Multi-environment QTL analysis (QEI)
The strength of QTL dependence on the environment was tested afterward in a second step by identifying QTLs that significantly interact with the environment (QEI). Two multi-environment forward-backward models (5 & 6) were used to test at each marker position the effect of the marker x environment interaction.
\(y_{\text{ij}}=\ \mu+E_{j}+\sum_{p=1}^{8}{\alpha_{\text{kp}}*x_{\text{ikp}}\ }+\sum_{p=1}^{8}{\beta_{\text{kpj}}*x_{\text{ikp}}}+G_{i}+\varepsilon_{\text{ij}}\ \)(5)
\(y_{\text{ij}}=\ \mu+E_{j}+\sum_{p=1}^{8}{\beta_{\text{kpj}}*x_{\text{ikp}}}+G_{i}+\ \varepsilon_{\text{ij}}\)(6)
For model (5) and (6), \(y_{\text{ij}}\) represents the phenotype (mean value per genotype and per environment), \(E_{j}\) reflects the fixed environment effect; \(\alpha_{\text{kp}}\) and \(\beta_{\text{kpj}}\)represent the main and interactive parental allelic effects (p )at marker k and in environment j for \(\beta_{\text{kpj}}\);\(x_{\text{ikp}}\) is the probability of the parental allele’s origin for the MAGIC line i ; \(G_{i}\) stands for a random genotype effect and the residual errors including a part of the GxE that is not explained by the detected QTLs are specific to each environment,\(\varepsilon_{\text{ij}}\) ~ N (0,σ2Rj ).
Significant QEI were declared in a two-step procedure. First, the main QTL and the QEI effects were tested separately in model (5). The QTL detection process was adapted from the script proposed by Giraud et al., (2017). Every marker showing a significant main QTL or QEI was added as a fixed cofactor and the significance of the remaining markers tested again until no more significant marker was found. All markers selected as cofactors were then jointly tested in the backward procedure and only significant QEI after the backward selection are reported. The second procedure used in model (6) to declare QEI consisted in a slight modification of model (5) where \(\beta_{\text{kpj}}\) represents this time the global (main + interactive) effect of the marker. It allowed the detection of markers that had a main QTL effect or QEI just below the threshold detection but whose global effect is significant when the two components are jointly tested. To determine the threshold level for QEI detection, permutation test were performed 1000 times on the adjusted means with the function sim.sightr of mpMap 2.0 R package (Huang and George, 2011).