• Mayur Pawar

Magic of mathematics in your genes

Updated: May 26, 2019

The importance of mathematics and statistics in genetics is well known. Perhaps less well known is the importance of these subjects in evolution. The main problem that Darwin saw in his theory of evolution by natural selection was solved by some simple mathematics. It is also not a coincidence that the re-writing of the Darwinian theory in Mendelian terms was carried largely by mathematical methods. In this article I discuss these historical matters and then consider more recent work showing how mathematical and statistical methods have been central to current genetical and evolutionary research.

A brief description of the Darwinian theory of evolution by natural selection is as follows. In his revolutionary book, generally called The Origin of Species, Darwin claimed that biological evolution arises by natural selection, operating on the variation that exists between the individuals in any biological population. The argument has four main components.

-First, the more fit individuals in the population leave disproportionately more offspring than the less fit individuals.

-Second, the offspring in large measure inherit the fitness of their parents.

-Third, and following from the first two points, the offspring generation is on average more fit than the parental generation.

-Finally fourth, as generation succeeds generation, the population steadily becomes more and more fit, and eventually the more fit types replace the less fit.

The changes in the population are often taken as being quite gradual, and may well be thought of as taking place over a time span of many thousands or even hundreds of thousands of years. This is a population level theory. There is no concept of the evolution of the individual. The individual himself does not evolve. It is the population that evolves as generation succeeds generation.

Clearly the existence of variation in the population is crucial to the argument. Without this variation there are no fitness differentials between individuals, and the selective process cannot proceed. Darwin considered variation from one person to another in physical, mental and other characteristics, but when his argument is recast in genetical terms, as it will be below, it will be necessary to measure variation at the genetic level, and to assess the extent to which his theory continues to hold when investigated at the level and in terms of the modern molecular notion of a gene.

The main problem with the theory, as it was presented by Darwin in 1859, was that at that time the hereditary mechanism was unknown. However, the nature of this mechanism is crucial to a complete understanding of the argument. Worse than this, insofar as any idea of a theory of heredity was known in 1859, the most prevalent theory was based on the idea that any characteristic of a child, for example his blood pressure, is a mixture or blending of that characteristic in that child’s parents, plus or minus some small deviation deriving from unknown random effects. It is easy to see that under this so-called “blending” theory an effective uniformity of any characteristic among the individuals in the population will soon arise, so that after no more than about ten or twenty generations there will be essentially no individual- to-individual variation in any characteristic available for natural selection to act on. This difficulty was raised soon after the Darwinian theory was put forward, and was recognized immediately as a major criticism of the theory, and (unfortunately) Darwin amended later editions of his book in the light of it. To his dying day Darwin did not know of the resolution of the “variation preserving” difficulty.

Clearly some modification of the argument is necessary, since we do not observe, in present-day populations, the uniformity of characters that the blending theory predicts. However, any modification to the blending theory would probably require the assumption that the characteristics of children do not closely resemble those of the parents, and would thus remove one of the main underpinnings of the Darwinian theory.

The above discussion brings us to Mendel. Mendel’s work appeared in 1866, seven years after the appearance of The Origin of Species. It was in effect unread, and its importance unappreciated, until it was rediscovered in 1900. It led, however, to the solution of the maintenance of variation puzzle that Darwin could not solve.

Mendel made the first and basic steps in elucidating the hereditary mechanism. As is well known, he considered seven characters in peas, each of which happened to have a simple genetic basis. For example, he found that seed color, either green or yellow, is determined by the genes at a single gene locus, at which arose the “green” and the “yellow” alleles. We will call the green/yellow gene concept the billiard ball paradigm — a gene is seen under this paradigm as being either a green or a yellow billiard ball, with no known internal structure. In the first half of the 20th century the billiard ball paradigm.

