Team:UFAM Brazil/Modeling

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<tr><td colspan="3" align="justify"><p>
<tr><td colspan="3" align="justify"><p>
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<p>For better comprehension and organization of the mathematics modeling, each experimental situation of our will be analyzed separately:
 +
<a href="https://2014.igem.org/Team:UFAM_Brazil/Biosensor">Biosensoring</a>,
 +
<a href="https://2014.igem.org/Team:UFAM_Brazil/Bioaccumulation">Bioaccumulation</a> e
 +
<a href="https://2014.igem.org/Team:UFAM_Brazil/Bioremediation">Bioremediation</a>. </p>
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The Hg<sup>2+</sup> ions uptake and entrance from the cell exterior to the interior system and its subsequent reduction to Hg<sup>0</sup> species is a complex process that has several steps for its realization. This procedure aims to transform the mercury ion in a volatile element (Hg<sup>0</sup>), able to passively diffuse through the membranes to the cell exterior.
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</td></tr>
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<tr><td colspan="3" align="justify">
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<h3 align="center">BIOACCUMULATION AND BIOREMEDIATION</h3>
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 +
<p>
 +
The system for capturing of ions Hg<sup>2+</sup> into the cell and it consequent reduction for Hg<sup>0</sup> it is a complex process that has several steps for its realization. Such procedure aims to transform mercury ion in a volatile element (Hg<sup>0</sup>) capable of passive diffusion through membrane to cell’s exterior.
</p>
</p>
<p>
<p>
-
According to Figure 1, the uptake and reduction system operates basically with merP, merT, merC, merF and merA proteins, being the first a periplasmic protein that binds to mercury ion for carrying to one of its transporters (merT, merC or merF), which are located in the inner membrane, and release Hg<sup>2+</sup> inside the cells. Then, Hg<sup>2+</sup> undergoes the merA enzyme action, which can reduce Hg<sup>2+</sup> to Hg<sup>0</sup>, thereby enables its passive exist by membrane.
+
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 Hg<sup>2+</sup> 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.  
</p>
</p>
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</td></tr>
</td></tr>
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<tr><td  colspan="3" align="center"><p> Figure 01: Mer genes action in bacterial cell</p></td>
 
</tr>
</tr>
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<tr><td colspan="3" align="justify"><p> For the purpose of simplifying such processes, these are written in a sequence of chemical reactions, considering the enzyme complexes formation of mer substrates and proteins were always in equilibrium.</p></td>
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<tr><td colspan="3" align="justify"><p> 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.</p></td>
</tr>
</tr>
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<img src="https://static.igem.org/mediawiki/2014/a/a9/Img2.png" width="500">
<img src="https://static.igem.org/mediawiki/2014/a/a9/Img2.png" width="500">
</td></tr>
</td></tr>
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<tr><td  colspan="3" align="center"><p> Picture 01: Mer gene’s action on bacterium cell</p></td></tr>
<tr><td colspan="3" align="justify">
<tr><td colspan="3" align="justify">
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<p style="margin-left:50px">• Hg<sup>2+</sup> OUT – Mercury concentration outside the cell</p>
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<p style="margin-left:150px">• Hg<sup>2+</sup> OUT – Mercury concentration outside the cell</p>
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<p style="margin-left:50px" • merP, merT, merC e merF – Respective proteins concentrations</p>
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<p style="margin-left:150px" • merP, merT, merC e merF – Respective proteins concentrations</p>
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<p style="margin-left:50px">• CI – Complex concentration Hg2+-merP</p>
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<p style="margin-left:150px">• CI – Complex concentration Hg2+-merP</p>
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<p style="margin-left:50px">• CII – Complex concentration Hg2+-merP-merT</p>
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<p style="margin-left:150px">• CII – Complex concentration Hg2+-merP-merT</p>
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<p style="margin-left:50px">• CIII – Complex concentration Hg2+-merP-merC</p>
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<p style="margin-left:150px">• CIII – Complex concentration Hg2+-merP-merC</p>
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<p style="margin-left:50px">• CIV – Complex concentration Hg2+-merP-merF</p>
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<p style="margin-left:150px">• CIV – Complex concentration Hg2+-merP-merF</p>
</td>
</td>
</tr>
</tr>
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<tr><td  colspan="3" align="center"><p>So, as discussed earlier, assuming the reaction I in equilibrium:</p></td>
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<tr><td  colspan="3" align="center"><p>As stated before, all this supposing that the reaction I is in balance.</p></td></tr>
<tr><td  colspan="3" align="center">
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<img src="https://static.igem.org/mediawiki/2014/2/24/UFAM_Brazil_img4.png" width="350">
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<img src="https://static.igem.org/mediawiki/2014/2/28/UFAM_Brazil_img8.png" width="250">
</td></tr>
</td></tr>
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<tr><td colspan="3" align="justify"><p>Simplifing, it was assumed that only one Hg2+ transporter was active, for this example, the merT protein. Calculating the elements formation speeds involved in the mercury ions transport into the cell:</p></td>
+
<tr><td colspan="3" align="justify"><p>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:</p></td></tr>
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</tr>
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 +
 
