The distribution of the coalescence time and the number of pairwise nucleotide differences in the "isolation with migration" model

This paper is concerned with the “isolation with migration” model, where a panmictic ancestral population gave rise to a symmetric n-island model, time τ ago. Explicit analytical expressions are derived for the probability density function of the coalescence time of a pair of genes sampled at random from the same subpopulation or from different subpopulations, and for the probability distribution of the number of pairwise nucleotide differences.     

Pairwise differences under a general model of population subdivision

A number of different migration and isolation models of population subdivision have been studied. In this paper I analyse a general model of two populations derived from a common ancestral population at some time in the past. The two populations may exchange migrants, but they may also be completely isolated from each other. I derive the expectation and variance of the number of differences between two sequences sampled from the two populations. These are then compared to the corresponding results from two other much-used models: equilibrium migration and complete isolation.     

Impact of ancestral populations on postzygotic isolation in allopatric speciation

Postzygotic isolation evolves due to an accumulation of substitutions (potentially deleterious alleles in hybrids) in populations that have become geographically isolated. These potentially deleterious alleles might also be maintained in ancestral populations before geographic isolation. We used an individual-based model to examine the effect of the genetic state of an ancestral population on the evolution of postzygotic isolation after geographic isolation of a population. The results showed that the number of loci at which degenerative alleles are fixed in an ancestral population at equilibrium significantly affects the evolutionary rates of postzygotic isolation between descendant allopatric populations. Our results suggest that: (1) a severe decrease in population size (e.g., less than ten individuals) is not necessary for the rapid evolution of postzygotic isolation (e.g., <10,000 generation); (2) rapid speciation can occur when there is a large difference in the equilibrium number of accumulated degenerative alleles between ancestral and descendant populations; and (3) in an ancestral population maintained at a small effective population size for a long period of time, postzygotic isolation rarely evolves if back mutations that restore the function of degenerative alleles are limited.     

Distinguishing migration from isolation: a Markov chain Monte Carlo approach

A Markov chain Monte Carlo method for estimating the relative effects of migration and isolation on genetic diversity in a pair of populations from DNA sequence data is developed and tested using simulations. The two populations are assumed to be descended from a panmictic ancestral population at some time in the past and may (or may not) after that be connected by migration. The use of a Markov chain Monte Carlo method allows the joint estimation of multiple demographic parameters in either a Bayesian or a likelihood framework. The parameters estimated include the migration rate for each population, the time since the two populations diverged from a common ancestral population, and the relative size of each of the two current populations and of the common ancestral population. The results show that even a single nonrecombining genetic locus can provide substantial power to test the hypothesis of no ongoing migration and/or to test models of symmetric migration between the two populations. The use of the method is illustrated in an application to mitochondrial DNA sequence data from a fish species: the threespine stickleback (Gasterosteus aculeatus).     

An approximate likelihood for genetic data under a model with recombination and population splitting

We describe a new approximate likelihood for population genetic data under a model in which a single ancestral population has split into two daughter populations. The approximate likelihood is based on the ‘Product of Approximate Conditionals’ likelihood and ‘copying model’ of Li and Stephens [Li, N., Stephens, M., 2003. Modeling linkage disequilibrium and identifying recombination hotspots using single-nucleotide polymorphism data. Genetics 165 (4), 2213–2233]. The approach developed here may be used for efficient approximate likelihood-based analyses of unlinked data. However our copying model also considers the effects of recombination. Hence, a more important application is to loosely-linked haplotype data, for which efficient statistical models explicitly featuring non-equilibrium population structure have so far been unavailable. Thus, in addition to the information in allele frequency differences about the timing of the population split, the method can also extract information from the lengths of haplotypes shared between the populations. There are a number of challenges posed by extracting such information, which makes parameter estimation difficult. We discuss how the approach could be extended to identify haplotypes introduced by migrants.     

