Sequential Markov coalescent algorithms for population models with demographic structure

We analyse sequential Markov coalescent algorithms for populations with demographic structure: for a bottleneck model, a population-divergence model, and for a two-island model with migration. The sequential Markov coalescent method is an approximation to the coalescent suggested by McVean and Cardin, and by Marjoram and Wall. Within this algorithm we compute, for two individuals randomly sampled from the population, the correlation between times to the most recent common ancestor and the linkage probability corresponding to two different loci with recombination rate R between them. These quantities characterise the linkage between the two loci in question. We find that the sequential Markov coalescent method approximates the coalescent well in general in models with demographic structure. An exception is the case where individuals are sampled from populations separated by reduced gene flow. In this situation, the correlations may be significantly underestimated. We explain why this is the case.     

Exact computation of coalescent likelihood for panmictic and subdivided populations under the infinite sites model

Coalescent likelihood is the probability of observing the given population sequences under the coalescent model. Computation of coalescent likelihood under the infinite sites model is a classic problem in coalescent theory. Existing methods are based on either importance sampling or Markov chain Monte Carlo and are inexact. In this paper, we develop a simple method that can compute the exact coalescent likelihood for many data sets of moderate size, including real biological data whose likelihood was previously thought to be difficult to compute exactly. Our method works for both panmictic and subdivided populations. Simulations demonstrate that the practical range of exact coalescent likelihood computation for panmictic populations is significantly larger than what was previously believed. We investigate the application of our method in estimating mutation rates by maximum likelihood. A main application of the exact method is comparing the accuracy of approximate methods. To demonstrate the usefulness of the exact method, we evaluate the accuracy of program Genetree in computing the likelihood for subdivided populations.     

The coalescence time of sampled genes in the structured coalescent model

The aim of this article is to investigate the distribution of the coalescence time (T) for sampled genes in the structured coalescent. We obtain some exact solutions for small samples and approximate distributions for n sampled genes in strong and weak migration. We also conduct computer simulation to evaluate efficiencies of these approximations and show the dependency of the distribution of the coalescence time on the geographical structure and the intensity of migration. In a panmictic population, we prove that the conditional distribution of the coalescence time given the number of segregating sites (S) among sampled genes is given by the weighted mean of the convolution of gamma distributions. We also study the joint distribution of T and S in the structured coalescent model and show some exact solutions.     

An application of the central limit theorem to coalescence times in the structured coalescent model with strong migration

The structured coalescent describes the ancestral relationship among sampled genes from a geographically structured population. The aim of this article is to apply the central limit theorem to functionals of the migration process to study coalescence times and population structure. An application of the law of large numbers to the migration process leads to the strong migration limit for the distributions of coalescence times. The central limit theorem enables us to obtain approximate distributions of coalescence times for strong migration. We show that approximate distributions depend on the population structure. If migration is conservative and strong, we can define a kind of effective population size N _e ^*, with which the entire population approximately behaves like a panmictic population. On the other hand, the approximate distributions for nonconservative migration are qualitatively different from those for conservative migration. And the entire population behaves unlike a panmictic population even though migration is strong.     

Differential population structuring and demographic history of two closely related fish species, Japanese sea bass (Lateolabrax japonicus) and spotted sea bass (Lateolabrax maculatus) in Northwestern Pacific

The Quaternary cold periods in the Northwestern Pacific are thought to have heavily influenced the amount and distribution of intraspecific genetic variation in marine fishes. To estimate the demographic history and genetic structure of Lateolabrax maculatus and L. japonicus in the Northwestern Pacific, 256 individuals were sampled from 19 localities throughout the distribution range of the two species. Mitochondrial DNA variation was analyzed using DNA sequence data from the cytochrome b gene and control region. Nucleotide diversity was much higher in L. japonicus (0.030) than in L. maculatus (0.012). The demographic history of the two species was examined using neutrality tests and mismatch distribution analyses and results indicated Pleistocene population expansion in both species. Estimates of population expansion time suggested earlier population expansion in L. japonicus than in L. maculatus. Molecular variance analyses showed differential genetic structuring for these two closely related species. The results indicated that L. japonicus is panmictic throughout its range. In contrast, populations of L. maculatus showed statistically significant levels of genetic structuring. Pattern of isolation by distance was observed in L. maculatus, suggesting that L. maculatus is in genetic equilibrium. In contrast, L. japonicus did not exhibit isolation by distance.     

