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Model for T7 RNA polymerase diffusion

Summary

The T7 RNA polymerase diffusion construct is one of the central point of our project. In this design, the T7 RNA polymerase is used both as an auto-amplifier and as the signal being transmitted. Because the T7 promoter is supposed to be orthogonal to both B.subtilis and E.coli, we used it in several of our other construct as an amplifier. We hoped its orthogonality would reduce leakiness and that the T7 RNA polymerase would prove to be an excellent amplifier for very low signals. It would however give us only an ON/OFF response in an individual cell, not a quantitative one.

You will find below the results for a simple simulation. All of our components are in one cell. Between t=7500s and t=12500s, IPTG is added to the cell, lifting repression by LacI. RFP is the reporter for the emitting construct and GFP for the receiving construct.

Our simulation exhibits the behaviour we expected. The model shows that in our experimental conditions the cell should produce a significant response in a reasonable time (approximately 20 to 30 minutes). Once started, the auto-amplification loop can not be stopped, as we can see with the GFP staying at its maximum even when RFP levels decrease. The major limit of this model could come from leakiness of promoters. If the pT7 is not as reliable as we thought, the auto-amplification loop could trigger itself all the time.

With our parameters, the model is extremely reactive to T7 RNA polymerase. Having only one T7 RNA polymerase in a cell is sufficient to start the autoamplification loop. This is not surprising with a Hill coefficient of 1 and a dissociation constant of 4.8 molecules per cell for pT7.

Design

The T7 diffusion design is our first original construct. In this design, the T7 RNA polymerase acts both as the signal transmited and as the amplifier (auto-amplification). T7 RNA polymerase is produced in the emitter cell, then diffuses through the nanotubes and arrives in the receiver cell. In this cell, the T7 RNA polymerase activates on a pT7 promoter. Behind this promoter, we put a gene coding for the same T7 RNA polymerase (amplifier) and for GFP (reporter).

This construction was put in two different settings. One is what we just described, where the emitting gene network is in one cell and the receiving gene network is in another. In the other construction, everything is in one cell. We use the second construct as a control to really see the impact of the cell-to-cell communication on the behaviour of the cells. Because of our parameters, only the "all in one cell" was relevant from a modeling point of view since we do not have detailed data on the nanotubes and only one T7 RNA polymerase triggers the system (see discussion for more details).

We ran our models for those two configurations. We used a steady flow of signaling molecules in the receiver cell for the "one emitting cell - one receiving cell" construction. You can find our justifications about this assumption here.

Model

LacI

We use LacI as a repressor for the emitter gene construct. LacI repression can be cancelled by IPTG. This way we can induce production of RFP and T7' by puttig IPTG on the cells.

Inactivated LacI can not repress the pLAC promoter anymore. Note that we consider that the reaction between IPTG and LacI fires without any delay. This assumption is justified by the fact that this reaction is much faster than any other in our gene network.

Emitter gene construct - T7'

The emitter gene construct is modeled by the following equations:

The reporter for the emitter gene construct (RFP) is modeled by the following equations:

Receiver and amplification gene construct - T7''

The receiver and amplification gene construct is modeled by the following equations:

The reporter for the receiver and amplification gene construct (GFP) is modeled by the following equations:

Parameters

This design relies on the T7 RNA polymerase (which is noted T7) both as the signaling molecule going through the nanotubes and as the auto-amplification system when it acts on the pT7 promoter. In our equations however, we chose to distinguish these functions.

• T7' represents the signaling T7 RNA polymerase
• T7'' represents is the auto-amplifying molecule

The parameters used in this model are:

Parameter	Description	Value	Unit	Reference
	Active LacI concentration (LacI which is not inactivated by IPTG)	NA	molecules per cell	Notation convention
	IPTG concentration	NA	molecules per cell	Notation convention
	Inactived LacI concentration	NA	molecules per cell	Notation convention
	Total LacI concentration	TBD	molecules per cell	Steady state for equation
	T7 RNA polymerase (emitter, T7') concentration	NA	molecules per cell	Notation convention
	mRNA associated with T7' concentration	NA	molecules per cell	Notation convention
	T7 RNA polymerase (auto-amplification, T7'') concentration	NA	molecules per cell	Notation convention
	mRNA associated with T7'' concentration	NA	molecules per cell	Notation convention
	GFP concentration	NA	molecules per cell	Notation convention
	mRNA associated with GFP concentration	NA	molecules per cell	Notation convention
	RFP concentration	NA	molecules per cell	Notation convention
	mRNA associated with RFP concentration	NA	molecules per cell	Notation convention
	Maximal production rate of pVeg promoter (constitutive)	???	molecules.s-1 or pops	Estimated
	Maximal production rate of pLac promoter	0.02	molecules.s-1 or pops	Estimated
	Maximal production rate of pT7 promoter	0.02	molecules.s-1 or pops	Estimated
	Dissociation constant for IPTG to LacI	1200	molecules per cell	Aberdeen 2009 wiki
	Dissociation constant for LacI to LacO (pLac)	700	molecules per cell	Aberdeen 2009 wiki
	Dissociation constant for T7 RNA polymerase to pT7	4.8	molecules per cell	Estimated ADD EXPLANATION FIND MATHIAS SOURCE
	Translation rate of proteins	1	s-1	Estimated ADD EXPLANATION
	Dilution rate in exponential phase	2.88x10-4	s-1	Calculated with a 40 min generation time. See explanation
	Degradation rate of mRNA	2.88x10-3	s-1	Uri Alon (To Be Confirmed)
	Degradation rate of GFP	10-4	s-1	http://bionumbers.hms.harvard.edu/bionumber.aspx?s=y&id=105188&ver=2&hlid=56398 (half life 110 min)
	Degradation rate of RFP	10-4	s-1	Estimated equal to GFP degradation rate
	Delay due tT7 RNA polymerase production and maturation	300	s	http://mol-biol4masters.masters.grkraj.org/html/Prokaryotic_DNA_Replication13-T7_Phage_DNA_Replication.htm
	Delay due GFP production and maturation	360	s	http://bionumbers.hms.harvard.edu/bionumber.aspx?s=y&id=102972&ver=8
	Delay due RFP production and maturation	360	s	Estimated equal to GFP delay (similar molecules)
	Delay due to mRNA production	30	s	http://bionumbers.hms.harvard.edu/bionumber.aspx?s=y&id=104902&ver=5&hlid=58815 2kb/(50b/s) --> approximation: all our contructs are around 2kb

Results & discussions

We launched this simulation in matlab and obtained the following results:

The behaviour of the cell is as expected. The IPTG input removes the repression on the mRNA T7' production, which then is translated in T7'. This T7' polymerase activates the mRNA T7'' production. Finally, this last mRNA is translated into T7''.

This is exactly the results we wanted. The T7 RNA polymerase acts both as a transmission molecule and an amplifier. Once the pT7 is activated it auto-amplifies itself and gives us a clear result.

The IPTG imput is here theoritical. We can not in an experiment remove the IPTG from the medium. However, this input signal is an excellent way to understand the way the system behaves. After IPTG disappears, we can see the levels of mRNA T7' and T7' decreasing as expected since they are regulated by pHyperSpank. On the other hand, mRNA T7'' and T7'' regulated by pT7 are not affected.

Because of the time scale it is hard to realize, but all delays have been correctly implemented and the order of appearance/disparitions of each product is what we expected. For instance, mRNA appear 30 seconds after the conditions for initiation of transcription are met.

The reporter proteins RFP and GFP follow closely the behaviour of the T7 RNA polymerase they are working on. Note that the slight difference between T7 RNA polymerase and reporter protein maximum levels of expression is due to RFP and GFP degradation rate. This does not however pose a problem for following the behaviour of cells, as expected.

With our parameters, the model is extremely reactive to T7 RNA polymerase. Having only one T7 RNA polymerase in a cell is sufficient to start the autoamplification loop. This is not surprising with a Hill coefficient of 1 and a dissociation constant of 4.8 molecules per cell for pT7. Since we know that a single molecule takes only a few second to find the pT7 promoter(see our hypotheses), it is of no consequence if our parameters are correct.

Limits

The most obvious limit is that we supposed the pT7 promoter to be not leaky at all, since it needs very little T7 RNA polymerase to be activated. If the leak is too important in pratice the model and the design might need some adjustments.

Most parameters are well defined, but promoter strengths tend to be quite difficult to find or to evaluate. In this model, changing moderately these strengths does not impact much on the overall behaviour of the system. It could be troubling however in our experiments with two cells if very few T7 RNA polymerase pass through the nanotubes.

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