Genealogy of neutral genes in two partially isolated populations

Gene genealogy in two partially isolated populations which diverged at a given time t in the past and have since been exchanging individuals at a constant rate m is studied based upon an analytic method for large t and a simulation method for any t. Particular attention is paid to the conditions under which neutral genes sampled from populations are mono-, para-, and polyphyletic in terms of coalescence (divergence) times of genes. It is shown tha the probability of monophyly is high if M = 2Nm less than 0.5 and T = t/(2N) greater than 1, where N is the size of ancestral and descendant haploid populations, in which case most gene genealogies are likely to be concordant with the population relatedness. This probbility decreases as the sample size of genes increases. On the other hand, the case where the probability of monophyly is low will be either that of M greater than 1 and any T or that of M less than 1 and T less than 1, but the clear distinction between these conditions appears very difficult to make. These results are also examined if the gene genealogy is reconstructed from nucleotide differences. It is then shown that the results based upon coalescence times remain valid if the number of nucleotide differences between any pair of genes is not much smaller than 10. To observe such large nucleotide differences in small populations and therefore infer a reliable gene genealogy, we must examine a fairly long stretch of DNA sequences.     

Estimating Ancestral Population Parameters

The expected numbers of different categories of polymorphic sites are derived for two related models of population history: the isolation model, in which an ancestral population splits into two descendents, and the size-change model, in which a single population undergoes an instantaneous change in size. For the isolation model, the observed numbers of shared, fixed, and exclusive polymorphic sites are used to estimate the relative sizes of the three populations, ancestral plus two descendent, as well as the time of the split. For the size-change model, the numbers of sites segregating at particular frequencies in the sample are used to estimate the relative sizes of the ancestral and descendent populations plus the time the change took place. Parameters are estimated by choosing values that most closely equate expectations with observations. Computer simulations show that current and historical population parameters can be estimated accurately. The methods are applied to DNA data from two species of Drosophila and to some human mitochondrial DNA sequences.     

Inferring number of populations and changes in connectivity under the n-island model

Inferring the demographic history of species is one of the greatest challenges in populations genetics. This history is often represented as a history of size changes, ignoring population structure. Alternatively, when structure is assumed, it is defined a priori as a population tree and not inferred. Here we propose a framework based on the IICR (Inverse Instantaneous Coalescence Rate). The IICR can be estimated for a single diploid individual using the PSMC method of Li and Durbin (2011). For an isolated panmictic population, the IICR matches the population size history, and this is how the PSMC outputs are generally interpreted. However, it is increasingly acknowledged that the IICR is a function of the demographic model and sampling scheme with limited connection to population size changes. Our method fits observed IICR curves of diploid individuals with IICR curves obtained under piecewise stationary symmetrical island models. In our models we assume a fixed number of time periods during which gene flow is constant, but gene flow is allowed to change between time periods. We infer the number of islands, their sizes, the periods at which connectivity changes and the corresponding rates of connectivity. Validation with simulated data showed that the method can accurately recover most of the scenario parameters. Our application to a set of five human PSMCs yielded demographic histories that are in agreement with previous studies using similar methods and with recent research suggesting ancient human structure. They are in contrast with the view of human evolution consisting of one ancestral population branching into three large continental and panmictic populations with varying degrees of connectivity and no population structure within each continent.Subject terms: Population genetics, Biological models, Population genetics     

Geographical invariance and the strong-migration limit in subdivided populations

Invariance under population subdivision and the strong-migration limit are investigated for digenic samples in neutral models. The monoecious, diploid population is subdivided into a finite number of panmictic colonies that exchange gametes. The backward migration matrix is arbitrary, but time independent and ergodic (i.e., irreducible and aperiodic). Results are derived for the distribution of the place and time of coalescence, for the probability of identity in the model of infinitely many alleles, and for the distribution of the number of nucleotide differences in the model of infinitely many sites without recombination. Received: 5 October 1999 / Revised version: 1 February 2000 / Published online: 4 July 2000     

Genetic diversity in Dactylorhiza majalis subsp. majalis populations (Orchidaceae) of northern Poland

