Team:UANL Mty-Mexico/Modeling

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<p><br><h2><a name="Transport"></a>Transport and accumulation</p></h2></br>
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<ul class="nav nav-tabs">
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  <li class="active"><a href="#Introduction">Introduction</a></li>
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  <li><a href="#Types_of_RNAT">Types of RNAT</a></li>
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  <li><a href="#Project">Project</a></li>
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  <li class="dropdown">
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    <a class="dropdown-toggle" data-toggle="dropdown" href="#">
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      Circuit <span class="caret"></span>
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    </a>
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    <ul class="dropdown-menu">
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        <li><a href="#Circuit">Circuit</a></li>
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        <li><a href="#Results_overview">Results overview</a></li>
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    </ul>
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  </li>
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  <li><a href="#Conclusions">Conclusions</a></li>
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  <li><a href="#Perspectives">Perspectives</a></li>
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  <li><a href="#References">References</a></li>
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</ul>
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<p>Before us, team iGEM Groningen 2009 made a model for an arsenic accumulator at the population level; that is, they set some ODEs that represent the change on the total intracellular arsenic (considering not a single cell, but the whole culture, or more exactly, the total cell volume) with respect to time. Nevertheless, as the precise value for some parameters were unavailable, specially for the ArsB effect, part of their model remains aparameterized and they perform a quasi-steady state analysis.</p>
 
