<div class="art-postcontent">

<h2 class="art-postheader">Modelling</h2>

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    <td><div id="toctitle">

        <h2>Contents</h2>

      </div>

      <ul>

        <li class="toclevel-1"><a href="#Mathematical_modelling_page"><span class="tocnumber"></span> <span class="toctext">Mathematical modelling: introduction</span></a>

             <li class="toclevel-2"><a href="#The importance of the mathematical model"><span class="tocnumber">1</span> <span class="toctext">The importance of mathematical modelling</span></a></li>

             <li class="toclevel-2"><a href="#Equations_for_gene_networks"><span class="tocnumber">2</span> <span class="toctext">Equations for gene networks</span></a></li>

            <ul>

              <li class="toclevel-3"><a href="#Hypothesis"><span class="tocnumber">2.1</span> <span class="toctext">Hypotheses</span></a></li>

              <li class="toclevel-3"><a href="#Equations_1_and_2"><span class="tocnumber">2.2</span> <span class="toctext">Equations (1) and (2)</span></a></li>

              <li class="toclevel-3"><a href="#Equation_3"><span class="tocnumber">2.3</span> <span class="toctext">Equation (3)</span></a></li>

               <li class="toclevel-3"><a href="#Equation_4"><span class="tocnumber">2.4</span> <span class="toctext">Equation (4)</span></a></li>

            </ul>

             <li class="toclevel-2"><a href="#Table_of_parameters"><span class="tocnumber">3</span> <span class="toctext">Table of parameters</span></a></li>

             <ul>

              <li class="toclevel-3"><a href="#CV"><span class="tocnumber">3</span> <span class="toctext">Table of parameter CV</span></a></li>

            </ul>

            <li class="toclevel-2"><a href="#Parameter_estimation"><span class="tocnumber">4</span> <span class="toctext">Parameter estimation</span></a></li>

            <ul>

              <li class="toclevel-3"><a href="#Ptet_&_Plux"><span class="tocnumber">4.1</span> <span class="toctext">pTet & pLux</span></a></li>

              <li class="toclevel-3"><a href="#introduction_to_T9002"><span class="tocnumber">4.2</span> <span class="toctext">T9002 introduction</span></a></li>

              <li class="toclevel-3"><a href="#Enzymes"><span class="tocnumber">4.3</span> <span class="toctext"> AiiA & LuxI</span></a></li>

              <li class="toclevel-3"><a href="#N"><span class="tocnumber">4.4</span> <span class="toctext">N</span></a></li>

              <li class="toclevel-3"><a href="#Degradation_rates"><span class="tocnumber">4.5</span> <span class="toctext">Degradation rates</span></a></li>

            </ul>

            <li class="toclevel-2"><a href="#Simulations"><span class="tocnumber">5</span> <span class="toctext">Simulations</span></a></li>

            <li class="toclevel-1"><a href="#Sensitivity_Analysis"><span class="tocnumber">6</span> <span class="toctext">Sensitivity Analysis of the steady state of enzyme expression in exponential phase</span></a></li>

            <ul>

              <li class="toclevel-2"><a href="#Steady state of enzyme expression"><span class="tocnumber">6.1</span> <span class="toctext">Steady state of enzyme expression</span></a></li>

              <li class="toclevel-2"><a href="#Sensitivity analysis"><span class="tocnumber">6.2</span> <span class="toctext">Sensitivity analysis</span></a></li>

            </ul>

            <li class="toclevel-1"><a href="#References"><span class="tocnumber">7</span> <span class="toctext">References</span></a></li>

          </ul>

        </li>

      </ul>

       </ul></td>

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<a name="Mathematical_modelling_page"></a>

<h1><span class="mw-headline"> <b>Mathematical modelling: introduction</b> </span></h1>

<div style='text-align:justify'>

  <p>Mathematical modelling plays a central role in Synthetic Biology, due to its ability to serve as a crucial link between the concept and realization of a biological circuit: what we propose in this page is a mathematical modelling approach to the entire project, which has proven extremely useful before and after the "wet lab" activities.</p>

