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

• Lux-AHL: 30C6HSL is an acyl homoserine lactone which mainly binds to LuxR.

• LuxR: constitutively expressed regulator protein that can bind Lux-AHL and stimulate transcription of Bxb1.

• RLux: LuxR and Lux-AHL complex which can dimerise.

• DRLux: Dimerised 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.

• Las-AHL: 30C12HSL is an acyl homoserine lactone which mainly binds to LasR.

• LasR: constitutively expressed regulator protein that can bind Las-AHL and stimulate transcription of ΦC31.

• RLas: LasR and Las-AHL complex which can dimerise.

• DRLas: Dimerised 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

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

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.

$$\frac{d[Lux-AHL]}{dt} = k_{-RLux}[R_{Lux}]-k_{RLux}[Lux-AHL][LuxR]-d_{Lux-AHL}[Lux-AHL]\\$$ $$\frac{d[LuxR]}{dt} = \alpha_{LuxR} -k_{RLux}[Lux-AHL][LuxR] + k_{-RLux}[RLux] - d_{LuxR}[LuxR] \\$$ $$\frac{d[RLux]}{dt} = k_{RLux}[Lux-AHL][LuxR] - k_{-RLux}[RLux] - 2 k_{DRLux} [RLux]^2 + 2 k_{-DRLux} [DRLux] - d_{RLux} [RLux] \\ $$ $$\frac{d[DRLux]}{dt} = k_{DRLux} [RLux]^2 - k_{-DRLux} [DRLux] - d_{DRLux} [DRLux] \\$$

Characterization

Data

Assumptions

Parameter fitting

Range of validity of the assumptions

Characterization

Data

Assumptions

Parameter fitting

Range of validity of the assumptions