Hardy-Weinberg Law for Three Alleles: The Trinomial Expansion Explained

The Hardy-Weinberg law is a fundamental principle in population genetics, providing a mathematical model to predict genotype frequencies from allele frequencies under ideal conditions. While most examples focus on two alleles, many genes in natural populations have more than two alleles. So, how is the Hardy-Weinberg law expressed when a gene has three alleles?

The Hardy-Weinberg Principle and Multiple Alleles

For a gene with two alleles (p and q), the Hardy-Weinberg equilibrium is represented by the binomial expansion:

^{(p+q)2=p2+2pq+q2(p+q)2=p2+2pq+q2}

Here, p2p2 and q2q2 represent the frequencies of the two homozygotes, and $2pq$ represents the frequency of the heterozygote.

The Trinomial Expansion for Three Alleles

When a gene has three alleles (let’s say p, q, and r), the Hardy-Weinberg law is represented by the trinomial expansion:

^{(p+q+r)2=p2+q2+r2+2pq+2pr+2qr(p+q+r)2=p2+q2+r2+2pq+2pr+2qr}

• p2p2, q2q2, and r2r2 are the frequencies of the three possible homozygotes.

• $2pq$, $2pr$, and $2qr$ are the frequencies of the three possible heterozygotes.

This formula allows geneticists to calculate the expected genotype frequencies for any population in Hardy-Weinberg equilibrium, regardless of the number of alleles at a locus.

Why Not Other Expansions?

• (p + q + r)^3 is not used for diploid organisms, as it would represent a triploid scenario.

• (p + q + r) is simply the sum of allele frequencies, which must always equal 1, but it does not describe genotype frequencies.

• (p + q)^2 is correct only for two alleles, not three.

Practical Example

If the frequencies of the three alleles are:

• $p=0.5$

• $q=0.3$

• $r=0.2$

The expected genotype frequencies would be:

• Homozygotes: p2=0.25p2=0.25, q2=0.09q2=0.09, r2=0.04r2=0.04

• Heterozygotes: $2pq=0.3$, $2pr=0.2$, $2qr=0.12$

Conclusion

For a gene with three alleles, the Hardy-Weinberg law is represented as:

^{(p+q+r)2(p+q+r)2}

Correct answer: (1) (p + q + r)^2