# Team:Stockholm/Modeling

## Modeling

Our main goal was to estimate how the osmolarity in the cell's environment correlates with the signal from GFP production. For that we decided to divide our system into four main models:

## Parts 1 & 2: OmpR phosphorylation and OmpC/OmpF translation

We wanted to insert the EnvZ receptor into a bacterial cell to activate a signal cascade including four main parts (OmpR production, OmpC translation, quorum sensing and GFP expression). For the first two parts, OmpR and OmpC production, we used already published research to find equations describing the main reactions and their rate constants^{}^{}^{}. We designed those models with the help of SimBiology software, a plug-in installed in MatLab. The designing was done by drawing them qualitatively, choosing the kinetics and adding all the constants. We then put the two parts together. Finally, we designed the OmpC production-reduction EnvZ switch depending on the osmolarity level.

The way to design a model is to define each compartment, in this case the cell membrane and the cytoplasm. Each reactant is a chemical entity that is connected to another reactant by a corresponding chemical reaction. Each reaction has a specific kinetic rate. The kinetic law Mass Action is selected if the kinetics is unknown.

Figure 1: Model for OmpF production starting from the EnvZ receptor (left image) at low osmolarity levels. EnvZ phosphorylates OmpR (OmpR-P, right image) and activates OmpF production upon reacting with the binding sites F1, F1F2 and F1F2F3.

Figure 2: Model for OmpC production starting from the EnvZ receptor (left image) at high osmolarity levels. EnvZ phosphorylates OmpR (right image) and activates OmpC production upon reacting with the binding sites C1, C1C2 and C1C2C3. OmpF is degraded simultaneously when the osmolarity is high.

Since we wanted our final product to be OmpC or OmpF (depending on the surrounding osmolarity), we chose to make two models for each scenario. This was done in order to get a better understanding of how the osmolarity changes the endpoint. In our case, it was the osmolarity-dependent GFP expression^{}.

After the model was designed and a simulation created, we could change the initial data and study how each component is codependent.

Figure 3: Simulation of low osmolarity. The horizontal axis corresponds to the time [s] and the vertical axis corresponds to concentration [µM].

Figure 4: Simulation of high osmolarity. The horizontal axis corresponds to the time [s] and the vertical axis corresponds to concentration [µM].

## Part 3: release of quorum sensing (QS) molecules

At first we decided to use the same procedure as with the previous parts. But, after a lot of trial and error, we contacted the Modeling Representative from the Technion 2014 iGEM, Ittai Rubinstein, and got several tips on how to proceed. One of them was to use MatLab instead of SimBiology. This could be done by deriving our own equations from the SimBiology model. Most of the biological systems can be easily converted into a series of Ordinary Differential Equations (ODEs). A steady state solution of those ODEs is the simplest model that can be produced^{}. For the time dependency and feedback loop inclusion, which is present in our model, we actually needed to find a numerical solution for the ODEs, since a steady state is not sufficient to reflect the biological environment. Generally, the variant of the Euler’s method could be adequate, especially for ODEs of higher degrees^{}.

To estimate the degree of error approximated and time-dependent Fokker-Plank Partial Differential Equation was used (Fokker-Plank PDE)^{}. Eventually, quorum-sensing signaling could also be modeled by the use of averaging effect that is separated from the intracellular modeling. This would let us demonstrate the ability of the system to reduce the noise.

Our team got all the ideas and procedures involving mathematical concepts and Generalized Promoter Binding from the representation of iGEM Technion 2014. The file with detailed description of the mathematical concepts prepared by this team can be found in the following file:

Mathematical concepts by iGEM Technion 2014

Nevertheless, as a team, we decided to follow the SimBiology modeling that is represented in the Figures 5 and 6. We designed the Part 3 of the modeling using the BHL and LuxR quorum sensing systems^{}.

Figure 5: Reaction schematic of the quroum-sensing molecule BHL. It starts with BHL entering the cell and binding to RhlR to form the "Complex". In this model the "Complex" additionally binds to rhll to produce Rhll.

Figure 6: Schematic of the quorum-sensing molecule OHHL. There are two different outcomes depending on the cell density level. In case of high cell density the transcription of LuxR is hindered, as opposed to in low cell density.

