Q.With the help of an algebraic equation, how did Hardy-Weinberg explain that in a given population the frequency of occurrence of alleles of a gene is supposed to remain the same through generations? OR Although a prokaryotic cell has no defined nucleus, yet DNA is not scattered throughout the cell. Explain.

Hardy-Weinberg: in a large, undisturbed population, $p^2 + 2pq + q^2 = 1$ and allele frequencies ($p$, $q$) stay constant across generations (genetic equilibrium). OR: Prokaryotic DNA is held in place with proteins as a coiled nucleoid, so it is not scattered.

Concept (main). The Hardy-Weinberg principle states that in a population the frequency of alleles of a gene and of genotypes remains constant from generation to generation, if the population is stable and there is no disturbing evolutionary force.

Why the frequency stays constant. The total genetic makeup of a population (its gene pool) is constant, and the sum of all allelic frequencies equals 1. If mating is random and no factors (mutation, gene flow/migration, genetic drift, natural selection) disturb it, the proportions simply reshuffle each generation without changing.

Algebraic explanation. Let the frequency of allele A be $p$ and of allele a be $q$. Then $p + q = 1$. Individuals in the population combine these alleles, and using the binomial expansion $(p + q)^2 = 1$ gives:

$p^2 + 2pq + q^2 = 1$

where $p^2$ = frequency of homozygous AA, $2pq$ = frequency of heterozygous Aa, and $q^2$ = frequency of homozygous aa. These values remain the same generation after generation — a state called genetic equilibrium. When they change, it indicates that evolution is occurring.

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