<p>After considering the effect of their metallothioneins (As-binding proteins), GlpF, ArsB and ArsR, they ended with the following time derivative:</p>
<p>After considering the effect of their metallothioneins (As-binding proteins), GlpF, ArsB and ArsR, they ended with the following time derivative:</p>
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\end{equation}
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<p></br></p>
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<p><a name="Introduction"><h3>Introduction</h3></a><hr></p>
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<div class="justified">
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RNA thermometers (RNATs) are RNA sequences that range from 40 to more than a 100 nucleotides commonly found in the 5' untranslated region of some genes and that regulate in <i>cis</i> their translation without the need of other factors [<a href="http://www.ncbi.nlm.nih.gov/pubmed/22421878">Kortmann and Narberhaus, (2012)</a>; <a href="http://www.ncbi.nlm.nih.gov/pubmed/20009504">Narberhaus, (2009)</a>]. These RNAT sequences show certain three dimensional structures, some of which interact with the ribosome binding site (RBS) of their regulated genes and hinders the proccessivity of the ribosome complex at certain temperatures. The dynamics of the formation of these structures is temperature dependent and is the basis of the regulation of the translation rate of a given transcript [<a href="http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1478195/">Chowdhury, S., <i>et al</i>.,(2006)</a>; <a href="http://www.ncbi.nlm.nih.gov/pubmed/16438677">Narberhaus, F., <i>et al</i>.,(2006)</a>].</p>
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<p align="justify" class="margin">Functional RNAT have been found in different organisms, mainly pathogenic bacteria, and many others have been predicted in almost everyfrom a number of bioinformatic studies. They have been found to regulate the expression of virulence factors, heat and cold shock response factors and even proteins involved the development of some bacteriophages.</p>
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<p align="justify" class="margin">Their apparent widespread presence in living organisms has made RNATs attractive for some applications, specially the ones related to the replacement of chemical inducers and for the development of new drugs.</p>
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<p align="justify" class="margin">However, from the experience of those who have been working extensively with RNAT in the later years, the accurate bioinformatic prediction of functional RNAT has proven to be an exceptionally difficult task; the reasons for this are pointed to be the poor sequence conservation observed among RNATs and the gaps in our current understanding of the RNAT function, their structural diversity and the effect of other regulatory sequences far from the RBS region [<a href="http://www.ncbi.nlm.nih.gov/pubmed/22421878">Kortmann and Narberhaus, (2012)</a>; <a href="http://www.ncbi.nlm.nih.gov/pubmed/17647020">Waldminghaus, <i>et al</i>., (2007)</a>].
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<p align="justify" class="margin">The discovery of new RNATs has relied on a mixed approach that involves bioinformatics and experimental validation, as well as approaches that involve mutational libraries, synthetic constructions and directed evolution.</p>
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<p align="justify" class="margin">Even when the naturally found RNATs usually regulate the expression of transcription factors, the synthetic constructions made so far have focused mainly to characterize the effect of a given RNAT using a reporter protein (LacZ or a fluorescent protein) directly downstream of a RNAT. In our work, we intend to prove that RNATs can also be employed to effectively regulate the expression of transcription factors in synthetic circuits and point at possible applications for the circuit topologies that would be made feasible with this new kind of synthetic regulatory device.</p></div>
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<p>Where <i>As(III)<sub>in</sub></i> is the total intracellular arsenic; </i> <i>ArsR<sub>As</sub></i>, <i>MBPArsR<sub>As</sub></i>, <i>fMT<sub>As</sub></i>, <i>ArsB<sub>As</sub></i> and <i>GlpF<sub>As</sub></i> are the arsenic bound proteins; <i>n<sub>f</sub></i> is the Hill coefficient for the interaction between As and fMT; <i>k<sub>1</sub></i> and <i>k<sub>2</sub></i> are the kinetic constants for the interaction between As and ArsB and GlpF, respectively; finally, <i>Vs/Vc</i> represents the proportion between the total solution volume (Vs) and the total cell volume (Vc).</p>
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<div id="title">
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<a name="Types_of_RNAT"><h3>Types of RNA thermometers&nbsp;&nbsp;<a href="#" class="btn btn-info"><font color="#fff">Back to top</font></a></h3></div>
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<div id="paragraph">
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<p align="justify" class="margin">Although RNATs show almost no sequence similarity among them, a number of structural features can be used to classify them. Here we enlist the most described RNATs structural groups described to date [<a href="http://www.ncbi.nlm.nih.gov/pubmed/22421878">Kortmann and Narberhaus, (2012)</a>]:</p>
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<ol class="margin" align="justify">
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<li><b>ROSE.-</b> ROSE stands for "Regulation Of heat Shock Expression". ROSE elements are 60 to >100 nucleotide sequences found upstream of heat shock proteins. They have been found to be conserved in alpha and gamma-proteobacteria. Among the structural features of the ROSE element family are: a) their folding in 2 to 4 stemloop structures; b) a short conserved sequence (UU/CGCU) near the Shine-Dalgarno sequence; and c) the presence of a number of non-cannonical base interactions (the G83-G94 pair; a triple bair among U96-C80-C81; the U79-U97; and the interaction of the AUG codon and C71, G72 and U73. Functional ROSE elements have been found in <i>E. coli</i> (<i>rpoH</i> and <i>ibpA</i>) and <i>B. japonicum</i> (<i>hspA</i>).</li>
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</br><li><b>FourU elements.-</b> these elements are characterized by a short motif composed of four uridines that pair with the Shine-Dalgarno region and is embedded in a hairpin that shows temperature-induced conformational changes. FourU elements have only one A-G non-cannonical base interaction. Among the structural features that characterize FourU elements are a) the A-G pair and b) the G34-C46 pair that regulates melting. Functional FourU elements have been described in <i>Salmonella</i> (<i>agsA</i>) and <i>Yersina pseudotuberculosis</i> (<i>lcrF</i>).</li>
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</br><li><b>Synechocystis <i>hsp17</i> element.-</b> with a length of 46 nucleotides, this is the shortest RNAT described so far. The distinctive structural features essential for the function of this element are a) the pairing of a UCCU sequence with the AGGA in the Shine-Dalgarno sequence and b) the presence of two loops in its stems.</li>
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</br><li><b>Coding region spanning RNATs.-</b> RNATs are not exclusively found in the 5'UTR of genes; they can also span into the coding region and even be intergenic. Functional coding region spanning RNATs have been found in <i>E. coli</i> (<i>rpoH</i>), phage lambda (<i>cIII</i>) and <i>Lysteria monocytogenes</i> (<i>prfA</i>).</li>
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</br><li><b>Cold shock RNATs.-</b> cold shock RNATs also depend on the dynamics of the folding of different loops, but in contrast to heat shock RNATs, the conformation that prevents the binding of the ribosome is found at high temperatures, while at low temperatures, the RNAT folds into a conformation that allows for the ribosome to proceed. An example of a cold shock RNAT is the element found upstream and inside the coding region of <i>E. coli</i> gene <i>cspA</i>.</li>
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</ol>
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</div>
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<div id="title">
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<a name="Project"><h3>The <i>ThermoColi</i> project&nbsp;&nbsp;<a href="#" class="btn btn-info"><font color="#fff">Back to top</font></a></h3></div>
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<p><br><h2><a name="core_model"></a>Core model</h2><br></p>
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<div id="subtitle"><a name="Circuit"><h4>Circuit description&nbsp;&nbsp;<a href="#" class="btn btn-info"><font color="#fff">Back to top</font></a></h4></div>
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<div id="subtitle"><a name="Results_overview"><h4>Results overview&nbsp;&nbsp;<a href="#" class="btn btn-info"><font color="#fff">Back to top</font></a></h4></div>
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<p>We built upon their model and made the following modifications, which we'll call the "core modifications" from now on:</p>
 
