The idea behind this model was to ensure our Receiver->Sender design is viable and Quorum Sensing will trigger GFP production. It is a simple model that explores the production of GFP protein in the Receiver cell due to AHL-activated LuxR.

Since the Receivers do not produce AHL, the surrounding AHL concentration is not affected by the Receiver itself. Thus self-stimulation is out of the picture and we can assume the amount of AHL is solely due to the "background".

We use the following equations (Hill function) to model how GFP production is triggered^[1]:

$$\frac{d[AHL-LuxR]}{dt} = k_{+}[AHL][LuxR]-k_{-}[AHL-LuxR]$$ $$\frac{d[GFP]}{dt} = \frac{[AHL-LuxR]^n}{[AHL-LuxR]^n + a^n}V_{max} - \gamma{}[GFP] $$

• where

  [AHL]	- AHL concentration
  [LuxR]	- LuxR concentration
  [AHL-LuxR]	- AHL+LuxR complex concentration
  [GFP]	- GFP concentration
  Vmax	- maximum GFP production
  a	- dissociation constant
  n	- Hill coefficient
  γ	- GFP degradation constant

The simplicity of the system allows for a few more assumptions to be made. As the Receivers are getting their AHL from the background we assume to be constant, that means that [AHL-LuxR] will very quickly saturate as the background reaches QS activation. Thus we can conclude that:

$$ \frac{d[AHL-LuxR]}{dt}=0 $$

Following this line of thought, that means GFP will also be produced at a constant rate and will gradually build up in the medium as the cell culture grows.

$$ \frac{d[GFP]}{dt} = 0 \implies [GFP] = \frac{1}{\gamma{}} \frac{[AHL-LuxR]^{n}V_{max}}{[AHL-LuxR]^{n} + a^{n}}$$ $$ \gamma{} = \frac{\ln{2}}{1800s} $$

So our model predicts a steady GFP production as a response to QS and the curve that GFP production follows depends mostly on the efficiency of the promoter.

Fig.1 Range GFP response curves according to Vmax

References

[1] James, Sally et al. “Luminescence Control in the Marine Bacterium Vibro fischeri: An Analysis of the Dynamics of lux Regulation.” JMB 2000: 296, 1127-1137