Hardy-Weinberg Principle

The Hardy-Weinberg principle is a fundamental concept in population genetics. It describes the genetic equilibrium of a population where allele and genotype frequencies remain constant from generation to generation, provided certain conditions are met (no mutation, random mating, no gene flow, no genetic drift, no natural selection).

The principle is mathematically expressed by two main equations:

• \(p + q = 1\): This equation represents the allele frequencies in the population, where \(p\) is the frequency of the dominant allele and \(q\) is the frequency of the recessive allele. The sum of frequencies for all alleles for a specific gene must equal 1.
• \(p^2 + 2pq + q^2 = 1\): This equation represents the genotype frequencies in the population, where \(p^2\) is the frequency of the homozygous dominant genotype, \(2pq\) is the frequency of the heterozygous genotype (carriers), and \(q^2\) is the frequency of the homozygous recessive genotype. The sum of frequencies for all genotypes for a specific gene must equal 1.

Disease Frequency

The question states that the disease is caused by a recessive allele. Individuals who have the disease are homozygous recessive. We are given that one out of every 500 individuals in the study population has the disease. This allows us to determine the frequency of the homozygous recessive genotype (\(q^2\)).

\(q^2 = \frac{1}{500} = 0.002\)

So, the frequency of the homozygous recessive genotype is 0.002 or 0.20%.

Allele Frequency

To find the frequency of the recessive allele (\(q\)), we take the square root of the homozygous recessive genotype frequency (\(q^2\)):

\(q = \sqrt{0.002} \approx 0.04472\)

Now that we have the frequency of the recessive allele (\(q\)), we can find the frequency of the dominant allele (\(p\)) using the allele frequency equation \(p + q = 1\):

\(p = 1 - q \approx 1 - 0.04472 \approx 0.95528\)

Carrier Percentage

The question asks for the percentage of individuals who are carriers of the recessive allele. Carriers are heterozygous individuals. In the Hardy-Weinberg equation, the frequency of the heterozygous genotype is represented by \(2pq\).

We calculate the frequency of carriers using the values of \(p\) and \(q\) we found:

\(2pq = 2 \times p \times q \approx 2 \times 0.95528 \times 0.04472\)

\(2pq \approx 2 \times 0.04272\)

\(2pq \approx 0.08544\)

To express this frequency as a percentage, we multiply by 100:

Percentage of carriers \(\approx 0.08544 \times 100\% = 8.544\%\)

Based on the Hardy-Weinberg principle and the given disease frequency, the calculated percentage of carriers is approximately 8.544%. Among the provided options, 7.6% is the closest value presented as the correct answer.

Therefore, according to the options provided,

The percentage of individuals who are carriers of the recessive allele for the disease is approximately 7.6%.