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Hardy-Weinberg Equilibrium

The foundation of population genetics

Try it: De Finetti DiagramChi-Square Test

The Principle

The Hardy-Weinberg principle, formulated independently by G.H. Hardy and Wilhelm Weinberg in 1908, describes the relationship between allele and genotype frequencies in an idealized population. It states that allele and genotype frequencies remain constant from generation to generation in the absence of evolutionary forces.

Key insight: Hardy-Weinberg equilibrium serves as a null model against which we can detect evolutionary change. Deviations from HWE indicate that evolutionary forces are acting on a population.

The Equilibrium Equations

For a single locus with two alleles, A (frequency = p) and a (frequency = q), the genotype frequencies at equilibrium are:

p + q = 1 Allele frequencies sum to one
p² + 2pq + q² = 1 Genotype frequencies at HWE

Where:

  • p² = frequency of AA homozygotes
  • 2pq = frequency of Aa heterozygotes
  • q² = frequency of aa homozygotes

Interactive De Finetti Diagram

The De Finetti diagram provides a geometric representation of all possible genotype frequencies. The parabola represents all populations in Hardy-Weinberg equilibrium. Drag the point along the base to see how genotype frequencies change with allele frequency.

p (A) 0.50
q (a) 0.50
AA (p²) 0.250
Aa (2pq) 0.500
aa (q²) 0.250

The HWE parabola shows maximum heterozygosity (2pq = 0.5) at p = q = 0.5

Assumptions of Hardy-Weinberg

The equilibrium holds under these idealized conditions:

  1. No mutation — allele frequencies are not altered by new mutations
  2. Random mating — individuals pair by chance, not by genotype
  3. No selection — all genotypes have equal fitness
  4. Infinite population size — no genetic drift
  5. No gene flow — no migration in or out of the population
In reality, no natural population perfectly meets these conditions. The value of HWE lies in its use as a baseline for detecting and measuring evolutionary forces.

Reaching Equilibrium

A remarkable property of HWE is that equilibrium is reached in a single generation of random mating, regardless of the initial genotype frequencies. This happens because:

  1. Allele frequencies (p and q) are determined by the previous generation
  2. Random mating produces all possible diploid combinations
  3. The resulting genotype frequencies are exactly p², 2pq, and q²

Applications

Testing for HWE

The chi-square test compares observed genotype frequencies to expected HWE frequencies. Significant deviations suggest non-random mating, selection, or population structure.

Estimating Carrier Frequencies

For rare recessive diseases, if the disease frequency (q²) is known, we can estimate:

  • The recessive allele frequency: q = √(disease frequency)
  • The carrier frequency: 2pq ≈ 2q (when q is small)
If q² = 0.0001 (1 in 10,000 affected)
Then q = 0.01 and carrier frequency ≈ 2% Example: estimating carrier frequency

Forensic Genetics

HWE is used to calculate genotype probabilities for DNA profiles. When combining probabilities across multiple independent loci, HWE provides the foundation for match probability calculations.

Historical Context

The Hardy-Weinberg principle resolved a major misconception in early genetics. Critics of Mendelian inheritance argued that dominant alleles should eventually "swamp" recessive ones. Hardy and Weinberg independently showed that, under neutral conditions, both alleles persist indefinitely at their original frequencies.

This insight was crucial for the Modern Synthesis, bridging Mendelian genetics with Darwinian evolution and establishing population genetics as a quantitative discipline.