Neutral and Stable Equilibria of Genetic Systems and The Hardy-Weinberg Principle: Limitations of the Chi-Square Test and Advantages of Auto-Correlation Functions of Allele Frequencies

TL;DR

This paper challenges the traditional use of the Hardy-Weinberg principle and Chi-Square tests in population genetics, proposing that allele frequency auto-correlation functions provide a better understanding of genetic dynamics and equilibrium.

Contribution

It demonstrates the limitations of Chi-Square tests for Hardy-Weinberg equilibrium and introduces auto-correlation functions as a more effective tool for analyzing genetic stability.

Findings

Chi-Square tests can give false positives and negatives in population analysis.

Under Hardy-Weinberg conditions, populations evolve neutrally and do not reach observable equilibrium.

Relaxing random mating leads to asymptotic stable equilibrium characteristics.

Abstract

Since the foundations of Population Genetics the notion of genetic equilibrium (in close analogy to Classical Mechanics) has been associated to the Hardy-Weinberg (HW) Principle and the identification of equilibrium is currently assumed by stating that the HW axioms are valid if appropriate values of Chi-Square (p<0.05) are observed in experiments. Here we show by numerical experiments with the genetic system of one locus/two alleles that considering large ensembles of populations the Chi-Square test is not decisive and may lead to false negatives in random mating populations and false positives in nonrandom mating populations. As a result we confirm the logical statement that statistical tests can not be used to deduce if the genetic population is under the HW conditions. Furthermore, we show that under the HW conditions populations of any finite size evolve in time according to what can be identified as neutral dynamics to which the very notion of equilibrium is unattainable for any practical purpose. Therefore, under the HW conditions equilibrium properties are not observable. We also show that by relaxing the condition of random mating the dynamics acquires all the characteristics of asymptotic stable equilibrium. As a consequence our results show that the question of equilibrium in genetic systems should be approached in close analogy to non-equilibrium statistical physics and its observability should be focused on dynamical quantities like the typical decay properties of the allelic auto correlation function in time. In this perspective one should abandon the classical notion of genetic equilibrium and its relation to the HW proportions and open investigations in the direction of searching for unifying general principles of population genetic transformations capable to take in consideration these systems in their full complexity.