  <p>Thus, immediately at the beginning, when there was little knowledge, a mathematical model based on a system of differential equations was derived and implemented using a set of reasonable values of model parameters, to validate the feasibility of the project. Once this became clear, starting from the characterization of each simple subpart created in the wet lab, some of the parameters of the mathematical model were estimated thanks to several ad-hoc experiments we performed within the iGEM project (others were derived from literature) and they were used to predict the final behaviour of the whole engineered closed-loop circuit. This approach is consistent with the typical one adopted for the analysis and synthesis of a biological circuit, as exemplified by <a href="#Pasotti"><i><b>Pasotti L</b> et al. 2011.</i></a></p>

  <p>After a brief overview on the importance of the mathematical modelling approach, we deeply analyze the system of equations, underlining the role and function of the parameters involved.</p>

  <p>Experimental procedures for parameter estimation are discussed and, finally, a different type of circuit is presented. <font color="red">Simulations were performed, using <em>ODEs</em> with MATLAB and used to explain the difference between a closed-loop control system model and an open one.</font>

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</p>

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<a name="The importance of the mathematical model"></a>

<h2> <span class="mw-headline"> <b>The importance of mathematical modelling</b> </span></h2>

<div style='text-align:justify'>

  <p>Mathematical modelling reveals fundamental in the challenge of understanding and engineering complex biological systems. Indeed, these are characterized by a high degree of interconnection among the single constituent parts, requiring a comprehensive analysis of their behavior through mathematical formalisms and computational tools.</p>

  <p>Synthetically, we can identify two major roles concerning mathematical models: </p>

    <li><b>Simulation</b>: mathematical models  allow to analyse complex system dynamics and to reveal the relationships between the involved variables, starting from the knowledge of the single subparts behavior and from simple hypotheses of their interconnection. <a href="#Endler">(<i><b>Endler L</b> et al. 2009</i>)</a></li>

    </p>

    <p>

     <li><b>Knowledge elicitation</b>: mathematical models summarize into a small set of parameters the results of several experiments (parameter identification), allowing a robust comparison among different experimental conditions and  providing an efficient way to synthesize knowledge about biological processes. Then, through the simulation process, they make possible the re-usability of the knowledge coming from different experiments, engineering complex systems from the composition of its constituent subparts under appropriate experimental/environmental conditions <a href="#Braun">(<i><b>Braun D</b> et al. 2005</i></a>;<a href="#Canton"> <b>Canton B</b> et al 2008</a>).</li>

    </p>

  </ul>

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<br>

<a name="Equations_for_gene_networks"></a>

<h2> <span class="mw-headline"> <b>Equations for gene networks</b> </span></h2>

<p>Below is provided the system of equations of our mathematical model. </p>

   <img alt="" src="https://static.igem.org/mediawiki/2011/4/43/Model_new.jpg" class="thumbimage" height="55%" width="80%">

<center>

   <img alt="" src="https://static.igem.org/mediawiki/2011/5/5e/Circuito_finale.jpg" class="thumbimage" width="87%">

</center>

<center>

   <img alt="" src="https://static.igem.org/mediawiki/2011/c/c7/QS_system_synthetic_circuit.png" class="thumbimage" width="85%">

</center>

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<a name="Hypothesis"></a>

<h4> <span class="mw-headline"> <b>Hypotheses of the model</b> </span></h4>

<table class="data">

     <td><div style='text-align:justify'> <em> <b>HP₁</b>: in equation (2) only HSL is considered as inducer, instead of the complex LuxR-HSL.

        This is motivated by the fact that our final device offers a constitutive LuxR production due to the upstream constitutive promoter P&lambda;. Assuming LuxR is abundant in the cytoplasm, we can understand this simplification of attributing pLux promoter induction only by HSL. <br>

        <br>

        <b>HP₂</b>: in system equation, LuxI and AiiA amounts are expressed per cell. For this reason, the whole equation (3), except for the

        term of intrinsic degradation of HSL, is multiplied by the number of cells N, due to the property of the lactone to diffuse freely inside/outside bacteria. <br>

        <br>

        this is the reason why the strength of the complex promoter-RBS is expressed in Arbitrary Units [AUr]. <br>

        <b>HP₄</b>: considering the exponential growth, the enzymes AiiA and LuxI concentration is supposed to be constant, because their production is equally compensated by dilution. </em> </div></td>