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<OL TYPE = "1">
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<div id="title">
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<LI>We assume that ArsB is non functional, so that the only protein affecting As transport is GlpF.
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<a name="Conclusions"><h3>Conclusions&nbsp;&nbsp;<a href="#" class="btn btn-info"><font color="#fff">Back to top</font></a></h3></div>
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<LI>GlpF effect is masked by the population level kinetics.
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<LI>We assume that the intracellular As concentration and the GlpF effect at the population level (that is, considering total cell volume) are homogeneously distributed and should be the same as in a single cell.
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<LI>The protein MBPArsR is not present in our system, so the variable <i>MBPArsR<sub>As</sub></i> is not considered for our model.
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</OL></br>
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<p>The next equation shows the application of those modifications:</p>
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<div id="title">
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<a name="Perspectives"><h3>Perspectives&nbsp;&nbsp;<a href="#" class="btn btn-info"><font color="#fff">Back to top</font></a></h3></div>
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<div id="title">
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<p>
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<a name="References"><h3>References&nbsp;&nbsp;<a href="#" class="btn btn-info"><font color="#fff">Back to top</font></a></h3></div>
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\begin{equation}
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</body>
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\large\frac{\mathrm{d[As(III)in] } }{\mathrm{d} x} = -ArsR_{As} -n_{f}\cdot fMT_{As} + (\frac{V_s}{V_c}) \cdot V_{max} \cdot \frac{As_{ex}}{K_{t}+As_{ex}}
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\end{equation}
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<br>
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<p>Now, let us introduce a variable called <i>As<sub>TOTALin</sub></i>, which represents the total amount of arsenic inside a cell (recall core modification number 2), whether free or bound to whatever protein. Let's also call <i>As<sub>FREEin</sub></i> the amount of free intracellular arsenic and <i>As<sub>BOUNDin</sub></i> the protein-bound As</i>. In this way, <i>As<sub>TOTALin</sub></i> can be represented as follows:</br>
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\begin{equation}
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\large[As_{TOTALin}] = [As_{FREEin}] + [As_{BOUNDin}]
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\end{equation} </br></p>
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<br><p>In the iGEM Groningen 2009 model, <i>As<sub>in</sub></i> represents the free intracellular arsenic, as this variable has negative terms related to the binding of As to proteins; to avoid further confusions, we'll establish this equivalence as follows: <i>As<sub>FREEin</sub></i>, so that equation 2 changes to:
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</p>
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<br>
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\begin{equation}
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\large[As_{FREEin}] = [As_{in(iGEM Groningen 2009)}]
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\end{equation} </br></p>
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</br>
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<p>If we further analyze the <i>As<sub>BOUNDin</sub></i> variable, considering that in our system only ArsR and a methalothionein (which we'll simply call MT) are being expressed, then equation 5 turns to be:</p>
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<br>
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\begin{equation}
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\large\large[As_{TOTALin}] = h_{2}[ArsR|As] + h_{3}[MT|As]
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\end{equation} </br></p>
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</br>
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<p>Where <i>h<sub>2</sub></i> and <i>h<sub>3</sub></i> are the Hill coefficients for the interaction between As and ArsR and MT, respectively.
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Finally, taking into consideration the work of team Cambridge 2009 and assuming that equilibrium is reached quickly, we can describe the formation <i>k<sub>ArsR|As</sub></i> and <i>k<sub>MT|As</sub></i> as follows:
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<br>
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\begin{equation}
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\large \frac{d[ArsR|As]}{dt} = (ArsR_{TOTAL}) - (\frac{[ArsR_{TOTAL}]}{1+(\frac{[As_{FREEin}])}{k_{[ArsR|As]}})^{h_2}} - \delta_{ArsR|As}[ArsR|As]
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\end{equation}
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<br>
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\begin{equation}
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\large \frac{d[ArsR_{FREE}]}{dt} = (\frac{[ArsR_{TOTAL}]}{1+(\frac{[As_{FREEin}])}{k_{[ArsR|As]}})^{h_2}} - \delta_{ArsR|As}[ArsR|As]
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\end{equation}
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<br><p>
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\begin{equation}
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\large \frac{d[MT|As]}{dt} = (MT_{TOTAL})-(\frac{[MT_{TOTAL}]}{1+(\frac{[As_{FREEin}])}{k_{[MT|As]}})^{h_3}} - \delta_{MT|As}[MT|As]
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\end{equation}
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<br><p>
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\begin{equation}
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\large \frac{d[MT_{FREE}]}{dt} = (\frac{[MT_{TOTAL}]}{1+(\frac{[As_{FREEin}])}{k_{[MT|As]}})^{h_3}} - \delta_{MT|As}[MT|As]
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\end{equation}
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</br></p>
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<p>In equations 5 to 7, <i>k<sub>ArsR|As</sub></i> and <i>k<sub>MT|As</sub></i> are the kinetic constants for the interaction of arsenic with ArsR and MT, respectively, using a different nomenclature as in equations 1 and 2; here, in equations 6 and 7, the binding of two molecules is represented as "moleculeA|moleculeB". The indexes <i>h2</i> and <i>h3</i> are the Hill coefficients for the interaction between arsenic and ArsR and MT, respectively. The deltas are the degradation constants for the protein|As complexes. The unbound As that results from complex degradation then goes to the <i>As<sub>FREEin</sub></i> pool and is ready to bind again available ArsR or MT.</p>
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<p><br><h3><a name="ODEANT"></a>ODEs</h3></p><hr align="center" width="33%"/></br></hr>
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<p>The core modifications and equation 6 allow us to propose a set of ODEs that describe the change of the concentrations of intracellular As, ArsR|As, MT|As and the unbound protein species. </p>
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<br><p><b>Core model ODEs</b></p>
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<p><b><i>mRNAs
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</p></b></i>
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\begin{equation}
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\large \frac{d[mRNA_{ArsR}]}{dt} = \alpha _{mArsR}\cdot (pro_{ars})\cdot(\frac{k_{D1}^{h_{1}}}{k_{D1}^{h_{1}}+[ArsR]^{h_{1}}})- \delta _{mRNA_{ArsR}}[mRNA_{ArsR}]
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\end{equation} </br></p>
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\begin{equation}
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\large \frac{d[mRNA_{MT}]}{dt} = \alpha _{mMT}\cdot(pro_{cons})- \delta _{mRNA_{MT}}[mRNA_{MT}]
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\end{equation} </br></p>
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<p><b><i>Proteins
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</p></b></i>
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\begin{equation}
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\large \frac{d[ArsR]}{dt} = \alpha _{pArsR}\cdot[mRNA_{ArsR}]- \delta _{ArsR}[ArsR] - [ArsR|As]
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\end{equation} </br></p>
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\begin{equation}
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\large \frac{d[MT]}{dt} = \alpha _{pMT}\cdot[mRNA_{MT}]- \delta _{MT}[MT] - [MT|As]
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\end{equation} </br></p>
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<p><b><i>Proteins with arsenic
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</p></b></i>
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<p>See equations 7 and 8</p>
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<p><b><i>Arsenic
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</p></b></i>
