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Chapter 23: Problem 9

A trait determined by an X-linked dominant allele shows 100 percent penetrance and is expressed in 36 percent of the females in a population. Assuming that the population is in Hardy-Weinberg equilibrium, what proportion of the males in this population express the trait?

Short Answer

Expert verified
81.3% of the males express the trait.

Step by step solution

01

Understanding X-linked Dominant Traits

In X-linked dominant inheritance, females have two X chromosomes and thus have two possible alleles, while males have only one X chromosome. Therefore, females can be homozygous dominant, heterozygous, or homozygous recessive. Males can only be hemizygous dominant or recessive. Since the allele has 100% penetrance, any individual with the allele will express the trait.
02

Relating to Hardy-Weinberg Equilibrium

Under Hardy-Weinberg equilibrium, allele frequencies in the population remain constant. Given the allele frequency for females and the trait is expressed in 36% of females, we interpret this as the phenotypic frequency representing the sum of homozygous dominant (XAXA) and heterozygous (XAXa) females expressing the trait due to complete penetrance.
03

Calculating Frequency of the Dominant Allele in Females

Let p be the frequency of the dominant allele (XA) and q be the frequency of the recessive allele (Xa). The proportion of females expressing the trait can be given by: p^2 + 2pq = 0.36. As q = 1 - p, substitute q in the equation, leading to p^2 + 2p(1-p) = 0.36, which simplifies to 3p^2 - 2p - 0.36 = 0.
04

Solving the Quadratic Equation

To solve 3p^2 - 2p - 0.36 = 0, use the quadratic formula: p = [ -b 卤 sqrt(b^2 - 4ac) ] / 2a, with a = 3, b = -2, c = -0.36. Compute the discriminant first: (-2)^2 - 4(3)(-0.36) = 4 + 4.32 = 8.32.
05

Applying the Quadratic Formula

Apply the quadratic formula: p = [2 卤 sqrt(8.32)] / 6. Calculate sqrt(8.32) 鈮� 2.88. Therefore, p = (2 卤 2.88) / 6 leads to two solutions. Choose the positive feasible solution: p = (2 + 2.88) / 6 鈮� 0.813.
06

Calculating Proportion of Males Expressing the Trait

Since males are either XA (expressing the trait) or Xa, and since the allele is dominant with full penetrance, the proportion of males showing the trait depends solely on p. Hence, about 81.3% of males express the trait, as p = 0.813 represents the prevalence of the XA allele among males.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Hardy-Weinberg equilibrium
The Hardy-Weinberg equilibrium is a fundamental principle in genetics used to describe the genetic variation in a population. It posits that the genetic makeup will remain constant over generations in the absence of evolutionary influences. There are five major conditions for a population to be in Hardy-Weinberg equilibrium:
  • Large population size to minimize genetic drift.
  • No mutation altering allele frequencies.
  • No migration, thus preventing gene flow.
  • Random mating to ensure every individual has an equal chance of reproduction.
  • No natural selection affecting allele frequencies.
When these conditions are met, allele and genotype frequencies stay constant, making it easier to predict the genetic traits in a population. This principle is crucial in calculating expected genetic variation like the expression of X-linked traits.
100% penetrance
Penetrance refers to the likelihood that a person carrying a particular gene variant will express the trait associated with it. When a trait exhibits 100% penetrance, it means that every individual with the allele will express the trait, regardless of their environmental factors or genetic background.
In the case of the X-linked dominant trait from the exercise, this means that if an individual possesses even one copy of the dominant allele, they will show the trait. This simplifies calculations because you don鈥檛 need to account for variability in expression, which might complicate understanding of the gene's impact on the trait.
Allele frequency
Allele frequency is a measure of how common a particular allele is in a population. In the context of the exercise, it is crucial for determining how many individuals in the population express a given trait.
In a population at equilibrium, the sum of the frequencies of all alleles for a particular gene must equal 1. This is represented mathematically as:
  • If p is the frequency of the dominant allele and q is the frequency of the recessive allele, then:
\[ p + q = 1 \]
This relationship is used to calculate the frequency of genetic traits, such as the X-linked dominant trait in females, which involves both the homozygous dominant and heterozygous expressions for the trait.
Quadratic formula in genetics
In genetics, the quadratic formula is often used when dealing with equations that arise from allele frequency calculations. These equations usually result from setting up models based on Hardy-Weinberg equilibrium where you need to solve for allele frequencies.
The quadratic formula is given by:
  • \( p = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)
In our exercise, we had a quadratic equation from the expression of the trait in females, \( 3p^2 - 2p - 0.36 = 0 \). By using the quadratic formula, we were able to find the value of \( p \), the frequency of the dominant allele, which was essential to determine the proportion of males expressing the trait.

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Most popular questions from this chapter

Because individuals with cystic fibrosis die before they can reproduce, the coefficient of selection against them is \(s=1 .\) Assume that heterozygous carriers of the recessive mutant allele responsible for this disease are as fit as wild-type homozygotes and that the population frequency of the mutant allele is 0.02 (a) Predict the incidence of cystic fibrosis in the population after one generation of selection. (b) Explain why the incidence of cystic fibrosis hardly changes even with \(s=1\)

The frequency of newborn infants homozygous for a recessive lethal allele is about 1 in 25,000 . What is the expected frequency of carriers of this allele in the population?

A population is segregating three alleles, \(A_{1}, A_{2},\) and \(A_{3}\) with frequencies \(0.2,0.5,\) and \(0.3,\) respectively. If these alleles are selectively neutral, what is the probability that \(A_{2}\) will ultimately be fixed by genetic drift? What is the probability that \(A_{3}\) will ultimately be lost by genetic drift?

In a survey of moths collected from a natural population, a researcher found 51 dark specimens and 49 light specimens. The dark moths carry a dominant allele, and the light moths are homozygous for a recessive allele. If the population is in Hardy-Weinberg equilibrium, what is the estimated frequency of the recessive allele in the population? How many of the dark moths in the sample are likely to be homozygous for the dominant allele?

In Drosophila the ruby eye phenotype is caused by a recessive, X-linked mutant allele. The wild-type eye color is red. A laboratory population of Drosopbila is started with 25 percent ruby-eyed females, 25 percent homozygous red-eyed females, 5 percent ruby-eyed males, and 45 percent red-eyed males. (a) If this population mates randomly for one generation, what is the expected frequency of ruby-eyed males and females? (b) What is the frequency of the recessive allele in each of the sexes?
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