Problem 7 - Biology Concepts and Applications

Question

Mendel crossed a true-breeding tall, purple-flowered pea plant with a true- breeding dwarf, white-flowered plant. All \(\mathrm{F}_{1}\) plants were tall and had purple flowers. If an \(\mathrm{F}_{1}\) plant self-fertilizes, then what is the probability that a randomly selected \(\mathrm{F}_{2}\) offspring will be heterozygous for the genes specifying height and flower color?

Step-by-Step Solution

Answer

The probability is \(\frac{1}{4}\) or 25%.

Step 1: Understand the Genetic Cross

In Mendel's experiment, a true-breeding tall, purple-flowered pea plant (TT, PP) is crossed with a true-breeding dwarf, white-flowered plant (tt, pp). "True-breeding" means both plants were homozygous for each trait, so the offspring (F1 generation) will all have the genotype TtPp, showing the dominant traits of tall and purple flowers.

Step 2: Determine F1 Genotype Possibilities

The phenotype of all F1 plants will be tall and purple because of the dominance of the T (tall) and P (purple) alleles over t (dwarf) and p (white). Thus, all F1 progeny are heterozygous for both traits with the genotype TtPp.

Step 3: Set up the Punnett Square for F1 Self-Fertilization

When the F1 (TtPp) plants self-fertilize, each parent can contribute the alleles T or t for height and P or p for flower color. The Punnett square must include all combinations: TP, Tp, tP, and tp, each pairing with these same combinations from the second parent, resulting in 16 possible genotype combinations for F2.

Step 4: Identify Heterozygous Genotypes

A heterozygous offspring for both traits has the genotype TtPp. From the Punnett square, we must count how many of the 16 outcomes are TtPp. Genotypes that count are those with exactly one T and one t, as well as one P and one p. TtPp is the only combination that is heterozygous for both traits.

Step 5: Calculate the Probability

There are 4 combinations of the genotype TtPp in the 16 squares of the Punnett square (rows Tt x columns Pp in each case). Therefore, the probability of a random F2 offspring being heterozygous for both genes is \(\frac{4}{16} = \frac{1}{4}\).

Key Concepts

Punnett Square

One of the foundational tools used in genetic studies is the Punnett square. It is a simple chart that helps predict the possible genotypes of offspring in a genetic cross. For instance, when Mendel's true-breeding tall purple-flowered plants were crossed with dwarf white-flowered plants, the resulting \( F_1 \) generation all had a genotype of TtPp, since tall and purple traits are dominant.
To check further generations, like the \( F_2 \) offspring, a Punnett square can be set up to visualize the combinations of alleles that result from self-fertilization. It involves listing all possible gametes each parent can produce and combining them in a grid to see all potential genotypic outcomes of the progeny.
The Punnett square offers a visual method to identify possible combinations of alleles and their associated traits.

Dominant and Recessive Traits

In the study of Mendelian genetics, traits are often classified as either dominant or recessive. Dominant traits are visible in the phenotype even when only one dominant allele is present in the genotype. In contrast, a recessive trait only appears phenotypically when both alleles are recessive.
In Mendel's experiment, tall plants (T) and purple flowers (P) are the dominant traits, while dwarf plants (t) and white flowers (p) are recessive. The dominance of an allele is significant because it dictates the physical appearance (phenotype) of the organism when different genotype combinations occur.
This principle explains why all \( F_1 \) plants were tall and purple, as they inherited a dominant allele from each trait.

Genotype and Phenotype

Genotype refers to the genetic makeup of an organism, specifically the alleles present at a given loci. Meanwhile, phenotype is the observable physical or biochemical characteristics of an organism as dictated by the genotype and the environment.
In Mendel's cross, the \( F_1 \) plants all had the genotype of TtPp for both height and flower color. However, the phenotype was characterized by tallness and purple flowers because these are dominant over their recessive counterparts.
Understanding the distinction between genotype and phenotype helps explain how organisms can carry recessive traits (like t and p) without actually expressing them in their physical appearance.

Heterozygous

When an organism has two different alleles for a specific gene, it is said to be heterozygous. In Mendel's pea plant experiment, the \( F_1 \) generation appeared as heterozygous for both height and flower color with the genotype TtPp.
Being heterozygous for these traits means carrying one dominant and one recessive allele. This combination is crucial in offspring, as it contributes to genetic diversity and variation, allowing traits to be present without always being expressed phenotypically.
Heterozygous individuals can produce gametes (sperm or eggs) with different allele combinations, leading to a variety of possible genotypes and phenotypes in subsequent generations, as shown in the \( F_2 \) generation.

Probability in Genetics

Probability plays a crucial role in predicting genetic outcomes. Mendelian genetics often use probability to estimate the likelihood of inheriting particular traits.
In the \( F_2 \) generation of Mendel's peas, probability helps us understand the chance of offspring being heterozygous (TtPp) for both traits. By determining the percentage of Punnett square outcomes that result in this genotype, we calculate the probability as \( \frac{1}{4} \) or 25%.
Probability is a key tool in genetic predictions, providing insights into inheritance patterns and helping geneticists forecast trait distributions in populations.