Problem 38 - Calculus for Biology and Medicine

Question

Cystic fibrosis is an autosomal recessive disease, which means that two copies of the gene must be mutated for a person to be affected. Assume that two unaffected parents who each carry a single copy of the mutated gene have a child. What is the probability that the child is affected?

Step-by-Step Solution

Answer

The probability is 25%.

Step 1: Understanding Genetic Terminology

An autosomal recessive disease requires two copies of a mutated gene for an individual to be affected. Each parent can carry one copy without exhibiting symptoms, known as carriers.

Step 2: Establishing Genotypes of Parents

Each parent, being a carrier, has one normal allele (N) and one mutated allele (n), making their genotype Nn.

Step 3: Using a Punnett Square to Determine Offspring Genotypes

A Punnett square can help visualize the possible genotypes of the child. Each parent has a genotype of Nn, so the four potential combinations for their child are: NN, Nn, nN, nn.

Step 4: Identifying Affected Genotype

Since the disease is autosomal recessive, the genotype nn is required for the child to be affected by cystic fibrosis.

Step 5: Calculating Probability of Affected Child

From the Punnett square, only 1 out of the 4 combinations (nn) leads to the child being affected. Hence, the probability is \( \frac{1}{4} \) or 25%.

Key Concepts

Autosomal Recessive Inheritance

In genetics, certain traits and disorders are inherited in specific patterns. Autosomal recessive inheritance is one such pattern where two copies of an abnormal gene must be present for a trait or disorder to be expressed. In simpler terms, a person needs to inherit the mutated gene from both parents to be affected by the disorder. An example of this pattern is cystic fibrosis, which requires that both contributed genes (alleles) be the mutated form to cause the disease. If both parents are carriers, meaning they each have one normal gene and one mutated gene, they typically show no symptoms. However, they can pass on the mutated gene to their offspring. In autosomal recessive conditions, it's essential to understand that a single mutated copy without a corresponding pair doesn't lead to the disease. Both parents need to contribute the mutated allele for the effects to manifest in the child.

Understanding the Punnett Square

The Punnett square is a helpful tool in genetics for visualizing how genes combine from parents to offspring. It simplifies the process of predicting the probability of inheriting particular genotypes based on parental genetic makeup. To use a Punnett square, draw a grid. Place one parent's alleles along the top and the other parent's along the side. In the case where both parents are carriers (Nn), the Punnett square shows all possible combinations in the offspring:

• NN - inherits two normal alleles
• Nn - inherits one normal and one mutated allele
• nN - also inherits one normal and one mutated allele
• nn - inherits two mutated alleles

Each box in the square represents a possible genotype in their offspring. The Punnett square helps to quickly visualize genetic outcomes and understand genotype distributions.

Probabilities in Genetics

Probability is a key concept when predicting genetic traits in offspring. It uses statistical principles to determine how likely it is for a specific genetic trait to appear.Using the Punnett square for our example, the probability of having an affected child is determined by the number of affected combinations over the total possibilities. With the parental genotypes of Nn, Nn, one combination in the Punnett square leads to the affected genotype (nn).Thus, the probability of the child having cystic fibrosis is calculated as follows:\[ \text{Probability} = \frac{\text{Number of affected combinations}}{\text{Total combinations}} = \frac{1}{4} = 25\% \]This means there's a 25% chance that the child will inherit the disease.

Genotypes and Their Impact

Genotypes refer to the genetic makeup of an organism in terms of its alleles. For autosomal recessive diseases like cystic fibrosis, the specific combinations of alleles determine whether a child will be affected. The genotype notation

• "NN" stands for two normal alleles, meaning the child is unaffected and not a carrier.
• "Nn" or "nN" indicates the child is a carrier but unaffected, similar to the parents.
• "nn" represents the affected condition, where the child displays symptoms of cystic fibrosis.

Understanding these combinations allows us to predict the chances of different genetic outcomes, helping individuals and families plan for potential health management scenarios. The genotype is foundational in understanding genetic diseases and inheritance.