# Team:Paris Bettencourt/Modeling/ComS diffusion

### From 2011.igem.org

# Model for ComS diffusion system

## Summary

The model in 5 bullet points:

- ComK/ComS system requires around 500 ComS molecules to be activated
- Adequate response time between 30 min and 1 hour before having a visible fluorescence signal
- Promoter strengths need to be experimentaly evaluated in
*B.subtilis* - Leakage might create a very noisy behaviour for fluorescence (YFP)
- ComK/ComS systems has never been used in exponential phase. All parameters are therefore assumptions

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 |

References

*An excitable gene regulatory circuit induces transient cellular differentiation*, Süel GM, Garcia-Ojalvo J, Liberman LM, Elowitz MB, Nature. 2006 Mar 23;440(7083):545-50.*Architecture-Dependent Noise Discriminates Functionally Analogous Differentiation Circuits*, Çağatay, Tolga and Turcotte, Marc and Elowitz, Michael B. and Garcia-Ojalvo, Jordi and Süel, Gürol M, Cell, 2009 139 (3). pp. (Supplementary Data)*An Introduction to Systems Biology: Design Principles of Biological Circuits*, Uri Alon

## 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*.