Distribution theories for genetic line of least resistance and evolvability measures

Quantitative genetic theory on multivariate character evolution predicts that a population’s response to directional selection tends to happen around the major axis of the genetic covariance matrix G —the so-called genetic line of least resistance. Inferences on the genetic constraints in this sense have traditionally been made by measuring the angle of deviation of evolutionary trajectories from the major axis, or more recently by calculating the amount of genetic variance—the Hansen–Houle evolvability—available along the trajectories. However, there have not been clear practical guidelines on how these quantities can be interpreted, especially in a high-dimensional space. This study summarizes pertinent distribution theories for relevant quantities, pointing out that they can be written as ratios of quadratic forms in evolutionary trajectory vectors by taking G as a parameter. For example, a beta distribution with appropriate parameters can be used as a null distribution for squared cosine of the angle of deviation from a major axis or subspace. More general cases can be handled with the probability distribution of ratios of quadratic forms in normal variables. Apart from its use in hypothesis-testing, this latter approach could potentially be used as a heuristic tool for looking into various selection scenarios like directional and/or correlated selection as parameterized with mean and covariance of selection gradients.