Wright's Hierarchical F-Statistics.

This perspective article offers a meditation on FST and other quantities developed by Sewall Wright to describe the population structure, defined as any departure from reproduction through random union of gametes. Concepts related to the F-statistics draw from studies of the partitioning of variation, identity coefficients, and diversity measures. Relationships between the first two approaches have recently been clarified and unified. This essay addresses the third pillar of the discussion: Nei's GST and related measures. A hierarchy of probabilities of identity-by-state provides a description of the relationships among levels of a structured population with respect to genetic diversity. Explicit expressions for the identity-by-state probabilities are determined for models of structured populations undergoing regular inbreeding and recurrent mutation. Levels of genetic diversity within and between subpopulations reflect mutation as well as migration. Accordingly, indices of the population structure are inherently locus-specific, contrary to the intentions of Wright. Some implications of this locus-specificity are explored.

Allele frequency spectra in structured populations: Novel-allele probabilities under the labelled coalescent.

We address the effect of population structure on key properties of the Ewens sampling formula. We use our previously-introduced inductive method for determining exact allele frequency spectrum (AFS) probabilities under the infinite-allele model of mutation and population structure for samples of arbitrary size. Fundamental to the sampling distribution is the novel-allele probability, the probability that given the pattern of variation in the present sample, the next gene sampled belongs to an as-yet-unobserved allelic class. Unlike the case for panmictic populations, the novel-allele probability depends on the AFS of the present sample. We derive a recursion that directly provides the marginal novel-allele probability across AFSs, obviating the need first to determine the probability of each AFS. Our explorations suggest that the marginal novel-allele probability tends to be greater for initial samples comprising fewer alleles and for sampling configurations in which the next-observed gene derives from a deme different from that of the majority of the present sample. Comparison to the efficient importance sampling proposals developed by De Iorio and Griffiths and colleagues indicates that their approximation for the novel-allele probability generally agrees with the true marginal, although it may tend to overestimate the marginal in cases in which the novel-allele probability is high and migration rates are low.

Inductive determination of allele frequency spectrum probabilities in structured populations.

We present a method for inductively determining exact allele frequency spectrum (AFS) probabilities for samples derived from a population comprising two demes under the infinite-allele model of mutation. This method builds on a labeled coalescent argument to extend the Ewens sampling formula (ESF) to structured populations. A key departure from the panmictic case is that the AFS conditioned on the number of alleles in the sample is no longer independent of the scaled mutation rate (θ). In particular, biallelic site frequency spectra, widely-used in explorations of genome-wide patterns of variation, depend on the mutation rate in structured populations. Variation in the rate of substitution across loci and through time may contribute to apparent distortions of site frequency spectra exhibited by samples derived from structured populations.

Evolution of the sex ratio and effective number under gynodioecy and androdioecy.

We address the evolution of effective number of individuals under androdioecy and gynodioecy. We analyze dynamic models of autosomal modifiers of weak effect on sex expression. In our zygote control models, the sex expressed by a zygote depends on its own genotype, while in our maternal control models, it depends on the genotype of its maternal parent. Our analysis unifies full multi-dimensional local stability analysis with the Li-Price equation, which for all its heuristic appeal, describes evolutionary change over a single generation. We define a point in the neighborhood of a fixation state from which a single-generation step indicates the asymptotic behavior of the frequency of a modifier allele initiated at an arbitrary point near the fixation state. A concept of heritability appropriate for the evolutionary modification of sex emerges from the Li-Priceframework. We incorporate our theoretical analysis into our previously-developed Bayesian inference framework to develop a new method for inferring the viability of gonochores (males or females) relative to hermaphrodites. Applying this approach to microsatellite data derived from natural populations of the gynodioecious plant Schiedea salicaria and the androdioecious killifish Kryptolebias marmoratus, we find that while female and hermaphrodite S. salicaria appear to have similar viabilities, male K. marmoratus appear to survive to reproductive age at less than half the rate of hermaphrodites.

A Bayesian Approach to Inferring Rates of Selfing and Locus-Specific Mutation.

We present a Bayesian method for characterizing the mating system of populations reproducing through a mixture of self-fertilization and random outcrossing. Our method uses patterns of genetic variation across the genome as a basis for inference about reproduction under pure hermaphroditism, gynodioecy, and a model developed to describe the self-fertilizing killifish Kryptolebias marmoratus. We extend the standard coalescence model to accommodate these mating systems, accounting explicitly for multilocus identity disequilibrium, inbreeding depression, and variation in fertility among mating types. We incorporate the Ewens sampling formula (ESF) under the infinite-alleles model of mutation to obtain a novel expression for the likelihood of mating system parameters. Our Markov chain Monte Carlo (MCMC) algorithm assigns locus-specific mutation rates, drawn from a common mutation rate distribution that is itself estimated from the data using a Dirichlet process prior model. Our sampler is designed to accommodate additional information, including observations pertaining to the sex ratio, the intensity of inbreeding depression, and other aspects of reproduction. It can provide joint posterior distributions for the population-wide proportion of uniparental individuals, locus-specific mutation rates, and the number of generations since the most recent outcrossing event for each sampled individual. Further, estimation of all basic parameters of a given model permits estimation of functions of those parameters, including the proportion of the gene pool contributed by each sex and relative effective numbers.

