Aspects of optimality behavior in population genetics theory

Abstract

Optimality principles are central to many areas of the physical sciences, and often the simplest way of finding the evolutionary behavior of some dynamical system is by finding that path satisfying some optimality criterion. This paper discusses two aspects of the evolutionary paths followed by gene frequencies under natural selection as derived by optimality principles.

The first, due to Svirezhev, is that when fitnesses depend on the genes at a single locus only, and random mmating occurs, the evolutionary paths of gene frequencies, as determined by natural selection, minimize a functional which can be thought of as the sum of a kinetic and a potential energy. The second principle applies when fitness depends on all loci in the genome and random mating does not necessarily occur. The set of gene frequencies start at some point p in gene frequency space, and, some time later, under natural selection, are at some point q. There is a natural non-euclidean metric in the space of gene frequencies, and with this metric the distance from p to q is some value d. Then of all points in gene frequency space at distance d from p, the point q corresponding to natural selection maximizes the so-called partial increase in mean fitness, a central concept in a recent interpretation of the Fundamental Theorem of Natural Selection.