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<br>

<br>

<a name="Equations_1_and_2"></a>

<h4> <span class="mw-headline"> <b>Equations (1) and (2)</b> </span></h4>

<div style='text-align:justify'>

   <p>Equations (1) and (2) have identical structure, differing only in the parameters involved. They represent the synthesis, degradation and diluition of both the enzymes in the circuit, LuxI and AiiA, respectively in the first and second equation: in each of them both transcription and translation processes have been condensed. The mathematical formulation is analogous to the one used by <a href="#Pasotti"><i><b>Pasotti L</b> et al. 2011</i></a>, Suppl. Inf., even if we do not take LuxR-HSL complex formation into account, as explained below.</p>

   <p>These equations are composed of 2 parts:</p>

   <ol>

     <li>The first term describes, through Hill's equation, the synthesis rate of the protein of interest (either LuxI or AiiA) depending on the concentration of the inducer (anhydrotetracicline -aTc- or HSL respectively), responsible for the activation of the regulatory element composed of promoter and RBS. In the parameter table (see below), &alpha; refers to the maximum activation of the promoter, while &delta; stands for its leakage activity (this means that the promoter is slightly active even if there is no induction). In particular, in equation (1), the almost entire inhibition of pTet promoter is given by the constitutive production of TetR by our MGZ1 strain. In equation (2), pLux is almost inactive  in the absence of the complex LuxR-HSL. Furthermore, in both equations k stands for the dissociation constant of the promoter from the inducer (respectively aTc and HSL in eq. 1 and 2), while &eta; is the cooperativity constant.

      </p>

       <p>

    <li>The second term in equations (1) and (2) is in turn composed of 2 parts. The former one (<em>&gamma;</em>*LuxI or <em>&gamma;</em>*AiiA respectively) describes, with an exponential decay, the degradation rate per cell of the protein. The latter (&mu;*(Nmax-N)/Nmax)*LuxI or &mu;*(Nmax-N)/Nmax)*AiiA, respectively) takes into account the dilution factor against cell growth which is related to the cell replication process.

       </p>

   </ol>

<a name="Equation_3"></a>

<h4> <span class="mw-headline"> <b>Equation (3)</b> </span></h4>

<p>Here the kinetics of HSL is modeled, through enzymatic reactions either related to the production or the degradation of HSL. This equation is composed of 3 parts: </p>

<ol>

<p>

<li> The first term represents the production of HSL due to LuxI expression. We modeled this process with a saturation curve in which V_max is the HSL maximum transcription rate, while k_M,LuxI is LuxI dependent half-saturation constant.

  </p>

  <p>

<li> The second term represents the degradation of HSL due to the AiiA expression. Similarly to LuxI, k_cat represents the maximum degradation per unit of HSL concentration, while k_M,AiiA is the concentration at which AiiA dependent HSL degradation rate is (k_cat*HSL)/2. The formalism is similar to that found in the Supplementary Information of <a href="#Danino"><i><b>Danino T</b> et al 2010.</i></a></font>

  </p>

   <p>

<li> The third term (&gamma;_HSL*HSL) is similar to the corresponding ones present in the first two equations and describes the intrinsic protein degradation.

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  </p>

  <br>

  <a name="Equation_4"></a>

   <h4> <span class="mw-headline"> <b>Equation (4)</b> </span></h4>

  <div style='text-align:justify'>This is the typical cells growth equation, depending on the rate &mu; and the maximum number N_max of cells per well reachable <a href="#Pasotti">(<i><b>Pasotti L</b> et al. 2009</i>).</a></div>

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   <br>

   <a name="Table_of_parameters"></a>

  <h2> <span class="mw-headline"> <b>Table of parameters and species</b> </span></h2>

   <center>

     <table class="data">

         <td class="row"><b>Parameter & Species</b></td>

         <td class="row"><b>Description</b></td>

         <td class="row"><b>Unit of Measurement</b></td>

         <td class="row"><b>Value</b></td>

         <td class="row">&alpha;_{p_Tet}</td>

         <td class="row">maximum transcription rate of pTet (dependent on <a href="#RBS">RBSx</a> efficiency)</td>

