Biology · Introductory biology · Concept

Hardy–Weinberg equilibrium: p² + 2pq + q²

Hardy–Weinberg equilibrium is the baseline model of population genetics. For one gene with two alleles at frequencies p and q, where p + q = 1, random mating gives the genotypes AA, Aa and aa in the proportions p², 2pq and q², and those proportions stay the same every generation while the Hardy–Weinberg conditions hold. A sample that departs from them shows that at least one condition fails: mating may not be random, or selection, migration, mutation or drift may be at work.

Two alleles, two frequencies

Take one gene with two alleles, A and a. Let p be the fraction of all copies of the gene in the population that are A, and q the fraction that are a. Every copy is one or the other, so p + q = 1. These are allele frequencies: they count gene copies, not individuals.

p + q = 1

Count gene copies to find p

A diploid individual carries two copies of the gene, so N individuals carry 2N copies. Each AA individual adds two A copies and each Aa adds one, so p is the number of A copies over all copies. Then q = 1 − p.

p = \frac{2 N_{A A} + N_{A a}}{2 N}

Random mating predicts the genotypes

If gametes combine at random, an offspring receives A from each parent with probability p, so AA has frequency p² and aa has frequency q². A heterozygote can form two ways, A from the mother and a from the father or the reverse, so Aa has frequency 2pq. The three add to 1 because (p + q)² = 1.

p^{2} + 2 p q + q^{2} = 1

Equilibrium after one generation

From any starting mix of genotypes, one round of random mating produces the p², 2pq and q² proportions for an autosomal gene, and they stay there while the conditions hold. Allele frequencies do not drift toward one half on their own: a rare allele stays rare.

The five conditions

The prediction assumes random mating with respect to this gene, no mutation, no migration (gene flow), no natural selection, and a population large enough that genetic drift is negligible. No real population meets all five exactly. Hardy–Weinberg equilibrium is a null model: a clear departure from it shows that at least one condition fails, but not which one.

From a recessive trait to carriers

With complete dominance only aa individuals show the recessive phenotype, so their frequency estimates q². Its square root is q, and 2pq estimates the frequency of carriers. This route assumes the population is in equilibrium, so it cannot also test equilibrium.

q = \sqrt{q^{2}}, 2 p q = 2 (1 - q) q

\begin{array}{c} q = \sqrt{q^{2}} \\ 2 p q = 2 (1 - q) q \end{array}

Testing a sample with chi-square

When all three genotypes can be counted, compare the observed counts with the expected counts Np², N·2pq and Nq² using a chi-square goodness-of-fit test. Three genotype classes give 3 − 1 = 2 degrees of freedom, and estimating p from the same counts uses up one more, which leaves 1.

df = 3 - 1 - 1 = 1

Common mistakes

Taking the frequency of aa individuals as q: it estimates q², so q is its square root.
Counting individuals instead of gene copies when finding p: each heterozygote carries one copy of each allele.
Using 2 degrees of freedom in a three-genotype test when p comes from the same counts: the answer is 1.
Reading a good fit as proof that every condition holds: a test can only fail to detect a departure.

Key terms

Hardy–Weinberg equilibrium: The baseline genetics of a population that isn’t evolving: random mating and no selection, mutation, migration or drift. With two alleles, the genotype frequencies are then p², 2pq and q².
Allele frequency: The share of all copies of a gene in a population that are one particular allele: p = (copies of allele A) ÷ (total allele copies). Count allele copies, two per diploid individual, not individuals.
Genotype frequency: The share of individuals in a population with a particular genotype, such as AA. It differs from allele frequency, because each individual carries two allele copies.
Genetic carrier: Someone with one copy of the allele for a recessive condition who does not have the condition, such as Aa. A carrier can pass the allele to their children.
Genetic drift: Random changes in allele frequencies from one generation to the next, just by chance in which individuals reproduce. Its effect is strongest in small populations.
Gene flow: Movement of alleles between populations when individuals or their gametes migrate. It changes allele frequencies whether or not selection favors those alleles, and it makes populations more alike.
Natural selection: Individuals with heritable traits that suit their environment tend to survive and reproduce more, so those traits become more common. Which traits help depends on the environment; evolution has no goal.
Chi-square goodness-of-fit test: A test of whether observed counts in categories match the counts a model expects, using χ² = Σ(observed − expected)²/expected. It needs expected counts that aren’t too small, usually at least 5 each.

Work through an example

A DNA test of 200 fish at one gene finds 90 AA, 80 Aa and 30 aa. Find the allele frequencies p and q, and the genotype counts Hardy–Weinberg equilibrium predicts for a sample of this size.

Find allele frequencies from genotype counts →

Test Hardy–Weinberg equilibrium with chi-square →

Find the carrier frequency of a recessive allele →

Sources and scope

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OpenStax Biology 2e — Population evolution

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Open the Hardy–Weinberg tool Open worked example on a board Hardy–Weinberg equation in Biology Reference

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