The goal of this part was only to check the speed of the
diffusion. We use the same model as the 2011 IGEM team of Grenoble, which is based on the Bangalore 2007 IGEM team model. We first have to add the diffusion terms in the equations thus we get:
Because of the temporal and special derivatives, we could not use a classic matlab solver to solve this set of equations. The approximation we used consisted in dividing the space into a grid:
It is the finite difference method. We thus get:
with h=lx/N, where lx is the length of the grid and N is the number of points of discretization along x. By using the same approximation on yi, and assuming that ly=lx, and that we have the same number of points of discretization, we get:
In addition, we also use the finite difference method on the time scale. Thus, we get:
where p=∆t/Nt where ∆t is the interval of time we work in and Nt the number of points of our time discretization.
With these approximations, we could solve the equations. Here are the scripts we used for the simulations.
First visual simulation
Here we give a video, just in the goal to illustrate the diffusion, you can find the scripts here. We model the diffusion in a petri dish. We put a drop of cAMP in the middle of the dish, and we can check how it diffuses through the bacteria. What we can see on the video is the evolution of adenylate cyclase. We give the evolution of the system during 280 minutes. Here is the colors scale, representing the intensity of the signal, and the link to the video:
Here we give the temporal evolution of Ca at the drop of cAMP.
Then we give the temporal evolution of Ca at 0,5 cm of the drop of cAMP:
We notice that it takes 100 minutes more than at the drop.
Conclusion:
If we use a test tube to make the detection, as soon as one bacterium is turned on (as we can get a signal), we will have to wait 100 minutes to have the whole test tube green.
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First visual simulation
Quorum sensing video 
Temporal evolution
