A fast indirect method to compute functions of genomic relationships concerning genotyped and ungenotyped individuals, for diversity management

Abstract

BackgroundPedigree-based management of genetic diversity in populations, e.g., using optimal contributions, involves computation of the {mathbf{Ax}} type yielding elements (relationships) or functions (usually averages) of relationship matrices. For pedigree-based relationships {mathbf{A}}, a very efficient method exists. When all the individuals of interest are genotyped, genomic management can be addressed using the genomic relationship matrix {mathbf{G}}; however, to date, the computational problem of efficiently computing {mathbf{Gx}} has not been well studied. When some individuals of interest are not genotyped, genomic management should consider the relationship matrix {mathbf{H}} that combines genotyped and ungenotyped individuals; however, direct computation of {mathbf{Hx}} is computationally very demanding, because construction of a possibly huge matrix is required. Our work presents efficient ways of computing {mathbf{Gx}} and {mathbf{Hx}}, with applications on real data from dairy sheep and dairy goat breeding schemes.ResultsFor genomic relationships, an efficient indirect computation with quadratic instead of cubic cost is {mathbf{x}} = {mathbf{Z}}left( {{mathbf{Z^{prime}x}}} right)/k, where Z is a matrix relating animals to genotypes. For the relationship matrix {mathbf{H}}, we propose an indirect method based on the difference between vectors {mathbf{Hx}} - {mathbf{Ax}}, which involves computation of {mathbf{Ax}} and of products such as {mathbf{Gw}} and {mathbf{A}}_{22}^{ - 1} {mathbf{w}}, where {mathbf{w}} is a working vector derived from {mathbf{x}}. The latter computation is the most demanding but can be done using sparse Cholesky decompositions of matrix {mathbf{A}}^{ - 1}, which allows handling very large genomic and pedigree data files. Studies based on simulations reported in the literature show that the trends of average relationships in {mathbf{H}} and {mathbf{A}} differ as genomic selection proceeds. When selection is based on genomic relationships but management is based on pedigree data, the true genetic diversity is overestimated. However, our tests on real data from sheep and goat obtained before genomic selection started do not show this.ConclusionsWe present efficient methods to compute elements and statistics of the genomic relationships {mathbf{G}} and of matrix {mathbf{H}} that combines ungenotyped and genotyped individuals. These methods should be useful to monitor and handle genomic diversity.