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Team:ETH Zurich/modeling/whole
From 2014.igem.org
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 |
|---|---|
| Lux-AHL | 30C6-HSL 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 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. |
| Las-AHL | 30C12-HSL 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 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 Lux-AHL from SAM and ACP. |
| mRNALasI | mRNA for LasI which is produced when the cell are ON. |
| LasI | Enzyme catalysing the production of Las-AHL from SAM and ACP. |
Reactions
$$ \begin{align} &\rightarrow LuxR \\ Lux-AHL+LuxR & \leftrightarrow RLux\\ RLux+RLux &\leftrightarrow DRLux\\ DRLux+P_{luxOFF} & \leftrightarrow P_{luxON}\\ P_{luxON}&\rightarrow P_{luxON}+mRNA_{Bxb1}\\ mRNA_{Bxb1}&\rightarrow Bxb1\\ Bxb1 + Bxb1 &\leftrightarrow DBxb1 \\ DBxb1 + SI_{Bxb1} & \leftrightarrow SA_{Bxb1}\\ Lux-AHL &\rightarrow \\ LuxR &\rightarrow \\ RLux &\rightarrow\\ DRLux &\rightarrow\\ mRNA_{Bxb1} &\rightarrow\\ Bxb1 &\rightarrow\\ DBxb1 &\rightarrow\\ \\ &\rightarrow LasR \\ Las-AHL+LasR & \leftrightarrow RLas \\ RLas+RLas & \leftrightarrow DRLas\\ DRLas+P_{LasOFF} & \leftrightarrow P_{LasON}\\ P_{LasON}&\rightarrow P_{LasON}+mRNA_{\phi C31}\\ mRNA_{\phi C31}&\rightarrow \phi C31\\ \phi C31 + \phi C31 &\leftrightarrow D\phi C 31 \\ D\phi C 31 + SI_{\phi C31} & \leftrightarrow SA_{\phi C31}\\ Las-AHL &\rightarrow \\ LasR &\rightarrow \\ RLas &\rightarrow\\ DRLas &\rightarrow\\ mRNA_{\phi C31} &\rightarrow \\ \phi C31 &\rightarrow \\ D\phi C31 &\rightarrow \\ \\ SA_{Bxb1}+SA_{Bxb1}+T_{on,i}& \rightarrow T_{offBxb1}+ SF_{Bxb1}+SF_{Bxb1}\\ T_{offBxb1} &\rightarrow T_{offBxb1} + mRNA_{GFP} + mRNA_{LasI} \\ SA_{\phi C31}+SA_{\phi C31}+T_{on,i}& \rightarrow T_{off\phi C31}+SF_{\phi C31}+SF_{\phi C31}\\ T_{off\phi C31} &\rightarrow T_{off\phi C31} + mRNA_{GFP} + mRNA_{LasI} \\ SA_{Bxb1}+SA_{Bxb1}+T_{off,\phi C31}& \rightarrow T_{on,f}+SF_{Bxb1}+SF_{Bxb1}\\ SA_{\phi C31}+SA_{\phi C31}+T_{offBxb1}& \rightarrow T_{on,f}+SF_{\phi C31}+SF_{\phi C31}\\ mRNA_{GFP} &\rightarrow GFP\\ mRNA_{LasI} &\rightarrow LasI\\ mRNA_{GFP} &\rightarrow \\ mRNA_{LasI}&\rightarrow\\ GFP &\rightarrow \\ LasI&\rightarrow\\ \\ S_{LasI}+LasI & \rightarrow Las-AHL\\ \end{align}$$
Equations
Applying mass action kinetic laws, we obtain the following set of differential equations. $$\begin{align*} \frac{d[Lux-AHL]}{dt} &= k_{-RLux}[R_{Lux}]-k_{RLux}[Lux-AHL][LuxR]-d_{Lux-AHL}[Lux-AHL]\\ \frac{d[LuxR]}{dt} &= \alpha_{LuxR} -k_{RLux}[Lux-AHL][LuxR] + k_{-RLux}[RLux] - d_{LuxR}[LuxR] \\ \frac{d[RLux]}{dt} &= k_{RLux}[Lux-AHL][LuxR] - k_{-RLux}[RLux] - 2 k_{DRLux} [RLux]^2 + 2 k_{-DRLux} [DRLux] - d_{RLux} [RLux] \\ \frac{d[DRLux]}{dt} &= k_{DRLux} [RLux]^2 - k_{-DRLux} [DRLux] - d_{DRLux} [DRLux] \\ \frac{d[P_{LuxON}]}{dt} &= k_{P_{LuxON}} [P_{LuxOFF}][DRLux] - k_{-P_{LuxON}} [P_{LuxON}]\\ \frac{d[mRNA_{Bxb1}]}{dt} &= L_{P_{Lux}} + k_{mRNA_{Bxb1}} [P_{LuxON}] - d_{mRNA_{Bxb1}} [mRNA_{Bxb1}]\\ \frac{d[Bxb1]}{dt} &= k_{mRNA_{Bxb1}} [mRNA_{Bxb1}] - d_{Bxb1}[Bxb1]\\ \end{align*}$$
The same holds true for the Las system.
Dynamics
Ideal case
With Leakiness
With Leakiness and Crosstalk
