Maximisation Principles in Evolutionary Biology
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MAXIMISATION PRINCIPLES IN EVOLUTIONARY BIOLOGY A. W. F. Edwards 1 INTRODUCTION Maximisation principles in evolutionary biology (or, more properly, extremum principles, so as to include minimisation as well) fall into two classes — those that seek to explain evolutionary change as a process that maximises fitness in some sense, and those that seek to reconstruct evolutionary history by adopting the phylogenetic tree that minimises the total amount of evolutionary change. Both types have their parallels in the physical sciences, from which they originally drew their inspiration. There are important differences, however, and the naive introduction of extremum principles into biology has tended to obscure rather than clarify the underlying processes and estimation procedures, often leading to controversy. To understand the difficulties it is necessary first to note the use of extremum principles in physics. 2 EXTREMUM PRINCIPLES IN PHYSICS Extremum principles exert a peculiar fascination in science. The classic physics example is from optics, where Fermat in the 17th century established that when a ray of light passes from a point in one medium through a plane interface to a point in another medium with a different refractive index its route is the one which minimises the time it takes for the light to travel. That is, on various assumptions about the speed of light in the different media, he showed that this solution corresponded to Snel’s existing Sine Law of refraction. As a matter of fact modern books on optics find it necessary to frame Fermat’s principle rather differently, not using the notion of a minimum at all, but the point for us is that in its original form it solves at least some optical path problems by minimisation. The mathematics works out just right, but that is not to agree with Fermat that ‘nature is economical’. In the 18th and 19th centuries mathematical physics unearthed a whole range of extremum principles of great practical value because they provided a unifying point of view which gave results that are mathematically identical to the older approaches. Ideas like potential energy and its minimisation led to the notions Handbook of the Philosophy of Science. Philosophy of Biology Volume editors: Mohan Matthen and Christopher Stephens General editors: Dov M. Gabbay, Paul Thagard and John Woods c 2007 Elsevier B.V. All rights reserved. 336 A. W. F. Edwards of potentials and fields of force in gravitational and electromagnetic theory. The great 18th century developments in the calculus not only provided the necessary mathematical tools but also encouraged thinking in terms of maxima and minima. No-one supposed any longer that nature was economical. Rather, there was an implicit appeal to the classical concept of parsimony, that the simplest mathe- matical approach to any problem should be preferred, and this often involved an extremum principle. Ockham’s razor cut away the unnecessary mathematics but offered no explanation of nature. A quite different kind of extremum principle grew up more-or-less simultane- ously in statistics in these centuries, the most famous example being the method of least squares originating in the work of Legendre and Gauss. Far from being a way of solving problems with mathematical economy, as is Fermat’s principle, it is a method for selecting among probability models that one which most closely fits some observed data (in Gauss’s case, the observations of cometary positions). There is an element of arbitrariness in the choice of the criterion to be minimised (here the sum of the squares of the residual errors unexplained by the model) but subsequent research in statistical inference has produced plenty of justifications, the most dominant of which is Fisher’s method of maximum likelihood. Both these types of extremum principle have their analogues in evolutionary studies. First, we shall look at attempts to model evolution as a process that maximises ‘fitness’. 3 EVOLUTION AS FITNESS-MAXIMISATION. (1) FISHER’S ‘FUNDAMENTAL THEOREM’ An English proverb reminds us that ‘Nothing succeeds like success’ but it was not until 1930 that anyone put figures to it. In The Genetical Theory of Natural Selection published that year R. A. Fisher introduced his ‘Fundamental Theorem of Natural Selection’. We shall discuss it in more detail in due course, but for the moment we note that its basis is a simple ‘growth-rate’ theorem: ‘The rate of increase in the growth-rate of a subdivided population is proportional to the variance in growth-rates’. It is obvious that if the subdivisions of a population are growing at different rates then those subdivisions with the fastest rates of growth will increasingly dominate so that the overall growth-rate will increase. (‘Growth’ could here be negative