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Team:ETH Zurich/modeling/whole
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S_{LasI}+LasI & \rightarrow LasAHL\\ | S_{LasI}+LasI & \rightarrow LasAHL\\ | ||
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Revision as of 23:49, 12 October 2014
Whole cell model
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.
Chemical Species
| Name | Description |
|---|---|
| LuxAHL | 30C6-HSL is an acyl homoserine lactone which mainly binds to LuxR. |
| 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. |
| mRNABxb1 | 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 mainly binds to LasR. |
| 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. |
| mRNAΦC31 | 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. |
| SIBxb1 | Inactive DNA binding site for Bxb1. No dimer is bound to this site. |
| SABxb1 | Active DNA binding site for Bxb1. A dimer is bound to this site. |
| SFBxb1 | Flipped DNA binding site for Bxb1. The site generated after recombination by integrase. |
| SIΦC31 | Inactive DNA binding site for ΦC31. No dimer is bound to this site. |
| SAΦC31 | Active DNA binding site for ΦC31. A dimer is bound to this site. |
| SFΦC31 | Flipped DNA binding site for ΦC31. The site generated after recombination by integrase. |
| Ton,i | The number of terminators which are blocking the transcription of GFP and LuxI/LasI initially. |
| ToffBxb1 | The number of terminators turned off by recombination due to Bxb1. Favours the transcription of GFP and LuxI. |
| ToffΦC31 | The number of terminators turned off by recombination due to ΦC31. Favours the transcription of GFP and LuxI. |
| Ton,f | The number of terminators blocking the transcription of GFP and LuxI/LasI after recombination by Bxb1 and ΦC31. No transcription. |
| mRNAGFP | mRNA for Green fluorescent protein which is produced when the cells are ON. |
| GFP | Green fluorescent protein which is produced when the cells are ON. |
| mRNALuxI | mRNA for LuxI which is produced when the cells are ON. |
| LuxI | Enzyme catalysing the production of LuxAHL from SAM and ACP. |
| mRNALasI | mRNA for LasI which is produced when the cell are ON. |
| LasI | Enzyme catalysing the production of LasAHL from SAM and ACP. |
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
