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| | <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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| - | <tr>
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| - | <td>
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| - | <div id="toctitle">
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| - | <h2>Sommaire</h2>
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| - | </div>
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| - | <ul>
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| - | <A HREF="https://2012.igem.org/Team:ULB-Brussels/Team#1. Introduction"> 1. Introduction </A>
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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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| - | </ul>
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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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| - | </ul>
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| - | </td>
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| - | </tr>
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| - | </table>
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| - | <h2><A NAME="1. Introduction"> 1. Introduction </A></h2>
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| - | <p> 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> 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> 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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| - | <br></br>
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| - | <h2><A NAME="2. Modeling the competition experiment"> 2. Modeling the competition experiment</A></h2>
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| - | <p> 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> 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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| - | <br></br>
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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> 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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| - | superscripted letters will stand for the corresponding antibiotics.
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| - | <img id="logo" src="https://static.igem.org/mediawiki/2012/a/aa/Model_1.PNG" height="65%" width="65%">
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| - | <p> 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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| - | <img id="logo" src="https://static.igem.org/mediawiki/2012/5/55/Model_2.PNG" height="65%" width="65%">
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| - | <p> 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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| - | <img id="logo" src="https://static.igem.org/mediawiki/2012/6/6f/Model_3.PNG" height="65%" width="65%">
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| - | <br></br>
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| - | <h3><A NAME="2.2. Natural selection?"> 2.2. Natural selection?</A></h3>
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| - | <p> 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 ( ) will be those
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| - | with the best chances of reproduction, thus leading to the emergence of the bacteria with an optimal
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| - | gene order.
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| - | <p> However, we will see in this section that natural selection can only happen on the basis of the
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| - | immunity properties of the different subpopulations, so that, within subpopulations <em>Bi</em> (resp. <em>Ci</em>),
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| - | the most immune ones will emerge, - which is unfortunately not interesting for us as we initially
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| - | aimed to select the most productive ones. We first rigorously prove this fact in a simplified context,
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| - | and then discuss numerical simulations of our model in the general case.
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| - | <br></br>
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| - | <A NAME="2.2.1. Neglecting the Microcin diffusion"> 2.2.1. Neglecting the Microcin diffusion</A>
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| - | <p> In this paragraph, we neglect the diffusion effects of the Microcins: in other words, we stop distinguishing between Microcins inside and outside bacteria.
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| - | <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>
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| - | <p> We now discuss the impact of different production and immunity rates on the time evolution of
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| - | the solutions of general system (2). Since finding solutions analytically seems beyond possibilities,
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| - | we use <em>Mathematica 8</em> to solve the system numerically (with biologically reasonable values of the
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| - | parameters) in the special case when there are only two different gene orders per type (<em>Bi</em>, <em>Ci</em>,
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| - | i = 1; 2): see Figures 1-8 for a few numerical simulations of the relationship between populations <em>B1</em>,
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| - | <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
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| - | value.
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| - | <p>  In Figures 1-7, both populations B (resp. C) have the same immunity but different production rates T . We immediately observe that both populations B (resp. C) have exactly the same behaviour,
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| - | so that absolutely no difference can be seen on the figures (on each graph, only two double curves
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| - | can be seen: red/blue for the two populations B and green/yellow for the two populations C).
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| - | More precisely, numerical simulations show that both populations B (resp. C) behave as one single
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| - | population with production rate
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| - | <h2><A NAME="4. Appendix"> 4. Appendix</A></h2>
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| - | <br></br>
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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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| - | <br></br>
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| - | <h3><A NAME="4.2. Parameter estimation"> 4.2. Parameter estimation</A></h3>
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| - | <br></br>
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| | <h2></h2> | | <h2></h2> |
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