Question

Assume a population of garden peas in genetic equilibrium in which the frequencies of the genes for full pods (F) and constricted pods (f) are \(0.6\) and \(0.4\), respectively. If this population is allowed only to reproduce by self-fertilization for three generations, what will be the distribution of the three genotypes by the third generation of self-fertilization?

Short answer

After three generations of self-fertilization in a population of garden peas in genetic equilibrium, the distribution of the three genotypes will be: Homozygous dominant (FF) - 57%, Heterozygous (Ff) - 6%, and Homozygous recessive (ff) - 37%.

Step-by-step solution

Understanding Hardy-Weinberg equilibrium

The Hardy-Weinberg equilibrium principle states that gene frequencies remain constant from generation to generation in a population if there is random mating, no mutation, no selection, no gene flow (migration), and the population size is infinitely large. In this exercise, we assume that the population is in genetic equilibrium, so we can use the Hardy-Weinberg principle. The equation for the Hardy-Weinberg equilibrium is: \(p^2 + 2pq + q^2 = 1\), where p is the frequency of the F gene, q is the frequency of the f gene, p^2 is the frequency of the homozygous dominant genotype (FF), 2pq is the frequency of the heterozygous genotype (Ff), and q^2 is the frequency of the homozygous recessive genotype (ff).

Calculate initial genotype frequencies

We are given the gene frequencies in the population: p (frequency of F) = 0.6 and q (frequency of f) = 0.4. Now, we can calculate the initial genotype frequencies: - FF: \(p^2 = (0.6)^2 = 0.36\) - Ff: \(2pq = 2 * 0.6 * 0.4 = 0.48\) - ff: \(q^2 = (0.4)^2 = 0.16\)

Self-fertilization for three generations

To find the genotype frequencies after three generations of self-fertilization, we will simulate mating for each genotype with itself and sum the results for FF, Ff, and ff genotypes. We will perform this operation for each generation. To self-fertilize a genotype, we use the binomial expansion: - (FF) x (FF) => (1)FF - (Ff) x (Ff) => (0.25)FF + (0.50)Ff + (0.25)ff - (ff) x (ff) => (1)ff

Calculate genotype frequencies after one generation

After one generation, the new genotype frequencies are: - FF: 0.36 + 0.25(0.48) = 0.48 - Ff: 0.50(0.48) = 0.24 - ff: 0.16 + 0.25(0.48) = 0.28

Calculate genotype frequencies after two generations

After two generations, the new genotype frequencies are: - FF: 0.48 + 0.25(0.24) = 0.54 - Ff: 0.50(0.24) = 0.12 - ff: 0.28 + 0.25(0.24) = 0.34

Calculate genotype frequencies after three generations

After three generations, the new genotype frequencies are: - FF: 0.54 + 0.25(0.12) = 0.57 - Ff: 0.50(0.12) = 0.06 - ff: 0.34 + 0.25(0.12) = 0.37

Interpret the results

After three generations of self-fertilization, the distribution of the three genotypes in the garden pea population is: - Homozygous dominant (FF): 57% - Heterozygous (Ff): 6% - Homozygous recessive (ff): 37% This is the final distribution of the genotypes after the third generation of self-fertilization.

Key concepts

Genotype Frequencies

Genotype frequencies refer to the proportion of different genetic combinations in a population. In the context of Hardy-Weinberg equilibrium, these frequencies help us understand the genetic makeup of a population over time.

Using the Hardy-Weinberg principle, we can calculate these frequencies from gene frequencies. If we know the frequency of a dominant allele (like F for full pods in peas) and a recessive allele (like f for constricted pods), we can express the distribution of genotypes mathematically as:
- Homozygous dominant (\(p^2\), like FF)- Heterozygous (\(2pq\), like Ff)- Homozygous recessive (\(q^2\), like ff)

For example, with allele frequencies of 0.6 for F and 0.4 for f, the genotype frequencies are:

• FF: \(0.36\) or 36%
• Ff: \(0.48\) or 48%
• ff: \(0.16\) or 16%

Understanding these frequencies is crucial for predicting how traits like plant pod shape will persist in a population.

Self-Fertilization

Self-fertilization, or selfing, is a form of reproduction where a plant fertilizes itself. This practice is common in many plant species, including garden peas.

When self-fertilization occurs, there's no exchange of genetic material between different plants, leading to significant changes in genotype frequencies over generations. By the second and third generations of self-fertilization, the heterozygous genotype (Ff) tends to decrease substantially because selfing primarily promotes homozygosity.

Let's illustrate this. After the first round of self-fertilization, the offspring of heterozygous (Ff) plants will have a genotype distribution of:

• 25% FF
• 50% Ff
• 25% ff

After three generations, more homozygous individuals (both FF and ff) appear compared to heterozygous ones, as seen by the reduction in Ff frequency from 48% to just 6%. Self-fertilization, therefore, leads to rapid changes in genetic compositions within populations.

Population Genetics

Population genetics is the branch of biology that studies genetic differences within and between populations. It enables us to understand evolutionary changes.

By examining both stable and changing genetic compositions over time, scientists can infer how species might evolve or respond to environmental changes. Concepts like genetic drift, selection, mutation, and migration interact with principles like the Hardy-Weinberg equilibrium to highlight genetic stability or shifts.

The garden pea exercise offers a snapshot of population genetics in action by exploring self-fertilization's effects on genotype frequencies. Starting with Hardy-Weinberg equilibrium, which assumes no evolution without external pressures, we can observe how self-fertilization disrupts genetic balance over time, showcasing real-world applications of these powerful genetic principles. These insights are foundational in conservation biology, agriculture, and understanding human genetics.