Dissecting High-Dimensional Phenotypes with Bayesian Sparse Factor Analysis of Genetic Covariance Matrices

Abstract

Quantitative genetic studies that model complex, multivariate phenotypes are important for both evolutionary prediction and artificial selection. For example, changes in gene expression can provide insight into developmental and physiological mechanisms that link genotype and phenotype. However, classical analytical techniques are poorly suited to quantitative genetic studies of gene expression where the number of traits assayed per individual can reach many thousand. Here, we derive a Bayesian genetic sparse factor model for estimating the genetic covariance matrix (G-matrix) of high-dimensional traits, such as gene expression, in a mixed-effects model. The key idea of our model is that we need consider only G-matrices that are biologically plausible. An organism’s entire phenotype is the result of processes that are modular and have limited complexity. This implies that the G-matrix will be highly structured. In particular, we assume that a limited number of intermediate traits (or factors, e.g., variations in development or physiology) control the variation in the high-dimensional phenotype, and that each of these intermediate traits is sparse – affecting only a few observed traits. The advantages of this approach are twofold. First, sparse factors are interpretable and provide biological insight into mechanisms underlying the genetic architecture. Second, enforcing sparsity helps prevent sampling errors from swamping out the true signal in high-dimensional data. We demonstrate the advantages of our model on simulated data and in an analysis of a published Drosophila melanogaster gene expression data set.

The evolution of phenotypic integration: How directional selection reshapes covariation in mice

“…This is a robust method for estimating high dimensional G‐ and P‐matrices from limited samples proposed by Runcie and Mukherjee (2013) and implemented in MATLAB (2013) by the authors. We removed differences due to age, generation, and sex by using these groups as fixed effects in the mixed model.…”

The Phenome-Wide Distribution of Genetic Variance

“…Gene expression traits lie at the interface between genotype and phenotype and might underlie evolutionary diversifications in other phenotypes (Britten and Davidson 1971;King and Wilson 1975;Carroll 2008;Wittkopp and Kalay 2012). While there will clearly be ways in which individuals vary that are not captured by variation in gene expression, expression traits nevertheless represent a broad range of biological functions and have been shown to be associated with responses to selection in the field (e.g., McGraw et al 2011;Whitehead et al 2011;Pespeni et al 2013) and to be genetically correlated with fitness measures under laboratory conditions (e.g., Rest et al 2013;Runcie and Mukherjee 2013). Gene expression therefore provides perhaps the best opportunity, given current technologies, to explore the distribution of genetic variance in very-highdimensional phenotypes.…”

“…Several approaches have recently been advanced that attempt to accommodate large numbers of traits measured within standard quantitative genetic experimental designs (Meyer and Kirkpatrick 2005;Stone and Ayroles 2009;McGraw et al 2011;Runcie and Mukherjee 2013). The most generalizable of these approaches are likely to be those that result in the estimation of a reduced-rank G, where the number of dimensions with genetic variance is constrained to be fewer than the number of traits measured and thus fewer parameters need to be estimated.…”

“…The most generalizable of these approaches are likely to be those that result in the estimation of a reduced-rank G, where the number of dimensions with genetic variance is constrained to be fewer than the number of traits measured and thus fewer parameters need to be estimated. The Bayesian sparse factor (BSF) approach of Runcie and Mukherjee (2013) in particular promises to be a versatile approach for identifying sparse trait combinations that explain substantial proportions of the observed genetic variance. As a consequence of the Bayesian framework, the implementation of BSF is quite complex and requires a number of simplifying assumptions, the most important of which are that (1) the eigenvectors of the resulting G matrix have few traits contributing to them (the sparse assumption) and (2) the residual matrix is also of a reduced rank.…”

EvolQG - An R package for evolutionary quantitative genetics

“…The EvolQG function CalculateMatrix() uses R’s lm() model object to calculate variance-covariance matrices adjusting for the proper degrees of freedom in a simple fixed-effects MANCOVA. More complicated methods may be used to obtain G -matrices, such as an animal model or a mixed model 31, 55 , and these can be used for further analysis using EvolQG .…”