Coalescent theory of the ψ directionality index

The ψ directionality index was introduced by Peter and Slatkin (Evolution 67: 3274-3289, 2013) to infer the direction of range expansions from single-nucleotide polymorphism variation. Computed from the joint site frequency spectrum for two populations, ψ uses shared genetic variants to measure the difference in the amount of genetic drift experienced by the populations, associating excess drift with greater distance from the origin of the range expansion. Although ψ has been successfully applied in natural populations, its statistical properties have not been well understood. In this article, we define Ψ as a random variable originating from a coalescent process in a two-population demography. For samples consisting of a pair of diploid genomes, one from each of two populations, we derive expressions for moments E[Ψk] for standard parameterizations of bottlenecks during a founder event. For the expectation E[Ψ], we identify parameter combinations that represent distinct demographic scenarios yet yield the same value of E[Ψ]. We also show that the variance V[Ψ] increases with the time since the bottleneck and bottleneck severity, but does not depend on the size of the ancestral population; the ancestral population size affects ψ computed from many biallelic loci only through its contribution to the total number of loci available for the computation. Finally, we analyze the values of E[Ψ] computed from existing demographic models of Drosophila melanogaster and compare them with empirically computed ψ. Our work builds the foundation for theoretical treatments of the ψ index and can help in evaluating its behavior in empirical applications.