Q24

3 marksShort AnswerSection C

(a) How is Hardy-Weinberg expression $(p^2 + 2pq + q^2) = 1$ derived? [2]

(b) List any two factors that disturb the genetic equilibrium. [1]

Evolution

Hardy-Weinberg principle and genetic equilibrium

Official Answer

The expression is simply the binomial expansion of the two allele frequencies, which together must total 1.

(a) Derivation of p² + 2pq + q² = 1:

In a population, consider a gene with two alleles; let p = frequency of the dominant allele (A) and q = frequency of the recessive allele (a).
Since these are the only alleles, $p + q = 1$ .
Under random mating, gametes combine randomly, so genotype frequencies are given by the binomial expansion (p + q)²:
$(p + q)^2 = p^2 + 2pq + q^2 = 1$
Here p² = frequency of homozygous dominant (AA), 2pq = frequency of heterozygous (Aa), and q² = frequency of homozygous recessive (aa).
These allele and genotype frequencies stay constant generation after generation — this constancy is genetic equilibrium.

(b) Any two factors that disturb genetic equilibrium:

Natural selection
Genetic drift
(also acceptable: gene migration/gene flow, mutation, genetic recombination)

Hardy-Weinberg principleallele frequency p qp + q = 1binomial expansiongenetic equilibriumgenetic driftnatural selectiongene flow

Marking Scheme

1½ mark: defining p and q as allele frequencies with $p + q = 1$ .
21 mark: applying binomial expansion $(p + q)^2 = p^2 + 2pq + q^2 = 1$ under random mating.
3½ mark: identifying $p^2 = AA$ , $2pq = Aa$ , $q^2 = aa$ (homozygous dominant, heterozygous, homozygous recessive).
4½ + ½ mark: any two disturbing factors — gene migration/gene flow, genetic drift, mutation, recombination, natural selection.

Hint

Start from $p + q = 1$ , then square it: $(p + q)^2 = p^2 + 2pq + q^2 = 1$ ; disturbances = selection, drift, migration, mutation, recombination.

Quick Oral Answer

Taking allele frequencies p and q with $p + q = 1$ , random mating means the genotype frequencies follow the binomial expansion $(p + q)^2 = p^2 + 2pq + q^2 = 1$ , representing AA, Aa and aa; this genetic equilibrium is disturbed by factors such as natural selection and genetic drift.

Analysis & Explanation

Concept — Hardy-Weinberg principle:

The principle states that in a large, randomly-mating population, allele and genotype frequencies remain constant from generation to generation — the population is in genetic equilibrium — provided no evolutionary forces act.

The derivation (step logic):

Two alleles A and a have frequencies p and q, and since they are the only options, $p + q = 1$ .
Random mating = random union of gametes, which is modelled by squaring the allele frequencies: $(p + q)^2 = p^2 + 2pq + q^2$ .
Term meanings: $p^2 = AA$ , $2pq = Aa$ , $q^2 = aa$ , and their sum equals 1.

Exam trap:

You must start from $p + q = 1$ and clearly state that the expression is the binomial expansion of (p + q)² — jumping straight to p² + 2pq + q² without this loses marks.
Label each term (AA, Aa, aa); an unlabelled expansion is incomplete.

The five disturbing factors (list any two):

Gene migration (gene flow)
Genetic drift
Mutation
Genetic recombination
Natural selection

Real-world link:

When the equilibrium is disturbed — say by strong natural selection or genetic drift in a small population — allele frequencies change over generations, and that change in frequency is precisely what we call evolution (microevolution).

Common Mistakes

1Writing the final expression without first stating $p + q = 1$ and that it is the binomial expansion of $(p + q)^2$ — the derivation steps carry the marks.
2Not labelling the terms — $p^2 = AA$ , $2pq = Aa$ , $q^2 = aa$ must be identified.
3Listing 'small population size' alone as a factor instead of naming genetic drift, or naming vague factors not among the five (gene flow, drift, mutation, recombination, natural selection).

Interesting Facts

The principle was derived independently in 1908 by the English mathematician G. H. Hardy and the German physician Wilhelm Weinberg — a rare case of two fields reaching the same law at once.

Hardy-Weinberg equilibrium is a 'null model': real populations rarely meet all its conditions, and the very deviations from it are how biologists detect evolution in action.

The sum of allele frequencies always equalling 1 is why q² (the frequency of the recessive homozygote) can be used to estimate carrier frequency (2pq) for recessive genetic disorders in a population.

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Frequently Asked Questions

How is the Hardy-Weinberg expression $p^2 + 2pq + q^2 = 1$ derived?

Let p and q be the frequencies of the two alleles A and a, so $p + q = 1$ . Under random mating, gametes unite randomly, and genotype frequencies are given by the binomial expansion $(p + q)^2 = p^2 + 2pq + q^2 = 1$ , where p² is AA, 2pq is Aa and q² is aa.

What does genetic equilibrium mean?

Genetic equilibrium means that the allele and genotype frequencies of a population remain constant from generation to generation, as long as no evolutionary forces disturb them. This is the state described by the Hardy-Weinberg principle.

Which factors disturb Hardy-Weinberg equilibrium?

Five factors can disturb it: gene migration (gene flow), genetic drift, mutation, genetic recombination and natural selection. Any of these changes allele frequencies over generations, which is how evolution occurs.