Problem 14 - Calculus for Biology and Medicine

Question

Write down a formula for the population size, \(N_{t}\), as a function of time, \(t\). A strain of bacteria reproduces asexually every 24 minutes. That is, every 24 minutes, each bacterial cell splits into two cells. If, initially, there is 1 bacterium, how long will it take until there are 512 bacteria?

Step-by-Step Solution

Answer

It will take 216 minutes to reach 512 bacteria.

Step 1: Understanding Doubling Time

The bacteria population doubles every 24 minutes. This means if you start with one bacterium, at 24 minutes you will have 2, at 48 minutes you will have 4, and so on.

Step 2: Setting up the formula

To find the population after a given number of time intervals, we need to calculate how many times the bacteria double. If initially there is one bacterium, and it doubles every 24 minutes, the formula for the population size after time \( t \) minutes is \( N_t = 1 \cdot 2^{\frac{t}{24}} \).

Step 3: Solving for 512 bacteria

We need to find \( t \) such that \( N_t = 512 \). Using the formula: \( 512 = 1 \cdot 2^{\frac{t}{24}} \). This simplifies to \( 2^{\frac{t}{24}} = 512 \). Since \( 512 = 2^9 \), we have \( \frac{t}{24} = 9 \).

Step 4: Calculating time

Multiply both sides by 24 to solve for \( t \): \( t = 24 \times 9 = 216 \).

Key Concepts

Population Dynamics

Population dynamics is the study of how populations change over time. It involves understanding the factors that affect population sizes, growth rates, and interactions within an ecosystem.
In the context of bacterial growth, we often study how bacteria populations increase rapidly under ideal conditions. This is because bacteria can reproduce quickly, especially when resources such as nutrients are abundant.

• Bacteria replicate by a process called binary fission, where one cell divides into two cells.
• This rapid reproduction can lead to exponential growth, where the population size multiplies at a consistent rate over equal time intervals.

The growth observed in bacterial populations can help us understand general principles of population dynamics, which also apply to other organisms including animals and plants, albeit over different timescales and under different conditions.

Bacterial Reproduction

Bacterial reproduction occurs through a process known as binary fission. This method is simple and quick, making it highly efficient for rapid population increase.
In binary fission, a single bacterial cell duplicates its DNA and then divides into two identical cells, each with a copy of the original DNA.

• This process typically takes a short amount of time, such as 20-30 minutes under optimal conditions.
• Because it results in the doubling of the population, it's easy to understand how bacteria can rapidly achieve large population sizes.

Bacterial reproduction is an example of exponential growth, where each generation doubles the population of the previous generation. This is much faster than many other organisms, leading to widespread bacterial presence in various environments.

Doubling Time

In exponential growth, the doubling time is the amount of time it takes for a population to double in size. For bacteria, this time period is often very short due to their rapid reproduction.
In our example, the doubling time is 24 minutes, which means that every 24 minutes, the number of bacteria doubles.
This is calculated using the formula: \( N_t = N_0\times 2^{t/T_d} \), where \( N_0 \) is the initial population size, \( t \) is time, and \( T_d \) is the doubling time.

• Understanding doubling time helps us quantify how quickly a population is growing, which is critical in both natural ecosystems and human-managed systems like fermentations or medical microbiology.
• It is a key concept in assessing the speed and implications of bacterial spread.

By calculating the doubling time, we can predict population sizes at different time points and manage environmental impacts accordingly.