Three-locus systems impose additional constraints on pairwise disequilibria

Combinations of allele frequencies and pairwise linkage disequilibrium terms, each of which is permissible at the two-locus level, may not always be permissible at the three-locus level. These additional constraints on the possible maximum and minimum values for the pairwise disequilibrium terms are formally determined and numerically analyzed. In some cases, the three-locus constraints on a pairwise disequilibrium (D) may be equivalent to the usual two-locus constraints, while in others, the positive or negative range may be restricted. This can result in situations where the allowable values of D are limited to only positive or only negative values up to the extreme case where there is only a single admissible value. No additional restrictions are placed on pairwise disequilibrium values when four loci are considered, other than those imposed by the three-way combinations containing the two loci of interest. A new measure of normalized pairwise linkage disequilibrium, allowing for the three-locus constraints, is defined and illustrated by an application to data from the human histocompatibility antigen (HLA) system. An analogous normalized three-way disequilibrium measure is also formulated.