Covariance of pairwise differences on a multi-species coalescent tree and implications forF_ST

Abstract

The multi-species coalescent (MSC) provides a theoretical foundation for modern phylogenetics and comparative population genetics. Its theoretical properties have been heavily studied but there are still aspects of the MSC that are largely unknown, including the covariances in pairwise coalescence times, which are fundamental for understanding the properties of statistics that combine data from multiple species, such as the fixation index ( F ST ). The major contribution of this study is the derivation and implementation of exact expressions for the covariances of pairwise coalescence times under phylogenetic models with piecewise constant changes in population size, assuming no gene flow after species divergence. We use these expressions to derive the variance in average pairwise differences within and between populations. We then derive approximations for the expectation and bias of a sequence-based estimator of F ST , a commonly used genetic measurement of population differentiation, when it is applied to a non-recombining region of the genome. We show that the estimator of F ST is generally biased downward. A freely available software package is provided, STCov, to calculate the mean, variances and covariances in coalescence times presented here under user-defined piecewise-constant species trees. This article is part of the theme issue ‘Celebrating 50 years since Lewontin's apportionment of human diversity’.

Celebrating 50 years since Lewontin's apportionment of human diversity

“…A second important approach to F ST considers the behaviour of F ST estimators under specific evolutionary models. Guerra & Nielsen [12] perform a study in this tradition. After arriving at expressions for the covariance of pairwise coalescence times under a general framework, they consider the behaviour of estimators of Slatkin's formulation of F ST [13], which casts F ST in terms of a ratio involving coalescence times for pairs of lineages randomly drawn from the same subpopulation versus pairs of lineages randomly drawn from the total population.…”