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 <li class="active"><a href="#Introduction">Introduction</a></li>

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       <li><a href="#Model">Model Conditions</a></li>
       <li><a href="#Fixed_temperature_model">Fixed Temperature Model</a></li>
       <li><a href="#Dynamic">Dynamic Model</a></li>

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     Gaussian Function Fitting<span class="caret"></span></a>

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       <li><a href="#Gaussian Function">Gaussian Function</a></li>
       <li><a href="#Biological Antecedents">Biological Antecedents </a></li>
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     Fitting Results<span class="caret"></span></a>

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               <li><a href="#RNAT Comparison">RNAT Comparison</a></li>
       <li><a href="#Parameters">Parameters</a></li>
       <li><a href="#M12 Special Case">M12 special Case</a></li>
<li><a href="#References">References</a></li>
<li><a href="#Appendix">Appendix</a></li>


<b><h2>Math Model</b></h2>

<div class="title"> <p><a name="Introduction"><h3>Introduction</a></h3><hr></p></div> <div class="justified"> <p>Mathematical models that represent the dynamic behavior of biological systems are a quite prolific field of work and are pillar for Systems Biology. A number of deterministic and stochastic formalisms have been developed at different abstraction levels that range from the molecular to the population levels.</p> <div id="title"> <p><a name="Model Description"><h3>Model Description&nbsp;&nbsp;<a href="#" class="btn btn-info">Back to top</a></h3><hr></p></div> <p>We present a model for the relation between time, temperature and the change in fluorescence (measured in Relative Fluorescent Units or RFUs) of an <i>E. coli</i> culture that harbors a genetic construction where a fluorescent protein is under control of a RNAT.</p> <div id="title"> <p><a name="Model"><h4>Model Conditions&nbsp;&nbsp;<a href="#" class="btn btn-info">Back to top</a></h4><hr></p></div>

\begin{equation} \large F_{R} = \frac{F_{sample}}{F_{standard}} \end{equation}

<p>where F<sub>sample</sub> is the OD<sub>600</sub>-normalized fluorescence emited by a sample, while F<sub>standard</sub> is the OD<sub>600</sub>-normalized fluorescence measurement for the corresponding standard culture (again, BBa_E1010 for RFP and BBa_E0040 for GFP).</p>

<center>\(F_{R} - F_{Rst} = e^{C} \ e^{- \delta {t}} \)</center>

<div id="title"> <p><a name="Fixed_temperature_model"><h4>Fixed Temeperature Model&nbsp;&nbsp;<a href="#" class="btn btn-info">Back to top</a></h4><hr></p></div> <div id="title"> <p><a name="Dynamic"><h4>Dynamic Model&nbsp;&nbsp;<a href="#" class="btn btn-info">Back to top</a></h4><hr></p></div> <p>In <a href="" target="3">Shah and Gilchrist, (2010)</a>, it was found that the probability of openness of a ribosome binding site (RBS) of an mRNA with respect to temperature, fits well into a logistic equation. However, the authors did not find significant differences in the behaviour of known RNATs and non-RNAT elements and admit that RBS openness cannot be assumed to be directly correlated to translational activity. Therefore, their RBS-melting probability equation would not be recommendable to be used directly in gene expression models for RNATs.</p> <div id="title"> <p><a name="Gaussian Function Fitting"><h3>Gaussian Function Fitting&nbsp;&nbsp;<a href="#" class="btn btn-info">Back to top</a></h3><hr></p></div> <div id="title"> <p><a name="Gaussian Function"><h4>Gaussian Function</h4></a><hr></p></div> <div id="title"> <p><a name="Biological Antecedents"><h4>Biological Antecedents </h4></a><hr></p></div> <div id="title"> <p><a name="Justification"><h4>Justification</h4></a><hr></p></div> <div id="title"> <p><a name="Fitting Results"><h3>Fitting Results&nbsp;&nbsp;<a href="#" class="btn btn-info">Back to top</a></h3><hr></p></div> <div id="title"> <p><a name="RNAT Comparison"><h4>RNAT Comparison</h4></a><hr></p></div> <div id="title"> <p><a name="Parameters"><h4>Parameters</h4></a><hr></p></div> <div id="title"> <p><a name="M12 Special Case"><h4>M12 Special Case</h4></a><hr></p></div> <div id="title"> <p><a name="References"><h3>References&nbsp;&nbsp;<a href="#" class="btn btn-info">Back to top</a></h3><hr></p></div> <ol> <li>ShahP ,Gilchrist MA(2010)Is Thermosensing Property of RNA Thermometers Unique? <i>PLoS ONE</i>,5(7):e11308.doi:10.1371/journal.pone.0011308. <li>H. A. Von Fircks, T. Verwijst,(1993)Plant Viability as a Function of Temperature Stress(The Richards Function Applied to Data from Freezing Tests of Growing Shoots <i>Plant Physio</i> ,103(1):125–130. <li>Hoops S, <i>et al.</i> (2010)COPASI–a COmplex PAthway SImulator <i>Bioinformatics</i> ,22,3067-3074,2006, <li>COPASI Documentation 2.Steady State calculation(2013,May 16).Retrieved from<il> <li>Ting Chen, <i>et al.</i> Modeling gene expression with differential equations (1999) <i>Pacic Symposium of Biocomputing</i> <li>Gaussian function(Consulted on 2013,September 27)Retrieved from </ol>

<div id="title"> <p><a name="Appendix"><h3>Appendix&nbsp;&nbsp;<a href="#" class="btn btn-info">Back to top</a></h3><hr></p></div>

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