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. All the simulations here are described for a red cell .i.e. a cell producing LasI and GFP when it is ON.

The four cases with different combinations of input for a red cell.

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.
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.
DBxb1	Dimerized form of Bxb1. We chose to use a common connotation for both homodimers, DBxb1a and DBxb1b.
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.
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.
DΦC31	Dimerized form of ΦC31. We chose to use a common connotation for both homodimers, DΦC31a and DΦC31b.
SI_Bxb1	Inactive DNA binding site for Bxb1. No dimer is bound to this site.
SA_Bxb1	Active DNA binding site for Bxb1. A dimer is bound to this site.
SF_Bxb1	Flipped DNA binding site for Bxb1. The site generated after recombination by integrase.
SI_ΦC31	Inactive DNA binding site for ΦC31. No dimer is bound to this site.
SA_ΦC31	Active DNA binding site for ΦC31. A dimer is bound to this site.
SF_ΦC31	Flipped DNA binding site for ΦC31. The site generated after recombination by integrase.
T_on,i	The number of terminators which are blocking the transcription of GFP and LuxI/LasI initially.
T_offBxb1	The number of terminators turned off by recombination due to Bxb1. Favours the transcription of GFP and LuxI.
T_offΦC31	The number of terminators turned off by recombination due to ΦC31. Favours the transcription of GFP and LuxI.
T_on,f	The number of terminators blocking the transcription of GFP and LuxI/LasI after recombination by Bxb1 and ΦC31. No transcription.
mRNA_GFP	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.
mRNA_LuxI	mRNA for LuxI which is produced when the cells are ON.
LuxI	Enzyme catalysing the production of LuxAHL from SAM and ACP.
mRNA_LasI	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 ΦC31 is reduced to a hill function of RLux an RLas respectively. 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}^2 + [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}^2 + [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

The dynamics of flipping of the terminators when the cell receives 0.5 nM LuxAHL and 0 nM of LasAHL and produces GFP and LasI

In the case above, the cell receives only 0.5 nM of LuxAHL and no LasAHL but the cell produces LasI upon flipping of the terminator. The LasI produced in turn catalyses the production of LasAHL. The LasAHL activates P_Las thus producing ΦC31. ΦC31 results in flipping of the terminator again thus, turning OFF the system. The dynamics of T_oni, T_offBxb1, T_offΦC31 and T_onf is summarised in the figure above.

The four cases for the whole cell under ideal conditions

Under ideal conditions, there is no leakiness or cross-talk. The figure shows the production of GFP as a function of four different combinations of inputs in an ideal whole cell. No GFP is produced when there is no LasAHL and LuxAHL or when both are present. GFP is produced only when one of the input AHLs are present, thus emulating an XOR system. In the second case, the rate of production of GFP reduces after three hours. This could be attributed to the delayed production of LasAHL from LasI.

Predicted XOR behaviour for the whole cell model without leakiness

The figure above, shows the predicted XOR gate performance by a single cell model with no basal leakiness for different concentrations of LasAHL and LuxAHL.

With Leakiness

No inputs and only basal leakiness

The figure above summarises the predicted effect of basal leakiness on the flipping of the terminator. The basal leakiness results in production of Bxb1 and ΦC31 which result in flipping of the terminator. In this case, since the cell produces LasI there is increased production of LasAHL. The LasAHL produced induces the production of ΦC31 which further causes flipping of all terminators flanked by the ΦC31 sites. Thus, by 200 minutes almost all ΦC31 sites are inactive and the cell will stay ON.

The four cases for the whole cell under ideal conditions

From our experimental data, we observed some basal leakiness for P_Lux and P_Las even after using riboregulators. From the model we see that, this small basal leakiness is amplified downstream. The basal leakiness results in production of integrases which further act on the XOR module and cause the switching of the terminator. Thus, there is GFP and LasI produced, and the LasI produced further catalyses the production of more LasAHL. Thus, we observe some GFP even without inputs.

However, if we measure the fluorescence at around 150 mins, we observe a good and acceptable XOR behaviour. Therefore, one of the solutions we propose is to kill or freeze the cells in each row after 3 hours.

Initially from our model we observed that the feedback was rapid and hence, the amplification was much higher. However, from literature ^[9] we see that the XOR module is relatively slow. We were able to correct this by modelling transcription and translation steps. The delay introduced seems more reliable although we do not have our own experimental data to validate the same.

Predicted XOR behaviour for the whole cell model with leakiness

Team:ETH Zurich/modeling/whole

From 2014.igem.org