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Frequencies of two alleles at a single locus We already know that when two individuals heterozygous for the same gene e.g. Aa are crossed (Aa x Aa), they segregate in a ratio 1AA : 2Aa : 1aa. This ratio can be obtained by simple expansion of binomial (A + a)= 1^{2} AA + 2Aa + \aa. In this case, frequencies of A and a are the same i.e. 0.5. However, in a population the frequencies of two alleles belonging to a particular gene are not always the same and therefore, in a random mating population the above ratio of 1 : 2 : 1 is not always available. Suppose 'p' represents the frequency of allele 'A' and 'q' represents the frequency of V (where p + q = 1), then these frequencies 'p' and 'q' can also be realised in the form of gametes carrying 'A' and 'a'. In other words, the proportion of gametes carrying 'A' will be 'p' and the proportion of gametes carrying 'a' will be 'q'. The probabilities of individuals obtained from random mating from such a population can be obtained as shown in Figure 46.1. It can be seen in the figure that the probability of homozygous dominant (AA)obtained through random mating would be 'pthat of heterozygous (^{2}', Aa)individuals would be '2pq' and that of homozygous recessive (aa)would be 'q^{2}'. This leads to the formulation of Hardy-Weinberg Equation : p(^{2} AA)' + 2pq (Aa) + q(^{2} aa).
It is possible to prove algebraically that a population having genotypic frequencies expressed as pwill be in equilibrium. In doing so, as shown in Table 46.1, there are six possible types of matings using three genotypes. Taking the example of alleles ^{2} + 2pq + q^{2} 'A' and 'a', the possible matings will be AA x AA, AA x Aa, AA x aa, Aa x Aa, Aa x aa and aa x aa. Out of these six possible matings. whenever genotypes of the two parents differ, the mating can take place in two different ways using either of the genotypes as female parent. Since frequencies of different parental genotypes are known from the equilibrium equation, frequencies of progeny can be obtained as depicted in Table 46.2. It will be seen from the table that the progeny will still exhibit the same proportions between three genotypes as shown in Hardy-Weinberg Equation i.e. p^{2} + 2pq + q^{2}.
One can see from this ratio that although the genotype frequencies have changed, but gene frequencies did not change. This can be illustrated by calculating the gene frequencies from the above genotypic ratio. In this case ' T'will be equal to .36 (TT) + ½ x .48 (Tt)= .60 and 't' will be equal to .16 (tt) + ½ x .48 (Tt) = 0.40. However, it can be shown that genotypic ratio 0.36 TT + .48Tt + .16tt will give the same genotypic ratio in the following generation. This demonstrates that after one generation of random mating, equilibrium in the genotypic ratio can be achieved and this will follow the Hardy-Weinberg Equation. In view of the above, following conclusions can be made : (i) In a large panmictic population, where different genotypes are equally viable, gene frequencies of a particular generation will be similar to and depend upon gene frequencies of previous generations and not upon its genotype frequencies, (ii) The frequencies of different genotypes produced due to random mating will depend upon the gene frequencies and will reach an equilibrium after one. single generation of random mating. |