Let's have look at mathematical implication of Meldelism of random mating case -

Our focus throughout is on diploid organisms, such as Man, in which any individual receives half his genetic composition from his mother and half from his father. We consider only autosomal loci: the sex-linked case involves minor differences from the analysis below. Consider some specific gene locus on some specific chromosome, which we call locus “A”. In general, genes of many different types might occur at this locus. Suppose for the moment that only two types of genes can arise at this locus, which we call the A1 allele and the A2 allele. (The words “gene” and “allele” often become confused in the literature. Here we adhere to the concept that a gene is a physical entity while the word “allele” refers to a gene type. Despite this we often use the expression “gene frequency” instead of the more logical “allele frequency”, since the term “gene frequency” has become embedded in the literature.) There are three possible genotypes, A1A1, A1A2 and A2A2. As mentioned above, the Darwinian theory is a population- level theory, and it is therefore necessary to consider the population frequencies of these genotypes, and how they change with time. We start with arbitrary frequencies in generation 1, as shown in the table below, and consider what happens in succeeding generations under the assumptions of random mating, no mutation, no selective differences, and indeed no complicating elements of any kind. It is easy to see that the genotype frequencies in generations 1, 2 and 3 are as given below:

A1A1 A1A2 A2A2

frequency, generation 1 X11 2X12 X22

frequency, generation 2 X^2 2x(1 – x) (1 – x)^2

frequency, generation 3 X^2 2x(1 – x) (1 – x)^2

In this table x = X11 + X12, so that x is the frequency of the allele A1 in generation 1. The entries in this table show three important features. First, the genotype frequencies attained in generation 2 are of a binomial form, with x, the frequency of A1 in generation 1, being the parameter of this distribution. Second, elementary calculations show that the frequency of A1 in generation 2 is also x. Finally, the genotype frequencies achieved in generation 2 continue to hold in generation 3, and hence also hold in all future generations. This final observation shows that there is no tendency for the variation in the population to be dissipated. These elementary calculations, first made independently by Hardy and Weinberg in 1908, just a few years after Mendelism was re-discovered, show that under a Mendelian hereditary system, Darwinism is saved: the variation needed for the operation of the Darwinian theory is preserved. A little bit of mathematics has gone a long way!

The “Hardy-Weinberg”, or binomial, form of the genotype frequencies in the above table, from generation 2 onwards, arise (in a randomly mating population) even when selective differences between genotypes arise, provided that genotype frequencies are taken at the time of conception of any generation. Thus Hardy-Weinberg frequencies will be used in the analysis of the selective case below, it then being understood that all frequencies are taken at this stage of the life cycle. In Genetics courses the emphasis is often placed mainly on the binomial form of the Hardy-Weinberg frequencies, which, conveniently, depend on the single quantity x. The importance of the permanence of these frequencies (at least in cases where there is no selection) to the Darwinian paradigm is often not even mentioned. Yet this permanence is the true implication of the Hardy-Weinberg scheme. A similar result hold for haploid populations: again, since the gene is the unit of transmission, genetic variation is preserved in these populations also. Of course genetic variation might eventually be lost through the action of selection or random drift (discussed below), but the time-scales for changes brought about by these is much longer than that appropriate to the blending theory.

In this course, we actually learn more about ‘daily life’ than ‘mathematics’. We did not do the endless calculation, did not prove endless theorems and did not solve endless equations. What we did was finding, learning and improving . During this whole week, I work hard to find the mathematics hidden in our daily lives, learn from other students from our class, and never stop trying to improve ourselves. From the team work I learnt how to communicate with others, how to work efficiently and how to be patient. From the presentation I learnt how to consider problems in different ways, how to give a better presentation, and to respect the labors of others. I used to feel cool when I understood the math theory that others could not underst and at all. But I gradually realized that people do not care what you can do only in mathematics field, but they care about what you can do to make the world better. There are several name of mathematicians in my text book my mum has never heard about. But people always remember the ones who devoted themselves to the society even though the ones who did not get high education. I do not expect to let people remember me, but I want to make something in the world better. When I sat down and looked back the whole week, I got the thinking above. I hope I can keep the idea in my mind: never stop trying figure out things, never stop improving ourselves and never forget to take others’ shoes.

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