Problem 64 - Calculus for Biology and Medicine

Question

Homeostasis Sterner and Elser (2002) studied the relationship between the amount of nitrogen in an animal's body and the amount of nitrogen present in the food that it eats. Many animals maintain homeostasis (balance), that is, they control their own nitrogen content. As the amount of nitrogen present in their food increases, the amount of nitrogen in the animal's body increases more slowly. If the amount of nitrogen in the animal is \(N\) and the amount of nitrogen in its food is \(F\), Sterner and Elser argue that: $$ \frac{1}{N} \frac{d N}{d t}=\frac{\sigma}{F} \frac{d F}{d t} $$ where \(\sigma\) is a constant. (a) Show that if \(\sigma=1\), then \(N \propto F ;\) that is, the nitrogen content of the animal increases in proportion to its food. This is called absence of homeostasis. (b) If \(\sigma=0\), then \(N\) is a constant, independent of \(F\). This is called homeostasis (the animal maintains a balanced amount of nitrogen, independent of its food). (c) Show that if \(0<\sigma<1\), then, if \(F\) doubles, \(N\) also increases but by a factor less than 2 .

Step-by-Step Solution

Answer

(a) If \(\sigma = 1\), \(N \propto F\). (b) If \(\sigma = 0\), \(N\) is constant. (c) If \(0 < \sigma < 1\), if \(F\) doubles, \(N\) increases by less than 2x.

Step 1: Understanding the Equation

We are given the equation \( \frac{1}{N} \frac{d N}{d t} = \frac{\sigma}{F} \frac{d F}{d t} \). This equation relates the rate of change of nitrogen content in an animal \(N\) with the rate of change of nitrogen content in its food \(F\). The variable \(\sigma\) is a constant that modulates the relationship.

Step 2: Solving part (a): Absence of Homeostasis

To show that \(N \propto F\) when \(\sigma = 1\), substitute \(\sigma = 1\) into the equation: \[ \frac{1}{N} \frac{d N}{d t} = \frac{1}{F} \frac{d F}{d t} \] This simplifies to: \[ \frac{d N}{d t} = \frac{N}{F} \frac{d F}{d t} \] This is a separable differential equation. Rearranging gives: \[ \frac{d N}{N} = \frac{d F}{F} \] Integrate both sides: \[ \ln N = \ln F + C \] Exponentiating both sides leads to: \[ N = C' F \] Which shows \(N\) is proportional to \(F\). Thus, absence of homeostasis occurs when \(\sigma = 1\).

Step 3: Solving part (b): Full Homeostasis

To show \(N\) is constant for \(\sigma = 0\), substitute \(\sigma = 0\) into the equation: \[ \frac{1}{N} \frac{d N}{d t} = 0 \] This implies: \[ \frac{d N}{d t} = 0 \] The solution to this differential equation is \(N = \text{constant}\). Hence, \(N\) is independent of \(F\), demonstrating full homeostasis.

Step 4: Solving part (c): Partial Homeostasis

We need to prove that if \(0 < \sigma < 1\), then \(N\) increases by a factor less than 2 if \(F\) doubles. Replace \(\sigma\) in the equation: \[ \frac{d N}{d t} = \frac{\sigma N}{F} \frac{d F}{d t} \] The separable differential equation becomes: \[ \frac{d N}{N} = \sigma \frac{d F}{F} \] Integrating both sides results in: \[ \ln N = \sigma \ln F + C \] Exponentiating gives: \[ N = C' F^{\sigma} \] If \(F\) doubles to \(2F\), then: \[ N(2F) = C' (2F)^{\sigma} = C' 2^{\sigma} F^{\sigma} \] Since \(0 < \sigma < 1\), \(2^{\sigma} < 2\). Thus, \(N\) increases by a factor less than 2.

Key Concepts

Differential Equations

Differential equations are mathematical tools used to describe how things change over time. In our exercise, the differential equation given is \( \frac{1}{N} \frac{d N}{d t} = \frac{\sigma}{F} \frac{d F}{d t} \). This equation helps us understand the change in nitrogen levels, both in the animal's body and in its food.
Differential equations are composed of derivatives, which show how a certain quantity, like nitrogen here, changes over time. Solving these equations requires integrating, which means finding what function has a given derivative.
In Step 2 and 3 of the exercise, we solve the differential equation in different scenarios of the constant \(\sigma\). Understanding how to manipulate and integrate these equations can reveal how the nitrogen content adjusts in various homeostatic conditions.

Proportional Relationships

When two quantities have a proportional relationship, it means they change at the same rate. In the context of our exercise, when \(\sigma = 1\), we find that \(N \propto F\). This means if the nitrogen in the food increases, the nitrogen in the animal's body increases in a direct proportion.
This is demonstrated by the equation \(N = C' F\), where \(C'\) is a constant. Such relationships are linear, where the change in one variable directly affects the other in a predictable and straightforward way.
In biology, proportional relationships reflect situations where there is no regulation to maintain balance, often termed as an absence of homeostasis. Understanding these relationships helps in predicting how an organism reacts to changes in its environment.

Nitrogen Balance

Nitrogen balance is a critical aspect of animal physiology, determining health and growth. The exercise explores how animals regulate their nitrogen intake based on their food. Achieving nitrogen balance involves the body adjusting its nitrogen content in response to changes, maintaining ideal internal conditions.
In the equations discussed, nitrogen balance is evaluated under different values of \(\sigma\). When \(\sigma = 0\), nitrogen in the animal remains constant regardless of food changes, which is full homeostasis. It highlights the body's ability to stabilize nitrogen content within.
Understanding nitrogen balance is key in ecology and nutrition, ensuring animals get enough nitrogen for proteins and nucleic acids while avoiding excess that could be harmful.

Partial Homeostasis

Partial homeostasis is a state between full regulation and no regulation. It is demonstrated in our exercise with \(0 < \sigma < 1\), where the nitrogen in the animal's body does increase with food intake but at a diminished rate.
The equation \(N = C' F^{\sigma}\) encapsulates this relationship. When the food nitrogen doubles, \(N\) increases by less than double, reflecting the body's ability to partially control nitrogen balance.
Partial homeostasis suggests a flexible biological mechanism, allowing animals to adjust to varying food availability. This concept is useful in understanding how organisms thrive in fluctuating environments, maintaining enough control to prevent extremes that can disrupt other body functions.