Comparing G: multivariate analysis of genetic variation in multiple populations

Abstract

The additive genetic variance-covariance matrix (G) summarizes the multivariate genetic relationships among a set of traits. The geometry of G describes the distribution of multivariate genetic variance, and generates genetic constraints that bias the direction of evolution. Determining if and how the multivariate genetic variance evolves has been limited by a number of analytical challenges in comparing G-matrices. Current methods for the comparison of G typically share several drawbacks: metrics that lack a direct relationship to evolutionary theory, the inability to be applied in conjunction with complex experimental designs, difficulties with determining statistical confidence in inferred differences and an inherently pair-wise focus. Here, we present a cohesive and general analytical framework for the comparative analysis of G that addresses these issues, and that incorporates and extends current methods with a strong geometrical basis. We describe the application of random skewers, common subspace analysis, the 4th-order genetic covariance tensor and the decomposition of the multivariate breeders equation, all within a Bayesian framework. We illustrate these methods using data from an artificial selection experiment on eight traits in Drosophila serrata, where a multi-generational pedigree was available to estimate G in each of six populations. One method, the tensor, elegantly captures all of the variation in genetic variance among populations, and allows the identification of the trait combinations that differ most in genetic variance. The tensor approach is likely to be the most generally applicable method to the comparison of G-matrices from any sampling or experimental design.

Figure 1

Matrix representation of a fourth-order tensor. The top-left quadrant of the matrix contains the (co)variances of the variances (for example, Σ_{ii, kk}). The bottom-right quadrant of the matrix contains the (co)variances of the covariances (for example, Σ_{ij, kl}). Finally, the upper-right and lower-left quadrants contain the covariances of the variances and covariances (for example, Σ_{ii, kl}).

Figure 2

Results from method 1. (a) An example of a random vector that identified a significant difference in additive genetic variation among populations. (b) An example of a random vector that resulted in overlapping posterior distributions of additive genetic variation among populations.

Figure 3

Results from method 2. (a) Eigenvalues of H for a comparison of the first four eigenvectors of each G matrix. (b) Eigenvalues of H for a comparison of the minimum number of eigenvalues required to account for 90% of the variation of the posterior mean observed G. Shaded regions of the plots indicate eigenvectors of H > min(k_i).

Figure 4

Results from method 3. (a) Variance accounted for by each eigentensor (α) for the observed G and the null G. (b, c) The additive genetic variance in each population in the direction of e₁₁ and e₂₁, respectively.

Figure 5

Results from method 4. (a,b) The posterior means and 95% HPD intervals for the predicted response to selection for each population for the two traits (lc2 and lc6) for which there were significant among-population differences in Δz (indicated by asterisk). (c, d) The contribution of g_max to differences in Δz for lc2 and lc6 in the populations where Δz differed significantly. (e,f) The contribution of the remaining eigenvectors of G to the predicted differences in responses to selection. Note the change in scale for the y axis between c, d vs e, f.