Three pairs of genes with two alleles each (A₁ and A₂, B₁ and B₂, and C₁ and C₂) control the height of a plant. The alleles of these genes have an additive relationship: Each copy of alleles A₁, B₁, and C₁ contributes 6 cm to plant height, and each copy of alleles A₂, B₂, and C₂ contributes 3 cm. Identify all possible genotypes for plants with an expected height of 33 cm.

Verified step by step guidance

1. Step 1: Understand the problem. The height of the plant is determined by three pairs of genes (A, B, and C), each with two alleles. The alleles A₁, B₁, and C₁ contribute 6 cm each, while A₂, B₂, and C₂ contribute 3 cm each. The total height is the sum of contributions from all alleles in the genotype.
2. Step 2: Write the general formula for the plant height. Let x₁, x₂, and x₃ represent the number of A₁, B₁, and C₁ alleles, respectively. Similarly, let y₁, y₂, and y₃ represent the number of A₂, B₂, and C₂ alleles. The total height can be expressed as: $\mathrm{Height}=\; 6(x₁\; +\; x₂\; +\; x₃)\; +\; 3(y₁\; +\; y₂\; +\; y₃)$. Note that the sum of x and y for each gene pair must equal 2 (e.g., x₁ + y₁ = 2 for the A gene pair).
3. Step 3: Substitute the target height into the formula. The problem specifies a plant height of 33 cm. Substitute this value into the formula: $33\; =\; 6(x₁\; +\; x₂\; +\; x₃)\; +\; 3(y₁\; +\; y₂\; +\; y₃)$. Simplify the equation to find the total contribution of the alleles: $33\; =\; 6(x₁\; +\; x₂\; +\; x₃)\; +\; 3(6\; -\; x₁\; -\; x₂\; -\; x₃)$, since the total number of alleles is fixed at 6.
4. Step 4: Solve for the number of A₁, B₁, and C₁ alleles. Simplify the equation further: $33\; =\; 6(x₁\; +\; x₂\; +\; x₃)\; +\; 18\; -\; 3(x₁\; +\; x₂\; +\; x₃)$. Combine like terms: $33\; =\; 18\; +\; 3(x₁\; +\; x₂\; +\; x₃)$. Subtract 18 from both sides: $15\; =\; 3(x₁\; +\; x₂\; +\; x₃)$. Divide by 3: $x₁\; +\; x₂\; +\; x₃\; =\; 5$. This means the total number of A₁, B₁, and C₁ alleles is 5.
5. Step 5: Identify all possible genotypes. Since x₁ + x₂ + x₃ = 5, distribute the 5 alleles among the three gene pairs (A, B, and C). For each distribution, calculate the corresponding y-values (y₁, y₂, y₃) using the relationship x + y = 2 for each gene pair. For example, if x₁ = 2, x₂ = 2, and x₃ = 1, then y₁ = 0, y₂ = 0, and y₃ = 1. Repeat this process for all valid combinations of x-values to identify all possible genotypes.

Key Concepts

Genotype and Alleles

A genotype refers to the genetic constitution of an organism, specifically the alleles it possesses for a particular gene. In this case, each gene has two alleles (e.g., A₁ and A₂), and the combination of these alleles determines the plant's traits, such as height. Understanding how these alleles interact is crucial for predicting phenotypic outcomes.

Additive Gene Action

Additive gene action occurs when the effects of different alleles contribute cumulatively to a trait. In this scenario, each allele contributes a specific amount to the plant's height: A₁, B₁, and C₁ add 6 cm, while A₂, B₂, and C₂ add 3 cm. This means that the total height can be calculated by summing the contributions of the alleles present in the genotype.

Phenotypic Expression

Phenotypic expression is the observable physical or biochemical characteristics of an organism, resulting from the interaction of its genotype with the environment. In this question, the expected height of 33 cm is the phenotype that needs to be achieved through various combinations of the alleles from the three gene pairs, highlighting the relationship between genotype and phenotype.