Calculating plasticity: environmental and genotype-by-environment contribution
Plastic variation is defined by Scheiner and Goodnight as the variation due to environment (E) and genotype-by-environment (GxE) interactions (Scheiner & Goodnight, 1984). Again, we use the pooled data to estimate within genotype variation (microenvironmental & measurement error). To calculate each, we used a Bayesian mixed model regression analysis in R using the rstanarm v. 2.19.2 package(Goodrich, Gabry, Ali, & Brilleman, 2019) via the following equation.
\begin{equation} {\left(4\right)Y}_{\text{igG}}=\ {\alpha_{G}+\alpha_{g}+\ \alpha}_{G:g}+\varepsilon_{\text{igG}}\nonumber \\ \end{equation}
The model calculates the variation within the random effects of Environment (G or Garden), Genotype (g), and GxE (G:g, or Garden:genotype) (Table S3). We then use these variances, estimated as the mean of 6,000 random draws from the posterior distribution of equation (4), to calculate the contribution of an individuals’ phenotype due to plasticity, also known as the S indexv(Scheiner & Lyman, 1989).
\begin{equation} \left(5\right)\ S=(\sigma_{E}^{2}\ +\ \sigma_{\text{GxE}}^{2}\ )/\ (\sigma_{G}^{2}+\ \sigma_{\text{GxE}}^{2}+\ \sigma_{E}^{2}+\ \sigma_{e}^{2})\nonumber \\ \end{equation}
We then used these properties to calculate the proportion of plasticity due to environment versus genotype-by-environment interactions (Table 2).
\begin{equation} (6)\ \sigma_{\text{Plasticity}}^{2}=(\sigma_{E}^{2}\ +\ \sigma_{\text{GxE}}^{2}\ )\nonumber \\ \end{equation}
We build this model separately from our heritability model because of the way H2 and S are defined in the literature. Plasticity (S) estimates require us to separate variation due to GxE interactions from genetic variation. However, GxE interactions would be partially captured under the umbrella of genetic variation in our heritability model. Conversely, our heritability model also examines the variation due to population in order to calculate Qst, which is partially captured by the G and GxE random effect terms from our plasticity model. In order to accurately parse the subtle differences in how heritability and plasticity define genetic variation, we run two separate models.
We also used Relative Distance Plasticity Index (RDPI) as a measure of genotypic plasticity, which is a more general way of calculating plasticity that doesn’t rely on assumptions of the underlying distribution of the data(Valladares et al., 2006) .
\begin{equation} \left(7\right)\ RDPI=\ \sum{\ \frac{\left|X_{\text{Clatskanie}}-\ X_{\text{Corvallis}}\right|}{\max\left(\ X_{\text{Clatskanie}},\ X_{\text{Corvallis}}\right)\ }\ /\ N}\nonumber \\ \end{equation}
RDPI measures the absolute difference in genetic trait values between genotypes grown in two different environments, then normalizes that measure by the maximum of the two values. All of these measures are then summed and divided by the number of samples to get the final average RDPI metric.