Question

A pastry chef accidentally inoculated a cream pie with six \(S\). aureus cells. If \(S\). aureus has a generation time of 60 minutes, how many cells would be in the cream pie after 7 hours?

Short answer

768 S. aureus cells after 7 hours.

Step-by-step solution

Determine the number of generations

To find out how many times the bacterial population doubles, calculate the number of generations. Since the generation time is 60 minutes and the total time is 7 hours, convert hours to minutes: \(7 \text{ hours} \times 60 \text{ minutes/hour} = 420 \text{ minutes}\). The number of generations is then \(\frac{420 \text{ minutes}}{60 \text{ minutes/generation}} = 7 \text{ generations}\).

Use the doubling formula

The formula to calculate the final number of cells after a certain number of generations is \(N = N_0 \times 2^n\), where \(N\) is the final number of cells, \(N_0\) is the initial number of cells, and \(n\) is the number of generations. Initially, there are 6 cells: \(N_0 = 6\), and from Step 1, \(n = 7\).

Substitute values and calculate

Substitute the known values into the formula: \(N = 6 \times 2^7\). Calculate \(2^7\), which equals 128. Then multiply this by 6: \(N = 6 \times 128 = 768\).

Interpret the result

After performing the multiplication, the population of \(S\), aureus cells in the cream pie after 7 hours is 768 cells.

Key concepts

Generation Time

The concept of generation time is crucial in understanding bacterial growth. It refers to the time it takes for a bacterial population to double in number. Each species of bacteria has its own characteristic generation time which can vary depending on environmental conditions such as temperature, nutrient availability, and pH.
For many bacteria, the generation time can be quite short; for instance, under optimal conditions, some bacteria can double every 20 minutes. However, in our scenario, the bacteria in question—Staphylococcus aureus—has a generation time of 60 minutes.
This means that every hour, given suitable conditions, the number of Staphylococcus aureus cells would double.

Cell Doubling

Cell doubling is a straightforward yet powerful concept in the study of microbiology. It describes how populations of cells increase exponentially under favorable conditions. Each cycle of doubling can dramatically increase the population size. Here's how it works:
- Imagine starting with a single bacterial cell. After one generation time, that single cell divides into two.
- After another generation time passes, each of those two cells will have divided, resulting in four cells.
As such, every new generation can lead to a significant increase in the total number of cells.
In practice, this exponential growth can be calculated using the formula: N = N_0 \times 2^n, where \(N_0\) is the initial number of cells, \(2^n\) represents the number of times the population has doubled, and \(n\) is the number of generations.

Staphylococcus aureus

Staphylococcus aureus is a type of bacteria commonly found on the skin and in the noses of healthy individuals. It's known for being both a harmless resident of the human microbiota and, in some cases, a cause of severe infections.
This dual role makes understanding its growth under various conditions important, such as in food products where it can lead to food poisoning if left unchecked.
In the context of our exercise, Staphylococcus aureus accidentally introduced into a cream pie can rapidly multiply, especially if stored at room temperature, due to suitable conditions for its growth.
Knowing its generation time helps in predicting its population size over time, focusing on preventing potential outbreaks linked to this bacterium.

Bacterial Population Calculation

Calculating bacterial populations over time is essential in microbiology, especially to assess growth under different conditions. The exercise provided an opportunity to see how simple calculations can be used to predict the population size of bacteria.
The formula used was: \[ N = N_0 \times 2^n \] where:

• \(N\) represents the final number of cells.
• \(N_0\) stands for the initial number of cells.
• \(n\) is the number of generations.

In the scenario with Staphylococcus aureus in the cream pie, we started with 6 cells. Given a 7-hour period, and with a generation time of 60 minutes, there were 7 doubling events: \(n = 7\).
These calculations show how, beginning with only 6 cells, the population can explode to 768 cells in just 7 hours, highlighting the rapid multiplication potential of bacteria.