Modeling

Our main goal was to estimate how the EnvZ concentration correlates with the signal from GFP production. For that we decided to divide our system into four main models:

OmpR production and phosphorylation beginning with the EnvZ receptor

OmpC translation

Release of quorum sensing (QS) molecules

GFP expression for detection.

Parts 1 & 2: OmpR phosphorylation and OmpC 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 . With the help of SimBiology software, a plug-in installed in MatLab, we designed these models by drawing them qualitatively, choosing the kinetics and adding all the constants. We then put the two parts together. Then 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 substance and the reactant molecules connect them. The substances are given their corresponding initial values. The kinetic law Mass Action is selected if the kinetics is unknown.

Since we wanted our final product to be OmpC and not OmpF, we chose to make two models for each scenario to get a better understanding of how the osmolarity changes the endpoint, in our case being 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.

Part 3: release of quorum sensing (QS) molecules

At first we decided to use the same procedure as with the previous parts. But with 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 time dependency and feedback loop inclusion that is present in our model, we actually need 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 could be utilized (Fokker-Plank PDE). Eventually, quorum-sensing signaling could be also modeled by use of averaging effect that is separated from the in-cellular modeling, to demonstrate the ability of the system to reduce the noise.

All the information on the ideas and procedures involving mathematical concepts of the MatLab modeling and Generalized Promoter Binding our team got from the representation of the 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 next part of the modeling part of the project. We designed the Part 3 of the modeling using the BHL and LuxR quorum sensing systems

Part 4: GFP expression

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

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.

References

#	Reference
1	Srividhya KV, Krishnaswamy S. A simulation model of Escherichia coli osmoregulatory switch using E-CELL system. BMC Microbiology. 2004, 4:44.
2	Sadelain, M., Brentjens, R., & Rivière, I. (2013). The basic principles of chimeric antigen receptor design. Cancer discovery, 3(4), 388-398.
3	PaParkinson, J. S. (2010). Signaling mechanisms of HAMP domains in chemoreceptors and sensor kinases. Annual review of microbiology, 64, 101-122.
4	Eigenbrot, C., Ultsch, M., Dubnovitsky, A., Abrahmsén, L., & Härd, T. (2010). Structural basis for high-affinity HER2 receptor binding by an engineered protein. Proceedings of the National Academy of Sciences, 107(34), 15039-15044.
5	Levskaya, A., Chevalier, A. A., Tabor, J. J., Simpson, Z. B., Lavery, L. A., Levy, M., ... & Voigt, C. A. (2005). Synthetic biology: engineering Escherichia coli to see light. Nature, 438(7067), 441-442.
6	Schmidt, C. (2009). Urine biomarkers may someday detect even distant tumors. Journal of the National Cancer Institute, 101(1), 8-10.
7	Collin, M., and Schuch, R. (2009). Bacterial Sensing and Signalling.
8	Morten Källberg, Haipeng Wang, Sheng Wang, Jian Peng, Zhiyong Wang, Hui Lu & Jinbo Xu. Template-based protein structure modeling using the RaptorX web server. Nature Protocols 7, 1511–1522, 2012.