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From 2013.igem.org

Contents 1 Modeling TMAO system 1.1 Labels and conventions 1.2 First Modeling 1.3 Second Modeling

Modeling TMAO system

The TMAO system requires several reactions steps. Thus, we modeled it in order to obtain the production rate of activated TorR. Since the activated TorR enables the reporter's protein transcription, our model could be simplified in just one equation.

Labels and conventions

C_E: The enzymatic complex formed by TorT+TorS+TMAO, which reacts with TorR

C: intermediate component of the final enzymatic process

P: Product of the enzymatic reaction

C_TS: Complex formed by TorT and TorS

C_TSM: Complex formed by TorT+TorS and one molecule of TMAO

C_TSM: Complex formed by TorT and TMAO

First Modeling

This is the simplest modeling process to be considered in our study. The product of the enzymatic process is collapsed in only one reaction that was placed at equilibrium.

Imposing the equilibrium in the first reaction we have:

The value of C_E is the initial enzyme concentration for the subsequent enzymatic reactions.

Regarding the enzymatic reaction:

The following relation holds E₀ = C_E + C. And as usual imposing the steady state condition:

With C_E = E₀ - C

Solving for C:

with

In conclusion, we find:

Second Modeling

It was used the same pattern of reactions as before, but with a standpoint of "essential activation" (the first reaction is no longer considered to equilibrium). Some enzymes need to be activated to bind the substrate. In our case, the enzyme is activated by TorS and activators are TMAO and TorT. The rate of product P appearance depends on the activators concentration. The equation that varies with respect to the previous case is:

The total concentration of the enzyme is:

Then we considered and imposed and

The system of equations also comes from:

Defining and replacing we have:

In conclusion the rate of product appearance results

We have made simulations in order to see if our model can distinguish between different concentrations of TMAO (input) and how would be the activation of the transcription factor TorR (output). In general, it seems that the system will reach a saturation limit, except for very low concentrations of TMAO, but the appearing of the product is faster for higher concentrations of TMAO.

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