Hardy–Weinberg equilibrium
Hardy–Weinberg equilibrium gives the genotype proportions expected from allele frequencies under random mating and a standard set of idealised population-genetic assumptions.
Start with allele frequencies
Let
\[p=P(A),\qquad q=P(a),\qquad p+q=1.\]These are allele frequencies, not genotype frequencies.
Random union of gametes
Under random mating, the allele contributed by one gamete is paired independently with the allele contributed by the other gamete. Therefore
\[P(AA)=p\times p=p^2,\]\[P(Aa)=p\times q+q\times p=2pq,\]\[P(aa)=q\times q=q^2.\]Hence the expected genotype frequencies are
\[\boxed{AA:p^2,\qquad Aa:2pq,\qquad aa:q^2}.\]Why the frequencies sum to one
\[p^2+2pq+q^2=(p+q)^2=1.\]Thus the three genotype classes form a complete probability distribution.
A numerical example
If \(p=0.6\), then \(q=0.4\). Hardy–Weinberg proportions are
\[f_{AA}=0.6^2=0.36,\]\[f_{Aa}=2(0.6)(0.4)=0.48,\]\[f_{aa}=0.4^2=0.16.\]These sum to one:
\[0.36+0.48+0.16=1.\]Recovering the allele frequency
The genotype proportions reproduce the original allele frequency:
\[p'=p^2+\frac12(2pq)=p^2+pq=p(p+q)=p.\]Similarly,
\[q'=q^2+pq=q.\]So random mating rearranges alleles into genotype combinations but does not by itself change their allele frequencies.
Classical assumptions
| Assumption | Role |
|---|---|
| Random mating with respect to the locus | Allows genotype probabilities to be formed from random pairing of gametes. |
| No selection | Genotypes do not contribute differentially because of fitness differences. |
| No mutation | Alleles are not converted into one another. |
| No migration | Allele frequencies are not altered by gene flow from other populations. |
| Effectively infinite population | Removes random changes in allele frequency caused by genetic drift. |
These assumptions define an idealised reference model. Real populations need not satisfy them exactly for Hardy–Weinberg calculations to remain useful.
Why one generation of random mating matters
Suppose the parental genotype frequencies are not in Hardy–Weinberg proportions but the gamete pool has allele frequencies \(p\) and \(q\). If gametes unite randomly, the resulting zygotes have frequencies
\[p^2,\quad 2pq,\quad q^2.\]Thus, for a simple autosomal locus under the model assumptions, random mating produces Hardy–Weinberg genotype proportions in one generation.
Equilibrium does not mean nothing happens
Individuals are still born, reproduce and die. The word equilibrium means that the relevant population-level frequencies remain unchanged under the specified model, not that biological activity stops.
Expected genotype counts
For a sample of \(N\) diploid individuals, the expected Hardy–Weinberg genotype counts are
\[E[N_{AA}]=Np^2,\]\[E[N_{Aa}]=2Npq,\]\[E[N_{aa}]=Nq^2.\]Observed counts can then be compared with these expectations.
Departure from Hardy–Weinberg proportions
If observed genotype frequencies differ from \(p^2,2pq,q^2\), at least one assumption or modelling condition may not describe the data adequately. Possible explanations include non-random mating, population subdivision, selection, genotyping error or other biological structure.
Inbreeding as an example of non-random mating
Inbreeding increases homozygosity relative to random mating. A common parameterisation using an inbreeding coefficient \(F\) is
\[f_{AA}=p^2+Fpq,\]\[f_{Aa}=2pq(1-F),\]\[f_{aa}=q^2+Fpq.\]For \(F>0\), heterozygosity is reduced while the allele frequencies remain \(p\) and \(q\).
Population subdivision
Even if separate subpopulations are individually close to Hardy–Weinberg proportions, pooling groups with different allele frequencies can produce an apparent heterozygote deficit. This is one reason population structure must be considered when interpreting deviations.
Hardy–Weinberg as a null model
The equilibrium is best viewed as a baseline. It answers the question:
What genotype frequencies would we expect if alleles paired randomly and the specified evolutionary forces did not alter their frequencies?
Observed departures can then motivate richer models.
What Hardy–Weinberg does not say
It does not say that evolution never occurs, that all real populations must have these proportions, that every deviation is caused by natural selection, or that allele frequencies can be inferred without sampling uncertainty.
Transition to evolutionary forces
Hardy–Weinberg provides the neutral reference relationship between allele and genotype frequencies. The following lessons deliberately relax its assumptions. The next lesson combines mutation with selection and asks how the balance between those processes can maintain an allele in a population.