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\begin{equation}
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\large\frac{\mathrm{d[As_{FREEin}] } }{\mathrm{dt}} = (\frac{V_s}{V_c} \cdot V_{max} \cdot \frac{As_{e}}{K_{t}+As_{e}} + h_2 \delta _{ArsR|As}[ArsR|As] + h_3 \delta _{MT|As}[MT|As])
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\end{equation}
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<p><br><h3><a name="parametersANT"></a>Parameters</h3></p><hr align="center" width="33%"/></br>
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<br><p><b>Transcription Rates</b></p>
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<p>Based on the arithmetic average of five experimental references for the mean transcription rate in E. coli, we used in our model a value of 48.18 nt/s, or 2890.8 nt/min [<a href="http://www.ncbi.nlm.nih.gov/pmc/articles/PMC209006/">Gotta, Miller Jr and French (1991),</a>, <a href="http://www.ncbi.nlm.nih.gov/pubmed/7514589/">Vogel and Jensen (1994)</a>, <a href="https://ctbp.ucsd.edu/qbio/beemer96.pdf"> Bremer and Dennis, (1996) in Neidhardt, <i>et al</i>.</a>,<a href="http://openwetware.org/wiki/Computational_Biology/Gene_Expression_modeling/Stochastic_Approach"> Gene Expression Modelling</a>] for this parameter. Assuming that transcription for all genes occurs at this speed, and based on the supposition that 1 nM equals one molecule per cell in E. coli (<a href="http://bionumbers.hms.harvard.edu/Includes/KeyNumbersLinks.pdf">Bionumbers</a>), we propose the following equation to estimate the maximum transcription rate for every transcriptional unit in the model:
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</p>
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<p>
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<i>Maximum transcription rate = Average transcription speed (2890.8 nt/min)*Number of copies of the plasmid</i>
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</p>
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<p> The size of a transcriptional unit takes into account the CDS plus 3' and 5' untranslated regions.
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</p>
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<br><p><b>Translation Rates</b></p>
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<p>A similar process was followed to calculate the maximum translation rates for all the proteins in our model. Using an average translation rate of 19 aa/s, or 1140 aa/min [<a href="https://ctbp.ucsd.edu/qbio/beemer96.pdf">Bremer and Dennis, (1996) in Neidhardt, <i>et al</i>.</a>], and assuming that all translation in our system works at the same speed, the maximum translation rate can be written as:
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</p>
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<p><i>Maximum translation rate = Average translation rate (1400 aa/min⁡〖)/〗 [Protein size (aa)]*RBS strenght(Assumed as 1 due to lack of data)</i>
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</p>
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<br><p><b>mRNA and Protein degradation rates</b></p>
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<p>The degradation rates for all mRNAs were obtained based on the half-lives (in minutes) and the cell division rates, and expressed as the sum of the actual degradation rate <i>ln(2)/half life</i> and the dilution rate (Ln(2)/cell duplication time, 30 minutes). When the half-lives of the mRNAs used in our system were not available in the literature, we assumed them to be the average half-life of mRNA in E. coli, 6.8 minutes, <a href="http://www.ncbi.nlm.nih.gov/pmc/articles/PMC420366/">Selinger, GW, <i>et al</i>. (2003)</a>.
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</p>
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<br><p>According to <a href="http://openwetware.org/images/c/c2/The_N-end_Rule_Pathway_of_Protein_Degradation.pdf">Varshavsky, (1997)</a> and <a href="http://www.ncbi.nlm.nih.gov/pubmed/1962196">Tobias <i>et al.</i>, (1991)</a> (8), when the N-terminal aminoacid of a protein in E. coli is K, R, L, F, Y or W, the half-life of that protein will be as short as 2 minutes; otherwise, it will be greater than 10 hours. Since none of the proteins in our system begins with any of the aminoacids listed above, and because the term Ln(2)/600 has a value very close to zero, the actual degradation rate for the protein will not be taken into consideration, except when the half-life is available in the literature. Thus, the protein degradation rates will be equal to the dilution rate (Ln(2)/cell duplication time, 30 minutes).
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</p>
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<br><p><b>Cell volume and solution volume</b></p>
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The total cell volume was fixed in all simulations 0.0073 mL and the solution volume was 1.1 mL
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<br><hr align="center" width="33%"/></br>
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<br><p><b>Parameter table</b></p>
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<br>
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<center>
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<table border="1" cellspacing="1" cellpadding="1">
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  <tbody>
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    <tr  style="background-color:#99CCFF">
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      <td valign="top">
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        <p>Parameter</p>
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      </td>
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      <td valign="top">
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        <p>Description</p>
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      </td>
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      <td valign="top">
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        <p>Value</p>
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      </td>
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      <td valign="top">
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        <p>References</p>
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      </td>
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    </tr>
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  <tr  border="1" style="background-color:#CCCCCC">
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      <td valign="top">
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        <p>α<sub>mArsR</sub></p>
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      </td>
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      <td valign="top">
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        <p>Maximal transcription rate of ArsR</p>
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      </td>
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      <td valign="top">
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        <p>3.74 nM/min</p>
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      </td>
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      <td valign="top">
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        <p>Assumptions</p>
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      </td>
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    </tr>
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  <tr  border="1" style="background-color:#CCCCCC">
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      <td valign="top">
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        <p>α<sub>mMT</sub></p>
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      </td>
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      <td valign="top">
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        <p>Maximal transcription rate of MT</p>
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      </td>
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      <td valign="top">
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        <p>5.08 nM/min</p>
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      </td>
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      <td valign="top">
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        <p>Assumptions</p>
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      </td>
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    </tr>
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  <tr  border="1" style="background-color:#CCCCCC">
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      <td valign="top">
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        <p>α<sub>pArsR</sub></p>
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      </td>
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      <td valign="top">
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        <p>Maximal translation rate of ArsR</p>
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      </td>
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      <td valign="top">
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        <p>9.74 nM/min</p>
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      </td>
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      <td valign="top">
+
-
        <p>Assumptions</p>
+
-
      </td>
+
-
    </tr>
+
-
 
