General derivation of the sets of pedigrees with the same kinship coefficients

Abstract

Quantification of kinships between two individuals using unlinked autosomal markers rests upon the identity-by-descent (IBD) probabilities among their four alleles at a locus because they determine the algebraic expressions of the joint genotypic probabilities. Nevertheless, some pedigrees share the same IBD probabilities and are therefore indistinguishable using those markers. Examples of these pedigrees were previously described, such as the case of half-siblings, grandparent-grandchild and avuncular, but a general analysis has not been attempted. The aim of this study is to present a systematic and mathematically supported framework where considering unlinked autosomal markers complete sets of indistinguishable pedigrees linking two non-inbred individuals are generally derived. In our work, complete sets of pedigrees with the same IBD partitions are formally established and mathematically treated, considering kinships linking any pair of non-inbred individuals, whether they are related just maternally or paternally, or both. Moreover, general expressions for IBD partitions, and consequently for joint genotypic probabilities, are derived considering a simple counting rule based on two 'atom' pedigrees: parent-child and full-siblings. Besides the theoretical formalization of the problem, the developed framework has potential applications in forensics as well as in breeding strategies design and in conservation studies.