Joint Inference of Migration and Reassortment Patterns for Viruses with Segmented Genomes

The structured coalescent allows inferring migration patterns between viral subpopulations from genetic sequence data. However, these analyses typically assume that no genetic recombination process impacted the sequence evolution of pathogens. For segmented viruses, such as influenza, that can undergo reassortment this assumption is broken. Reassortment reshuffles the segments of different parent lineages upon a coinfection event, which means that the shared history of viruses has to be represented by a network instead of a tree. Therefore, full genome analyses of such viruses are complex or even impossible. Although this problem has been addressed for unstructured populations, it is still impossible to account for population structure, such as induced by different host populations, whereas also accounting for reassortment. We address this by extending the structured coalescent to account for reassortment and present a framework for investigating possible ties between reassortment and migration (host jump) events. This method can accurately estimate subpopulation dependent effective populations sizes, reassortment, and migration rates from simulated data. Additionally, we apply the new model to avian influenza A/H5N1 sequences, sampled from two avian host types, Anseriformes and Galliformes. We contrast our results with a structured coalescent without reassortment inference, which assumes independently evolving segments. This reveals that taking into account segment reassortment and using sequencing data from several viral segments for joint phylodynamic inference leads to different estimates for effective population sizes, migration, and clock rates. This new model is implemented as the Structured Coalescent with Reassortment package for BEAST 2.5 and is available at https://github.com/jugne/SCORE.     

Corridors for migration between large subdivided populations, and the structured coalescent

We study the ancestral genetic process for samples from two large, subdivided populations that are connected by migration to, from, and within a small set of subpopulations, or demes. We consider convergence to an ancestral limit process as the numbers of demes in the two large, subdivided populations tend to infinity. We show that the ancestral limit process for a sample includes a recent instantaneous adjustment to the sample size and structure followed by a more ancient process that is identical to the usual structured coalescent, but with different scaled parameters. This justifies the application of a modified structured coalescent to some hierarchically structured populations.     

Phylodynamic Inference for Structured Epidemiological Models

Coalescent theory is routinely used to estimate past population dynamics and demographic parameters from genealogies. While early work in coalescent theory only considered simple demographic models, advances in theory have allowed for increasingly complex demographic scenarios to be considered. The success of this approach has lead to coalescent-based inference methods being applied to populations with rapidly changing population dynamics, including pathogens like RNA viruses. However, fitting epidemiological models to genealogies via coalescent models remains a challenging task, because pathogen populations often exhibit complex, nonlinear dynamics and are structured by multiple factors. Moreover, it often becomes necessary to consider stochastic variation in population dynamics when fitting such complex models to real data. Using recently developed structured coalescent models that accommodate complex population dynamics and population structure, we develop a statistical framework for fitting stochastic epidemiological models to genealogies. By combining particle filtering methods with Bayesian Markov chain Monte Carlo methods, we are able to fit a wide class of stochastic, nonlinear epidemiological models with different forms of population structure to genealogies. We demonstrate our framework using two structured epidemiological models: a model with disease progression between multiple stages of infection and a two-population model reflecting spatial structure. We apply the multi-stage model to HIV genealogies and show that the proposed method can be used to estimate the stage-specific transmission rates and prevalence of HIV. Finally, using the two-population model we explore how much information about population structure is contained in genealogies and what sample sizes are necessary to reliably infer parameters like migration rates.     

