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Passive diffusion model

Summary

Our passive diffusion model uses a random walker approach to create a simulation of a population of particles randomly moving from one cell to another via a nanotube. We ran this simulation for different size of nanotubes, different molecules, and two different simple collisions models. The results of those simulations shows that passive diffusion alone can give us a very efficient transfer in only a few seconds.

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 collision 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. That would not be a problem except our simulation counts particle that pass through th exit of the nanotube. This phenomenon 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). Here, we modelize diffusion for GFP with the "wait and see" collision model.

Our 3D simulation, the actual model, gave us the following results:

This first graph represents the evolution of diffusion speed for differents size of nanotubes. We ran this simulation for 3 size of molecules

Two other interesting informations are the comparison between the two models of collision used ("wait and see" and "restart"), and the average relative error, which helped us to choose the right number of molecules for simulating valid models.

Conclusions

What we obtain for passive diffusion is several orders of magnitude slower than for assisted diffusion, but it remains really fast compared to cell division. This means that we can not measure diffusion time directly through genetic network response. The time for significant GFP diffusion for instance is under a minute when response time for a genetic network is around an hour. We therefore need to focus on molecule sizes, natures and numbers in our experiments, not on diffusion time.

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 for the same parameter.

Data

You can find the full dataset for this page here:

• Full data
• Data for the error calculus
• Data for collision models comparison
• Data for nanotube size comparison