Latent mutations in the ancestries of alleles under selection.

We consider a single genetic locus with two alleles A1 and A2 in a large haploid population. The locus is subject to selection and two-way, or recurrent, mutation. Assuming the allele frequencies follow a Wright-Fisher diffusion and have reached stationarity, we describe the asymptotic behaviors of the conditional gene genealogy and the latent mutations of a sample with known allele counts, when the count n1 of allele A1 is fixed, and when either or both the sample size n and the selection strength |α| tend to infinity. Our study extends previous work under neutrality to the case of non-neutral rare alleles, asserting that when selection is not too strong relative to the sample size, even if it is strongly positive or strongly negative in the usual sense (α→-∞ or α→+∞), the number of latent mutations of the n1 copies of allele A1 follows the same distribution as the number of alleles in the Ewens sampling formula. On the other hand, very strong positive selection relative to the sample size leads to neutral gene genealogies with a single ancient latent mutation. We also demonstrate robustness of our asymptotic results against changing population sizes, when one of |α| or n is large.

The grapheme-valued Wright-Fisher diffusion with mutation.

In Athreya et al. (2021), models from population genetics were used to define stochastic dynamics in the space of graphons arising as continuum limits of dense graphs. In the present paper we exhibit an example of a simple neutral population genetics model for which this dynamics is a Markovian diffusion that can be characterized as the solution of a martingale problem. In particular, we consider a Markov chain in the space of finite graphs that resembles a Moran model with resampling and mutation. We encode the finite graphs as graphemes, which can be represented as a triple consisting of a vertex set (or more generally, a topological space), an adjacency matrix, and a sampling (Borel) measure. We equip the space of graphons with convergence of sample subgraph densities and show that the grapheme-valued Markov chain converges to a grapheme-valued diffusion as the number of vertices goes to infinity. We show that the grapheme-valued diffusion has a stationary distribution that is linked to the Griffiths-Engen-McCloskey (GEM) distribution. In a companion paper (Greven et al. 2023), we build up a general theory for obtaining grapheme-valued diffusions via genealogies of models in population genetics.

Another view of sequential sampling in the birth process with immigration.

We explore properties of the family sizes arising in a linear birth process with immigration (BI). In particular, we study the correlation of the number of families observed during consecutive disjoint intervals of time. Letting S(a, b) be the number of families observed in (a, b), we study the expected sample variance and its asymptotics for p consecutive sequential samples [Formula: see text], for [Formula: see text]. By conditioning on the sizes of the samples, we provide a connection between [Formula: see text] and p sequential samples of sizes [Formula: see text], drawn from a single run of a Chinese Restaurant Process. Properties of the latter were studied in da Silva et al. (Bernoulli 29:1166-1194, 2023. https://doi.org/10.3150/22-BEJ1494 ). We show how the continuous-time framework helps to make asymptotic calculations easier than its discrete-time counterpart. As an application, for a specific choice of [Formula: see text], where the lengths of intervals are logarithmically equal, we revisit Fisher's 1943 multi-sampling problem and give another explanation of what Fisher's model could have meant in the world of sequential samples drawn from a BI process.

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.

Ancestral inference from haplotypes and mutations.

We consider inference about the history of a sample of DNA sequences, conditional upon the haplotype counts and the number of segregating sites observed at the present time. After deriving some theoretical results in the coalescent setting, we implement rejection sampling and importance sampling schemes to perform the inference. The importance sampling scheme addresses an extension of the Ewens Sampling Formula for a configuration of haplotypes and the number of segregating sites in the sample. The implementations include both constant and variable population size models. The methods are illustrated by two human Y chromosome datasets.

Simulating the component counts of combinatorial structures.

This article describes and compares methods for simulating the component counts of random logarithmic combinatorial structures such as permutations and mappings. We exploit the Feller coupling for simulating permutations to provide a very fast method for simulating logarithmic assemblies more generally. For logarithmic multisets and selections, this approach is replaced by an acceptance/rejection method based on a particular conditioning relationship that represents the distribution of the combinatorial structure as that of independent random variables conditioned on a weighted sum. We show how to improve its acceptance rate. We illustrate the method by estimating the probability that a random mapping has no repeated component sizes, and establish the asymptotic distribution of the difference between the number of components and the number of distinct component sizes for a very general class of logarithmic structures.

Recurrent mutation in the ancestry of a rare variant.

Recurrent mutation produces multiple copies of the same allele which may be co-segregating in a population. Yet, most analyses of allele-frequency or site-frequency spectra assume that all observed copies of an allele trace back to a single mutation. We develop a sampling theory for the number of latent mutations in the ancestry of a rare variant, specifically a variant observed in relatively small count in a large sample. Our results follow from the statistical independence of low-count mutations, which we show to hold for the standard neutral coalescent or diffusion model of population genetics as well as for more general coalescent trees. For populations of constant size, these counts are distributed like the number of alleles in the Ewens sampling formula. We develop a Poisson sampling model for populations of varying size and illustrate it using new results for site-frequency spectra in an exponentially growing population. We apply our model to a large data set of human SNPs and use it to explain dramatic differences in site-frequency spectra across the range of mutation rates in the human genome.

Inferring the Allelic Series at QTL in Multiparental Populations.

Multiparental populations (MPPs) are experimental populations in which the genome of every individual is a mosaic of known founder haplotypes. These populations are useful for detecting quantitative trait loci (QTL) because tests of association can leverage inferred founder haplotype descent. It is difficult, however, to determine how haplotypes at a locus group into distinct functional alleles, termed the allelic series. The allelic series is important because it provides information about the number of causal variants at a QTL and their combined effects. In this study, we introduce a fully Bayesian model selection framework for inferring the allelic series. This framework accounts for sources of uncertainty found in typical MPPs, including the number and composition of functional alleles. Our prior distribution for the allelic series is based on the Chinese restaurant process, a relative of the Dirichlet process, and we leverage its connection to the coalescent to introduce additional prior information about haplotype relatedness via a phylogenetic tree. We evaluate our approach via simulation and apply it to QTL from two MPPs: the Collaborative Cross (CC) and the Drosophila Synthetic Population Resource (DSPR). We find that, although posterior inference of the exact allelic series is often uncertain, we are able to distinguish biallelic QTL from more complex multiallelic cases. Additionally, our allele-based approach improves haplotype effect estimation when the true number of functional alleles is small. Our method, Tree-Based Inference of Multiallelism via Bayesian Regression (TIMBR), provides new insight into the genetic architecture of QTL in MPPs.

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.