Correlation measures for linkage disequilibrium within and between populations

Correlation statistics can be used to measure the amount of linkage disequilibrium (LD) between two loci in subdivided populations. Within populations, the square of the correlation of gene frequencies, r2, is a convenient measure of LD. Between populations, the statistic rirj, for populations i and j, measures the relatedness of LD. Recurrence relationships for these two parameters are derived for the island model of population subdivision, under the assumptions of the linked identity-by-descent (LIBD) model in which correlation measures are equated to probability measures. The recurrence relationships closely predict the build-up of r2 and rirj following population subdivision in computer simulations. The LIBD model predicts that a steady state will be reached with r2 equal to 1/[1+4Nec(1+(k-1)rho)], where k is the number of island populations, Ne is the effective local population (island) size, and rho measures the ratio of migration (m) to recombination (c) and is equal to m/[c(k-1)+m]. For low values of m/c, rho=0, and E(r2) is equal to 1/(1+4Nec). For high values of m/c, rho=1, and E(r2) is equal to 1/(1+4kNec). The value of rirj following separation eventually settles down to a steady state whose expectation, E(rirj), is equal to E(r2) multiplied by rho. Equations predicting the change in rirj values are applied to the separation of African (Yoruba - YRI) and non-African (European - CEU) populations, using data from Hapmap. The primary data lead to an estimate of separation time of less than 1000 generations if there has been no migration, which is around one-third of minimum current estimates. Ancient rather than recent migration can explain the form of the data.