Bacterial Concentration

The basic mathematical model for bacterial chemotaxis is the Keller-Segal equations (1) of chemotaxis:

Equation 1: Keller-Segal equation

where:

u[M] is the bacteria concentration.
v[M] is the chemo-repellent concentration.
k₁[m²/s] is the bacteria diffusion coefficient.
k₂[m²/M * s] is the bacteria chemotactic coefficient.
k₃[M/s] is bacteria life and death coefficient.
x[m] is the distance in the channel.
t[s] is time.

The basic assumptions of the chemotaxis model are:

-k₂ describes how sensitive is the bacteria to changes in chemo-repellent concentration. In other words, as its negative value decreases the bacteria will react more violently to same repellent exposure.

-Because of the geometric properties of the channel, this is approximately a one dimensional problem.

-We ran our tests in a short time scale (t<30[min]) so we presume that the change in concentration of bacteria due to life and death is negligible, k₃=0.

-The chemo-repellent concentration is known for every x and t.

-There are no changes in the flux of bacterial concentration at the start (a) and at end of the channel (b).

Mathematical Model

Under these assumptions the equation takes the form:

Equation 2: The Bacterial concentration problem.

The above partial differential equation cannot be solved analytically, so we must turn to numerical analysis tools. The implicit Euler method is one of the most basic numerical methods for the solution of ordinary and partial differential equations. This method is guaranteed to be stable and gives us the solution for the entire space in a single point in time.

Using the following discretization:

Equation 3: Bacteria concentration problem – discrete form.

And we can write it as follows:

Equation 4: Bacteria concentration problem – Final form

Given the initial condition:

Equation 5: Bacteria concentration problem- initial condition.

And the boundary conditions which are translated to the following discrete conditions:

Equation 6: Bacteria concentration problem- boundary conditions

With I being the final value of i – index of location.
The above conditions keep the flux of bacteria zero on both sides of the chip as occurs in the actual chip. Hence, no bacteria enter or exit the chip.
The above equation was entered into the Thomas-Three-Diagonal algorithm for solving matrix equations, giving us the solution for the entire space of the problem in a specific point in time. By advancing in time as we solve the equation at each time point we get the solution for the bacterial concentration for every x,t.

Model results

We ran the chemo-repellent concentration equation in Matlab - Our code can be found here.
The parameters used:

Table 1: Parameters for chemotaxis model

Please notice, the results are normalized to enable us to show them on the same scale.

The results are as follows:

Graph 1: Results of bacterial chemotaxis model

Model conclusions:

- The peak of bacterial concentration is caused by the bacterial chemotactic response, moving away from the chemo-repellent, and concentrating.
- The "wave" of bacterial concentration starts moving relatively fast, and slows down quickly. This is due to the change in repellent concentration. The "wave" converges to ~7[mm].
- The bacteria react significantly less to a normalized repellent concentration of less than ~3[mm]. This is approximately where the two graphs intersect.
- As the concentration of repellent goes down, the bacteria are less reactive. This continues as the bacteria's diffusion speed surpasses the chemotaxis rate. In other words, more bacteria move away from the concentration peak than into it for t>15[mim].
- Projecting this on the chip color experiment, we can predict there will be three shades of color: weak where the bacteria moved from (low concentration), strong where the bacteria moved to (high concentration) and on the far end, unchanged as the bacteria were not exposed to the repellent.

Comparison to Experiments

The experiment ran as the setup shown in the "Overview" section: The channel was filled with bacteria in motility buffer and then the sample was inserted. We used engineered E. coli with a S.Tar PctA receptor taken from a plate and suspended in motility buffer. The chemo-repellent used is TCE.

Table 2: Substance for chemotaxis experiment

As expected, a visible cluster of strong dark blue has formed next to a lighter shade due to chemotactic activity. Furthermore, the distance the bacteria passed is only a few millimeter as the model predicted.
The time scale does not line up: The color darkens as the experiment continues. This will probably be corrected by using more accurate diffusion and chemotactic coefficients.
Some of the inconsistencies between the model and the experiment (like the uneven cluster of colored bacteria) can be explained by problems loading the chemo-repellent as shown in the chemo-repellent concentration experiment.

Conclusions

Results

This model predicts the overall behavior of our system. In our experiments we were able to show that the concentration of the repellent acts as we expected in terms of changes in the diffusion limit's velocity. When compared to the bacterial concentration, again, the experiments showed a similar behavior as the numeric solution of Keller-Segal equation.

As explained before, this model requires further fitting to get more accurate results. Not only by using more accurate coefficients, but with improving the system itself.

Future development

We would like to improve the model and design new ones based on it. First, develop models for the different coefficients in the Keller-Segal equation: bacteria diffusion coefficient, chemotactic coefficient and bacteria life and death. Theoretically, those coefficients can be controlled by changing some parameters like the number of flagella or receptors a cell has, or even manipulating the biological tracks of the bacteria. This will enable us to get even more accurate results. Secondly, to build a library of different receptors and ligands. Thirdly, expanding our model so it could predict movement of bacteria in different geometric constructs such as funnels or U bends. This research can be the basis of a prototype commercial device.

As for the first part, we worked with iGEM Freiburg (2) in developing a function for the chemotactic coefficient. Also, we designed a new assay for experimentally measuring it.

This assay, named Trap & Track, is a novel way to detect chemotaxis on the nanometric level. By using it we can measure the exact repellent concentration that induces chemotaxis and calculate the chemotactic coefficient accurately for every material. A detailed explanation about the assay can be found here.

As for the second part, the S.Tar system enables us to change the receptors a bacterium has and by that, change the materials it repels from. In the future, this system can be expanded to control the efficiency of these receptors and even control other aspects of the chemotaxis pathway such as the flagella.

The third and final part is to improve the device itself. We designed a new fluidic chip and fabricated it in different methods. This new design will give us a more controlled diffusion by cancelling out most of the flow and fixing the diffusion source. Also, by changing the geometry of the channel, the bacteria concentration will increase and cause a more noticeable signal. This will improve the accuracy of the experiments we run, and in turn, our overall model.