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
(πT-πS)/πT; where
πT is SNP diversity across all individuals and
πS is weighted within-population SNP diversity (Evans et
al., 2014).