Introduction

In our project, we want to stop the movement of C.crescentus to image a clear photo. So we need to know whether we have stopped those excited boys. One problem about this is, how to define and measure the parameter of the C.crescentus' motion?

Analysis

Here we make the hypothesis below.

1. the solution is uniform, which means the composition of every part in the solution is the same
2. the number of the bacteria would not change in short time.
3. the motion of the C.crescentus is random and they can be regarded as free gas.

And we suppose the distance one bacterium moved in time $t$ is $l$. Then the diffusion coefficient D can be defined as
$$D=l^2/(2t)$$

As you can guess, we are going to use $D$ to describe the motion of C.crescentus in stead of velocity. The reason is that the object of our study is a large crowd of bacteria, and it is more suitable to use diffusion coefficient as the parameter of the system.

In fact, $D$ describe the diffusion rate of bacteria. It's easy to predict that after we give the signal "stop", $D$ would decline because the C.crescentus can not get the help from the rotation of their flagellum and the holdfast would block their movement.

But, how can we measure the value of the diffusion coefficient?

Getting the help from Einstein

In early years of last century, Einstein focused on the motion of free gas, and got the famous Einstein Relation: $$Ave(x^2)=2Dt$$ where $x$ is the distance one bacterium moved in time $t$, and $Ave(x^2)$ is the average value of x square. The relation shows that $Ave(x^2)$ is in direct proportion to the time $t$.

With this theory, we can develop a method to measure the value of $D$. Firstly we use micro camera to record the motion of bacteria. And then we focus on several bacteria's movement locus to count $Ave(x^2)$ with time $t$. Using these statistic data we can plot the relation between $Ave(x^2)$ and $2t$, and the slope of the plot is what we want, the value of D.

Results

We use Mathematics to simulate 500-step movements of 5 bacteria, the image above shows their locus (in different color). ALl the bacteria "moved" randomly to different directions. Than we calculate square of the distance from original point, plot it with steps $ n $

From the figure we can conclude that different diffusion coefficients correspond to different slopes of the fitting line. So we can develop a experimental method to evaluate the parameter of the system.

When we get the value of diffusion coefficients before and after we gave the signal STOP, we can compare them to see how our paths woke. What's more, with D, we can estimate how far a bacterium can move in a particular period of time, and predict how clear our image would be.

From a quantitative perspective, this work upholds the idea of using C.crescentus instead of E. Coli as chassis to make the image more clear.

• This work is chiefly done by Hongda Jiang, with the assistance of Fangming Xie.
• This article is written by Hongda Jiang, edited by Fangming Xie.