Problem 33 - Calculus for Biology and Medicine

Question

Investigate the advantage of dimensionless variables. Suppose a bacterium divides every 20 minutes, which we call the characteristic time scale and denote by \(T\). Let \(t\) be the time elapsed since the beginning of an experiment that involves this bacterium. Define \(z=t / T .\) Find \(z\) if \(t=120\) minutes, and show that \(z\) has the same value if \(t\) and \(T\) are measured in units of hours instead.

Step-by-Step Solution

Answer

The dimensionless variable \(z = 6\), in both minutes and hours.

Step 1: Understand the Problem

We are asked to define a dimensionless variable \(z\) using time \(t\) and a characteristic time scale \(T\). We then solve for \(z\) given \(t = 120\) minutes and \(T = 20\) minutes.

Step 2: Set Up the Equation for the Dimensionless Variable

The dimensionless variable \(z\) is defined as \(z = \frac{t}{T}\). This means that we convert the original time \(t\) into a form independent of any units.

Step 3: Solve for z in Minutes

Given \(t = 120\) minutes and \(T = 20\) minutes, substitute these values into the equation: \[z = \frac{120}{20} = 6\]. This shows that in terms of minutes, \(z = 6\).

Step 4: Convert Units to Hours and Recalculate

Convert the times \(t\) and \(T\) from minutes to hours. - Since \(t = 120\) minutes, convert to hours: \(t = \frac{120}{60} = 2\) hours.- Since \(T = 20\) minutes, convert to hours: \(T = \frac{20}{60} = \frac{1}{3}\) hours.Now substitute into the dimensionless variable equation: \[z = \frac{2}{\frac{1}{3}} = 2 \times 3 = 6\]. This confirms that \(z = 6\) in hours as well.

Step 5: Conclusion

The dimensionless variable \(z\) retains the same value of \(6\) whether \(t\) and \(T\) are measured in minutes or hours, demonstrating that \(z\) eliminates dependency on the choice of time units.

Key Concepts

Characteristic Time Scale

In the study of dynamic systems, a characteristic time scale often plays a crucial role. It's essentially the time frame in which a particular process occurs or evolves. For the problem with our bacterium, the characteristic time scale, denoted by \( T \), is the time it takes for one division to occur, which is 20 minutes. This concept helps us understand at what rate a process happens and allows us to compare different processes by normalizing time relative to the characteristic time.
By employing a characteristic time scale, you can simplify complex biological or physical phenomena. In mathematical terms, it serves as a baseline or benchmark, enabling you to track changes, such as bacterial growth, that happen over time. Thus, the characteristic time scale becomes an essential parameter in making time dimensionless, helping us analyze and compare processes regardless of the unit of measurement.

Unit Conversion

Unit conversion is a method of changing one unit of measure to another. It's a necessary skill when dealing with dimensionless variables or any physical quantities. For the exercise in question, converting minutes to hours and vice versa can deeply affect calculations if not done correctly.
First, understand the relationship between minutes and hours:

• 1 hour = 60 minutes
• 1 minute = 1/60 hour

Through these relationships, you can convert any time given in minutes to hours and ensure consistency across calculations. In the exercise, both time \( t \) and the characteristic time scale \( T \) were converted from minutes to hours to verify that the dimensionless variable \( z \) remains invariant regardless of unit changes. Proper unit conversion simplifies comparison and assures the dimensionless variable maintains its consistency.

Bacterial Growth

Bacterial growth typically follows an exponential pattern, characterized by consistent and rapid multiplication over time. In our exercise, this growth is modeled by noting the time it takes for bacteria to divide, known as the generation time, which is \( T = 20 \) minutes.
Bacteria grow by binary fission, where one cell divides into two, and this rate of division is encapsulated by dividing time through characteristic time scale \( T \). With a dimensionless approach, we ignore units, focusing on how many times they divide rather than the elapsed time.
Understanding bacterial growth with the help of dimensionless variables allows researchers to generalize findings across different experiments, compare different bacterial species, and apply this knowledge effectively in fields like microbiology and medicine.

Mathematical Modeling

Mathematical modeling involves using mathematical expressions and equations to represent and study real-world systems. In our exercise, we use the model \( z = \frac{t}{T} \) to transform time-dependent bacterial growth into a dimensionless form.
Such modeling is crucial because it allows for a clear analysis of dynamics without worrying about the units of measurement. By employing the concept of dimensionless variables, you can simplify complex systems, making mathematical analysis and comparisons more straightforward.
Through mathematical modeling, researchers can predict behaviors, analyze relationships among variables, and gain valuable insights into processes like bacterial growth. This method is extensively used across various fields including biology, physics, and engineering, to describe phenomena concisely and effectively manage changing variables.