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The Mathematical Theory of Selection, Recombination & Mutation. By Reinhard Bürger. John Wiley & Sons, Chichester, 2000. £65.00. ISBN 0 471 98653 4. 409 pp.

Modelling the effects of selection in multilocus systems has long been a preoccupation of theoretical population geneticists. The conclusions derived from the models have implications for many important questions in evolutionary biology, including the maintenance of variation in quantitative traits, the speed of response of populations to selection, and the evolution of sex and recombination. During the last 20 years or so, there have been substantial advances in this area, many of which are rather daunting to the non-mathematician like myself. Reinhard Bürger has himself contributed signficantly to these advances. Together with the recent book by Freddy Christiansen, this book provides a comprehensive and authoritative review of the field, and will be an invaluable resource for researchers and advanced students in evolutionary genetics. It is not, however, an easy read, since it provides detailed treatments of some of the most advanced aspects of the subject.

Bürger starts with a straightforward chapter dealing with the partitioning of genetic variance due to a single locus, in both random-mating and non-random mating populations, as well as the basic mathematics of drift and single-locus selection theory. Chapter 2 deals with standard two-locus selection models, and goes on to develop variance partitioning in a multilocus context. This is then incorporated into an analysis of selection in multilocus systems, with a focus on the simplifications arising when selection is weak in relation to recombination, and leading up to Nagylaki’s generalization of the Fundamental Theorem of Natural Selection. Chapter 3 deals with the theory of mutation and selection in multiallelic systems. This leads on to the core of the book, Chapters 4 and 5, which deal, respectively, with mutation-selection equilibria for quantitative traits in an asexual (i.e. effectively single locus) population, and with the effects of selection, mutation and recombination on quantitative traits affected by arbitrary numbers of loci. Only a very dedicated reader would want to follow the details required to derive the major results, but it is easy enough to understand the main conclusions. Most biologists will be comforted to learn that the standard breeder’s equation, relating selection response to heritability and selection differential, emerges more or less unscathed, at least as an approximation.

Chapter 6 examines in detail the problem of the maintenance of variation in quantitative traits under stabilizing selection; Chapter 7 provides a wide-ranging survey of the data on the genetics of quantitative traits, and of further elaborations to the theoretical models, notably the consequences of various type of pleiotropy, and the expected change in genetic variance with directional selection under various models. It seems that it is still hard to evaluate the contribution of mutation-selection balance vs. balancing selection to variability in quantitative traits. One suspects that this old problem will only be solved by direct studies of the population genetics of the loci themselves, which will require their identification at the molecular level rather than more mathematics. Nonetheless, the models described here provide a fundamental underpinning to our understanding of the properties of quantitative genetic variation.