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Passive diffusion model

Introduction

Ours fisrt 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 shperical 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 stochastical 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 emittor cell, diffuse though the tube and meet a target in the receiver cell.

Model description

	First we design a simple représentation of the model with 2 boxes for cells and a tube between. 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 mondelize 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 witch 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 modelisation 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 acount of :

• Movement of particle : each particle have a random movment at each row.
• colision with cell membrane, nanotube and self colision. We have 2 model of colision :
  • Random restart model
  • Wait and see model

Random restart

In this colision model, when a particle colide with an other object it is reset to a random position without other particle on. This model is statisticaly correct because of the random definition of the particle movment.

Wait and see

In this colision model, when a particle colide with an other object it stay in place, waiting the next row, so the molecule lost one movment, but it stay a credible modelisation because of the random definition of the movment. This model have some problem due to the time lost by doing nothig with the particle. This model is actually only theoric, the java program taking more than 2 houre to calculate the movment.

How do we calculate the real time ?

A row correspond to the movment frome one site to an other, so in reality it take

We optain the realtime of diffusion of 10 molecules throw the nanotube with :

finaly 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.

Results

results extracted from java output

Conclusion

Actualy, because of the realy short difference between passive end acrive difus, we can' say what is th nanotube type of usion. 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 hypotesis.

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