<tr><td  colspan="3" align="center">
<tr><td  colspan="3" align="center">
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<img src="https://static.igem.org/mediawiki/2014/e/ea/UFAM_Brazil_Img5.png" width="350">
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<img src="https://static.igem.org/mediawiki/2014/2/24/UFAM_Brazil_img4.png" width="350">
</td></tr>
</td></tr>
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<tr><td colspan="3" align="center"><p>Supposing, again, the complex formation CII in balance:</p></td>
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<tr><td colspan="3" align="center"><p>Assuming CII  complex formation in equilibrium, again:</p></td>
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</tr>
<tr><td  colspan="3" align="center">
<tr><td  colspan="3" align="center">
<img src="https://static.igem.org/mediawiki/2014/f/f8/UFAM_Brazil_img6.png" width="350">
<img src="https://static.igem.org/mediawiki/2014/f/f8/UFAM_Brazil_img6.png" width="350">
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<tr><td  colspan="1" align="center"><p>However,</p></td>
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<tr><td  colspan="1" align="center"><p>Although,</p></td></tr>
<tr><td  colspan="3" align="center">
<tr><td  colspan="3" align="center">
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</td></tr>
</td></tr>
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<tr><td  colspan="1" align="center"><p>Therefore,</p></td>
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<tr><td  colspan="1" align="center"><p>Therefore,</p></td></tr>
<tr><td  colspan="3" align="center">
<tr><td  colspan="3" align="center">
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</td></tr>
</td></tr>
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<tr><td  colspan="1" align="center"><p>Thus, Hg<sup>2+</sup> uptake into the cell speed, assuming functionality of only merT, is given by:</p></td>
+
<tr><td  colspan="1" align="center"><p>This way the speed of Hg<sup>2+</sup> uptake to the cell interior, supposing only merT function, is given by:</p></td></tr>
<tr><td  colspan="3" align="center">
<tr><td  colspan="3" align="center">
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</td></tr>
</td></tr>
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<tr><td  colspan="1" align="center"><p>Similarly, for exclusive functionality to merC or merF it is given:</p></td>
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<tr><td  colspan="1" align="center"><p>Finally, the absolute Hg2+ uptake from bacteria on intervals 0 until a general time T is given by the integration of the equation above:</p></td></tr>
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 +
 
