The strong-migration limit for the genealogical process in geographically structured populations

In this article we assume that the entire population is subdivided into a finite number of panmictic colonies, each of which consists of a respective number of haploid individuals. We also assume that random genetic drift occurs in each colony and migration among colonies, which is independent of time and ergodic. We study the genealogical process of sampled genes from geographically structured populations. We prove that if the actual total population number is replaced by the effective population number, the mean coalescence time converges to that in a panmictic population in the strong migration limit. We also obtain the geographical distribution of the common ancestor.     

The coalescent in two colonies with symmetric migration

Kingman's coalescent process is extended to two colonies with symmetric migration. The mean waiting time until a sample of genes taken from two colonies coalesces to a common ancestor is obtained. The final step in the waiting time before the process is absorbed at 1 is observed to have an intriguing behaviour. The distribution of this final waiting time converges to the known distribution of the corresponding waiting time in the case of a single population as the migration rate tends to zero. The mean, however, does not converge. The waiting time until a sample has two common ancestors is modeled as a function of the migration rate. Finally bounds for the expected waiting time for the two colonies to have j > 1 ancestors are derived.     

Simulation of carbon reserve dynamics in Microcystis and its influence on vertical migration with Yoyo model

Blue-green algae control their buoyancy depending upon the surrounding conditions. This process is essential for Cyanobacteria development and can account for their dominance in eutrophic waters in summer. In order to determine the main regulating factors of those movements, we developed a mechanistic and deterministic model, based on differential equations, that simulates the vertical migration of Microcystis sp. In Microcystis, buoyancy regulation results from the dynamics of the carbohydrate reserve metabolism during photosynthesis. These fundamental processes are modelled daily by this vertical 1-D model named Yoyo. It describes the movement of colonies with different sizes in response to variations of environmental conditions. This paper presents the model sensitivity analysis. We individually investigated the role of light and temperature upon algal migration with colonies of two different diameters. Under a daily light cycle and a temperature of 20 degrees C, the model described vertical migration on a 48 h rhythm in colonies with a 300-micron diameter.     

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.     

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.     

When and where was the most recent common ancestor?

 The structured coalescent is investigated for single-locus, digenic samples in the diffusion limit of the unidimensional stepping-stone model for homogeneous, isotropic migration and random genetic drift. Let T denote the scaled time to the most recent common ancestor (MRCA) of the two genes, and let Z designate the scaled deviation of the position of the MRCA from the average position of the two genes. The joint probability density of T and Z is evaluated explicitly. Both the marginal and conditional distributions of T have infinite expectation, as does the marginal distribution of Z. Conditioned on T = τ, the distribution of Z is Gaussian with mean zero and variance 2τ. The main results are extended to anisotropic migration. The results establish the existence of and define in the diffusion limit a retrospective stochastic process for digenic samples in one spatial dimension. Received: 1 May 2001 / Revised version: 2 September 2001 / Published online: 8 February 2002     

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.     

Small-world networks decrease the speed of Muller's ratchet

Muller's ratchet is an evolutionary process that has been implicated in the extinction of asexual species, the evolution of non-recombining genomes, such as the mitochondria, the degeneration of the Y chromosome, and the evolution of sex and recombination. Here we study the speed of Muller's ratchet in a spatially structured population which is subdivided into many small populations (demes) connected by migration, and distributed on a graph. We studied different types of networks: regular networks (similar to the stepping-stone model), small-world networks and completely random graphs. We show that at the onset of the small-world network - which is characterized by high local connectivity among the demes but low average path length - the speed of the ratchet starts to decrease dramatically. This result is independent of the number of demes considered, but is more pronounced the larger the network and the stronger the deleterious effect of mutations. Furthermore, although the ratchet slows down with increasing migration between demes, the observed decrease in speed is smaller in the stepping-stone model than in small-world networks. As migration rate increases, the structured populations approach, but never reach, the result in the corresponding panmictic population with the same number of individuals. Since small-world networks have been shown to describe well the real contact networks among people, we discuss our results in the light of the evolution of microbes and disease epidemics.     

Organelle Gene Diversity under Migration, Mutation, and Drift: Equilibrium Expectations, Approach to Equilibrium, Effects of Heteroplasmic Cells, and Comparison to Nuclear Genes

We developed stochastic population genetic theory for mitochondrial and chloroplast genes, using an infinite alleles model appropriate for molecular genetic data. We considered the effects of mutation, random drift, and migration in a finite island model on selectively neutral alleles. Recurrence equations were obtained for the expectation of gene diversities within zygotes, within colonies, and between colonies. The variables are number and sizes of colonies, migration rates, sex ratios, degree of paternal transmission, number of germ line cell divisions, effective number of segregating organelle genomes, and mutation rate. Computer solutions of the recurrence equations were used to study the approach to equilibrium. Gene diversities equilibrate slowly, while GST, used to measure population subdivision, equilibrates rapidly. Approximate equilibrium equations for gene diversities and GST can be obtained by substituting Neo and me, simple functions of the numbers of breeding or migrating males and females and of the degree of paternal transmission, for the effective numbers of genes and migration rates in the corresponding equations for nuclear genes. The approximate equations are not valid when the diversity within individuals is large compared to that between individuals, as is often true for the D-loop of animal mtDNA. We used the exact equations to verify that organelle genes often show more subdivision than nuclear genes; however, we also identified the range of breeding and migrating sex ratios for which population subdivision is greater for nuclear genes. Finally, we show that gene diversities are higher for nuclei than for organelles over a larger range of sex ratios in a subdivided population than in a panmictic population.     

