates between subpopulations but is not strongly influenced by the number of alleles possible at a locus. The variance of homozygosity within a subpopulation, given the value of expected homozygosity, is approximately equal to the value of the variance of homozygosity given by Stewart's formula for a single population. If the population is presumed to be panmictic, but is actually subdivided, and the gametes are sampled at random from the total population, the apparent variance of homozygosity depends on the number of alleles possible. With small migration rates and K large, the apparent variance of homozygosity is much smaller than in a single population with the same expected homozygosity. However, when K is small, the variance of homozygosity is approximately given by Stewart's formula. The transient behavior of the variance of homozygosity shows that a large number of generations may be required to approach equilibrium values.     

Effective Size and F-Statistics of Subdivided Populations. I. Monoecious Species with Partial Selfing

Assuming discrete generations and autosomal inheritance involving genes that do not affect viability or reproductive ability, we have derived recurrence equations for the inbreeding coefficient and coancestry between individuals within and among subpopulations for a subdivided monoecious population with arbitrary distributions of male and female gametes per family, variable pollen and seed migration rates, and partial selfing. From the equations, formulas for effective size and expressions for F-statistics are obtained. For the special case of a single unsubdivided population, our equations reduce to the simple expressions derived by previous authors. It is shown that population structure (subdivision and migration) is important in determining the inbreeding coefficient and effective size. Failure to recognize internal structures of populations may lead to considerable bias in predicting effective size. Inbreeding coefficient, coancestry between individuals within and among subpopulations accrue at different and variable rates over initial generations before they converge to the same asymptotic rate of increase. For a given population, the smaller the pollen and seed migration rates, the more generations are required to attain the asymptotic rate and the larger the asymptotic effective size. The equations presented herein can be used for the study of evolutionary biology and conservation genetics.     

Effective Size and F-Statistics of Subdivided Populations. II. Dioecious Species

Assuming discrete generations and autosomal inheritance involving genes that do not affect viability or reproductive ability, we have derived recurrence equations for the inbreeding coefficient and coancestry between individuals within and among subpopulations for a subdivided monoecious population with arbitrary distributions of male and female gametes per family, variable pollen and seed migration rates, and partial selfing. From the equations, formulas for effective size and expressions for F-statistics are obtained. For the special case of a single unsubdivided population, our equations reduce to the simple expressions derived by previous authors. It is shown that population structure (subdivision and migration) is important in determining the inbreeding coefficient and effective size. Failure to recognize internal structures of populations may lead to considerable bias in predicting effective size. Inbreeding coefficient, coancestry between individuals within and among subpopulations accrue at different and variable rates over initial generations before they converge to the same asymptotic rate of increase. For a given population, the smaller the pollen and seed migration rates, the more generations are required to attain the asymptotic rate and the larger the asymptotic effective size. The equations presented herein can be used for the study of evolutionary biology and conservation genetics.     

Effective size and F-statistics of subdivided populations for sex-linked loci.

For a population subdivided into an arbitrary number (s) of subpopulations, each consisting of different numbers of separate sexes, with arbitrary distributions of family size and variable migration rates by males (dm) and females (df), the recurrence equations for inbreeding coefficient and coancestry between individuals within and among subpopulations for a sex-linked locus are derived and the corresponding expressions for asymptotic effective size are obtained by solving the recurrence equations. The usual assumptions are made which are stable population size and structure, discrete generations, the island migration model, and without mutation and selection. The results show that population structure has an important effect on the inbreeding coefficients in any generation, asymptotic effective size, and F-statistics. Gene exchange among subpopulations inhibits inbreeding in initial generations but increases inbreeding in later generations. The larger the migration rate, the greater the final inbreeding coefficients and the smaller the effective size. Thus if the inbreeding coefficient is to be restricted to a specific value within a given number of generations, the appropriate population structure (the values of s, dm, and df) can be obtained by using the recurrence equations. It is shown that the greater the extent of subdivision (large s, small dm and df), the larger the effective size. For a given subdivided population, the effective size for a sex-linked locus may be larger or smaller than that for an autosomal locus, depending on the sex ratio, variance and covariance of family size, and the extend of subdivision. For the special case of a single unsubdivided population, our recurrence equations for inbreeding coefficient and coancestry and formulas for effective size reduce to the simple expressions derived by previous authors.     