Genealogical histories in structured populations.

In genealogies of genes sampled from structured populations, lineages coalesce at rates dependent on the states of the lineages. For migration and coalescence events occurring on comparable time scales, for example, only lineages residing in the same deme of a geographically subdivided population can have descended from a common ancestor in the immediately preceding generation. Here, we explore aspects of genealogical structure in a population comprising two demes, between which migration may occur. We use generating functions to obtain exact densities and moments of coalescence time, number of mutations, total tree length, and age of the most recent common ancestor of the sample. We describe qualitative features of the distribution of gene genealogies, including factors that influence the geographical location of the most recent common ancestor and departures of the distribution of internode lengths from exponential.

Effects of polymorphism for locally adapted genes on rates of neutral introgression in structured populations.

Adaptation to local conditions within demes balanced by migration can maintain polymorphisms for variants that reduce fitness in certain ecological contexts. Here, we address the effects of such polymorphisms on the rate of introgression of neutral marker genes, possibly genetically linked to targets of selection. Barriers to neutral gene flow are expected to increase with linkage to targets of local selection and with differences between demes in the frequencies of locally adapted alleles. This expectation is borne out under purifying and disruptive selection, regimes that promote monomorphism within demes. In contrast, overdominance within demes induces minimal barriers to neutral introgression even in the face of very large differences between demes in the frequencies of locally adapted alleles. Further, segregation distortion, a phenomenon observed in a number of interspecific hybrids, can in fact promote transmission by migrants to future generations at rates exceeding those of residents.

Sex-specific incompatibility generates locus-specific rates of introgression between species.

Disruption of interactions among ensembles of epistatic loci has been shown to contribute to reproductive isolation among various animal and plant species. Under the Bateson-Dobzhansky-Muller model, such interspecific incompatibility arises as a by-product of genetic divergence in each species, and the Orr-Turelli model indicates that the number of loci involved in incompatible interactions may "snowball" over time. We address the combined effect of multiple incompatibility loci on the rate of introgression at neutral marker loci across the genome. Our analysis extends previous work by accommodating sex specificity: differences between the sexes in the expression of incompatibility, in rates of crossing over between neutral markers and incompatibility loci, and in transmission of markers or incompatibility factors. We show that the evolutionary process at neutral markers in a genome subject to incompatibility selection is well approximated by a purely neutral process with migration rates appropriately scaled to reflect the influence of selection targeted to incompatibility factors. We confirm that in the absence of sex specificity and functional epistasis among incompatibility factors, the barrier to introgression induced by multiple incompatibility factors corresponds to the product of the barriers induced by the factors individually. A new finding is that barriers to introgression due to sex-specific incompatibility depart in general from multiplicativity. Our partitioning of variation in relative reproductive rate suggests that such departures derive from associations between sex and incompatibility and between sex and neutral markers. Concordant sex-specific incompatibility (for example, greater impairment of male hybrids or longer map lengths in females) induces lower barriers (higher rates of introgression) than expected under multiplicativity, and discordant sex-specific incompatibility induces higher barriers.

Ancestral population genomics: the coalescent hidden Markov model approach.

With incomplete lineage sorting (ILS), the genealogy of closely related species differs along their genomes. The amount of ILS depends on population parameters such as the ancestral effective population sizes and the recombination rate, but also on the number of generations between speciation events. We use a hidden Markov model parameterized according to coalescent theory to infer the genealogy along a four-species genome alignment of closely related species and estimate population parameters. We analyze a basic, panmictic demographic model and study its properties using an extensive set of coalescent simulations. We assess the effect of the model assumptions and demonstrate that the Markov property provides a good approximation to the ancestral recombination graph. Using a too restricted set of possible genealogies, necessary to reduce the computational load, can bias parameter estimates. We propose a simple correction for this bias and suggest directions for future extensions of the model. We show that the patterns of ILS along a sequence alignment can be recovered efficiently together with the ancestral recombination rate. Finally, we introduce an extension of the basic model that allows for mutation rate heterogeneity and reanalyze human-chimpanzee-gorilla-orangutan alignments, using the new models. We expect that this framework will prove useful for population genomics and provide exciting insights into genome evolution.

