Unattainable regions in the space of genetic relationships

Artikel i vetenskaplig tidskrift, 2026

The relationship between two pedigree members may be summarized by their nine condensed identity coefficients(Δ1,⋯,Δ9), also known as Jacquard coefficients. These give the expected relative frequencies of the possible patterns of identity by descent between the alleles carried by the individuals at an autosomal locus. The collection J of all such vectors, taken over all pairs in all pedigrees, forms a subset of the probability simplex S8⊂R9. Despite decades of constant use and recurring interest in the Jacquard coefficients, remarkably little is known about the geometry of J. For instance, a long-standing open question is whether J is dense in S8, or, conversely, whether there exists full-dimensional unattainable regions not realizable by any pedigree. In the case of non-inbred individuals, this was famously resolved by Thompson, who identified the unattainable region bounded by a quadratic equation in the two-dimensional simplex of such relationships. In this paper we develop a general framework for constructing restrictions in the identity coefficients of any number of individuals. Applied to the pairwise case, we find several explicit quadratic inequalities in the nine Jacquard coefficients, each defining a full-dimensional unattainable region. One of these contains the known region unattainable by non-inbred relationships. To our knowledge, our results provide the first examples of full-dimensional holes in the space of pairwise relationships, arising purely from Mendelian constraints.