Directional heritability and the geometry of multivariate constraint

Abstract Evolutionary responses to selection depend on how additive genetic variance is distributed across trait combinations. We focus on directional heritability—the fraction of phenotypic variance that is additive genetic along a given selection gradient—and treat its distribution across directions as a central object for describing multivariate constraint. Using a geometric transformation that rescales trait space so that phenotypic variance is isotropic, we show that directional heritability becomes a quadratic form in a whitened genetic matrix G ∗ = P −1/2 GP −1/2 . Under uniformly distributed selection directions, the squared coefficient of variation of directional heritability satisfies CV 2 ( h 2 ) = (2/( p + 2)) V rel ( G ∗ ), where V rel ( G ∗ ) is the relative eigenvalue variance of the whitened matrix and p is the number of traits. Simulations show that alignment between the eigenvector systems of G and P has a larger effect on the spread of directional heritability than correlation strength. Analyses of 55 empirical G – P matrix pairs from 11 studies reveal wide variation across biological systems in how often selection encounters low-heritability directions: over two thirds of the populations examined had more than 25% of phenotypic directions with h 2 < 0.25. The eigenvalue spectrum of G ∗ provides a sufficient summary for characterising how matrix geometry shapes evolutionary constraint on the phenotypic scale.