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 strai can then only react to the externalm 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 problems:

Figure 1: Diffusion of AHL through colonies.

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.

Figure 2: Bacterial full adder system in wells.

Inoculation of cultures

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

But is this a problem as dead cells would not respond either to the lights?????

The ideal distance between well

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 takes place to pass from one colony to another.

Modelling steps

Diffusion reminder

Figure 3: Evolution of AHL concentration versus distance.

Figure 3 shows that the largest amount of AHL is between 2.5 and 20 mm. This information is important but we need to develop a model based on time.

Experimentation

We realized many experiences in order to measure how AHL diffuse into agar medium. We used a particular bacteria, Chromobacterium violaceum, which is capable to detect presence of AHL by producing a purple pigment, the violacein.

Figure 4 shows petri dishes containing eight colonies of bacteria placed at 5, 10, 15, 20, 25, 30, 35 and 40 mm from the center of the box. With this experience we can track the diffusion of the AHL. Also we can see that the colonies are in a spiral disposition. This typical disposition allow us to avoid the problem of utilisation of AHL of the n-1 colony to the n colony.

Figure 4: Photographs at 25h of petri dishes containing each 8 Chromobacterium violaceum colonies from 5 mm to 40 mm.

Figure 5 represent the evolution of AHL diffusion versus time.

Figure 5: Evolution of AHL diffusion into petri dish.

Analytical Model

After the creation of our first model above which allowed us to find relevant answer for our system, we thought that it term of modeling we weren’t really satisfied. That is why we developed a new model based on equation of continuity in cylindrical coordinates. The general equation in our case is the following. Equation of continuity:

Numerical Model