Team:UFAM Brazil/Modeling

For better comprehension and organization of the mathematics modeling, each experimental situation of our will be analyzed separately: Biosensoring, Bioaccumulation e Bioremediation.

BIOACCUMULATION AND BIOREMEDIATION

The system for capturing of ions Hg²⁺ into the cell and it consequent reduction for Hg⁰ it is a complex process that has several steps for its realization. Such procedure aims to transform mercury ion in a volatile element (Hg⁰) capable of passive diffusion through membrane to cell’s exterior.

According to Picture 1, the capturing and reduction system work, basically, with proteins merP, merT, merC, merF and merA, where the first one a periplasmic protein, that binds to mercury ion to carry it to one of the transporters (merT, merC ou merF), that are localized at the inner membrane, and release Hg²⁺ to intracellular. Then it will get under MerA enzyme action that is capable of reduce Hg2+ into Hg0, making it possible to get out passively through cell membrane.

In order to simplify such processes, those were written as a sequence of chemical reactions where it was considered that the formation of complexes enzyme-substrate and mer proteins were in balance.

Picture 01: Mer gene’s action on bacterium cell

• Hg²⁺ OUT – Mercury concentration outside the cell

• CI – Complex concentration Hg2+-merP

• CII – Complex concentration Hg2+-merP-merT

• CIII – Complex concentration Hg2+-merP-merC

• CIV – Complex concentration Hg2+-merP-merF

As stated before, all this supposing that the reaction I is in balance.

To keep it simple, it is supposed that just one of Hg2+ transporters to cell interior was active, in this example, the protein merT. Calculating formation speed of the elements involved in ion transport to cell interior:

Supposing, again, the complex formation CII in balance:

Although,

Therefore,

This way the speed of Hg²⁺ uptake to the cell interior, supposing only merT function, is given by:

Finally, the absolute Hg2+ uptake from bacteria on intervals 0 until a general time T is given by the integration of the equation above:

Similarly, it is for exclusive function of merC or merF:

Now considering the reduction process of Hg2+ into Hg0, we have the general equation:

Supposing the balance for complex CV formation:

Finally, the mercury ion reduction speed is given by the expression:

The quantity of reduced mercury on time 0 until a general time T is given by:

BIOSENSORING

On our project, experiments related to biosensoring are based on the idea of allowing bacteria growth of our construction (RTP + GFP) in medium containing different mercury concentrations and, by this, analyze fluorescence levels derived from GFP protein expression.

In order to simplify, we’re considering that Hg²⁺ uptake is mediated only by merT transporter, by the following equation:

Considering that Hg²⁺ is a GFP transcription factor, we assume that the transcriptional mRNA activation speed is given by:

mRNA transcription rate is given by:

And finally, GFP synthesis rate in function of mRNA ratees:

Beside the model values to quantify GFP cell production, it’s necessary to study models to quantify photons’ emission by GFP. Photons emission’s rate equation is given by the function of the constants, that are quantum yield (α), medium volume (V), Avogrado number (A) and GFP synthesis rate.

Photons emission rate is given by the equation:

Although, it is already known, GFP formation rate in function of mercury uptake rates and mRNA transcription rate from Hg2+ transcription factor, so, we have:

Photons production rate is directly proportional to light intensity measured by the spectrofluorometer, which measures the experiment data.

To know how many photons are released in a given time T, it is just necessary to do:

For better comprehension and organization of the mathematics modeling, each experimental situation of our will be analyzed separately: Biosensoring, Bioaccumulation e Bioremediation.

Predictive model for protein expression

We use the GFP data for the factors concentration and time. Data was provided for specific experiments and we plotted the graphs for time vs concentration.

From this plot we infer that the equation that fitted our model would be and exponential equation. We transferred the data to the Mathematical software and organized the factors as time=x1 and concentration=x2.

Using a non-linear regression for the observe patterns. First we plot a 3d graph for the analyzed data and then we fed a model of the form.

Where GFPi is the data that we obtained, IC is the initial mercury concentration and t is the time.

Therefore, our equation for the fitted model is

With this equation we can infer the intensity of GFP given any time or any initial concentration of mercury, then using the equation reported by JCBRAFF(BBa_E0040) which is

Where y = GFP intensity and x = GFP concentration in nanomolar, we can infer the amount of GFP protein produced by the cell and finally have the ability to predict the concentration of our bioremediation proteins since the concentration of GFP is the same that this proteins. By the equation:

Since the GFPi is given in µg/ml, we have to divide the equation for the Mol weight of Hg.