Evaluation of GxE and heritability
Data were first analyzed separately in each environment to remove outliers and correct for spatial heterogeneity within the environment. The model (1) below was applied to test for micro-environmental variation within the greenhouse where \(y_{\text{ijjk}}\) represents the phenotype of the individual i , located in row j and position k in the greenhouse; \(\mu\) is the overall mean;\(C_{i}\) and \(L_{i}\) represent the fixed effect of control lines and the random effect of the MAGIC lines, respectively. In this model,\(t_{i}\) is an index of 0 or 1, defined to distinguish between control and MAGIC lines; \(\varepsilon_{\text{ijk}}\) is the random residual error.
\(y_{\text{ijjk}}=\ \mu+\ C_{i}.t_{i}+\ L_{i}.\left(1-t_{i}\right)+\ R_{j}+P_{k}+\ \varepsilon_{\text{ijk}}\ \)(1)
For every trait where row (\(R_{j}\)) and/or position (\(P_{k}\)) effects were significant, required corrections were applied by removing the BLUP of the significant effects from the raw data. Corrected data were gathered and used in model (2) in order to estimate the broad-sense heritability (\(H^{2}\)) and the proportion of variance associated to the GxE (\(\text{prop.}{\sigma^{2}}_{\text{GxE}}\)).
\(y_{\text{ij}}=\ \mu+E_{j}+C_{i}.t_{i}+\text{CxE}_{\text{ij}}.t_{i}+\ L_{i}.\left(1-t_{i}\right)+\ \text{LxE}_{\text{ij}}.\left(1-t_{i}\right)+\ \varepsilon_{\text{ij}}\)(2)
In model (2), \(y_{\text{ij}}\ \)represents the phenotype of the individual i , in environment j ; \(\mu\), \(C_{i},\)\(L_{i}\) and the \(t_{i}\) index are as described in model (1);\(\text{CxE}_{\text{ij}}\) and \(\text{LxE}_{\text{ij}}\) are the fixed control lines x environment interaction effect and the random MAGIC lines x environment interaction effect, respectively. Within a given environment, random residuals error terms were assumed to be independent and identically distributed with a variance specific to each environment. From this model, the proportion of the total genotypic and GxE variance explained by the model was calculated as the following formula:\(\text{prop.}{\sigma^{2}}_{\text{GxE}}=\ \frac{{\sigma^{2}}_{\text{LxE}}}{({\sigma^{2}}_{L}+{\sigma^{2}}_{\text{LxE}})}\). The significance of GxE was tested with a likelihood ratio test (at 5% level) between the models with and without GxE. The broad-sense heritability at the whole design level (\(H^{2}\)) was derived from variance components of model (2) and calculated as the following:\(H^{2}=\ \frac{{\sigma^{2}}_{L}}{{{(\sigma}^{2}}_{L}+\frac{{\sigma^{2}}_{\text{LxE}}}{\text{nb.E}}+\ \frac{{\sigma^{2}}_{E}}{\text{nb.R}}})\), where \({\sigma^{2}}_{L}\) and \({\sigma^{2}}_{\text{LxE}}\) are the variance components associated to the MAGIC lines and MAGIC lines x environment interaction effects, respectively. Here nb.E andnb.R represent the number of environments (e.g. 12 for FW) and the average number of replicates over the whole design;\({\sigma^{2}}_{E}\ \)is the average environmental variance (i.e.\(\sum{{\sigma^{2}}_{\text{Ej}}/}\text{nb.E}\)).