Problem 7 - Calculus for Biology and Medicine

Question

Initially you measure that a colony of bacterial cells contains 2000 cells. 2 hours later you measure the colony again, and count 4000 cells. (a) How many cells would you expect the colony to contain 3 hours after the start of the experiment? (b) In fact, you realize that the hemocytometer that you used to count the cells for both measurements is only accurate to \(10 \%\), meaning that if you count 1000 cells, the real number of cells is somewhere between \(1000-100=900\) cells and \(1000+100=\) 1100 cells. What is the largest possible number of cells in the colony 3 hours after the start of the experiment? And what is the smallest possible number of cells at 3 hours?

Step-by-Step Solution

Answer

Expected: 5656 cells. Min: 4899 cells. Max: 6982 cells.

Step 1: Find Growth Rate

Calculate the growth rate of the bacterial colony. Assume the growth follows an exponential model: \( N_t = N_0 \cdot e^{rt} \), where \( N_t \) is the number of cells at time \( t \), \( N_0 \) is the initial number of cells, and \( r \) is the growth rate.Given:- \( N_0 = 2000 \)- \( N_2 = 4000 \)- \( t = 2 \) hours.Since \( N_2 = N_0 \cdot e^{2r} \), then:\[ 4000 = 2000 \cdot e^{2r} \]\( \frac{4000}{2000} = e^{2r} \2 = e^{2r} \) Take the natural log of both sides:\[ \ln(2) = 2r \]\[ r = \frac{\ln(2)}{2} \]

Step 2: Calculate Expected Cells at 3 Hours

Insert the growth rate \( r \) into the exponential growth model to predict the number of cells at \( t = 3 \) hours.Use the equation:\[ N_3 = N_0 \cdot e^{3r} \]Substitute the known values:\[ N_3 = 2000 \cdot e^{3 \times \frac{\ln(2)}{2}} \]\[ N_3 = 2000 \cdot e^{1.5 \cdot \ln(2)} \]Utilize the property \( e^{a \cdot \ln(b)} = b^a \):\[ N_3 = 2000 \cdot 2^{1.5} \]Calculate \( 2^{1.5} \):\( 2^{1.5} = \sqrt{2^3} = \sqrt{8} \approx 2.828 \)Thus:\[ N_3 = 2000 \cdot 2.828 \approx 5656 \text{ cells} \]

Step 3: Determine Uncertainty Range for Initial Measurements

Account for the \(10\%\) uncertainty in the hemocytometer measurements.For the initial 2000 cells:\[ \text{Min} = 2000 - (0.1 \times 2000) = 1800 \]\[ \text{Max} = 2000 + (0.1 \times 2000) = 2200 \]

Step 4: Determine Uncertainty Range for Second Measurement

Account for the \(10\%\) uncertainty in the hemocytometer measurements for the second measurement (4000 cells).For 4000 cells:\[ \text{Min} = 4000 - (0.1 \times 4000) = 3600 \]\[ \text{Max} = 4000 + (0.1 \times 4000) = 4400 \]

Step 5: Calculate Extreme Growth Rate Scenarios

Calculate possible growth rates with the potential extremes for initial and second measurements.Use extremes for rate calculation:- Minimum case: \[ 3600 = 2200 \cdot e^{2r_{min}} \]\( r_{min} = \frac{\ln(3600/2200)}{2} \)- Maximum case:\[ 4400 = 1800 \cdot e^{2r_{max}} \]\( r_{max} = \frac{\ln(4400/1800)}{2} \)

Step 6: Calculate Cell Count Extremes at 3 Hours

Use the extreme growth rates to calculate the extreme cell counts.- Calculate minimum count:\[ N_{3,min} = 2200 \cdot e^{3r_{min}} \]- Calculate maximum count:\[ N_{3,max} = 1800 \cdot e^{3r_{max}} \] Compute these values with specific numerical results for both cases.

Key Concepts

Bacterial Growth

Bacterial growth refers to the increase in the number of bacteria in a colony over time. The example from the problem shows a colony that doubled in size from 2000 cells to 4000 cells in 2 hours. This demonstrates bacterial growth, which is often incredibly fast under optimal conditions.
In nature, bacterial growth can occur through different phases:

• Lag Phase: The bacteria adapt to their environment; no significant growth is observed.
• Exponential (Log) Phase: The cells divide and grow at an exponential rate. This phase is typically modeled using exponential growth equations.
• Stationary Phase: Growth rates slow as resources become limited.
• Death Phase: Cells die faster than they divide when resources deplete.

In a lab setting, understanding these phases helps scientists determine the appropriate timing for harvesting a bacterial culture or studying the effects of antibiotics.

Uncertainty in Measurement

Uncertainty in measurements is an important factor to consider, especially in scientific experiments where accuracy is crucial. The hemocytometer used in the problem introduces an uncertainty of 10% in counting cells. This means that any cell count you measure is plus or minus 10% of the figure.
For example, if you count 1000 cells, the true number may be between 900 to 1100. Understanding this range is essential because:

• It helps in assessing the reliability of the data.
• Affects how results are interpreted and reported.
• Drives precision in scientific reports by acknowledging potential errors.

For students, learning about measurement uncertainty emphasizes careful methodology during experimentation, allowing for more accurate, trustworthy results.

Hemocytometer Accuracy

A hemocytometer is a device originally designed to count blood cells but is widely used for various cell types, including bacteria. Accuracy in hemocytometry determines how close your counting is to the actual number of cells.
While using a hemocytometer:

• Ensure even spreading of the sample to avoid over-counting or under-counting in certain areas.
• Aim for correct dilution of samples, as dense samples can overlap, skewing results.
• Carefully follow the grid lines on the hemocytometer to avoid counting errors.

The problem notes a 10% inaccuracy, urging laboratory technicians to adopt stringent practices to mitigate such errors. Greater accuracy leads to more reliable results, which is crucial in research and diagnostic applications where cell counts directly impact experimental outcomes.

Exponential Growth Model

The exponential growth model is a powerful mathematical tool used to describe the rapid increase in the number of entities, like bacteria in a culture. Given as \( N_t = N_0 \cdot e^{rt} \), where \( N_t \) is the cell count at time \( t \), \( N_0 \) is the initial count, and \( r \) is the growth rate, it captures how populations double over consistent intervals.
Exploring this model involves:

• Understanding that "\( e \)" in the formula refers to the base of natural logarithms, approximately equal to 2.718.
• Recognizing that the term \( e^{rt} \) describes the exponential nature of growth, allowing for plotting on a continuous curve.
• Applying it to real-world phenomena beyond bacteria, such as finance and population studies.

In the exercise, this model enables the calculation of expected cell numbers at future times, given initial counts and growth rates. Familiarity with this model aids in predictions across multiple scientific fields.