Our goal with this model was to analyse our system in detail so that we learn more about its spacial features. Namely we wanted to see if co-localised cells will be able to distinguish each other from far away cells through quorum sensing and if so under what conditions. This was essential to our design as initially we envisioned that cells will communicate and create an AND-gate type of response in presence of according stimuli.

Since Quorum Sensing(QS) occurs due to diffusion of molecules in the medium, we naturally decided to employ the diffusion equation.

Diffusion Equation (more precisely Fick's 2nd Law)¹

$$\frac{\partial{C}}{\partial{t}}=D\nabla^2{C}$$ $$C - concentration$$ $$D - diffusion\:constant$$

However, in our system there are dedicated Sender and Receiver cells, which is different from natural QS. Sender cells can only produce and diffuse AHLs into the medium without the ability to react to it and Receiver cells can only react to AHLs without the ability to produce it. This arrangement made our system particularly interesting as we will discuss further down.

• Initial Assumptions

  • Constant Production of AHL from Sender cells
  • Receivers react to the AHL concentration around them
  • E. Coli cell-shape symmetry is taken into account

Initially we wanted to solve the equation in a particular case, so that we gain some intuition for the situation. Thus, in a big enough medium after some time T we can say that

$$\frac{\partial{C}}{\partial{t}}=0 \implies D\nabla^2{C}=0$$

Finally, assuming spherical coordinates we get: $$C(R)=\frac{K}{R}$$ for some constant K.

Now if we look at the Gaussian around a Sender cell, we find: $$\phi=-D\nabla{C}$$ which directly leads to a solution of the form $$C(R)=\frac{Q}{4\pi{}DR}$$ where Q is the production of AHLs by the Sender per unit time.

Knowing this we were able to simulate Senders and look at the resulting AHL concentration in the medium

Fig.1: Concentration potential around single sender (top view)

Fig.2: Concentration potential around single sender (side view)

Fig.3: Concentration potential in medium with many senders

From Fig.3 you can see that the overall "background" concentration in the medium is very uniform regardless of cell distribution. This quick observation made us investigate further and calculate precisely if Receiver cells will be able to tell the difference between local high concentration and simply a lot of Senders far away in the medium.

• Consider the following situation:
  • Medium of 1 ml volume
  • A spherical shell of radius 150 μm (roughly 100 cell radii) of cells concentrated in the middle of that volume
  • Assume the concentrated shell has 3 orders of magnitude higher concentration of cells per volume
  $$\frac{V_{shell}}{V_{volume}}=10^{3}$$

Evaluating the difference in contribution to the overall background of AHLs due to the shell and the surrounding volume of cells gives surprising results. $$ C(V_{volume}-V_{shell})=\iiint_{V_{diff}} \ \frac{q_{i}}{4\pi{}Dr_{i}}\,dV $$ If we compare this difference to the total concentration, we see that its contribution is insignificant (~6 orders of magnitude smaller). This realisation played a crucial role in the following adjustment of our assay design.

It was obvious at this point that our diagnostic method cannot rely on co-localization of cells through QS. This led to the adjustment of isolating the binding of Sender and Receiver cells to their respective epitopes into two separate phases and then recombining them for QS response. Essentially, QS will only occur if cells have previously bound to an epitope so that the can be transferred to the final "tube". If they have not bound anything they will be excluded from the final culture.

Moreover, further analysis revealed that if cells are bound to a surface through surface-mounted epitopes, QS response will be achieved earlier and will be stronger. This is due to the spacial symmetry as the "flux" of AHLs can thus only diffuse in a semi-sphere.

This was further confirmed experimentally by comparing QS cells bound by Poly-L-Lysine on a plate to QS cells free in medium.

Fig.4 GFP fluorescence of surface bound cells (triangle) and freely suspended cells (circle).

Overall, immobilised T9002/K1090000 pairs exhibited a higher rate of GFP response, and higher absolute GFP response after 7hours than pairs free in suspension(Fig.4).

References

[1] Jean Philibert, One and a Half Century of Diffusion: Fick, Einstein, before and beyond, Diffusion Fundamentals 2, 2005 1.1–1.10

[2] Ward, J.P., J.R. King and A.J. Koerber. “Mathematical modelling of quorum sensing in bacteria.” IMA Journal of Mathematics Applied in Medicine and Biology 2001: 18, 263-292