## Mathematics and the Genome: Mathematics and Classical Genetics (1900-1953)

3. Mathematics and Classical Genetics (1900-1953)

Another classical contribution of mathematics to genetics is known as the Hardy-Weinberg Equations. G.H. Hardy (1877-1947) was a mathematician at Cambridge and Wilhem Weinberg (1862-1937) was a medical doctor in Stuttgart.They carried out their work independently. The story is that R. C. Punnett raised a genetical question with Hardy at the Cambridge University faculty club in 1908. Reginald Punnett (1875-1967) is known for the Punnett square, which is a diagram that is useful in understanding the results of mating organisms with different possible genotypes. Punnett gave a lecture on Mendelian heredity. In the audience was George Udney Yule (1871-1951), who claimed to Punnett that if a particular allele was dominant that the frequency of this allele would grow until it reached .5; thereafter one would have stability which was consistent with the standard Mendelian observation that phenotypes were seen in the ratio of 3 to 1. Punnett thought this reasoning was not correct and took the problem to Hardy. Yule denies this account. Hardy, using some simple probability theory and the assumption that a trait was governed by the alleles A and a, then derived the result that an equilibrium between the genotypes AA, Aa, and aa would be reached under the assumption that there was random mating of an infinite population, no migration, no mutation, and no natural selection. Under these assumptions one can show that the frequencies of the alleles A and a do not change with time. Furthermore, if p and q represent the frequency of A and a respectively (where p + q = 1), then the frequencies of the genotypes AA, Aa, and aa will be p2, 2pq, and q2, respectively. Thus, genotype frequencies can be predicted from allele frequencies, and though one might think that a trait that one sees rarely in a population would die out after many generations, under the assumptions of the Hardy-Weinberg model this would not be the case.

During this period there was a strong association between statistics and genetics. Three of the giants in this work (but whose involvement with mathematics was not limited completely to statistical matters) were J.B.S. Haldane (1892-1964), R.H. Fisher (1890-1962), and Sewell Wright (1889-1988).These men made contributions to the part of genetics, often referred to as population genetics which deals with the complex issues of genotype involving many alleles, over long periods of time, and under different modes of mating. To give but one example of ways that this research profited mathematics and applied areas as well, look at the work that Fisher did while at the Rothamsted Agricultural Station. He pioneered the use of mathematical structures now known as block designs, which can be used to sort out the complex effects that contribute to the yields and characteristics of different plants. Block designs are structures with points (varieties) and blocks with the property that each block contains the same number of points and each point lies on the same number of blocks and any two points lie in exactly (lambda) blocks. Block designs are intensively investigated both within mathematics and statistics, where there are part of the broader domain of experimental design. (There is a well known result called Fisher's Inequality.)

Another major thread of investigation during this period was to reconcile views about unchanging units of inheritance (genes) and Darwin's ideas concerning evolution. Much work had to be done to understand the ways that variation in different individuals within a species and between species was the consequence of the basic mechanism (that Mendel had shown but with additional change due to mutation in individual genes) and the consequences of changes in chromosomal structure. Unfortunately, a dispute concerning the ideas of Francis Galton, Karl Pearson, W. F. R. Weldon and William Bateson delayed the realization that the ideas of Mendel and Darwin were not antithetical.

Another area of investigation was to understand the genetic consequence of breeding between isolated populations which might become separated from another population, and thereby show different characteristics. To do this a theory of genetic drift was studied, as well as the consequences of different types of inbreeding. All of this work helped to make sense of the very complex empirical situation both in natural populations and ones that were produced by animal and plant breeders. Mathematicians and statisticians continue to make significant contributions to classical genetics. However, with the birth of molecular biology a whole new branch of genetics emerged, molecular genetics. Crick and Watson's development of the double helix model can be taken as a milestone of the transition to new horizons, even as work continues along classical lines to this day.

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