We analysed 16 populations of Dactylorhiza majalis subsp. majalis from northern Poland, simultaneously utilizing both morphological and molecular data. Genetic differentiation was examined using five microsatellite loci, and morphological variation was assessed for 23 characters. At the species level, our results showed a moderate level of genetic diversity (A = 6.00; A_e = 1.86; H_o = 0.387; F_IS = 0.139) which varied between the studied populations (A = 2.60–4.20; A_e = 1.68–2.39; H_o = 0.270–0.523; F_IS = ?0.064–0.355). A significant excess of homozygotes was detected in five population, while excess of heterozygotes was observed in four populations, but the latter values were statistically insignificant. Moderate, but clear between population genetic differentiation was found (F_ST = 0.101; p < 0.001). Considering pairwise‐F_ST and number of migrants among populations, we recognized three population groups (I, II, III), where the first could be further divided into two subgroups (Ia, Ib). These three groups differed with respect to gene flow values (N_m = 0.39–1.12). The highest number of migrants per generation was noticed among populations of subgroup Ia (8.58), indicative of a central panmictic population with free gene flow surrounded by peripatric local populations (Ib) with more limited gene flow. Geographic isolation, habitat fragmentation and limited seed dispersal are inferred to have caused limitations to gene flow among the three indicated population groups. There was a significant correlation between the morphological and genetic distance matrices. A weak but significant pattern of isolation by distance was also observed (r = 0.351; p < 0.05).     

The effect of migration during the divergence

The mean and variance of the number of nucleotide differences were obtained when the ancestral population diverged with migration. The number of nucleotide differences obtained indicates that not only the migration rate but also the period of migration has influence on a population structure. According to the migration rate and the period of migration, populations behave approximately as a single unit, diverged and isolated populations, two populations under equilibrium, or none of them. When sigma m(t) is about one, the variance of the number of nucleotide differences becomes large, where sigma m(t) is the sum of the migration rate for the period of migration. The distribution of the estimated divergence time was also obtained using computer simulations. It was found that the divergence time can be explained by sigma m(t). That is, the divergence time is mostly estimated as the time when sigma m(t) is less than 1.     

Distinguishing Migration from Isolation Using the Variance of Pairwise Differences

Two demographic scenarios are considered: two populations with migration and two populations that have been completely isolated from each other for some period of time. The variance of the number of differences between pairs of sequences in a single sample is studied and forms the basis of a test of the isolation model. The migration model is one possible alternative to isolation. The isolation model is rejected when the proposed test statistic, which involves the variances of pairwise difference within and between populations, is larger than some critical value. The power and realized significance of the test are investigated using simulations, and an example using mitochondrial DNA illustrates its application.     

Recent colonization by a coastal plant of inland habitats at an ancient freshwater lake,Lake Biwa: multilocus sequencing and a demographic history of Lathyrus japonicus (Fabaceae)

Ancient lakes have been recognized as “long‐term isolated islands” in terrestrial ecosystems. Lake Biwa, one of the few ancient lakes that formed around 4 million years ago, harbors many coastal species that commonly inhabit seashores. The beach pea, Lathyrus japonicus, is a typical coastal species of this freshwater lake, where morphological, physiological, and genetic differentiations have been reported between Biwa and coastal populations. Whether Biwa populations were isolated for long periods throughout Pleistocene climatic oscillations and subsequent range shifts is unclear. We assessed population genetic structure and demography of beach pea in this ancient freshwater lake using the sequences of eight nuclear loci. The results of STRUCTURE analyses showed evidence of admixture between Biwa and coastal populations, reflecting recent gene flow. The estimated demographic parameters implemented by the isolation with migration model (IM model) revealed a recent divergence (postglacial period) of Biwa populations, with some gene flow from Biwa to coastal populations. In addition, Biwa populations were significantly smaller in size than the ancestral or coastal populations. Our study suggests that a Holocene thermal maximum, when transgression could allow seeds from coastal plants to access Lake Biwa, was involved in the origin of the Biwa populations and their genetic divergence. Thus, coastal populations might have migrated to Lake Biwa relatively recently. Our study concluded that ancestral migrants in Lake Biwa were derived from small founding populations and accelerated genetic isolation of Biwa populations during short‐term isolation.     