## Part 4: GFP expression

The GFP production is the last part of our model. By modeling the GFP production we would hopefully, in theory, be able to estimate the EnvZ concentration by looking at their correlation.

## Conclusions

Design and optimization of our model proved that:

## Appendix: Equations for SimBiology

### Model 1: EnvZ to OmpC (high osmolarity)

(1)* K1*cell1.EnvZP*cell1.OmpR - K_1*cell1.EnvZPOmpR
*

(2)* Kt*cell1.EnvZPOmpR
*

(3)* K2*cell1.EnvZ*[cell(High osmolarity)].OmpRP - K_2*cell1.EnvZOmpRP
*

(4)* Kp*cell1.EnvZOmpRP
*

(5)* Kk*cell1.EnvZ*cell1.OmpR - K_k*cell1.OmpR*cell1.EnvZP
*

(6)* Kc1*[cell (High osmolarity)].C1OmpRP
*

(7)* Kc1c2c3Omprp*[cell (High osmolarity)].C1C2C3OmpRPKt*cell1.EnvZPOmpR
*

(8)* Kc1c2c3omprp*[cell (High osmolarity)].C1C2C3*[cell (High osmolarity)].OmpRP
*

(9)* Kouoiui*[cell (High osmolarity)].C1C2OmpRP
*

(10)* Kc1c2omprp*[cell (High osmolarity)].C1C2*[cell (High osmolarity)].OmpRP
*

(11)* Kc1omprp*[cell (High osmolarity)].C1*[cell (High osmolarity)].OmpRP
*

(12)* KOmpF*[cell (High osmolarity)].F1F2F3F4OmpRP
*

(13)* KF1F2F3F4omprp*[cell (High osmolarity)].F1F2F3*[cell (High osmolarity)].OmpRP
*

### Model 2: EnvZ to OmpF (low osmolarity)

(14)* kF1F2F3Omprp*[cell (Low osmolarity)].F1F2F3*[cell (Low osmolarity)].OmpRP_1
*

(15)* kF1F2Omprp*[cell (Low osmolarity)].F1F2*[cell (Low osmolarity)].OmpRP_1K2*cell1.EnvZ*[cell (High osmolarity)].OmpRP - K_2*cell1.EnvZOmpRP
*

(16)* kF1Omprp*[cell (Low osmolarity)].F1*[cell (Low osmolarity)].OmpRP_1
*

(17)* kOmpF*[cell (Low osmolarity)].F1F2F3OmpRP
*

(18)* kOmpF*[cell (Low osmolarity)].F1F2OmpRP
*

(19)* kOmpF*[cell (Low osmolarity)].F1OmpRP
*

(20)* Kk*[cell 2].EnvZ*[cell 2].OmpR - K_k*[cell 2].OmpR*[cell 2].EnvZPkouoiui*[cell (High osmolarity)].C1C2OmpRP
*

(21)* Kp*[cell 2].EnvZOmpRP
*

(22)* K2*[cell 2].EnvZ*[cell (Low osmolarity)].OmpRP_1- K_2*[cell 2].EnvZOmpRP
*

(23)* Kt*[cell 2].EnvZPOmpR
*

(24)* K1*[cell 2].EnvZP*[cell 2].OmpR - K_1*[cell 2].EnvZPOmpR
*

### Model 3: quorum sensing for BHL molecule

(25)* k1*rhlR
*

(26)* k2*RhlR
*

(27)* k3*BHL*RhlI - k4*Complex
*

(28)* k4*RhlR
*

(29)* k5*rhlI*Complex
*

(30)* k6*Complex
*

(31)* Va2*[BHL outside cell]- d2*BHL
*

### Model 4: quorum sensing for LuxR molecule

(32)* kA*LuxR*OHHL - k_A*[LuxR-complex]
*

(33)* k1*LuxI
*

(34)* k2*[LuxR-complex]*[Lux promoter]
*

(35)* k3*OHHL
*

(36)* k4*LuxI
*

(37)* k5*[LuxR transcribed]
*

The rate constant in each model is not related to one from another model.