+
-
  <tr  border="1" style="background-color:#CCCCCC">
+
-
      <td valign="top">
+
-
        <p>α<sub>pMT</sub></p>
+
-
      </td>
+
-
      <td valign="top">
+
-
        <p>Maximal translation rate of MT</p>
+
-
      </td>
+
-
      <td valign="top">
+
-
        <p>6.33 nM/min</p>
+
-
      </td>
+
-
      <td valign="top">
+
-
        <p>Assumptions</p>
+
-
      </td>
+
-
    </tr>
+
-
 
+
-
  <tr  border="1" style="background-color:#CCCCCC">
+
-
      <td valign="top">
+
-
        <p>δ<sub>mRNAArsR</sub></p>
+
-
      </td>
+
-
      <td valign="top">
+
-
        <p>Degradation rate of ArsR mRNA</p>
+
-
      </td>
+
-
      <td valign="top">
+
-
        <p>2.16x10<sup>-1</sup> min<sup>-1</sup></p>
+
-
      </td>
+
-
      <td valign="top">
+
-
        <p>Assumptions, <a href="http://genome.cshlp.org/content/13/2/216.long">Selinger, <i>et al.</i> (2003)</p>
+
-
      </td>
+
-
    </tr>
+
-
 
+
-
  <tr  border="1" style="background-color:#CCCCCC">
+
-
      <td valign="top">
+
-
        <p>δ<sub>mRNAMT</sub></p>
+
-
      </td>
+
-
      <td valign="top">
+
-
        <p>Degradation rate of MT mRNA</p>
+
-
      </td>
+
-
      <td valign="top">
+
-
        <p>1.25x10<sup>-1</sup> min<sup>-1</sup></p>
+
-
      </td>
+
-
      <td valign="top">
+
-
        <p>Assumptions</p>
+
-
      </td>
+
-
    </tr>
+
-
 
+
-
  <tr  border="1" style="background-color:#CCCCCC">
+
-
      <td valign="top">
+
-
        <p>δ<sub>ArsR</sub></p>
+
-
      </td>
+
-
      <td valign="top">
+
-
        <p>Degradation rate of ArsR</p>
+
-
      </td>
+
-
      <td valign="top">
+
-
        <p>2.31x10<sup>-2</sup> min<sup>-1</sup></p>
+
-
      </td>
+
-
      <td valign="top">
+
-
        <p>Assumptions</p>
+
-
      </td>
+
-
    </tr>
+
-
 
+
-
  <tr  border="1" style="background-color:#CCCCCC">
+
-
      <td valign="top">
+
-
        <p>δ<sub>MT</sub></p>
+
-
      </td>
+
-
      <td valign="top">
+
-
        <p>Degradation rate of MT</p>
+
-
      </td>
+
-
      <td valign="top">
+
-
        <p>2.31x10<sup>-2</sup> min<sup>-1</sup></p>
+
-
      </td>
+
-
      <td valign="top">
+
-
        <p>Assumptions</p>
+
-
      </td>
+
-
    </tr>
+
-
 
+
-
   
+
-
<tr  border="1" style="background-color:#CCCCCC">
+
-
      <td valign="top">
+
-
        <p>pro<sub>ars</sub></p>
+
-
      </td>
+
-
      <td valign="top">
+
-
        <p>Concentration of ars promoter</p>
+
-
      </td>
+
-
      <td valign="top">
+
-
        <p>aprox. 1*plasmid copy (nM)</p>
+
-
      </td>
+
-
      <td valign="top">
+
-
        <p>Assumptions</p>
+
-
      </td>
+
-
    </tr>
+
-
 
+
-
<tr  border="1" style="background-color:#CCCCCC">
+
-
      <td valign="top">
+
-
        <p>pro<sub>cons</sub></p>
+
-
      </td>
+
-
      <td valign="top">
+
-
        <p>Concentration of constitutive promoter</p>
+
-
      </td>
+
-
      <td valign="top">
+
-
        <p>aprox. 1*plasmid copy (nM)</p>
+
-
      </td>
+
-
      <td valign="top">
+
-
        <p>Assumptions</p>
+
-
      </td>
+
-
    </tr>
+
-
 
