Question

Consider a population in which the frequency of allele \(A\) is \(p=0.7\) and the frequency of allele \(a\) is \(q=0.3,\) and where the alleles are codominant. What will be the allele frequencies after one generation if the following occurs? (a) \(w_{A A}=1, w_{A a}=0.9,\) and \(w_{a a}=0.8\) (b) \(w_{A A}=1, w_{A a}=0.95,\) and \(w_{a a}=0.9\) (c) \(w_{A A}=1, w_{A a}=0.99, w_{a a}=0.98\) (d) \(w_{A A}=0.8, w_{A a}=1, w_{a a}=0.8\)

Short answer

Question: In a population of individuals with the following genotype frequencies: A A = 49%, A a = 42%, and a a = 9%, consider the four given scenarios of fitness values for each genotype (A A, A a, and a a): (a) 1, 0.9, 0.8, (b) 1, 0.95, 0.9, (c) 1, 0.99, 0.98, and (d) 0.8, 1, 0.8. Calculate the allele frequencies after one generation for each scenario. Answer: The allele frequencies after one generation for each scenario are: (a) A: 0.737, a: 0.263 (b) A: 0.719, a: 0.281 (c) A: 0.707, a: 0.293 (d) A: 0.692, a: 0.308

Step-by-step solution

Calculate genotype frequencies

We will use the Hardy-Weinberg equation to find the genotype frequencies in the initial generation: $$p^2 + 2pq + q^2 = 1$$ $$A A: (0.7)^2 = 0.49$$ $$A a: 2(0.7)(0.3) = 0.42$$ $$a a: (0.3)^2 = 0.09$$ Step 2: Calculate average fitness for each scenario

Calculate average fitness

We will use the following formula to compute average fitness for each scenario: $$w = (f_{A A} \times w_{A A}) + (f_{A a} \times w_{A a}) + (f_{a a} \times w_{a a})$$ We will now calculate this for each scenario. - (a): $$w = (0.49 \times 1) + (0.42 \times 0.9) + (0.09 \times 0.8)$$ $$w = 0.913$$ - (b): $$w = (0.49 \times 1) + (0.42 \times 0.95) + (0.09 \times 0.9)$$ $$w = 0.958$$ - (c): $$w = (0.49 \times 1) + (0.42 \times 0.99) + (0.09 \times 0.98)$$ $$w = 0.985$$ - (d): $$w = (0.49 \times 0.8) + (0.42 \times 1) + (0.09 \times 0.8)$$ $$w = 0.864$$ Step 3: Calculate allele frequencies in the next generation

Calculate next-generation allele frequencies

For the next-generation allele frequencies, we can use the following formula: $$p'=\frac{p^2w_{A A}+pqw_{A a}}{w}$$ $$q'=\frac{pqw_{A a}+q^2w_{a a}}{w}$$ - (a): $$p'=\frac{(0.49 \times 1)+(0.42 \times 0.9)}{0.913}=0.737$$ $$q'=1-p'=0.263$$ - (b): $$p'=\frac{(0.49 \times 1)+(0.42 \times 0.95)}{0.958}=0.719$$ $$q'=1-p'=0.281$$ - (c): $$p'=\frac{(0.49 \times 1)+(0.42 \times 0.99)}{0.985}=0.707$$ $$q'=1-p'=0.293$$ - (d): $$p'=\frac{(0.49 \times 0.8)+(0.42 \times 1)}{0.864}=0.692$$ $$q'=1-p'=0.308$$ So, the allele frequencies for each scenario after one generation are (rounded to the nearest thousandth): (a): \(A: 0.737,\) \(a: 0.263\) (b): \(A: 0.719,\) \(a: 0.281\) (c): \(A: 0.707,\) \(a: 0.293\) (d): \(A: 0.692,\) \(a: 0.308\)

Key concepts

Genotype Frequencies

Understanding genotype frequencies is foundational in studying population genetics and the Hardy-Weinberg equilibrium. These frequencies describe the proportion of the various genotypes within a population. In the context of the given exercise, genotype frequencies pertain to the combinations of alleles for a genetic locus: AA, Aa, and aa.

Using the Hardy-Weinberg principle, we expect the genotype frequencies to be represented by the terms of the binomial expansion of \( (p + q)^2 \) for a population in equilibrium. However, as fitness values are introduced, which represent the reproductive success of each genotype, these frequencies can shift, reflecting natural selection. Through calculations such as the ones in the exercise, students can observe how selective pressures can alter genotype frequencies and thus change genetic structure over generations.

Allele Frequencies

On the other hand, allele frequencies refer to how common an allele is in the population, typically denoted as ‘p’ for the dominant allele and ‘q’ for the recessive allele. These two frequencies must always add up to 1, as they represent all the possible variants of a gene in the population. During the exercise, allele frequencies were recalculated after applying fitness values to account for differential survival and reproduction, demonstrating how allele frequencies can change from one generation to the next due to selection. Thus, the concept of allele frequencies is crucial in understanding the genetic variation within a population and predicting its future genetic composition.

Average Fitness

Average fitness of a population—denoted as ‘w’ in genetic equations—is a measure that encompasses the reproductive success of all genotypes weighted by their respective frequencies. This measure, fundamentally entwined with the concept of natural selection, gives insight into the adaptive landscape of a population. In the context of the exercise, calculating the average fitness under different scenarios showed how different survival and reproduction rates among the genotypes influence the overall adaptive capacity of the population. Furthermore, it illuminates how more fit genotypes can increase in frequency over time, thereby driving evolutionary change. The computation of average fitness thus merges genotype frequencies and individual fitness into a composite metric reflecting the evolutionary vitality of the population.