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Covariances between arbitrary relatives with linkage and epistasis in the case of linkage disequilibrium.

In this study the new concept of link relationship is developed for two loci. There is a link relationship between two nonhomologous genes if they derive by descent from the same ancestral gamete belonging to a reference gametic generation. By considering together the link and identity by descent relationships, new coefficients of dependence between genes of one or two zygotes are defined. They are probabilities of having groups of genes at a pair of loci from one or two zygotes such that there are particular link and identity by decent relationships amongst them. Consideration of the states of dependence between the genes of one and of two zygotes at one or two loci leads to new parameters for the expressions of the mean, variance and the covariance under multiallelism and epistjasis limited to a pair of loci. The case of random mating is considered without restriction on epistasis. With second degree statistics for inbred relatives, the epistasis is limited to the additive X additive interaction. Some simplifying assumptions are envisaged, with some applications particularly in the case of biallelism.