Site frequency spectra from genomic SNP surveys.

Genomic survey data now permit an unprecedented level of sensitivity in the detection of departures from canonical evolutionary models, including expansions in population size and selective sweeps. Here, we examine the effects of seemingly subtle differences among sampling distributions on goodness of fit analyses of site frequency spectra constructed from single nucleotide polymorphisms. Conditioning on the observation of exactly two alleles in a random sample results in a site frequency spectrum that is independent of the scaled rate of neutral substitution (theta). Other sampling distributions, including conditioning on a single mutational event in the sample genealogy or randomly selecting a single mutation from a genealogy with multiple mutations, have distinct site frequency spectra that show highly significant departures from the predictions of the biallelic model. Some aspects of data filtering may contribute to significant departures of site frequency spectra from expectation, apart from any violation of the standard neutral model.

Importance sampling for the infinite sites model.

Importance sampling or Markov Chain Monte Carlo sampling is required for state-of-the-art statistical analysis of population genetics data. The applicability of these sampling-based inference techniques depends crucially on the proposal distribution. In this paper, we discuss importance sampling for the infinite sites model. The infinite sites assumption is attractive because it constraints the number of possible genealogies, thereby allowing for the analysis of larger data sets. We recall the Griffiths-Tavaré and Stephens-Donnelly proposals and emphasize the relation between the latter proposal and exact sampling from the infinite alleles model. We also introduce a new proposal that takes knowledge of the ancestral state into account. The new proposal is derived from a new result on exact sampling from a single site. The methods are illustrated on simulated data sets and the data considered in Griffiths and Tavaré (1994).

The evolutionary forest algorithm.

MOTIVATION: Gene genealogies offer a powerful context for inferences about the evolutionary process based on presently segregating DNA variation. In many cases, it is the distribution of population parameters, marginalized over the effectively infinite-dimensional tree space, that is of interest. Our evolutionary forest (EF) algorithm uses Monte Carlo methods to generate posterior distributions of population parameters. A novel feature is the updating of parameter values based on a probability measure defined on an ensemble of histories (a forest of genealogies), rather than a single tree. RESULTS: The EF algorithm generates samples from the correct marginal distribution of population parameters. Applied to actual data from closely related fruit fly species, it rapidly converged to posterior distributions that closely approximated the exact posteriors generated through massive computational effort. Applied to simulated data, it generated credible intervals that covered the actual parameter values in accordance with the nominal probabilities. AVAILABILITY: A C++ implementation of this method is freely accessible at http://www.isds.duke.edu/~scl13

Likelihoods from summary statistics: recent divergence between species.

We describe an importance-sampling method for approximating likelihoods of population parameters based on multiple summary statistics. In this first application, we address the demographic history of closely related members of the Drosophila pseudoobscura group. We base the maximum-likelihood estimation of the time since speciation and the effective population sizes of the extant and ancestral populations on the pattern of nucleotide variation at DPS2002, a noncoding region tightly linked to a paracentric inversion that strongly contributes to reproductive isolation. Consideration of summary statistics rather than entire nucleotide sequences permits a compact description of the genealogy of the sample. We use importance sampling first to propose a genealogical and mutational history consistent with the observed array of summary statistics and then to correct the likelihood with the exact probability of the history determined from a system of recursions. Analysis of a subset of the data, for which recursive computation of the exact likelihood was feasible, indicated close agreement between the approximate and exact likelihoods. Our results for the complete data set also compare well with those obtained through Metropolis-Hastings sampling of fully resolved genealogies of entire nucleotide sequences.

The evolutionary dynamics of self-incompatibility systems.

Self-incompatible flowering plants reject pollen that expresses the same mating specificity as the pistil (female reproductive tract). In most plant families, pollen and pistil mating specificities segregate as a single locus, the S locus. In at least two self-incompatibility systems, distinct pollen and pistil specificity genes are embedded in an extensive nonrecombining tract. To facilitate consideration of how new S locus specificities arise in systems with distinct pollen and pistil genes, we present a graphical model for the generation of hypotheses. It incorporates the evolutionary principle that nonreciprocal siring success (cross-pollinations between two plants produce seeds in only one direction) tends to favor the rejecting partner. This model suggests that selection within S-allele specificity classes could accelerate the rate of nonsynonymous (amino acid-changing) substitutions, with periodic selective sweeps removing segregating variation within classes. Accelerated substitution within specificity classes could also promote the origin of new S-allele specificities.

Evolution under tight linkage to mating type.

