Problem 57 - Algebra and Trigonometry with Analytic Geometry

Question

The basic source of genetic diversity is mutation, or changes in the chemical structure of genes. If a gene mutates at a constant rate \(m\) and if other evolutionary forces are negligible, then the frequency \(F\) of the original gene after \(t\) generations is given by \(F=F_{0}(1-m)^{t}\), where \(F_{0}\) is the frequency at \(t=0\). (a) Solve the equation for \(t\) using common logarithms. (b) If \(m=5 \times 10^{-5}\), after how many generations does \(F=\frac{1}{2} F_{0}\) ?

Step-by-Step Solution

Answer

Approximately 13,847 generations are needed.

Step 1: Understand the Equation

The equation given is \( F = F_0(1-m)^t \), where \( F_0 \) is the initial frequency, \( m \) is the mutation rate, \( F \) is the frequency after \( t \) generations, and \( t \) is the number of generations. We'll solve for \( t \) using logarithms.

Step 2: Solve for t using Logarithms

First, divide both sides by \( F_0 \) to isolate the exponential term: \( \frac{F}{F_0} = (1-m)^t \). Now, apply the logarithm to both sides to solve for \( t \): \[ \log_{10}\left(\frac{F}{F_0}\right) = t \cdot \log_{10}(1-m).\] Now, solve for \( t \): \[ t = \frac{\log_{10}\left(\frac{F}{F_0}\right)}{\log_{10}(1-m)}.\]

Step 3: Substitute Known Values for Part b

For part (b), we have \( m = 5 \times 10^{-5} \) and \( \frac{F}{F_0} = \frac{1}{2} \). Substitute these values into the equation for \( t \): \[ t = \frac{\log_{10}\left(\frac{1}{2}\right)}{\log_{10}(1 - 5 \times 10^{-5})}.\]

Step 4: Calculate Logarithmic Values

Compute \( \log_{10}\left(\frac{1}{2}\right) \approx -0.3010 \). Note that \( \log_{10}(1 - 5 \times 10^{-5}) \) requires using a calculator because \( 1 - 5 \times 10^{-5} \) is close to 1. Using a calculator, find \( \log_{10}(0.99995) \approx -2.1739 \times 10^{-5} \).

Step 5: Compute the Number of Generations

Substitute the calculated logarithmic values into the equation for \( t \): \[ t = \frac{-0.3010}{-2.1739 \times 10^{-5}}. \] Simplify to find \( t \approx 13846.54 \). Therefore, approximately 13,847 generations are required for the frequency to reduce to half.

Key Concepts

Mutation Rate

Genetic diversity arises primarily due to mutations, which are changes in the genetic material of an organism. The mutation rate, denoted as \( m \), is a crucial factor in this process. This rate measures how often a gene undergoes a mutation per generation. For example, if we have a mutation rate of \( m = 5 \times 10^{-5} \), it implies that out of every 100,000 times a gene is replicated, approximately 5 mutations might occur.
This is a small number, but over many generations, even a low mutation rate can significantly alter gene frequencies within a population. Consequently, the mutation rate helps geneticists understand and predict how quickly genetic variations might emerge and spread. Understanding mutation rates is important in evolutionary biology, as it affects the genetic makeup and adaptability of organisms. A higher mutation rate can lead to more genetic diversity, whereas a low mutation rate might preserve the status quo, limiting diversity.

Frequency Calculation

When examining genetic diversity, frequency calculation is a method used to determine how often a specific gene variant appears within a population over generations. Initially, we have an original frequency \( F_0 \), which is the starting point before genetic mutations occur.
The frequency \( F \) after \( t \) generations can be calculated using the formula \( F = F_0 (1-m)^t \). This equation accounts for the reduction in the original gene's frequency due to ongoing mutations with each successive generation.

• First, you divide the final frequency by the initial frequency to isolate the effect of the mutation rate over time.
• Express this relationship in terms of logarithms to solve for the number of generations \( t \) needed for a specific change in frequency, as shown by \( t = \frac{\log_{10}(F/F_0)}{\log_{10}(1-m)} \).

By performing these calculations, scientists can predict how quickly a new mutation might become prevalent or how long it will take for original genes to diminish in frequency within a population.
This approach is essential for understanding evolutionary processes and estimating the genetic stability or adaptability of a species over time.

Exponential Decay

Exponential decay describes a process in which a quantity decreases at a rate proportional to its current value. In genetics, this concept applies to the frequency of an original gene as mutations occur over generations. As time passes, the proportion of the original gene is reduced, following the equation \( F = F_0(1-m)^t \), where \( (1-m) \) is the decay factor.
This mathematical model indicates that the gene frequency does not simply decrease linearly; instead, it starts to decrease rapidly and then more slowly as fewer individuals carry the original gene. A clear example is when the gene frequency must drop to half of its original value:

• Determine how many generations (\( t \)) are needed for \( F = \frac{1}{2}F_0 \).
• Using logarithmic calculations, solve \( t = \frac{-0.3010}{-2.1739 \times 10^{-5}} \) to find approximately 13,847 generations.

This illustrates that even a small mutation rate, such as \( 5 \times 10^{-5} \), can lead to significant changes over many generations, showcasing the power of exponential decay in the transformation of genetic frequencies.