Question

Given a population in genetic equilibrium in which the initial gene frequency of \(\mathrm{d}\) is \(0.2\); assume the rate of mutation (u) of \(D \rightarrow d\) to be \(4.1 \times 10^{-5}\), and the rate of back mutation (v) of \(\mathrm{d} \rightarrow \mathrm{D}\) to be \(2.5 \times 10^{-7}\) (a) If the above rates of mutation are introduced into the population, what will be the change in \(q\) (frequency of d) in the first generation? (b) Assuming that the above rates continue over time, what will be the value of q at mutational equilibrium?

Short answer

(a) The change in the frequency of allele d in the first generation is approximately \(3.275 \times 10^{-5}\). (b) Assuming that the mutation rates continue over time, the value of q at mutational equilibrium is approximately 0.994.

Step-by-step solution

Understanding the given data

The initial frequency of allele d (q) is given as 0.2. Then the initial frequency of allele D (p) is 1 - 0.2 = 0.8, as p + q = 1. The rate of mutation (u) of D to d is given as \(4.1 \times 10^{-5}\) and the rate of back mutation (v) of d to D is given as \(2.5 \times 10^{-7}\).

Calculate the change in q (Δq) after the first generation

Recall that Δq is the change in the frequency of allele d after the first generation. Δq = Frequency of new d alleles - Frequency of d alleles lost Δq = (Rate of mutation from D to d) × (Initial frequency of D) - (Rate of back mutation from d to D) × (Initial frequency of d) Δq = (u × p) - (v × q) Now, substitute the given values to find Δq. Δq = \((4.1 \times 10^{-5}) \times 0.8 - (2.5 \times 10^{-7}) \times 0.2\)

Compute the change in q (Δq) after the first generation

Δq = \((4.1 \times 10^{-5}) \times 0.8 - (2.5 \times 10^{-7}) \times 0.2 = 3.28 \times 10^{-5} - 5 \times 10^{-8}\) Δq = \(3.275 \times 10^{-5}\) (approx) So, the change in the frequency of allele d after the first generation is approximately \(3.275 \times 10^{-5}\). (a) Answer: The change in \(q\) (frequency of allele d) in the first generation is approximately \(3.275 \times 10^{-5}\).

Derive the formula to calculate the equilibrium frequency of allele d (q)

At mutational equilibrium, the rates of mutation (u) and back mutation (v) are equal. p × u = q × v We know, p + q = 1. Then, p = 1 - q. Now, substitute p = 1 - q into the balance equation: \( (1 - q) \times u = q \times v \)

Solve for q at mutational equilibrium

Now, solve the equation \((1 - q) \times u = q \times v\) for the frequency of allele d (q) at mutational equilibrium. \( q \times v + q \times u = u \) \( q (v + u) = u \) \( q = \frac{u}{(v + u)} \) Substitute the given values for u and v: \( q = \frac{4.1 \times 10^{-5}}{(2.5 \times 10^{-7} + 4.1 \times 10^{-5})} \)

Compute the equilibrium frequency of allele d (q)

\( q = \frac{4.1 \times 10^{-5}}{(2.5 \times 10^{-7} + 4.1 \times 10^{-5})} \) \( q \approx 0.994 \) (b) Answer: Assuming that the mutation rates continue over time, the value of q at mutational equilibrium is approximately 0.994.

Key concepts

Genetic Equilibrium

Genetic equilibrium refers to a state in which the allele frequencies in a population remain constant over time. This concept is fundamental in population genetics because it provides a baseline from which we can understand how evolutionary forces such as mutation, selection, migration, and genetic drift cause changes in the gene pool of a population.

For a population to be at genetic equilibrium, certain conditions must be met: there should be no mutations, no gene flow between different populations, random mating must occur, the population size must be very large (to minimize drift), and no selection should be acting on any particular allele.
When these conditions are met, the allele frequencies are stable from generation to generation and the population is said to be in Hardy-Weinberg equilibrium.

Allele Frequency

The term allele frequency, sometimes known as gene frequency, is the proportion of all copies of a particular gene in a population that are of a given allele type. If we were to look at a gene locus where two alleles are possible, for example, allele A and allele a, we would express the frequency of allele A as 'p' and the frequency of allele a as 'q'. Our aim is usually to determine the proportion of the population that carries these alleles.

For example, if we have a large population where 60% of the alleles at a given gene locus are A and 40% are a, we would say that the allele frequency of A (p) is 0.6, and the allele frequency of a (q) is 0.4. It's important to keep in mind that allele frequencies are used to predict genotype frequencies under the Hardy-Weinberg equilibrium.

Mutation Rate

Mutation rate is a measure of the frequency at which random mutations occur in a given gene or organism over time. It's a critical parameter in genetics as it can indicate the rate at which a genetic sequence is changing, potentially affecting the genetic diversity of a population.

In population genetics, the mutation rate may refer to the probability of a gene mutating from one allele to another in a single generation. Understanding mutation rates is crucial when researchers study genetic disorders, evolutionary biology, and even when calculating the mutational equilibrium in a population.

Back Mutation

Back mutation or reverse mutation is the process by which an allele that has undergone mutation reverses to its original state, or a state that restores the original function that was lost due to the forward mutation. Although typically less frequent than forward mutations, back mutations play an essential role in maintaining genetic variation within a population.

In an exercise where mutation and back mutation rates are given, they become integral to calculating the new equilibrium of allele frequencies after these mutations are introduced. For instance, back mutations can counteract the effect of forward mutations and vice versa, leading to a new balance or 'mutational equilibrium'.

Hardy-Weinberg Equilibrium

The Hardy-Weinberg equilibrium is a principle that states that allele and genotype frequencies in a population will remain constant from generation to generation in the absence of other evolutionary influences. These include mutation, gene flow, genetic drift, and natural selection, as mentioned previously. In a Hardy-Weinberg equilibrium, the genotype frequencies can be predicted based on the allele frequencies using the equation p^2 + 2pq + q^2 = 1 where p^2 represents the frequency of the homozygous dominant genotype, 2pq the frequency of the heterozygous genotype, and q^2 the frequency of the homozygous recessive genotype.

This formula is a cornerstone of population genetics and helps us understand how genetic variation is expressed and maintained within a population. However, real-life populations rarely meet all the criteria needed to maintain Hardy-Weinberg equilibrium, thereby leading to evolution.