Question

A locus at Hardy-Weinberg equilibrium in a diploid organism has n alleles. The maximum heterozygosity (i.e., proportion of heterozygotes) for this locus is

• A. n
• B. 1/n
• C. 1 – (1/n)
• D. 1 – n

Hint:

This is a classic case of complementary gene interaction, where both genes are required for the expression of the trait (purple colour).

Answer 1

The Correct Option is C

Step 1: Express heterozygosity in terms of allele frequencies.
Let the \(n\) allele frequencies be \(p_1,\ldots,p_n\) with \(\sum_i p_i = 1\). Under HWE, the expected heterozygosity \[ H \;=\; 1 - \sum_{i=1}^{n} p_i^2 . \]

Step 2: Maximize \(H\) subject to \(\sum p_i=1\).
Maximizing \(H\) is equivalent to minimizing \(\sum p_i^2\). By symmetry/convexity (or by Lagrange multipliers), the sum of squares is minimized when all \(p_i\) are equal: \[ p_i = \frac{1}{n}\quad \forall i. \]

Step 3: Evaluate \(H\) at the optimum.
\[ H_{\max} = 1 - \sum_{i=1}^{n}\left(\frac{1}{n}\right)^2 = 1 - n\cdot \frac{1}{n^2} = 1 - \frac{1}{n}. \] Final Answer:
\[ \boxed{\,1-\frac{1}{n}\,} \]