Team:ULB-Brussels/Modeling

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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

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 !align="center"|[[Team:ULB-Brussels/Parts|Parts]]
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+!align="center"|[[Team:ULB-Brussels/Conclusion|Conclusion & Perspectives]]
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 <br></br>
-<center><font color="#000000"; size="100"> Team ULB-Brussels, modelisation of our </font></center>
+<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>
-<table id="toc" class="toc">
-<tr>
-<td>
-<div id="toctitle">
-<h2>Sommaire</h2>
-</div>
-<ul>
+<balise id="hautdepage"></balise>
-<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>
+<p><font size="+3"> An online version of the report can be found <a href="http://www.mediafire.com/view/?ubp5sobban0b9l6">here</a>.</font>
-<p><A HREF="https://2012.igem.org/Team:ULB-Brussels/Team#2. Modeling the competition experiment"> 2. Modeling the competition experiment</A>
-<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>
-<p><A HREF="https://2012.igem.org/Team:ULB-Brussels/Team#2.2. Natural selection?"> 2.2. Natural selection? </A></ul>
-<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>
-<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>
-<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>
-</ul>
-<p><A HREF="https://2012.igem.org/Team:ULB-Brussels/Team#3. New experiment and conclusion"> 3. New experiment and conclusion </A>
-<p><A HREF="https://2012.igem.org/Team:ULB-Brussels/Team#4. Appendix"> 4. Appendix </A>
-<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>
-<p><A HREF="https://2012.igem.org/Team:ULB-Brussels/Team#4.2. Parameter estimation"> 4.2. Parameter estimation </A>
-</ul>
-</td>
-</tr>
-</table>
-<h2><A NAME="1. Introduction"> 1. Introduction </A></h2>
-<p>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Since complex biological pathways are used in an industrial way in order to produce molecules of
-interest, it has become crucial to understand and, above all, optimize these pathways. However,
-biological systems are so complex that it is sometimes impossible to have a complete understanding
-of the reactions and mechanisms of the different pathways. The idea of our project is to solve this
-optimization problem by using the integron platform { which represents a natural genetic optimiza-
-tion tool in bacteria { and putting in competition different populations with different gene orders,
-so that the population(s) with the optimal order(s) will be naturally selected with time.
-<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
-two natural antibiotics: Microcin C7 and Microcin B17. The first one inhibits a tRNA synthetase
-(thus inhibits protein synthesis and, as a consequence, cell division), and the second inhibits a gyrase
-(thus provokes inhibition of DNA replication and eventually cell death). We might then expect that
-natural selection occurs, so that the optimal gene order(s) finally emerge.
-<p>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;In the sequel, we model this competition experiment, and try to see in what sense and in what
-conditions natural selection could happen.
-<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>
 <br></br>
+<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>
-<h2><A NAME="2. Modeling the competition experiment"> 2. Modeling the competition experiment</A></h2>
-<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,
-the bacterial populations producing these antibiotics will be denoted by <em>Bi</em> and <em>Cj</em> , respectively,
-where the indices i and j run through all different gene cassette orders.
-<p>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;We consider the experiment where all these populations are put in competition together. In our
-model, for the sake of simplicity, we will simply consider that Microcin <em>B</em> causes the production of
-some protein complexes that provoke cell death (bactericidal), while Microcin <em>C</em> inhibits cell division
-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
-used phenomenologically to better describe the situation.
 <br></br>
-<h3><A NAME="2.1. Notations and mathematical model"> 2.1. Notations and mathematical model</A></h3>
+<center>
-<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
+Here are the several graphs that did not appear on the reading version above.
-superscripted letters will stand for the corresponding antibiotics.
-&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%">
-<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
-different possible gene orders for the antibiotics production gene cassettes):
-&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%">
-<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
-half-lives that are much larger than the experiment time, we may neglect the corresponding terms.
-If we also neglect the saturation effect in the population growth (which is natural if the experiment
-is carried out in exponential phase), we then get the following simpler system:
-&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%">
 <br></br>
+<img id="logo" src="https://static.igem.org/mediawiki/2012/6/6a/Graphe_1.PNG" >
-<h3><A NAME="2.2. Natural selection?"> 2.2. Natural selection?</A></h3>
+<img id="logo" src="https://static.igem.org/mediawiki/2012/3/3f/Graphe_2.PNG" >
-<p>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;We could hope that putting together bacteria with all the different gene orders leads to a natural
-selection regime, meaning that the subpopulations with the best offensive and/or defensive charac-
-teristics (     ) will be those
-with the best chances of reproduction, thus leading to the emergence of the bacteria with an optimal
-gene order.
-<p>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;However, we will see in this section that natural selection can only happen on the basis of the
-immunity properties of the di�erent subpopulations, so that, within subpopulations <em>Bi</em> (resp. <em>Ci</em>),
-the most immune ones will emerge, - which is unfortunately not interesting for us as we initially
-aimed to select the most productive ones. We �rst rigorously prove this fact in a simpli�ed context,
-and then discuss numerical simulations of our model in the general case.
 <br></br>
+<font size="+0"> Figures 1-8</font>
-<A NAME="2.2.1. Neglecting the Microcin diffusion"> 2.2.1. Neglecting the Microcin diffusion</A>
-<p>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;In this paragraph, we neglect the di�usion e�ects of the Microcins: in other words, we stop distinguishing between Microcins inside and outside bacteria.
-<A NAME="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>
-<p>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;We now discuss the impact of di�erent production and immunity rates on the time evolution of
-the solutions of general system (2). Since �nding solutions analytically seems beyond possibilities,
-we use <em>Mathematica 8</em> to solve the system numerically (with biologically reasonable values of the
-parameters) in the special case when there are only two di�erent gene orders per type (<em>Bi</em>, <em>Ci</em>,
-i = 1; 2): see Figures 1-8 for a few numerical simulations of the relationship between populations <em>B1</em>,
-<em>B2</em> (blue and red), as well as <em>C1</em> and <em>C2</em> (green and yellow). All populations have the same initial
-value.
-<p>&nbsp;&nbsp;&nbsp;&nbsp;&nbspIn Figures 1-7, both populations B (resp. C) have the same immunity but di�erent production rates � . We immediately observe that both populations B (resp. C) have exactly the same behaviour,
-so that absolutely no di�erence can be seen on the �gures (on each graph, only two double curves
-can be seen: red/blue for the two populations B and green/yellow for the two populations C).
-More precisely, numerical simulations show that both populations B (resp. C) behave as one single
-population with production rate
-<h2><A NAME="4. Appendix"> 4. Appendix</A></h2>
 <br></br>
+<img id="logo" src="https://static.igem.org/mediawiki/2012/2/2e/Graphe_3.PNG" >
-<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>
 <br></br>
+<font size="+0"> Figure 9</font>
-<h3><A NAME="4.2. Parameter estimation"> 4.2. Parameter estimation</A></h3>
+<br></br>
+<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>
 <br></br>
 <h2></h2>
+</center>
 <br></br>