too, but we describe the simplest case.) In full: If a population consists of independent groups each with its own growth- rate (the factor by which it grows in a given time interval), the change in the population growth-rate in this interval is equal to the variance in growth-rates amongst the groups, divided by the population growth- rate. [Edwards, 1994, 2002a] To prove this, let a population be subdivided into k parts in proportions pi (i = 1,2, ..., k)andlettheith part change in size by a factor wi in unit time Maximisation Principles in Evolutionary Biology 337 (its ‘growth-rate’). At the end of the unit time interval the new proportion of the ith part will be pi = piwi/w, where w =Σpiwi is the overall growth rate. 2 The new overall growth rate w =Σpiwi is therefore equal to Σpiwi /w and the 2 2 change in the overall growth-rate, w − w,is(Σpiwi − w )/w. But the numerator of this expression is simply the variance of the wi by definition (say v), being their expected squared value minus their squared expected value. Thus w − w = v/w and the theorem is proved. The constant of proportionality, 1/w, is of course just a scaling factor: dividing both sides by w leads to w/w − 1=v/w2, showing that the proportionate change is equal to the square of the coefficient of variation. A continuous version of the theorem is easily obtained by proceeding to a limit, the growth in each sub-population then being exponential. It is to Adam Smith’s An Inquiry into the Nature and Causes of the Wealth of Nations [1776] that we must turn for the germ of the idea behind the Fundamental Theorem. In Book IV: Of Systems of Political Economy he wrote: As every individual, therefore, endeavours as much as he can both to employ his capital in the support of domestic industry, and so to direct that industry that its produce may be of the greatest value; every individual necessarily labours to render the annual revenue of the society as great as he can. He generally, indeed, neither intends to promote the public interest, nor knows how much he is promoting it. By preferring the support of domestic to that of foreign industry, he intends only his own security; and by directing that industry in such a manner as its produce may be of the greatest value, he intends only his own gain, and he is in this, as in many other cases, led by an invisible hand to promote an end which was no part of his intention. Nor is it always the worse for the society that it was no part of it. By pursuing his own interest he frequently promotes that of the society more effectually than when he really intends to promote it. As Sober [1984] remarked, ‘The Scottish economists offered a non-biological model in which a selection process improves a population as an unintended con- sequence of individual optimization’, and he saw this as one of the influences in the formation of Darwin’s views on the effects of natural selection. We can only speculate on whether Fisher was familiar with these ideas in the 1920s; if so, a possible source would have been the economist J. Maynard Keynes, whose ATrea- tise on Probability was published in 1921 and whom Fisher had known since his undergraduate days [Box, 1978]. He will, however, certainly have read Darwin’s comment: ‘The larger and more dominant groups thus tend to go on increasing in size; and they consequently supplant many smaller and feebler groups’ [Darwin, 1859, 428]. The theorem exactly captures the vague notion that the more variable are the growth-rates the more quickly will the most rapidly-growing parts of the popula- tion (or sectors of the economy) come to dominate the rest. Eventually dominance by the fastest will be complete, all variability in growth-rates will have vanished, 338 A. W. F. Edwards and no further increase in the overall growth-rate can occur. It is as though the process of change, or evolution, consumes the variability. At the time when Fisher entered on his researches the importance of heritable variation as the raw material on which natural selection acts was not well under- stood, and the prevailing intellectual atmosphere was one of doubt as to whether Darwinian natural selection could account fully for evolutionary change. Further- more, noone had yet successfully quantified this variation mathematically. Indeed, not until Francis Galton had statisticians learnt to regard biological variability as intrinsically interesting, in contrast to variability in physics and astronomy which they had been accustomed to think of as ‘error’. Karl Pearson, in his many papers on evolution, had pursued the mathematical study of biological variation, but it was the young Fisher who saw the overwhelming advantages of working not with the standard deviation but its square, which he christened the variance in his pa- per The correlation between relatives on the supposition of Mendelian inheritance [Fisher, 1918a].