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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

	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.
	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

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. 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.

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. This model is stastistically correct because of the random definition of the particle movement.

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.

How do we calculate the real time ?

A row correspond to the movement from one site to an other, so in reality it take

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

Results

results extracted from java output for 100 molecules and 10 row. the average time for one molecule is calculated from 10 molecule passing fromm cell1 to cell2

Molecules name	T7	tRNA	Insulin	GFP	Glucose
number of row	37870.95	30980.83	60813.87	65580.56
realTime(s)	0.021087233522727282	0.4520549124513621	0.001689274166666667	0.014573457777777778
result file	T7	tRNA	Insulin	GFP	Glucose

Conclusion

Actually, because of the really short difference between passive end active diffusion, we can' say what is the nanotube type of fusion. the difference between those two model can be interpreted as noise in simulations.. Maybe a more precise method would provide better distinction between those two hypothesis.