Team:Colombia/Deterministic

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Deterministic model

Finding the mean behaviour of the system...

The first step is to develop a deterministic model based in differential equations. With the ordinary differential equations we can describe quantitatively the system at molecular level.

Using the law of mass action is possible to model most of the metabolic networks. For each substance of the system, we express the changes of concentration trough time. We consider processes related with the export and import of a molecule into the cell, the production by a gene or a chemical reaction. All these terms depend on reaction kinetics that could be expressed as a simple multiplication or a complex expression like a Hill equation.

Signal transduction →

The receptor CqsS acts as a kinase when there is no auto inducer (CAI). It phosphorylates LuxU, which transfers its phosphate to LuxO. And in our case LuxO phosphorylated activates Pqrr promoter.

Once the inducer is in the media the receptor changes to a phosphase (CqsSp) mode and the flow is reversed. LuxU is unphosphorylated and it removes the phosphate from LuxO. In consequence Pqrr promoter is repressed.

Colombia Rxn1.png
Colombia Rxn2.png
Colombia Rxn3.png
Colombia Rxn4.png
Colombia Rxn5.png


Taking this process into account and given the fact the phosphorylation process can been described as an enzymatic process; we used the Mass Action Law and the Michaelis-Menten Kinetics in order to obtain the mathematical expressions that describe the signal transduction:


EqA.png


EqB.png

Equations a and b describe the behavior of the receptor acting as a kinase (CqsS) and as a phosphase (CqsSP). The kinase is produced at a constant rate alpha CS, it is involved in Reaction 1 and has a decay rate of Gamma CS. The phosphase is also involved in Reaction 1 and has its own decay rate.


EqC.png


EqD.png


EqE.png


EqF.png


To describe the phosphorylation cascade we based our expressions on the model that Liu et al. did in 2012: LuxU and LuxO are involved in the enzymatic Reactions 2-5, both are produced by a constitute promoter at a rate a_u and a_o respectively and each molecule has its own decay rate, taking this into account Equations c-f were made.


Response →

LuxO controls Pqrr promoter. Downstream of this promoter there is the repressor TetR of the tetracycline promoter (pTet), which in our design is downstream the repressor and, controls de production of the colored response and the activator of the promoter TetA.


EqG.png


EqH.png


LuxO controls the production of TetR and it can be expressed with a Hill type equation where LuxO phosphorylated is the activator. The promoter has a small constitutive production rate alpha tr and TetR has a decay rate of gamma tr; also TetR is consumed when forms a dimer in order to repress pTet.

EqI.png


EqJ.png


Equations i and j describe the production of the response and the pTet activator. As in all the preview equation, both proteins have a constitutive production and a decay rate. In our design there is a promoter (pTet) with a unique operator sequence (TetO) where the repressor (TetR2) and the activator (tTA) bind (TetR2 and tTA have the same binding domain). For that reason it was necessary to propose a mathematical expression in which the repressor and the activator compete for the binding site.