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Passive diffusion model

Summary

Introduction

While working on the assisted diffusion model, we quickly realized that it may not fully explain the GFP diffusion observed in the original paper. We therefore investigated passive diffusion.

We want to know if passive diffusion can, in theory, explain the transfer behaviour of the Dubey/Ben-Yehuda paper. We created a simulation to have an estimation of the order of magnitude of passive diffusion time through the nanotubes.

We used a model similar to the stochastic diffusion model presented in our assumptions. Our methodology and our results are presented below.

Model description

Dividing the cell

The core of the model is to divide the cell in occupation sites. Each of these occupation sites is roughly the size of the particle that we want to study. For instance if the particle has a length of 10nm, we will divide the cell in occupation sites of 10nm by 10 nm by 10 nm.

When we want to study one particle we therefore model the cell as a 3D matrix(M*M*M). Its size depends of the particle characteristic size. We use the following parameters:

• V volume of the cell (for B.subtilis 10^-18 m³)
• characteristic size of the particle (m)

We therefore divided our cell in occupation sites, giving us the size of the 3D matrix we used in out simulation:

Random walker movements

We now have the layout for modeling our cells and the nanotube connecting between them. How does a particle moves in such a matrix? We were able to evaluate the characteristic transition time from one occupation site to one of its direct neighbours. Using D, diffusion coefficient of the particle (m².s^-1), we have the characteristic time associated with the transition: [3]

The simulation is then simple. We put a certain number of those particles in an emitter cell connected to an empty receiver cell via a nanotube. At each time step (corresponding to the characteristic transition time of the particle), the particles move from the occupation site where they are to one of its neighbour. This is a synchronous model where all the particles move at the same time. We repeat this process and count the particles going through the nanotubes and arriving in the receiver cell.

We also had to take into account collisions between particles, with the cell membrane or with the nanotube membrane. We used two configurations for dealing with such problems.

Wait and see

In this collision model, when a particle collides with another object, it stays in its place and waits for the next time step. This model resulted in massive simulation problems. It took several hours to run our java programme for a few hundred time steps. This problem can be explained by the fact that particles kept colliding in the nanotube and prevented other molecules from entering the nanotube by colliding with them. We therefore had to do a lot of debugging before successfully implementing this idea.

Restart

In this collision model, when a particle collides with another object, it is reset to its starting position in the first cell. This model is stastistically correct because of the random definition of the particle movement. We can compare the "teleportation" of one particle to its degradation and the synthesis of a new one. This is the model we used in our simulation.

The behaviour of the particles regarding collisions is probably somewhere between these two assumptions. The restart for instance probably gives us a longer diffusion time.

Our simulation is particle-specific because it relies on parameters specific to a particle both for the spatial grid and the characteristic time.

"Ping-pong" particles

We first assumed that particles, once they enter the receiver cell, would not go back in the nanotubes. This proved to be false. Particles passing in the receiver cell exhibit a "ping-pong" behaviour. They tend to go back and forth from one cell to the tube entrance. This could account for up to half of the particles passing the entrance (or exit). We adapted our software so that the count would be correct even with those "ping-pong" particles.

Our java software to simulate this model is downloadable here. Each simulation is done with a certain base amount of molecules, (1000 in most of the next example). All the molecules of the simulation will start in one cell and move randomly until at least a certain amount decided by the user (50 in most of the example) molecules pass through the nanotube into the other cell. At that point our "counter" annonces that a significant number of molecules has passed the tube.

Results

We ran simulations changing three major parameters:

• Nanotube size
• Type of collision (see above)
• Nature of the particle

This last parameter allowed us to compare theoritical diffusion times for molecules of different sizes and natures. The characteristic sizes and diffusion coefficients are summed up below:

We ran a quick 2D simplifed version of our model to give you an idea of how it works (the timescale might not be coherent with results from our 3D model):

Our 3D simulation gave us the following results:

Conclusions

TO BE CHANGED LATER

Introduction

Our first calculations of the diffusion of molecules 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 speed of molecule transfer through nanotubes is compatible with passive diffusion hypothesis.

We try to propose several ideas about the origin of the molecule transfer, to know if it is faster than the diffusion or imply more sophisticated models (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).

About the passive diffusion model

Earlier, we did some math to calculate the speed of the diffusion inside a cell. But we were in spherical coordinates, that is to say a mathematically 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 molecules 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 formula, we obtain the model size M of our cell

For example : for glucose, if we want to have a size of 1*1*1 the matrix representing the cell needs 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 molecules in 10 rows. All the 100 molecules of the simulation will start in one cell and move randomly until at least 10 molecules pass through the nanotube to the other cell.

We use a synchronous model so that all 100 molecules move simultaneously. At each row (step of execution), all the molecules move randomly from their site to one of the 26 other possible positions.

this simulation takes into account :

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

andom restart

In this collision model, when a particle collides with another object, it is reset to a random position not occupied by another particle in the first cell. This model is stastistically correct because of the random definition of the particle movement. We can compare the teleportation of one particle to its degradation and synthesis of a new one.

Wait and see

In this collision model, when a particle collides with another object, it stays in place, waiting for the next row, so the molecule loses one movement, but it stays a credible model because of the random definition of the movement. This model has some problems due to the time lost by doing nothing with the particle. This model is actually only theoretic, the java program taking more than 2 hours to calculate the movement. This problem can be explained by the fact that particles keep colliding in the nanotube and stop other molecules which enter the nanotube by colliding with them.

How do we calculate the real time ?

A row corresponds to the movement from one site to another. The time it takes without boundary 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 simulations 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 representation is just for you to have an idea of diffusion and it doesn't use the diffusion equation used by the java program. In this representation, particles move linearly and are subjected to random turbulences.

Results

Results extracted from java have an output for 100 molecules and 10 rows. The average time for one molecule is calculated with the results 10 molecules passing from cell 1 to cell 2

Considering "ping-pong" particles as discret elements

Molecules name	T7	tRNA	Insulin	GFP	Glucose
number of row	37870.95	30980.83	60813.87	65580.56	1039227.4
realTime(s)	2.11E-2	4.52E-1	1.69E-3	1.46E-2	2.89E-4
result file	T7	tRNA	Insulin	GFP	Glucose

without counting "ping-pong" particles

Molecules name	T7	tRNA	Insulin	GFP	Glucose
number of row	80144.21	58873.76	267651.7	227912.07	TIMEOUT
realTime(s)	4.46E-2	8.59E-1	7.43E-3	5.06E-2	TIMOUT
result file	T7	tRNA	Insulin	GFP	Glucose

Conclusion

Looking to those results, it seems to be a linear function :

the k coefficient found is about 5000 which is really different from the 1/6 coefficient of

Because of the lack of experimental results, we can't compare those results to the real time calculated in vivo. So we can't know if it is passive or active transport.

This software can be extended for other molecules, but diffusion coefficent is an ambiguous data subject do debate in papers, so simulating for a molecule is subject to error due to different definitions from a paper to an other.

In a first implementation of the software, we decide that particles moving from cell1 to nanotube and coming back next time are occasional, and don't influence the model. It turned out to be false, and particle playing ping-pong represent more than 1/2 of all the molecules passing through the nanotube. We modified the first software, so now we don't take them into account.