Problem 6 - Calculus for the Life Sciences: A Modeling Approach Volume I

Question

Consider two alleles \(A\) and \(a\) at a locus of a random mating population and the fractions of \(A A, A a\) and aa zygotes that reach maturity and mate are in the ratio \(1+s_{1}: 1: 1+s_{2}\) where \(s_{1}\) and \(s_{2}\) can be positive, negative, or zero, but \(s_{1} \geq-1\) and \(s_{2} \geq-1 .\) The balance function is $$ \begin{aligned} F(p)=\left(1+s_{1}\right) p^{2}+2 p q+\left(1+s_{2}\right) q^{2} &=\left(1+s_{1}\right) p^{2}+2 p(1-p)+\left(1+s_{2}\right)(1-p)^{2} \\ &=1+s_{1} p^{2}+s_{2}(1-p)^{2} \end{aligned} $$ where \(p\) and \(q\) are the frequencies of \(A\) and \(a\) among the zygotes. a. Sketch the graphs of \(F\) and find the values \(\hat{p}\) of \(p\) in [0,1] for which \(F(p)\) is a maximum for 1\. \(s_{1}=0.2\) and \(s_{2}=-0.3\). 2\. \(s_{1}=0\) and \(s_{2}=-0.2\) 3\. \(s_{1}=-0.2\) and \(s_{2}=-0.3\). 4\. \(s_{1}=0.2\) and \(s_{2}=0.3\) b. Suppose that \(s_{1}+s_{2} \neq 0\) and \(0 \leq s_{2} /\left(s_{1}+s_{2}\right) \leq 1 .\) Is it true that \(\hat{p} s_{2} /\left(s_{1}+s_{2}\right)\) is the value of \(p\) in [0,1] for which \(F(p)\) is a maximum?

Step-by-Step Solution

Answer

Only condition 3 gives a valid \(\hat{p} = 0.6\). The statement in part b is true when conditions are met.

Step 1: Understanding the Problem

The problem involves analyzing the behavior of the function \(F(p)\) which represents the probability distribution of alleles in a population under given selection pressures \(s_1\) and \(s_2\). We need to investigate the conditions for which \(F(p)\) attains its maximum typically by finding the derivative and equating to zero to find the critical points.

Step 1: Formulate the Function

Given: \[ F(p) = 1 + s_1p^2 + s_2(1-p)^2 \]Expand this: \[ F(p) = 1 + s_1p^2 + s_2(1-2p+p^2) \] Simplify to: \[ F(p) = 1 + (s_1 + s_2)p^2 - 2s_2p + s_2 \]Combine like terms to get:\[ F(p) = (s_1 + s_2)p^2 - 2s_2p + (1 + s_2) \]

Step 2: Calculate Derivative

To find maxima, we set the derivative of \(F\) with respect to \(p\) to zero: \[ \frac{dF}{dp} = 2(s_1 + s_2)p - 2s_2 \]Set \( \frac{dF}{dp} = 0 \) and solve for \(p\):\[ 2(s_1 + s_2)p = 2s_2 \]\[ p = \frac{s_2}{s_1 + s_2} \]

Step 3: Determine Maximum Points for Given Conditions

For different values of \(s_1\) and \(s_2\):1. \(s_1=0.2, s_2=-0.3\) yields \(p = \frac{-0.3}{0.2 - 0.3} = 3\) (not feasible as \(p\) must be in \([0,1]\))2. \(s_1=0, s_2=-0.2\) yields \(p = \frac{-0.2}{0 - 0.2} = 1\)3. \(s_1=-0.2, s_2=-0.3\) yields \(p = \frac{-0.3}{-0.5} = 0.6\)4. \(s_1=0.2, s_2=0.3\) yields \(p = \frac{0.3}{0.5} = 0.6\)

Step 4: Analyze the Condition and Verify It

Given: \(0 \leq \frac{s_2}{s_1 + s_2} \leq 1\), this implies that \(p\) lies within feasible range \([0,1]\).If \(s_1 + s_2 eq 0\), indeed \(\hat{p} = \frac{s_2}{s_1 + s_2} \) correctly represents the maximum based on our derived formula since we ensured the derivative equals zero at that point, thus it's a sufficient condition for maximization within given limits.

Key Concepts

Allele Frequency

Allele frequency is the proportion of a specific allele of a gene in a population. In population genetics, allele frequencies help us understand the genetic variation within the population. For example, considering alleles \(A\) and \(a\), if \(p\) is the frequency of allele \(A\), then \(q = 1 - p\) is the frequency of allele \(a\).

These frequencies are crucial in predicting the genetic makeup of future generations. In our exercise, we use the function \(F(p)\) to analyze allele distribution among \(AA, Aa,\) and \(aa\) zygotes. Varying selection pressures on these genotypes, represented by parameters \(s_1\) for \(AA\) and \(s_2\) for \(aa\), influence allele frequency changes over time. In particular, understanding their impact through calculations of critical values supports insights into evolutionary processes.

Natural Selection

Natural selection is a fundamental concept of evolutionary biology that explains how traits become more or less common in a population over generations. It results from differential survival and reproduction of individuals due to variations in phenotype. In our scenario, natural selection is represented by the parameters \(s_1\) and \(s_2\), influencing the fitness of different zygotes.

Fitness, in this context, reflects the ability of zygotes carrying specific allele combinations to mature and reproduce. If \(s_1\) and \(s_2\) are both positive, alleles \(A\) and \(a\) are beneficial. If negative, they're deleterious, affecting the prevalence of these alleles over time. Hence, understanding these selection coefficients helps in forecasting how natural selection will shape allele frequencies via the balance function, \(F(p)\).

Critical Points Analysis

Critical points analysis is a method to determine specific values within a function where maximum or minimum values occur. In population genetics, this is crucial for predicting stable allele frequencies. To find these points, we differentiate the balance function \(F(p)\) in terms of \(p\), leading us to critical points when the derivative equals zero.

By solving \[ \frac{dF}{dp} = 2(s_1 + s_2)p - 2s_2 = 0 \],
we locate the point \(p = \frac{s_2}{s_1 + s_2}\). This value tells us where the function \(F(p)\) might have a maximum. Substituting values for \(s_1\) and \(s_2\) from our different conditions, we determine feasible \(p\) values that reflect potential equilibria in allele frequencies under specific selection pressures.

Mathematical Modeling

Mathematical modeling in population genetics involves using mathematical equations to represent biological processes influencing allele frequencies. The balance function \(F(p)\) is a model incorporating selection coefficients \(s_1\) and \(s_2\) to explore genetic variations under natural selection.

This model illustrates how different levels of selection pressure alter allele distribution in a population. By expanding and differentiating this function, we study the equilibrium conditions—reflecting how populations might stabilize genetically over time. Such models allow scientists and students to predict real-world genetic dynamics, providing valuable insights into the evolutionary fate of alleles based on mathematical principles.