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.

Bursts of coalescence within population pedigrees whenever big families occur.

We consider a simple diploid population-genetic model with potentially high variability of offspring numbers among individuals. Specifically, against a backdrop of Wright-Fisher reproduction and no selection, there is an additional probability that a big family occurs, meaning that a pair of individuals has a number of offspring on the order of the population size. We study how the pedigree of the population generated under this model affects the ancestral genetic process of a sample of size two at a single autosomal locus without recombination. Our population model is of the type for which multiple-merger coalescent processes have been described. We prove that the conditional distribution of the pairwise coalescence time given the random pedigree converges to a limit law as the population size tends to infinity. This limit law may or may not be the usual exponential distribution of the Kingman coalescent, depending on the frequency of big families. But because it includes the number and times of big families, it differs from the usual multiple-merger coalescent models. The usual multiple-merger coalescent models are seen as describing the ancestral process marginal to, or averaging over, the pedigree. In the limiting ancestral process conditional on the pedigree, the intervals between big families can be modeled using the Kingman coalescent but each big family causes a discrete jump in the probability of coalescence. Analogous results should hold for larger samples and other population models. We illustrate these results with simulations and additional analysis, highlighting their implications for inference and understanding of multilocus data.

TRAILS: Tree reconstruction of ancestry using incomplete lineage sorting.

Genome-wide genealogies of multiple species carry detailed information about demographic and selection processes on individual branches of the phylogeny. Here, we introduce TRAILS, a hidden Markov model that accurately infers time-resolved population genetics parameters, such as ancestral effective population sizes and speciation times, for ancestral branches using a multi-species alignment of three species and an outgroup. TRAILS leverages the information contained in incomplete lineage sorting fragments by modelling genealogies along the genome as rooted three-leaved trees, each with a topology and two coalescent events happening in discretized time intervals within the phylogeny. Posterior decoding of the hidden Markov model can be used to infer the ancestral recombination graph for the alignment and details on demographic changes within a branch. Since TRAILS performs posterior decoding at the base-pair level, genome-wide scans based on the posterior probabilities can be devised to detect deviations from neutrality. Using TRAILS on a human-chimp-gorilla-orangutan alignment, we recover speciation parameters and extract information about the topology and coalescent times at high resolution.

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.

Evaluating the structure-coefficient theorem of evolutionary game theory.

In order to accommodate the empirical fact that population structures are rarely simple, modern studies of evolutionary dynamics allow for complicated and highly heterogeneous spatial structures. As a result, one of the most difficult obstacles lies in making analytical deductions, either qualitative or quantitative, about the long-term outcomes of evolution. The "structure-coefficient" theorem is a well-known approach to this problem for mutation-selection processes under weak selection, but a general method of evaluating the terms it comprises is lacking. Here, we provide such a method for populations of fixed (but arbitrary) size and structure, using easily interpretable demographic measures. This method encompasses a large family of evolutionary update mechanisms and extends the theorem to allow for asymmetric contests to provide a better understanding of the mutation-selection balance under more realistic circumstances. We apply the method to study social goods produced and distributed among individuals in spatially heterogeneous populations, where asymmetric interactions emerge naturally and the outcome of selection varies dramatically, depending on the nature of the social good, the spatial topology, and the frequency with which mutations arise.

Assessing the performance of qpAdm: a statistical tool for studying population admixture.

qpAdm is a statistical tool for studying the ancestry of populations with histories that involve admixture between two or more source populations. Using qpAdm, it is possible to identify plausible models of admixture that fit the population history of a group of interest and to calculate the relative proportion of ancestry that can be ascribed to each source population in the model. Although qpAdm is widely used in studies of population history of human (and nonhuman) groups, relatively little has been done to assess its performance. We performed a simulation study to assess the behavior of qpAdm under various scenarios in order to identify areas of potential weakness and establish recommended best practices for use. We find that qpAdm is a robust tool that yields accurate results in many cases, including when data coverage is low, there are high rates of missing data or ancient DNA damage, or when diploid calls cannot be made. However, we caution against co-analyzing ancient and present-day data, the inclusion of an extremely large number of reference populations in a single model, and analyzing population histories involving extended periods of gene flow. We provide a user guide suggesting best practices for the use of qpAdm.

