Question

A gene \(C\) mutates to \(c\) with a frequency of \(2 \times 10^{-6}\) per generation. There are no other forces acting on these alleles and mating is random. How many generations are needed to increase the frequency of gene c from 2 percent to 3 percent?

Short answer

It would take approximately 5102 generations to increase the frequency of gene c from 2% to 3%.

Step-by-step solution

Understand the given information

We are given the following: - The mutation rate of gene C to c is \(2 \times 10^{-6}\) per generation. - The initial frequency of gene c is 2% or 0.02. - The desired final frequency of gene c is 3% or 0.03. - There are no other forces acting on these alleles, and mating is random.

Apply the allele frequency equation

The rate of change in allele frequency due to mutation can be represented using the equation: \[\Delta q = \mu p\] where, \(\Delta q\) = change in frequency of the mutant allele c, \(\mu\) = mutation rate from the wild-type allele C to the mutant allele c, and \(p\) = frequency of the wild-type allele C.

Calculate the frequency of the wild-type allele C

Given that the initial frequency of the mutant allele c is 0.02, the frequency of the wild-type allele C can be represented as: \[p = 1 - q\] \[p = 1 - 0.02\] \[p = 0.98\]

Calculate the change in frequency of the mutant allele c

Now we can use the allele frequency equation and the values of \(\mu\) and \(p\) to find the change in frequency of the mutant allele c in one generation: \[\Delta q = \mu p\] \[\Delta q = (2 \times 10^{-6}) \times 0.98\] \[\Delta q \approx 1.96 \times 10^{-6}\]

Calculate the number of generations required

To find the number of generations needed to increase the frequency of the mutant allele c from 0.02 to 0.03, we can use the equation: \[\text{Total change in frequency} = \text{Change in frequency per generation} \times \text{Number of generations}\] \[\text{Number of generations} = \frac{\text{Total change in frequency}}{\text{Change in frequency per generation}}\] The total change in frequency of gene c needed is 0.03 - 0.02 = 0.01. Therefore, we have: \[\text{Number of generations} = \frac{0.01}{1.96 \times 10^{-6}}\] \[\text{Number of generations} \approx 5102\] Therefore, it would take approximately 5102 generations to increase the frequency of gene c from 2% to 3%.

Key concepts

Allele Frequency

Allele frequency refers to how often a particular allele appears within a given population. Think of it like counting how many times a specific card appears in a deck. In our exercise, we're looking at how often gene c shows up, compared to its parent gene, C. Allele frequency is typically expressed as a decimal or a percentage.
For instance, if in our population the frequency of allele c is 2%, that means if there were 100 alleles sampled, 2 would be type c. Calculating allele frequency helps scientists understand genetic variation within a population. To find this, you can use the equation:

• \( q = \frac{\text{Number of c alleles}}{\text{Total number of alleles}} \)

Here, q represents the frequency of the allele you're interested in tracking. In the exercise, we start with an allele frequency of c at 0.02 and aim for 0.03. Adjusting allele frequencies over generations helps study how populations evolve and adapt.

Gene Mutation

Gene mutation is a change in the sequence of nucleotides in the DNA of an organism. It can lead to a new allele forming from an existing gene. In our example, gene C is mutating into gene c.
This mutation happens at a rate of \(2 \times 10^{-6}\) per generation. The mutation rate is an essential factor because it tells us how often these changes occur and helps in predicting allele frequency shifts over time. Mutations can occur spontaneously during DNA replication or be induced by environmental factors.
Often, mutations can lead to advantageous traits, disadvantages, or be neutral in terms of their effects on organisms. The mutation from C to c is happening randomly without any selective pressure or other forces influencing it. This simplicity helps us focus solely on the mathematical aspect of mutations influencing allele frequency.

Population Genetics

Population genetics examines the distribution and changes of allele frequency in populations. It's like looking at how card frequencies change in a massive deck over time. This branch of genetics considers random mating, genetic drift, and mutations, among other factors.
The exercise assumes random mating and no other forces, such as selection or migration, acting on the population—aligning perfectly with the primary concern of population genetics: understanding how and why allele frequencies change. With random mating, every individual has an equal chance of passing on their alleles. This simplifies predictions about genetic variation.
Key insights of population genetics:

• It utilizes models to predict changes in allele frequencies over generations, like in our exercise, where we calculated the number of generations needed to increase allele frequency.
• It helps describe how genetic variation is maintained or altered over time, crucial for understanding evolution.
• It combines mathematical models with biological data to depict the "genetic health" of a population.

By understanding population genetics, scientists can predict how large-scale forces disturb genetic equilibriums. In our problem, we've used these fundamentals to measure a straightforward mutation scenario.