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The basic assumptions of the chemotaxis model are: | The basic assumptions of the chemotaxis model are: |
Revision as of 11:35, 18 October 2016
Introduction
The goal of this model is to describe the processes inside the Flash Lab system:
- Change in the concentration of chemo-repellent.
- Change in the concentration of bacteria.
This model is based on the Keller – Segal equation of chemotaxis (1) in a one dimensional problem (Thin channel).
It's important to notice that this model can show the overall behavior and not exact values. The final system is supposed to detect a variety of materials in many different unknown solvents, each of them has its own diffusion properties. Also, some aspects such as working conditions (temperature, humidity etc.) might change in widespread commercial use and affect the results. Taking those into account, further fitting will be necessary.
Chemo-Repellent Concentration
Model
The basic assumptions of the model for the chemo-repellent are:
There are no forces except diffusion:
-Chemo-repellent concentration in the sample is relatively low and does not cause osmotic pressure.
-The changes in pressure due to loading the sample is negligible.
-No other significant external forces (for example, moving the chip while in use).
The bacteria do not consume the chemo-repellent and its concentration does not change with time. This is not case with chemo-attractants.
We expect to detect small proteins and molecules (the materials bacterial receptors bind to). The diffusion coefficient for such materials is about 10-9[m2/s].
Because of the geometric properties of the channel and the expected diffusion coefficient, this is approximately a half-infinite one dimensional problem.
Initial condition: no chemo-repellent is present in the chip at time zero.
Boundary condition: at infinite distance the concentration is zero and the there is conservation of dissolved material mass (c).
We modelled the change in concentration of the chemo-repellent based on a "Top Hat Function" solution for the diffusion problem:
v[M] is the chemo-repellent concentration,
D[m2/s] is the chemo-repellent diffusion coefficient,
N[mol] is the number of repellent atoms,
A[m2] is the cut section of the channel,
x[m] is the distance on the channel,
t[s] is time.
The solution for this problem is:
In our problem, we want the diffusion to start from. Also, we are interested only in the positive part of the solution:
Model Results
We ran the chemo-repellent concentration equation in Matlab - Our code can be found here. The parameters used are:
*This is the diffusion coefficient for potassium permanganate (see "Comparison to Experiment")
** h = (Sample_volume)/(Reservoir_cut_section).
The change in distance of the diffusion limit between 0 to 15 minutes, is relatively big. As the time passes the diffusion limit's speed lowers significantly and the concentration, becomes more linear.
Comparison to Experiment
Most diffusion experiments need a dedicated system that is based on the diffusion of an isotope or a fluorescent material that can be detected easily and very precisely. In this case, we chose a more basic system given that this is only a preliminary testing as our goal is showing that the overall system behaves as we expect.
The experiment ran as shown in the "Introduction" section: The channel was filled with bacteria in motility buffer and then the sample was inserted. We replaced the motility buffer with water and the chemo-repellent with potassium permanganate in the following amounts:
Motility buffer is mostly water (~98%) and can be modelled by it. Potassium permanganate is a salt with a known diffusion limit and acts as most of the materials we want detect using our system (small molecules). Also, it has a very distinct pink color in low concentration which makes diffusion limit visible.
We ran the experiment 4 times, with a standard ruler to measure the distance of the diffusion limit.
As expected by the mathematical model, the diffusion limit starts moving relatively fast and its speed decreases rapidly. The difference in distance between the models to the experiment can be explained by:
The actual diffusion limit is in too low concentration of potassium permanganate to be seen in the naked eye. If the visible concentration is about 0.000015 [M] the experimental results line up with the model (graph 3).
The ruler is a crude measuring tool. Its mistake is ± 0.5 [mm].
Difficulties loading the sample in a uniform way, especially in low volumes. Mistakes in loading the sample inside the bacterial fluid and not on, or sticking the drop of sample to one of the entry slot walls will cause uneven diffusion.
Graph 3: Comparison of diffusion model (c=0.00015[M]) to experiment.
Bacterial Concentration
The basic mathematical model for bacterial chemotaxis is the Keller-Segal equations of chemotaxis:
Equation 4: Keller-Segal equation
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
and 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 bacteria concentration at the start (a) and at end of the channel (b).
Bacterial Concentration
The basic mathematical model for bacterial chemotaxis is the Keller-Segal equations of chemotaxis:
Outlook
References:
1. KELLER, Evelyn F.; SEGEL, Lee A. Model for chemotaxis. Journal of theoretical biology, 1971, 30.2: 225-234.