# Team:INSA Toulouse/contenu/project/modelling

### From 2013.igem.org

# Modelling

The full adder was tailored taking into account the diffusion of the carry from the bit n to the bit n+1. Evidently, the molecule should reach the n+1 colony prior the calculation step! Therefore, we have modeled the diffusion of AHL through the agar plate. The model would then help us determine the spacing between the different *E. calculus* colonies and the time necessary between two bits counting.

For the modelling, we used a strain of Chromobacterium violaceum deleted in the gene producing AHL. This strain can then only react to the external presence of AHL, coloring nicely with violacein, a violet pigment.

*N*-acetyl Homoserine lactone diffusion in agar medium

*N*-acyl Homoserine Lactone (AHL), 3-oxohexanoyl-homoserine lactone was chosen as the biological messenger in our system.
Let's imagine a simple system. A petri dish containing colonies equidistant from each other. The lights provide the information for the addition to perform, and the expression and diffusion of AHL from one colony to another allows the carry propagation.

This system seems quite simple but nevertheless raises a certain number of questions:

- Can a colony produce enough AHL to induce a response on the n+1 colony?
- Do the colonies have to be inoculated all at once or progressively during the calculation step?
- What is the ideal distance between the colonies? How can we avoid excessive AHL diffusion that would reach the colony n+1 but also n+2, n+3 etc.

### Production of AHL

To overcome the problem of the amount of AHL required for a rapid diffusion of the messenger, we also imagined a system in which liquid precultures may be deposited. A higher cell density would be obtained as well as a greater production of AHL.

### Inoculation of cultures

A gradual inoculation of wells during the addition process avoids the direct interference between the wells. Furthermore, progressive inoculation would lower the problem of cells dying on the plate that could not respond anymore to the AHL messenger.

### The ideal distance between wells

In order to find the ideal distance between two colonies, we searched a model that would calculate how does AHL diffuse into the medium and how long does the diffusion process take place to pass from one colony to another.

### Diffusion reminder

The graph above represents cylinder containing bacterias. Bacterias can produce AHL to send a message to another well. Here we can imagine that AHL diffuse into the medium. In order to introduce the theory of diffusion we can realize a simple model, with stationary state conditions.

In fact, we can establish a mass balance on AHL over a thickness of delta(r) :

Here we can introduce the Fick’s law for diffusion.

With this equation we can establish the concentration profile (figure 3) of the diffusion of AHL. This equation is only valid in our case of well geometry. However, this model does not depend on time, that’s why we must try to develop a more complex system in order to model the diffusion of AHL into LBagar medium with time.

# What modelling has tought us.

If you are not afraid of maths, you can go to the Modelling results
Otherwise here is a brief summary of our main findings:

- the best model for diffusion is a system where the supply of AHL is not infinite and basically that will be the case in *E. calculus*. Two elements supports this theory : i) production of AHL is linked to growing. When cells growing will stop, production of AHL will stop too; ii )production of AHL is also linked to presence of the general inducer in the medium (which is not permanent)

- in order to avoid that AHL goes to n+1 and n+2 (and many other) colonies, AHL must be taken up and destroyed by the n+1 colony. That should also be the case as the n+1 colony will pump up AHL and destroy it.

So basically, the design would be OK provided that we could have extracted the right parameters for AHL diffusion starting from an *E. calculus* strain instead of pure AHL.

Now that you have had the conclusions, you can go to the modelling results, there are very nice animated pictures at the end!!!