Problem 36 - Calculus for Biology and Medicine

Question

We discuss the inheritance of red-green color blindness. Color blindness is an X-linked inherited disease. A woman who carries the color blindness gene on one of her \(X\) chromosomes, but not on the other, has normal vision. \(A\) man who carries the gene on his only \(X\) chromosome is color blind. If a woman with normal vision who carries the color blindness gene on one of her \(X\) chromosomes has a child with a man who has normal vision, what is the probability that their child will be color blind?

Step-by-Step Solution

Answer

The probability their child will be color blind is 25%.

Step 1: Understand the Problem

We need to determine the probability of a child being color blind when a woman with normal vision, but carrying the color blindness gene, has a child with a man who has normal vision.

Step 2: Define the Genetic Details

Color blindness is X-linked, meaning the gene is located on the X chromosome. Let's denote: - X^C for a chromosome carrying the color blindness gene - X^N for a normal chromosome The woman has genotype X^C X^N, and the man has genotype X^N Y.

Step 3: Determine the Possible Combinations

List the possible genetic combinations for each child. The woman's possible contributions are X^C or X^N, and the man's contributions are X^N or Y. Thus, the children's possible genotypes are: 1. X^C X^N (girl, carrier, normal vision) 2. X^N X^N (girl, normal vision) 3. X^C Y (boy, color blind) 4. X^N Y (boy, normal vision)

Step 4: Calculate the Probability

We are interested in children who are color blind. Only the X^C Y combination results in color blindness and is applicable only to boys. Calculate the probability: - Probability of getting X^C Y is 1/4 or 25%, since there are 4 equally likely combinations.

Key Concepts

X-linked Inheritance

In genetics, X-linked inheritance refers to genes located on the X chromosome, one of the two sex chromosomes in humans. Genes on the X chromosome can exhibit different patterns of inheritance compared to genes on autosomes (non-sex chromosomes). This is primarily because females have two X chromosomes (XX), while males have one X and one Y chromosome (XY). Since males have only one X chromosome, any gene located on an X chromosome will fully express its trait in males, whether it is recessive or dominant.

Here are a few key points about X-linked inheritance:

• Women are more often carriers for X-linked conditions because they have two X chromosomes. A recessive gene can be masked by the normal gene on the other X chromosome.
• Men are more likely to express an X-linked recessive condition since they have only one X chromosome, and if it possesses the condition gene, it will be expressed.
• Parents pass X-linked traits to their offspring through their X chromosomes. Sons receive their X chromosome from their mother, while daughters receive one X chromosome from each parent.

Understanding X-linked inheritance allows for predictions about the likelihood of offspring inheriting certain genetic conditions, such as color blindness, which is X-linked recessive.

Color Blindness

Color blindness, most often an X-linked recessive condition, impacts the ability to distinguish between certain colors. Red-green color blindness is the most common form. Due to its genetic basis:

• Color blindness is similar to many X-linked recessive conditions, where males are primarily affected because they have a single X chromosome.
• Females, with two X chromosomes, can be carriers if they inherit one affected X chromosome (X^C X^N), displaying normal vision if the other X chromosome is normal.
• For a male to have color blindness, he must inherit an affected X chromosome from his mother, as his X is not contributed by his father.

This particular mode of inheritance makes it more prevalent in men than women, explaining the relationship between being a carrier and actually exhibiting the condition. The genetic make-up (genotype) of the parents is thus critical in determining whether their children might inherit color blindness.

Probability Calculation

To calculate the probability of a child being color blind in this scenario, we consider the genetic makeup of the parents. We know:

• The mother is a carrier with one normal and one color blindness gene on her X chromosomes (genotype X^C X^N).
• The father has normal vision and contributes either a normal X chromosome or a Y chromosome (genotype X^N Y).

The possible genetic combinations for their children are:

• X^C X^N (girl, carrier, normal vision)
• X^N X^N (girl, normal vision)
• X^C Y (boy, color blind)
• X^N Y (boy, normal vision)

To determine the probability of having a color blind child, we look at each possible outcome:

• There are 4 equally likely genetic combinations in total.
• Only one combination (X^C Y) results in color blindness and this combination results in a male child being affected.
• Thus, the probability of having a color blind child is 1 out of 4 or 25%.

Understanding how to compute these probabilities will help when evaluating genetic risks in families.