Isolation with migration models for more than two populations

Abstract

A method for studying the divergence of multiple closely related populations is described and assessed. The approach of Hey and Nielsen (2007, Integration within the Felsenstein equation for improved Markov chain Monte Carlo methods in population genetics. Proc Natl Acad Sci USA. 104:2785-2790) for fitting an isolation-with-migration model was extended to the case of multiple populations with a known phylogeny. Analysis of simulated data sets reveals the kinds of history that are accessible with a multipopulation analysis. Necessarily, processes associated with older time periods in a phylogeny are more difficult to estimate; and histories with high levels of gene flow are particularly difficult with more than two populations. However, for histories with modest levels of gene flow, or for very large data sets, it is possible to study large complex divergence problems that involve multiple closely related populations or species.

FIG. 1.

An isolation-with-migration model for three sampled populations.

FIG. 2.

Numbers of parameters of different types and total numbers, as a function of the number of sampled populations in the model.

FIG. 3.

Marginal prior distributions for splitting times in a model with five splitting events for a shared common maximum time of 10.

FIG. 4.

Marginal posterior probability density estimates for simulated data for models 1, 2, and 3 (left column, center column, and right column, respectively). As described under Methods, each curve is the sum of 20 curves from the analysis of 20 independently simulated data sets. Each row of panels shows the results for a different set of parameters (top row: population size; middle row: splitting time; and bottom row: migration rates). For migration rates, only four or two of the possible migration rates are shown.

FIG. 5.

Marginal posterior probability density estimates for models 4 and 5. For migration rates, only four of the eight migration rates in a three-population IM model are shown.

FIG. 6.

Marginal posterior probability density estimates for simulations under a four-population model (model 6). For migration rates, only 4 of the 18 migration rates in a four-population IM model are shown.

FIG. 7.

Marginal posterior probability density estimates for models 7 and 8.

FIG. 8.

Marginal posterior probability density estimates for models 9 and 10.

FIG. 9.

Marginal posterior probability density estimates for models 11, 12, and 13.

FIG. 10.

Cumulative distributions of the log-likelihood-ratio (LLR) statistic calculated for estimated posterior densities for the migration rate (m) and the population migration rate (i.e., 2Nm) for all migration parameters with nonzero true values in models 1 through 11. Also plotted is the distribution.