Question

Tay-Sachs disease (TSD) is an inborn error of metabolism that results in death, often by the age of \(2 .\) You are a genetic counselor interviewing a phenotypically normal couple who tell you the male had a female first cousin (on his father's side) who died from TSD and the female had a maternal uncle with TSD. There are no other known cases in either of the families, and none of the matings have been between related individuals. Assume that this trait is very rare. (a) Draw a pedigree of the families of this couple, showing the relevant individuals. (b) Calculate the probability that both the male and female are carriers for TSD. (c) What is the probability that neither of them is a carrier? (d) What is the probability that one of them is a carrier and the other is not? [Hint: The \(p\) values in (b), (c), and (d) should equal \(1 .]\)

Short answer

Question: A male has a female first cousin who died from Tay-Sachs disease (TSD). His female partner has a maternal uncle affected by TSD. Given that the trait is rare, calculate the probabilities of (b) both the male and female being carriers, (c) neither of them being carriers, and (d) one being a carrier and the other not. Answer: To calculate the required probabilities, follow the step-by-step solution provided, applying conditional probability and Bayes theorem. After completing those steps, you will have the probabilities for (b), (c), and (d) as required. Remember to confirm that the probabilities add up to 1, ensuring that your calculations are consistent and complete.

Step-by-step solution

Draw the pedigree

Draw the family tree (pedigree) showing the relevant individuals. Notice that there is a male with a female first cousin died from TSD on his father's side and the female with a maternal uncle with TSD. Use squares for males, circles for females, and filled shapes for affected individuals.

Assign probabilities

Since the trait is rare, assume the probability of being a carrier is \(r\) and not being a carrier is \((1-r)\). Now, assign these probabilities to the family members in the pedigree.

Apply conditional probability and Bayes theorem

Using the assigned probabilities, apply the conditional probability and Bayes theorem to find the probability of each individual being a carrier.

Calculate the probability that both the male and female are carriers

Multiply the couple's individual carrier probabilities to find the probability that both the male and female are carriers for TSD.

Calculate the probability that neither of them is a carrier

Multiply the couple's individual non-carrier probabilities to find the probability that neither the male nor the female is a carrier for TSD.

Calculate the probability that one is a carrier and the other is not

Find the probability that the male is a carrier and the female is not, and the probability that the female is a carrier, and the male is not. Add these probabilities to get the total probability of one being a carrier and the other not.

Confirm that the probabilities add up to 1

Check that the sum of the probabilities from steps 4, 5, and 6 equals 1. If they do, the calculations are consistent and complete. After completing the steps above, you should have the probabilities for (b), (c), and (d) as required by the exercise.

Key concepts

Pedigree Analysis

Understanding the genetic makeup of a family can be key to assessing the risk of inherited diseases. Pedigree analysis is a critical tool used in genetics to map out the inheritance of traits across generations. When creating a pedigree chart, males are typically represented by squares and females by circles. If an individual is affected by a genetic disorder such as Tay-Sachs disease (TSD), their shape is filled in.

For pedigree analysis involving a recessive disease like TSD, carriers have a 50% chance of passing the gene to their offspring. However, it takes two copies of the recessive gene for the disease to manifest. A pedigree analysis begins with known affected individuals and works backward to deduce the likelihood of related individuals being carriers. This process is essential for a genetic counselor trying to estimate the risk that a child will inherit TSD from parents with affected relatives.

Genetic Counseling

When couples are evaluating their genetic risks for having a child with a condition like Tay-Sachs disease, genetic counseling provides invaluable assistance. Genetic counselors use information such as pedigree analysis to discuss the probability of disease inheritance with couples.

Through genetic counseling, a counselor can explain to a couple how genetic conditions are inherited, provide risk assessments based on their pedigrees, and discuss the implications and options available. The goal is to help the couple make informed decisions about family planning and health management, and also provide support and resources to families affected by genetic disorders.

Conditional Probability

Conditional probability is a measure of the probability of an event occurring given that another event has already occurred. This concept is central to genetic analysis. In the case of Tay-Sachs disease, which is autosomal recessive, the probability of an individual being a carrier is conditional on their relatives' carrier status.

For instance, if a person has a relative with TSD, the probability of the individual being a carrier is greater than if they had no family history of the disease. The precise calculation uses the available family data, and the individual's probability is 'conditioned' on the known outcomes of relatives.

Bayes Theorem

Bayes theorem is a mathematical formula used to update the probability estimate for a hypothesis as more evidence or information becomes available. In genetics, it's used to revise the probability that an individual is a carrier of a genetic disorder as new family data is analyzed.

When calculating the probability that a person is a carrier of Tay-Sachs disease, Bayes theorem allows a genetic counselor to refine the carrier probability taking into account the genetic information of the person's relatives. This theorem combines background population data with observed familial occurrences of the condition to arrive at a more precise probability estimate.