+
-
<tr  border="1" style="background-color:#CCCCCC">
+
-
      <td valign="top">
+
-
        <p>K<sub>D1</sub></p>
+
-
      </td>
+
-
      <td valign="top">
+
-
        <p>Dissociation constant for the interaction of ArsR and pro<sub>ars</sub></p>
+
-
      </td>
+
-
      <td valign="top">
+
-
        <p>330 nM</p>
+
-
      </td>
+
-
      <td valign="top">
+
-
        <p><a href="https://2009.igem.org/Team:Groningen/Modelling/Arsenic">Groningen 2009</p>
+
-
      </td>
+
-
    </tr>
+
-
 
+
-
<tr  border="1" style="background-color:#CCCCCC">
+
-
      <td valign="top">
+
-
        <p>K<sub>D1</sub></p>
+
-
      </td>
+
-
      <td valign="top">
+
-
        <p>Dissociation constant for the interaction of ArsR and As</p>
+
-
      </td>
+
-
      <td valign="top">
+
-
        <p>6000 nM</p>
+
-
      </td>
+
-
      <td valign="top">
+
-
        <p><a href="https://2009.igem.org/Team:Groningen/Modelling/Arsenic">Groningen 2009</p>
+
-
      </td>
+
-
    </tr>
+
-
 
+
-
<tr  border="1" style="background-color:#CCCCCC">
+
-
      <td valign="top">
+
-
        <p>K<sub>D1</sub></p>
+
-
      </td>
+
-
      <td valign="top">
+
-
        <p>Dissociation constant for the interaction of MT and As</p>
+
-
      </td>
+
-
      <td valign="top">
+
-
        <p>6000 nM</p>
+
-
      </td>
+
-
      <td valign="top">
+
-
        <p>Unknown; taken same value as K<sub>D2</sub></p>
+
-
      </td>
+
-
    </tr>
+
-
 
+
-
<tr  border="1" style="background-color:#CCCCCC">
+
-
      <td valign="top">
+
-
        <p>k<sub>t</sub></p>
+
-
      </td>
+
-
      <td valign="top">
+
-
        <p>Kinetic constant for the transport of extracellular arsenic</p>
+
-
      </td>
+
-
      <td valign="top">
+
-
        <p>27.21 µM</p>
+
-
      </td>
+
-
      <td valign="top">
+
-
        <p><a href="http://partsregistry.org/Part:BBa_K190028:Design">Groningen 2009</p>
+
-
      </td>
+
-
    </tr>
+
-
 
+
-
<tr  border="1" style="background-color:#CCCCCC">
+
-
      <td valign="top">
+
-
        <p>h<sub>1</sub></p>
+
-
      </td>
+
-
      <td valign="top">
+
-
        <p>Hill coefficient for the interaction between ArsR and pro<sub>ars</sub></p>
+
-
      </td>
+
-
      <td valign="top">
+
-
        <p>2</p>
+
-
      </td>
+
-
      <td valign="top">
+
-
        <p><a href="https://2009.igem.org/Team:Groningen/Modelling/Arsenic">Groningen 2009</p>
+
-
      </td>
+
-
    </tr>
+
-
<tr  border="1" style="background-color:#CCCCCC">
+
-
      <td valign="top">
+
-
        <p>h<sub>2</sub></p>
+
-
      </td>
+
-
      <td valign="top">
+
-
        <p>Hill coefficient for the interaction of ArsR and As</p>
+
-
      </td>
+
-
      <td valign="top">
+
-
        <p>1</p>
+
-
      </td>
+
-
      <td valign="top">
+
-
        <p><a href="http://digitalcommons.wayne.edu/dissertations/AAI9715913/">Wei-Ping (1996)</p>
+
-
      </td>
+
-
    </tr>
+
-
 
+
-
<tr  border="1" style="background-color:#CCCCCC">
+
-
      <td valign="top">
+
-
        <p>h<sub>3</sub></p>
+
-
      </td>
+
-
      <td valign="top">
+
-
        <p>Hill coefficient for the interaction of MT and As</p>
+
-
      </td>
+
-
      <td valign="top">
+
-
        <p>6</p>
+
-
      </td>
+
-
      <td valign="top">
+
-
        <p><a href="http://pubs.acs.org/doi/pdf/10.1021/ja062914c">Ngu and Stillman (2006)</p>
+
-
      </td>
+
-
    </tr>
+
-
 
+
-
<tr  border="1" style="background-color:#CCCCCC">
+
-
      <td valign="top">
+
-
        <p>V<sub>S</sub>/V<sub>C</sub></p>
+
-
      </td>
+
-
      <td valign="top">
+
-
        <p>Relation between total solution volume(V<sub>S</sub>) and total cell volume (V<sub>C</sub>) </p>
+
-
      </td>
+
-
      <td valign="top">
+
-
        <p>6.64x10<sup>-3</sup></p>
+
-
      </td>
+
-
      <td valign="top">
+
-
        <p><a href="http://partsregistry.org/Part:BBa_K190028:Design">Groningen 2009</p>
+
-
      </td>
+
-
    </tr>
+
-
 
