Department of Earth Sciences, University of Cambridge, Downing Street, Cambridge, CB2 3EQ, UK.
J Math Biol. 2023 May 22;86(6):95. doi: 10.1007/s00285-023-01930-8.
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.
理论预测，加性遗传协方差([Formula: see text])矩阵决定了一个群体在短期（in）内对定向选择的反应能力——汉森-豪尔意义上的可进化性——通常通过称为可进化性度量的某些标量指数来量化和比较。通常，人们感兴趣的是获得所有可能选择梯度的这些度量的平均值，但这些平均度量的大多数显式公式尚不清楚。以前的作者要么依赖于汉森-豪尔方法的近似值，而其准确性通常是未知的，要么依赖于蒙特卡罗评估（包括随机串分析），而蒙特卡罗评估必然涉及随机波动。本研究利用二次型的比，提出了新的、精确的条件可进化性、自主性、响应性、灵活性、响应差异和响应相关性的平均表达式。新的表达式是涉及矩阵参数的最高阶的带状和不变多项式的无穷级数，并且可以作为它们的部分和进行数值评估，对于某些度量，已知误差界限。只要这些部分和在合理的计算时间和内存内数值收敛，它们将取代以前的近似方法。此外，还推导了选择梯度服从一般正态分布时平均度量的新表达式，将这些度量的适用性扩展到更广泛的选择制度类别。
