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

Answer: To find these probabilities, use the calculated values from the step-by-step solution: 1. Probability that both are carriers. 2. Probability that neither of them is a carrier. 3. Probability that one of them is a carrier and the other is not. Note: The probabilities must be calculated using the specific prevalence rate and pedigree information given in the problem.

Step-by-step solution

Understand the given information

We know that Tay-Sachs disease (TSD) is a very rare inborn error of metabolism, inherited recessively. This means it requires two copies of the affected gene to be expressed. In this case, we have a phenotypically normal couple, and the male has a first cousin (father's side) who died from TSD, while the female has a maternal uncle with TSD. Now, we will draw the pedigree.

Draw the pedigree

We can represent individuals as squares (males) and circles (females). For affected patients, we'll use filled shapes. - The male had a first cousin (father's side) who died from TSD - The female had a maternal uncle with TSD. 1 2 3 4 ⬛⬛ ⬜⬜ ⬜⬜ ⬛⬛ | | 1.1 ⬜⬜---⬜⬜ 2.1 3.1 ⬛⬛---⬜⬜ 4.1 | | | | | | ⬜⬜---⬜⬜ ⬜⬜---⬛⬜ ⬛⬛---⬜⬜ 1.2 2.2 3.2 4.2 To indicate carriers, we will fill half the shape.

Calculate the probability that both the male and female are carriers

We can use Bayes' theorem for this step. Since TSD is rare, we'll use a prevalence rate of 0.0001 (q). We assume that the normal allele frequency (p) is 1-q. • Male carrier probability: P(Heterozygote | TSD cousin) = P(TSD cousin | Heterozygote) * P(Heterozygote) / P(TSD cousin) • Female carrier probability: P(Heterozygote | TSD uncle) = P(TSD uncle | Heterozygote) * P(Heterozygote) / P(TSD uncle) Calculating these probabilities: - P(TSD cousin) = 1/16 (from pedigree) - P(Heterozygote) = 2pq = 2(1-q)q - P(TSD cousin | Heterozygote) = 1/2 - P(TSD uncle) = 1/4 (from pedigree) - P(TSD uncle | Heterozygote) = 1/2 Plugging in the values and solving for the carrier probability: Male: P(Heterozygote | TSD cousin) = (1/2) * 2(1-q)q / (1/16) Female: P(Heterozygote | TSD uncle) = (1/2) * 2(1-q)q / (1/4) Probability both are carriers = P(Male is carrier) * P(Female is carrier)

Calculate the probability that neither of them is a carrier

Probability that neither of them is a carrier will be equal to the product of the probabilities of each not being a carrier. P(neither is carrier) = P(Male is not carrier) * P(Female is not carrier)

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

Probability that one of them is a carrier and the other is not is equal to the sum of the probabilities of each combination (male carrier-female non-carrier and male non-carrier-female carrier). P(one is carrier, other is not) = P(Male is carrier and Female is not) + P(Male is not and Female is carrier) Note that the sum of probabilities for step 3, 4, and 5 should equal 1.

Key concepts

Inborn Error of Metabolism

Tay-Sachs disease (TSD) is classically known as an inborn error of metabolism, which is a group of disorders that involve genetic defects in the body's biochemical pathways. These disorders result from various mutations that affect specific enzymes necessary for metabolism, thereby hindering the body's ability to break down and process different substances.

In the case of TSD, the enzyme hexosaminidase A is absent or not functional due to a genetic mutation. The disease is characterized by a buildup of a fatty substance called GM2 ganglioside in the nerve cells of the brain. This accumulation leads to progressive neurological deterioration, with affected children often passing away by the age of 2-4 years.

Understanding the mechanisms of this disease assists us in grasping the complex nature of genetic disorders and the importance of metabolic pathways in maintaining healthy bodily functions.

Autosomal Recessive Inheritance

Autosomal recessive inheritance is a pattern of genetic transmission. This means that an individual must inherit two copies of an affected gene, one from each parent, to express the condition. In this pattern, the parents of an affected individual are often carriers, possessing one copy of the mutated gene and one normal gene, showing no signs of the condition themselves.

Carriers have a 25% chance of passing the disease to their offspring if they mate with another carrier. In a pedigree, such as that for Tay-Sachs disease, carriers’ symbols are typically indicated by half-filled shapes, while affected individuals are represented by fully filled shapes. An understanding of this inheritance pattern is crucial for genetic counseling and risk assessment within families.

Genetic Pedigree Analysis

Genetic pedigree analysis serves as a visual tool for evaluating the inheritance patterns of traits, conditions, or diseases within a family. Pedigrees utilize standardized symbols to denote gender and relationship, with squares representing males and circles representing females. A line connecting a square and circle represents a mating, and their offspring are connected via vertical lines.

In the context of Tay-Sachs disease, pedigree analysis can identify carriers and affected individuals within a family history. By determining whose siblings or parents might be carriers, health professionals can estimate the likelihood of the disease manifesting in future generations. Pedigree analysis becomes a cornerstone in predicting genetic risks and guiding decisions surrounding family planning and disease management.

Bayes' Theorem in Genetics

Bayes' theorem is a principle of probability that allows us to update the probability that a hypothesis is true based on new evidence. In genetic counseling, Bayes' theorem is instrumental in calculating the likelihood that an individual is a carrier of a genetic disorder, given their family history and the prevalence of the disease in the population.

For instance, if we conjecture that an individual might be a carrier of Tay-Sachs disease based on their family pedigree, we can revise this probability by incorporating the known frequency of the disease within the general population and the presence of the disease within their family. This theorem plays a vital role in delivering more accurate genetic counseling to individuals, enabling better-informed health decisions.

Carrier Probability Calculation

Carrier probability calculation involves determining the likelihood of an individual carrying a recessive allele for a genetic condition. These calculations take into account factors such as the individual's family history, the inheritance pattern of the disorder, and the frequency of the mutated gene in the population.

In the exercise concerning Tay-Sachs disease, the carrier probabilities for a male with an affected cousin and a female with an affected uncle would be determined by using Bayes' theorem. We calculate the probability of an individual being a carrier given the evidence of affected relatives and combine that with the pre-existing population frequency of carriers. From these probabilities, we can deduce the associated risks for their offspring and answer complex genetic counseling questions with greater precision.