+
-
<tr  border="1" style="background-color:#CCCCCC">
+
-
      <td valign="top">
+
-
        <p>V<sub>max</sub></p>
+
-
      </td>
+
-
      <td valign="top">
+
-
        <p>Vmax for the Michaelis-Menten equation that describes As transport</p>
+
-
      </td>
+
-
      <td valign="top">
+
-
        <p>3.186 µM/sL</p>
+
-
      </td>
+
-
      <td valign="top">
+
-
        <p><a href="http://partsregistry.org/Part:BBa_K190028:Design">Groningen 2009</p>
+
-
      </td>
+
-
    </tr>
+
-
 
+
-
 
+
-
  </tbody>
+
-
</table>
+
-
</center>
+
-
 
+
-
<p><br><h3><a name="simulations"></a>Simulations</h3></p><hr align="center" width="33%"/></br>
+
-
 
+
-
<p>For the simulations of the ODEs, we built a Simulink model. First, we set the extracellular As to zero and ran a simulation with the initial conditions for all variables set to zero, as well; then, we interpolated the graphs obtained for the variables ArsR and MT (both in protein and mRNA) and determined the following initial conditions for further simulations:
+
-
<p><OL TYPE="2">
+
-
<LI>mRNAs: ArsR = 2.1058 nM; MT = 24.6324 nM
+
-
</LI>
+
-
<LI>Proteins: ArsR = 887.7 nM; MT = 6748.4978 nM
+
-
</LI>
+
-
</OL>
+
-
</p>
+
-
<p>We also set all the numerical integrators to have lower saturation limits equal to zero, to be in concordance with the mass conservation law. Here are the results for simulations at extracellular As set to 0, 0.1, 1, 5 and 10 μM (0, 100, 1000, 5000, 10000 nM).
+
-
</p>
+
-
<table class="image" align="center">
+
-
<caption align="bottom"><b>Figure 1.</b>Time vs total bound arsenic</caption>
+
-
<tr><td><img src="https://static.igem.org/mediawiki/2012/4/40/TOTALBOUND_AS_MOD_UANL2012.jpg" style="width:800px;"></td></tr>
+
-
</table>
+
-
<br>
+
-
 
+
-
<table class="image" align="center">
+
-
<caption align="bottom"><b>Figure 2.</b>Extracellular arsenic, total intracellular arsenic and free intracellular arsenic vs time</caption>
+
-
<tr><td><img src="https://static.igem.org/mediawiki/2012/6/6e/AS_EXT_TINT_FREEINT_UANL2012.jpg" style="width:800px;"></td></tr>
+
-
</table>
+
-
<br>
+
-
 
+
-
 
+
-
 
+
-
<p>The color code is the same in figures 1 and 2. In the first one, we show the simulations for the five different extracellular arsenic concentration; in the second one, we see the simulated dynamics of the total internal arsenic and the free internal arsenic.
+
-
</p>
+
-
 
+
-
<p>Remember that we assume that the arsenic concentration in the total cell volume should be equally distributed; in consequence, the concentration inside a cell should be the same as the one at the total cell volume.
+
-
</p>
+
-
 
+
-
<p><br><h3><a name="copynumber"></a>MT plasmid copy number effect</h3></p><hr align="center" width="33%"/></br>
+
-
 
+
-
<p>For the simulations presented in figures 1 and 2, the copy plasmid of each plasmid is assumed to be equal to 1; also, we assume that the concentration of one molecule in the volume of a cell should be approximately equal to 1 nM, so that "(1 nM)*(plasmid copy number)" should give a result the concentration of a promoter.</p>
+
-
<p>Here we show simulations varying the copy number of the plasmid containing MT (we used copy numbers 1, 5, 10 and 100) at 500 nM extracellular arsenic.
+
-
</p>
+
-
 
+
-
<table class="image" align="center">
+
-
<caption align="bottom"><b>Figure 3.</b> Total captured As vs time at different plasmid copy number for MT </caption>
+
-
<tr><td><img src="https://static.igem.org/mediawiki/2012/3/38/Intracellular_As_%28nM%29_at_four_different_plasmid_copy_number_for_rhMT_with_series_label_UANL2012.jpg" style="width:800px;"></td></tr>
+
-
</table>
+
-
<br>
+
-
 
+
-
 
+
-
<p><br><h2><a name="Simulink_TNA"></a>Simulink model: Transport and accumulation</p></h2></br>
+
-
 
+
-
<p>You can find our Simulink model for the transport and accumulation module <a href="https://dl.dropbox.com/u/105944108/Transport_and_acumulation_model_David2.mdl">here</a>.</p>
+
-
</div>
+
-
<div class="sidebar2">
+
-
    <ul class="nav">
+
-
<a href="#Transport"><li><b>>>Transport and accumulation</b></a></li>
+
-
<a href="#core_model"><li><i>Core model</i></a></li>
+
-
<a href="#ODEANT"><li><i>ODEs</i></a></li>
+
-
<a href="#parametersANT"><li><i>Parameters</i></a></li>
+
-
<a href="#simulations"><li><i>Simulations</i></a></li>
+
-
<a href="#copynumber"><li><i>Copy number effect</i></a></li>
+
-
<a href="#Simulink_TNA"><li><i>Simulink model: Transport and accumulation</i></a></li>
+
-
<a href="https://2012.igem.org/Team:UANL_Mty-Mexico/Modeling/Biosensor"><li>Biosensor</a></li>
+
-
<a href="https://2012.igem.org/Team:UANL_Mty-Mexico/Modeling/Silica_binding"><li>Silica binding</a></li>
+
-
 
