Exact expressions and numerical evaluation of average evolvability measures for characterizing and comparing $$\textbf{G}$$ matrices

Abstract

Theory predicts that the additive genetic covariance ($$\textbf{G}$$ G ) matrix determines a population’s short-term (in)ability to respond to directional selection—evolvability in the Hansen–Houle sense—which is typically quantified and compared via certain scalar indices called evolvability measures. Often, interest is in obtaining the averages of these measures across all possible selection gradients, but explicit formulae for most of these average measures have not been known. Previous authors relied either on approximations by the delta method, whose accuracy is generally unknown, or Monte Carlo evaluations (including the random skewers analysis), which necessarily involve random fluctuations. This study presents new, exact expressions for the average conditional evolvability, average autonomy, average respondability, average flexibility, average response difference, and average response correlation, utilizing their mathematical structures as ratios of quadratic forms. The new expressions are infinite series involving top-order zonal and invariant polynomials of matrix arguments, and can be numerically evaluated as their partial sums with, for some measures, known error bounds. Whenever these partial sums numerically converge within reasonable computational time and memory, they will replace the previous approximate methods. In addition, new expressions are derived for the average measures under a general normal distribution for the selection gradient, extending the applicability of these measures into a substantially broader class of selection regimes.

Distribution theories for genetic line of least resistance and evolvability measures

“…When p = 2, the distribution of e under the uniform distribution of x is a scaled beta distribution (Appendix A). The distributions can be multimodal, as previously mentioned by Watanabe (2023a) as a possibility. These multimodal cases are where inspection of the density, rather than just the mean and range, is most relevant.…”

“…(An equivalent of the former was called projected variance in Hunt (2007).) The expressions are (Hansen & Houle, 2008; Watanabe, 2023a) where G − is a generalized inverse of G , which equals the ordinary inverse G − 1 when G is nonsingular (see, e.g., Schott, 2016, chapter 5). The condition x ∉ R ( G ) in (9) is relevant only when G is singular.…”

“…These can be replaced by the analytic probability distributions of these quantities as ratios of quadratic forms, assuming the multivariate normality of x , for greater accuracy and enhanced power. Moments of e and c in this general condition can be obtained as described in Watanabe (2023a).…”

“…By denoting with tr ( · ) being matrix trace and the partition of H T µ corresponding to that of the rows of above, the expression of Forchini (2005, after correcting some errors) is Here, Γ ( · ) is the gamma function, ( · ) k denotes Pochhammer’s symbol, ( a ) k = a ( a +1) … ( a + k− 1) (with ( a ) 0 = 1), | · | denotes matrix determinant, and 2 F 1 ( a, b ; c ; · ) denotes the (Gauss) hyperge-ometric function. are the top-order invariant polynomials of two matrix arguments of the ( i, j )-th degree; these are certain scalar-valued functions (homogeneous polynomials) of matrix elements (see Watanabe, 2023a, appendix A for a brief introduction). In the central case ( µ = 0 p ), the expression simplifies to that of Forchini (2002, theorem 4).…”

“…The expression can be obtained by inserting GW g max G and G 2 into A and B , respectively, in (18). Watanabe (2023a) stated expressions for other mean evolvability measures using the same theory.…”