How to Calculate Genotype Frequencies for Three Alleles Using Hardy-Weinberg Equilibrium

Population genetics often deals with more than two alleles at a single gene locus. The Hardy-Weinberg principle can be extended to calculate the expected genotype frequencies for any number of alleles, providing a powerful tool for understanding genetic structure in populations.

Problem Overview

• Alleles: a, b, and c

• Frequencies: a = 0.5, b = 0.2, c = 0.3

• Question: What are the expected frequencies of genotypes ab and bc at Hardy-Weinberg equilibrium?

Step 1: Use the Hardy-Weinberg Formula for Multiple Alleles

For three alleles with frequencies p, q, and r, the Hardy-Weinberg genotype frequencies are given by the multinomial expansion:

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

• p, q, r: frequencies of alleles a, b, c respectively

• 2pq: frequency of genotype ab

• 2qr: frequency of genotype bc

Step 2: Plug in the Values

Let:

• p (a) = 0.5

• q (b) = 0.2

• r (c) = 0.3

Frequency of genotype ab:

$$2pq=2\times 0.5\times 0.2=2\times 0.1=0.2$$

Frequency of genotype bc:

$$2qr=2\times 0.2\times 0.3=2\times 0.06=0.12$$

Step 3: Match with the Provided Options

• (1) 0.1 and 0.06

• (2) 0.2 and 0.15

• (3) 0.2 and 0.12

• (4) 0.04 and 0.09

The correct answer is (3) 0.2 and 0.12.

Why This Calculation Matters

• Genetic diversity: Knowing genotype frequencies helps assess genetic variation in a population.

• Breeding programs: Predicts the likelihood of offspring genotypes.

• Disease studies: Important for calculating carrier and affected rates in populations with multiple alleles.

Conclusion

In a population at Hardy-Weinberg equilibrium with allele frequencies a = 0.5, b = 0.2, and c = 0.3, the expected frequencies of genotypes ab and bc are 0.2 and 0.12, respectively.