Modeling Additive × Environment and Additive × Additive × Environment Using Genetic Covariances of Relatives of Wheat Genotypes

Abstract

In self‐pollinated species, the variance–covariance matrix of breeding values of the genetic strains evaluated in multienvironment trials (MET) can be partitioned into additive effects, additive × additive effects, and their interaction with environments. The additive relationship matrix A can be used to derive the additive × additive genetic variance–covariance relationships among strains, Ã. This study shows how to separate total genetic effects into additive and additive × additive and how to model the additive × environment interaction and additive × additive × environment interaction by incorporating variance–covariance structures constructed as the Kronecker product of a factor‐analytic model across sites and the additive (A) and additive × additive relationships (Ã), between strains. Two CIMMYT international trials were used for illustration. Results show that partitioning the total genotypic effects into additive and additive × additive and their interactions with environments is useful for identifying wheat (Triticum aestivum L.) lines with high additive effects (to be used in crossing programs) as well as high overall production. Some lines and environments had high positive additive × environment interaction patterns, whereas other lines and environments showed a different additive × additive × environment interaction pattern.

Association Analysis of Historical Bread Wheat Germplasm Using Additive Genetic Covariance of Relatives and Population Structure

“…Two kinds of linear mixed models (LMM1 and LMM2) were used to fit data from g lines, s sites, and r replicates (at each site) for modeling association of phenotypic traits with m markers. LMM1 is similar to some of the models proposed by Smith et al (2002) for modeling GE and by Crossa et al (2006) and Burgueño et al (2007) for modeling GE with matrix A. In LMM2, markers are included as covariates for modeling the fixed effects of marker 3 environment interaction in a manner similar to the partition proposed in the factorial regression model (van Eeuwijk et al 1996).…”

“…The VCV matrix G, which combines the main effect of lines and GE, can be represented as G ¼ S g 5A, where 5 is the Kronecker product operator, and the jth diagonal element of the s 3 s matrix S g is the additive genetic variance s 2 aj within the jth site, and the jjth element is the additive genetic covariance r jj9 s aj s aj9 between sites j and j9; thus, r ij9 is the correlation of additive genetic effects between sites j and j9. The VCV matrix G was modeled using the factor analytical model (Smith et al 2002;Crossa et al 2006;Burgueño et al 2007).…”

“…A variant of LMM1 was proposed by Burgueño et al (2007) for partitioning the total genetic effect (g) of a line into additive (a) and additive 3 additive (i) effects, as in Oakey et al (2006). The covariance between individuals due to additive 3 additive effects in matrix notation is (A # A) ¼Ã (where # is the elementwise multiplication operator).…”

“…Using linear mixedmodel methodology, a genetic covariance matrix can be estimated, and best linear unbiased predictors (BLUPs) can be obtained. BLUPs for simultaneously modeling GE and incorporating information on additive genetic variation have been previously suggested by Crossa et al (2006) and Burgueño et al (2007).…”

Prediction of Genetic Values of Quantitative Traits in Plant Breeding Using Pedigree and Molecular Markers

“…Animal breeders have used this model for predicting breeding values either in a mixed model (best linear unbiased prediction, BLUP) (Henderson 1984) or in a Bayesian framework (Gianola and Fernando 1986). More recently, plant breeders have incorporated pedigree information into linear mixed models for predicting breeding values (Crossa et al 2006Oakey et al 2006;Burgueño et al 2007;Piepho et al 2007).…”