Introductory RemarksAuthor(s): J. F. C. KingmanSource: Proceedings of the Royal Society of London. Series B, Biological Sciences, Vol. 219, No.1216, Mathematical Genetics (Oct. 22, 1983), pp. 221-222Published by: The Royal SocietyStable URL: http://www.jstor.org/stable/35847 .

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Proc. R. Soc. Lond. B 219, 221-222 (1983) 221 Printed in Great Britain

Introductory remarks

BY J. F. C. KINGMAN, F.R.S.

Science and Engineering Research Council, Polaris House, North Star Avenue, Swindon SN2 lET, U.K.

Mathematics is about the systematic exploitation, by abstract and symbolic means, of regularities or patterns observed in the world. These regularities may be precise and predictable, like the Keplerian motions of the planets, or they may be merely statistical tendencies discerned with difficulty in apparently chaotic

phenomena. The latter is the more usual in the biological sciences, but genetics stands

somewhat apart because of the rather simple 'laws' of heredity postulated by Mendel. These make it possible to bring mathematical arguments to bear on

genetical problems, to clarify the consequences of biological hypotheses and to

suggest experiments which will discriminate between them. Because it combines biological importance with intrinsic mathematical elegance,

the Mendelian theory of inheritance has attracted the attention of distinguished scientists whose mathematical abilities were matched by biological perception, and the heroic phase of the subject will always be associated with the names of R. A. Fisher, J. B. S. Haldane and Sewall Wright. It is appropriate that the

centrepiece of this meeting of the Royal Society is to be the eleventh Fisher Memorial Lecture, to mark the magisterial contributions that he made to mathematical genetics.

But science moves on, and in genetics it has done so in such a way as to pose quite new challenges to the mathematician. In this respect the dynamics of

planetary motion form a poor analogy. When 'God said, "let Einstein be!"' He required only a small perturbation of the Newtonian analysis of the solar system. Modern experimental genetics, however, and above all the insights of molecular

biology, have changed the level of description at which the scientist must operate. It was almost as true for Fisher as it was for Mendel that a gene was an abstraction, a label attached to a chromosome locus. Today we know the alphabet in which these labels are written, and we are even beginning to read their language. Moreover the mechanisms by which these messages become corrupted in the reproductive process are more and more understood.

Thus whereas the classical geneticist dealt with loci at which only a few alleles could exist, we can now contemplate (and sometimes observe) a very large number, and complex forms of mutation between them. This produces very difficult mathematical problems, even for simple situations like the maintenance by selective differences at a single locus of a stable polymorphism. On the other hand, it sometimes leads to simplification, as in the very beautiful 'infinitely many alleles' theory of neutral mutation.

In analysing such special models, the mathematician lays himself open to the criticism that biological reality is much more complex than his calculations admit.

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J. F. C. Kingman

This criticism misunderstands the role of models in scientific argument. A biologist will often argue as if a certain situation were dominated by a few key factors. If his argument is cast in mathematical form, these implicit simplifications are

exposed as initial assumptions, and the chain of reasoning can be tested at each link. If the conclusions are qualitatively unacceptable or quantitatively at variance with experiment, one or more of the assumptions must be at fault. Thus the model makes explicit, and turns the spotlight on to, basic biological preconceptions.

Another related function is that of 'the model as null hypothesis'. Suppose for instance we have some data which appear to suggest that a particular selective mechanism is at work. It may be useful to construct a neutral model, not because we believe that selective differences are absent, but because if the data are

compatible with a neutral model, they probably contain no cogent evidence for or against any particular pattern of selection.

The use of mathematics in genetics is not however confined to the construction and analysis of particular models. As experimental data become richer, more

searching statistical techniques are needed to make full use of them. We look for inspiration to Fisher the statistician as well as to Fisher the geneticist.

The papers at this meeting well reflect the range of problems in genetics to which mathematics may contribute, as well as the different approaches which different

aspects of the subject require. No one pretends that the mathematician holds the

key to genetics, but the subject would be the poorer without him.

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