Foundations of mathematical genetics (2nd edn).

Foundations of Mathematical Genetics (2nd edn). Anthony W. F. Edwards. Cambridge University Press, Cambridge. 2000. Pp. 121. Price £12.95, paperback. ISBN 0 521 77544 2. This book is

essentially a reprint of the first edition (published 1977) but with the important addition of a final chapter on ‘Fisher’s Fundamental Theorem of natural selection’. The book’s scope is

much narrower than its title implies. It gives a detailed mathematical analysis of selection models with discrete generations of random mating and constant genotypic viabilities. Successive

chapters are devoted to analysing models for the following genetic systems: two alleles at a single locus; multiple alleles; sex-linkage; and two diallelic loci. The treatment is entirely

mathematical: theorems are stated and rigorously proved. Apart from the final chapter on the recent interpretation of Fisher’s Fundamental Theorem, the rest of the book concerns material

most of which had been published before 1970. There is little discussion of the biological justification for the models or how they may be used to estimate selection parameters from

observational data. In spite of its purely mathematical approach, the book carries an important message, still widely ignored, for all evolutionary biologists. Great emphasis is placed on

conditions for equilibrium and changes in mean viability. The chapter on many alleles at a single locus gives results all evolutionary biologists should be familiar with, even if the proofs,

set out in an elegant matrix algebra, are passed by. Edwards gives rigorous proof that mean viability always increases at a multiallelic locus with constant viabilities. Provided this

represents an ‘internal’ equilibrium (where a number of different alleles remain in the population), it will be a point of globally stable equilibrium. This is the most general model for

which proof has been obtained that a population ‘climbs an adaptive peak’ to a point of maximum fitness. Even for the simplest two-locus, two-allele model with constant viability, Moran

(1964) proved that mean viability does not maximize: counter examples can easily be constructed showing decreasing mean viability. Yet still, the textbooks — for example, in the Open

University textbook _Evolution_ (Skelton, 1993) — show populations climbing adaptive peaks. But it is precisely when there is more than one peak, implying strong interaction, that fitness

does maximize. Edwards shows that if the viabilities at the two loci are additive, viability does then maximize in this model. From an evolutionary biologists’ point of view, this is a

trivial and uninteresting result: the loci are essentially independent. The existence of two adaptive peaks would imply strong interaction between the loci: alleles at one locus must

determine the viabilities of alleles at the other. However, in no case have I been able to construct a diagram like that in Skelton (1993) with two _internal_ peaks: if two peaks exist, they

are always at the corners of the two-dimensional diagram of gene frequencies. I conjecture there is never more than one internal peak. Formal proof that populations do climb adaptive peaks

has never, at least so far, dissuaded evolutionary biologists from taking the ascent for granted. Refutation of this seductive but erroneous metaphor of the evolutionary process could

usefully have been given a far greater emphasis in Edwards’ book. Edwards’ final chapter contains a brief proof of Fisher’s Fundamental Theorem, based on his much longer discussion in

_Biological Reviews_ (Edwards, 1994). As it is now understood, the theorem concerns a partial change in fitness — the change in ‘the breeding value in fitness’. This is the change in that

component in fitness which each allele carries to the next generation, not the total change in mean fitness. In this restricted sense, the theorem is exact. Although Hardy–Weinberg

frequencies are not assumed, the proof does depend on allelic frequencies being passed unchanged through the mating system to the next generation. This would only be true if matings between

genotypes do not vary in fertility, for example in random mating. It could also hold for some very restrictive cases of assortative mating in which matings are strictly monogamous and equal

in fertility. But whenever matings are polygynous or vary in fertility the change in gene frequency due to natural selection will then be changed again by sexual selection. If so, the proof

of the Fundamental Theorem fails. Edwards gives a simple and clear proof of the theorem and what it asserts. Anybody — that is to say almost everybody – who has been baffled by Fisher’s

chapter on the Fundamental Theorem in _The Genetical Theory of Selection_ should now be able to understand what Fisher was trying to say. REFERENCES * Edwards, A. W. F. (1994). The

fundamental theorem of natural selection. _Biol Rev_, 69: 443–474. Article  CAS  Google Scholar  * Moran, P. A. P. (1964). On the nonexistence of adaptive topographies. _Ann Hum Genet Lond_,

27: 383–393. Article  CAS  Google Scholar  * Skelton, P. W. (1993). _Evolution_. Addison-Wesley, Wokingham. Google Scholar  Download references AUTHOR INFORMATION AUTHORS AND AFFILIATIONS *

Emmanuel College, Cambridge, CB2 3AP, UK Peter O’donald Authors * Peter O’donald View author publications You can also search for this author inPubMed Google Scholar RIGHTS AND PERMISSIONS

Reprints and permissions ABOUT THIS ARTICLE CITE THIS ARTICLE O’donald, P. Foundations of Mathematical Genetics (2nd edn).. _Heredity_ 84, 620–621 (2000).

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