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.

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

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 condition. In fact, we can establish a mass balance on AHL over a thickness of Δr :

In fact, we can establish a mass balance on AHL over a thickness of Δr :

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

Figure 3: Evolution of AHL concentration versus distance.

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

With this experience we can represent on figure 6 the surface of AHL diffusion (surface is a disk) versus time in order to find the coefficient of diffusion of AHL into LBagar. The directing coefficient of the regression line give us the order of magnitude of coefficient of diffusion, we find an order of 10-8 m²/s. This coefficient is an important value to develop analytical and numerical modelling.

Figure 6: Determination of coefficient of diffusion of AHL into LBagar medium.

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:

In our case we have: No reaction No evolution in z and θ direction dwA/dr=0 because of the mass equation (Dρ/Dt=0) with ρ=cste

After simplification

After simplification and changing the mass in concentration we obtained:

To solve this equation we first thought to use the variables separation but we faced different issues, but we found after some more research a new solution which fit better with our system. This solution is corresponding to a Dirac impulsion. This is not the perfect solution but this is the closest analytical solution to our system. The only difference with our system is that the boundary condition for C(0;0), is not exactly the same. For the Dirac impulsion C(0;0)->∞, but this approximation can be acceptable for our system considering that our AHL concentration is high at the beginning of the experiment and the loading is really small.This is our analytical solution:

In this case as we consider our system as a dirac impulsion system, we cannot be focused on the concentration value (because C(0;0)-> ∞). That is why here, we are working with normalized concentration value. The point of interest with this model is the evolution of the concentration in time and space and also the determination of the diffusion coefficient D. We tested different diffusivity coefficient to find which one allow the model to fit better with the experimentation. We found that coefficient of diffusivity of AHL into LBagar medium is around: D = 1.10-8 m²/s On this 1st chart, we can see for a given distance from the petri dish center the evolution of the concentration. Close to the center the concentration increase really fast and goes down more slowly. More the radius is important more the concentration increase is weak.

On this second chart we can see at different time the concentration profile along the petri dish. This graph is relevant to compare at different time how the concentration profile is. For instance at 70min the concentration is still high at r=0 but AHL reached 25 mm, whereas at 20min at r=0 the concentration is 2 times higher but the AHL only reached 10mm.

This model fit well in term of evolution with our experimentation. Thanks to this model we can approximately predict the diffusion of AHL into the LB agar. But we have to be careful here because we don’t have the real concentration values and for high time the analytical model isn’t corresponding to the reality.

Numerical Model

The last step of our modeling was to solve on a numerical way our diffusion problem. The aim was to solve the same differential equation presented in the analytical solution part. But here as we solved it with a computer we could use the right boundary condition, and also added a reaction term. Therefore we developed a numerical solution of AHL diffusion into LB agar medium with the software Comsol. We just needed to enter all the physical parameter of our petri dish and the AHL diffusivity. We can see on this first animation the diffusion of AHL into a petri dish with colonies but without reaction term. We note that the concentration of AHL becomes progressively homogeneous in the medium. This model is closest to the reality than the analytical one because here the boundary conditions are the experimental boundary condition: C(0,0)=cste C(r=R,t->∞)= cste

On these second and third animations we tried to simulate what could be our final system with AHL diffusion from a well 1 to other wells. These following two animations are complementary. The first shows the diffusion of the AHL from well 1, the second allows us to visualize the reach of the AHL in wells 2 and 3.

We can note that the AHL easily reached the well 2, but also 3 after some time. The passage of the AHL from a well n to well n+1 without reaching the point n+2 is really a key point. It is certainly difficult to implement such a system. In addition, it is imperative that the production of AHL is not continuous, because the concentration in the petri reach all the wells. We need a short pulse in time for transferring the AHL only in well 2. It also requires that the AHL is consumed in well 2 to limit the diffusion in well 3.