         <td class="row">[(AUr/min)/cell]</td>

         <td class="row">-</td>

         <td class="row">&delta;_{p_Tet}</td>

         <td class="row">leakage factor of promoter pTet basic activity</td>

         <td class="row">[-]</td>

         <td class="row">-</td>

         <td class="row">&eta;_{p_Tet}</td>

         <td class="row">Hill coefficient of pTet</td>

         <td class="row">[-]</td>

         <td class="row">-</td>

         <td class="row">k_{p_Tet}</td>

         <td class="row">dissociation constant of aTc from pTet</td>

        <td class="row">[1/min]</td>

         <td class="row">-</td>

        <td class="row">[1/min]</td>

         <td class="row">-</td>

        <td class="row">[1/min]</td>

         <td class="row">-</td>

        <td class="row">[1/(min*cell)]</td>

        <td class="row">-</td>

   </center>

   <div align="justify"><b><a name="RBS">NOTE</a></b>

    <p>In order to  better investigate the range of dynamics of each subpart, every promoter has been studied with 4 different RBSs, so as to develop more knowledge about the state variables in several configurations of RBS' efficiency <a href="#Salis">(<i><b>Salis HM</b> et al. 2009</i>)</a>. Hereafter, referring to the notation "RBSx" we mean, respectively, <a href="http://partsregistry.org/Part:BBa_B0030">RBS30</a>, <a href="http://partsregistry.org/Part:BBa_B0031">RBS31</a>, <a href="http://partsregistry.org/Part:BBa_B0032">RBS32</a>, <a href="http://partsregistry.org/Part:BBa_B0034">RBS34</a>. </p>

  </div>

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  <center>

     <table class="data">

         <td class="row"><b>Parameter & Species</b></td>

         <td class="row"><b>BBa_B0030</b></td>

         <td class="row"><b>BBa_B0031</b></td>

         <td class="row"><b>BBa_B0032</b></td>

         <td class="row"><b>BBa_B0034</b></td>

         <td class="row">&alpha;_{p_Tet}</td>

         <td class="row">3.7</td>

         <td class="row">ND</td>

         <td class="row">12</td>

         <td class="row">5.94</td>

         <td class="row">&delta;_{p_Tet}</td>

         <td class="row">91.61</td>

         <td class="row">>>100</td>

         <td class="row">>100</td>

         <td class="row">40.59</td>

         <td class="row">&eta;_{p_Tet}</td>

         <td class="row">23.72</td>

         <td class="row">>>100</td>

         <td class="row">57.62</td>

         <td class="row">47.6</td>

         <td class="row">k_{p_Tet}</td>

        <td class="row">k_{p_Lux}</td>

         <td class="row">27.5</td>

         <td class="row">25.81</td>

         <td class="row">8.46</td>

         <td class="row">17.86</td>

   <center>

    <img alt="" src="https://static.igem.org/mediawiki/2011/5/58/Scell.jpg" class="thumbimage" width="45%">

   </center>

   <p>However, also Relative Promoter Unit (RPU, <a href="#Kelly"><i><b>Kelly JR</b> et al. 2009</i></a>) has been calculated as a ratio of S_cell of promoter of interest and the S_cell of <a href="http://partsregistry.org/Part:BBa_J23101">BBa_J23101</a> (reference to <a href="#Hypothesis"><span class="toctext"><b><em>HP₃</em></b></span></a>).</p>

  <center>

    <img alt="" src="https://static.igem.org/mediawiki/2011/2/26/Box2_new.jpg" class="thumbimage" height="48%" width="120%">

  </center>

  <div style='text-align:justify'>

    <p>As shown in the figure, &alpha;, as already mentioned, represents the protein maximum synthesis rate, which is reached, in accordance with Hill equation, when the inducer concentration tends to infinite, and, more practically, when the inducer concentration is sufficiently higher than the dissociation constant. Meanwhile the product &alpha;*&delta; stands for the leakage activity (at no induction), liable for protein production (LuxI and AiiA respectively) even in the absence of inducer.</p>

     <p>The paramenter &eta; is the Hill's cooperativity constant and it  affects the ripidity of transition from the lower and upper boundary of the curve relating S_cell to the inducer concentration.

      Lastly, k stands for the semi-saturation constant and, in case of &eta;=1, it indicates the concentration of substrate at which half the synthesis rate is achieved. <br>

       <img alt="" src="https://static.igem.org/mediawiki/2011/c/c2/T9002.jpg" class="thumbimage" width="110%">

     <p>This is a biosensor which receives HSL concentration as input and returns GFP intensity (more precisely S_cell) as output.<a href="#Canton"> (<i><b>Canton</b> et al. 2008</i>).</a> According to this, it is necessary to understand the input-output relationship: so, a T9002 "calibration" curve is plotted for each test performed.</p>