Hardy & Weinberg

The law of Hardy & Weinberg is the base for population genetics. It describes how different alleles of a gene stay is a population over the generations. The starting point of this law is the notion of allele frequency.

Allele frequency is the frequency of a certain allele of a gene in a population. In a population of all non spangle birds the allele frequency of the Sp allele is 0%. In a population of all double factor spangles the allele frequency of the Sp allele is 100%. In a population of all single factor spangles the allele frequency of the Sp allele is 50%.

Let us take a population of 50 birds: 5 double factor spangles, 9 single factor spangles en 36 non spangles.
In this population the allele frequency of the Sp allele is:
p = 5/50 * 100% + 9/50 * 50% + 36/50 * 0% = 19%
The allele frequency of the Sp⁺ allele is:
q = 5/50 * 0% + 9/50 * 50% + 36/50 * 100% = 81%
The sum of the allele frequencies (p + q) is 1 (or 100%) because these are the only two alleles of the gene in the population.

What happens with the allele frequencies in this population over time? Because we're doing population genetics we will no longer look at the individual pairs but we will look at the population. So we are no longer interested in the gametes of the individual birds, but in the gametes produced in the population and in what proportion.

The gametes produced in the population are Sp and Sp⁺. The proportions of these gametes are of course the allele frequencies:
Proportion allele Sp = p
Proportion allele Sp⁺ = q

Filling in the Punnet square:

Sp
(p) Sp⁺
(q)

Sp
(p) Sp/Sp
(p²) Sp⁺/Sp
(p * q)

Sp⁺
(q) Sp⁺/Sp
(p * q) Sp⁺/Sp⁺
(q²)

The offspring and the proportions of the genotypes are:
Sp/Sp: double factor spangle (p²)
Sp⁺/Sp: single factor spangle (2pq)
Sp⁺/Sp⁺: non spangle (q²)

If p + q = 1, then p² + 2pq + q² = 1.

The allele frequency of the Sp allele in the offspring is:
p² * 100% + 2pq * 50% + q² * 0% = p² + pq
Unleashing mathematics onto it and remembering p + q = 1:
p² + pq = p * (p + q) = p

The allele frequencies in the offspring are just the same as in the original population. On top of that there is a relation between the allele frequencies and the proportion of the genotypes. This is the law of Hardy & Weinberg:
In a population with just two alleles of a gene, being a and A, where p and q are the respective allele frequencies (p + q = 1), the allele frequencies are constant. Once the population is in equilibrium the proportions of the genotypes are:
aa: p²
Aa: 2pq
AA: q²
These proportions of genotypes is called the Hardy & Weinberg equilibrium.