Past,present, future: tracking and simulating genetic differentiation over time in a closed metapopulation system

Genetic differentiation plays an essential role in the assessment of metapopulation systems of conservation concern. Migration rates affect the degree of genetic differentiation between subpopulations, with increasing genetic differentiation leading to increasing extinction risk. Analyses of genetic differentiation repeated over time together with projections into the future are therefore important to inform conservation. We investigated genetic differentiation in a closed metapopulation system of an obligate forest grouse, the Western capercaillie Tetrao urogallus, by comparing microsatellite population structure between a historic and a recent time period. We found an increase in genetic differentiation over a period of approximately 15 years. Making use of forward simulations accounting for population dynamics and genetics from both time periods, we explored future genetic differentiation by implementing scenarios of differing migration rates. Using migration rates derived from the recent dataset, simulations predicted further increase of genetic differentiation by 2050. We then examined effects of two realistic yet hypothetical migration scenarios on genetic differentiation. While isolation of a subpopulation led to overall increased genetic differentiation, the re-establishment of connectivity between two subpopulations maintained genetic differentiation at recent levels. Our results emphasize the importance of maintaining connectivity between subpopulations in order to prevent further genetic differentiation and loss of genetic variation. The simulation set-up we developed is highly adaptable and will aid researchers and conservationists alike in anticipating consequences of conservation strategies for metapopulation systems.

     

Synchronous dynamics and rates of extinction in spatially structured populations

We explore extinction rates using a spatially arranged set of subpopulations obeying Ricker dynamics. The population system is subjected to dispersal of individuals among the subpopulations as well as to local and global disturbances. We observe a tight positive correlation between global extinction rate and the level of synchrony in dynamics among thesubpopulations. Global disturbances and to a lesser extent, migration, are capable of synchronizing the temporal dynamics of the subpopulations over a rather wide span of the population growth rate r. Local noise decreases synchrony, as does increasing distance among the subpopulations. Synchrony also levels off with increasing r: in the chaotic region, subpopulations almost invariably behave asynchronously. We conclude that it is asynchrony that reduces the probability of global extinctions, not chaos as such: chaos is a special case only. The relationship between global extinction rate, synchronous dynamics and population growth rate is robust to changes in dispersal rates and ranges.     

Estimating parameters of speciation models based on refined summaries of the joint site-frequency spectrum

Understanding the processes and conditions under which populations diverge to give rise to distinct species is a central question in evolutionary biology. Since recently diverged populations have high levels of shared polymorphisms, it is challenging to distinguish between recent divergence with no (or very low) inter-population gene flow and older splitting events with subsequent gene flow. Recently published methods to infer speciation parameters under the isolation-migration framework are based on summarizing polymorphism data at multiple loci in two species using the joint site-frequency spectrum (JSFS). We have developed two improvements of these methods based on a more extensive use of the JSFS classes of polymorphisms for species with high intra-locus recombination rates. First, using a likelihood based method, we demonstrate that taking into account low-frequency polymorphisms shared between species significantly improves the joint estimation of the divergence time and gene flow between species. Second, we introduce a local linear regression algorithm that considerably reduces the computational time and allows for the estimation of unequal rates of gene flow between species. We also investigate which summary statistics from the JSFS allow the greatest estimation accuracy for divergence time and migration rates for low (around 10) and high (around 100) numbers of loci. Focusing on cases with low numbers of loci and high intra-locus recombination rates we show that our methods for the estimation of divergence time and migration rates are more precise than existing approaches.     

Colonization of the Americas: Disease Ecology and the Paleoindian Lifestyle

The disease barrier hypothesis is one long-standing explanation of the temporal discrepancies between the initial colonization of North and South America. The model postulates an epidemiological barrier that prohibited or slowed the initial migration into South America during the Late Pleistocene. Using data from ethnographically documented hunter–gatherers, the theoretical foundations of the hypothesis are explored. In addition, likely demographic effects to colonizing populations are postulated and compared to disease-response mechanisms in foraging societies. Based on identified disease conditions deemed necessary to maintain a prohibitive barrier, it is suggested that disease transmission rates in the initial colonizing populations of the New World were likely extremely limited and insufficient to support the disease barrier concept.     

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.