91Ó°ÊÓ

A dominantly inherited genetic disease is identified over several generations of a large family. However, about half the families have dominant disease with complete penetrance, whereby if a parent is affected there is a \(50 \%\) probability that any one offspring will be affected. Similarly, about half the families have dominant disease with reduced penetrance, whereby if a parent is affected there is a \(25 \%\) probability that any one offspring will be affected. Suppose in a particular family one parent and two of the two offspring are affected.What is the probability that the mode of transmission for this particular family is dominant with complete penetrance? Is this a prior probability or a posterior probability?

Step by step solution

01

Understand the Concepts

Complete penetrance means that if a parent is affected by a genetic disease, there is a 50% chance that each child will also be affected if it's a dominant trait. Reduced penetrance means that the probability decreases to 25%. Penetrance refers to the probability of expressing the phenotype given the genotype.

02

Identify the Given Information

We are given that there are two types of penetrance associated with the genetic disease: complete penetrance (50% chance) and reduced penetrance (25% chance). In the family of interest, one parent and both offspring are affected.

03

Establish the Probabilities for Each Mode

The probability of the observed family configuration under complete penetrance is \((0.5)^2 = 0.25\), as both children need to independently inherit the disease. Under reduced penetrance, the probability would be \((0.25)^2 = 0.0625\).

04

Use Bayes' Theorem to Calculate the Posterior Probability

Let \( P(C) = 0.5 \) be the prior probability of complete penetrance mode, and \( P(R) = 0.5 \) be the prior probability of reduced penetrance mode. The likelihoods are \( P(Data|C) = 0.25 \) and \( P(Data|R) = 0.0625 \). The posterior probability of complete penetrance is:\[P(C|Data) = \frac{P(Data|C) \cdot P(C)}{P(Data|C) \cdot P(C) + P(Data|R) \cdot P(R)} = \frac{0.25 \times 0.5}{0.25 \times 0.5 + 0.0625 \times 0.5} = \frac{0.125}{0.125 + 0.03125} = \frac{0.125}{0.15625} = 0.8.\]

05

Interpret the Result

The probability that the mode of transmission for this family is dominant with complete penetrance, given their configuration, is 80%. This probability is a posterior probability because it is calculated after observing the particular family's affected status.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Dominant Genetic Disease

A dominant genetic disease is one where a single copy of a mutant gene inherited from one parent is sufficient to cause the disease.
In genetic terms, a trait or disorder that shows dominance means it manifests in individuals generating a visible characteristic even if only one parent passes down the variant.
This is quite different from recessive diseases where two copies of the mutant gene, one from each parent, are required for the disease to occur.

• Dominant disorders often have a 50% chance of being passed on, given that only one allele (gene variant) is needed.
• Due to their genetic mechanism, these conditions often appear consistently in each generation of a family.
• Examples include Huntington's Disease and Marfan Syndrome.

The presence and frequency of a dominant disease within families can be influenced by either complete or reduced penetrance, adding layers to predicting genetic outcomes.

Complete Penetrance

Complete penetrance describes situations where the presence of a gene guarantees that associated trait or disorder will manifest.
In other words, when complete penetrance is at play, there is a 100% chance of the phenotype being expressed if the genotype consists of at least one dominant allele.

• For dominantly inherited diseases, this translates into an affected individual passing the trait to about half of their offspring.
• Complete penetrance aids in making genetic predictions clearer as it follows straightforward Mendelian rules.

Using Bayes' Theorem in genetics, particularly when dealing with complete penetrance, helps calculate likelihoods and understand inheritance patterns. For a family where both parents and offspring are affected, the straightforward probability calculations rely on the certainty that the disease will be expressed whenever the gene is present.

Reduced Penetrance

Reduced penetrance represents variability in genetic expression.
It occurs when individuals carry a gene for a dominant trait or disorder yet exhibit no symptoms.
In cases of reduced penetrance, inheriting a mutant gene does not always guarantee that the trait will be visible.

• This phenomenon implies a lower probability—often less than the expected 50%—that an offspring will express the disorder if a parent is affected.
• Environmental factors, other genetic variables, and lifestyle may influence whether the trait manifests.

Using genetic probability calculations, such as those with Bayes' Theorem, involves considering both complete and reduced penetrance.
This model helps predict the nuances in how genetic diseases might appear within and across generations.