Problem 43 - Calculus for Biology and Medicine

Question

Selection at a Single Locus We consider one locus with two alleles, \(A_{1}\) and \(A_{2}\), in a randomly mating diploid population. That is, each individual in the population is either of type \(A_{1} A_{1}, A_{1} A_{2}\), or \(A_{2} A_{2}\). We denote by \(p(t)\) the frequency of the \(A_{1}\) allele and by \(q(t)\) the frequency of the \(A_{2}\) allele in the population at time \(t\). Note that \(p(t)+q(t)=1\). We denote the fitness of the \(A_{i} A_{j}\) type by \(w_{i j}\) and assume that \(w_{11}=1, w_{12}=1-s / 2\), and \(w_{22}=1-s\), where \(s\) is a nonnegative constant less than or equal to \(1 .\) That is, the fitness of the heterozygote \(A_{1} A_{2}\) is halfway between the fitness of the two homozygotes, and the type \(A_{1} A_{1}\) is the fittest. If \(s\) is small, we can show that, approximately, $$\frac{d p}{d t}=\frac{1}{2} s p(1-p) \quad \text { with } p(0)=p_{0}$$ (a) Use separation of variables and partial fractions to find the solution of \((8.49)\). (b) Suppose \(p_{0}=0.1\) and \(s=0.01\); how long will take until \(p(t)=0.5 ?\) (c) Find \(\lim _{t \rightarrow \infty} p(t)\). Explain in words what this limit means.

Step-by-Step Solution

Answer

(a) Solved differential equation gives \( p(t) = \frac{\frac{p_0}{1-p_0}e^{\frac{s}{2}t}}{1 + \frac{p_0}{1-p_0}e^{\frac{s}{2}t}} \). (b) Time to reach \( p(t) = 0.5 \) is about 439.32 units. (c) \( \lim_{t \to \infty} p(t) = 1 \), meaning \( A_1 \) allele will be fixed.

Step 1: Use separation of variables

To solve the differential equation \( \frac{d p}{d t}=\frac{1}{2} s p(1-p) \) by separation of variables, start by rewriting it as \( \frac{d p}{p(1-p)} = \frac{s}{2} \, dt \). This separates the variables \(p\) and \(t\), allowing us to integrate both sides separately.

Step 2: Integrate using partial fractions

Rewrite the left-hand side \( \frac{1}{p(1-p)} = \frac{1}{p} + \frac{1}{1-p} \). Integrate both sides: \( \int \left( \frac{1}{p} + \frac{1}{1-p} \right) dp = \int \frac{s}{2} \, dt \). On integrating, we obtain \( \ln|p| - \ln|1-p| = \frac{s}{2}t + C \), where \(C\) is the constant of integration.

Step 3: Solve for the constant of integration

Apply the initial condition \(p(0) = p_0\), substituted into \( \ln\left(\frac{p}{1-p}\right)= \frac{s}{2}t + C \), to solve for \(C\). With \(t=0\), \( \ln\left(\frac{p_0}{1-p_0}\right) = C \). Substitute \(p_0 = 0.1\) to get \( C = \ln\left(\frac{0.1}{0.9}\right) \).

Step 4: Find the general solution

Substitute \(C\) back into the integrated equation from Step 2: \( \ln\left(\frac{p}{1-p}\right) = \frac{s}{2}t + \ln\left(\frac{p_0}{1-p_0}\right) \). Rearrange to find \(p\): \( \frac{p}{1-p} = \frac{p_0}{1-p_0}e^{\frac{s}{2}t} \). Solve for \(p\) to get \( p(t) = \frac{\frac{p_0}{1-p_0}e^{\frac{s}{2}t}}{1 + \frac{p_0}{1-p_0}e^{\frac{s}{2}t}} \).

Step 5: Calculate time to reach \( p(t) = 0.5 \)

Set \( p(t) = 0.5 \) and substitute into the expression from Step 4: \( \frac{0.5}{0.5} = \frac{0.1}{0.9}e^{\frac{0.01}{2}t} \). Simplifying, we get \( e^{\frac{0.01}{2}t} = 9 \). Solve for \(t\) to find the time required: \( t = \frac{2}{0.01} \ln(9) \approx 439.32 \).

