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Model for tRNA amber diffusion system

Summary

The tRNA amber diffusion design is the one using the smallest molecule we want to diffuse through nanotubes: tRNA. The emitter cell produces this tRNA amber which is transfered through nanotubes and serves to translate the so-called mRNA amber (mRNA for T7 RNA polymerase with two amber mutations). This mRNA amber can be translated only in presence of our mutant tRNA and is therefore our receptor construct. Having two amber mutations in the T7 RNA polymerase gene in the receiver cell should prevent any leakiness from the receiver construct. The auto-amplification is obtained through the same T7 auto-amplification loop used in the T7 RNA polymerase diffusion design.

The behaviour expected is to have an ON/OFF switch activated when a certain number of tRNA amber have entered the cell. To be certain this activation was possible, we ran additionnal simulation of tRNA diffusion in one cell. Our conclusion was that a delay of less than 10 minutes might be observed in the receiver cell if very few tRNA amber arrive in the receiver cell. You can find more about this sub-model in the link below:

The tRNA amber diffusion and the translation of mRNA strand with two amber codon models

You will find below the results for a simple simulation with every part of the design in one cell. 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.

Matlab simulation for the tRNA amber construct (all in one cell, only reporters molecules and IPTG input)

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 after the tRNA amber appears in the cell). 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.

Even though the timescale for tRNA amber diffusion in the receiver cell is reasonable, we tried several possible stable states of tRNA amber input for the receiver construct. This stable state hypothesis for tRNA amber is justified by the recycling of tRNA in the cells.

Our MATLAB files are available here: download.

Design

The tRNA amber diffusion design is the design using the smallest molecule we want to diffuse through nanotubes: tRNA. The emitter construct produces tRNA amber a tRNA which is complementary to amber stop codons but transports an amino-acid tyrosine instead of stopping translation. tRNA amber is then transfered through the nanotubes and into the receiver cell.

In the receiver cell, a T7 RNA polymerase amber gene constitutively produces the so-called mRNA amber (mRNA for T7 RNA polymerase with two amber mutations) which is our receiver construct. This mRNA amber can be translated into functional T7 RNA polymerase only in presence of our mutant tRNA amber. We note the functional polymerases expressed from this gene T7 amber.

The amplifier construct is then a T7 auto-amplifying loop identical to the one in the T7 RNA polymerase diffusion design. The T7 amber activates this loop which is also where our reporter GFP is expressed.

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.

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 tRNA amber by adding IPTG on the cells.

Inactivated LacI can not repress the pHs 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 - tRNA amber

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 gene construct - T7 amber

The receiver gene construct is modeled by the following equations:

You will remark that activated mRNA amber is noted mRNA^* amber. This corresponds simply to mRNA_amber being translated into functional T7 RNA polymerase using two tRNA amber. Since we have hints that tRNA amber are used pretty quickly in the translation process (see tRNA diffusion in one cell) we used a very simple equation to model this:

Amplifier gene construct - T7 auto-amplification loop

The amplifier gene construct is modeled by the following equations:

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

Parameters

This design relies on tRNA amber as the signaling molecule going through the nanotubes.

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
	Inactivated LacI concentration	NA	molecules per cell	Notation convention
	Total LacI concentration	10000	molecules per cell	Steady state for equation, assumed realistic
	tRNA amber concentration	NA	molecules per cell	Notation convention
	mRNA amber concentration	NA	molecules per cell	Notation convention
	Translated mRNA amber concentration	NA	molecules per cell	Notation convention
	T7 amber concentration	NA	molecules per cell	Notation convention
	mRNA associated with T7 (non amber, amplification) concentration	NA	molecules per cell	Notation convention
	T7 (non amber, amplification) 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)	0.02	molecules.s-1 or pops	Estimated, see the justification
	Maximal production rate of pHs promoter	0.02	molecules.s-1 or pops	Estimated, see the justification
	Maximal production rate of pT7 promoter	0.02	molecules.s-1 or pops	Estimated, see the justification
	Dissociation constant for IPTG to LacI	1200	molecules per cell	Aberdeen 2009 wiki
	Dissociation constant for LacI to LacO (pHs)	700	molecules per cell	Aberdeen 2009 wiki
	Dissociation constant for T7 RNA polymerase to pT7	10	molecules per cell	We used the classic assumption 1nM=1 molecule per cell and [1]
	Hill coefficient for LacI/IPTG interaction	2		Aberdeen 2009 wiki
	Hill coefficient for LacI/pHyperSpank interaction	2		Aberdeen 2009 wiki
	Translation rate of proteins	0.9	s-1	Estimated, see the justification
	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	[3]
	Degradation rate of GFP	10-4	s-1	BioNumbers
	Degradation rate of RFP	10-4	s-1	Estimated equal to GFP degradation rate
	Delay due to T7 RNA polymerase production and maturation	300	s	[2]
	Delay due to GFP production and maturation	360	s	BioNumbers
	Delay due to RFP production and maturation	360	s	Estimated equal to GFP delay (similar molecules)
	Delay due to mRNA production	30	s	BioNumbers with an approximation: all our contructs are around 1-2kb

Results & discussions

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

Matlab simulation for the tRNA amber construct (all in one cell, click to enlarge)

The behaviour of the cell is as expected. The IPTG input removes the repression on the tRNA amber production, which then is used in the translation of mRNA amber into T7 RNA polymerase. This T7 amber 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 amber and T7 amber 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/disparition 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 gene 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 10 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

As with the T7 model, 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 (see our justification for our choices of promoter strengths). 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 tRNA pass through the nanotubes.