Estimating multilocus linkage disequilibria

The state of a diploid population segregating for two alleles at each of n loci is described by 2(2n) genotype frequencies, or equivalently, by allele frequencies and by multilocus moments or cumulants of various orders. Thes e measures of linkage disequilibrium cannot usually be determined, both bec ause one cannot tell whether a gene came from the maternal or paternal game te, and because such a large number of parameters cannot be estimated even from large samples. Simplifying assumptions must therefore be made. This pa per sets out methods for estimating multilocus genotype frequencies which a re appropriate for unlinked neutral loci, and for populations that are ulti mately derived by mixing of two source populations. In such a hybrid popula tion, all multilocus associations depend primarily on the number of loci in volved that derive from the maternal genome, and the number derived from th e paternal genome. Allele frequencies may differ across loci, and the contr ibution of each locus to multilocus associations may be scaled by the diffe rence in allele frequency between source populations for that locus (delta p less than or equal to 1). For example, the cumulant describing the associ ation between genes i, j, k from the maternal genome, and genes i, l from t he paternal genome is kappa(i,j,k,i*l*), = delta p(i)(2) delta p(j) delta p (k) delta p(l) kappa(3,2). The state of the population is described by n al lele frequencies; n divergences, delta p; and by a symmetric matrix of cumu lants, kappa(J,K) (J=0 ,..., n, K=0 ,..., n). Expressions for these cumulan ts under short- and long-range migration are given. Two methods for estimat ing the cumulants are described: a simple method based on multivariate mome nts, and a maximum likelihood procedure, which uses the Metropolis algorith m. Both methods perform well when tested against simulations with two or fo ur loci.