Problem 31 - Calculus for Biology and Medicine

Question

Suppose that two parents are of genotype \(A a\). What is the probability that their offspring is of genotype \(A a ?\) (Assume Mendel's first law.)

Step-by-Step Solution

Answer

The probability that their offspring is of genotype \(A a\) is \(\frac{1}{2}\) or 50%.

Step 1: Understand Mendel's First Law

Mendel's first law, also known as the Law of Segregation, states that each organism carries two alleles for a given trait and that these alleles separate during the formation of gametes. Therefore, each gamete carries only one allele for each trait. In this problem, each parent has the genotype \(A a\).

Step 2: Determine Possible Alleles in Gametes

Each \(A a\) parent can contribute either allele \(A\) or allele \(a\). This results in the following possible gametes produced by each parent: \(A\) and \(a\).

Step 3: Set Up a Punnett Square

To find the possible genotypes of the offspring, set up a Punnett square. The rows represent one parent's gametes, and the columns represent the other parent's gametes:\[\begin{array}{c|c|c} & A & a \hlineA & AA & Aa \hlinea & Aa & aa \\end{array}\]The Punnett square shows the possible genotypes: \(AA\), \(Aa\), \(Aa\), and \(aa\).

Step 4: Calculate Probability of Genotype \(Aa\)

The Punnett square shows two boxes with genotype \(Aa\) out of four total boxes. Thus, the probability that the offspring will be \(Aa\) is the number of \(Aa\) outcomes divided by the total number of outcomes: \(\frac{2}{4} = \frac{1}{2}\).

Key Concepts

Genotype Aa

The genotype "Aa" is a representation of an individual's genetic makeup for a specific trait. It consists of two alleles: one from each parent. In genotype Aa, one allele is dominant (represented by "A") and the other is recessive (represented by "a"). This combination means that the dominant trait will be expressed in the phenotype, while the recessive trait will remain hidden unless paired with another recessive allele.

When considering Mendel's First Law, the genotype Aa plays a crucial role. During gamete formation, these alleles are randomly segregated, ensuring genetic diversity. Each parent with an Aa genotype can pass on either one of these alleles to their offspring. Therefore, understanding these allelic combinations helps us predict possible outcomes in their progeny.

Punnett square

The Punnett square is a powerful tool used in genetics to predict the probable genetic makeup of offspring from parental genotypes. It is a simple grid structure that allows us to visually represent how alleles from each parent combine during fertilization. Here's how it works:

• Each row and column represents the possible gametes produced by each parent.
• By crossing the gametes, we see all potential allele combinations the offspring can inherit.

For example, in our scenario with parents both having genotype Aa, a Punnett square helps map out their potential offspring genotypes:

• Each parent can contribute either allele 'A' or allele 'a'.
• The resulting combinations form the genotypes: AA, Aa, and aa.
• The Punnett square shows these combinations clearly, making it easier to calculate probabilities.

Law of Segregation

The Law of Segregation is one of the foundational principles of genetics, introduced by Gregor Mendel through his work with pea plants. It states that during the formation of gametes, the two alleles for a trait separate, or segregate, from each other. Each resulting gamete then carries only one allele for each trait- either the maternal or paternal allele.

This principle explains why offspring may not always inherit the exact genetic makeup of their parents. For example, if both parents are heterozygous with genotype Aa, due to the Law of Segregation, each gamete is equally likely to carry allele A or allele a. This random segregation is what adds to the genetic variation observed in each new generation and is the foundation of the predictable patterns seen in heredity, as shown through tools like the Punnett square.