Maximisation Principles in Evolutionary Biology 3415 EVOLUTION AS FITNESS-MAXIMISATION. (3) WHAT REMAINS TO
BE SAIDIf it is true that the course of evolution cannot in general be described in terms
of a potential function like those that exist in physics, is it possible to salvage
anything from the idea at all in the context of evolution?
First, a distinction must be made between complete ‘dynamic’ theories involv-
ing a potential function that can fully describe the processes of change, and weaker
‘static’ theories of maximisation which merely describe stationary points of a sys-
tem in terms of the maximisation of some non-decreasing function. The latter class
includes the basic multiallelic single-locus model of population genetics in which
stable equilibria are at a maximum of the mean fitness, as proved by a number of
authors around 1960 (see [Edwards, 2000a], for the history). However, since the
proof is specific to the problem no general theory emerges.
Somewhere between the dynamic and the static theories there is the possibility
that changes can be described by a variational principle, but not one that involves
a potential function. This turns out to be the case for the above model provided
a transformed gene-frequency space is adopted, as follows.
Svirezhev [1972] stated that in the continuous-time model for selection at a
multiallelic locus the application of Fisher’s angular transformation sin−^1
√
pto
the gene frequencies results in (1) the direction of the gene-frequency change being
given by the steepest direction on the functionWand (2) the magnitude of the
change being proportional to the square root of the additive genetic variance, all
with respect to the new coordinate system. This does not in itself define a potential
function, though the case of only two alleles exhibits an interesting atypical feature
in the discrete-generation case because the square root of the additive genetic
variance is exactly proportional to the slope (taken positively) of the mean fitness
Win the transformed space [Edwards, 2000b]. Thus the magnitude of the change
is indeed proportional to the slope andWis then a potential function.
It is possible to pursue this line of approach using the Riemannian distance
Σ[(dpi)^2 /pi] implied by Fisher’s transformation, but it requires advanced mathe-
matics and the reader is referred to the books by Svirezhev and Passekov [1990],
Hofbauer and Sigmund [1998] and B ̈urger [2000]. However, in a more accessible
account Ewens [1992] placed the result in the context of an optimization principle
originally proposed by Kimura [1958] and developed by Ewens to prove that nat-
ural selection changes the gene frequencies so as to maximise the partial increase
in the mean fitness subject to a constraint on the Riemannian distance between
new and old gene frequencies. This is indeed a maximisation principle, but does
not go so far as to establish a potential function.
We thus see that a naive description of evolution as a process that tends to
increase fitness is misleading in general, and hill-climbing metaphors are too crude
to encompass the complexities of Mendelian segregation and other biological phe-
nomena. In some particular cases there is a deep structure involving variational
principles, but it is principally of mathematical interest only. Fisher’s Fundamen-