Model

The Quorum sensing module is mainly involved in receiving signals from the sender cells. The sender cells secrete some signalling molecules (inducers) which bind to the regulator molecules in the receiver cells, thus activating the transcription of certain genes. The model for this module is presented below.

Chemical Species

Reactions

For the Lux system

$$ \begin{align} &\rightarrow LuxR \\ Lux-AHL+LuxR & \leftrightarrow RLux\\ RLux+RLux &\leftrightarrow DRLux\\ DRLux+P_{luxOFF} & \leftrightarrow P_{luxON}\\ P_{luxON}&\rightarrow P_{luxON}+mRNA_{Bxb1}\\ mRNA_{Bxb1}&\rightarrow Bxb1\\ AHL &\rightarrow \\ LuxR &\rightarrow \\ RLux &\rightarrow\\ DRLux &\rightarrow\\ mRNA_{Bxb1} &\rightarrow\\ Bxb1 &\rightarrow \end{align}$$

For the Las system

\begin{align} &\rightarrow LasR \\ Las-AHL+LasR & \leftrightarrow RLas \\ RLas+RLas & \leftrightarrow DRLas\\ DRLas+P_{LasOFF} & \leftrightarrow P_{LasON}\\ P_{LasON}&\rightarrow P_{LasON}+mRNA_{\phi C31}\\ Las-AHL &\rightarrow \\ LasR &\rightarrow \\ RLas &\rightarrow\\ DRLas &\rightarrow\\ mRNA_{\phi C31} &\rightarrow \\ \phi C31 &\rightarrow \\ \end{align}

Differential Equations

Applying mass action kinetic laws, we obtain the following set of differential equations. $$\begin{align*} \frac{d[Lux-AHL]}{dt} &= k_{-RLux}[R_{Lux}]-k_{RLux}[Lux-AHL][LuxR]-d_{Lux-AHL}[Lux-AHL]\\ \frac{d[LuxR]}{dt} &= \alpha_{LuxR} -k_{RLux}[Lux-AHL][LuxR] + k_{-RLux}[RLux] - d_{LuxR}[LuxR] \\ \frac{d[RLux]}{dt} &= k_{RLux}[Lux-AHL][LuxR] - k_{-RLux}[RLux] - 2 k_{DRLux} [RLux]^2 + 2 k_{-DRLux} [DRLux] - d_{RLux} [RLux] \\ \frac{d[DRLux]}{dt} &= k_{DRLux} [RLux]^2 - k_{-DRLux} [DRLux] - d_{DRLux} [DRLux] \\ \frac{d[P_{LuxON}]}{dt} &= k_{P_{LuxON}} [P_{LuxOFF}][DRLux] - k_{-P_{LuxON}} [P_{LuxON}]\\ \frac{d[mRNA_{Bxb1}]}{dt} &= L_{P_{Lux}} + k_{P_{LuxON}} [P_{LuxOFF}][DRLux] - k_{-P_{LuxON}} [P_{LuxON}]\\ \frac{d[Bxb1]}{dt} &= k_{mRNA_{Bxb1}} [mRNA_{Bxb1}] - d_{Bxb1}[Bxb1]\\ \end{align*}$$

The same holds true for the Las system.