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.

Characterization: K_mLux

Data

For the Quorum sensing module we used established experimentally determined parameters for the rate of formation of RLux (reference). Since, in the literature the other parameters were estimated or fitted to their data, we decided to determine the parameters specific to our system. Hence, we used our data for the remaining parameters. Our data was mainly a transfer function of normalized GFP concentration as a function of input Lux-AHL concentrations. (link to data)

Assumptions

From literature, we found that DRLux is specific to DNA and the dissociation constant is low (k_m = 0.1nM) {Reference}. Therefore, we used quasi steady state approximation (QSSA) for the rate of formation of P_LuxON. Further, we assumed that the dimerization of LuxR is quick and hence assumed QSSA for this equation as well.

Parameter fitting

Range of validity of the assumptions

Characterization

Data

Assumptions

Parameter fitting

Range of validity of the assumptions