The consequences of switching strategies in a two-player iterated survival game.

We consider two-player iterated survival games in which players are able to switch from a more cooperative behavior to a less cooperative one at some step of an n-step game. Payoffs are survival probabilities and lone individuals have to finish the game on their own. We explore the potential of these games to support cooperation, focusing on the case in which each single step is a Prisoner's Dilemma. We find that incentives for or against cooperation depend on the number of defections at the end of the game, as opposed to the number of steps in the game. Broadly, cooperation is supported when the survival prospects of lone individuals are relatively bleak. Specifically, we find three critical values or cutoffs for the loner survival probability which, in concert with other survival parameters, determine the incentives for or against cooperation. One cutoff determines the existence of an optimal number of defections against a fully cooperative partner, one determines whether additional defections eventually become disfavored as the number of defections by the partner increases, and one determines whether additional cooperations eventually become favored as the number of defections by the partner increases. We obtain expressions for these switch-points and for optimal numbers of defections against partners with various strategies. These typically involve small numbers of defections even in very long games. We show that potentially long stretches of equilibria may exist, in which there is no incentive to defect more or cooperate more. We describe how individuals find equilibria in best-response walks among n-step strategies.

Genomic architecture and introgression shape a butterfly radiation.

We used 20 de novo genome assemblies to probe the speciation history and architecture of gene flow in rapidly radiating Heliconius butterflies. Our tests to distinguish incomplete lineage sorting from introgression indicate that gene flow has obscured several ancient phylogenetic relationships in this group over large swathes of the genome. Introgressed loci are underrepresented in low-recombination and gene-rich regions, consistent with the purging of foreign alleles more tightly linked to incompatibility loci. Here, we identify a hitherto unknown inversion that traps a color pattern switch locus. We infer that this inversion was transferred between lineages by introgression and is convergent with a similar rearrangement in another part of the genus. These multiple de novo genome sequences enable improved understanding of the importance of introgression and selective processes in adaptive radiation.

Bayesian Estimation of Population Size Changes by Sampling Tajima's Trees.

The large state space of gene genealogies is a major hurdle for inference methods based on Kingman's coalescent. Here, we present a new Bayesian approach for inferring past population sizes, which relies on a lower-resolution coalescent process that we refer to as "Tajima's coalescent." Tajima's coalescent has a drastically smaller state space, and hence it is a computationally more efficient model, than the standard Kingman coalescent. We provide a new algorithm for efficient and exact likelihood calculations for data without recombination, which exploits a directed acyclic graph and a correspondingly tailored Markov Chain Monte Carlo method. We compare the performance of our Bayesian Estimation of population size changes by Sampling Tajima's Trees (BESTT) with a popular implementation of coalescent-based inference in BEAST using simulated and human data. We empirically demonstrate that BESTT can accurately infer effective population sizes, and it further provides an efficient alternative to the Kingman's coalescent. The algorithms described here are implemented in the R package phylodyn, which is available for download at https://github.com/JuliaPalacios/phylodyn.

Some topics in theoretical population genetics: Editorial commentaries on a selection of Marc Feldman's TPB papers.

This article consists of commentaries on a selected group of papers of Marc Feldman published in Theoretical Population Biology from 1970 to the present. The papers describe a diverse set of population-genetic models, covering topics such as cultural evolution, social evolution, and the evolution of recombination. The commentaries highlight Marc Feldman's role in providing mathematically rigorous formulations to explore qualitative hypotheses, in many cases generating surprising conclusions.

A two-player iterated survival game.

