Difference between revisions of "Team:Technion Israel/Model"
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Because of the geometric properties of the channel and the expected diffusion coefficient, this is approximately a half-infinite one dimensional problem. | Because of the geometric properties of the channel and the expected diffusion coefficient, this is approximately a half-infinite one dimensional problem. | ||
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Revision as of 10:25, 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).
Equation 1: "Top Hat"" 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:
Equation 2: General solution for chemo-repellent diffusion problem.
Graph 1: Chemo-repellent concentration: top hat diffusion.
In our problem, we want the diffusion to start from . Also, we take into account only the positive distance:
Equation 3: Chemo-repellent concentration.
Model Predictions
We ran the chemo-repellent concentration equation in matlab (The code is in ap-pendix). The parameters used:
Table 1: Parameters for diffusion model.
The output for different times:
Graph 2: Model for repellent concentration.
The change in value of the diffusion limit between times 0 to 15 minutes, is relatively big. As the time passes the change lowers.
* This is the diffusion coefficient for potassium permanganate (see "Comparison to Experiment").
** h = (Sample_volume)/(Reservoir_cut_section).
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 pre-cisely. In this case, as explained in the opening, our goal is
showing that the overall system behaves as we expect.
The experiment ran as shown in the "Introduction" section where we replaced the bacterial medium with water and
the repellent with potassium permanganate in the following amounts:
Table 2: Substance for diffusion experiment.
Motility buffer is mostly water (98%) and can be modelled by it. Potassium perman-ganate is a salt with a known diffusion
limit and acts as most of the materials we want detect using our system. Also, it has a very distinct pink color in low
concentra-tion, so the diffusion limit can be seen easily.
We ran the experiment 4 times, with a standard roller to measure the distance of the diffusion limit.
Fig. 1: Diffusion of potassium permanganate in water in different times (enhanced picture).
As expected by the mathematical model, the diffusion limit starts moving relatively fast and its speed decreases rapidly.
The difference in distance between the model to the experiment (average of 2.5[mm] fro, T=0) 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 con-centration is about 0.000015 [M] the experiments results lines up with the model (Graph 2.2).
- The roller 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 2.3: Comparison of diffusion model to experiment.
Results
Outlook
References:
1. KELLER, Evelyn F.; SEGEL, Lee A. Model for chemotaxis. Journal of theoretical biology, 1971, 30.2: 225-234.
