Model

The whole cell model is the combination of the Quorum sensing, Integrase and XOR modules. The model shows the behaviour of the a single cell in response to incoming signals. The model enables us to understand the effect of leakiness, cross-talk and their combinations on the whole system.

Reactions

$$ \begin{align} \emptyset&\rightarrow LuxR \\ LuxAHL+LuxR & \leftrightarrow RLux\\ RLux+RLux &\leftrightarrow DRLux\\ DRLux+P_{luxOFF} & \leftrightarrow P_{luxON}\\ P_{luxON}&\rightarrow P_{luxON}+mRNA_{Bxb1}\\ mRNA_{Bxb1}&\rightarrow Bxb1\\ Bxb1 + Bxb1 &\leftrightarrow DBxb1 \\ DBxb1 + SI_{Bxb1} & \leftrightarrow SA_{Bxb1}\\ LuxAHL &\rightarrow \emptyset\\ LuxR &\rightarrow \emptyset\\ RLux &\rightarrow \emptyset\\ DRLux &\rightarrow \emptyset\\ mRNA_{Bxb1} &\rightarrow \emptyset\\ Bxb1 &\rightarrow \emptyset\\ DBxb1 &\rightarrow \emptyset\\ \\ \emptyset&\rightarrow LasR \\ LasAHL+LasR & \leftrightarrow RLas \\ RLas+RLas & \leftrightarrow DRLas\\ DRLas+P_{LasOFF} & \leftrightarrow P_{LasON}\\ P_{LasON}&\rightarrow P_{LasON}+mRNA_{\phi C31}\\ mRNA_{\phi C31}&\rightarrow \phi C31\\ \phi C31 + \phi C31 &\leftrightarrow D\phi C 31 \\ D\phi C 31 + SI_{\phi C31} & \leftrightarrow SA_{\phi C31}\\ LasAHL &\rightarrow \emptyset\\ LasR &\rightarrow \emptyset\\ RLas &\rightarrow \emptyset\\ DRLas &\rightarrow \emptyset\\ mRNA_{\phi C31} &\rightarrow \emptyset\\ \phi C31 &\rightarrow \emptyset\\ D\phi C31 &\rightarrow \emptyset\\ \\ SA_{Bxb1}+SA_{Bxb1}+T_{on,i}& \rightarrow T_{offBxb1}+ SF_{Bxb1}+SF_{Bxb1}\\ T_{offBxb1} &\rightarrow T_{offBxb1} + mRNA_{GFP} + mRNA_{LasI} \\ SA_{\phi C31}+SA_{\phi C31}+T_{on,i}& \rightarrow T_{off\phi C31}+SF_{\phi C31}+SF_{\phi C31}\\ T_{off\phi C31} &\rightarrow T_{off\phi C31} + mRNA_{GFP} + mRNA_{LasI} \\ SA_{Bxb1}+SA_{Bxb1}+T_{off,\phi C31}& \rightarrow T_{on,f}+SF_{Bxb1}+SF_{Bxb1}\\ SA_{\phi C31}+SA_{\phi C31}+T_{offBxb1}& \rightarrow T_{on,f}+SF_{\phi C31}+SF_{\phi C31}\\ mRNA_{GFP} &\rightarrow GFP\\ mRNA_{LasI} &\rightarrow LasI\\ mRNA_{GFP} &\rightarrow \emptyset\\ mRNA_{LasI}&\rightarrow \emptyset\\ GFP &\rightarrow \emptyset\\ LasI&\rightarrow \emptyset\\ \\ S_{LasI}+LasI & \rightarrow LasAHL\\ \end{align}$$

Equations

The following equations were used for the whole cell model. The production of Bxb1 and C31 is reduced to a hill function. For details check Quorum sensing module.

The equation is for a system which produces LasI and GFP as outputs. The LasI catalyses the production of LasAHL from SAM and ACP. Since, we do not know the amount of ACP or SAM present in the cell, we assume the rate of production of LasAHL to depend only on LasI concentration.

