Hardy-Weinberg Equilibrium

Assumptions

Infinitely large population
Random mating (no population structure)
No change in allele frequency between generations. This means:
- No mutation/selection/drift
- No migration
  If these assumptions are true, then after one generation the relationship between allele frequencies and genotype frequencies will follow a certain ratio.

Initial allele frequencies

We're looking at a locus with the following possible genotypes: AA, AB, BB.

If we measure the genotype frequency through sequencing or something, we can calculate the allele frequencies for A ( $p$ ) and B ( $q$ ) with the following equation, where $P_{A A}$ is the genotype frequency of AA:

p = P_{A A} + \frac{1}{2} P_{A B}

This is basically just counting the number of A alleles - the AA homozygote frequency plus half the heterozygote frequency. Then $q = 1 - p$ .

Genotype frequencies in the next generation

Now that we know the allele frequencies, which we assume don't change between generations, we use the following equations to predict the expected genotype frequencies in the next generation:

$P_{A A}^{'} = p^{2}$
$P_{A B}^{'} = 2 p q$
$P_{B B}^{'} = q^{2}$
In other words:

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

If the measured genotype frequencies in the next generation are significantly different than what is expected, it means the population is not in H-W, so one of our assumptions must not be true. This might mean:

Allele frequencies have changed, due to:
- Mutation/selection/drift
- Population Genetics - Migration between populations (gene flow)
There is Population Structure
- Non-random mating,
- Your samples are from different populations
The population isn't large enough to be considered "infinitely large" - this value depends on other species characteristics and statistical concepts.

See Population Genetics Assignment for a (sort of) practical example of this.

!mathematical-methods-lecturenotes, p.6

Jensen's Inequality

When measuring allele frequencies from herbarium collections, their heterozygosity doesn't often match H-W. This is because of Jensen's Inequality when the averages from different populations are lower than their H-W expectations by a predictable, measureable amouont.