Team:ULB-Brussels/Modeling

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!align="center"|[[Team:ULB-Brussels/Parts|Parts]]
!align="center"|[[Team:ULB-Brussels/Parts|Parts]]
!align="center"|[[Team:ULB-Brussels/Modeling|Modeling]]
!align="center"|[[Team:ULB-Brussels/Modeling|Modeling]]
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!align="center"|[[Team:ULB-Brussels/Conclusion|Conclusion & Perspectives]]
!align="center"|[[Team:ULB-Brussels/Safety|Safety]]
!align="center"|[[Team:ULB-Brussels/Safety|Safety]]
!align="center"|[[Team:ULB-Brussels/Previous|Older wiki's]]
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<center><font color="#000000"; size="100"> Team ULB-Brussels, modelisation of our </font></center>
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<center><font color="#000000"; size="100"> Team ULB-Brussels, modeling of our </font></center>
<br></br><p><center><font color="#000000"; size="100"> project! </font></center></p>
<br></br><p><center><font color="#000000"; size="100"> project! </font></center></p>
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<table id="toc" class="toc">
 
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<div id="toctitle">
 
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<h2>Sommaire</h2>
 
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<A HREF="https://2012.igem.org/Team:ULB-Brussels/Team#1. Introduction"> 1. Introduction&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </A>
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<p><font size="+3"> An online version of the report can be found <a href="http://www.mediafire.com/view/?ubp5sobban0b9l6">here</a>.</font>
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<p><A HREF="https://2012.igem.org/Team:ULB-Brussels/Team#2. Modeling the competition experiment"> 2. Modeling the competition experiment</A>
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<ul><p><A HREF="https://2012.igem.org/Team:ULB-Brussels/Team#2.1. Notations and mathematical model"> 2.1. Notations and mathematical model </A>
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<p><A HREF="https://2012.igem.org/Team:ULB-Brussels/Team#2.2. Natural selection?"> 2.2. Natural selection? </A></ul>
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<ul><p><A HREF="https://2012.igem.org/Team:ULB-Brussels/Team#2.2.1. Neglecting the Microcin diffusion"> 2.2.1. Neglecting the Microcin diffusion </A>
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<p><A HREF="https://2012.igem.org/Team:ULB-Brussels/Team#2.2.2. Qualitative discussion of the solutions of (2) in the general case"> 2.2.2. Qualitative discussion of the solutions of (2) in the general case </A>
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<p><A HREF="https://2012.igem.org/Team:ULB-Brussels/Team#2.2.3. Conclusion: no natural selection"> 2.2.3. Conclusion: no natural selection </A>
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<p><A HREF="https://2012.igem.org/Team:ULB-Brussels/Team#3. New experiment and conclusion"> 3. New experiment and conclusion </A>
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<p><A HREF="https://2012.igem.org/Team:ULB-Brussels/Team#4. Appendix"> 4. Appendix </A>
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<ul><p><A HREF="https://2012.igem.org/Team:ULB-Brussels/Team#4.1. Determining analytically the asymptotical behaviour of the solutions of (1)"> 4.1 Determining analytically the asymptotical behaviour of the solutions of (1) </A>
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<p><A HREF="https://2012.igem.org/Team:ULB-Brussels/Team#4.2. Parameter estimation"> 4.2. Parameter estimation </A>
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<h2><A NAME="1. Introduction"> 1. Introduction </A></h2>
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<p>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Since complex biological pathways are used in an industrial way in order to produce molecules of
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interest, it has become crucial to understand and, above all, optimize these pathways. However,
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biological systems are so complex that it is sometimes impossible to have a complete understanding
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of the reactions and mechanisms of the different pathways. The idea of our project is to solve this
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optimization problem by using the integron platform { which represents a natural genetic optimiza-
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tion tool in bacteria { and putting in competition different populations with different gene orders,
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so that the population(s) with the optimal order(s) will be naturally selected with time.
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<p>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;As a proof of concept, we will try to optimize the order of the genes governing the production of
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two natural antibiotics: Microcin C7 and Microcin B17. The first one inhibits a tRNA synthetase
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(thus inhibits protein synthesis and, as a consequence, cell division), and the second inhibits a gyrase
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(thus provokes inhibition of DNA replication and eventually cell death). We might then expect that
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natural selection occurs, so that the optimal gene order(s) finally emerge.
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<p>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;In the sequel, we model this competition experiment, and try to see in what sense and in what
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conditions natural selection could happen.
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<td><a href=""#hautdepage""><img id="logo" src="http://www.clker.com/cliparts/9/2/8/c/1216180855712705788claudita_home_icon.svg.hi.png" height="40px" width="40px" align="right"></a>
