Problem 5 - Campbell Biology

Question

A fruit fly population has a gene with two alleles, \(A 1\) and \(A 2\) . Tests show that 70\(\%\) of the gametes produced in the population contain the \(A 1\) allele. If the population is in Hardy-Weinberg equilibrium, what proportion of the flies carry both \(A 1\) and \(A 2 ?\) \begin{equation}{ (A) }0.7 \quad \text { (B) } 0.49 \quad \text { (C) } 0.42 \quad \text { (D) } 0.21 \end{equation}

Step-by-Step Solution

Answer

0.42 (C)

Step 1 - Understanding Hardy-Weinberg Equilibrium

In a population in Hardy-Weinberg equilibrium, allele and genotype frequencies remain constant from generation to generation. The equation used is: \[ p^2 + 2pq + q^2 = 1 \]where \( p \) and \( q \) are the frequencies of the two alleles.

Step 2 - Identifying Allele Frequencies

Here we are given the frequency of allele \( A1 \), which is 70\( \% \) or 0.7. Therefore, \( p = 0.7 \). Since there are only two alleles, the frequency of \( A2 \) would be: \[ q = 1 - p = 1 - 0.7 = 0.3 \]

Step 3 - Calculating Heterozygous Proportion

The proportion of flies that carry both \( A1 \) and \( A2 \) (heterozygous genotype \( A1A2 \)) is represented by the term \( 2pq \). So, we calculate: \[ 2pq = 2 \times 0.7 \times 0.3 = 0.42 \]

Step 4 - Conclusion

The calculated proportion of flies carrying both \( A1 \) and \( A2 \) is 0.42, which matches with option (C).

Key Concepts

allele frequencies

Allele frequencies describe how common an allele is in a population. They are usually expressed as a proportion or percentage. For example, if you have a gene with two alleles, say, \(A1\) and \(A2\), the frequency of \(A1\) is denoted as \(p\) and the frequency of \(A2\) is denoted as \(q\). These frequencies can be determined through genetic testing or observation.
According to the Hardy-Weinberg principle, the sum of the allele frequencies for a gene with two alleles is always equal to 1. This means:

• \( p + q = 1 \)

In this exercise, we are given the frequency of the \(A1\) allele as 0.7, thus:
\( p = 0.7 \). Since there are only two alleles, we calculate \( q \) as:
\( q = 1 - p = 1 - 0.7 = 0.3 \).
Once you have the allele frequencies, you can use them to predict the frequencies of different genotypes under Hardy-Weinberg equilibrium.

genotype frequencies

Genotype frequencies represent how often specific combinations of alleles (genotypes) appear in a population. Under Hardy-Weinberg equilibrium, these frequencies are calculated using the binomial expansion theorem. The equation used is:
\( p^2 + 2pq + q^2 = 1 \)
In this equation:

• \( p^2 \) represents the frequency of the homozygous dominant genotype (\(A1A1\))
• \( 2pq \) represents the frequency of the heterozygous genotype (\(A1A2\))
• \( q^2 \) represents the frequency of the homozygous recessive genotype (\(A2A2\))

For our problem, we are specifically interested in the heterozygous genotype frequency, which is calculated as \(2pq\). Knowing that \( p = 0.7 \) and \( q = 0.3 \), we can substitute these values into the equation:
\( 2pq = 2 \times 0.7 \times 0.3 = 0.42 \).
So, the frequency of the heterozygous genotype \(A1A2\) in this population is 0.42 or 42\( \% \).

heterozygous genotype

The heterozygous genotype is an important concept in genetics. It occurs when an individual carries two different alleles for a particular gene—one from each parent. In this exercise, we are looking at the genotype \(A1A2\), where one allele is \(A1\) and the other is \(A2\).
The proportion of a population that carries the heterozygous genotype can provide insights into genetic diversity and potential evolutionary changes. Using the Hardy-Weinberg principle, the frequency of the heterozygous genotype can be calculated with the following formula:
\( 2pq \).
To find this frequency, simply multiply the frequency of one allele by the frequency of the other and multiply the result by 2. This gives 42\(\%\) in our example for the \(A1A2\) genotype.
Understanding heterozygous genotypes helps geneticists determine how traits are passed down and can be useful in studying populations for traits and diseases.