Hardy-Weinberg Equilibrium Analysis

The question describes a population of plants in Hardy-Weinberg equilibrium. We are given the frequency of recessive homozygotes and the total population size, and we need to find the numbers of homozygous dominant and heterozygous plants.

Understanding Hardy-Weinberg Principle

The Hardy-Weinberg principle describes the genetic makeup of a population that is not evolving. It provides a baseline to compare with real populations. The core equations are:

• Allele frequencies: $p + q = 1$
• Genotype frequencies: $p^2 + 2pq + q^2 = 1$

Where:

• $p$ is the frequency of the dominant allele (let's say R for red).
• $q$ is the frequency of the recessive allele (let's say r for white).
• $p^2$ is the frequency of the homozygous dominant genotype (RR, red flowers).
• $2pq$ is the frequency of the heterozygous genotype (Rr, pink flowers, assuming incomplete dominance as suggested by "pink").
• $q^2$ is the frequency of the homozygous recessive genotype (rr, white flowers).

Calculating Allele Frequencies

We are given that 40% of the plants are recessive homozygotes (rr) and produce white flowers (WF). In Hardy-Weinberg terms, this means the frequency of the recessive genotype ($q^2$) is 0.40.

So, $q^2 = 0.40$.

To find the frequency of the recessive allele ($q$), we take the square root of $q^2$:

$q = \sqrt{0.40} \approx 0.6324$

Now we can find the frequency of the dominant allele ($p$) using the equation $p + q = 1$:

$p = 1 - q = 1 - 0.6324 = 0.3676$

Calculating Genotype Frequencies

Using the calculated allele frequencies ($p$ and $q$), we can find the frequencies of the other genotypes ($p^2$ and $2pq$).

• Frequency of homozygous dominant (RR, RF): $p^2 = (0.3676)^2 \approx 0.13513976 \approx 0.1351$
• Frequency of heterozygous (Rr, PF): $2pq = 2 \times 0.3676 \times 0.6324 \approx 0.46494992 \approx 0.4649$
• Frequency of homozygous recessive (rr, WF): $q^2 = 0.40$ (as given)

Let's quickly check if the genotype frequencies sum up to 1: $p^2 + 2pq + q^2 = 0.1351 + 0.4649 + 0.40 = 1.0000$. The frequencies add up correctly (with minor rounding differences).

Calculating Number of Plants for Each Genotype

The total number of individuals in the population is 14000. To find the number of plants for each genotype, we multiply the total population size by the frequency of each genotype.

• Number of homozygous dominant (RR, RF) plants = Frequency of RR $\times$ Total Population = $p^2 \times 14000 \approx 0.1351 \times 14000 \approx 1891.4$
• Number of heterozygous (Rr, PF) plants = Frequency of Rr $\times$ Total Population = $2pq \times 14000 \approx 0.4649 \times 14000 \approx 6508.6$
• Number of homozygous recessive (rr, WF) plants = Frequency of rr $\times$ Total Population = $q^2 \times 14000 = 0.40 \times 14000 = 5600$

Rounding to the nearest whole number for the number of plants:

• Homozygous dominant (RF): Approximately 1891 plants.
• Heterozygous (PF): Approximately 6509 plants (or 6508 depending on exact rounding points).
• Homozygous recessive (WF): 5600 plants.

Let's check the total number of plants: $1891 + 6509 + 5600 = 14000$. This matches the total population size.

Comparing with Options

We calculated approximately 1891 RF plants and 6509 PF plants. Let's look at the options provided:

Option 2 closely matches our calculated values for RF (1891) and PF (6508). The slight difference in the PF number (6508 vs 6509) is likely due to rounding during the intermediate calculations.