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.\(\sum{{\sigma^{2}}_{\text{Ej}}/}\text{nb.E}\)).