Problem 2 - Calculus for the Life Sciences: A Modeling Approach Volume I

Question

For the following experiments, determine the independent variable and the dependent variable, and draw a simple graph or give a brief verbal description (your best guess) of the function relating the two. a. A rabbit population size is a function of the number of coyotes in the region. b. An agronomist, interested in the most economical rate of nitrogen application to corn, measures the corn yield in test plots using eight different levels of nitrogen application. c. An enzyme, E, catalyzes a reaction converting a substrate, \(\mathrm{S},\) to a product \(\mathrm{P}\) according to $$ \mathrm{E}+\mathrm{S} \rightleftharpoons \mathrm{ES} \rightleftharpoons \mathrm{E}+\mathrm{P} $$ Assume enzyme concentration, [E], is fixed. A scientist measures the rate at which the product P accumulates at different concentrations, [S], of substrate. d. A scientist titrates a \(0.1 \mathrm{M}\) solution of \(\mathrm{HCl}\) into \(5 \mathrm{ml}\) of an unknown basic solution containing litmus (litmus causes the color of the solution to change as the pH changes).

Step-by-Step Solution

Answer

a) Independent: coyotes, Dependent: rabbits. b) Independent: nitrogen, Dependent: yield. c) Independent: [S], Dependent: rate. d) Independent: HCl volume, Dependent: pH.

Step 1: Identify Variables for Experiment a

For the rabbit population-size experiment, the independent variable is the number of coyotes since this is what is manipulated or observed as a factor. The dependent variable is the rabbit population size, which is expected to change in response to the varying number of coyotes.

Step 2: Describe Relationship for Experiment a

As the number of coyotes increases, the size of the rabbit population is likely to decrease due to increased predation. This relationship can be visualized as a decreasing function where the x-axis represents the number of coyotes and the y-axis represents the rabbit population size.

Step 3: Identify Variables for Experiment b

In the corn yield experiment, the independent variable is the level of nitrogen application since it is varied across different test plots. The dependent variable is the corn yield, which is measured in response to different concentrations of nitrogen.

Step 4: Describe Relationship for Experiment b

The relationship between nitrogen application and corn yield is likely to be a curve that initially rises due to increased yield with more nitrogen, but eventually levels off or even declines if too much nitrogen hinders growth. The graph would likely show nitrogen levels on the x-axis and corn yield on the y-axis.

Step 5: Identify Variables for Experiment c

For the enzyme-catalyzed reaction, the independent variable is the concentration of substrate \( [S] \), as it is varied in the experiment. The dependent variable is the rate of product formation, which depends on the substrate concentration.

Step 6: Describe Relationship for Experiment c

The enzyme-catalyzed reaction typically follows Michaelis-Menten kinetics, where the rate of product formation increases and then levels off as substrate concentration increases. This can be represented as a hyperbolic curve when plotting substrate concentration on the x-axis and reaction rate on the y-axis.

Step 7: Identify Variables for Experiment d

In the titration experiment, the independent variable is the volume of HCl added, as it is controlled by the scientist. The dependent variable is the pH change of the basic solution, which is measured as more HCl is added.

Step 8: Describe Relationship for Experiment d

The relationship between the volume of HCl added and the pH of the solution is usually an S-shaped curve. Initially, the pH remains relatively stable, then changes rapidly as the solution approaches neutral pH due to successful neutralization, and finally, becomes stable again when completely titrated.

Key Concepts

Rabbit Population Dynamics

In the study of rabbit population dynamics, the interaction between predators and prey plays a crucial role. Here, the independent variable is the number of coyotes in the region. This is because the coyote population is what we track or change to see its effect. The dependent variable is the size of the rabbit population, which varies in response to the number of coyotes.

As we increase the coyote population, the rabbit population tends to decrease. This inverse relationship often results from increased predation; the more predators there are, the fewer rabbits survive. If you were to plot this relationship on a graph, the x-axis would represent the number of coyotes, and the y-axis would represent the number of rabbits. You would likely see a downward sloping line, indicating the decrease in rabbit population as coyote numbers rise.

Corn Yield Experiment

The corn yield experiment revolves around determining the right amount of nitrogen application needed for optimal corn growth. Here, the independent variable is the amount of nitrogen applied to the corn plots. This is the factor that you manipulate to observe changes in corn yield, which is the dependent variable.

Initially, as the level of nitrogen application increases, corn yield also increases. However, after reaching a certain point, further increasing nitrogen can lead to negative effects, like nutrient leaching or plant damage, causing the yield to level off or decline. On a graph, nitrogen levels would be on the x-axis and corn yield on the y-axis. The curve you would observe starts upwards as nitrogen improves growth but then flattens or drops when application becomes excessive.

Enzyme-Catalyzed Reaction

Enzyme-catalyzed reactions are fundamental in biochemistry, often described by Michaelis-Menten kinetics. In this scenario, the concentration of the substrate \( [S] \) is your independent variable. You vary this concentration to see how it affects the reaction rate, which is the dependent variable.

As substrate concentration increases, the rate of product formation also increases, but only up to a certain point. Once the enzyme molecules are saturated with substrate, the reaction rate hits a maximum and levels off, resulting in a hyperbolic curve. This shape illustrates how the reaction progresses and plateaus at high substrate levels when plotted with substrate concentration on the x-axis and reaction rate on the y-axis.

Titration Curve Analysis

Titration curves are important for understanding how solutions behave during titration processes. In this experiment, the volume of HCl added to the solution is the independent variable. Meanwhile, the pH of the solution, which changes in response to the acid added, is the dependent variable.

As you begin adding HCl to the basic solution, there might be minimal change in pH. This is because the base initially neutralizes the acid effectively. However, as more HCl is added, a tipping point is reached where the pH experiences a rapid change, leading the solution toward neutrality. Once neutralized, further addition of HCl alters the pH minimally. This relationship typically forms an S-shaped curve, with the x-axis representing the volume of HCl and the y-axis displaying the changing pH. Such a curve is indicative of the buffering action and eventual equivalence point of the solution.