Epistasis and covariance: how gene interaction translates into genomic relationship

Models based on additive marker effects and on epistatic interactions can be translated into genomic relationship models. This equivalence allows to perform predictions based on complex gene interaction models and reduces computational effort significantly. In the theory of genome-assisted prediction, the equivalence of a linear model based on independent and identically normally distributed marker effects and a model based on multivariate Gaussian distributed breeding values with genomic relationship as covariance matrix is well known. In this work, we demonstrate equivalences of marker effect models incorporating epistatic interactions and corresponding mixed models based on relationship matrices and show how to exploit these equivalences computationally for genome-assisted prediction. In particular, we show how models with epistatic interactions of higher order (e.g., three-factor interactions) translate into linear models with certain covariance matrices and demonstrate how to construct epistatic relationship matrices for the linear mixed model, if we restrict the model to interactions defined a priori. We illustrate the practical relevance of our results with a publicly available data set on grain yield of wheat lines growing in four different environments. For this purpose, we select important interactions in one environment and use this knowledge on the network of interactions to increase predictive ability of grain yield under other environmental conditions. Our results provide a guide for building relationship matrices based on knowledge on the structure of trait-related gene networks.