Bayesian inference of local trees along chromosomes by the sequential Markov coalescent

Inferring Past Effective Population Size from Distributions of Coalescent Times

Inferring and understanding changes in effective population size over time is a major challenge for population genetics. Here we investigate some theoretical properties of random-mating populations with varying size over time. In particular, we present an exact solution to compute the population size as a function of time, [Formula: see text], based on distributions of coalescent times of samples of any size. This result reduces the problem of population size inference to a problem of estimating coalescent time distributions. To illustrate the analytic results, we design a heuristic method using a tree-inference algorithm and investigate simulated and empirical population-genetic data. We investigate the effects of a range of conditions associated with empirical data, for instance number of loci, sample size, mutation rate, and cryptic recombination. We show that our approach performs well with genomic data (≥ 10,000 loci) and that increasing the sample size from 2 to 10 greatly improves the inference of [Formula: see text] whereas further increase in sample size results in modest improvements, even under a scenario of exponential growth. We also investigate the impact of recombination and characterize the potential biases in inference of [Formula: see text] The approach can handle large sample sizes and the computations are fast. We apply our method to human genomes from four populations and reconstruct population size profiles that are coherent with previous finds, including the Out-of-Africa bottleneck. Additionally, we uncover a potential difference in population size between African and non-African populations as early as 400 KYA. In summary, we provide an analytic relationship between distributions of coalescent times and [Formula: see text], which can be incorporated into powerful approaches for inferring past population sizes from population-genomic data.

Resolving Evolutionary Relationships in Closely Related Species with Whole-Genome Sequencing Data

Using genetic data to resolve the evolutionary relationships of species is of major interest in evolutionary and systematic biology. However, reconstructing the sequence of speciation events, the so-called species tree, in closely related and potentially hybridizing species is very challenging. Processes such as incomplete lineage sorting and interspecific gene flow result in local gene genealogies that differ in their topology from the species tree, and analyses of few loci with a single sequence per species are likely to produce conflicting or even misleading results. To study these phenomena on a full phylogenomic scale, we use whole-genome sequence data from 200 individuals of four black-and-white flycatcher species with so far unresolved phylogenetic relationships to infer gene tree topologies and visualize genome-wide patterns of gene tree incongruence. Using phylogenetic analysis in nonoverlapping 10-kb windows, we show that gene tree topologies are extremely diverse and change on a very small physical scale. Moreover, we find strong evidence for gene flow among flycatcher species, with distinct patterns of reduced introgression on the Z chromosome. To resolve species relationships on the background of widespread gene tree incongruence, we used four complementary coalescent-based methods for species tree reconstruction, including complex modeling approaches that incorporate post-divergence gene flow among species. This allowed us to infer the most likely species tree with high confidence. Based on this finding, we show that regions of reduced effective population size, which have been suggested as particularly useful for species tree inference, can produce positively misleading species tree topologies. Our findings disclose the pitfalls of using loci potentially under selection as phylogenetic markers and highlight the potential of modeling approaches to disentangle species relationships in systems with large effective population sizes and post-divergence gene flow.

Assessing Differences Between Ancestral Recombination Graphs

Ancestral recombination graphs (ARGs) represent the history of portions of a genome with recombination. Attempts to infer ARGs have been hampered by the lack of an ARG comparison metric which could be used to measure how well inference succeeded. We propose a simple ARG comparison framework based on averaging standard tree comparison measures across either all sites or variable sites only. Using simulated data, we show that this framework, instantiated with an appropriate tree comparison measure, can distinguish better from worse inferences of an ARG.

The SMC' is a highly accurate approximation to the ancestral recombination graph

Two sequentially Markov coalescent models (SMC and SMC') are available as tractable approximations to the ancestral recombination graph (ARG). We present a Markov process describing coalescence at two fixed points along a pair of sequences evolving under the SMC'. Using our Markov process, we derive a number of new quantities related to the pairwise SMC', thereby analytically quantifying for the first time the similarity between the SMC' and the ARG. We use our process to show that the joint distribution of pairwise coalescence times at recombination sites under the SMC' is the same as it is marginally under the ARG, which demonstrates that the SMC' is, in a particular well-defined, intuitive sense, the most appropriate first-order sequentially Markov approximation to the ARG. Finally, we use these results to show that population size estimates under the pairwise SMC are asymptotically biased, while under the pairwise SMC' they are approximately asymptotically unbiased.

A renewal theory approach to IBD sharing

A long genomic segment inherited by a pair of individuals from a single, recent common ancestor is said to be identical-by-descent (IBD). Shared IBD segments have numerous applications in genetics, from demographic inference to phasing, imputation, pedigree reconstruction, and disease mapping. Here, we provide a theoretical analysis of IBD sharing under Markovian approximations of the coalescent with recombination. We describe a general framework for the IBD process along the chromosome under the Markovian models (SMC/SMC'), as well as introduce and justify a new model, which we term the renewal approximation, under which lengths of successive segments are independent. Then, considering the infinite-chromosome limit of the IBD process, we recover previous results (for SMC) and derive new results (for SMC') for the mean number of shared segments longer than a cutoff and the fraction of the chromosome found in such segments. We then use renewal theory to derive an expression (in Laplace space) for the distribution of the number of shared segments and demonstrate implications for demographic inference. We also compute (again, in Laplace space) the distribution of the fraction of the chromosome in shared segments, from which we obtain explicit expressions for the first two moments. Finally, we generalize all results to populations with a variable effective size.