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Team:Paris Bettencourt/Modeling/Diffusion
From 2011.igem.org
Passive diffusion model
Introduction
Ours first calculations of the diffusion of the molecule inside a cell shows that the diffusion inside a cell is a very fast process (see the bottom of this page). It takes from 10 seconds to one minute for a component of the cell to have the average movement in the order of magnitude of the size of the cell. We wanted to explore further about the speed the passive diffusion can be achieved for a molecule to pass with passive diffusion, to see if the author is right or not in his statement.
On the other hand, we try to propose several ideas about the origin of the motion if this motion is indeed faster than the diffusion (see toward assisted diffusion). We also propose some experiments to verify these models.
The designs had been devised so that we can try to measure the speed of the diffusion through the tubes (see the Design & concepts page). We aimed to show if the diffusion is indeed a passive diffusion, and the author is wrong, or, if we have to think about more sophisticated models.
About the passive diffusion model
Earlier, we did some maths to calculate the speed of the diffusion inside a cell. But we were in spherical coordinates, that is to say a mathematics friendly conditions.Here, we have to deal with a more complex geometry, so we had to be helped by a computer.
The essence of the model remains the same than the stochastic motion approach, but in this space we introduce boundary limits with the shape of two cells linked by a tube. We observe the passage of the molecule through the tube and we calculate the time taken for a molecule (like a transcription factor) to leave the emitter cell, diffuse though the tube and meet a target in the receiver cell.
Model description
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First we design a simple representation of the model with 2 boxes for cells and a tube between them. Each cell is designed like a 3D matrix(M*M*M) witch is proportional to the studied molecule. |
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We define the matrix size using our hypotesis about diffusion. So we create the cell matrix using this formula : 
We want to model each particle with a pixel, so each occupation site need to be 1px large. We calculate N, and with this other formula, we obtain the model size M of our cell 
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For example : for the glucose, if we want to have a size of 1*1*1 the matrix representing the cell need to be 1000*1000*1000 and the nanotube which is 1/10 large of the cell will be 100*100*200.
We define the size of cells for each molecule of the hypothesis array :
We develop a java software to simulate this model (downloadable here). each simulation is done on 100 molecule in 10 row. All the 100 molecules of the simulation will start in one cell and move randomly until at least 10 molecule pass throw the nanotube to the other cell.
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We use a synchronous model so all the 100 molecules move simultaneously. at each row (step of execution), all the molecule move randomly from there site to one of the 26 other possible position. |
this simulation take account of :
- Movement of particle : each particle have a random movement at each row.
- collision with cell membrane, nanotube and self collision. We have 2 model of collision :
- Random restart model
- Wait and see model
Random restart
In this collision model, when a particle collides with an other object it is reset to a random position without other particle on in the first cell. This model is stastistically correct because of the random definition of the particle movement. We can compare the teleport of one particle to the death of one and the synthesis to a new one.
Wait and see
In this collision model, when a particle collide with an other object it stay in place, waiting the next row, so the molecule lost one movement, but it stays a credible model because of the random definition of the movement. This model have some problem due to the time lost by doing nothing with the particle. This model is actually only theoric, the java program taking more than 2 hours to calculate the movement. this problem can be explain by the fact that particle stay coliding in the nanotube and stop other molecules witch enter the nanotube by coliding them too
How do we calculate the real time ?
A row correspond to the movement from one site to an other. the time it take without boudary limitation is 
[1]
We obtain the real time of diffusion of 10 molecules through the nanotube with :
finally we obtain the time for diffusion from a cell to an other with :
For more accuracy, we do 10 simulation for each type of molecule, and we take the average time of those 10.
Maya modeling
this model is mapped in Maya for a user friendly aspect. This graphic reprsentaton is just for an idea of diffusion and it doesn't use the diffusion equation used by the java program. In this representation, particles move lineary and are subjected to random turbulences.
Results
results extracted from java output for 100 molecules and 10 row. the average time for one molecule is calculated from 10 molecule passing from cell1 to cell2
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Conclusion
Looking to those result, it seems that speed diffusion is proportional to the square of particle size. those time are really short, so, we can't clearly define if it is active or passive diffusion. Because of the lake of experimental result, we can't compare those result to real time calculated in vitro. This software can be extended for other molecule, but diffusion coefficent and molecule size is a weird data subject do debat in papers, so simulating for a molecule is subject to error due to different definition from a paper to an other.
References
- Diffusion-based Channel Characterization in Molecular Nanonetworks. Llatser, I., Alarcón, E. and Pierobon, M., to appear in Proc. of the 1st IEEE International Workshop on Molecular and Nano Scale Communication (MoNaCom), held in conjunction with IEEE INFOCOM, Shanghai (China), April 2011