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<tr><td  colspan="3" align="center">
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<img src="https://static.igem.org/mediawiki/2014/7/72/UFAM_Brazil_imgNova1.png" width="400">
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</td></tr>
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<tr><td  colspan="1" align="center"><p>Similarly, it is for exclusive function of merC or merF:</p></td></tr>
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<tr><td  colspan="3" align="center">
<tr><td  colspan="3" align="center">
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</td></tr>
</td></tr>
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<tr><td  colspan="1" align="center"><p>Now taking into account the reduction process of Hg2+ to Hg0, we have this general equation:</p></td>
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<tr><td  colspan="3" align="center">
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<img src="https://static.igem.org/mediawiki/2014/b/b8/UFAM_Brazil_imgNova2.png" width="400">
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</td></tr>
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<tr><td  colspan="3" align="center">
<tr><td  colspan="3" align="center">
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</td></tr>
</td></tr>
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<tr><td  colspan="3" align="center">
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<img src="https://static.igem.org/mediawiki/2014/5/53/UFAM_Brazil_imgNova3.png" width="400">
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</td></tr>
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<tr><td  colspan="1" align="center"><p>Now considering the reduction process of Hg2+ into Hg0, we have the general equation:</p></td></tr>
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<tr><td  colspan="1" align="center"><p>Assuming CV complex formation in equilibrium:</p></td>
 
