Problem 24 - Calculus for Biology and Medicine

Question

Suppose that you have a batch of red- and white-flowering pea plants, and suppose also that all three genotypes \(C C, C c\), and \(c c\) are equally represented in the batch. You pick one plant at random and cross it with a white-flowering pea plant. What is the probability that the offspring will have red flowers?

Step-by-Step Solution

Answer

The probability is 0.5.

Step 1: Determine Genotypes of Offspring

Consider each possible genotype of the randomly picked plant from the batch. The possible genotypes are \(C C\), \(C c\), and \(c c\). If you cross each with a white-flowering plant (whose genotype is \(c c\)), you'll have different outcomes for their offspring:- **\(C C\) crossed with \(c c\):** Offspring will be \(C c\) and will have red flowers. - **\(C c\) crossed with \(c c\):** Offspring can be either \(C c\) (red flowers) or \(c c\) (white flowers), with a 50% chance for each. - **\(c c\) crossed with \(c c\):** Offspring will be \(c c\) and will have white flowers.

Step 2: Calculate Probabilities of Each Genotype

Since all genotypes \(C C\), \(C c\), and \(c c\) are equally represented in the batch, the probability of picking any specific genotype is \(\frac{1}{3}\).

Step 3: Calculate Probability of Red Flowers Given Each Genotype

Now calculate the probability of having red flowers depending on each genotype:- If \(C C\) is picked, all offspring have red flowers: Probability \(= 1.0\).- If \(C c\) is picked, half of the offspring have red flowers: Probability \(= 0.5\).- If \(c c\) is picked, none of the offspring have red flowers: Probability \(= 0.0\).

Step 4: Use Total Probability Theorem

To find the total probability of the offspring having red flowers, use the law of total probability:\[ P(\text{Red}) = P(\text{Red} | C C) \cdot P(C C) + P(\text{Red} | C c) \cdot P(C c) + P(\text{Red} | c c) \cdot P(c c) \]Substituting the known values gives:\[ P(\text{Red}) = (1.0 \cdot \frac{1}{3}) + (0.5 \cdot \frac{1}{3}) + (0 \cdot \frac{1}{3}) \]\[ P(\text{Red}) = \frac{1}{3} + \frac{1}{6} + 0 = \frac{1}{2} \]

Step 5: Conclusion

After calculating through all the steps, the probability that the offspring will have red flowers is \(\frac{1}{2}\), or 0.5.

Key Concepts

Genotype

Genotype refers to the genetic makeup of an organism. It is the combination of alleles inherited from the parents. In our pea plant scenario, we have three potential genotypes:

• \(C C\): Two dominant alleles, resulting in red flowers.
• \(C c\): One dominant and one recessive allele, which also results in red flowers because the dominant allele masks the effect of the recessive one.
• \(c c\): Two recessive alleles, leading to white flowers.

Genotype determines phenotype, which is the observable characteristic, like flower color. Understanding genotype helps predict the traits of future generations. Each genotype in our example contributes differently to the potential traits of offspring due to the combination of alleles.

Offspring

Offspring are the progeny or new individuals produced by a living organism. In the context of our exercise, the offspring we are interested in are the new pea plants resulting from the cross between a randomly chosen plant (with one of three genotypes) and a consistent white-flowering plant with genotype \(c c\).

The genotype of the offspring depends on the genetic makeup of both parents:

• From a \(C C\) plant crossed with \(c c\): All offspring receive a \(C\) allele from the \(C C\) plant, leading to a \(C c\) genotype and resultant red flowers.
• From a \(C c\) plant crossed with \(c c\): Each offspring has a 50% chance of being \(C c\) (red) or \(c c\) (white).
• From a \(c c\) plant crossed with another \(c c\): All offspring will be \(c c\) with white flowers.

Understanding the genotype of offspring helps predict the distribution of traits in subsequent generations.

Mendelian inheritance

Mendelian inheritance is a set of principles regarding the transmission of inherited characteristics from one generation to another, named after Gregor Mendel, who pioneered this field. It involves the understanding that traits are inherited through dominant and recessive alleles as Mendel observed with pea plants.

Key principles include:

• Law of Segregation: During gamete formation, allele pairs separate so each gamete carries only one allele for each gene.
• Law of Independent Assortment: Alleles for different traits segregate independently, so the inheritance of one trait generally doesn't affect another.

In our pea plant exercise, the laws of Mendelian inheritance help explain the expected ratios of flower colors in the offspring. By knowing that the red flower color is dominant and white is recessive, we see how the genotypes of the parents influence the flower color of the offspring. These observations illustrate how Mendel's principles translate genetics into observable patterns in progeny.