Problem 27 - Calculus for Biology and Medicine

Question

Metabolism West, Brown, and Enquist (1997) argued that because of the distribution of blood vessels through mammalian bodies, the energy needs \(E\) of mammals increase with the \(3 / 4\) power of their mass, \(M ;\) i.e., $$ E=c M^{3 / 4} $$ for some constant \(c\). As a mammal grows, \(M\) increases. Show how \(d E / d t\) is related to \(d M / d t\) according to the theory of West, Brown, and Enquist.

Step-by-Step Solution

Answer

The relationship is \(\frac{dE}{dt} = \frac{3c}{4} M^{-1/4} \frac{dM}{dt}\).

Step 1: Start with the Given Equation

The relationship between energy needs \(E\) and mass \(M\) of a mammal is given by the equation \(E = c M^{3/4}\), where \(c\) is a constant.

Step 2: Differentiate Both Sides with Respect to Time

To find how the energy needs change over time with the mass, differentiate both sides of the equation with respect to time \(t\). This step involves applying the chain rule.\[ \frac{dE}{dt} = \frac{d}{dt}(c M^{3/4}) \]

Step 3: Apply the Chain Rule

Using the chain rule for differentiation, we have:\[ \frac{dE}{dt} = c \times \frac{d}{dt}(M^{3/4}) \]According to the chain rule, this becomes:\[ \frac{dE}{dt} = c \times \left(\frac{3}{4} M^{-1/4}\right) \frac{dM}{dt} \]

Step 4: Simplify the Expression

Combine terms to complete the differentiation process:\[ \frac{dE}{dt} = \frac{3c}{4} M^{-1/4} \frac{dM}{dt} \] This demonstrates the relationship between \(\frac{dE}{dt}\) and \(\frac{dM}{dt}\).

Key Concepts

Understanding Differentiation

Differentiation is a fundamental concept in calculus that revolves around finding the rate at which a quantity changes. We use differentiation when we want to understand how a function's value changes as its inputs change. In simple terms, it's about calculating the slope of the tangent to the curve of a function at any point. This process allows us to determine how quickly something is increasing or decreasing at any given moment.

In our example, the curve represents the relationship between a mammal's mass and its energy needs. We are interested in how these energy needs change as the mammal's mass changes over time. This involves taking the derivative of the given relation, which describes the energy needs in terms of mass. By differentiating, we move from understanding static relationships to dynamic ones, analyzing change rather than just position.

Exploring the Chain Rule

The chain rule is an essential differentiation technique, especially when dealing with composite functions. A composite function can be thought of as a function within another function, and the chain rule provides a way to differentiate them.

Let's consider our equation: \( E = c M^{3/4} \). Here, our function can be broken down into the outer function \( E(x) = c x \) and the inner function \( x = M^{3/4} \). The chain rule tells us how to differentiate such nested functions by first differentiating the outer function with respect to the inner one, and then multiplying by the derivative of the inner function.

• Differentiate the outer function: \( \frac{d}{dx} (c x) = c \).
• Differentiate the inner function: \( \frac{d}{dt} (M^{3/4}) = \frac{3}{4} M^{-1/4} \frac{dM}{dt} \).

The final step combines these results: \( \frac{dE}{dt} = c \times \frac{3}{4} M^{-1/4} \frac{dM}{dt} \). This highlights the role each function plays in the rate of change, demonstrating their intertwined relationship.

Applications in Biological Modeling

Biological modeling is a fascinating application of calculus, often used to describe how organisms develop and interact with their environment. Models like the one proposed by West, Brown, and Enquist help us translate biological phenomena into mathematical terms, making them easier to analyze and predict. In this case, the model describes the energy needs of mammals based on their mass.

As mammals grow, their mass increases, thereby changing their energy requirements. West, Brown, and Enquist's model suggests that these energy needs increase with the 3/4 power of their mass, capturing a universal pattern across different species. By understanding how energy needs change over time with mass, scientists can make informed predictions about a mammal's growth and health.

• Provides a mathematical framework for understanding growth patterns.
• Helps in predicting energy needs, food requirements, and even potential lifespan.
• Plays a critical role in ecology, conservation efforts, and health management.

This application of calculus illustrates how mathematics is not just a theoretical exercise but a practical tool in solving real-world problems, offering insights into the complex biological systems.