Taxonomical status and phylogenetic relations between the “thomasi” and “atticus” chromosomal races of the underground vole Microtus thomasi (Rodentia, Arvicolinae)

Laboratory crosses among wild caught individuals of the chromosomal races “atticus” and “thomasi”, were performed to analyze the degree of interracial reproductive isolation. The fertility of the studied specimens was evaluated by taking into consideration the reproductive success, the litter size and performing comparative histological examination of the testicular material. All studied populations were submitted to classical cytogenetic and mitochondrial analysis (cytochrome b gene), providing new evidences to the potential phylogenetic relations and taxonomical status of the two chromosomal races. The previously described “atticus” populations are divided in two genetically distinct, geographically and reproductively isolated lineages (2.9% total and 2.4% net divergence), which probably derived from different glacial refugia of Southern Greece. Here, we suggest that the lineage, consisting of the populations from Attiki and Evia Island, should be distinguished as a valid species, named Microtus atticus, including the two chromosomal races “atticus” and “evia”. On the contrary, the ex-“atticus” populations from North Peloponnesus belong to the same mitochondrial lineage with the other Microtus thomasi populations and should be considered as a chromosomal polymorphism inside the chromosomal race “thomasi”.     

Population genetic structure of the catadromous Australian bass from throughout its range

One hatchery-derived impoundment and nine wild riverine populations of the catadromous percoid Australian bass Macquaria novemaculeata were examined at six polymorphic allozyme loci to reveal population genetic structure. Subtle genetic differences among many populations indicated that fish did not interact as a panmictic unit, but rather conformed to an isolation by distance pattern of population structure. The hatchery-derived Glenbawn Dam sample differed from the riverine populations in both average heterozygosity and number of polymorphic loci, demonstrating the need for caution in supplementation programs to rehabilitate wild riverine populations.     

Mathematical consequences of the genealogical species concept

A genealogical species is defined as a basal group of organisms whose members are all more closely related to each other than they are to any organisms outside the group ("exclusivity"), and which contains no exclusive group within it. In practice, a pair of species is so defined when phylogenies of alleles from a sample of loci shows them to be reciprocally monophyletic at all or some specified fraction of the loci. We investigate the length of time it takes to attain this status when an ancestral population divides into two descendant populations of equal size with no gene exchange, and when genetic drift and mutation are the only evolutionary forces operating. The number of loci used has a substantial effect on the probability of observing reciprocal monophyly at different times after population separation, with very long times needed to observe complete reciprocal monophyly for a large number of loci. In contrast, the number of alleles sampled per locus has a relatively small effect on the probability of reciprocal monophyly. Because a single mitochondrial or chloroplast locus becomes reciprocally monophyletic much faster than does a single nuclear locus, it is not advisable to use mitochondrial and chloroplast DNA to recognize genealogical species for long periods after population divergence. Using a weaker criterion of assigning genealogical species status when more than 50% of sampled nuclear loci show reciprocal monophyly, genealogical species status depends much less on the number of sampled loci, and is attained at roughly 4-7 N generations after populations are isolated, where N is the historically effective population size of each descendant. If genealogical species status is defined as more than 95% of sampled nuclear loci showing reciprocal monophyly, this status is attained after roughly 9-12 N generations.     

DNA diversity in the studies of genetic distance among isolated human populations in bosnia

Different degrees of isolation found in various part of Bosnia and Herzegovina may be induced by various factors. Bjelasnica-Treskavica region, located around 40 kilometers southwest from Sarajevo — capital of Bosnia and Herzegovina, is highly specific in that way. We chose three isolated communities: Dejcici, Bobovica and Lukomir for the study of genetic structure of isolated human populations. Based on general data three relative degrees of isolation/openness among the villages have been presumed as follows: first (lower-Dejcici), second (middle — Bobovica) and third (higher — Lukomir) 15 short tandem repeat (STR) loci and hypervariable region of mtDNA were chosen as a markers for study of population structure. Microsatellite allele frequencies, and mtDNA molecular diversity of Heterozigosity and coefficient of gene differentiation across all observed STR loci were estimated. Also, gene and nucleotide diversity of observed mtDNA regions were obtained. Genetic distance between three populations was calculated using method of Reynolds et al. (1983). For analysis of interpopulation relationship based on polymorphism of HV I and HV II region, estimation of pairwise differences was used. results of this research showed consistence with initial hypothesis on divergence based on socio-cultural factors.     