Impacts of seed and pollen flow on population genetic structure for plant genomes with three contrasting modes of inheritance

The classical island and one-dimensional stepping-stone models of population genetic structure developed for animal populations are extended to hermaphrodite plant populations to study the behavior of biparentally inherited nuclear genes and organelle genes with paternal and maternal inheritance. By substituting appropriate values for effective population sizes and migration rates of the genes concerned into the classical models, expressions for genetic differentiation and correlation in gene frequency between populations can be derived. For both models, differentiation for maternally inherited genes at migration-drift equilibrium is greater than that for paternally inherited genes, which in turn is greater than that for biparentally inherited nuclear genes. In the stepping-stone model, the change of genetic correlation with distance is influenced by the mode of inheritance of the gene and the relative values of long- and short-distance migration by seed and pollen. In situations where it is possible to measure simultaneously Fst for genes with all three types of inheritance, estimates of the relative rates of pollen to seed flow can be made for both the short- and long-distance components of migration in the stepping-stone model.     

Maximum Likelihood Implementation of an Isolation-with-Migration Model with Three Species for Testing Speciation with Gene Flow

We implement an isolation with migration model for three species, with migration occurring between two closely related species while an out-group species is used to provide further information concerning gene trees and model parameters. The model is implemented in the likelihood framework for analyzing multilocus genomic sequence alignments, with one sequence sampled from each of the three species. The prior distribution of gene tree topology and branch lengths at every locus is calculated using a Markov chain characterization of the genealogical process of coalescent and migration, which integrates over the histories of migration events analytically. The likelihood function is calculated by integrating over branch lengths in the gene trees (coalescent times) numerically. We analyze the model to study the gene tree-species tree mismatch probability and the time to the most recent common ancestor at a locus. The model is used to construct a likelihood ratio test (LRT) of speciation with gene flow. We conduct computer simulations to evaluate the LRT and found that the test is in general conservative, with the false positive rate well below the significance level. For the test to have substantial power, hundreds of loci are needed. Application of the test to a human-chimpanzee-gorilla genomic data set suggests gene flow around the time of speciation of the human and the chimpanzee.     

Bayesian estimation of recent migration rates after a spatial expansion

Approximate Bayesian computation (ABC) is a highly flexible technique that allows the estimation of parameters under demographic models that are too complex to be handled by full-likelihood methods. We assess the utility of this method to estimate the parameters of range expansion in a two-dimensional stepping-stone model, using samples from either a single deme or multiple demes. A minor modification to the ABC procedure is introduced, which leads to an improvement in the accuracy of estimation. The method is then used to estimate the expansion time and migration rates for five natural common vole populations in Switzerland typed for a sex-linked marker and a nuclear marker. Estimates based on both markers suggest that expansion occurred <10,000 years ago, after the most recent glaciation, and that migration rates are strongly male biased.     

Structured coalescent processes from a modified Moran model with large offspring numbers

Structured coalescent processes are derived for the finite island model under a migration mechanism that conserves the subpopulation sizes. The underlying population model is a modified Moran model in which the reproducing individual can have very many offspring with some probability. Convergence to a structured coalescent process results when assuming that migration follows a coalescent timescale which can be much shorter than the usual Wright–Fisher timescale. Three different limit processes are possible depending on the coalescent timescale, two of which allow multiple mergers of ancestral lines. The expected time to most recent common ancestor, and the expected total size of the genealogy, of balanced and unbalanced samples can be very similar, even when migration is low, if the coalescent process allows multiple mergers. The expected total size increases almost linearly with sample size in some cases. The results have implications for inference about genetic population structure.     

The genetic signature of rapid range expansions: How dispersal,growth and invasion speed impact heterozygosity and allele surfing

As researchers collect spatiotemporal population and genetic data in tandem, models that connect demography and dispersal to genetics are increasingly relevant. The dominant spatiotemporal model of invasion genetics is the stepping-stone model which represents a gradual range expansion in which individuals jump to uncolonized locations one step at a time. However, many range expansions occur quickly as individuals disperse far from currently colonized regions. For these types of expansion, stepping-stone models are inappropriate. To more accurately reflect wider dispersal in many organisms, we created kernel-based models of invasion genetics based on integrodifference equations. Classic theory relating to integrodifference equations suggests that the speed of range expansions is a function of population growth and dispersal. In our simulations, populations that expanded at the same speed but with spread rates driven by dispersal retained more heterozygosity along axes of expansion than range expansions with rates of spread that were driven primarily by population growth. To investigate surfing we introduced mutant alleles in wave fronts of simulated range expansions. In our models based on random mating, surfing alleles remained at relatively low frequencies and surfed less often compared to previous results based on stepping-stone simulations with asexual reproduction.     

