Problem 46 - Calculus for Biology and Medicine

Question

We sometimes talk about functions increasing at accelerating (or decelerating) rates. To clarify what we mean by these terms, we will consider some specific examples. (a) First, the Monod growth function: $$ r(N)=\frac{N}{2+N} \quad N \geq 0 $$ (i) Calculate \(r(0.1)\) and \(r(0.2) .\) How much does \(r(N)\) change when \(N\) is increased by \(0.1\), from \(N=0.1 ?\) (ii) Now calculate \(r(2)\) and \(r(2.1) .\) Show that \(r\) increases less from when \(N\) is initially 2 and is increased by \(0.1\) than when \(N\) is initially \(0.1\) and is increased by \(0.1\). (iii) Calculate \(r(4)\) and \(r(4.1) .\) Show that \(r\) increases less when \(N\) is initially 4 than it does when \(N\) is 2 initially. \(r\) is increasing with \(N\), but the effect of increasing \(N\) by the same amount is smaller for \(N=4\) than \(N=2\), and smaller for \(N=2\) than \(N=0.1 .\) So we say that \(r(N)\) is increasing at a decelerating rate. (b) Now consider the growth function $$ s(N)=\frac{N^{2}}{4+N^{2}} \quad N \geq 0 $$ This is an example of a sigmoidal function. (i) Plot \(s(N)\) and \(r(N)\) on the same axes. Show that \(s(N)\) and \(r(N)\) have the same range. (ii) Compare the increases in \(s(N)\) when \(N\) is increased by \(0.1\), from $$ \begin{array}{llll} 0 \text { to } 0.1,2 & \text { to } 2.1, & \text { and } & 4 \text { to } 4.1 . \end{array} $$ Since \(s\) increases more from 2 to \(2.1\) than from 0 to \(0.1\), we say that s increases at an accelerating rate between \(N=0\) and \(N=2\). (iii) Show that the increase of \(s(N)\) is decelerating for large \(N\).

Step-by-Step Solution

Answer

r(N) increases at a decelerating rate, s(N) at an accelerating then decelerating rate.

Step 1: Calculate r(0.1) and r(0.2)

We will first calculate the value of the Monod growth function when \( N = 0.1 \) and \( N = 0.2 \). \[ r(0.1) = \frac{0.1}{2 + 0.1} = \frac{0.1}{2.1} \approx 0.0476 \] \[ r(0.2) = \frac{0.2}{2 + 0.2} = \frac{0.2}{2.2} \approx 0.0909 \] Now, calculate the change in \( r(N) \) from \( N=0.1 \) to \( N=0.2 \): \( r(0.2) - r(0.1) = 0.0909 - 0.0476 = 0.0433 \).

Step 2: Calculate r(2) and r(2.1)

Next, compute the values for \( N = 2 \) and \( N = 2.1 \). \[ r(2) = \frac{2}{2 + 2} = \frac{2}{4} = 0.5 \] \[ r(2.1) = \frac{2.1}{2 + 2.1} = \frac{2.1}{4.1} \approx 0.5122 \] Calculate the change: \( r(2.1) - r(2) = 0.5122 - 0.5 = 0.0122 \).

Key Concepts

Growth Functions

Growth functions are mathematical equations that describe how a quantity changes over time or in response to some other variable. These functions are widely used in biology to model the growth of populations, cells, or microorganisms.

A well-known example in microbiology is the Monod growth function, which is used to describe the relationship between microbial growth rate and nutrient concentration. The general form is:

• As the substrate concentration increases, the growth rate also increases but at a decreasing rate.
• The Monod function is an example of a saturation curve - initially steep, then leveling off as resources become limiting.

Understanding growth functions is crucial in fields like ecology, pharmacology, and even economics, because it helps in predicting how systems behave under different conditions.

Sigmoidal Function

A sigmoidal function is a type of mathematical function that describes a characteristic "S" shaped curve. These functions are frequently used in biological sciences to model scenarios where things increase slowly at first, then more rapidly, before leveling off again.

• A common example of a sigmoidal function is the logistic growth model, which is applied to model populations which grow rapidly initially, then slow, and finally stabilize at a carrying capacity.
• The growth function provided in the exercise, described by \( s(N)=\frac{N^{2}}{4+N^{2}} \), is also a sigmoidal function. Initially, growth accelerates rapidly, then as resources become limited, growth slows.

Sigmoidal functions are pivotal in depicting balanced growth dynamics, capturing nuances better than linear models.

Increasing and Decelerating Rates

Understanding the concept of increasing and decelerating rates is key when analyzing growth functions.

"Increasing" means the function is rising as the input variable, such as time or resource availability, grows. However, "decelerating" means that while the function continues to increase, the rate of increase slows down over time.

• For example, in the Monod growth function \( r(N)=\frac{N}{2+N} \), while the growth rate \( r(N) \) increases as \( N \) increases, the amount of growth per unit increase in \( N \) decreases.
• As a system approaches its limits due to resource constraints or other factors, growth inevitably slows down, reflecting diminishing returns.

This concept illustrates why initial growth is often rapid, but over time, additional input yields less and less of an increase in output.

Calculus in Biology

Calculus is an essential part of modern biology, providing the mathematical foundation for modeling change within biological systems. It allows biologists and researchers to describe complex dynamics in living organisms with precision.

• In biology, calculus is used to model growth rates, population dynamics, and other changes over time.
• Differential equations, a fundamental tool in calculus, help describe how systems evolve continually.
• Understanding how a function's rate of change varies (its derivative) provides insight into the system's behavior.

The connection between calculus and biology extends to genetics, epidemiology, and many other fields, as it helps uncover underlying principles governing life processes. Through calculus, biologists can develop more accurate models to predict potentially critical outcomes, like population booms or the spread of disease.