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). 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.