Problem 11 - Calculus for Biology and Medicine

Question

Write down a formula for the population size, \(N_{t}\), as a function of time, \(t\). 11\. A strain of bacteria reproduces asexually every hour. That is, every hour, each bacterial cell splits into two cells. If, initially, there is one bacterium, find the number of bacterial cells after 1 hour, 2 hours, 3 hours, 4 hours, and 5 hours.

Step-by-Step Solution

Answer

The population size formula is \( N_t = 2^t \). After 1, 2, 3, 4, and 5 hours, the populations are 2, 4, 8, 16, and 32, respectively.

Step 1: Understand the Problem

The bacteria replicate by splitting into two every hour. Initially, there is one bacterium at time 0. We want to find the number of bacteria after each hour up to 5 hours. This describes an exponential growth pattern.

Step 2: Define the Formula Structure

Since each bacterium splits into two every hour, the number of bacteria doubles each hour. The general formula for exponential growth can be written as: \[ N_t = N_0 imes (r)^t \]where \( N_t \) is the population size at time \( t \), \( N_0 \) is the initial population size, and \( r \) is the growth factor per time unit. In this case, \( r = 2 \).

Step 3: Formulate the Specific Expression

Given \( N_0 = 1 \) (the initial number of bacteria) and \( r = 2 \) (the growth factor as each bacterium doubles every hour), we can specifically write:\[ N_t = 1 imes 2^t \]Hence, \( N_t = 2^t \).

Step 4: Calculate Population Size at Each Time Point

Using the formula \( N_t = 2^t \), calculate for each hour:- After 1 hour: \( N_1 = 2^1 = 2 \)- After 2 hours: \( N_2 = 2^2 = 4 \)- After 3 hours: \( N_3 = 2^3 = 8 \)- After 4 hours: \( N_4 = 2^4 = 16 \)- After 5 hours: \( N_5 = 2^5 = 32 \)

Key Concepts

Bacterial Reproduction

Bacterial reproduction, particularly in species that reproduce asexually, is a fascinating process that involves a single organism creating a copy of itself. In the context of our example, the bacteria replicate by splitting into two every hour, which is a method known as binary fission.
During binary fission:

• A single bacterium, the parent cell, doubles its genetic material.
• It divides into two genetically identical offspring cells.
• These daughter cells grow independently and prepare to divide again in the next cycle.

This cycle can repeat very rapidly, which is why bacterial populations can explode into large numbers in a short period. This rapid increase is a classic example of exponential growth, characterized by a constant doubling period.

Population Dynamics

Population dynamics is the study of how and why the size of populations changes over time. In the exponential growth of bacteria, we observe a simple and direct model of population dynamics.
When studying bacterial growth, the initial population size and growth rate play crucial roles. With our bacteria example:

• The initial number of bacteria: 1 bacterium
• Time taken for each reproduction cycle: 1 hour
• Reproduction rate: doubling every hour

These variables help us understand how the bacterial population multiplies. As the bacteria reproduce, the population size at each time point can be calculated using the formula for exponential growth. The larger the initial population or the faster the generation time (shorter cycles), the quicker the population size will expand.

Growth Factor

The growth factor is a key concept in understanding exponential growth. It describes how much the population increases during each time period. In our bacteria example, the growth factor is 2, meaning the population doubles every hour.
For bacterial growth:

• The growth factor remains constant over time.
• Each hour, the existing number of bacteria is multiplied by the growth factor of 2.
• This consistent multiplication leads to a predictable exponential increase in population size.

Because of the constant growth factor, predicting bacterial population using the formula \( N_t = N_0 \times r^t \) becomes straightforward. Here, \( N_0 \) is the initial size, \( t \) is time, and \( r \) is the growth factor. This formula gives us a powerful tool to predict population sizes without needing to manually count each bacterial cell as time progresses.