$$\begin{align*} \frac{d[LuxAHL]}{dt} &= k_{-RLux}[R_{Lux}]-k_{RLux}[LuxAHL][LuxR]-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] - d_{RLux} [RLux] \\ \frac{d[mRNA_{Bxb1}]}{dt} &= L_{P_{Lux}} + \frac{K_{mRNA_{Bxb1}}[RLux]^2}{K_{mLux} + [RLux]^2 }- d_{mRNA_{Bxb1}} [mRNA_{Bxb1}]\\ \frac{d[Bxb1]}{dt} &= k_{Bxb1} [mRNA_{Bxb1}] -2 k_{DBxb1}[Bxb1]^2+ 2 k_{-DBxb1}[DBxb1] - d_{Bxb1}[Bxb1]\\ \frac{d[DBxb1]}{dt}&= k_{DBxb1}[Bxb1]^2-k_{-DBxb1}[DBxb1]-k_{SABxb1}[DBxb1][SI_{Bxb1}]+k_{-SABxb1}[SA_{Bxb1}]-d_{DBxb1}[DBxb1]\\ \frac{d[SI_{Bxb1}]}{dt}&=-k_{SABxb1}[DBxb1][SI_{Bxb1}]+k_{-SABxb1}[SA_{Bxb1}]\\ \frac{d[SA_{Bxb1}]}{dt}&=k_{SABxb1}[DBxb1][SI_{Bxb1}]-k_{-SABxb1}[SA_{Bxb1}] - 2 k_{ToffBxb1} [SA_{Bxb1}]^2 [T_{on,i}] - 2 k_{-Toff\phi C31} [SA_{Bxb1}]^2 [T_{off\phi C31}]\\ \frac{d[SF_{Bxb1}]}{dt}&= 2 k_{ToffBxb1} [SA_{Bxb1}]^2 [T_{on,i}] + 2 k_{-Toff\phi C31} [SA_{Bxb1}]^2 [T_{off\phi C31}]\\ \\ \frac{d[LasAHL]}{dt} &= k_{-RLas}[R_{Las}]-k_{RLas}[LasAHL][LasR]-d_{LasAHL}[LasAHL] + k_{LasAHL} [LasI]\\ \frac{d[LasR]}{dt} &= \alpha_{LasR} -k_{RLas}[LasAHL][LasR] + k_{-RLas}[RLas] - d_{LasR}[LasR] \\ \frac{d[RLas]}{dt} &= k_{RLas}[LasAHL][LasR] - k_{-RLas}[RLas] - d_{RLas} [RLas] \\ \frac{d[mRNA_{\phi c31}]}{dt} &= L_{P_{Las}} + \frac{K_{mRNA_{\phi C31}}[RLas]^2}{K_{mLas} + [RLas]^2 } - d_{mRNA_{\phi c31}} [mRNA_{\phi c31}]\\ \frac{d[\phi c31]}{dt} &= k_{\phi c31} [mRNA_{\phi c31}] -2 k_{D\phi c31}[\phi c31]^2+ 2 k_{-D\phi c31}[D\phi c31] - d_{\phi c31}[\phi c31]\\ \frac{d[D\phi c31]}{dt}&= k_{D\phi c31}[\phi c31]^2-k_{-D\phi c31}[D\phi c31]-k_{SA\phi c31}[D\phi c31][SI_{\phi c31}]+k_{-SA\phi c31}[SA_{\phi c31}]-d_{D\phi c31}[D\phi c31]\\ \frac{d[SI_{\phi c31}]}{dt}&=-k_{SA\phi c31}[D\phi c31][SI_{\phi c31}]+k_{-SA\phi c31}[SA_{\phi c31}]\\ \frac{d[SA_{\phi C31}]}{dt}&=k_{SA\phi C31}[D\phi C31][SI_{\phi C31}]-k_{-SA\phi C31}[SA_{\phi C31}] - 2 k_{Toff\phi C31} [SA_{\phi C31}]^2 [T_{on,i}] - 2 k_{-ToffBxb1} [SA_{\phi C31}]^2 [T_{offBxb1}]\\ \frac{d[SF_{\phi C31}]}{dt}&= 2 k_{Toff\phi C31} [SA_{\phi C31}]^2 [T_{on,i}] + 2 k_{-ToffBxb1} [SA_{\phi C31}]^2 [T_{offBxb1}]\\ \\ \frac{d[T_{on,i}]}{dt}&= -k_{ToffBxb1}[T_{on,i}][SA_{Bxb1}]^2 -k_{Toff\phi C31} [T_{on,i}][SA_{\phi C31}]^2 \\ \frac{d[T_{offBxb1}]}{dt}&=k_{ToffBxb1} [T_{on,i}][SA_{Bxb1}]^2 - k_{-ToffBxb1} [T_{offBxb1}] [SA_{\phi C31}]^2\\ \frac{d[T_{off\phi C31}]}{dt}&=k_{Toff\phi C31} [T_{on,i}][SA_{\phi C31}]^2 - k_{-Toff\phi C31} [T_{offBxb1}] [SA_{Bxb1}]^2\\ \frac{d[T_{on,f}]}{dt}&= k_{-ToffBxb1} [T_{offBxb1}] [SA_{\phi C31}]^2 + k_{-Toff\phi C31} [T_{offBxb1}] [SA_{Bxb1}]^2\\ \\ \frac{d[mRNA_{LasI}]}{dt}&= k_{mRNA_{LasI}} (\frac{[T_{offBxb1}]}{\theta + [T_{offBxb1}]} + \frac{[T_{off\phi C31}]}{\theta + [T_{off\phi C31}]}) - d_{mRNA_{LasI}}[mRNA_{LasI}]\\ \frac{d[LasI]}{dt}&= k_{LasI} [mRNA_{LasI}] - d_{LasI}[LasI]\\ \frac{d[mRNA_{GFP}]}{dt}&= k_{mRNA_{GFP}} (\frac{[T_{offBxb1}]}{\theta + [T_{offBxb1}]} + \frac{[T_{off\phi C31}]}{\theta + [T_{off\phi C31}]}) - d_{mRNA_{GFP}}[mRNA_{GFP}]\\ \frac{d[GFP]}{dt}&= k_{GFP} [mRNA_{GFP}] - d_{GFP}[GFP]\\ \end{align*}$$

Dynamics

Ideal case

With Leakiness

With Leakiness and Crosstalk