Question

Dentinogenesis imperfecta is a tooth disorder involving the production of dentin sialophosphoprotein, a bone-like component of the protective middle layer of teeth. The trait is inherited as an autosomal dominant allele located on chromosome 4 in humans and occurs in about 1 in 6000 to 8000 people. Assume that a man with dentinogenesis imperfecta, whose father had the disease but whose mother had normal teeth, married a woman with normal teeth. They have six children. What is the probability that their first child will be a male with dentinogenesis imperfecta? What is the probability that three of their six chil- dren will have the disease?

Short answer

Answer: The probability that the first child will be a male with dentinogenesis imperfecta is 1/4, and the probability that three out of their six children will have dentinogenesis imperfecta is 5/16.

Step-by-step solution

Analyze the inheritance pattern of dentinogenesis imperfecta

The given problem states that dentinogenesis imperfecta is inherited as an autosomal dominant allele. This means that an individual only needs one copy of the affected allele, paired with a normal allele, to express the condition. Here, the father has dentinogenesis imperfecta, so his genotype will be Dd (D represents the dominant allele for dentinogenesis imperfecta and d represents the normal, recessive allele). The mother has normal teeth, implying her genotype is dd.

Calculate the probability of having a male child with dentinogenesis imperfecta

To get a male child with dentinogenesis imperfecta, we first need to find out the probability of having a male child, which is 1/2. Then, we need to calculate the probability of the male child having dentinogenesis imperfecta by creating a Punnett square to show the possible genotypes for their children: D d d \|/ \|/ /dd Dd dd The probability of having a child with dentinogenesis imperfecta (Dd genotype) is 1/2. Combining these probabilities, the chance of having a male child with dentinogenesis imperfecta is: 1/2 (probability of a male child) * 1/2 (probability of having dentinogenesis imperfecta) = 1/4

Calculate the probability of three out of six children having dentinogenesis imperfecta using the binomial probability formula

The binomial probability formula is: P(k) = C(n, k) * p^k * (1-p)^{n-k} Where: - n is the total number of children - k is the number of children with dentinogenesis imperfecta - p is the probability of each child having dentinogenesis imperfecta - C(n, k) is the binomial coefficient, or the number of ways to choose k successes from n trials In this scenario, n = 6, k = 3, and p = 1/2 (since each child has a 1/2 chance of inheriting the Dd genotype from their parents). C(6,3) = 6! / (3! * (6-3)!) = 20 Plugging these values into the binomial probability formula, we get: P(3) = 20 * (1/2)^3 * (1/2)^{6-3} = 20 * (1/8) * (1/8) = 20/64 = 5/16 So, the probability of three out of their six children having dentinogenesis imperfecta is 5/16.

Key concepts

Dentinogenesis Imperfecta

Dentinogenesis imperfecta is a genetic condition affecting the teeth. It complicates the development of dentin, which is a crucial layer in tooth structure. The condition is linked to a dominant allele on chromosome 4, meaning only one copy of the affected gene can lead to the disorder. Individuals with this condition often have teeth that may appear discolored, with a blue-gray or amber-brown hue.

• The structure of the teeth is compromised, leading to issues like wear, breakage, and, in severe cases, even tooth loss.
• Dentinogenesis imperfecta occurs in approximately 1 in 6,000 to 8,000 people, highlighting its rarity.
• When assessing genetic risks, it’s key to identify this trait as autosomal dominant, where an individual with one affected parent has a 50% chance of inheriting the condition.

Understanding the inheritance pattern aids in predicting the likelihood of the disorder in offspring, which brings us to the tools used for genetic probability calculations.

Punnett Square

The Punnett square is a simple tool used to predict the possible gene combinations in offspring from their parents. In the case of dentinogenesis imperfecta, it helps determine the probabilities of the disorder being passed to children. Let’s break this down:

• A parent with dentinogenesis imperfecta, being heterozygous Dd, contributes either the dominant (D) or recessive (d) allele.
• A parent with normal teeth, being homozygous recessive (dd), contributes the recessive (d) allele.
• By plotting these on a Punnett square, you find the potential genotypes for their children: Dd or dd.

With these genotypes mapped out:

• There's a 50% chance of the children inheriting Dd (dentinogenesis imperfecta) and 50% dd (normal).

Performing such calculations not only guides family planning but also supports understanding of genetic diseases in broader genetic studies.

Binomial Probability

The binomial probability formula is especially useful in genetics for predicting the occurrence of a trait over multiple offspring. For the given problem about dentinogenesis imperfecta and six children, it is applied to find the probability that exactly three inherit the condition.
Here’s how it works:

• Total number of children ( n ) = 6
• Desired number with the condition ( k ) = 3
• Probability of one child inheriting the condition ( p ) = 1/2

Using the formula: $$P(k)=C(n,k)\times p^k\times (1-p{)}^{n-k}$$

• The binomial coefficient C(6, 3) calculates how many ways you can select 3 children from 6, which equals 20.
• Solving the equation gives you the probability: $$P(3)=20\times (1/8)\times (1/8)=5/16$$
• This indicates there is a 5/16 chance that exactly 3 out of their 6 children will have dentinogenesis imperfecta.

These probabilities are vital for genetic counseling and help in understanding how likely it is for a genetic trait to recur in families.