Estimating relatedness between individuals in general populations with a focus on their use in conservation programs

Relatedness estimators are widely used in genetic studies, but effects of population structure on performance of estimators, criteria to evaluate estimators, and benefits of using such estimators in conservation programs have to date received little attention. In this article we present new estimators, based on the relationship between coancestry and molecular similarity between individuals, and compare them with existing estimators using Monte Carlo simulation of populations, either panmictic or structured. Estimators were evaluated using statistical criteria and a diversity criterion that minimized relatedness. Results show that ranking of estimators depends on the population structure. An existing estimator based on two-gene and four-gene coefficients of identity performs best in panmictic populations, whereas a new estimator based on coancestry performs best in structured populations. The number of marker alleles and loci did not affect ranking of estimators. Statistical criteria were insufficient to evaluate estimators for their use in conservation programs. The regression coefficient of pedigree relatedness on estimated relatedness (beta2) was substantially lower than unity for all estimators, causing overestimation of the diversity conserved. A simple correction to achieve beta2 = 1 improves both existing and new estimators. Using relatedness estimates with correction considerably increased diversity in structured populations, but did not do so or even decreased diversity in panmictic populations.     

The joint allele frequency spectrum of multiple populations: a coalescent theory approach

The allele frequency spectrum is a series of statistics that describe genetic polymorphism, and is commonly used for inferring population genetic parameters and detecting natural selection. Population genetic theory on the allele frequency spectrum for a single population has been well studied using both coalescent theory and diffusion equations. Recently, the theory was extended to the joint allele frequency spectrum (JAFS) for three populations using diffusion equations and was shown to be very useful in inferring human demographic history. In this paper, I show that the JAFS can be analytically derived with coalescent theory for a basic model of two isolated populations and then extended to multiple populations and various complex scenarios, such as those involving population growth and bottleneck, migration, and positive selection. Simulation study is used to demonstrate the accuracy and applicability of the theoretical model. The coalescent theory-based approach for the JAFS can characterize the demographic history with comprehensive statistical models as the diffusion approach does, and in addition gains several novel advantages: the computational complexity of calculating the JAFS with coalescent theory is reduced, and thus it is feasible to analytically obtain the JAFS for multiple populations; the hitchhiking effect can be efficiently modeled in coalescent theory, enabling the development of methodologies for detecting selection via multi-population polymorphism data. As an alternative to the diffusion approximation approach, the coalescent theory for the JAFS also provides a foundation for population genetic inference with the advent of large-scale genomic polymorphism data.     

Environmental variation, fluctuating selection and genetic drift in subdivided populations

Although there have many studies of the population genetical consequences of environmental variation, little is known about the combined effects of genetic drift and fluctuating selection in structured populations. Here we use diffusion theory to investigate the effects of temporally and spatially varying selection on a population of haploid individuals subdivided into a large number of demes. Using a perturbation method for processes with multiple time scales, we show that as the number of demes tends to infinity, the overall frequency converges to a diffusion process that is also the diffusion approximation for a finite, panmictic population subject to temporally fluctuating selection. We find that the coefficients of this process have a complicated dependence on deme size and migration rate, and that changes in these demographic parameters can determine both the balance between the dispersive and stabilizing effects of environmental variation and whether selection favors alleles with lower or higher fitness variance.     

Effective population size and population subdivision in demographically structured populations

A fast-timescale approximation is applied to the coalescent process in a single population, which is demographically structured by sex and/or age. This provides a general expression for the probability that a pair of alleles sampled from the population coalesce in the previous time interval. The effective population size is defined as the reciprocal of twice the product of generation time and the coalescence probability. Biologically explicit formulas for effective population size with discrete generations and separate sexes are derived for a variety of different modes of inheritance. The method is also applied to a nuclear gene in a population of partially self-fertilizing hermaphrodites. The effects of population subdivision on a demographically structured population are analyzed, using a matrix of net rates of movement of genes between different local populations. This involves weighting the migration probabilities of individuals of a given age/sex class by the contribution of this class to the leading left eigenvector of the matrix describing the movements of genes between age/sex classes. The effects of sex-specific migration and nonrandom distributions of offspring number on levels of genetic variability and among-population differentiation are described for different modes of inheritance in an island model. Data on DNA sequence variability in human and plant populations are discussed in the light of the results.     

