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Chapter 22: Problem 10

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 HardyWeinberg assumptions are met?

Short Answer

Expert verified
Based on the given observed genotype frequencies and calculated expected genotype frequencies, determine if the population of cattle is in Hardy-Weinberg equilibrium and predict the distribution of genotypes in the next generation.

Step by step solution

01

1. Determine observed genotype frequencies

The exercise provides us with the observed genotype frequencies in the population: - Red (RR): \(36\%\) - Roan (Rr): \(48\%\) - White (rr): \(16\%\)
02

2. Calculate allele frequencies

We can use these observed genotype frequencies to calculate the frequency of each allele (R and r) in the population by applying the formula: \(p = \frac{2n(RR) + n(Rr)}{2n}\) (frequency of R) \(q = \frac{2n(rr) + n(Rr)}{2n}\) (frequency of r), where n is the total number of individuals in the population. Let's plug in the observed genotype frequencies: \(p = \frac{2(0.36) + 0.48}{2} = \frac{1.2}{2} = 0.6\) \(q = \frac{2(0.16) + 0.48}{2} = \frac{0.8}{2} = 0.4\) This means \(60\%\) of the alleles in the population are R and \(40\%\) are r.
03

3. Calculate expected genotype frequencies

Using the Hardy-Weinberg principle, we can determine the expected frequency of each genotype as follows: \(RR = p^2\) (Expected frequency of RR) \(Rr = 2pq\) (Expected frequency of Rr) \(rr = q^2\) (Expected frequency of rr) Plug in the allele frequencies we calculated above: \(RR = (0.6)^2 = 0.36\) \(Rr = 2(0.6)(0.4) = 0.48\) \(rr = (0.4)^2 = 0.16\)
04

4. Compare observed and expected genotype frequencies

Now, we can compare the expected frequencies we calculated above with the observed frequencies we were given to determine if the population is in Hardy-Weinberg equilibrium: Observed: - Red (RR): \(36\%\) - Roan (Rr): \(48\%\) - White (rr): \(16\%\) Expected (based on Hardy-Weinberg equilibrium): - Red (RR): \(36\%\) - Roan (Rr): \(48\%\) - White (rr): \(16\%\) The observed and expected genotype frequencies are the same, so the population is in Hardy-Weinberg equilibrium.
05

5. Predict genotype distribution in the next generation

If the population remains in Hardy-Weinberg equilibrium, then the frequencies of the genotypes will remain constant from generation to generation, and the distribution will be the same in the next generation: - Red (RR): \(36\%\) - Roan (Rr): \(48\%\) - White (rr): \(16\%\)

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