We describe an iterated game between two players, in which the payoff is to survive a number of steps. Expected payoffs are probabilities of survival. A key feature of the game is that individuals have to survive on their own if their partner dies. We consider individuals with hardwired, unconditional behaviors or strategies. When both players are present, each step is a symmetric two-player game. The overall survival of the two individuals forms a Markov chain. As the number of iterations tends to infinity, all probabilities of survival decrease to zero. We obtain general, analytical results for n-step payoffs and use these to describe how the game changes as n increases. In order to predict changes in the frequency of a cooperative strategy over time, we embed the survival game in three different models of a large, well-mixed population. Two of these models are deterministic and one is stochastic. Offspring receive their parent's type without modification and fitnesses are determined by the game. Increasing the number of iterations changes the prospects for cooperation. All models become neutral in the limit (n→∞). Further, if pairs of cooperative individuals survive together with high probability, specifically higher than for any other pair and for either type when it is alone, then cooperation becomes favored if the number of iterations is large enough. This holds regardless of the structure of pairwise interactions in a single step. Even if the single-step interaction is a Prisoner's Dilemma, the cooperative type becomes favored. Enhanced survival is crucial in these iterated evolutionary games: if players in pairs start the game with a fitness deficit relative to lone individuals, the prospects for cooperation can become even worse than in the case of a single-step game.

Public goods games in populations with fluctuating size.

Many mathematical frameworks of evolutionary game dynamics assume that the total population size is constant and that selection affects only the relative frequency of strategies. Here, we consider evolutionary game dynamics in an extended Wright-Fisher process with variable population size. In such a scenario, it is possible that the entire population becomes extinct. Survival of the population may depend on which strategy prevails in the game dynamics. Studying cooperative dilemmas, it is a natural feature of such a model that cooperators enable survival, while defectors drive extinction. Although defectors are favored for any mixed population, random drift could lead to their elimination and the resulting pure-cooperator population could survive. On the other hand, if the defectors remain, then the population will quickly go extinct because the frequency of cooperators steadily declines and defectors alone cannot survive. In a mutation-selection model, we find that (i) a steady supply of cooperators can enable long-term population survival, provided selection is sufficiently strong, and (ii) selection can increase the abundance of cooperators but reduce their relative frequency. Thus, evolutionary game dynamics in populations with variable size generate a multifaceted notion of what constitutes a trait's long-term success.

A non-zero variance of Tajima's estimator for two sequences even for infinitely many unlinked loci.

The population-scaled mutation rate, θ, is informative on the effective population size and is thus widely used in population genetics. We show that for two sequences and n unlinked loci, the variance of Tajima's estimator (θˆ), which is the average number of pairwise differences, does not vanish even as n→∞. The non-zero variance of θˆ results from a (weak) correlation between coalescence times even at unlinked loci, which, in turn, is due to the underlying fixed pedigree shared by gene genealogies at all loci. We derive the correlation coefficient under a diploid, discrete-time, Wright-Fisher model, and we also derive a simple, closed-form lower bound. We also obtain empirical estimates of the correlation of coalescence times under demographic models inspired by large-scale human genealogies. While the effect we describe is small (Varθˆ∕θ2≈ONe-1), it is important to recognize this feature of statistical population genetics, which runs counter to commonly held notions about unlinked loci.

Population structure and coalescence in pedigrees: Comparisons to the structured coalescent and a framework for inference.

Contrary to what is often assumed in population genetics, independently segregating loci do not have completely independent ancestries, since all loci are inherited through a single, shared population pedigree. Previous work has shown that the non-independence between gene genealogies of independently segregating loci created by the population pedigree is weak in panmictic populations, and predictions made from standard coalescent theory are accurate for populations that are at least moderately sized. Here, we investigate patterns of coalescence in pedigrees of structured populations. We find that the pedigree creates deviations away from the predictions of the structured coalescent that persist on a longer timescale than in the case of panmictic populations. Nevertheless, we find that the structured coalescent provides a reasonable approximation for the coalescent process in structured population pedigrees so long as migration events are moderately frequent and there are no migration events in the recent pedigree of the sample. When there are migration events in the recent sample pedigree, we find that distributions of coalescence in the sample can be modeled as a mixture of distributions from different initial sample configurations. We use this observation to motivate a maximum-likelihood approach for inferring migration rates and mutation rates jointly with features of the pedigree such as recent migrant ancestry and recent relatedness. Using simulation, we show that our inference framework accurately recovers long-term migration rates in the presence of recent migration events in the sample pedigree.

