The effects of subdivision on the genetic divergence of populations and species

Abstract. An island model of migration is used to study the effects of subdivision within populations and species on sample genealogies and on between-population or between-species measures of genetic variation. The model assumes that the number of demes within each population or species is large. When populations (or species), connected either by gene flow or historical association, are themselves subdivided into demes, changes in the migration rate among demes alter both the structure of genealogies and the time scale of the coalescent process. The time scale of the coalescent is related to the effective size of the population, which depends on the migration rate among demes. When the migration rate among demes within populations is low, isolation (or speciation) events seem more recent and migration rates among populations seem higher because the effective size of each population is increased. This affects the probability of reciprocal monophyly of two samples, the chance that a gene tree of a sample matches the species tree, and relative likelihoods of different types of polymorphic sites. It can also have a profound effect on the estimation of divergence times.     

Estimating Effective Population Size and Mutation Rate from Sequence Data Using Metropolis-Hastings Sampling

We present a new way to make a maximum likelihood estimate of the parameter 4N(e)μ (effective population size times mutation rate per site, or θ) based on a population sample of molecular sequences. We use a Metropolis-Hastings Markov chain Monte Carlo method to sample genealogies in proportion to the product of their likelihood with respect to the data and their prior probability with respect to a coalescent distribution. A specific value of θ must be chosen to generate the coalescent distribution, but the resulting trees can be used to evaluate the likelihood at other values of θ, generating a likelihood curve. This procedure concentrates sampling on those genealogies that contribute most of the likelihood, allowing estimation of meaningful likelihood curves based on relatively small samples. The method can potentially be extended to cases involving varying population size, recombination, and migration.     

The joint effects of background selection and genetic recombination on local gene genealogies

Background selection, the effects of the continual removal of deleterious mutations by natural selection on variability at linked sites, is potentially a major determinant of DNA sequence variability. However, the joint effects of background selection and genetic recombination on the shape of the neutral gene genealogy have proved hard to study analytically. The only existing formula concerns the mean coalescent time for a pair of alleles, making it difficult to assess the importance of background selection from genome-wide data on sequence polymorphism. Here we develop a structured coalescent model of background selection with recombination and implement it in a computer program that efficiently generates neutral gene genealogies for an arbitrary sample size. We check the validity of the structured coalescent model against forward-in-time simulations and show that it accurately captures the effects of background selection. The model produces more accurate predictions of the mean coalescent time than the existing formula and supports the conclusion that the effect of background selection is greater in the interior of a deleterious region than at its boundaries. The level of linkage disequilibrium between sites is elevated by background selection, to an extent that is well summarized by a change in effective population size. The structured coalescent model is readily extendable to more realistic situations and should prove useful for analyzing genome-wide polymorphism data.     

Evolution of the human immunodeficiency virus envelope gene is dominated by purifying selection

The evolution of the human immunodeficiency virus (HIV-1) during chronic infection involves the rapid, continuous turnover of genetic diversity. However, the role of natural selection, relative to random genetic drift, in governing this process is unclear. We tested a stochastic model of genetic drift using partial envelope sequences sampled longitudinally in 28 infected children. In each case the Bayesian posterior (empirical) distribution of coalescent genealogies was estimated using Markov chain Monte Carlo methods. Posterior predictive simulation was then used to generate a null distribution of genealogies assuming neutrality, with the null and empirical distributions compared using four genealogy-based summary statistics sensitive to nonneutral evolution. Because both null and empirical distributions were generated within a coalescent framework, we were able to explicitly account for the confounding influence of demography. From the distribution of corrected P-values across patients, we conclude that empirical genealogies are more asymmetric than expected if evolution is driven by mutation and genetic drift only, with an excess of low-frequency polymorphisms in the population. This indicates that although drift may still play an important role, natural selection has a strong influence on the evolution of HIV-1 envelope. A negative relationship between effective population size and substitution rate indicates that as the efficacy of selection increases, a smaller proportion of mutations approach fixation in the population. This suggests the presence of deleterious mutations. We therefore conclude that intrahost HIV-1 evolution in envelope is dominated by purifying selection against low-frequency deleterious mutations that do not reach fixation.     