<tr><td  colspan="3" align="center">
<tr><td  colspan="3" align="center">
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<img src="https://static.igem.org/mediawiki/2014/2/2b/UFAM_Brazil_img13.png" width="400">
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<img src="https://static.igem.org/mediawiki/2014/2/2b/UFAM_Brazil_img13.png" width="600">
</td></tr>
</td></tr>
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<tr><td  colspan="1" align="center"><p>Finally, the reduction rate of mercury ion is given by:</p></td>
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<tr><td  colspan="1" align="center"><p>Supposing the balance for complex CV formation: </p></td></tr>
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<tr><td  colspan="3" align="center">
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<img src="https://static.igem.org/mediawiki/2014/b/bc/UFAM_Brazil_img14.png" width="350">
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</td></tr>
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<tr><td  colspan="1" align="center"><p>Finally, the mercury ion reduction speed is given by the expression:</p></td></tr>
<tr><td  colspan="3" align="center">
<tr><td  colspan="3" align="center">
<img src="https://static.igem.org/mediawiki/2014/d/d4/UFAM_Brazil_img15.png" width="400">
<img src="https://static.igem.org/mediawiki/2014/d/d4/UFAM_Brazil_img15.png" width="400">
</td></tr>
</td></tr>
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<tr><td  colspan="1" align="center"><p>The quantity of reduced mercury on time 0 until a general time T is given by:</p></td></tr>
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<tr><td  colspan="3" align="center">
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<img src="https://static.igem.org/mediawiki/2014/7/74/UFAM_Brazil_img16.png" width="400">
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</td></tr>
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<tr><td  colspan="1" align="justify">
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<h3 align="center">BIOSENSORING</h3>
 +
 +
<p>On our project, experiments related to <i>biosensoring</i> 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.</p>
 +
 +
<p>In order to simplify, we’re considering that Hg<sup>2+</sup> uptake is mediated only by merT transporter, by the following equation:</p>
 +
 +
</td>
 +
</tr>
 +
 +
<tr><td  colspan="3" align="center">
 +
<img src="https://static.igem.org/mediawiki/2014/2/29/UFAM_Brazil_img17.png" width="400">
 +
</td></tr>
 +
 +
<tr><td  colspan="1" align="center"><p>Considering that Hg<sup>2+</sup> is a GFP transcription factor, we assume that the transcriptional mRNA activation speed is given by:</p></td></tr>
 +
 +
<tr><td  colspan="3" align="center">
 +
<img src="https://static.igem.org/mediawiki/2014/6/69/UFAM_Brazil_img18.png" width="400">
 +
</td></tr>
 +
 +
 +
<tr><td  colspan="1" align="center"><p>mRNA transcription rate is given by:</p></td></tr>
 +
 +
<tr><td  colspan="3" align="center">
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<img src="https://static.igem.org/mediawiki/2014/7/72/UFAM_Brazil_img19.png" width="400">
 +
</td></tr>
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 +
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<tr><td  colspan="1" align="center"><p>And finally, GFP synthesis rate in function of mRNA ratees:</p></td></tr>
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 +
<tr><td  colspan="3" align="center">
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<img src="https://static.igem.org/mediawiki/2014/b/ba/UFAM_Brazil_2014_Img20.png" width="400">
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</td></tr>
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<tr><td  colspan="1" align="justify"><p>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.</p></td></tr>
 +
 +
<tr><td  colspan="1" align="center"><p>Photons emission rate is given by the equation:</p></td></tr>
 +
 +
<tr><td  colspan="3" align="center">
 +
<img src="https://static.igem.org/mediawiki/2014/f/f8/UFAM_Brazil_2014_Img21.png" width="300">
 +
</td></tr>
 +
 +
<tr><td  colspan="1" align="justify"><p>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:</p></td></tr>
 +
 +
<tr><td  colspan="3" align="center">
 +
<img src="https://static.igem.org/mediawiki/2014/c/c3/UFAM_Brazil_2014_Img22.png" width="400">
 +
</td></tr>
 +
 +
 +
<tr><td  colspan="1" align="justify">
 +
 +
<p>Photons production rate is directly proportional to light intensity measured by the spectrofluorometer, which measures the experiment data.</p>
 +
 +
<p>To know how many photons are released in a given time T, it is just necessary to do:</p>
 +
</td></tr>
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 +
<tr><td  colspan="3" align="center">
 +
<img src="https://static.igem.org/mediawiki/2014/c/c1/UFAM_Brazil_2014_Img23.png" width="400">
 +
</td></tr>
 +
 +
<tr><td colspan="3" align="justify"><p>
 +
<p>For better comprehension and organization of the mathematics modeling, each experimental situation of our will be analyzed separately:
 +
<a href="https://2014.igem.org/Team:UFAM_Brazil/Biosensor">Biosensoring</a>,
 +
<a href="https://2014.igem.org/Team:UFAM_Brazil/Bioaccumulation">Bioaccumulation</a> e
 +
<a href="https://2014.igem.org/Team:UFAM_Brazil/Bioremediation">Bioremediation</a>. </p>
 +
 +
</td></tr>
 +
 +
 +
 +
<tr><td colspan="3" align="justify">
 +
<h3 align="center">Predictive model for protein expression</h3>
 +
 +
<p>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. </p>
 +
 +
<p align="center"><img src="https://static.igem.org/mediawiki/2014/e/ec/UFAM_BRAZIL_2014_mod1.png" width="500"></p>
 +
 +
 +
<p>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. </p>
 +
 +
<p align="center"><img src="https://static.igem.org/mediawiki/2014/e/e4/UFAM_BRAZIL_2014_mod2.png" width="500"></p>
 +
 +
 +
<p>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.</p>
 +
 +
<p align="center"><img src="https://static.igem.org/mediawiki/2014/0/0e/UFAM_BRAZIL_2014_mod3.png" width="150"></p>
 +
 +
 +
<p>Where GFPi is the data that we obtained, IC is the initial mercury concentration and t is the time. </p>
 +
 +
<p>Therefore, our equation for the fitted model is</p>
 +
 +
 +
<p align="center"><img src="https://static.igem.org/mediawiki/2014/d/d0/UFAM_BRAZIL_2014_mod4.png" width="350"></p>
 +
 +
 +
<p>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 </p>
 +
 +
 +
<p align="center"><img src="https://static.igem.org/mediawiki/2014/3/3c/UFAM_BRAZIL_2014_mod5.png" width="150"></p>
 +
 +
 +
<p>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:</p>
 +
 +
<p align="center"><img src="https://static.igem.org/mediawiki/2014/a/a8/UFAM_BRAZIL_2014_mod6.png" width="300"></p>
 +
 +
 +
<p>Since the GFPi is given in µg/ml, we have to divide the equation for the Mol weight of Hg.</p>
 +
 +
 +
<p align="center"><img  src="https://static.igem.org/mediawiki/2014/8/80/UFAM_BRAZIL_2014_mod7.png" width="350"></p>
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 +
</td></tr>
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 +

Latest revision as of 17:09, 17 October 2014

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.