Efficient computation of the additive relationship matrix and its inverse in self-breeding individuals

Abstract

Inbreeding increases homozygosity and therefore additive relationships within and among related individuals. The main cause of inbreeding is breeding of related individuals in which case the inbreeding coefficient (Fs) = 0.5asd where asd is the additive relationship among the individual's parents. An extreme case of inbreeding is the self-breeding occurring in plant inbred lines such as those generated by multiple generations of self-pollination, or by double haploid production. In the case of selfing generations, Fs = 1 - 0.5n where n is the number of selfing generations. If the parents of an individual are related and then it is self-bred, both sources for inbreeding should be accounted for in the progeny and Fs = 1 - 0.5n + 0.5n (0.5asd). In order to perform Best Linear Unbiased Prediction (BLUP), accurate calculation of the additive relationship coefficients matrix (A) or its inverse (A-1) is needed depending on the solving algorithm, or for single-step genomic BLUP where the submatrix A22 is used. Current methods to calculate A accounting for selfing generations require the expansion of the pedigree, which is computationally inefficient. Furthermore, freely available algorithms for setting up A-1 without generating A and inverting it do not contemplate selfing generations. The objective of this work was to develop efficient methods for calculating A and A-1 matrices accounting for inbreeding in self-bred individuals. Existing algorithms were adapted to account for selfing generations in A that require less memory than existing methods, and algorithms for the direct construction of A-1 accounting for inbreeding and selfing where developed in R. These algorithms are freely available at https://github.com/minesrebollo. In the future, these methods will be tested in large datasets and their performance will be reported.