Prospects for genomic selection in conifer breeding: a simulation study of Cryptomeria japonica

Abstract

Genomic selection (GS) can be a powerful technology in conifer breeding because conifers have long generation intervals, protracted evaluation times, and high costs of breeding inputs. To elucidate the potential of GS for conifer breeding, we simulated 60-year breeding programs in Cryptomeria japonica with and without GS. In conifers, the rapid decay of linkage disequilibrium (LD) can constitute a severe barrier to application of GS. For overcoming that barrier, we proposed an idea to leverage a seed orchard system, which has been used commonly in conifers, because some degree of LD exists in progenies derived from the limited number of elite trees in a seed orchard. The base population used for simulations consisted of progenies from 25 elite trees. Results show that GS breeding (GSB) done without model updating outperformed phenotypic selection breeding (PSB) during the first 30 years, but the genetic gain achieved over the 60 years was smaller in GSB than in PSB. However, GSB with model updating outperformed PSB over the 60 years. The genetic gain achieved over the 60 years of GSB with model updating was nearly twice that of PSB. Advantages of GSB over PSB prevailed, even for a low heritability polygenic trait. The number of markers necessary for efficient GS was a realistic level (e.g., one in every 1 cM), although higher marker density engendered higher accuracy of selection. These results suggest that GS can be useful in C. japonica breeding. Updating of the prediction model was, however, indispensable for attaining the large genetic gain.

Appendix: Parameter estimation in ridge regression

Regression coefficients of ridge regression are defined as the minimizer of the penalized sum of squares, as

$$ \frac{1}{n}\sum\limits_{{i = 1}}^n {{{\left( {{y_i} - \sum\limits_{{j = 1}}^p {{x_{{ij}}}{\beta_j}} } \right)}^2} + \lambda \sum\limits_{{j = 1}}^p {\beta_j^2} .} $$

Therein, n signifies the number of observations, p denotes the number of markers, y _i stands for a response for the ith observation, x _ij is the jth marker for the ith observation, β _j is the coefficient corresponding to x _ij , and λ represents a ridge parameter. The first term of the equation is the residual sum of squares of the regression. The second term of the expression is a “penalty”. The addition of this term ensures that the coefficients are determined not only by goodness-of-fit to the data, as quantified by the residual sum of squares, i.e., the first term, but also by the square of the magnitude of estimated regression coefficients, i.e., $\sum {\beta_j^2}$. The parameter λ controls the degree to which the large magnitude of the coefficients is penalized. For a given value of λ, estimates of regression coefficients are calculated as

$$ \widehat{\beta } = {\left( {{{\mathbf{X}}^{\rm{T}}}{\mathbf{X}} + n\lambda {\mathbf{I}}} \right)^{{ - 1}}}{{\mathbf{X}}^{\rm{T}}}{\hbox{y,}} $$

where X is a matrix whose element in the ith row and jth column is x _ij , and y is a column vector whose ith element is y _i ; I is an identity matrix.The value of the parameter λ was optimized through leave-one-out cross-validation to maximize the prediction accuracy. The predicted residual sum of squares (PRESS), which is the sum of squares of difference between y _i and the prediction of y _i (i.e., ${\widehat{y}_{{(i)}}}$, which is calculated by omitting the y _i from a training dataset), is calculable as a function of parameter λ:

$$ {\hbox{PRESS}}\left( \lambda \right) = \sum\limits_{{i = 1}}^n {{{\left( {{y_i} - {{\widehat{y}}_{{(i)}}}} \right)}^2} = {{\widetilde{{\mathbf{y}}}}^{\rm{T}}}\widetilde{{\mathbf{y}}},} $$

where

$$ \widetilde{{\mathbf{y}}} = {\left\{ {{\hbox{Diagonal}}\left( {{\mathbf{I}} - {\mathbf{A}}} \right)} \right\}^{{ - 1}}}\left( {{\mathbf{I}} - {\mathbf{A}}} \right){\mathbf{y}} $$

and

$$ {\mathbf{A}} = {\mathbf{X}}{\left( {{{\mathbf{X}}^{\rm{T}}}{\mathbf{X}} + \left( {n - 1} \right)\lambda {\mathbf{I}}} \right)^{{ - 1}}}{{\mathbf{X}}^{\rm{T}}}. $$

A one-dimensional grid search was performed to find the optimal value of parameter λ (i.e., the value minimizing PRESS(λ)) with 0.1 increments from λ = 0.1.