We present an implementation of a recently introduced method for estimating the allele-frequency spectrum under the diffusion approximation. For single-nucleotide polymorphism (SNP) frequency data from multiple populations, the method computes numerical solutions to the allele-frequency spectrum (AFS) under a complex model that includes population splitting events, migration, population expansion, and admixture. The solution to the diffusion partial differential equation (PDE) that mimics the evolutionary process is found by means of truncated polynomial expansions. In the absence of gene flow, our computation of frequency spectra yields exact results. The results are compared to those that use a finite-difference method and to forward diffusion simulations. In general, all the methods yield comparable results, although the polynomial-based approach is the most accurate in the weak-migration limit. Also, the economical use of memory attained by the polynomial expansions makes the study of models with four populations possible for the first time. The method was applied to a four-population model of the human expansion out of Africa and the peopling of the Americas, using the Environmental Genome Project (EGP) SNP database. Although our confidence intervals largely overlapped previous analyses of these data, some were significantly different. In particular, estimates of migration among African, European, and Asian populations were considerably lower than those in a previous study and the estimated time of migration out of Africa was earlier. The estimated time of founding of a human population outside of Africa was 52,000 years (95% confidence interval: 36,000-80,800 years).

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