A fast and reliable computational method for estimating population genetic parameters

The estimation of ancestral and current effective population sizes in expanding populations is a fundamental problem in population genetics. Recently it has become possible to scan entire genomes of several individuals within a population. These genomic data sets can be used to estimate basic population parameters such as the effective population size and population growth rate. Full-data-likelihood methods potentially offer a powerful statistical framework for inferring population genetic parameters. However, for large data sets, computationally intensive methods based upon full-likelihood estimates may encounter difficulties. First, the computational method may be prohibitively slow or difficult to implement for large data. Second, estimation bias may markedly affect the accuracy and reliability of parameter estimates, as suggested from past work on coalescent methods. To address these problems, a fast and computationally efficient least-squares method for estimating population parameters from genomic data is presented here. Instead of modeling genomic data using a full likelihood, this new approach uses an analogous function, in which the full data are replaced with a vector of summary statistics. Furthermore, these least-squares estimators may show significantly less estimation bias for growth rate and genetic diversity than a corresponding maximum-likelihood estimator for the same coalescent process. The least-squares statistics also scale up to genome-sized data sets with many nucleotides and loci. These results demonstrate that least-squares statistics will likely prove useful for nonlinear parameter estimation when the underlying population genomic processes have complex evolutionary dynamics involving interactions between mutation, selection, demography, and recombination.     

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.     

Extending coalescent theory to autotetraploids

We develop coalescent models for autotetraploid species with tetrasomic inheritance. We show that the ancestral genetic process in a large population without recombination may be approximated using Kingman's standard coalescent, with a coalescent effective population size 4N. Numerical results suggest that this approximation is accurate for population sizes on the order of hundreds of individuals. Therefore, existing coalescent simulation programs can be adapted to study population history in autotetraploids simply by interpreting the timescale in units of 4N generations. We also consider the possibility of double reduction, a phenomenon unique to polysomic inheritance, and show that its effects on gene genealogies are similar to partial self-fertilization.     

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.     

A coalescent-based estimator of admixture from DNA sequences

A variety of estimators have been developed to use genetic marker information in inferring the admixture proportions (parental contributions) of a hybrid population. The majority of these estimators used allele frequency data, ignored molecular information that is available in markers such as microsatellites and DNA sequences, and assumed that mutations are absent since the admixture event. As a result, these estimators may fail to deliver an estimate or give rather poor estimates when admixture is ancient and thus mutations are not negligible. A previous molecular estimator based its inference of admixture proportions on the average coalescent times between pairs of genes taken from within and between populations. In this article I propose an estimator that considers the entire genealogy of all of the sampled genes and infers admixture proportions from the numbers of segregating sites in DNA sequence samples. By considering the genealogy of all sequences rather than pairs of sequences, this new estimator also allows the joint estimation of other interesting parameters in the admixture model, such as admixture time, divergence time, population size, and mutation rate. Comparative analyses of simulated data indicate that the new coalescent estimator generally yields better estimates of admixture proportions than the previous molecular estimator, especially when the parental populations are not highly differentiated. It also gives reasonably accurate estimates of other admixture parameters. A human mtDNA sequence data set was analyzed to demonstrate the method, and the analysis results are discussed and compared with those from previous studies.     

A coalescent process with simultaneous multiple mergers for approximating the gene genealogies of many marine organisms

We describe a forward-time haploid reproduction model with a constant population size that includes life history characteristics common to many marine organisms. We develop coalescent approximations for sample gene genealogies under this model and use these to predict patterns of genetic variation. Depending on the behavior of the underlying parameters of the model, the approximations are coalescent processes with simultaneous multiple mergers or Kingman’s coalescent. Using simulations, we apply our model to data from the Pacific oyster and show that our model predicts the observed data very well. We also show that a fact which holds for Kingman’s coalescent and also for general coalescent trees–that the most-frequent allele at a biallelic locus is likely to be the ancestral allele–is not true for our model. Our work suggests that the power to detect a “sweepstakes effect” in a sample of DNA sequences from marine organisms depends on the sample size.     

