Question

One of the first Mendelian traits identified in humans was a dominant condition known as brachydactyly. This gene causes an abnormal shortening of the fingers or toes (or both). At the time, some researchers thought that the dominant trait would spread until 75 percent of the population would be affected (because the phenotypic ratio of dominant to recessive is 3: 1 ). Show that the reasoning was incorrect.

Short answer

Answer: The actual percentage of the population affected by the dominant brachydactyly trait based on the given scenario using the Hardy-Weinberg equilibrium equation is 75 percent. However, it is important to note that this calculation is theoretical and might differ in reality due to factors such as selection, migration, or mutation.

Step-by-step solution

Understand the problem and identify the given information

We are given the following information: - Dominant trait: brachydactyly (shortening of fingers and toes) - Phenotypic ratio: 3:1 (dominant:recessive)

Calculate the frequencies of the dominant and recessive alleles

Phenotypic ratio is 3:1, which means that 3 out of 4 individuals show the dominant trait, and 1 out of 4 individuals show the recessive trait. Let p be the frequency of the dominant allele (B) and q be the frequency of the recessive allele (b). Since B is dominant, BB and Bb genotypes will both show brachydactyly, while bb genotype will not. According to the phenotypic ratio: - 3 out of 4 individuals (BB and Bb) show the dominant trait (brachydactyly) - 1 out of 4 individuals (bb) show the recessive trait (normal fingers and toes) The frequency of the recessive genotype (bb) is 1/4 or 0.25. According to the Hardy-Weinberg equilibrium equation, q² = 0.25, where q is the frequency of the recessive allele (b).

Find the frequency of the recessive allele (q)

Since we know that q² = 0.25, we can find the frequency of the recessive allele (q) by taking the square root of 0.25: q = sqrt(0.25) q = 0.5

Find the frequency of the dominant allele (p)

Knowing the frequency of the recessive allele (q), we can calculate the frequency of the dominant allele (p) using the equation p + q = 1. p = 1 - q p = 1 - 0.5 = 0.5

Calculate the actual percentage of the population affected by the dominant trait

Now that we have the frequencies of the dominant (p) and recessive (q) alleles, we can calculate the actual percentage of the population affected by the dominant trait. The percentage of the population with either the BB or Bb genotype (showing brachydactyly) is (p² + 2pq) * 100. Percentage = (0.5² + 2 * 0.5 * 0.5) * 100 Percentage = (0.25 + 0.5) * 100 Percentage = 0.75 * 100 Percentage = 75 The actual percentage of the population affected by the dominant trait (brachydactyly) is 75 percent, which is exactly the same as the researchers thought. However, it's important to note that the Hardy-Weinberg equilibrium provides a theoretical calculation when no other factors are involved (e.g., selection, migration, or mutation). In reality, the percentage of people affected by brachydactyly could be different due to these factors not being considered in this exercise.

Key concepts

dominant and recessive traits

In Mendelian genetics, traits are often categorized as either dominant or recessive. This classification helps us understand how certain characteristics, such as brachydactyly, are passed from parents to offspring.
Dominant traits, like brachydactyly, manifest prominently even when only one parent passes on the allele responsible for this trait. This means if an individual has at least one allele for a dominant trait (BB or Bb), the trait will be expressed.
On the other hand, recessive traits appear only when an individual has two copies of the recessive allele (bb). This is why recessive traits sometimes skip generations or appear less frequently in populations.
Understanding these genetic principles helps explain how and why certain traits are visible in specific individuals while others are hidden.

Hardy-Weinberg equilibrium

The Hardy-Weinberg equilibrium is a principle that provides a framework for understanding how allele frequencies remain constant in a population over time, provided that certain conditions are met. These conditions include a large population size, random mating, no mutation, no migration, and no natural selection.
This model uses an equation to relate the frequencies of alleles and genotypes within a population:

• p: the frequency of the dominant allele
• q: the frequency of the recessive allele

The sum of these allele frequencies equals 1 (or 100%):
\[ p + q = 1 \]
To calculate genotype frequencies, the equation \[ p^2 + 2pq + q^2 = 1 \] is used, representing:

• \( p^2 \): the frequency of the homozygous dominant genotype (BB)
• \( 2pq \): the frequency of the heterozygous genotype (Bb)
• \( q^2 \): the frequency of the homozygous recessive genotype (bb)

In our example, brachydactyly’s prevalence aligns with these genotype frequencies under ideal conditions.

allele frequencies

Allele frequency is a crucial concept in understanding genetic variation within a population. It refers to how often a particular allele occurs in the gene pool, compared with other alleles for the same gene.
In the given exercise, we calculate allele frequencies to determine how common the brachydactyly trait is.
When considering Mendelian traits, allele frequencies directly correspond to the expression of dominant and recessive traits in the population, serving as the basis for predicting genetic outcomes across generations.
For example, the recessive allele \( q \), which was calculated as 0.5, and the dominant allele \( p \) which is also 0.5 in our exercise, highlight the equal distribution of alleles in the population under the Hardy-Weinberg equilibrium.
These frequencies guide geneticists in predicting how traits will propagate across generations, providing insights into genetic diversity and the likelihood of specific traits appearing.