+
-
 
+
-
</ul>
+
-
</div>
+
</html>
</html>
 +
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After considering the effect of their metallothioneins (As-binding proteins), GlpF, ArsB and ArsR, they ended with the following time derivative:


\begin{equation} \large\frac{\mathrm{d[As(III)in] } }{\mathrm{d} x} = -ArsR_{As}-MBPArsR_{As} -n_{f}\cdot fMT_{As} -k_{1} ArsB_{As} + \frac{k_{2}V_{s}GlpF_{As}}{V_{c}} \end{equation}


Introduction

RNA thermometers (RNATs) are RNA sequences that range from 40 to more than a 100 nucleotides commonly found in the 5' untranslated region of some genes and that regulate in cis their translation without the need of other factors [Kortmann and Narberhaus, (2012); Narberhaus, (2009)]. These RNAT sequences show certain three dimensional structures, some of which interact with the ribosome binding site (RBS) of their regulated genes and hinders the proccessivity of the ribosome complex at certain temperatures. The dynamics of the formation of these structures is temperature dependent and is the basis of the regulation of the translation rate of a given transcript [Chowdhury, S., et al.,(2006); Narberhaus, F., et al.,(2006)].

Functional RNAT have been found in different organisms, mainly pathogenic bacteria, and many others have been predicted in almost everyfrom a number of bioinformatic studies. They have been found to regulate the expression of virulence factors, heat and cold shock response factors and even proteins involved the development of some bacteriophages.

Their apparent widespread presence in living organisms has made RNATs attractive for some applications, specially the ones related to the replacement of chemical inducers and for the development of new drugs.

However, from the experience of those who have been working extensively with RNAT in the later years, the accurate bioinformatic prediction of functional RNAT has proven to be an exceptionally difficult task; the reasons for this are pointed to be the poor sequence conservation observed among RNATs and the gaps in our current understanding of the RNAT function, their structural diversity and the effect of other regulatory sequences far from the RBS region [Kortmann and Narberhaus, (2012); Waldminghaus, et al., (2007)].

The discovery of new RNATs has relied on a mixed approach that involves bioinformatics and experimental validation, as well as approaches that involve mutational libraries, synthetic constructions and directed evolution.

Even when the naturally found RNATs usually regulate the expression of transcription factors, the synthetic constructions made so far have focused mainly to characterize the effect of a given RNAT using a reporter protein (LacZ or a fluorescent protein) directly downstream of a RNAT. In our work, we intend to prove that RNATs can also be employed to effectively regulate the expression of transcription factors in synthetic circuits and point at possible applications for the circuit topologies that would be made feasible with this new kind of synthetic regulatory device.

Although RNATs show almost no sequence similarity among them, a number of structural features can be used to classify them. Here we enlist the most described RNATs structural groups described to date [Kortmann and Narberhaus, (2012)]:

  1. ROSE.- ROSE stands for "Regulation Of heat Shock Expression". ROSE elements are 60 to >100 nucleotide sequences found upstream of heat shock proteins. They have been found to be conserved in alpha and gamma-proteobacteria. Among the structural features of the ROSE element family are: a) their folding in 2 to 4 stemloop structures; b) a short conserved sequence (UU/CGCU) near the Shine-Dalgarno sequence; and c) the presence of a number of non-cannonical base interactions (the G83-G94 pair; a triple bair among U96-C80-C81; the U79-U97; and the interaction of the AUG codon and C71, G72 and U73. Functional ROSE elements have been found in E. coli (rpoH and ibpA) and B. japonicum (hspA).

  2. FourU elements.- these elements are characterized by a short motif composed of four uridines that pair with the Shine-Dalgarno region and is embedded in a hairpin that shows temperature-induced conformational changes. FourU elements have only one A-G non-cannonical base interaction. Among the structural features that characterize FourU elements are a) the A-G pair and b) the G34-C46 pair that regulates melting. Functional FourU elements have been described in Salmonella (agsA) and Yersina pseudotuberculosis (lcrF).

  3. Synechocystis hsp17 element.- with a length of 46 nucleotides, this is the shortest RNAT described so far. The distinctive structural features essential for the function of this element are a) the pairing of a UCCU sequence with the AGGA in the Shine-Dalgarno sequence and b) the presence of two loops in its stems.

  4. Coding region spanning RNATs.- RNATs are not exclusively found in the 5'UTR of genes; they can also span into the coding region and even be intergenic. Functional coding region spanning RNATs have been found in E. coli (rpoH), phage lambda (cIII) and Lysteria monocytogenes (prfA).

  5. Cold shock RNATs.- cold shock RNATs also depend on the dynamics of the folding of different loops, but in contrast to heat shock RNATs, the conformation that prevents the binding of the ribosome is found at high temperatures, while at low temperatures, the RNAT folds into a conformation that allows for the ribosome to proceed. An example of a cold shock RNAT is the element found upstream and inside the coding region of E. coli gene cspA.
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