New Routes to Phylogeography: A Bayesian Structured Coalescent Approximation

Phylogeographic methods aim to infer migration trends and the history of sampled lineages from genetic data. Applications of phylogeography are broad, and in the context of pathogens include the reconstruction of transmission histories and the origin and emergence of outbreaks. Phylogeographic inference based on bottom-up population genetics models is computationally expensive, and as a result faster alternatives based on the evolution of discrete traits have become popular. In this paper, we show that inference of migration rates and root locations based on discrete trait models is extremely unreliable and sensitive to biased sampling. To address this problem, we introduce BASTA (BAyesian STructured coalescent Approximation), a new approach implemented in BEAST2 that combines the accuracy of methods based on the structured coalescent with the computational efficiency required to handle more than just few populations. We illustrate the potentially severe implications of poor model choice for phylogeographic analyses by investigating the zoonotic transmission of Ebola virus. Whereas the structured coalescent analysis correctly infers that successive human Ebola outbreaks have been seeded by a large unsampled non-human reservoir population, the discrete trait analysis implausibly concludes that undetected human-to-human transmission has allowed the virus to persist over the past four decades. As genomics takes on an increasingly prominent role informing the control and prevention of infectious diseases, it will be vital that phylogeographic inference provides robust insights into transmission history.     

Extending the coalescent to multilocus systems: the case of balancing selection

Natural populations are structured spatially into local populations and genetically into diverse 'genetic backgrounds' defined by different combinations of selected alleles. If selection maintains genetic backgrounds at constant frequency then neutral diversity is enhanced. By contrast, if background frequencies fluctuate then diversity is reduced. Provided that the population size of each background is large enough, these effects can be described by the structured coalescent process. Almost all the extant results based on the coalescent deal with a single selected locus. Yet we know that very large numbers of genes are under selection and that any substantial effects are likely to be due to the cumulative effects of many loci. Here, we set up a general framework for the extension of the coalescent to multilocus scenarios and we use it to study the simplest model, where strong balancing selection acting on a set of n loci maintains 2n backgrounds at constant frequencies and at linkage equilibrium. Analytical results show that the expected linked neutral diversity increases exponentially with the number of selected loci and can become extremely large. However, simulation results reveal that the structured coalescent approach breaks down when the number of backgrounds approaches the population size, because of stochastic fluctuations in background frequencies. A new method is needed to extend the structured coalescent to cases with large numbers of backgrounds.     

Genomic diversity,population structure,and migration following rapid range expansion in the Balsam Poplar,Populus balsamifera

Rapid range expansions can cause pervasive changes in the genetic diversity and structure of populations. The postglacial history of the Balsam Poplar, Populus balsamifera, involved the colonization of most of northern North America, an area largely covered by continental ice sheets during the last glacial maximum. To characterize how this expansion shaped genomic diversity within and among populations, we developed 412 SNP markers that we assayed for a range‐wide sample of 474 individuals sampled from 34 populations. We complemented the SNP data set with DNA sequence data from 11 nuclear loci from 94 individuals, and used coalescent analyses to estimate historical population size, demographic growth, and patterns of migration. Bayesian clustering identified three geographically separated demes found in the Northern, Central, and Eastern portions of the species’ range. These demes varied significantly in nucleotide diversity, the abundance of private polymorphisms, and population substructure. Most measures supported the Central deme as descended from the primary refuge of diversity. Both SNPs and sequence data suggested recent population growth, and coalescent analyses of historical migration suggested a massive expansion from the Centre to the North and East. Collectively, these data demonstrate the strong influence that range expansions exert on genomic diversity, both within local populations and across the range. Our results suggest that an in‐depth knowledge of nucleotide diversity following expansion requires sampling within multiple populations, and highlight the utility of combining insights from different data types in population genomic studies.     

