Constraints and normalized measures for cytonuclear disequilibria

The full bounds are derived for cytonuclear disequilibria in two-locus systems with an arbitrary number of alleles at the cytoplasmic and nuclear markers. The associated marginal frequencies constrain the nonrandom associations between cytoplasmic alleles and nuclear genotypes in the same way that the allele frequencies constrain the linkage disequilibrium between two nuclear loci. Additional constraints are imposed on the nonrandom associations between cytoplasmic and nuclear alleles, however, by the marginal frequencies of nuclear genotypes carrying either two or no copies of the associated nuclear allele. These bounds are analysed and used to define normalized measures of cytonuclear disequilibria, whose practical utility is illustrated through applications to two sets of recent nuclear-mitochondrial data.