Recent large-scale sequencing studies of mating type loci in a number of organisms offer insight into the origin and evolution of these genomic regions. Extensive tracts containing genes with a wide diversity of functions typically cosegregate with mating type. Cases in which mating type determination entails complementarity between distinct transcription units may descend from systems in which close physical linkage facilitated the coordinated expression and cosegregation of the interacting genes. In response to the particular selection pressures associated with the maintenance of more than one mating type, this nucleus of low recombination may expand over evolutionary time, engulfing neighboring tracts bearing genes with no direct role in reproduction. This scenario is consistent with the present-day structure of some mating type loci, including regulators of homomorphic self-incompatibility in angiosperms (S-loci). Recombination suppression and enforced S-locus heterozygosity accelerate the accumulation of genetic load and promote genetic associations between S-alleles and degenerating genes in cosegregating tracts. This S-allele-specific load may influence the evolution of self-incompatibility systems.

On the evolutionary costs of self-incompatibility: incomplete reproductive compensation due to pollen limitation.

Pollen limitation affects plants with diverse reproductive systems and ecologies. In self-incompatible (SI) species, pollen limitation may preclude full reproductive compensation for prezygotic rejection of pollen. We present a model designed to explore the effects of incomplete reproductive compensation on evolutionary changes at a modifier locus that regulates the level of SI expression. Our results indicate that incomplete reproductive compensation greatly increases the evolutionary costs of SI, particularly in populations with low S-allele diversity. The evolutionary fate of modifiers of SI expression depends on the rate at which they are transmitted to future generations as well as the effects of SI on offspring number and quality. Partial SI expression can represent a stable condition rather than an evolutionarily transient state between full expression and full suppression. This unanticipated result provides the first theoretical support for the evolutionary stability of such mixed mating systems, the existence of which has recently been documented.

Maximum-likelihood estimation of rates of recombination within mating-type regions.

Features common to many mating-type regions include recombination suppression over large genomic tracts and cosegregation of genes of various functions, not necessarily related to reproduction. Model systems for homomorphic self-incompatibility (SI) in flowering plants share these characteristics. We introduce a method for the exact computation of the joint probability of numbers of neutral mutations segregating at the determinant of mating type and at a linked marker locus. The underlying Markov model incorporates strong balancing selection into a two-locus coalescent. We apply the method to obtain a maximum-likelihood estimate of the rate of recombination between a marker locus, 48A, and S-RNase, the determinant of SI specificity in pistils of Nicotiana alata. Even though the sampled haplotypes show complete allelic linkage disequilibrium and recombinants have never been detected, a highly significant deficiency of synonymous substitutions at 48A compared to S-RNase suggests a history of recombination. Our maximum-likelihood estimate indicates a rate of recombination of perhaps 3 orders of magnitude greater than the rate of synonymous mutation. This approach may facilitate the construction of genetic maps of regions tightly linked to targets of strong balancing selection.

A simple method for computing exact probabilities of mutation numbers.

We describe a method for the recursive computation of exact probability distributions for the number of neutral mutations segregating in samples of arbitrary size and configuration. Construction of the recursions requires only characterization of evolutionary changes as a Markov process and determination of one-step transition matrices. We address the pattern of nucleotide diversity at a neutral marker locus linked to a determinant of mating type. Under a reformulation of parameters, the method also applies directly to metapopulation models with island migration among demes. Characterization of complete probability distributions facilitates parameter estimation and hypothesis testing by likelihood- as well as moment-based approaches.

Patterns of variation within self-incompatibility loci.

Diverse self-incompatibility (SI) mechanisms permit flowering plants to inhibit fertilization by pollen that express specificities in common with the pistil. Characteristic of at least two model systems is greatly reduced recombination across large genomic tracts surrounding the S-locus, which regulates SI. In three angiosperm families, including the Solanaceae, the gene that controls the expression of gametophytic SI in the pistil encodes a ribonuclease (S-RNase). The gene that controls pollen SI expression is currently unknown, although several candidates have recently been proposed. Although each candidate shows a high level of polymorphism and complete allelic disequilibrium with the S-RNase gene, such properties may merely reflect tight linkage to the S-locus, irrespective of any functional role in SI. We analyzed the magnitude and nature of nucleotide variation, with the objective of distinguishing likely candidates for regulators of SI from other genes embedded in the S-locus region. We studied the S-RNase gene of the Solanaceae and 48A, a candidate for the pollen gene in this system, and we also conducted a parallel analysis of the regulators of sporophytic SI in Brassica, a system in which both the pistil and pollen genes are known. Although the pattern of variation shown by the pollen gene of the Brassica system is consistent with its role as a determinant of pollen specificity, that of 48A departs from expectation. Our analysis further suggests that recombination between 48A and S-RNase may have occurred during the interval spanned by the gene genealogy, another indication that 48A may not regulate SI expression in pollen.