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.     

On the size distribution of private microsatellite alleles

Private microsatellite alleles tend to be found in the tails rather than in the interior of the allele size distribution. To explain this phenomenon, we have investigated the size distribution of private alleles in a coalescent model of two populations, assuming the symmetric stepwise mutation model as the mode of microsatellite mutation. For the case in which four alleles are sampled, two from each population, we condition on the configuration in which three distinct allele sizes are present, one of which is common to both populations, one of which is private to one population, and the third of which is private to the other population. Conditional on this configuration, we calculate the probability that the two private alleles occupy the two tails of the size distribution. This probability, which increases as a function of mutation rate and divergence time between the two populations, is seen to be greater than the value that would be predicted if there was no relationship between privacy and location in the allele size distribution. In accordance with the prediction of the model, we find that in pairs of human populations, the frequency with which private microsatellite alleles occur in the tails of the allele size distribution increases as a function of genetic differentiation between populations.     

A separation-of-timescales approach to the coalescent in a continuous population

This article presents an analysis of a model of isolation by distance in a continuous, two-dimensional habitat. An approximate expression is derived for the distribution of coalescence times for a pair of sequences sampled from specific locations in a rectangular habitat. Results are qualitatively similar to previous analyses of isolation by distance, but account explicitly for the location of samples relative to the habitat boundaries. A separation-of-timescales approach takes advantage of the fact that the sampling locations affect only the recent coalescent behavior. When the population size is larger than the number of generations required for a lineage to cross the habitat range, the long-term genealogical process is reasonably well described by Kingman's coalescent with time rescaled by the effective population size. This long-term effective population size is affected by the local dispersal behavior as well as the geometry of the habitat. When the population size is smaller than the time required to cross the habitat, deep branches in the genealogy are longer than would be expected under the standard neutral coalescent, similar to the pattern expected for a panmictic population whose population size was larger in the past.     

Genetic and morphological variation of Rhizophora mangle (red mangrove) along the northern Pacific coast of Nicaragua

Red mangrove (Rhizophora mangle) dominates in tidally active areas of neotropical mangrove forests. Despite the ecological and economic importance of this species, we still know little about the genetic structure and diversity of its natural populations. In particular, Nicaraguan populations have not yet been investigated although they could be important for a better understanding of the evolutionary history of R. mangle in Mesoamerica. The aim of this study was to estimate the genetic and morphological variability of 4 populations sampled along the northwestern coast of Nicaragua. Microsatellite analysis showed higher levels of allelic diversity (30 alleles and a mean number of allele per locus per population = 6.42) than reported for the same species in other sites of tropical America. This variability could be attributed to the existence of a glacial refugium in the studied area and/or to repeated introgression among closely related species. Analysis of molecular variance (AMOVA) revealed that there was little but significant variation among the sampled sites suggesting that the studied populations cannot be considered as a single panmictic group, although they are closely related. This result was confirmed by the Bayesian analysis and UPGMA dendrogram showing three genetically distinct clusters. Bayesian analysis of migration rates showed that propagule dispersion associated with marine currents is an important factor shaping the observed genetic structure. The scatterplot from discriminant analysis indicated significant phenotypic differences between two groups, mainly related to specific leaf area variation. The morphological similarity between PR and PO was consistent with the results of molecular analysis, supporting the common origin of these two populations. Nevertheless, we did not find a significant statistical correlation between microsatellite and morphological data. In conclusion, the results reported here contribute to a better understanding of R. mangle genetic structure and would provide baseline information to guide land managers in implementing conservation initiatives in Nicaragua.