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Modeling TMAO system

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:

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.

Modeling IMA System

First Reaction - Cobalt ions bind to albumin

The cobalt ion binds to albumin protein, originating the cobalt-albumin complex. If the albumin is derived from a healthy patient, it will bind more cobalt ions than a albumin derived from a patient with heart disease. It happens because patients with heart conditions will have a isquemic modified albumin (IMA), which has less capability of binding to cobalt. Thus, the amount of free cobalt in the system will be determined by the presence of a normal or ischemic albumin in the sample.

Second Reaction - The entry of cobalt in the cell

There are four cobalt transporters in the cell. The rcnA (Uniprot code: P76425) and zntA (Uniprot code: P37617) transport the cobalt to outside the cell. The corA (Uniprot code: P0ABI4) and zupT (Uniprot code: P0A8H3) transport the cobalt to inside the cell.

The equation for the cobalt flux through these transporters is:

Third Reaction - Transcription

In our modeling, we considered the cobalt acting as a transcriptional activator in order to simplify our model.

The transcriptional activation are accessed by the equation:

The transcription and translation are accesed by the equations:

Model simulations

Figure - Modeling simulation of IMA system

The first simulation of our model showed that the outside concentration of cobalt reaches an equilibrium with the inside cobalt concentration of the bacterial cell. After a while, the Yellow Fluorescent Protein starts to appear.

Figure - Modeling simulation of IMA system with different concentrations of cobalt

Figure - Modeling simulation of IMA system with different concentrations of Albumin

Model optimization

We have adapted an Evolutionary Algorithm to seek for the constants that best fit our model to the experimental results.

When we have functions that have an nonlinear behaviour, seeking for global optimal value, can be challenging and many types of functions and mathematical models have no known methods for this purpose. In these cases, it is fairly common to use stochastic methods, like Evolutionary Algorithms, to find interesting results within functions and models.

An [http://en.wikipedia.org/wiki/Evolutionary_algorithm Evolutionary Algorithm] is a method in which each solution represents an individual and by crossing and mutating them, we can reach solutions that better fit our purpose.

An schema, about how the Evolutionary Algorithm used our optimization works, is presented below:

Figure - Evolutionary Algorithm schema [http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=6557822&isnumber=6557545 Coelho, et al, 2013]

To fit the constants to the modeling, we compared the curve in our modeling to the curve obtained from experimental data and tried to reduce the difference between the curves.

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