Question

In a population of cattle, the following color distribution was noted: \(36 \%\) red \((R R), 48 \%\) roan \((R r),\) and \(16 \%\) white \((r r) .\) Is this population in a Hardy-Weinberg equilibrium? What will be the distribution of genotypes in the next generation if the Hardy-Weinberg assumptions are met?

Short answer

Answer: In the next generation, the distribution of genotypes will be 36% red (RR), 48% roan (Rr), and 16% white (rr) if the Hardy-Weinberg assumptions are met.

Step-by-step solution

Calculating allele frequencies

To find the allele frequencies, we can use the information given about the genotypes frequencies. Since RR (red) has a frequency of 36% and Rr (roan) has a frequency of 48%, we can calculate the frequency of the R allele (p) as follows: \(p = \frac{2 * RR~frequency + Rr~frequency}{2 * Total~frequency} = \frac{2 * 0.36 + 0.48}{2} = 0.6\) Now, to find the frequency of the r allele (q), we can use the fact that the sum of p and q must equal 1: \(q = 1 - p = 1 - 0.6 = 0.4\)

Testing for Hardy-Weinberg equilibrium

Now that we have the allele frequencies, we can use the Hardy-Weinberg equation to test if the population is in equilibrium. The expected frequencies for the various genotypes under Hardy-Weinberg equilibrium would be as follows: Expected \(RR (red) = p^2 = (0.6)^2 = 0.36\) Expected \(Rr (roan) = 2pq = 2*0.6*0.4 = 0.48\) Expected \(rr (white) = q^2 = (0.4)^2 = 0.16\) These expected genotype frequencies match the given color distribution from the problem. Therefore, this population is in Hardy-Weinberg equilibrium.

Predicting the next generation's genotype frequencies

Since the population meets the assumptions of the Hardy-Weinberg equilibrium, we can expect it to remain in equilibrium in the next generation. This means that the genotype frequencies in the next generation will be the same as the current generation: Next generation \(RR (red) = 36\%\) Next generation \(Rr (roan) = 48\%\) Next generation \(rr (white) = 16\%\) So, the distribution of genotypes in the next generation will be \(36\%\) red (RR), \(48\%\) roan (Rr), and \(16\%\) white (rr) if the Hardy-Weinberg assumptions are met.

Key concepts

Allele Frequency

Allele frequency describes how common an allele is in a population. Alley frequency is crucial for studying Hardy-Weinberg Equilibrium. It allows us to determine the genetic composition and variability within a population.

To understand allele frequency better, imagine each organism as having two alleles for any gene. In a diploid organism, these alleles can be dominant or recessive. For our cattle exercise, we have alleles 'R' (for red) and 'r' (for white). Therefore, calculating the frequencies of each allele helps us predict the population's genetic structure.

The formula to calculate allele frequency is:

• For allele 'R', the frequency (p) is determined by the equation:\[ p = \frac{2 \times \text{frequency of } RR + \text{frequency of } Rr}{2 \times \text{total frequency}} \]
• For allele 'r', the frequency (q) can be simply obtained by subtracting 'p' from 1: \( q = 1 - p \)

Using these calculations, we find that in the cattle population, 'R' has a frequency of 0.6 and 'r' a frequency of 0.4. These are foundational for determining the genetic distribution under Hardy-Weinberg equilibrium.

Genotype Distribution

Genotype distribution is about understanding how different combinations of alleles are spread in a population. It helps to visualize how individual organisms express their genetic information.

In our cattle exercise, the genotypes are represented by combinations of 'R' and 'r' alleles: RR, Rr, and rr. The frequency of each genotype can be tested against Hardy-Weinberg Equilibrium expectations:

• For homozygous dominant (RR), the expected frequency is \( p^2 \)
• For heterozygous (Rr), it's \( 2pq \)
• And for homozygous recessive (rr), it's \( q^2 \)

These calculations serve two purposes: confirm if the population is in equilibrium, and predict future distributions.
The observed genotype frequencies of 36% (RR), 48% (Rr), and 16% (rr) perfectly match the expected values. Hence, the population's distribution aligns with Hardy-Weinberg equilibrium.

Population Genetics

Population genetics is a critical field that examines genetic variations within populations and how these change over time and generations. It employs principles like Hardy-Weinberg Equilibrium to provide insights into evolutionary processes and genetic diversity.

In a stable population, modeled by Hardy-Weinberg, allele and genotype frequencies remain constant over generations—assuming no evolutionary forces are acting. This means random mating, no natural selection, no genetic drift, no migration, and no mutation.
Understanding this equilibrium is essential as it works as a null model in genetics. Deviation from this model allows scientists to detect evolutionary influences such as selection or genetic drift.

For the cattle population studied, the genetic equilibrium was confirmed since the observed and expected frequencies match. This consistency signals that, at least for the traits studied, no disruptive evolutionary forces appear to be at play. This allows the prediction that subsequent generations will maintain the same genetic structure, which is vital for breeding programs or conservation efforts.