## Model

The Quorum sensing module is mainly involved in receiving signals from the sender cells. The sender cells produce some signaling molecules (inducers) which diffuse out of their membrane, then diffuse in receiver cells membrane, and bind to the regulator molecules in the receiver cells, thus activating the transcription of certain genes. In order to characterize the quorum sensing module with a transfer function, we consider different initial inputs of external AHL, and see how much output is produced, as it was done in the quorum sensing experiments.

As diffusion through the membrane is very fast^{[27]}, according to Fick's law of diffusion, internal and external concentration of AHL can always be considered as equal. This can also be observed in the diffusion model results. When an initial external AHL concentration is given, AHL diffuses into the cells very quickly (less than 20 seconds)^{[27]} until internal AHL concentration equals external concentration. Then as soon as some internal AHL is consumed in the cell, it is taken up again without affecting external concentration, because external volume is very high compared to internal volume. Therefore we can consider in this module that external AHL (which is equal to internal AHL) only degrades, with the rate of extracellular decay.

### Chemical Species

Name
| Description |
---|---|

LuxAHL
| 30C6-HSL is an acyl homoserine lactone which diffuses into the cell and mainly binds to LuxR. Here we consider internal concentrations of LuxAHL. |

LuxR
| Constitutively expressed regulator protein that can bind LuxAHL and stimulate transcription of Bxb1. |

RLux
| LuxR and LuxAHL complex which can dimerize. |

DRLux
| Dimerized form of RLux. |

mRNA
_{Bxb1} | mRNA of the Bxb1 integrase being transcribed by the Lux promoter. |

Bxb1
| Serine integrase that can fold into two conformations - Bxb1a and Bxb1b. We chose to use a common connotation for both conformations - Bxb1. |

LasAHL
| 30C12-HSL is an acyl homoserine lactone which diffuses into the cell and mainly binds to LasR. Here we consider internal concentrations of LasAHL. |

LasR
| Constitutively expressed regulator protein that can bind LasAHL and stimulate transcription of ΦC31. |

RLas
| LasR and LasAHL complex which can dimerize. |

DRLas
| Dimerized form of RLas. |

mRNA
_{ΦC31} | mRNA of the ΦC31 integrase being transcribed by the Lux promoter. |

ΦC31
| Serine integrase that can fold into two conformations - ΦC31a and ΦC31b. We chose to use a common connotation for both conformations - ΦC31. |

### Reactions

For the Lux system: $$ \begin{align} &\rightarrow LuxR \\ LuxAHL+LuxR & \leftrightarrow RLux\\ RLux+RLux &\leftrightarrow DRLux\\ DRLux+P_{luxOFF} & \leftrightarrow P_{luxON}\\ P_{luxON}&\rightarrow P_{luxON}+mRNA_{Bxb1}\\ mRNA_{Bxb1}&\rightarrow Bxb1\\ LuxAHL &\rightarrow \\ LuxR &\rightarrow \\ RLux &\rightarrow\\ DRLux &\rightarrow\\ mRNA_{Bxb1} &\rightarrow\\ Bxb1 &\rightarrow \end{align}$$

- For the Las system

\begin{align} &\rightarrow LasR \\ LasAHL+LasR & \leftrightarrow RLas \\ RLas+RLas & \leftrightarrow DRLas\\ DRLas+P_{LasOFF} & \leftrightarrow P_{LasON}\\ P_{LasON}&\rightarrow P_{LasON}+mRNA_{\phi C31}\\ mRNA_{\phi C31}&\rightarrow \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. $$\begin{align*} \frac{d[LuxAHL]}{dt} &= -d_{LuxAHL}[LuxAHL]\\ \frac{d[LuxR]}{dt} &= \alpha_{LuxR} -k_{RLux}[LuxAHL][LuxR] + k_{-RLux}[RLux] - d_{LuxR}[LuxR] \\ \frac{d[RLux]}{dt} &= k_{RLux}[LuxAHL][LuxR] - k_{-RLux}[RLux] - 2 k_{DRLux} [RLux]^2 + 2 k_{-DRLux} [DRLux] - d_{RLux} [RLux] \\ \frac{d[DRLux]}{dt} &= k_{DRLux} [RLux]^2 - k_{-DRLux} [DRLux] - d_{DRLux} [DRLux] \\ \frac{d[P_{LuxON}]}{dt} &= k_{P_{LuxON}} [P_{LuxOFF}][DRLux] - k_{-P_{LuxON}} [P_{LuxON}]\\ \frac{d[mRNA_{Bxb1}]}{dt} &= L_{P_{Lux}} + k_{mRNA_{Bxb1}} [P_{LuxON}] - d_{mRNA_{Bxb1}} [mRNA_{Bxb1}]\\ \frac{d[Bxb1]}{dt} &= k_{Bxb1} [mRNA_{Bxb1}] - d_{Bxb1}[Bxb1]\\ \end{align*}$$

The same holds true for the Las system.

**From the original set of reactions, we reduce the rate of production of mRNA _{Bxb1} to a Hill function of RLux instead of Mass action kinetics in terms of P_{LuxON} and P_{LuxOFF}. For more information please check the characterization section.**