An inbred strain of plants has a mean height of 24 cm. A second strain of the same species from a different geographic region also has a mean height of 24 cm. When plants from the two strains are crossed together, the F₁ plants are the same height as the parent plants. However, the F₂ generation shows a wide range of heights; the majority are like the P₁ and F₁ plants, but approximately 4 of 1000 are only 12 cm high and about 4 of 1000 are 36 cm high.
How many gene pairs are involved?

Verified step by step guidance

1. Step 1: Understand the problem context. We have two inbred strains with the same mean height (24 cm), and their F₁ offspring also have the same height. The F₂ generation shows variation, with most plants at 24 cm, but some at 12 cm and 36 cm, which are extreme phenotypes.
2. Step 2: Recognize that the F₁ uniformity and F₂ variation suggest that the trait is controlled by multiple gene pairs showing additive effects, where each gene pair contributes to the height.
3. Step 3: Note the extreme phenotypes in the F₂ generation occur at a frequency of about 4 in 1000 (0.004) for both the low (12 cm) and high (36 cm) heights. These extremes correspond to homozygous genotypes at all gene pairs for either the recessive or dominant alleles.
4. Step 4: Use the fact that for each gene pair, the probability of a homozygous genotype (either dominant or recessive) in the F₂ is 1/4. For n independent gene pairs, the probability of being homozygous at all loci is \( (\frac{1}{4})^n \). Set this equal to the observed frequency of the extreme phenotypes:
5. \[ (\frac{1}{4})^n = 0.004 \]
6. Step 5: Solve for n by taking the logarithm of both sides:
7. \[ n = \frac{\log(0.004)}{\log(\frac{1}{4})} \]
8. This will give the number of gene pairs involved in controlling the height.

Key Concepts

Polygenic Inheritance

Polygenic inheritance involves multiple gene pairs contributing additively to a single trait, such as height. This results in continuous variation and a range of phenotypes rather than discrete categories. The presence of extreme phenotypes in the F2 generation suggests several genes influence the trait.

F1 and F2 Generations in Genetic Crosses

The F1 generation is the first filial generation from crossing two parent strains, often showing uniform traits due to heterozygosity. The F2 generation results from selfing or intercrossing F1 individuals, revealing genetic segregation and recombination, which can produce a wider range of phenotypes.

Segregation and Recombination of Alleles

During gamete formation, alleles segregate independently, and recombination can create new allele combinations. In polygenic traits, this leads to a distribution of phenotypes in the F2, including rare extreme types, which helps estimate the number of gene pairs involved based on observed ratios.