Predicting Heterotic Hybrid Performance

Genotypic data was simulated consisting of 511 SNP markers in 40 inbred lines belonging to two heterotic groups (20 in each). Phenotypic data was simulated consisting of grain yield (GY) and plant height (PH) for the 40 parents and 100 out the 400 possible hybrids produced from the single-cross of both heterotic groups allowing for heterosis. Genotypes of the 40 parents were used to estimate the genomic relationship matrices as K = ZZ’/2Σp_i(1-p_i) [27 (link)] for both heterotic groups (K₁ and K₂), and the genomic relationship matrix for the 400 possible hybrids was obtained as the Kronecker product of the parental genomic relationship matrices K₁ ⊗ K₂ (K₃). Given that the phenotypic data for the possible crosses was not masked, the hybrids were predicted by estimating the BLUPs for general combining abilities in males and females (GCA_female, GCA_male) and specific combining abilities (SCA) of crosses along with their variance components (σ²_GCA1, σ²_GCA2, σ²_SCA). The model has the form:
$y=\mathit{X}\beta +{\mathit{Z}}_{\mathbf{1}}{u}_{GCA1}+{\mathit{Z}}_{\mathbf{2}}{u}_{GCA2}+{\mathit{Z}}_{\mathbf{3}}{u}_{SCA}+\epsilon$
The mixed model equations for this model are:
${\left[\begin{array}{cccc}\mathit{X}\prime {\mathit{R}}^{-1}\mathit{X}& \mathit{X}\prime {\mathit{R}}^{-1}{\mathit{Z}}_1& \mathit{X}\prime {\mathit{R}}^{-1}{\mathit{Z}}_2& \mathit{X}\prime {\mathit{R}}^{-1}{\mathit{Z}}_3\\ {\mathit{Z}}_1\prime {\mathit{R}}^{-1}\mathit{X}& {\mathit{Z}}_1\prime {\mathit{R}}^{-1}{\mathit{Z}}_1+{\mathit{G}}_1^{-1}& {\mathit{Z}}_1\prime {\mathit{R}}^{-1}{\mathit{Z}}_2& {\mathit{Z}}_1\prime {\mathit{R}}^{-1}{\mathit{Z}}_3\\ {\mathit{Z}}_2\prime {\mathit{R}}^{-1}\mathit{X}& {\mathit{Z}}_2\prime {\mathit{R}}^{-1}{\mathit{Z}}_1& {\mathit{Z}}_2\prime {\mathit{R}}^{-1}{\mathit{Z}}_2+{\mathit{G}}_2^{-1}& {\mathit{Z}}_2\prime {\mathit{R}}^{-1}{\mathit{Z}}_3\\ {\mathit{Z}}_3\prime {\mathit{R}}^{-1}\mathit{X}& {\mathit{Z}}_3\prime {\mathit{R}}^{-1}{\mathit{Z}}_1& {\mathit{Z}}_3\prime {\mathit{R}}^{-1}{\mathit{Z}}_2& {\mathit{Z}}_3\prime {\mathit{R}}^{-1}{\mathit{Z}}_3+{\mathit{G}}_3^{-1}\end{array}\right]}^{-1}\left[\begin{array}{c}\mathit{X}\prime {\mathit{R}}^{-1}y\\ {\mathit{Z}}_1\prime {\mathit{R}}^{-1}y\\ {\mathit{Z}}_2\prime {\mathit{R}}^{-1}y\\ {\mathit{Z}}_3\prime {\mathit{R}}^{-1}y\end{array}\right]=\left[\begin{array}{c}\beta \\ u_{GCA1}\\ u_{GCA2}\\ u_{SCA}\end{array}\right]$
where β is the vector of fixed effects, u_GCA1, u_GCA2, u_SCA are the BLUPs for GCA_female, GCA_male, and SCA effects, X and Zs are incidence matrices for fixed and random effects respectively, R is the matrix for residuals (here Iσ²_e), and G^-1₁, G^-1₂, G^-1₃ are the inverse of the variance-covariance matrices for random effects. The BLUPs u_GCA1, u_GCA2, u_SCA were used to predict the rest of the single-crosses as the sum of their respective GCA and SCA effects.
We fitted this model using the sommer package by specifying the incidence and variance-covariance matrices and using the AI and EM algorithms, given that EMMA method cannot estimate more than one variance component. The model could not be implemented in rrBLUP which is also limited to a single variance component. In the BGLR package the Reproducing kernel Hilbert space [RKHS] kernel was used, in ASReml and MCMCglmm the ‘ginverse’ argument was used to specify the variance-covariance structures, and in the regress package the multiple random effects model using the ZKZ’ kernel was fitted. EMMREML uses a similar syntax than sommer. Results from other software were compared with sommer. In addition, a five-fold cross validation was performed to calculate the prediction accuracy for plant height and grain yield in this population.
In order to show the advantage of fitting a model including dominance (SCA) compared to a pure additive models (GCA) with respect to the prediction ability for species displaying heterotic effects, two additional models were fitted including only GCA effects; 1) both parents having the same variance component and 2) each parent from a different heterotic group having a different variance component:
$\mathit{G}=\left[\begin{array}{c}{\mathit{K}\sigma }_u^2\end{array}\right]\mathrm{a}\mathrm{n}\mathrm{d}\mathit{G}=\left[\begin{array}{cc}{\mathit{K}}_1{\sigma }_{u1}^2& 0\\ 0& {\mathit{K}}_2{\sigma }_{u2}^2\end{array}\right]$
Models were compared with respect to their prediction ability after 500 runs of a five-fold cross validation for plant height and grain yield. Models were fitted using sommer with the default AI algorithm. In addition, heritability for both trait was estimated as; h² = (σ²_GCA1 + σ²_GCA2) / (σ²_GCA1 + σ²_GCA2 + σ²_e).