Stochasticity, complex spatial structure, and the feasibility of the shifting balance theory

Sewall Wright's shifting balance theory of evolution posits a mechanism by which a structured population may escape local fitness optima and find a global optimum. We examine a one-locus, two-allele model of underdominance in populations with differing spatial arrangements of demes, both analytically and with Monte Carlo simulations. We find that inclusion of variance in interpatch connectivities can significantly reduce the number of generations required for fixation of the more favorable allele relative to island and stepping-stone models. Although time to fixation increases with migration rate in all cases, the presence of one or two relatively isolated demes may reduce the number of generations by 80% or more. These results suggest that the shifting balance process may operate under less restrictive conditions than those found with a simple spatial arrangement of demes.     

The influence of spatial inhomogeneities on neutral models of geographical variation : I. Formulation

The influence of spatial variation in the carrying capacity and migration rate of a geographical barrier on the one-dimensional stepping-stone model is studied. The monoecious, diploid population is subdivided into an infinite linear array of panmictic colonies that exchange gametes. In each deme, the rate of self-fertilization is equal to the reciprocal of the number of individuals in that deme. Generations are discrete and nonoverlapping; the analysis is restricted to a single locus in the absence of selection; every allele mutates to new alleles at the same rate. In the diffusion approximation, a partial differential equation that incorporates spatial (and temporal) variation in the carrying capacity and migration rate is derived for the probability of identity. Transition conditions that simultaneously take into account discontinuities in the carrying capacity and migration rate are established: the probability of identity is continuous, but its partial derivatives are not, their ratio being a simple function of the carrying capacities and migrational variance on the two sides of the inhomogeneity. The partial derivatives of the probability of identity are continuous across a geographical barrier, whereas the probability of identity itself has a discontinuity proportional to the partial derivative at the barrier, the constant of proportionality being a measure of the difficulty of crossing the barrier.     

The distribution of the ancestral haplotype in finite stepping-stone models with population expansion

Recent extensive analyses of human DNA polymorphism reveal that the ancestral haplotype at various genetic loci occurs almost exclusively in African samples. We develop a coalescence-based simulation method in stepping-stone models with population expansion and examine the probability (P(A)) that the ancestral haplotype is found in African samples and the probability (Q(A)) that the most recent common ancestor of sampled genes occurs in Africa. These probabilities and other summary statistics are used to infer the human demographic history. It is shown that the high observed P(A) value cannot be explained simply by sampling bias. Rather, it suggests that the African population has been more strongly subdivided and isolated from each other than the non-African population and that there must have been some African populations which were not directly involved in the Out-of-Africa expansion in the late Pleistocene.     

The influence of spatial inhomogeneities on neutral models of geographical variation IV. Discontinuities in the population density and migration rate

The equilibrium structure of the infinite, one-dimensional stepping-stone model with coincident discontinuities in the population density and migration rate is investigated in the diffusion approximation. The monoecious, diploid population is subdivided into an infinite linear array of equally large, panmictic colonies that exchange gametes isotropically. The population density and the migration rate have a discontinuity at the origin, but are elsewhere uniform. Generations are discrete and nonoverlapping; the analysis is restricted to a single locus without selection; every allele mutates to new alleles at the same rate. The three dimensionless parameters in the theory are alpha=(rho(+)/rho(-))(2) (V(+)/V(-))(3/2), and beta(+/-)=4rho(+/-) 2uV(+/-), where rho(+) (rho(-)) and V(+) (V(-)) designate the population density and variance of gametic dispersion per generation to the right (left) of the discontinuity, respectively, and u denotes the mutation rate. The characteristic length on the right (left) is V(+)/(2u) (V(-)/(2u)). The probability of identity is continuous at the origin, but its partial derivatives have a discontinuity unless migration is conservative (rho(-) V(-)=rho(+) V(+)). At least for nonconservative migration, the probability of identity (including the expected homozygosity) can be nonmonotonic even if the migration rate is uniform and the population density is monotonic. Thus, there can be a nonmonotonic genetic response in a neutral model to a monotonic environment.     

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.     

Metapopulation genetic structure in the water vole, Arvicola terrestris, in NE Scotland

The genetic structure of nine colonies of the water vole, Arvicola terrestris , in one area of NE Scotland was studied. Non-destructive samples from 478 individuals (mostly immature animals) were typed for 12 microsatellites. Cases of Hardy-Weinberg disequilibrium were frequent within all colonies. The five colonies in the inland part of the study area also showed frequent cases of linkage disequilibrium. All colonies showed high levels of genetic diversity (unbiased H = 0.52-0.74). All five colonies sampled in successive years showed significant annual changes in genetic composition. All colonies showed genetic differentiation from each other, whether measured by average 6, pairwise 9 or pairwise Nei's genetic distance. The spatial pattern of genetic differentiation was consistent with either a stepping-stone model over the whole study area or an island model within the coastal and inland parts and an intervening barrier to gene flow. The study suggested that the genetic structure of colonies of A. terrestris often departs from the equilibrium states assumed by traditional mefhods for the study of gene flow, and that a parentage-based approach would be fruitful.