Empirical Bayes Estimation of Coalescence Times from Nucleotide Sequence Data.

We demonstrate the advantages of using information at many unlinked loci to better calibrate estimates of the time to the most recent common ancestor (TMRCA) at a given locus. To this end, we apply a simple empirical Bayes method to estimate the TMRCA. This method is both asymptotically optimal, in the sense that the estimator converges to the true value when the number of unlinked loci for which we have information is large, and has the advantage of not making any assumptions about demographic history. The algorithm works as follows: we first split the sample at each locus into inferred left and right clades to obtain many estimates of the TMRCA, which we can average to obtain an initial estimate of the TMRCA. We then use nucleotide sequence data from other unlinked loci to form an empirical distribution that we can use to improve this initial estimate.

Effects of the population pedigree on genetic signatures of historical demographic events.

Genetic variation among loci in the genomes of diploid biparental organisms is the result of mutation and genetic transmission through the genealogy, or population pedigree, of the species. We explore the consequences of this for patterns of variation at unlinked loci for two kinds of demographic events: the occurrence of a very large family or a strong selective sweep that occurred in the recent past. The results indicate that only rather extreme versions of such events can be expected to structure population pedigrees in such a way that unlinked loci will show deviations from the standard predictions of population genetics, which average over population pedigrees. The results also suggest that large samples of individuals and loci increase the chance of picking up signatures of these events, and that very large families may have a unique signature in terms of sample distributions of mutant alleles.

Leveraging Distant Relatedness to Quantify Human Mutation and Gene-Conversion Rates.

The rate at which human genomes mutate is a central biological parameter that has many implications for our ability to understand demographic and evolutionary phenomena. We present a method for inferring mutation and gene-conversion rates by using the number of sequence differences observed in identical-by-descent (IBD) segments together with a reconstructed model of recent population-size history. This approach is robust to, and can quantify, the presence of substantial genotyping error, as validated in coalescent simulations. We applied the method to 498 trio-phased sequenced Dutch individuals and inferred a point mutation rate of 1.66 × 10(-8) per base per generation and a rate of 1.26 × 10(-9) for <20 bp indels. By quantifying how estimates varied as a function of allele frequency, we inferred the probability that a site is involved in non-crossover gene conversion as 5.99 × 10(-6). We found that recombination does not have observable mutagenic effects after gene conversion is accounted for and that local gene-conversion rates reflect recombination rates. We detected a strong enrichment of recent deleterious variation among mismatching variants found within IBD regions and observed summary statistics of local sharing of IBD segments to closely match previously proposed metrics of background selection; however, we found no significant effects of selection on our mutation-rate estimates. We detected no evidence of strong variation of mutation rates in a number of genomic annotations obtained from several recent studies. Our analysis suggests that a mutation-rate estimate higher than that reported by recent pedigree-based studies should be adopted in the context of DNA-based demographic reconstruction.

Bayesian Nonparametric Inference of Population Size Changes from Sequential Genealogies.

Sophisticated inferential tools coupled with the coalescent model have recently emerged for estimating past population sizes from genomic data. Recent methods that model recombination require small sample sizes, make constraining assumptions about population size changes, and do not report measures of uncertainty for estimates. Here, we develop a Gaussian process-based Bayesian nonparametric method coupled with a sequentially Markov coalescent model that allows accurate inference of population sizes over time from a set of genealogies. In contrast to current methods, our approach considers a broad class of recombination events, including those that do not change local genealogies. We show that our method outperforms recent likelihood-based methods that rely on discretization of the parameter space. We illustrate the application of our method to multiple demographic histories, including population bottlenecks and exponential growth. In simulation, our Bayesian approach produces point estimates four times more accurate than maximum-likelihood estimation (based on the sum of absolute differences between the truth and the estimated values). Further, our method's credible intervals for population size as a function of time cover 90% of true values across multiple demographic scenarios, enabling formal hypothesis testing about population size differences over time. Using genealogies estimated with ARGweaver, we apply our method to European and Yoruban samples from the 1000 Genomes Project and confirm key known aspects of population size history over the past 150,000 years.