LAISHA Explains
Genetics and plant breeding

Hardy–Weinberg equilibrium

The rule that links allele and genotype frequencies in a population, its assumptions, how to test it, and how breeders use it to estimate carriers.

Exam essential6 min readUpdated
On this page
  1. Allele and genotype frequencies
  2. The principle
  3. The assumptions
  4. Estimating carriers of a recessive defect
  5. Testing whether a population is in equilibrium
  6. What changes allele frequencies
  7. Beyond two alleles
  8. Key points
  9. What the exams ask
  10. Questions people ask
  11. Sources
  12. Related on Laisha
  13. Discussion

The Hardy–Weinberg principle says that in a large, randomly mating population with no selection, mutation or migration, allele and genotype frequencies stay constant from generation to generation, and the genotype frequencies can be predicted from the allele frequencies alone.

It was published independently in 1908 by the English mathematician G. H. Hardy and the German physician Wilhelm Weinberg. It is the starting point of population genetics and animal and plant breeding, and Hardy–Weinberg calculations, especially estimating carriers of a recessive defect, are standard in genetics, animal breeding and veterinary exams.

Allele and genotype frequencies

Take one gene with two alleles, A and a, in a population.

  • The allele frequency of A, written p, is the proportion of all copies of the gene that are A. The frequency of a is q. Since there are only two alleles, p + q = 1.
  • The genotype frequencies are the proportions of AA, Aa and aa individuals.

Allele frequencies can always be calculated from genotype counts:

Allele frequencies from genotype counts
p = (2 × NAA + NAa) ⁄ (2N) and q = 1 − p

because each AA individual carries two A alleles, each Aa one, and the population of N individuals carries 2N copies.

The principle

If gametes combine at random, the chance that an egg and a sperm both carry A is p × p. That gives:

Hardy–Weinberg genotype frequencies
AA : Aa : aa = p2 : 2pq : q2, and p2 + 2pq + q2 = 1

The heterozygote term is 2pq because an Aa individual can arise two ways: A from the mother and a from the father, or the reverse. It is simply (p + q)2 expanded.

Two further results matter:

  • Equilibrium is reached in one generation of random mating for a gene on an ordinary chromosome, whatever the starting genotype frequencies.
  • Allele frequencies do not change on their own. Dominance does not make a dominant allele spread; a rare dominant allele stays rare unless something else acts on it.
Hardy–Weinberg genotype frequencies plotted against the frequency q of the recessive allele: the homozygous dominant frequency p squared falls, the homozygous recessive frequency q squared rises, and the heterozygote frequency 2pq peaks at 0.5 when p equals q; when a recessive allele is rare, almost all its copies are in carriers 2pq = 0.5 at q = 0.5 p² (AA) q² (aa) 2pq (Aa) 0.00.20.40.60.81.00.00.20.40.60.81.0Frequency of recessive allele a (q) Genotype frequency If q = 0.01 aa = 0.0001 1 in 10,000 Aa = 0.0198 about 1 in 50 carriers hold 99% of the a alleles
Genotype frequencies under Hardy–Weinberg equilibrium as the frequency of the recessive allele changes. Heterozygotes are most common when both alleles are equally frequent; when a recessive allele is rare, nearly all of its copies are hidden in carriers.Diagram: Laisha, original.

The assumptions

The equilibrium holds only when all of these are true. Each failure is one of the forces of evolution, and the principle is useful precisely because departures from it point to them.

  1. Large population: in small populations, chance changes allele frequencies (genetic drift).
  2. Random mating with respect to the gene: no inbreeding, no assortative mating.
  3. No selection: all genotypes survive and reproduce equally.
  4. No mutation changing one allele into the other.
  5. No migration (gene flow) bringing alleles in or out.
  6. Allele frequencies equal in males and females, and generations not overlapping (simplifying assumptions).

Breeding programmes break assumptions 2 and 3 on purpose: selection changes allele frequencies, and planned matings are not random.

Estimating carriers of a recessive defect

The most practical use of the principle is estimating how many apparently normal animals or plants carry a hidden recessive allele.

Worked example

Carriers of a recessive defect

In a cattle breed, a lethal recessive defect appears in 1 calf in 10,000. Assuming Hardy–Weinberg equilibrium, how common are carriers?

  1. Affected calves are aa: q2 = 1 ⁄ 10,000 = 0.0001.
  2. q = √0.0001 = 0.01, so p = 0.99.
  3. Carriers (Aa): 2pq = 2 × 0.99 × 0.01 = 0.0198, about 1 animal in 50.

The defect looks rare, but carriers are 198 times more common than affected calves, and 99 per cent of the defective alleles are in carriers. This is why breed societies test sires with DNA tests: one popular carrier bull, used widely through artificial insemination, can spread the allele quickly.

