Calculating the genetic contribution via heritable variation
We estimated components of variation across all the data from both gardens to calculate broad-sense heritability and Qst. To account of the lack of replication in Clatskanie, we combined measurements from both gardens into the model and thus used the pooled data to parse within genotype variation and make our genotypic estimates, or “BLUPs”. This within genotype variation was coded as residual variation in the model \((\varepsilon_{\text{Gipg}})\) and captures both microenvironmental variation and measurement error in the estimate. Thus, genotypic estimates from the model are the end result after microenvironmental and measurement error are removed (Figures 3 & Sx).
The spatial autocorrelation corrected data were used to parse variation in our nested hierarchical structure of population, genotype, and environment via the following equations:
\begin{equation} {\left(1\right)Y}_{\text{Gigp}}=\ {{\ ss}_{G}+\alpha_{G}+\ \alpha_{\text{Gp}}+\ \alpha}_{\text{Gpg}}+\varepsilon_{\text{Gipg}}\nonumber \\ \end{equation}
Our goal is to quantify within-garden genetic variation, we therefore parsed variation in branches grown in two different gardens (G) using the fixed effect \({\ ss}_{G}\). The parameters p for population (i.e. provenance of genotype), g for genotype, and i for theith individual tree sampled are all random effects. All branch data were modeled as gamma distributions using Bayesian mixed regression models via the package rstan v.2.18.2 in (T. S. D. Team, 2014). We chose to evaluate our model using Bayesian methods as it allowed us to fully quantify the uncertainty associated with uneven sampling across the gardens. The random effects outcomes (\(\alpha^{\prime}s\)) and fixed effect (\({\ ss}_{G})\ \)of equation (1) were estimated as the mean of 6,000 random draws from the posterior distribution (Table 1 & S3).
The resultant fixed and random effect estimates (\({{\ ss}_{G}+\alpha_{G}+\ \alpha_{\text{Gp}}+\ \alpha}_{\text{Ggp}})\)from equation 1 were then used as our genetic estimate for each genotype and are displayed in all graphical analyses (often referred to as a Best Linear Unbiased Prediction or BLUP). The variation parameters estimated from equation 1 were used to calculate broad-sense heritability, H2, and Qst of each of the traits. H2 was calculated for all traits in stems, branches, and roots, using the random effects variances from equation (1) as:
(2)\((\sigma_{\text{Genotype}}^{2}+\sigma_{\text{Population}}^{2})\ /\ (\sigma_{\text{Population}}^{2}+\sigma_{\text{Genotype}}^{2}+\ \sigma_{\text{Microenvironment}}^{2})\)
Population variance was included with Genotype as it is also representative of genetic differences between individuals. Genotype variance was taken as the variance among replicates and microenvironmental variance was taken as the residual variance of the model. Qst was calculated via the formula (Spitze, 1993; Whitlock & Gilbert, 2012):
(3)\(\sigma_{\text{Population}}^{2}\ /\ (2\sigma_{\text{Genotype}}^{2}+\ \sigma_{\text{Population}}^{2})\)
Fst was taken from a previous publication using the same genotypes and calculated in 1-kb windows as (πTS)/πT; where πT is SNP diversity across all individuals and πS is weighted within-population SNP diversity (Evans et al., 2014).