## Hardy Weinberg Calculations for Multiple Alleles at a Single Locus.

For a gene locus segregating more than two alleles, the frequency of each allele is the frequency of its homozygote plus 1/2 the sum of the frequencies of all heterozygotes in which it can occur.

### Example of calculation for a multiple allele system

Let's say we have a gene locus with three distinct molecular alleles, "X", "@" and "&". The same shorthand can be used, although there is no distinct dominance/recessiveness implied.
• The frequency of X = p (which also can be written as "f(X)")
• The frequency of @ = q (which also can be written as "f(@)")
• The frequency of & = r (which also can be written as "f(&)")
To calculate your expected genotype frequencies (if the population is in Hardy Weinberg equilibrium), simply include the additional frequency variable (r):

(p + q + r)2
which expands to...

p2 + 2pq + q2 + 2pr + 2qr + r2 = 1.0

The following genotypes are possible at this locus:

• XX (p2)
• X@ (2pq)
• @@ (q2)
• X& (2pr)
• @& (2qr)
• && (r2)
To calculate the expected genotypes if the population is in HW equilibrium, two of the alleles in homozygous condition (in this case, we'll choose @@ and &&).

Of 1000 individuals sequenced, 200 were @@ (0.2, or 20%) and 50 were && (0.05, or 5% of the population).

The square root of each value to will yield q and r:

• square root of q2 = 0.44
• square root of r2 = 0.22 And solve for p:

If p + q + r = 1.0, then 1.0 - 0.44 - 0.22 = p, or 0.34

• q = 0.45
• r = 0.22
• p = 0.33 Plug into the HW equation to calculate the expected relative genotype frequencies, based on homozygous allele frequencies:

p2 + 2pq + q2 + 2pr + 2qr + r2 = 1.0

0.332 + 2(0.33)(0.45) + 0.452 + 2(0.33)(0.22) + 2(0.45)(0.22) + 0.222

• XX (p2) = 0.11
• X@ (2pq) = 0.30
• @@ (q2) = 0.20
• X& (2pr) = 0.14
• @& (2qr) = 0.20
• && (r2) = 0.05

Which means that of your 1000 individuals (allowing for slight rounding errors):

• 110 should be XX
• 300 should be X@
• 200 should be @@
• 140 should be X&
• 200 should be @&
• 50 should be &&

...if the popuation is in HW equilibrium with respect to this locus.

Does your population differ significantly from the predicted values? Your population of 1000 is a sample size of ONE population, and it could be that your observed genotypic ratios are due to sampling error.

But if your ratios are significantly different from those predicted, the next question is to determine why.