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<embed src="http://data.axmag.com/data/201209/U59129_F130176/main.swf?page=81" quality="high" width="100%" height="650px"; scale="noscale" align="TL" salign="TL" allowFullScreen="true" type="application/x-shockwave-flash"></embed>
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<h2><A NAME="2. Modeling the competition experiment"> 2. Modeling the competition experiment</A></h2>
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<p>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;In the following, we write Microcins <em>B</em> and <em>C</em> for Microcins <em>B17</em> and <em>C7</em>, respectively. Further,
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the bacterial populations producing these antibiotics will be denoted by <em>Bi</em> and <em>Cj</em> , respectively,
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where the indices i and j run through all different gene cassette orders.
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<p>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;We consider the experiment where all these populations are put in competition together. In our
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model, for the sake of simplicity, we will simply consider that Microcin <em>B</em> causes the production of
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some protein complexes that provoke cell death (bactericidal), while Microcin <em>C</em> inhibits cell division
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of the protein complexes that allow the cellular division process. (bacteriostatic). Thus note that the quantities <em>AXi</em> and <em>DXi</em> have no biological meaning, but are
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used phenomenologically to better describe the situation.
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<h3><A NAME="2.1. Notations and mathematical model"> 2.1. Notations and mathematical model</A></h3>
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<p>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;The study of the different populations will be accomplished through the time evolution of the following dynamical quantities. Notice that subscribed letters will designate the given population, while
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Here are the several graphs that did not appear on the reading version above.
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superscripted letters will stand for the corresponding antibiotics.
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&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<img id="logo" src="https://static.igem.org/mediawiki/2012/a/aa/Model_1.PNG" height="65%" width="65%">
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<p>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Using these constants and dynamical variables, we can describe the biological competition experiment by the following differential equation system (where <em>X = B;C</em> and i runs through all the
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different possible gene orders for the antibiotics production gene cassettes):
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&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<img id="logo" src="https://static.igem.org/mediawiki/2012/5/55/Model_2.PNG" height="65%" width="65%">
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<p>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;This general model can be simplified if we suppose that populations <em>Xi</em> are completely immune to Microcin <em>X</em>, which is a totally reasonable assumption. Further, since bacteria and Microcins have
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half-lives that are much larger than the experiment time, we may neglect the corresponding terms.
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If we also neglect the saturation effect in the population growth (which is natural if the experiment
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is carried out in exponential phase), we then get the following simpler system:
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&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<img id="logo" src="https://static.igem.org/mediawiki/2012/6/6f/Model_3.PNG" height="65%" width="65%">
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<img id="logo" src="https://static.igem.org/mediawiki/2012/6/6a/Graphe_1.PNG" >
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<h3><A NAME="2.2. Natural selection?"> 2.2. Natural selection?</A></h3>
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<p>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;We could hope that putting together bacteria with all the different gene orders leads to a natural
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selection regime, meaning that the subpopulations with the best offensive and/or defensive charac-
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teristics
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<h2><A NAME="4. Appendix"> 4. Appendix</A></h2>
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<font size="+0"> Figures 1-8</font>
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<h3><A NAME="4.1. Determining analytically the asymptotical behaviour of the solutions of (1)"> 4.1. Determining analytically the asymptotical behaviour of the solutions of (1)</A></h3>
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<font size="+0"> Figure 9</font>
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<h3><A NAME="4.2. Parameter estimation"> 4.2. Parameter estimation</A></h3>
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<td><a href=""#hautdepage""><img id="logo" src="http://www.clker.com/cliparts/9/2/8/c/1216180855712705788claudita_home_icon.svg.hi.png" height="40px" width="40px" align="right"></a>
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Latest revision as of 23:30, 26 September 2012

Home Team Project Parts Modeling Conclusion & Perspectives Safety Older wiki's


 

Team ULB-Brussels, modeling of our


project!



An online version of the report can be found here.



Here are the several graphs that did not appear on the reading version above.



Figures 1-8



Figure 9