Exact expressions and numerical evaluation of average evolvability measures for characterizing and comparing matrices

Area of Science:

• Quantitative genetics
• Evolutionary biology
• Mathematical biology

Background:

• The additive genetic covariance matrix governs a population's short-term evolutionary response to selection (evolvability).
• Quantifying average evolvability measures across all selection gradients has been challenging due to a lack of explicit formulas.
• Previous methods relied on approximations or Monte Carlo simulations with inherent inaccuracies and fluctuations.

Purpose of the Study:

• To derive exact mathematical expressions for average evolvability measures, including conditional evolvability, autonomy, respondability, flexibility, response difference, and response correlation.
• To establish a more accurate and reliable method for quantifying evolutionary potential.
• To extend the applicability of these measures to a broader range of selection regimes.

Main Methods:

• Utilized the mathematical structure of evolvability measures as ratios of quadratic forms.
• Developed infinite series expansions involving zonal and invariant polynomials of matrix arguments.
• Derived new expressions for average measures under a general normal distribution for the selection gradient.

Main Results:

• Presented novel, exact formulas for average conditional evolvability, autonomy, respondability, flexibility, response difference, and response correlation.
• These formulas, expressed as infinite series, can be numerically evaluated using partial sums with known error bounds.
• Established new expressions applicable to a general normal distribution of selection gradients.

Conclusions:

• The derived exact expressions offer a significant advancement over previous approximate methods for quantifying evolutionary potential.
• These new formulas provide a more accurate and computationally feasible approach to understanding population responses to selection.
• The extended applicability to general normal distributions broadens the scope of evolutionary and quantitative genetic analyses.