Robustness of coalescent estimators to between-lineage mutation rate variation

Data from HIV and from human neoplastic cells can show substantial between-lineage mutation rate variation even within a single population. Such variation may affect estimators of population quantities such as Theta = 4N(e)mu. Using simulated DNA data, I measured the effect of rate variation on recovery of Theta by the summary-statistic estimator of Watterson (Watterson GA. 1975. On the number of segregating sites in genetical systems without recombination. Theor Popul Biol. 7:256-276) and the coalescent maximum likelihood algorithm LAMARC (Kuhner MK. 2006. LAMARC 2.0: maximum likelihood and Bayesian estimation of population parameters. Bioinformatics. Advance Access doi: 10.1093/bioinformatics/btk051). Watterson's estimator showed a downward bias, as expected, with high values of Theta. LAMARC's mean estimate was accurate for all tested values of Theta and rate variation except for a downward bias when rate variation was maximal (i.e., the slow rate was zero). LAMARC had consistently narrower confidence intervals (CIs) than Watterson's estimator. Both methods tended to reject the truth too often when rate variation was 8x or greater and independent among branches, as well as when variation was 4x or greater and correlated among branches. In the case of Watterson's estimate, this excess rejection was fully attributable to variation among genealogies in the amount of total branch length associated with the fast and slow rates. However, in the case of LAMARC, some excess rejection was still observed even when between-genealogy variation was taken into account. Both estimators are robust to modest rate variation; however, their use should be coupled with a statistical test to rule out extreme rate variation as the resulting CIs may not be reliable.     

A composite-likelihood approach for detecting directional selection from DNA sequence data

We present a novel composite-likelihood-ratio test (CLRT) for detecting genes and genomic regions that are subject to recurrent natural selection (either positive or negative). The method uses the likelihood functions of Hartl et al. (1994) for inference in a Wright-Fisher genic selection model and corrects for nonindependence among sites by application of coalescent simulations with recombination. Here, we (1) characterize the distribution of the CLRT statistic (Lambda) as a function of the population recombination rate (R=4Ner); (2) explore the effects of bias in estimation of R on the size (type I error) of the CLRT; (3) explore the robustness of the model to population growth, bottlenecks, and migration; (4) explore the power of the CLRT under varying levels of mutation, selection, and recombination; (5) explore the discriminatory power of the test in distinguishing negative selection from population growth; and (6) evaluate the performance of maximum composite-likelihood estimation (MCLE) of the selection coefficient. We find that the test has excellent power to detect weak negative selection and moderate power to detect positive selection. Moreover, the test is quite robust to bias in the estimate of local recombination rate, but not to certain demographic scenarios such as population growth or a recent bottleneck. Last, we demonstrate that the MCLE of the selection parameter has little bias for weak negative selection and has downward bias for positively selected mutations.     

