Problem 59 - Calculus for Biology and Medicine

Question

A number of traits are caused by recessive genes. The traits show up only in individuals who are homozygous (i.e., have two copies of the mutant gene). An individual with one normal and one mutant gene is a carrier, but does not exhibit the trait. Calculate each of the probabilites. Tay-Sachs disease is caused by a single pair of recessive genes. If both parents are carriers of the mutant gene, what is the likelihood that none of their four children will be affected?

Step-by-Step Solution

Answer

The probability that none of the four children are affected is 0.3164.

Step 1: Understanding the Problem

To find the probability that none of the four children will be affected by Tay-Sachs, we need to calculate the probability that each child is not affected. Since the parents are carriers, each child has a 25% chance of being homozygous recessive (affected). Therefore, the complementary probability of not being affected (either being homozygous dominant or a carrier) is 75%, i.e., the child does not exhibit the trait.

Step 2: Calculating the Single Event Probability

The probability that one child is not affected by Tay-Sachs is 0.75 because each child independently has a 75% chance of not being homozygous recessive and, therefore, not being affected by the disease.

Step 3: Finding the Probability for All Four Children

Since we're considering four independent events (the four children), we calculate the probability that none are affected by multiplying the single event probability for each child. This is calculated as: \( P( ext{none affected}) = (0.75)^4 \).

Step 4: Calculating the Final Probability

Substitute 0.75 for the probability that each child is unaffected into the formula: \( P( ext{none affected}) = (0.75)^4 = 0.3164 \).

Step 5: Rounding the Probability

The probability that none of their four children will be affected by Tay-Sachs, when rounded to four decimal places, is approximately 0.3164.

Key Concepts

Recessive Genes

In genetics, recessive genes play a crucial role in determining traits that are not as visibly dominant in our phenotype. A phenotype is the set of observable characteristics that result from the interaction of an organism's genotype (genetic makeup) with the environment. Recessive genes, unlike dominant genes, require both alleles present to be expressed in the phenotype.

Here's a clearer breakdown:

• **Homozygous recessive**: This occurs when an individual possesses two copies of the recessive allele (e.g., aa). The trait will be expressed.
• **Homozygous dominant**: This is when an individual has two copies of the dominant allele (e.g., AA). The dominant trait will be expressed.
• **Heterozygous (carrier state)**: When an individual has one dominant and one recessive allele (e.g., Aa), the recessive trait is not expressed, but the individual can pass the recessive gene to offspring.

In cases like Tay-Sachs disease, where an individual must inherit two recessive genes to express the trait, understanding whether a genetic trait is recessive or dominant helps us see the probability of inheriting specific conditions.

Probability

The concept of probability is essential for predicting genetic outcomes. When working with genetic problems involving recessive genes, probability helps determine the likelihood of different genetic combinations in offspring.

Key points about probability:

• Probability is expressed as a number between 0 and 1, where 0 indicates an impossible event and 1 indicates a certain event.
• The closer the probability is to 0, the less likely the event will occur; the closer it is to 1, the more likely it will happen.
• When dealing with multiple events, probabilities of independent events are multiplied to find the combined probability.

In the context of the given exercise, each child's probability of being unaffected by Tay-Sachs is 0.75, since there is a 75% chance they will have a dominant allele present. For four children, the combined probability of all of them being unaffected is calculated by raising 0.75 to the power of 4, resulting in roughly 31.64%.

Tay-Sachs Disease

Tay-Sachs disease is a genetic disorder caused by the absence of a vital enzyme called Hex-A, leading to severe nerve damage and eventual death. It's a tragic condition that significantly impacts the lives of those affected.

Vital information about Tay-Sachs:

• The disorder is autosomal recessive, meaning it occurs only when an individual inherits two copies of the defective gene.
• Although rare, it predominantly affects certain populations, like people of Ashkenazi Jewish descent.
• Carrier individuals, those with one normal and one mutant gene, do not show symptoms but can pass the gene to their children.

Understanding the genetic nature of Tay-Sachs and how recessive genes work helps in preventing the disease through genetic counseling and testing. By understanding how the probability mechanics of inheritance work, couples can make informed decisions about family planning and health management.