On multilocus population genetic structure: I. Description of gametic disequilibrium architecture

Abstract

A diploid population is analyzed with respect to the set of L autosomal loci. The attention is focused on the description of possible gametic disequilibria defined as deviations of the current population state from the reference state with the same allelic concentrations but with independent combination of alleles in gametes. The disequilibrum description is not based on any assumptions about microevolution factor pressures, generation overlapping, mating systems, etc. It is simply used to represent the current genetic structure of the population in terms of gametic disequilibria and allele concentrations. As the indicators of linkage disequilibrium, the basis vectors of subspaces of admissible deviations are chosen that were originally suggested as eigenvectors of a linearization matrix for the model of generalized nonepistatic selection in a series of papers by S. Karlin and U. Liberman. In the present paper, these characteristics are considered from a completely different, new point of view, namely, as the basis gametic disequilibria, and admissible deviations are decomposed into the basis ones. The disequilibrium hierarchical organization with respect to inducing marginal deviations from appropriate reference states on the locus subsets of different sizes is revealed. Each of the basis disequilibria is characterized by a set of loci K, the so-called deviation support. For a basis disequilibrium of a certain level of hierarchy, there is a threshold m (equal to the quantity of loci in K) for the number of loci which is necessary for that disequilibrium detection. Thus, it is required to analyze no less than m loci simultaneously to reveal the existing basis deviation from linkage equilibrium. The justification of the results is based on the use of linear algebra machinery, Kronecker multiplication, and linear subspace of admissible deviations. Properties of marginal deviations on various subsets of loci are considered in terms of gametic disequilibria.