Directional heritability and the geometry of multivariate constraint

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-1/2GP-1/2. Under uniformly distributed selection directions, the squared coefficient of variation of directional heritability satisfies CV2(h2) = (2/(p + 2)) Vrel(G*), where Vrel(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 h2 < 0.25. The eigenvalue spectrum of G* provides a sufficient summary for characterising how matrix geometry shapes evolutionary constraint on the phenotypic scale.