Q.(a) How is Hardy-Weinberg expression (p2 + 2pq + q2) = 1 derived ?

The Hardy-Weinberg expression describes allele and genotype frequencies in a population at genetic equilibrium. Consider a gene with two alleles, A (dominant) and a (recessive), where p is the frequency of A and q is the frequency of a. Since these are the only two alleles, p + q = 1.

When gametes combine randomly during reproduction, the probability of any genotype is the product of the individual allele frequencies:

• AA genotype forms when both gametes carry A: probability = p × p = p²
• Aa genotype forms in two ways (A from father, a from mother OR a from father, A from mother): probability = 2pq
• aa genotype forms when both gametes carry a: probability = q × q = q²

Since every individual must have one of these three genotypes, their frequencies sum to 1, giving us p² + 2pq + q² = 1. This is simply the algebraic expansion of (p + q)² = 1.

Factors that disturb genetic equilibrium include: …

Understanding the Hardy-Weinberg Principle

Before we derive the expression, picture a large population where individuals mate randomly with respect to a particular gene. Suppose this gene has two alleles: a dominant allele A with frequency p in the population, and a recessive allele a with frequency q. Since these are the only two alleles for this gene, their frequencies must add up to the whole: p + q = 1.

Now, when gametes combine during fertilization, they do so randomly. Each egg or sperm carries either A (with probability p) or a (with probability q). The beauty of the Hardy-Weinberg principle lies in recognizing that the genotype frequencies in the next generation can be predicted simply by considering all possible combinations of these gametes.

Deriving the Expression

Think of it as a genetic lottery. The frequency of any genotype is the product of the frequencies of the two alleles that form it.

Homozygous dominant (AA): An AA individual forms when an A sperm fertilizes an A egg. The probability of picking an A from the male gamete pool is p, and from the female pool is also p. So the frequency of the AA genotype = p × p = p².

Heterozygous (Aa): This genotype can arise in two ways — either an A sperm meets an a egg, or an a sperm meets an A egg. The first scenario has probability p × q, the second has probability q × p. Together, the frequency of Aa = pq + qp = 2pq.

Homozygous recessive (aa): An aa individual results when an a sperm fertilizes an a egg. Frequency = q × q = q².

Since every individual in the population must have one of these three genotypes, their frequencies must sum to 1:

p² + 2pq + q² = 1

This is simply the algebraic expansion of (p + q)², which equals 1 because p + q = 1.

Factors That Disturb Genetic Equilibrium

The Hardy-Weinberg equilibrium holds only under very specific conditions: no mutation, random mating, no gene flow, infinite population size, and no natural selection. In reality, populations rarely meet all these criteria. When any of these conditions is violated, allele frequencies change across generations — evolution occurs.

Two major factors that disturb this equilibrium are: …