Alkaptonuria is an infrequent autosomal recessive condition. It is first noticed in newborns when the urine in their diapers turns black upon exposure to air. The condition is caused by the defective transport of the amino acid phenylalanine through the intestinal walls during digestion. About 4 people per 1000 are carriers of alkaptonuria.
Sara and James had never heard of alkaptonuria and were shocked to discover that their first child had the condition. Sara's sister Mary and her husband, Frank, are planning to have a family and are concerned about the possibility of alkaptonuria in one of their children.
The four adults (Sara, James, Mary, and Frank) seek information from a neighbor who is a retired physician. After discussing their family histories, the neighbor says, 'I never took genetics, but I know from my many years in practice that Sara and James are both carriers of this recessive condition. Since their first child had the condition, there is a very low chance that the next child will also have it, because the odds of having two children with a recessive condition are very low. Mary and Frank have no chance of having a child with alkaptonuria because Frank has no family history of the condition.' The two couples each have babies and both babies have alkaptonuria.


What is the probability that the second child of Mary and Frank will have alkaptonuria?

Verified step by step guidance

1. Step 1: Understand the inheritance pattern of alkaptonuria. It is an autosomal recessive condition, meaning that an individual must inherit two copies of the defective allele (one from each parent) to express the condition. Carriers have one defective allele and one normal allele, and they do not exhibit symptoms.
2. Step 2: Analyze the information provided about Mary and Frank. Although Frank has no family history of the condition, this does not guarantee that he is not a carrier. To determine the probability of their child having alkaptonuria, we must first calculate the likelihood that both Mary and Frank are carriers.
3. Step 3: Calculate the probability that Mary is a carrier. Since Sara and James are both carriers, there is a 2/3 chance that Sara's sister Mary is also a carrier (assuming she does not have the condition herself). This is because, for siblings of carriers, the probabilities are distributed as 1/4 affected, 1/2 carrier, and 1/4 non-carrier, but we exclude the 'affected' category since Mary does not have the condition.
4. Step 4: Calculate the probability that Frank is a carrier. The problem states that about 4 people per 1000 are carriers of alkaptonuria in the general population. This means the probability of Frank being a carrier is 4/1000 or 0.004.
5. Step 5: Combine the probabilities. If both Mary and Frank are carriers, the probability of their child having alkaptonuria is 1/4 (since both parents must pass on the defective allele). Multiply the probabilities: (Probability Mary is a carrier) × (Probability Frank is a carrier) × (Probability of two carriers having an affected child).

Key Concepts

Autosomal Recessive Inheritance

Autosomal recessive inheritance occurs when a trait or disorder is expressed only when an individual has two copies of the recessive allele, one inherited from each parent. In the case of alkaptonuria, both parents must be carriers of the defective gene for their child to express the condition. If only one parent carries the allele, the child will not exhibit the disorder but may be a carrier.

Carrier Probability

A carrier is an individual who possesses one copy of a recessive allele but does not express the associated trait. In the context of alkaptonuria, since both Sara and James are carriers, there is a 25% chance with each pregnancy that their child will inherit both recessive alleles and express the condition. Understanding carrier probability is crucial for assessing the risk of genetic conditions in offspring.

Punnett Square

A Punnett square is a diagram used to predict the genetic makeup of offspring from two parents. It helps visualize the possible combinations of alleles that can result from a cross. For alkaptonuria, a Punnett square can illustrate the likelihood of their children being affected, carriers, or unaffected, providing a clear understanding of the genetic risks involved in the family planning of Mary and Frank.