Problem 39 - Algebra 2

Question

The color of a particular flower may be either red, white, or pink. If the flower has two red alleles \(R\) , the flower is red. If the flower has two white alleles \(w,\) the flower is white. If the flower has one allele of each color, the flower will be pink. In a lab, two pink flowers are mated and eventually produce 1000 offspring. How many of the 1000 offspring will be pink?

Step-by-Step Solution

Answer

500 offspring will be pink.

Step 1: Determine Parental Genotypes

Since both parent flowers are pink, their genotypes must each be 'Rw', with one allele for red and one for white.

Step 2: Set Up a Punnett Square

Draw a 2x2 grid to represent the possible allele combinations from the parents. Label the rows and columns with the alleles from each parent 'R' and 'w', respectively.

Step 3: Fill in the Punnett Square

In the Punnett square, combine each parent's alleles: - Top-left box: 'RR' (red) - Top-right box: 'Rw' (pink) - Bottom-left box: 'Rw' (pink) - Bottom-right box: 'ww' (white)

Step 4: Calculate Probability of Pink Offspring

By examining the Punnett square, find that two boxes out of the four possible combinations ('Rw' and 'Rw') result in pink flowers. This represents a 50% chance, as two out of four is simplified to a ratio of 1:2.

Step 5: Apply Probability to Total Offspring

If there is a 50% chance of getting a pink offspring and there are 1000 offspring, calculate: \[ \frac{1}{2} \times 1000 = 500 \] Therefore, 500 of the 1000 offspring will be pink.

Key Concepts

Punnett Square

A Punnett Square is a tool used by geneticists to predict the possible genetic outcomes when two organisms breed. It helps visualize how alleles, which are different forms of a gene, can combine during fertilization.

When you set up a Punnett Square, you create a grid. For a simple cross like the one in the flower example, you use a 2x2 grid. The parent's alleles go on the sides. For the pink flowers, you place the alleles 'R' and 'w' along the top and side of the grid. Each cell inside the square represents a potential genotype of the offspring.

• The top row shows all possible combinations for one parent.
• The side column shows all combinations for the other parent.

As you fill in the square, you combine the alleles from each parent. This not only helps in predicting the traits (like flower color), but also calculates the likelihood of each trait appearing.

Alleles

Alleles are different versions of a gene that determine specific traits. In our flower example, the two alleles in question are 'R' for red and 'w' for white.

Every individual inherits two alleles for each gene, one from each parent.

• If both alleles are the same, it is called "homozygous." For example, 'RR' for red flowers or 'ww' for white flowers.
• If the alleles are different, it is called "heterozygous," like 'Rw' for pink flowers.

Alleles interact in various ways to express a trait. The way these interactions occur, like in our flower color scenario, can be predicted with tools such as the Punnett Square.

Probability

Probability in genetics is the likelihood that a particular trait will be passed down from parent to offspring.

Using the Punnett Square, we can calculate these probabilities. In the flower example, each of the four boxes represents a potential genotype. Two of these boxes contained 'Rw', which led to pink flowers. Therefore, the probability of producing a pink flower is 2 out of 4, or 50%.

• Probability can help predict how many offspring will have a specific trait out of a large number of potential offspring.
• It's important to remember that probability is a chance function and not a guarantee.

By applying probability, we calculated that out of 1000 potential offspring, about 500 will be pink.

Genotype

A genotype is the genetic makeup of an organism, specifically relating to particular genes.

In our flower color example, genotypes such as 'RR', 'Rw', and 'ww' determine the flower's color. It includes all the genes responsible for any observable trait.

• 'RR' genotype results in red flowers.
• 'Rw' genotype results in pink flowers.
• 'ww' genotype results in white flowers.

The genotype provides the blueprint, while environmental conditions and biochemistry determine the phenotype, or physical expression, of traits.

Inheritance patterns

Inheritance patterns dictate how traits are passed from parents to their offspring.

For the flowers in our example, the inheritance pattern is called incomplete dominance. This means the combined alleles produce an intermediate trait, like pink flowers resulting from both red and white alleles.

• In simple dominance, one allele completely masks the other. For example, if red was dominant over white, all 'Rw' flowers would be red.
• In codominance, both alleles express equally, though that's not the case for the flower colors here.

Understanding inheritance patterns helps predict outcomes of breeding and comprehend biological diversity.