In quantitative genetics, evolvability of a trait is measured as the amount of genetic variance available to directional selection in a population. This concept has been generalized into multivariate phenotypes using the (additive) genetic covariance matrix (G matrix), so that various aspects of evolvability and genetic constraints are expressed as functions of a G matrix and selection gradient vector—the Hansen–Houle evolvability measures. Growing evidence for descriptive and predictive values of these evolvability measures foregrounds their relevance in investigations into the evolution of complex phenotypes. However, mathematical and statistical properties of the evolvability measures have not yet been fully appreciated, even in the specialized quantitative genetics literature.
In this talk, I will introduce some distribution theories on evolvability measures, in particular focusing on their probability distributions when the selection gradient vectors vary across directions in the trait space. Key in looking at statistical properties of evolvability measures is that they are mathematically expressed as ratios of quadratic forms in selection gradients. This form of statistics is frequently encountered in the statistical and econometric literature, from which we can borrow useful distribution theories. One of these is a mathematical toolkit called the zonal and invariant polynomials of matrix arguments, which enable evaluation of some relevant integrals over a (hyper)sphere.
On one hand, the distribution theories provide explicit expressions for average evolvability measures, for which only approximate evaluation methods were known in the literature. This enables us to quantify and compare evolvability and constraints accurately in biologically meaningful manners. On the other hand, theories also allows for drawing exact probability distributions for certain evolvability measures. These are useful in conducting a retrospective hypothesis test on the role of genetic constraints in empirical evolutionary trajectories. Importantly, these theories are not restricted to the simple case with spherically distributed selection gradients but can accommodate arbitrary mean and covariance for the selection gradients, thereby extending the scope of evolvability measures to various scenarios like directional and/or correlated selections.
Main results have been implemented in my R package qfratio available from the CRAN and GitHub repositories. Hopefully, the enhanced accessibility and expandability of evolvability measures stimulate future quantitative investigations into multivariate trait evolution.