Question

Consider a population of garden peas in which the genes \(\mathrm{F}\) for full pods and \(\mathrm{f}\) for constricted pods are segregating. Assuming that gene frequencies for this population are found to be: \(p\) (frequency of \(\mathrm{F})=0.7\) and \(\mathrm{q}\) (frequency of \(\mathrm{f})=0.3\) and that the population is in genetic equilibrium, what proportion of the progeny produced from matings of full-podded \(\times\) full-podded will be constricted- podded?

Short answer

About 4.41% of the progeny produced from matings of full-podded \(\times\) full-podded peas will be constricted-podded.

Step-by-step solution

Understand the genotypes

There are three possible genotypes for the peas: 1. Homozygous Dominant (FF): Full-podded peas with both dominant alleles 2. Heterozygous (Ff or fF): Full-podded peas with one dominant(F) and one recessive(f) allele 3. Homozygous Recessive (ff): Constricted-podded peas with both recessive alleles Since we are mating full-podded peas, we need to consider both Homozygous Dominant (FF) and Heterozygous (Ff).

Apply the Hardy-Weinberg equation

The Hardy-Weinberg equation states the following relationship for a population in genetic equilibrium: \[ p^2 + 2pq + q^2 = 1 \] where - \(p^2\) is the frequency of Homozygous Dominant genotype (FF) - \(2pq\) is the frequency of Heterozygous genotype (Ff) - \(q^2\) is the frequency of Homozygous Recessive genotype (ff) We are given that \(p = 0.7\) and \(q = 0.3\). We will use these values to calculate the genotype frequencies.

Calculate the genotype frequencies

Using the given \(p\) and \(q\) values, we can calculate the genotype frequencies as: FF frequency (Homozygous Dominant): \( p^2 = (0.7)^2 = 0.49 \) Ff frequency (Heterozygous): \( 2pq = 2(0.7)(0.3) = 0.42 \)

Calculate the proportion of constricted-podded progeny

Now we will perform the cross of full-podded peas, considering both FF and Ff genotypes, and calculate the proportion of constricted-podded (ff) progeny. Consider the following crosses and their resulting progenies: 1. FF x FF: (0.49)² = 0.2401 probability, resulting in 100% FF progeny 2. FF x Ff: (0.49)(0.42) = 0.2058 probability, resulting in 50% FF and 50% Ff progeny 3. Ff x FF: (0.42)(0.49) = 0.2058 probability, resulting in 50% FF and 50% Ff progeny 4. Ff x Ff: (0.42)² = 0.1764 probability, resulting in 25% FF, 50% Ff, and 25% ff progeny To find the proportion of constricted-podded (ff) progeny, we're only interested in the outcome of the last cross (Ff x Ff).

Determine the proportion of ff progeny

From the Ff x Ff cross, we have a 0.1764 probability of occurrence and 25% (0.25) of the progeny being ff. So, the proportion of constricted-podded (ff) progeny in the population is: Proportion of ff progeny = (0.1764)(0.25) = 0.0441 Thus, about 4.41% of the progeny produced from matings of full-podded \(\times\) full-podded peas will be constricted-podded.

Key concepts

Genotype Frequencies

In any population, the distribution of genotypes is crucial for understanding genetic variation and predicting offspring traits. Genotype frequencies refer to how often different genotypes occur in a population. For peas, as in our exercise, these genotypes are: Homozygous Dominant (\(\text{FF}\),) Heterozygous (\(\text{Ff or fF}\),) and Homozygous Recessive (\(\text{ff}\)).

Each of these genotypes corresponds to different physical characteristics. In our case, both \(\text{FF}\) and \(\text{Ff}\) genotypes lead to full-podded peas, while \(\text{ff}\) results in constricted-podded peas.

• The frequency of \(\text{FF}\): \(p^2\)
• The frequency of \(\text{Ff}\): \(2pq\)
• The frequency of \(\text{ff}\): \(q^2\)

Given our equation, these frequencies must sum up to \(1\). This distribution helps in predicting the likelihood of each genotype appearing in offspring, guiding breeders and scientists in selecting for specific traits. Thus, this analysis serves as the foundation for understanding inheritance patterns in population genetics.

Allele Frequencies

Allele frequencies give us insight into how common certain alleles are within a population. In this exercise, we looked at two alleles: \(\text{F}\),) the dominant allele for full pods, and \(\text{f}\),) the recessive allele for constricted pods. The frequency of an allele, such as \(p\) for \(\text{F}\),) and \(q\) for \(\text{f}\),) helps us track how genetic traits are expected to propagate through generations.

A key point is the Hardy-Weinberg principle, which assumes that, in a stable population without evolutionary pressures, allele frequencies remain constant. This "genetic equilibrium" forms the basis for predicting gene distributions in future generations.

• If \(p = 0.7\) and \(q = 0.3\), the sum \(p + q = 1\)

This balance is essential for evaluating probabilities of genotypes when calculating expected offspring traits. By maintaining track of allele frequencies, scientists can assess genetic diversity and infer evolutionary trends.

Mendelian Genetics

Mendelian genetics, named after Gregor Mendel, is the classic approach to understanding how traits are inherited through generations. It focuses on the distribution of alleles from parents to offspring using core principles like segregation and independent assortment.

The Hardy-Weinberg equilibrium, which was applied in our solution, directly connects to Mendel's principles by providing a mathematical framework for predicting genotype distributions over time when there's no external interference.

• Dominant and recessive allele interactions can be predicted using Punnett squares, demonstrating Mendel's First Law of Segregation.
• Mendelian ratios emerge from predictable allele behavior, such as the \(3:1\) ratio often seen in monohybrid crosses.

Even though nature introduces complexity, such as incomplete dominance and epistasis, the basic Mendelian model remains a cornerstone. Understanding these principles is key to solving problems involving genetic predictions, just like determining the proportion of constricted-podded peas from full-podded parents.