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In this article we will discuss about the principle of Hardy and Weinberg which requires five assumptions for explaining the equilibrium state of gene and genotype frequency.

It was the year 1908, when an English mathematician — G. H. Hardy — and a German physician, W. Weinberg independently discovered the principle concerned with the frequency of alleles in a population, which is now known as Hardy-Weinberg equilibrium principle.

This principle states that genotypes in a Mendelian population tend to establish an equilibrium with reference to each other and, at equilibrium, both allele and genotype frequencies remain constant from generation to generation. This equilibrium state occurs among diploid, sexually reproducing organisms with non-overlapping generation and in large, random, panmictic populations where no selection or other factors are present.

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**Therefore, the principle of Hardy and Weinberg requires 5 assumptions for explaining the equilibrium state of gene and genotype frequency, which are: **

(a) Individuals of each genotype must be as reproductively fit as those of any other genotype in the population;

(b) The population must consist of an infinitely large number of individuals;

(c) Random mating must occur throughout the population;

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(d) Individuals must not migrate into or out of the population;

(e) There must be mutation equilibrium.

Before going into the details of Hardy- Weinberg principle, first we consider the Mendelian principle of heredity in mathematical terms. The principle of segregation can be represented by the binomial expansion of (a+b)^{n}: where “a” is the probability that an event will occur and “b” is the probability that it will not occur.

The segregation of a single pair of alleles (A_{a}) in a monohybrid cross may also be represented by the simple expansion of (a+b)^{n} = (A + a)^{2} = 1AA + 2Aa + 1aa. Now, if we consider the frequency of these two alleles (A and a) in a population is p and q, respectively, then at equilibrium the frequencies of each genotype class is p^{2}(AA), 2pq(A_{a}), and q^{2}(aa). Frequency means the ratio of the actual number of individuals falling in single class to the total number of individuals. The genetic proportions of an equilibrium population are entirely determined by its allele frequencies.

**Following is the algebraic proof of genetic equilibrium for any two alleles in a population: **

So, when two alleles are involved, the p + q = 1, and, if we deduce it mathematically:

p + q =1, or, = p = 1 – q

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Now, if 1 – q is substituted by ‘p’ then all the relationship of the formula can be represented in terms of q which is:

(1 – q)^{2} + 2q(1 – q) + q^{2} =1

Therefore, if an allele ‘A’ has a frequency of 1-q and another allele ‘a’ has a frequency of q then the expected distribution of these alleles under panmictic conditions in succeeding generations may be calculated.

In this regard it should be remembered that dominance and recessiveness of the alleles do not directly influence allele frequency and dominance alone does not make an allele occur more frequently in the population. If some phenotype has a selective advantage over another, dominance could indirectly influence allele frequency.

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The above-mentioned equation p + q = 1 applies when only two autosomal alleles in a population occur at a given locus, but if the system includes more alleles, more symbols must be added to the equation.

**For example, in case of AB blood group which is controlled by three alleles — namely I ^{A}, I^{B} and i — the equation would be:**

p + q + r = 1

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or, p^{2} + q^{2} + r^{2} + 2pq + 2pr + 2qr (sum of the genotype frequency).

Now, if we consider the alleles which are present in the sex chromosomes then it shows some different frequency from that discussed above. This difference in frequency of alleles in sex chromosome is due to the arrangements of sex chromosomes in the two sexes.

**For example, if we consider two alleles (Aa) are present in X-chromosome then the genotypic values at equilibrium will be: **

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(a) For females: p^{2} + 2pq + q^{2} i.e. AA + 2Aa + aa

(Due to the double dose of X-chromosome)

(b) For males: p + q i.e. A + a

(Due to the single dose of X-chromo- some)

Sex-linked (X-linked) genes (alleles) shows criss-cross pattern because in human the X- chromosome is transmitted from a father through his daughters) to half of her sons (her father’s grandsons). This criss-cross mechanism of inheritance indicates that the allele frequency in males in any generation equals the frequency in females in the previous generation i.e. in mathematical term:

If m_{n} and f_{n} represent the frequency of an allele in males and females, respectively, in generation n, then

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m_{n} = f_{n-1}(because a male receives his X-chromosome from “his mother)

The frequency in females in any generation equals the average allele frequency to the previous generation, because females receive one X-chromosome from each parent, so:

F_{n} = 1/2 (m_{n-1} + f_{n-1}).

It is interesting to note that naturally- occurring human populations are not genotypically in equilibrium state i.e. human population does not always follow the Hardy- Weinberg principle.

If we consider the protein and enzyme polymorphism in human population then it would be clear to us that the degree of allelic variation may occur at different gene loci and although multiple allelism may be demonstrable there is one allele that can be regarded as the standard or normal form which is almost universally present, while others are extremely rare. This type of change in the allele frequency may ultimately affect the population causing population change or population dynamics.