Question

Hardy-Weinberg Law Common blood types are determined genetically by three alleles \(\mathrm{A}, \mathrm{B},\) and \(\mathrm{O} .\) (An allele is any of a group of possible mutational forms of a gene.) A person whose blood type is \(\mathrm{AA}, \mathrm{BB},\) or \(\mathrm{OO}\) is homozygous. A person whose blood type is \(\mathrm{AB}, \mathrm{AO},\) or \(\mathrm{BO}\) is heterozygous. The HardyWeinberg Law states that the proportion \(P\) of heterozygous individuals in any given population is \(P(p, q, r)=2 p q+2 p r+2 q r\) where \(p\) represents the percent of allele \(\mathrm{A}\) in the population, \(q\) represents the percent of allele \(\mathrm{B}\) in the population, and \(r\) represents the percent of allele \(\mathrm{O}\) in the population. Use the fact that \(p+q+r=1\) to show that the maximum proportion of heterozygous individuals in any population is \(\frac{2}{3}\)

Short answer

The maximum proportion of heterozygotes according to the Hardy-Weinberg Law is \(\frac{2}{3}\). This is the maximum proportion possible under the assumption of random mating and no other forces changing the allele frequencies.

Step-by-step solution

Analysis of the Variables

First, let's notice the variables used here: \(p, q, r\). These represent the percent of alleles \(\mathrm{A}, \mathrm{B},\) and \(\mathrm{O}\) in the population respectively. According to the Hardy-Weinberg Law, the proportion of heterozygous individuals (P) in the population can be represented by the equation \(P(p, q, r)=2 p q+2 p r+2 q r\).

Implement the Given Condition

We are given that the total sum of the percentages of the three alleles is equal to one, \(p+q+r=1\). This equation is a result of the law of total probability, stating that the sum of the probabilities of all possible events is 1.

Calculate the Derivative

The maximum value of \(P\), according to the Hardy-Weinberg Law, will be where the derivative of this function equals zero. We know the derivative of \(P\) is the first derivative of the function \(P'\). Let us calculate the derivative of the function \(P\) with respect to each variable (\(p, q, r\)), keeping other variables constant and setting them equal to zero.

Solve for the Variables

After getting the derivatives and setting them to zero, we can solve the system of equations obtained to find the values of \(p\), \(q\), and \(r\).

Second Derivative Test

To assure that the values of \(p\), \(q\), \(r\) obtained provides maximum value for \(P\), we perform the second derivative test. If the second derivative of the function is negative then the function has a relative maximum at that point.

Substitute the Values in the Equation

After verifying with second derivative test and getting the values of \(p\), \(q\), and \(r\), substitute these values into the equation of \(P\). The result obtained will be the maximum proportion of heterozygous individuals under this model.