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) \(w_{\mu}=1, w_{A a}=0.9, w_{a a}=0.8\) (b) \(w_{\mu}=1, w_{A a}=0.95, w_{a a}=0.9\) (c) \(w_{\mu}=1, w_{h a}=0.99, w_{a a}=0.98\) (d) \(w_{\mu}=0.8, w_{A a}=1, w_{a a}=0.8\)

Short answer

Answer: (a) A: 0.7105, a: 0.2895 (b) A: 0.7165, a: 0.2835 (c) A: 0.7202, a: 0.2798 (d) A: 0.6764, a: 0.3236

Step-by-step solution

Review Hardy-Weinberg equilibrium proportions

For a population in Hardy-Weinberg equilibrium with allele frequencies \(p\) and \(q\), the genotype frequencies are distributed as follows: \(AA: p^2\) \(Aa: 2 \times p \times q\) \(aa: q^2\) In this case: \(p^2=0.7^2=0.49\) \(2pq=2(0.7)(0.3)=0.42\) \(q^2=0.3^2=0.09\)

Calculate the average fitness

For each of the situations given (a, b, c, and d), we need to calculate the average fitness of the population. This can be calculated with the following equation: \(w_{avg} = w_{AA}p^2 + w_{Aa}(2pq) + w_{aa}q^2\) It is important to note that as the question has different notations for the w values, you need to pay attention to which notation is provided in each part of the exercise.

Calculate the new allele frequencies after one generation

The new allele frequencies after one generation can be calculated with the following equation: - For allele A: \(\frac{w_{AA}p^2 + 0.5(w_{Aa})(2pq)}{w_{avg}}\) - For allele a: \(\frac{w_{aa}q^2 + 0.5(w_{Aa})(2pq)}{w_{avg}}\) Now, let's address each situation one by one.

(a) \(w_{\mu}=1, w_{Aa}=0.9, w_{aa}=0.8\)

Calculate average fitness: \(w_{avg} = (1)(0.49) + (0.9)(0.42) + (0.8)(0.09) = 0.950\) Calculate new allele frequencies: \(p' = \frac{(1)(0.49) + 0.5(0.9)(0.42)}{0.950} = 0.7105\) \(q' = \frac{(0.8)(0.09) + 0.5(0.9)(0.42)}{0.950} = 0.2895\)

(b) \(w_{\mu}=1, w_{Aa}=0.95, w_{aa}=0.9\)

Calculate average fitness: \(w_{avg} = (1)(0.49) + (0.95)(0.42) + (0.9)(0.09) = 0.970\) Calculate new allele frequencies: \(p' = \frac{(1)(0.49) + 0.5(0.95)(0.42)}{0.970} = 0.7165\) \(q' = \frac{(0.9)(0.09) + 0.5(0.95)(0.42)}{0.970} = 0.2835\)

(c) \(w_{\mu}=1, w_{Aa}=0.99, w_{aa}=0.98\)

Calculate average fitness: (Using the right notation for this part) \(w_{avg} = (1)(0.49) + (0.99)(0.42) + (0.98)(0.09) = 0.990\) Calculate new allele frequencies: \(p' = \frac{(1)(0.49) + 0.5(0.99)(0.42)}{0.990} = 0.7202\) \(q' = \frac{(0.98)(0.09) + 0.5(0.99)(0.42)}{0.990} = 0.2798\)

(d) \(w_{\mu}=0.8, w_{Aa}=1, w_{aa}=0.8\)

Calculate average fitness: \(w_{avg} = (0.8)(0.49) + (1)(0.42) + (0.8)(0.09) = 0.890\) Calculate new allele frequencies: \(p' = \frac{(0.8)(0.49) + 0.5(1)(0.42)}{0.890} = 0.6764\) \(q' = \frac{(0.8)(0.09) + 0.5(1)(0.42)}{0.890} = 0.3236\) So after one generation, the allele frequencies are: (a) A: 0.7105, a: 0.2895 (b) A: 0.7165, a: 0.2835 (c) A: 0.7202, a: 0.2798 (d) A: 0.6764, a: 0.3236

Key concepts

Allele Frequency

When we discuss allele frequency, we're talking about how often a particular form of a gene appears within a given population. In genetics, alleles are the different versions of a gene that may exist. Allele frequencies are expressed as a proportion or a percentage of the entire population carrying a specific allele. For example, in a simple case with one gene that has two alleles, A and a, the sum of the frequencies of both alleles (represented as p for the frequency of allele A, and q for the frequency of allele a) always equals 1 in a population. This is stated mathematically as:

\(p + q = 1\)

The significance of understanding allele frequency lies in being able to predict how a population's genetic makeup is likely to change over time, which is a core concept in population genetics. The Hardy-Weinberg equilibrium provides a mathematical baseline for evaluating if, and how, actual populations differ from a theoretical, non-evolving population.

Genotype Frequency

Moving from alleles to individuals, genotype frequency is the proportion of a population that carries a specific combination of alleles. Genotypes are often represented using two letters, as each organism has two copies of each gene, one from each parent. In a diploid species, such as humans, common genotypes for a gene with two alleles, A and a, are AA, Aa, and aa. If we have a population that is in Hardy-Weinberg equilibrium, the genotype frequencies can be calculated using the allele frequencies, p and q:

• The frequency of homozygous for the dominant allele (AA) is \(p^2\).
• The frequency of heterozygous (Aa) is \(2pq\).
• The frequency of homozygous for the recessive allele (aa) is \(q^2\).

This model assumes random mating within the population and that allele frequencies remain constant (no mutation, migration, selection, or genetic drift), which is often not the case in natural populations. Nonetheless, it provides a foundational framework for understanding how these frequencies are expected to present in a population that is not subject to evolutionary forces.

Population Genetics

The broader field that ties these concepts together is population genetics, which studies the distribution of and change in allele frequencies under the influence of the four main evolutionary processes: mutation, selection, gene flow, and genetic drift. In the exercise provided, selection is the evolutionary process at play — individuals with certain genotypes have a different survival or reproduction rate (fitness), represented by the w values.

In context, the Hardy-Weinberg equilibrium acts as a null hypothesis for scientists studying population genetics. When real-world populations deviate from the equilibrium, it prompts investigation into which evolutionary forces are responsible. For students trying to understand these concepts, it helps to consider a non-evolving population as a baseline. Real populations can then be compared to this baseline to understand the action of natural selection, as shown by the varying w-values given in the exercise and how they affect genotype and allele frequencies over one generation.

Population genetics blends the study of Mendelian genetics (inheritance patterns) with Darwinian evolution, and it is critical for understanding how species adapt to their environment — both in the past and in the ongoing dance with changing conditions.