Step 6: Compute the limit as \( t \to \infty \)

Examine \( \lim_{t \to \infty} p(t) \) using the expression from Step 4: As \( t \to \infty \), \( e^{\frac{s}{2}t} \to \infty \), causing \( \frac{p_0}{1-p_0}e^{\frac{s}{2}t} \to \infty \). Therefore, \( p(t) \to 1 \).

Step 7: Interpret the limit

The limit \( \lim_{t\to\infty}p(t)=1 \) indicates the eventual fixation of the allele \( A_1 \) in the population, due to it having the highest fitness relative to \( A_2 \). This means over infinite time, the population will predominantly consist of homozygote \( A_1A_1 \) individuals, assuming no other evolutionary forces act.

Key Concepts

Allele Frequency

In population genetics, allele frequency refers to how common an allele is in a population. It is a crucial concept as it helps scientists understand the genetic diversity and structure of populations.
In a given population, the allele frequency of a particular allele is the ratio of copies of that allele to the total number of allele copies.
For example, in a population with two alleles, \(A_1\) and \(A_2\), the frequency of \(A_1\) is represented as \(p(t)\) and \(A_2\) as \(q(t)\). These frequencies can tell us a lot about the genetic variation within the population. They are mathematically related by the equation: \[ p(t) + q(t) = 1 \]This equation signifies that the total probability of all allele occurrences is 100%.

Changes in allele frequency are influenced by several factors including mutation, selection, genetic drift, migration, and recombination. For example, selection might increase the frequency of a beneficial allele (like \(A_1\)), while a deleterious allele might decrease.

Tracking allele frequencies over time allows researchers to observe evolutionary processes in action. This dynamic nature of allele frequencies is what provides insight into how populations evolve and adapt to their environments.

Selection at a Locus

Selection at a locus refers to how some alleles at a particular genetic location (locus) are favored over others, often due to their effects on the fitness of an organism.
The exercise here deals with a population with two alleles, \(A_1\) and \(A_2\), where each individual can have one of three genotypes: \(A_1A_1\), \(A_1A_2\), or \(A_2A_2\).
The fitness of these genotypes influences which alleles are passed on to the next generation.

In this example, the genotype \(A_1A_1\) is the fittest with a relative fitness of 1, meaning it is the most advantageous. The heterozygote \(A_1A_2\) has a slightly lower fitness of \(1 - \frac{s}{2}\), and the homozygote \(A_2A_2\) has even lower fitness, \(1 - s\).
This means that \(A_1\) has a selective advantage over \(A_2\). Such selective forces can shift allele frequencies over time, potentially leading to the fixation of the favored allele (here \(A_1\)) in the population.

The exercise demonstrates how selection at a single locus can make a significant impact on evolutionary outcomes, playing a vital role in shaping the genetic makeup of populations.

Differential Equations in Biology

Differential equations are mathematical tools used to describe the dynamics of continuous change. They are especially useful in biology for modeling natural processes that change over time, like population growth, gene frequency changes, or reaction rates.

In this context, the differential equation \( \frac{d p}{dt} = \frac{1}{2} s p(1-p) \) models the change in allele frequency over time under selection pressure. The variable \( p(t) \) represents the frequency of allele \( A_1 \).
This equation belongs to a class of models known as logistic growth models, which are particularly useful for capturing the competition dynamics commonly observed in biological systems.

Solving such an equation involves techniques like separation of variables and integration. This specific equation captures how, under natural selection, the frequency of a favored allele increases over time.
As these equations provide a continuous view of genetic changes over generations, they offer powerful insights into evolutionary biology processes.

• They help predict how fast a selective advantage will spread through a population.
• They also identify equilibrium states, such as fixation (\( p(t) \to 1 \)).
• In this way, differential equations act as a bridge between abstract mathematical theory and real-world biological phenomena.

By understanding these models, biologists can make predictions about future changes in allele frequencies or the potential outcomes of various evolutionary scenarios.