Principles of Genetics and Breeding

10.1.5 Proof of H-W Law

• Let us assume that there is one locus with two alleles A₁ and A₂ with frequencies pq, respectively.
• The possible genotypes are A₁A₁,A₁A₂ and A₂A₂, Let the genotype frequencies be P, H and Q respectively in the parents. and

There will be 2 types of gametes with respect to the locus, one type carrying A_{1 2} gene with a frequency ‘Q’ there is random mating among the parents. Hence, these gametes will unite in all possible combinations to produce the zygotes. The zygotic frequencies can be obtained using the laws of probability in the following way. gene having a frequency ‘P’ and other will be carrying A

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A₁A₂ (P) (Q)

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A₁ (p) A₁A₁(p²) A₁A₂(pq)

A₂(q) A₁A₂(pq) A₂A₂(q² )

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The zygotic frequency that is equivalent to the genotypic frequency of the progeny is fA₁A₁=p²,fA₁A₂=2pq and fA₂A₂=q².The gene frequencies in the progeny generation will be as follows.

fA₁=p²+1/2pq=p²+pq= p (p + q) = p

fA₂=q²+1/2pq=q²+ pq = q (p + q ) = q

Therefore, the gene frequency of the parent and the progeny generation is the same. As regards the genotype frequency, we assumed it to be P, H and Q in the parent generation.

In progeny generation it came out to be p², 2pq and q², respectively. If we calculate the gene and genotype frequency of the several generation progeny in the above manner, it will be evident that the genotype frequencies are p²,2pq and q² (same as the first-generation progeny). The gene frequencies will be again p and q. Thus, the law is proved.