Problem 36 - Calculus for Biology and Medicine

Question

In Problems \(36-37\), 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 that the child is color blind is 25%.

Step 1: Understand the Genetic Inheritance

Color blindness is caused by a recessive gene carried on the X chromosome. Women have two X chromosomes (XX) while men have one X and one Y chromosome (XY). A woman with one normal vision gene and one color-blindness gene on her X chromosomes is not color-blind, but a man with the color-blindness gene on his X chromosome is color-blind because he doesn't have a second X chromosome to counteract it.

Step 2: Identify the Genetic Makeup of the Parents

The woman has normal vision but carries the color blindness gene, so her genotype is XᴺXᶜ where Xᴺ represents a normal vision X chromosome and Xᶜ represents a color blindness X chromosome. The man has normal vision, so his genotype is XᴺY.

Step 3: Determine Possible Genetic Combinations for Children

To find the possible genetic combinations for their children, we set up a Punnett square. This allows us to see the potential genotypes for their offspring. The combinations from the parents' genotypes XᴺXᶜ (mother) and XᴺY (father) are XᴺXᴺ, XᴺXᶜ, XᴺY, and XᶜY.

Step 4: Calculate the Probability of Color Blindness

From the possible genotypes listed, XᶜY represents a color-blind boy, as he has the color blindness gene on his only X chromosome. Since there are four possible genotype outcomes (XᴺXᴺ, XᴺXᶜ, XᴺY, and XᶜY), and only one of these (XᶜY) results in a color-blind child, the probability is 1 out of 4, or 25%.

Key Concepts

X-linked inheritance

X-linked inheritance is an important genetic principle, especially when discussing traits tied to gender. This type of inheritance involves genes that are located on the X chromosome. In humans, females have two X chromosomes (XX), while males have one X and one Y chromosome (XY).

This difference in chromosome pairs between males and females is crucial for understanding X-linked traits. In X-linked inheritance:

• Females can be carriers if they have one normal and one affected X chromosome, but they often do not show symptoms.
• Males express the trait if their single X chromosome carries the gene, as they do not have a second, potentially normal, X to mask the gene.

This type of genetic transmission explains why certain conditions, like color blindness, appear more frequently in males than in females.

Punnett square

The Punnett square is a simple yet powerful tool used in genetics to predict the possible genotypes of offspring from parental genotypes. It is a grid that helps visualize allele combinations and the resulting probability of inheriting specific traits.

For example, when examining X-linked traits, such as color blindness:

• You place the alleles of one parent along the top and those of the other parent along the side of the square.
• Each box within the Punnett square represents a potential allele combination in the offspring, showing all possible genetic outcomes.

This allows parents and educators to determine the likelihood of a child inheriting a recessive condition, like color blindness, by assessing the resulting genotypes in the squares.

recessive gene

A recessive gene is a form of a gene that does not produce a trait unless both copies within the individual (one from each parent) are present. In the context of X-linked inheritance, a recessive gene on the X chromosome means that:

• If a female has one affected X, the dominant allele on the other X chromosome typically masks its effects.
• Males, having only one X chromosome, will express the trait if their X carries the recessive gene.

This biological principle explains why X-linked recessive conditions, like color blindness, are often more common in males.

color blindness probability

Color blindness probability can be calculated by identifying the genotypes of the parents and combining them using a Punnett square. In the given scenario, a mother who is a carrier and a father with normal vision are considered:

• A carrier mother is represented as XᴺXᶜ (One normal vision gene and one color blindness gene).
• A father with normal vision is represented as XᴺY.

When these genotypes combine in a Punnett square, the child's possible genotypes are XᴺXᴺ, XᴺXᶜ, XᴺY, and XᶜY. The XᶜY combination results in a color blind male, since males have only one X chromosome. Given the four potential outcomes, there is a 25% probability that any child from this couple will be a color-blind male.