Listeria, like all bacteria, emit small molecules as waste products or for signaling. Here we will consider a "race" between a Listeria bacterium (about 1 um across) and a chemical signal it emits while inside a much larger (~ 15 µm) mammalian cell. Suppose the Listeria bacterium, when at one end of the mammalian cell, emits a pulse of signaling molecules which begin diffusing through the mammalian cell. The Listeria then moves along a straight line from one end of the mammalian cell to the other. a. Write down the equation for the rms distance, assuming the diffusion is three-dimensional. Explain in words what the rms distance means (that is, what it tells us about a diffusion process).
Listeria, like all bacteria, emit small molecules as waste products or for signaling. Here we will consider a "race" between a Listeria bacterium (about 1 um across) and a chemical signal it emits while inside a much larger (~ 15 µm) mammalian cell. Suppose the Listeria bacterium, when at one end of the mammalian cell, emits a pulse of signaling molecules which begin diffusing through the mammalian cell. The Listeria then moves along a straight line from one end of the mammalian cell to the other. a. Write down the equation for the rms distance, assuming the diffusion is three-dimensional. Explain in words what the rms distance means (that is, what it tells us about a diffusion process).
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Question
![**Understanding Bacterial Movement and Diffusion in a Mammalian Cell**
**Introduction:**
Listeria, like all bacteria, emit small molecules as waste products or for signaling. This example considers a "race" between a Listeria bacterium (about 1 μm across) and a chemical signal it emits while inside a much larger (~15 μm) mammalian cell. Suppose the Listeria bacterium, when at one end of the mammalian cell, emits a pulse of signaling molecules which then begin diffusing through the mammalian cell. The Listeria then moves in a straight line from one end of the mammalian cell to the other.
**Problem and Parameters:**
1. **Equation for rms distance in three dimensions:**
- **Explanation:** The root mean square (rms) distance is a statistical measure of the average distance a particle will travel due to diffusion over a given time period in three-dimensional space. It is given by the formula:
\[
\text{rms distance} = \sqrt{6Dt}
\]
where \(D\) is the diffusion constant and \(t\) is time. The rms distance tells us how far on average a diffusing particle will travel from its origin. It provides insight into the efficiency and speed of the diffusion process.
2. **Calculation for signal molecules' diffusion time:**
- **Given:**
- **Diffusion constant (D):** 200 μm\(^2\)/s
- **Distance to travel:** 15 μm (diameter of the mammalian cell)
- **Calculation:**
Rearrange the equation for rms distance to solve for time (\(t\)):
\[
t = \frac{(\text{rms distance})^2}{6D} = \frac{(15 \, \mu m)^2}{6 \, \times \, 200 \, \mu m^2/s} = \frac{225 \, \mu m^2}{1200 \, \mu m^2/s} = \frac{225}{1200} \text{s} \approx 0.1875 \text{s}
\]
- **Conclusion:** It takes approximately 0.1875 seconds for the signaling molecules to diffuse across the mammalian cell.
**Graphical Representation:**
On the right, there is an image of a mamm](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F48bba008-5eee-4157-b489-6f90113c3fbd%2F650c9b91-9ab1-4a51-bdd7-1f209ff35aea%2Fy2a2osn_processed.png&w=3840&q=75)
Transcribed Image Text:**Understanding Bacterial Movement and Diffusion in a Mammalian Cell**
**Introduction:**
Listeria, like all bacteria, emit small molecules as waste products or for signaling. This example considers a "race" between a Listeria bacterium (about 1 μm across) and a chemical signal it emits while inside a much larger (~15 μm) mammalian cell. Suppose the Listeria bacterium, when at one end of the mammalian cell, emits a pulse of signaling molecules which then begin diffusing through the mammalian cell. The Listeria then moves in a straight line from one end of the mammalian cell to the other.
**Problem and Parameters:**
1. **Equation for rms distance in three dimensions:**
- **Explanation:** The root mean square (rms) distance is a statistical measure of the average distance a particle will travel due to diffusion over a given time period in three-dimensional space. It is given by the formula:
\[
\text{rms distance} = \sqrt{6Dt}
\]
where \(D\) is the diffusion constant and \(t\) is time. The rms distance tells us how far on average a diffusing particle will travel from its origin. It provides insight into the efficiency and speed of the diffusion process.
2. **Calculation for signal molecules' diffusion time:**
- **Given:**
- **Diffusion constant (D):** 200 μm\(^2\)/s
- **Distance to travel:** 15 μm (diameter of the mammalian cell)
- **Calculation:**
Rearrange the equation for rms distance to solve for time (\(t\)):
\[
t = \frac{(\text{rms distance})^2}{6D} = \frac{(15 \, \mu m)^2}{6 \, \times \, 200 \, \mu m^2/s} = \frac{225 \, \mu m^2}{1200 \, \mu m^2/s} = \frac{225}{1200} \text{s} \approx 0.1875 \text{s}
\]
- **Conclusion:** It takes approximately 0.1875 seconds for the signaling molecules to diffuse across the mammalian cell.
**Graphical Representation:**
On the right, there is an image of a mamm
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