A 3-inch plant was crossed with a 15-inch plant, and all F₁ plants were 9 inches. The F₂ plants exhibited a 'normal distribution,' with heights of 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, and 15 inches.
What ratio will constitute the 'normal distribution' in the F₂?

Verified step by step guidance

1. Step 1: Recognize that the problem involves quantitative traits, which are typically controlled by multiple genes (polygenic inheritance) and influenced by environmental factors. The 'normal distribution' of plant heights suggests a polygenic trait with additive effects.
2. Step 2: Determine the number of loci (genes) involved. The F₁ generation plants are intermediate in height (9 inches), indicating that the alleles from the two parent plants contribute additively to the trait. The F₂ generation shows a range of heights, suggesting multiple loci are involved.
3. Step 3: Use the formula for calculating the number of phenotypic classes in a polygenic trait: \( \text{Number of phenotypic classes} = \text{Number of loci} \times 2 + 1 \). The observed phenotypic classes (3 to 15 inches) total 13, which can help estimate the number of loci.
4. Step 4: Calculate the ratio of individuals in each phenotypic class. In polygenic inheritance, the phenotypic distribution follows a binomial expansion. For \( n \) loci, the ratio of phenotypes corresponds to the coefficients in Pascal's triangle for \( (a+b)^n \). Determine \( n \) based on the number of phenotypic classes.
5. Step 5: Assign the ratios to the phenotypic classes (heights) based on the binomial expansion. For example, if there are 3 loci, the distribution would follow \( (a+b)^6 \), resulting in specific ratios for each height class. This explains the 'normal distribution' observed in the F₂ generation.

Key Concepts

Normal Distribution

Normal distribution is a statistical concept where data points are symmetrically distributed around a mean, forming a bell-shaped curve. In genetics, traits often follow this distribution when influenced by multiple genes, leading to a range of phenotypes. The heights of the F₂ plants in this scenario suggest that multiple alleles contribute to the trait, resulting in a continuous variation.

Phenotypic Ratio

Phenotypic ratio refers to the relative frequency of different phenotypes in a population. In the context of the F₂ generation, the ratio of plant heights can be analyzed to determine how traits are inherited. For traits governed by multiple alleles, the phenotypic ratio can often approximate a specific pattern, such as 1:2:1 or 9:3:3:1, depending on the genetic interactions.

Quantitative Traits

Quantitative traits are characteristics that are influenced by multiple genes and environmental factors, resulting in a continuous range of phenotypes. In this case, the plant heights represent a quantitative trait, as they vary from 3 to 15 inches. Understanding how these traits are inherited and expressed is crucial for predicting the distribution of phenotypes in the F₂ generation.