Results: Modelling

Experimentation

We realized different experiments in order to measure how AHL diffuses into thez agar medium. We used a mutant strain Chromobacterium violaceum that cannot produce AHL. However, in the presence of AHL, the mutated strain starts producing a purple pigment: violacein.

Figure 1 shows a petri dish 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 experiment, we can study the diffusion of AHL. We also can see that colonies are disposed in spiral. This geometric disposition allows us to avoid the problem of utilisation of AHL of the n-1 colony to the n colony.

Figure 2 represents the evolution of AHL diffusion versus time.

Figure 2: Evolution of AHL diffusion into petri dish.

With this experiment, we can represent the surface of AHL diffusion on figure 3(surface is a disk) versus time in order to find the diffusion coefficient of AHL into LB agar. The regression coefficient of the straight line gives us an order of magnitude for the diffusion coefficient at 10^-8 m²/s. This coefficient is an important value to develop analytical and numerical modelling.

Figure 3: Determination of diffusion coefficient of AHL into LB agar medium.

Analytical Model

After the creation of our first model, which allowed us to find some relevant answers for our system, we thought that in terms of modelling we were not really satisfied. That is why we developed a new model based on equations of continuity in cylindrical coordinates. The general equation in our case is the following: Equation of continuity:

Bird, R. B., Stewart, W. E., & Lightfoot, E. N. (1960). Transport phenomena (Vol. 2). New York: Wiley.

In our case we have:
No reaction
No evolution in z and ? directions
dwA/dr=0 because of the mass equation (D?/Dt=0) with ?=cste After simplification:

After simplification and changing the mass in concentration we obtained:

To solve this equation we first thought to use the separation of variables method but found a new solution which fits better with our system. This solution corresponds 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)->8, 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 surface is really small. This is our analytical solution:

In this case as we consider our system as a dirac impulsion, we cannot focus on the concentration values (because C(0;0)-> 8). We are thus working with normalized concentration values. The interest of this model is the evolution of the concentration over time and space and also the determination of the diffusion coefficient D. We tested different diffusivity coefficients to find the one which allows the model with the best fit to our experimenations . We found that the coefficient of AHL diffusivity into LBagar medium is around: D = 1.10-8 m²/s On this first chart, we can see the evolution of the concentration for a given distance from the petri dish center. Close to the center the concentration increases really fast and goes down more slowly.

Figure 4: Evolution of concentration in time for analytic solution.

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

Figure 5: Evolution of concentration with radius for analytic solution.

This model fits well in terms 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 longer times the analytical model does not correspond 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 parameters of our petri dish and the AHL diffusivity. On this first animation, we can see 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 equivalent to the experimental boundary conditions: C(0,0)=cste C(r=R,t->8)= 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 reaches 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.