Inferring the Joint Demographic History of Multiple Populations: Beyond the Diffusion Approximation

Abstract

Understanding variation in allele frequencies across populations is a central goal of population genetics. Classical models for the distribution of allele frequencies, using forward simulation, coalescent theory, or the diffusion approximation, have been applied extensively for demographic inference, medical study design, and evolutionary studies. Here we propose a tractable model of ordinary differential equations for the evolution of allele frequencies that is closely related to the diffusion approximation but avoids many of its limitations and approximations. We show that the approach is typically faster, more numerically stable, and more easily generalizable than the state-of-the-art software implementation of the diffusion approximation. We present a number of applications to human sequence data, including demographic inference with a five-population joint frequency spectrum and a discussion of the robustness of the out-of-Africa model inference to the choice of modern population.

Keywords: demographic inference; diffusion approximation; joint allele frequency spectrum; moments equations.

Figure B1

Illustration of two types of multiple coalescence. Dots represent individuals, arrows represent shared descendance.

Figure 1

Moments predictions for finite-genome frequency spectrum for a population of constant size $N=10,000$ with reversible mutation rates $u=v=2\times {10}^{-6}$ (●), compared to theoretical expectations obtained by integrating equations 5.70 and 5.72 in Ewens (2004) (solid lines).

Figure 2

Inferred ancestral population sizes (width of blue lines) and migration rates (width of black arrow) for populations of African (YRI), Asian (CHB), and European (CEU) ancestry.

Figure 3

Inferred ancestral population sizes (width of blue lines) and migration rates (width of black arrow) for populations of African (YRI), Asian (CHB and JPT), and European (CEU) ancestry.