Problem 83 - Calculus for Biology and Medicine

Question

Some microbes regulate their growth according to a circadian clock. This clock means that their growth rate fluctuates predictably over the course one day. For example the filamentous fungus Neurospora crassa grows almost twice as fast at night-time than during the day. Gooch, Freeman and \mathrm{\\{} L a k i n - T h o m a s ~ ( 2 0 0 4 ) ~ m e a s u r e d ~ t h e ~ c h a n g e ~ o f ~ g r o w t h ~ r a t e ~ o v e r ~ time. If growth rate is measured in \(\mathrm{mm} /\) day then their data can be fit by the following relationship: $$ \frac{d L}{d t}=38.4+2.4 \cos (4 \pi t)-12 \sin (2 \pi t) $$ where \(L(t)\) is the total size of the fungus, measured in \(\mathrm{mm}\), and \(t\) is the time measured in hours. Calculate: (a) the total extra length added to the fungus between \(t=0\) and \(t=1\) hours. (b) the total extra length added to the fungus between \(t=0\) and \(t=12\) hours. (c) the total extra length added to the fungus between \(t=0\) and \(t=24\) hours.

Step-by-Step Solution

Answer

(a) 38.4 mm, (b) 460.8 mm, (c) 921.6 mm.

Step 1: Understanding the Problem

We are given a differential equation that describes the growth rate of a fungus, expressed as \( \frac{dL}{dt} = 38.4 + 2.4 \cos(4\pi t) - 12 \sin(2\pi t) \), where \(L(t)\) is the size and \(t\) is time in hours. We need to find the total growth over specific time intervals: 0 to 1 hour, 0 to 12 hours, and 0 to 24 hours.

Step 2: Set Up the Integral for Total Growth

The total extra length added (total growth) is found by integrating the growth rate over the given time interval. This is expressed mathematically as \( \int_{a}^{b} \frac{dL}{dt} \, dt \), where \(a\) and \(b\) are the start and end times.

Step 3: Calculate the Integral for Part (a)

For part (a), we find the integral from \(t = 0\) to \(t = 1\) hour:\[ \int_{0}^{1} \left(38.4 + 2.4 \cos(4\pi t) - 12 \sin(2\pi t) \right) \, dt \]Evaluating this gives us the result of the definite integral. Use integration rules for trigonometric functions to find:\[ [38.4t + 0.6\sin(4\pi t) + 6\cos(2\pi t)]_{0}^{1} = 38.4 \text{ mm} \]

Step 4: Calculate the Integral for Part (b)

For part (b), calculate the definite integral from \(t = 0\) to \(t = 12\) hours:\[ \int_{0}^{12} \left(38.4 + 2.4 \cos(4\pi t) - 12 \sin(2\pi t) \right) \, dt \]Evaluating gives:\[ [38.4t + 0.6\sin(4\pi t) + 6\cos(2\pi t)]_{0}^{12} = 460.8 \text{ mm} \]

Step 5: Calculate the Integral for Part (c)

For part (c), evaluate the definite integral from \(t = 0\) to \(t = 24\) hours:\[ \int_{0}^{24} \left(38.4 + 2.4 \cos(4\pi t) - 12 \sin(2\pi t) \right) \, dt \]The integration gives us:\[ [38.4t + 0.6\sin(4\pi t) + 6\cos(2\pi t)]_{0}^{24} = 921.6 \text{ mm} \]

Key Concepts

Differential Equation

A differential equation like \(\frac{dL}{dt} = 38.4 + 2.4 \cos(4 \pi t) - 12 \sin(2 \pi t)\) is a mathematical expression that relates a function to its derivative. This one describes how the growth rate \(\frac{dL}{dt}\) of the fungus changes over time. This means that the growth rate is not constant but fluctuates with the value of \(t\) due to the trigonometric components, especially because of terms like \(\cos(4 \pi t)\) and \(\sin(2 \pi t)\).

Differential equations are powerful tools in modeling dynamic systems, such as predicting the growth of flora and fauna. By interpreting these equations, scientists can understand how specific variables, like time, affect a system. The terms in this differential equation represent various aspects of growth: a constant term, \(38.4\), mirrors a basic growth rate. The cosine and sine parts mirror variations that could relate to factors like light availability or temperature.

Definite Integral

A definite integral gives you the accumulated change from one point to another over an interval. In this scenario, integrating the derived differential equation provides the total growth of the fungus over given time periods.

The formula \(\int_{a}^{b} \frac{dL}{dt} \, dt\) is utilized, where \(a\) and \(b\) are the starting and ending points of the time interval you’re observing. For example, calculating \(\int_{0}^{1} \) from the solution will tell you how much fungus grows in the first hour. By working through this integral, you examine the balance of the different growth factors – those constant and variable due to time. Understanding definite integrals is crucial in calculus as it helps connect the dots between rate of change in systems and the entire change over specified intervals. This approach applies to diverse fields, from physics to economics.

Trigonometric Functions

The presence of \(\cos(4 \pi t)\) and \(\sin(2 \pi t)\) in the differential equation indicates how trigonometric functions shape growth patterns. Trigonometric functions – sine and cosine – are periodic, which means they fluctuate in a predictable manner, akin to the predictable cycles of nature.

In the context of the fungus's growth, the functions are regulating the fluctuations experienced: \(\cos(4 \pi t)\) modifies the growth rate by adding periodic peaks every half hour (as \(4\pi\) suggests faster oscillations), and \(\sin(2 \pi t)\) subtracts from it in a similar rhythm. These could correspond to biological rhythms and environmental cycles mimicking day and night or temperature variations, as seen in circadian rhythms.

Trigonometric functions as part of growth models are vital, as they can capture the natural periodic behaviors of biological entities, making them an integral part of understanding vital patterns.

Growth Modeling

Growth modeling in this exercise helps us understand how organisms like Neurospora crassa follow growth patterns influenced by internal and external rhythms. By utilizing mathematical models, researchers can ascertain how much and how quickly an organism will develop under specific environmental circumstances.

The key aspect of growth modeling is to incorporate variables and functions into an equation that accurately represents the real-world data. Here, the equation with trigonometric functions maps the variable growth rate due to factors like circadian clocks. These models become predictive tools, allowing scientists and researchers to foresee future growth under varying conditions.

Moreover, understanding growth models can help in fields like agriculture and ecology, where predicting how plants will grow over time could lead to better optimization of resources. With mathematics acting as the guiding light, growth modeling offers clarity and precision to what might appear as confusing biological chaos.