Problem 32 - Calculus for Biology and Medicine

Question

Suppose that one parent is of genotype \(A A\) and the other is 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 of an offspring being genotype \(AA\) is \(\frac{1}{2}\).

Step 1: Identify Parental Genotypes

One parent has the genotype \(AA\) which means they can only contribute the \(A\) allele to the offspring. The other parent has the genotype \(Aa\), meaning they can contribute either an \(A\) or an \(a\) allele.

Step 2: Construct a Punnett Square

Draw a 2 by 2 grid. On the top, place the alleles of one parent (\(A\) and \(A\)). On the side, place the alleles of the other parent (\(A\) and \(a\)).

Step 3: Fill in the Punnett Square

For each cell in the grid, combine the allele from the top with the allele from the side. The possible combinations are: for the top-left, \(AA\); for the top-right, \(AA\); for the bottom-left, \(Aa\); for the bottom-right, \(Aa\).

Step 4: Count Genotype Combinations

In the Punnett Square, there are four possible outcomes: two of them are \(AA\) and two are \(Aa\).

Step 5: Calculate Probability

The probability of an offspring having the genotype \(AA\) is the ratio of the number of \(AA\) outcomes to the total number of outcomes. This ratio is \(\frac{2}{4}\). Simplify this to get \(\frac{1}{2}\).

Key Concepts

Mendel's First Law

Gregor Mendel, often called the father of genetics, introduced what we refer to as Mendel's First Law, also known as the Law of Segregation. This fundamental principle states that during the formation of reproductive cells or gametes, the pairs of alleles (different versions of a gene) segregate, meaning they randomly separate, so that each gamete receives just one allele from each pair. This ensures genetic variation, as it allows offspring to inherit one allele from each parent. In practical terms, if an organism possesses one pair of alleles, say

• AA (homozygous)
• or Aa (heterozygous)

then each gamete produced will contain one randomly selected allele from this pair. This law is observable when creating a Punnett Square, which visualizes the random combination of alleles from two parents.

Genotype Probability

Probability in genetics is the calculation used to predict the likelihood of an offspring having a specific genotype. When using a Punnett Square for this purpose, we can determine these probabilities visually. Each cell within the grid represents a potential combination of parental alleles that could form an offspring's genotype. In the original exercise:

• One parent was homozygous dominant, contributing two types of gametes, both carrying an A allele.
• The other parent was heterozygous, contributing one gamete carrying an A allele and one carrying an a allele.

When these gametes combine, the resulting genotypes can be visualized. There are four possible outcomes in this case – two outcomes yield the AA genotype and two yield the Aa genotype. Hence the probability of producing an offspring with the genotype AA equals the ratio of AA outcomes to total outcomes, which is 2 out of 4, simplifying to 0.5 or 50%.

Alleles

Alleles are the different forms of a gene that determine distinct traits which can be passed from parents to offspring. For example, they'll decide attributes like hair color, eye color, or in this case, a genetic trait expressed as A or a. Every individual inherits two alleles for each gene, one from each parent, forming a genotype. These alleles occupy the same gene locus or position on a chromosome. The combination of alleles in a genotype can result in different traits being expressed. Consider these key points:

• An allele can be dominant (A) or recessive (a).
• The dominant allele masks the expression of the recessive allele unless there are two recessive alleles (aa).

In the exercise,

• The homozygous dominant genotype (AA) and the heterozygous genotype (Aa) were considered.
• Since we inherit one allele from each parent, the combination of these alleles—not just the individual alleles themselves—determines the expressed trait.

Thus, understanding the role of alleles is crucial to grasp how genetic inheritance and variation occur.