Testing whether a population is in equilibrium

Observed genotype counts are compared with those expected from the population's own allele frequencies, using a chi-square test. Because one parameter (p) is estimated from the same data, the degrees of freedom are the number of genotype classes minus 1, minus 1 more: for three genotypes, 1 degree of freedom.

Worked example

A chi-square test for equilibrium

A herd of 500 cattle is typed for a gene with codominant alleles R and W: 196 RR, 208 RW and 96 WW.

  1. p(R) = (2 × 196 + 208) ⁄ 1,000 = 600 ⁄ 1,000 = 0.6; q(W) = 0.4.
  2. Expected: RR = 0.36 × 500 = 180; RW = 0.48 × 500 = 240; WW = 0.16 × 500 = 80.
  3. χ2 = (196 − 180)2 ⁄ 180 + (208 − 240)2 ⁄ 240 + (96 − 80)2 ⁄ 80 = 1.42 + 4.27 + 3.20 = 8.89.
  4. The 5 per cent table value at 1 degree of freedom is 3.841. Since 8.89 is larger, the herd is not in equilibrium.

There are too few heterozygotes and too many homozygotes. That pattern points to inbreeding, or to the herd being a mixture of groups with different allele frequencies (the Wahlund effect), rather than to selection on this gene.

What changes allele frequencies

Selection. Selection against a recessive allele, by culling affected (aa) individuals only, is slow when the allele is rare, because almost all copies hide in carriers. If affected individuals never breed, the frequency after t generations follows:

Complete selection against a recessive
qt = q0 ⁄ (1 + t q0), so t = 1 ⁄ qt − 1 ⁄ q0

Halving q from 0.01 to 0.005 takes 1 ⁄ 0.005 − 1 ⁄ 0.01 = 200 − 100 = 100 generations. Testing and removing carriers is far faster, which is why DNA tests transformed the control of recessive defects in livestock and dogs.

Inbreeding does not change allele frequencies but redistributes genotypes: heterozygotes fall to 2pq(1 − F), where F is the inbreeding coefficient, and each homozygote rises by Fpq. This is why inbreeding exposes recessive defects.

Mutation, migration and drift change frequencies too: migration quickly when populations differ, drift strongly in small populations such as rare breeds, zoo populations and small seed collections.

Beyond two alleles

The principle extends naturally. With three alleles of frequencies p, q and r (as in the ABO blood groups), genotype frequencies are the terms of (p + q + r)2. For genes on the X chromosome, males show allele frequencies directly (a recessive X-linked trait appears in males at frequency q and in females at q2), and equilibrium is approached over several generations rather than one.

Key points

  • p + q = 1; genotype frequencies p2 (AA), 2pq (Aa), q2 (aa); they sum to 1.
  • Equilibrium is reached after one generation of random mating and then stays, if the assumptions hold.
  • Assumptions: large population, random mating, no selection, mutation or migration.
  • Carriers of a rare recessive far outnumber affected individuals: at q = 0.01, 1 in 50 carriers versus 1 in 10,000 affected.
  • Test equilibrium with chi-square at 1 degree of freedom for a two-allele gene; a heterozygote deficit suggests inbreeding or population mixture.
  • Culling affected recessives alone reduces q slowly (t = 1⁄qt − 1⁄q0 generations); carrier testing is much faster.

What the exams ask

  • School-leaving and entrance papers: the equation, the assumptions, and simple calculations from q2 to carrier frequency.
  • Degree and research-fellowship entrances in genetics and animal breeding: allele frequencies from counts, chi-square tests for equilibrium, forces that disturb it, and the effect of inbreeding.
  • Veterinary and animal-breeding licensing examinations: control of recessive genetic defects, carrier testing, and selection response for single genes.

Described in general terms: the kind of examination, not a past paper. Laisha never reproduces exam questions.

Questions people ask

Does a dominant allele become more common over time?

Not by being dominant. Without selection or other forces, its frequency stays the same. Dominance only describes how the heterozygote looks.

Why can't I just cull affected animals to remove a recessive defect?

Because when the allele is rare almost all copies are in healthy-looking carriers. Culling only affected animals removes a tiny share of the alleles each generation. Testing and managing carriers is far more effective.

Do real populations ever meet the assumptions?

Not exactly, but many large, randomly mating populations come close for genes that are not under strong selection. The principle is used as a baseline: departures from it are what tell geneticists something is happening.

Sources

  1. Hardy, G. H. "Mendelian proportions in a mixed population." Science 28, 49–50 (1908).
  2. Falconer, D. S. and Mackay, T. F. C. Introduction to Quantitative Genetics, 4th edition. Longman, 1996. Chapters 1–3.
  3. Hartl, D. L. and Clark, A. G. Principles of Population Genetics, 4th edition. Sinauer, 2007.
  4. Nicholas, F. W. Introduction to Veterinary Genetics, 3rd edition. Wiley-Blackwell, 2010.
  5. Online Mendelian Inheritance in Animals (OMIA). University of Sydney, online database of inherited disorders.

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