Multivariate Trait Evolution: Models for the Evolution of the Quantitative Genetic G-Matrix on Phylogenies.

Genetic covariance matrices (G-matrices) are a key focus for research and predictions from quantitative genetic evolutionary models of multiple traits. There is a consensus among quantitative geneticists that the G-matrix can evolve through deep time. Yet, quantitative genetic models for the evolution of the G-matrix are conspicuously lacking. In contrast, the field of macroevolution has several stochastic models for univariate traits evolving on phylogenies. However, despite much research into multivariate phylogenetic comparative methods, analytical models of how multivariate trait matrices might evolve on phylogenies have not been considered. Here we show how three analytical models for the evolution of matrices and multivariate traits on phylogenies, based on Lie group theory, Riemannian geometry and stochastic differential (diffusion) equations, can be combined to unify quantitative genetics and macroevolutionary theory in a coherent mathematical framework. The models provide a basis for understanding how G-matrices might evolve on phylogenies, and we show how to fit models to data via simulation using Approximate Bayesian Computation. Such models can be used to generate and test hypotheses about the evolution of genetic variances and covariances, together with the evolution of the traits themselves, and how these might vary across a phylogeny. This unification of macroevolutionary theory and quantitative genetics is an important advance in the study of phenotypes, allowing for the construction of a synthetic quantitative theory of the evolution of species and multivariate traits over deep time. Lay SummaryWe unite Quantitative Genetics, the major mathematical theory of multivariate quantitative trait microevolution, with the mathematical theory of multivariate macroevolution. To do this, we allow the key component of quantitative genetic theory, the matrix of additive genetic variances and covariances (the G-matrix) to evolve along evolutionary trees. This is an advance because the G-matrix is assumed to be constant in quantitative genetics (for convenience), but it has been recognised that it evolves on macroevolutionary timescales (in deep time). Uniting Quantitative Genetics with macroevolutionary theory allows for a more complete mathematical description of Darwins theory of evolution, and allows for further testing of evolutionary hypotheses.