Team:Stockholm/Modeling
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 various 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. Down below is 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.
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 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.
Figure 3: Simulation of low osmolarity.
Figure 4: Simulation of high osmolarity.
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 advice on how to move forward. One of them was to use MatLab instead of Simbiology. This could be done by deriving our own equations from the model below. Most of the biological systems can be easily converted into a series of Ordinary Differential Equations (ODEs) with 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 enough. Generally, the variant of the Euler’s method could be sufficient, 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 y 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 can be found in the following file:
https://static.igem.org/mediawiki/2014/3/3d/Modeling-Everything_Ever.pdf
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.
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. 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.
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 kc1omprp*[cell (High osmolarity)].C1*[cell (High osmolarity)].OmpRP (11) kOmpF*[cell (High osmolarity)].F1F2F3F4OmpRP (13) kF1F2F3F4omprp*[cell (High osmolarity)].F1F2F3*[cell (High osmolarity)].OmpRP (14)
Equations: Model 2; EnvZ to OmpF (low osmolarity) kF1F2F3Omprp*[cell (Low osmolarity)].F1F2F3 (15)*[cell (Low osmolarity)].OmpRP_1 kF1F2Omprp*[cell (Low osmolarity)].F1F2*[cell (Low osmolarity)].OmpRP_1K2*cell1.EnvZ*[cell (High osmolarity)].OmpRP - K_2 (16)*cell1.EnvZOmpRP 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 (25) Equations: Model 3; quorum sensing molecule 1 (BHL) 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 (32) Equations: Model 4; quorum sensing molecule 1 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] (38) The rate constant in each model is not related to one from another model.