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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 adapte each cell to the molecule we want to modelise, so the molecul have a size of 1*1*1. so the matrix is divided in site of the size of a molecule.
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*4000 and the nanotube 100*100*600.
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.
We take acount of :

• Brownian movment of particle : each particle have a random movment at each row.
• colision : we have 2 model of colision :
  • if a molecule colide another object, it will reset to is start point.
  • if a molecule colide another object, it will stay at it's position for this row.
• colision with cell membrane, nanotube and self colision

for each type of molecule, we do 10 simulation and we take the average time of those 10.

How do we calculate the real time ?

A row correspond to the movment frome one site to an other, so in reality it take So we optain the realtime of diffusion ofr ten molecules throw the nanotube with : Rtime=*(number of row).
finaly we obtain th time for diffusion from a cell to an other with : Rtime/10.
this model is mapped in Maya for a user friendly aspect.

Results

Conclusion