Demographic history of a common pioneer tree,Zanthoxylum ailanthoides,reconstructed using isolation-with-migration model

The Japanese prickly ash, Zanthoxylum ailanthoides Siebold & Zucc. (Rutaceae), is one of the most common pioneer tree species that grow in disturbed areas, such as canopy gaps, in Japanese warm-temperate evergreen oak forests. The strong genetic structure of its current population might suggest long-term isolation of subpopulations by demographic events, in addition to ecological features of this species. Analyses of genome-wide nucleotide variation using model-based approaches are necessary to achieve a good understanding of the demographic history of a species. In this study, we analyzed the nucleotide variation in natural populations of Z. ailanthoides using a computer program that applied the isolation-with-migration model to quantitatively infer the demographic history. Nucleotide variations at 10 or 26 nuclear loci in six populations from a wide range of the species distribution were analyzed. The maximum likelihood estimate of the divergence time between the current populations in the two hypothetical refugia of Z. ailanthoides was approximately 24,000 years. Unexpectedly, the estimated size of the ancestral population was larger than the sizes of the two current populations. The results suggested a relatively recent divergence of these two populations and rapid formation of strong genetic structure among subpopulations. The large ancestral population may indicate a more complex demographic history during or after the last glacial period than the simple isolation-with-migration model implies.     

The Effects of Background and Interference Selection on Patterns of Genetic Variation in Subdivided Populations

It is well known that most new mutations that affect fitness exert deleterious effects and that natural populations are often composed of subpopulations (demes) connected by gene flow. To gain a better understanding of the joint effects of purifying selection and population structure, we focus on a scenario where an ancestral population splits into multiple demes and study neutral diversity patterns in regions linked to selected sites. In the background selection regime of strong selection, we first derive analytic equations for pairwise coalescent times and F_ST as a function of time after the ancestral population splits into two demes and then construct a flexible coalescent simulator that can generate samples under complex models such as those involving multiple demes or nonconservative migration. We have carried out extensive forward simulations to show that the new methods can accurately predict diversity patterns both in the nonequilibrium phase following the split of the ancestral population and in the equilibrium between mutation, migration, drift, and selection. In the interference selection regime of many tightly linked selected sites, forward simulations provide evidence that neutral diversity patterns obtained from both the nonequilibrium and equilibrium phases may be virtually indistinguishable for models that have identical variance in fitness, but are nonetheless different with respect to the number of selected sites and the strength of purifying selection. This equivalence in neutral diversity patterns suggests that data collected from subdivided populations may have limited power for differentiating among the selective pressures to which closely linked selected sites are subject.     

Genetic diversity and connectivity in a brooding reef coral at the limit of its distribution

Remote populations are predicted to be vulnerable owing to their isolation from potential source reefs, and usually low population size and associated increased extinction risk. We investigated genetic diversity, population subdivision and connectivity in the brooding reef coral Seriatopora hystrix at the limits of its Eastern Australian (EA) distribution and three sites in the southern Great Barrier Reef (GBR). Over the approximately 1270 km survey range, high levels of population subdivision were detected (global F_ST = 0.224), with the greatest range in pairwise F_ST values observed among the three southernmost locations: Lord Howe Island, Elizabeth Reef and Middleton Reef. Flinders Reef, located between the GBR and the more southerly offshore reefs, was highly isolated and showed the signature of a recent bottleneck. High pairwise F_ST values and the presence of multiple genetic clusters indicate that EA subtropical coral populations have been historically isolated from each other and the GBR. One putative first-generation migrant was detected from the GBR into the EA subtropics. Occasional long-distance dispersal is supported by changes in species composition at these high-latitude reefs and the occurrence of new species records over the past three decades. While subtropical populations exhibited significantly lower allelic richness than their GBR counterparts, genetic diversity was still moderately high. Furthermore, subtropical populations were not inbred and had a considerable number of private alleles. The results suggest that these high-latitude S. hystrix populations are supplemented by infrequent long-distance migrants from the GBR and may have adequate population sizes to maintain viability and resist severe losses of genetic diversity.