The ancestry of a sample of sequences subject to recombination

In this article we discuss the ancestry of sequences sampled from the coalescent with recombination with constant population size 2N. We have studied a number of variables based on simulations of sample histories, and some analytical results are derived. Consider the leftmost nucleotide in the sequences. We show that the number of nucleotides sharing a most recent common ancestor (MRCA) with the leftmost nucleotide is approximately log(1 + 4N Lr)/4Nr when two sequences are compared, where L denotes sequence length in nucleotides, and r the recombination rate between any two neighboring nucleotides per generation. For larger samples, the number of nucleotides sharing MRCA with the leftmost nucleotide decreases and becomes almost independent of 4N Lr. Further, we show that a segment of the sequences sharing a MRCA consists in mean of 3/8Nr nucleotides, when two sequences are compared, and that this decreases toward 1/4Nr nucleotides when the whole population is sampled. A measure of the correlation between the genealogies of two nucleotides on two sequences is introduced. We show analytically that even when the nucleotides are separated by a large genetic distance, but share MRCA, the genealogies will show only little correlation. This is surprising, because the time until the two nucleotides shared MRCA is reciprocal to the genetic distance. Using simulations, the mean time until all positions in the sample have found a MRCA increases logarithmically with increasing sequence length and is considerably lower than a theoretically predicted upper bound. On the basis of simulations, it turns out that important properties of the coalescent with recombinations of the whole population are reflected in the properties of a sample of low size.     

MICROEVOLUTIONARY PROCESSES GENERATE PHYLOGENOMIC DISCORDANCE AT ANCIENT DIVERGENCES

Stochastic population processes may cause differences between species histories and gene histories. These processes are assumed to only influence the most recent divergences in the tree of life; however, there may be underappreciated potential for microevolutionary processes to impact deep divergences. I used multispecies coalescent models to determine the impact of stochastic processes on deep phylogenomic histories. Here I show phylogenomic discordance between gene histories and species histories is expected at deep divergences for many eukaryotic taxa, and the probability of discordance increases with population size, generation time, and the number of species in the tree. Five eukaryotic clades (angiosperms, birds, harpaline beetles, mammals, and nymphalid butterflies) demonstrate significant discordance potential at divergences over 50 million years old, and this discordance potential is independent of the age of divergence. These findings demonstrate population processes acting over very short timescales will leave a lasting impact on genomic histories, even for divergence events occurring tens to hundreds of millions of years ago.     

Complex population dynamics and the coalescent under neutrality

Estimates of the coalescent effective population size N(e) can be poorly correlated with the true population size. The relationship between N(e) and the population size is sensitive to the way in which birth and death rates vary over time. The problem of inference is exacerbated when the mechanisms underlying population dynamics are complex and depend on many parameters. In instances where nonparametric estimators of N(e) such as the skyline struggle to reproduce the correct demographic history, model-based estimators that can draw on prior information about population size and growth rates may be more efficient. A coalescent model is developed for a large class of populations such that the demographic history is described by a deterministic nonlinear dynamical system of arbitrary dimension. This class of demographic model differs from those typically used in population genetics. Birth and death rates are not fixed, and no assumptions are made regarding the fraction of the population sampled. Furthermore, the population may be structured in such a way that gene copies reproduce both within and across demes. For this large class of models, it is shown how to derive the rate of coalescence, as well as the likelihood of a gene genealogy with heterochronous sampling and labeled taxa, and how to simulate a coalescent tree conditional on a complex demographic history. This theoretical framework encapsulates many of the models used by ecologists and epidemiologists and should facilitate the integration of population genetics with the study of mathematical population dynamics.     

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.     

Inference of past population expansion from the timing of coalescence events in a gene genealogy

We investigate the expected coalescent in populations growing exponentially. The distribution of expected times to coalescence events may show a linear relationship with a number of ancestral lineages, when the latter is subjected to the "epidemic transformation". However, in a number of viral populations, upward curves are created when the epidemically transformed number of ancestral lineages is plotted against time. We consider possible causes of such upward curves. These include the possibility that a curved line is created through a transformation failure due to a sample size that is too large. We suggest a new formula for predicting such failure. The second cause is a population size increasing at an accelerating rate. However, the combination of recent coalescent events and an upward curve is created by an accelerating population increase only under restricted conditions. Specifically, such a pattern is expected only when, were population growth not to have accelerated, the transformation would have failed anyway. The third cause of nonlinearity arises in the estimated coalescent, as distinct from the real coalescent, if the mutation rate is small. However, coalescence times estimated from data typically give a straight line following epidemic transformation, but the rate of exponential increase, or r value, will be underestimated.