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Exact expressions and numerical evaluation of average evolvability measures for characterizing and comparing [Formula: see text] matrices.

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Theory predicts that the additive genetic covariance ([Formula: see text]) 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.


Acknowledgements: The author would like to thank Thomas F. Hansen, Diogo Melo, the Associate Editor Reinhard Bürger, and an anonymous reviewer for constructive comments. This work was supported by the Newton International Fellowships by the Royal Society (NIF\R1\180520) and the Overseas Research Fellowships by the Japan Society for the Promotion of Science (202160529).


Evolutionary constraint, Phenotypic integration, Quadratic form, Quantitative genetics, Random skewers, Zonal polynomial, Algorithms, Monte Carlo Method

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J Math Biol

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Springer Science and Business Media LLC
Royal Society (Newton International Fellowships NIF\R1\180520)
Japan Society for the Promotion of Science (Overseas Research Fellowships 202160529)