Question

In 1865 , Gregor Mendel suggested a theory of inheritance based on the science of genetics. He identified heterozygous individuals for flower color that had two alleles (one \(\mathrm{r}=\) recessive white color allele and one \(\mathrm{R}=\) dominant red color allele). When these individuals were mated, \(3 / 4\) of the offspring were observed to have red flowers and \(1 / 4\) had white flowers. The table summarizes this mating; each parent gives one of its alleles to form the gene of the offspring. $$\begin{array}{lll} & \multicolumn{2}{c} {\text { Parent } 2} \\\\\hline \text { Parent 1 } & \mathrm{r} & \mathrm{R} \\\\\hline \mathrm{r} & \mathrm{rr} & \mathrm{rR} \\\\\mathrm{R} & \mathrm{Rr} & \mathrm{RR}\end{array}$$ We assume that each parent is equally likely to give either of the two alleles and that, if either one or two of the alleles in a pair is dominant (R), the offspring will have red flowers. a. What is the probability that an offspring in this mating has at least one dominant allele? b. What is the probability that an offspring has at least one recessive allele? c. What is the probability that an offspring has one recessive allele, given that the offspring has red flowers?

Short answer

Answer: The probability that an offspring has one recessive allele, given that the offspring has red flowers, is 0.5.

Step-by-step solution

Identify the combinations with at least one R allele

The dominant allele is represented by R. If we examine the table, we can find three possible combinations where there is at least one dominant allele. These are: rR, Rr, and RR.

Calculate the respective probabilities

Since each parent is equally likely to give either of the two alleles, we can calculate the probabilities of each of these combinations: - rR: probability = 0.5 * 0.5 = 0.25 - Rr: probability = 0.5 * 0.5 = 0.25 - RR: probability = 0.5 * 0.5 = 0.25

Sum the probabilities

To find the probability of having at least one dominant allele, add up the probabilities of the three combinations (rR, Rr, and RR): Probability(at least one R) = 0.25 + 0.25 + 0.25 = 0.75 b. Probability that an offspring has at least one recessive allele.

Identify the combinations with at least one r allele

Examine the table to find three possible combinations where there is at least one recessive allele. These are: rr, rR, and Rr.

Calculate the respective probabilities

As in part (a), calculate the probabilities of each combination: - rr: probability = 0.5 * 0.5 = 0.25 - rR: probability = 0.5 * 0.5 = 0.25 - Rr: probability = 0.5 * 0.5 = 0.25

Sum the probabilities

Find the probability of having at least one recessive allele by adding the probabilities of the three combinations (rr, rR, and Rr): Probability(at least one r) = 0.25 + 0.25 + 0.25 = 0.75 c. Probability that an offspring has one recessive allele, given the offspring has red flowers. Given the offspring has red flowers, this means it has either one or two dominant alleles.

Identify conditional combinations

The combinations that fulfill this condition are rR and Rr.

Find the conditional probability

Given the offspring has red flowers, the total probability in that situation is the sum of the probabilities of rR and Rr, which is 0.25 + 0.25 = 0.5. The probability that an offspring has one recessive allele (rR or Rr) given the offspring has red flowers is the probability of rR divided by the combined probability of rR and Rr: 0.25 / 0.5 = 0.5. Therefore, the probability that an offspring has one recessive allele, given that the offspring has red flowers, is 0.5.

Key concepts

Genetics

Genetics is the branch of biology that studies how traits are inherited through the laws of heredity. It is rooted in the groundbreaking work of Gregor Mendel, who, in 1865, laid the foundational theories for how traits are passed from parents to offspring. Mendel's experiments with pea plants revealed that traits are determined by pairs of alleles, with each parent contributing one allele to the offspring. His work paved the way for our understanding of genetic inheritance, DNA, and how these elements contribute to the diversity of living organisms.

In genetics, alleles are different versions of a gene that determine specific traits, like flower color in plants or eye color in humans. Mendel discovered that traits can be dominant or recessive, meaning the expression of certain traits could overshadow others when different alleles are present.

This discovery is invaluable, influencing fields from medicine to agriculture. By understanding genetics, we can predict how certain traits might be passed on in families, aid in the diagnosis of genetic disorders, and even engineer plants that are more resilient to pests.

Mendelian Inheritance

Mendelian Inheritance refers to the patterns of inheritance that are characteristic of organisms that reproduce sexually. Mendel's laws describe the way in which alleles segregate and assort into gametes. The key principles include the Law of Segregation and the Law of Independent Assortment.

• Law of Segregation: This principle states that alleles separate during the formation of gametes (eggs and sperm), meaning that each gamete only carries one allele for each gene. For example, a plant with one red allele (R) and one white allele (r) will produce gametes with either an R or an r allele, but not both.
• Law of Independent Assortment: This principle explains that genes for different traits can segregate independently during the formation of gametes. This means the inheritance of one trait is generally not dependent on the inheritance of another trait.

Mendelian inheritance patterns are seen when analyzing a monohybrid cross, like the example of flower color in Mendel's peas. Each flower-colored allele (R or r) is passed to the offspring, demonstrating how Mendel's principles predict the 3:1 ratio of dominant to recessive phenotypes in the second generation (F2). These discoveries provide the basis for inheritance patterns observed in many organisms today.

Alleles

Alleles are variants of a given gene that arise by mutation and exist at the same locus on a chromosome. Each individual inherits two alleles for each gene, one from each parent. Depending on their interaction, alleles can be categorized as dominant or recessive.

• Dominant Alleles: These alleles express their traits even when only one copy is present. In our flower example, the red color allele (R) is dominant. This means that an offspring with the combinations of either Rr (one red and one white) or RR (two red) will both result in red flowers.
• Recessive Alleles: These alleles require two copies to express their traits fully. In the pea plants, the white color allele (r) is recessive. An offspring needs to have the rr combination to display white flowers.

Understanding alleles and their interactions is crucial for predicting genetic outcomes. These principles not only apply to plant genetics but are universal, affecting all sexually reproducing organisms. By studying alleles, scientists can predict and understand how certain traits may be inherited and expressed in future generations.