Question

In a population of cattle, the following color distribution was noted: $36\mathrm{\%}$ red $(RR),48\mathrm{\%}$ roan $(Rr),$ and $16\mathrm{\%}$ white $(rr).$ 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

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

1. Determine observed genotype frequencies

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

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 $60\mathrm{\%}$ of the alleles in the population are R and $40\mathrm{\%}$ are r.

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: $RR=(0.6{)}^2=0.36$ $Rr=2(0.6)(0.4)=0.48$ $rr=(0.4{)}^2=0.16$

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): $36\mathrm{\%}$ - Roan (Rr): $48\mathrm{\%}$ - White (rr): $16\mathrm{\%}$ Expected (based on Hardy-Weinberg equilibrium): - Red (RR): $36\mathrm{\%}$ - Roan (Rr): $48\mathrm{\%}$ - White (rr): $16\mathrm{\%}$ The observed and expected genotype frequencies are the same, so the population is in Hardy-Weinberg equilibrium.

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): $36\mathrm{\%}$ - Roan (Rr): $48\mathrm{\%}$ - White (rr): $16\mathrm{\%}$

Key concepts

Understanding Genotype Frequencies

In the study of genetics, genotype frequencies refer to how often certain genetic combinations appear within a population. In our example with cattle, we have three observed genotypes: Red (RR), Roan (Rr), and White (rr). These genotypes are expressed as percentages of the population. The Hardy-Weinberg principle helps us understand if these frequencies suggest genetic stability.

• Red (RR): 36% of the population
• Roan (Rr): 48% of the population
• White (rr): 16% of the population

Genotype frequencies are key to determining whether a population is in Hardy-Weinberg equilibrium by comparing observed frequencies with expected ones.

Determining Allele Frequencies

Allele frequencies represent the relative proportions of different alleles (versions of a gene) in a population. To determine allele frequencies, we use the observed genotype frequencies:

- Allele R is found in both RR and Rr genotypes. The formula for the frequency of R, denoted as $p$, is: $$p=\frac{2n(RR)+n(Rr)}{2n}$$ Plugging in our values gives us $p=0.6$ or 60% for allele R.

- Allele r is present in both rr and Rr genotypes. The frequency of r, denoted as $q$, is: $$q=\frac{2n(rr)+n(Rr)}{2n}$$ This results in $q=0.4$ or 40% for allele r.

In Hardy-Weinberg equilibrium, the sum of allele frequencies $p+q$ must equal 1. This confirms the balance within the gene pool.

Exploring Population Genetics

Population genetics is the study of genetic variation across populations, and the Hardy-Weinberg equilibrium is a fundamental concept in this field. It describes a situation where allele and genotype frequencies remain constant over generations, assuming no evolutionary influences.

For a population to achieve this balance, certain conditions must be met:

• Random mating
• No mutations
• No natural selection
• Large population size
• No migration

In our cattle population, the match between observed and expected genotype frequencies confirms that the population is in equilibrium. Thus, the genotype distribution will remain stable in future generations as long as these conditions continue to be met. In practice, this principle helps scientists predict genetic trends and explore evolutionary patterns.