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Model for ComS diffusion system

Summary

The ComS diffusion design is a bit different from our other construct. In this model, the receiver is the natural ComK/ComS switch. We try to toggle it out of its "OFF" state (few ComK in the cell) by producing ComS in the emitter cell. This will hopefully toggle the switch, creating a pulse of ComK and its associated reporter protein. Because we used strains generously given by professor Elowitz we already had a nice set of fluorescent protein placed next to the correct genes (YFP reporting for ComK and CFP reporting for ComS). We added RFP to report on our emitter ComS, distiguishing it from the ComS used in the switch.

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 YFP for the receiving construct.

Matlab simulation for the ComS construct (all in one cell, only reporter proteins and IPTG input)

Unfortunately, YFP is too much expressed at a basal level. This means seeing a response of the system will be difficult. We will not be looking at the beginning of the fluorescence, just at its strengthening. This is a problem coming from the very nature of the switch. It relies heavily on ComK repression by MecA (see the design). Since YFP is not repressed by MecA, its concentration does not follow closely ComK concentration.

Our model shows the system works but it might be difficult to measure precisely its response.

We also know we might need as much as 500 ComS molecules going through the tubes to see a significant response. This is a good way to test the order of magnitude of the number of molecules that can pass through the tubes.

Our MATLAB files are available here: all in one cell, emitter and receiver.

Design

The ComK/ComS system is a complex natural toggle switch. We intend to pass enough ComS through the toggle switch to induce a response. The strains we used as receptors were kindly given by M.Elowitz and already had YFP and CFP genes installed to report on the productions of ComS and ComK. To find out more about this complex system, please look at the corresponding design.

In this system, ComS reported by RFP acts as the emitted molecule and the natural toggle switch is our receptor and our auto-amplifier.

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

Trigger - 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 - ComS

The emitter gene construct modeled by the following equations:

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

Receiver and amplification gene construct - ComK/ComS switch

The receiver and amplification gene construct is the natural ComK/ComS switch. The ComS gene is regulated in the same way as the in the emitter construct by ComK, but is not affected by LacI. It is modeled by the following equations:

The reporter for the receiver and amplification gene construct (YFP for ComK and CFP for ComS) is modeled by the following equations:

Parameters

This design relies on ComS 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
	Inactived LacI concentration	NA	molecules per cell	Notation convention
	Total LacI concentration	10000	molecules per cell	Steady state for equation, assumed realistic
	YFP concentration	NA	molecules per cell	Notation convention
	mRNA associated with YFP concentration	NA	molecules per cell	Notation convention
	CFP concentration	NA	molecules per cell	Notation convention
	mRNA associated with CFP 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
	ComS concentration	NA	molecules per cell	Notation convention
	mRNA associated with ComS concentration	NA	molecules per cell	Notation convention
	ComK concentration	NA	molecules per cell	Notation convention
	mRNA associated with ComK concentration	NA	molecules per cell	Notation convention
	Maximal production rate of pComK promoter	0.049	molecules.s-1 or pops	[1]
	Maximal production rate of pComS promoter	0.057	molecules.s-1 or pops	[1]
	Maximal production rate of pComG promoter	0.02	molecules.s-1 or pops	Estimated, see the justification
	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
	Basal production rate of pComK promoter	0.0028	molecules.s-1 or pops	[1]
	Basal production rate of pComG promoter	0.028	molecules.s-1 or pops	Is roughly one order of magnitude higher than production rate of pComK[2]
	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 ComK to pComK	110	molecules per cell	[1]
	Dissociation constant for ComK to pComS	100	molecules per cell	[1]
	ComK concentration for half maximal degradation	500	molecules per cell	[1]
	ComS concentration for half maximal degradation	50	molecules per cell	[1]
	Hill coefficient for LacI/IPTG interaction	2	NA	Aberdeen 2009 wiki
	Hill coefficient for LacI/pHyperSpank interaction	2	NA	Aberdeen 2009 wiki
	Hill coefficient for ComK/pComK and ComK/pComG (positive feedback) interaction	2	NA	[1]
	Hill coefficient for ComK/pComS (negative feedback) interaction	5	NA	[1]
	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
	Unrepressed degradation rate of ComK	1.4x10-3	s-1	[1]
	Unrepressed degradation rate of ComS	1.4x10-3	s-1	[1]
	Degradation rate of mRNA	2.88x10-3	s-1	[3]
	Degradation rate of RFP	10-4	s-1	Estimated equal to GFP degradation rate
	Delay due to RFP production and maturation	360	s	Estimated equal to GFP delay (similar molecules)
	Delay due to CFP production and maturation	360	s	Estimated equal to GFP delay (similar molecules)
	Delay due to YFP production and maturation	360	s	Estimated equal to GFP delay (similar molecules)
	Delay due to ComK production and maturation	300	s	Arbitrary
	Delay due to ComS production and maturation	300	s	Arbitrary
	Delay for ComS repression by ComK	714	s	[1]
	Delay due to mRNA production	30	s	BioNumbers with an approximation: all our contructs are around 1-2kb

Results & discussions

We had absolutely no results at first. The productions levels were so low no molecule was produced. We soon realized that all the parameters the simulation is based on were obtained in stationnary phase. We therefore had to put a dilution factor of 0 s^-1 just to see results.

We then launched the simulation in Matlab with the emitter and the receiver in the same cell and obtained the following results:

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

As you can easily see, YFP increases but its "OFF" state is already relatively high (around 15000 molecules per cell). Since a response of our system will not be translated by YFP appearing, this means the reporter protein YFP increase will be hard to detect. In practical terms, we may not be able to see the receiver reacting. The reason YFP does not follow closely the behaviour of ComK is that it is not actively repressed by the MecA enzyme. mRNA levels are not affected by this repression, and the leaking of the pComG gives us unfortunately a basal expression of YFP.

On the other hand, the other proteins behave as we expected. This is striking for ComS and ComK where a little ComS triggers a lot of ComK production by lifting the repression of MecA. This is encouraging so we tried to see how many external ComS coming trhough the nanotubes would be required to trigger the response of the system (even though we can not see it easily):

Matlab simulation for the ComS receiver construct (5 external ComS, click to enlarge)

Matlab simulation for the ComS receiver construct (50 external ComS, click to enlarge)

Matlab simulation for the ComS receiver construct (500 external ComS, click to enlarge)

Matlab simulation for the ComS receiver construct (5000 external ComS, click to enlarge)

The order of magnitude for reaction of the system is therefore 500 ComS molecules. This seems like a lot to pass through the nanotubes but it might work. When we will have tested the pVeg promoter we will be able to see how many ComS are produced in the emitter cell and estimate if this design can produce something. For the moment, our arbitrary promoter strength gives us the following production:

Matlab simulation for the ComS emitter construct (click to enlarge)

Limits

The first obvious limit is that we can not run the simulation in exponantial phase. The dilution factor in exponantial phase is not compatible with all the data we have that was found for stationnary phase. We hope that the increase in production due to cell activity in exponantial phase will compensate this but we have no way to predict it. This is a serious limit to the practical application of this model.

The second limitation is that the reaction of the system is an increase in YFP production, not the beginning of the production. Following precisely such an increase is much more complex than verifying its appearance, it will therefore be very difficult to see the response of the system.