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Brownian diffusion model

Introduction

In the original paper, the authors claim that the diffusion that is happening through the nanotube is an active process, because the obderved diffusion goes too fast regarding the speed that could be achieved with only passive brownian motion, they said. This statement intrigated us much, and we wanted to investigate further. Indeed, if the diffusion is active, it means that there is some specificity of the transporter for the transported object, and the transported coponent that had been observed so far have noting to do with the regular coponent of the cell.

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 coponent 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 wright 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 sophiticated models.

About the brownian 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 :

• Brownian movment 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

====================================== ====================================== Characteristic size of particles : -------------------------------------- Glucose : 1.0E-9 T7 : 1.0500000000000001E-8 tRNA : 1.5000000000000002E-8 Insulin : 5.0E-9 GFP : 6.000000000000001E-9 ====================================== Diffusion coefficient of particles : -------------------------------------- Glucose : 6.0E-10 T7 : 3.3E-11 tRNA : 2.5699999999999998E-12 Insulin : 1.5E-10 GFP : 2.7000000000000004E-11 ====================================== ====================================== timeout set to : 100000000 you choose : Glucose div : 9.999999999999999E8 part_sz : 1.0E-9 cellsz : 1.0E-18 mult1 : 1.0000000000000002E-27 result : 1000 cell id :0 div : 9.999999999999999E8 part_sz : 1.0E-9 cellsz : 1.0E-18 mult1 : 1.0000000000000002E-27 result : 1000 cell id :1 nanotube id :2 cells ok : cell1 : 0 1000 1000 1000 0 0 0 0 cells ok : cell2 : 1 1000 1000 1000 0 0 0 0 cells ok : nanotube : 2 200 100 100 200 -1 -1 -1 0 Connection... cells ok : cell1 : 0 1000 1000 1000 1000 999 500 500 100 cells ok : cell2 : 1 1000 1000 1000 1000 0 500 500 100 cells ok : nanotube : 2 200 100 100 200 -1 -1 -1 0 ====================================== ====================================== simulation 0 row : 0 time : 742140 row : 0 Realtime : 2.0615000000000005E-4 -------------------------------------- simulation 1 row : 1 time : 437512 row : 1 Realtime : 1.215312777777778E-4 -------------------------------------- simulation 2 row : 2 time : 726945 row : 2 Realtime : 2.0192927777777783E-4 -------------------------------------- simulation 3 row : 3 time : 1012357 row : 3 Realtime : 2.812104444444445E-4 -------------------------------------- simulation 4 row : 4 time : 329636 row : 4 Realtime : 9.156569444444446E-5 -------------------------------------- simulation 5 row : 5 time : 468676 row : 5 Realtime : 1.301878055555556E-4 -------------------------------------- simulation 6 row : 6 time : 626588 row : 6 Realtime : 1.7405227777777778E-4 -------------------------------------- simulation 7 row : 7 time : 376689 row : 7 Realtime : 1.0463602777777779E-4 -------------------------------------- simulation 8 row : 8 time : 486871 row : 8 Realtime : 1.3524216666666668E-4 -------------------------------------- simulation 9 row : 9 time : 738944 row : 9 Realtime : 2.0526238888888892E-4 -------------------------------------- ====================================== ====================================== average time for 10 row : 594636.25 -------------------------------------- average Realtime for 10 row : 1.6517673611111114E-4 ====================================== ======================================

Conclusion