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Hardy's "Small" Discovery Remembered.

Year: 2008
Type: article
Abstract: The most beautiful mathematics to Godfrey Harold Hardy was that which had no application. For Hardy, mathematics was purely for intellectual challenge. He justified the pursuit of pure mathematics with the argument that its very “uselessness” meant that it could not be used to cause harm. Hardy went so far as to describe applied mathematics as “ugly”, “trivial”, and “dull” [1]. Despite Hardy's aversion to applied mathematics, he had a profound impact on biology. Mathematicians tend not to realize his contribution, which was downplayed by Hardy himself. At the same time, biologists tend not to appreciate his mathematical brilliance. This note is intended to recognize G. H. Hardy's “little” discovery as a contribution to genetics and to revisit a classic paper [2] that has shaped the field for the past century. Godfrey Harold Hardy, a graduate of Trinity College, Cambridge, began his mathematics career in the early 1900s as a fellow at Trinity [3]. He lectured in mathematics for a number of years and published many papers of such significance that he was considered Britain's leading pure mathematician. Hardy was also responsible for bringing the Indian mathematical genius Srinivasa Ramanujan to England, where they published many papers together and developed the field of number theory. While visiting an ill Ramanujan on one occasion, Hardy mentioned that he had traveled in cab number 1729 and “hoped it was not an unfavorable omen” to which Ramanujan replied “...it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways” (it is expressible as 1729 = 13 + 123 or 93 + 103, now known as the Hardy-Ramanujan number). In 1908, Hardy published a paper in Science that changed the field of population genetics, entitled “Mendelian Proportions in a Mixed Population”. The findings were later known as the Hardy-Weinberg Law (Equilibrium) because the same principle was published by Wilhelm Weinberg in the same year [4]. This principle offered a simple solution for the question of how genetic diversity is maintained in a population. For Hardy, the law was trivial and obvious, and he was reluctant to acknowledge its applications. But one hundred years later, the Hardy-Weinberg Law remains a cornerstone of modern computational genetics. Preservation of genetic diversity in a population requires stability, or equilibrium, of the genotype distribution from one generation to the next. The following is an outline of Hardy's stability condition for Mendelian proportions [2]. This condition holds for a closed system, where the population mates randomly, or for purposes of simplicity, where every individual mates with every other individual once, and where each mating yields a single offspring, with no selection, mutation, migration, or death. Genotypes A1A1 A1A2 A2A2 2nd generation frequency A1A1 P11a×P11a P11a×2P12a P11a×P22a (P11+P12)2a2 A1A2 2P12a×P11a 2P12a×2P12a 2P12a×P22a 2(P11+P12)(P12+P22)a2 A2A2 P22a×P11a P22a×2P12a P22a×P22a (P12+P22)2a2 View it in a separate window Recall that in the mammalian genome, each gene is represented by two of many possible variants, called alleles. For example, the gene, or locus, for eye color can exist in many forms, including a “blue” allele coding for blue eyes and a “brown” allele coding for brown eyes. If we consider only the blue and brown alleles, an individual may have two copies of the blue allele (homozygous for the blue allele), one copy of each allele (heterozygous), or two copies of the brown allele (homozygous for the brown allele). An individual who is homozygous for the blue allele can be designated as having an “A1A1” genotype, while an individual who is homozygous for the brown allele is designated as having an “A2A2” genotype. Consider the following case for alleles A1 and A2, P11 = the number of individuals with genotype A1A1 (the homozygous A1 case) 2P12 = the number of individuals with genotype A1A2 or A2A1 (the heterozygous case) P22 = the number of individuals with genotype A2A2 (the homozygous A2 case) Therefore, we can write the first generation proportions of individuals as P11 : 2P12 : P22. Let the total number of individuals in the first generation be represented as a, where a = P11 + 2P12 + P22. The second generation proportions of individuals can be derived from the table seen above. Therefore, we can write the second generation proportions of individuals as (P11+P12)2:2(P11+P12)(P12+P22):(P12+P22)2 Let the total number of individuals in the second generation be represented as b, where b = (P11 + P12)2 + 2(P11 + P12)(P12 + P22) + (P12 + P22)2. Hardy's stability condition requires that the proportion of individuals with any given geno-type remains constant across generations and is therefore established as follows P11a=(P11+P12)2b (1) 2P12a=2(P11+P12(P12+P22)b (2) P22a=(P12+P22)2b (3) Solving the above system simultaneously yields P122=P11×P22, the stability condition for the two-allele case. This provides a null hypothesis for biologists investigating the distribution of genetic characteristics in a population. Since a2=(P11+2P12+P22)2=P112+4P12P11+4P122+2P11P22+4P12P22+P222=(P11+P12)2+2(P11+P12)(P12+P22)+(P12+P22)2=b, this demonstrates that a2 = b, as required by the assumption of a closed system. Let the haplotype frequency of allele A1 = p, and the haplotype frequency of allele A2 = q, recalling that a haplotype is a combination of alleles at multiple genetic loci that are transmitted together from one generation to the next. Accordingly, and p=2P11+2P122a, and q=2P22+2P122a, where homozygous individuals are counted twice, heterozygous individuals counted once, and the number of haplotypes is twice the population size. Hence, p=P11+P12a, and q=P22+P12a. Since p + q = 1, the modern population genetics interpretation, p2 + 2pq + q2 = 1 must hold, which states that Pr(A1A1) = p2, Pr(A1A2) = 2pq, and Pr(A2A2) = q2. We can extend the two allele case to a general case as follows. Under the same assumptions as above and additionally that a given gene includes k alleles, say A1, A2, · · · , Ak and Pij is the number of people with the genotype AiAj, where i, j can be any real number. The following generations will approach values in proportion to those suggested by counting all possibilities of mating (the way Hardy did). Thus, if in addition the values satisfy (or nearly satisfy) the stability condition Pij2=Pii×Pij, then we can assume that the probabilities for each genotype in each generation will be the same as the probabilities for each genotype in the preceding generation. That is, the proportion of individuals with each genotype will stay the same, while genetic diversity will be maintained in a predictable way. This formulation is the k allele analog of the Hardy-Weinberg Law. Certain violations of the closed system assumption lead to departure from Hardy-Weinberg Equilibrium. One such violation is non-random mating, where preferential mating according to genotype may occur in the population. We will not explore the mathematical details, but we can represent this case as follows: Pr(A1A1) = p2 + pqF,Pr(A1A2)2) = 2pq(1 − F), and Pr(A2A2) = p2 + pqF, where F is defined as the inbreeding=coe cient [5]. F can be thought of as the probability that two alleles are identical due to parents passing on the same allele to their progeny. It is therefore also a measure of the degree of parental relatedness. When 0 < F < 1, homozygosity in the population increases, which may reduce health and reproductive fitness. The modern interdisciplinary approach has gathered great minds to work on some of the most challenging problems in biology. Hardy's contribution has greatly influenced the growing field of computational genetics. Today, applied mathematics is a critical component of genetics and has the potential to revolutionize the field and profoundly impact modern medicine. If Hardy were alive today, it would be interesting to know whether he would join these minds or remain steadfast in his pursuit of pure mathematics. Despite his disdain for applied mathematics, Hardy was one of the greatest contributors to contemporary mathematical biology, and at this one hundred year anniversary, his “small” discovery will be remembered as such a great contribution.
Source: PubMed
Language: English
Open Access status: green
OpenAlex ID: W2412964563
FWCI: 2.885
Cites: 4
Cited by: 1
Related to: 10