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| | Université Libre de Bruxelles **/ --> | | Université Libre de Bruxelles **/ --> |
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| | <td width="50%"><section style="text-align: right"> | | <td width="50%"><section style="text-align: right"> |
| | <a href="https://2014.igem.org/Team:ULB-Brussels/Modelling/Population-Dynamics"><b> Pop Dyn ></b></a> | | <a href="https://2014.igem.org/Team:ULB-Brussels/Modelling/Population-Dynamics"><b> Pop Dyn ></b></a> |
| | <a href="https://2014.igem.org/Team:ULB-Brussels/Modelling/TA-System"><b> TA Sys ></b></a> | | <a href="https://2014.igem.org/Team:ULB-Brussels/Modelling/TA-System"><b> TA Sys ></b></a> |
| | <a href="https://2014.igem.org/Team:ULB-Brussels/Modelling/2A-Peptid"><b> 2A Pep ></b></a> | | <a href="https://2014.igem.org/Team:ULB-Brussels/Modelling/2A-Peptid"><b> 2A Pep ></b></a> |
| - | </section></td> | + | </section></td> --> |
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| | <!-- back + text --> | | <!-- back + text --> |
| | <tr style="background-color:rgb(204,214,234); box-shadow: 1px 1px 12px #555;"><td colspan="2"> | | <tr style="background-color:rgb(204,214,234); box-shadow: 1px 1px 12px #555;"><td colspan="2"> |
| | <p class="title"><font color="#002B9B"> | | <p class="title"><font color="#002B9B"> |
| - | Overview | + | <font color="#CCD6EA">'Now begins an</font> $Overview$ <font color="#CCD6EA">about Modelling'</font> |
| | </font></p> | | </font></p> |
| | </td></tr> | | </td></tr> |
| | </table> | | </table> |
| - | </br> | + | <br/> |
| - | <table style="background-color:rgb(204,214,234);" width="90%" align="center"> | + | <table style="background-color:#ebebeb;" width="90%" align="center"> |
| - | <tr style="background-color:rgb(246,246,246);"><td> | + | <tr style="background-color:rgb(245,245,245);"> |
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| | + | <section style="margin: -40px"></section> |
| | <section style="text-align: justify; margin: 50px"> | | <section style="text-align: justify; margin: 50px"> |
| - | <p>$\Igo$n this section, we will introduce the modelling part with an overview of our biological system and its expected effects with numerical simulations of bacterial populations. Structural components of the chosen plasmids will also be covered. In next sections, the mathematical procedure will be detailed.</p>
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| - | <p>Afterwards, these simulations will be compared with experimental results. In parallel, an estimation of the production of sub-populations in bioreactors will be made by coupling the recombinant protein with an essential protein and a numerical estimation will be runned. Now, we're continuing to perform it.</p></section>
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| - | <section style="text-align: justify; margin: 50px">
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| - | <p> The following diagram was drawn to identify the principal components of our biological system (i.e. the effects related to the sequences induced by a promotor, to the growth of bacteria, to the transcription and translation of a specific protein) :</p>
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| - | </section>
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| - | <section style="text-align: justify; margin: -45px"></section>
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| - | <section style="text-align: justify; margin: 50px">
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| - | <center>
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| - | <img src="https://static.igem.org/mediawiki/2014/e/e0/MightyColi-Sch-v2-Coli.png">
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| - | </center></p>
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| - | <font size="1"><b>Figure m1 </b>: This diagram illustrates the production of GFP fluorescent proteins controlled by PSK (post-segregational killing), in the case of TA CcdBA system. </font>
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| - | </section>
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| - | <section style="margin: -30px"></section>
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| - | </td></tr>
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| - | <tr><td><br/><br/></td></tr>
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| - | </table>
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| - | <section style="margin: 30px"></section>
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| - | <h1>Population Dynamics Model</h1>
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| - | <!-- Hypertxt -->
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| - | <section style="margin: -70px"></section>
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| - | <section style="text-align: right; margin: 70px">
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| - | <a href="https://2014.igem.org/Team:ULB-Brussels/Modelling/Population-Dynamics"><b> ~Pop~ </b></a> </section>
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| - | <section style="margin: -70px"></section>
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| - | <p>The growth of bacteria involves ...</p>
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| - | a Population Dynamics Model can be fitted in our system. Theorically, two approaches have been planned:
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| - | <section style="text-align: justify; margin: 50px"><p>
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| - | <h3>$1.1)$ $By$ $Probabilities$</h3>
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| - | When some new plasmids are introduced into the cytoplasm of E.Coli bacteria, it does not garantee that the daughter cells will contain it. Indeed, these plasmids can be lost after cell division or replication. In this regard, it is interesting to study a model based on the different possibilities of plasmid combinations in bacteria, as in the study of mutations in animals. A typical example of a similar case is found if we study the mutations of the eyes color in a family, by vertical genes transfer. IOf course, in the case of plasmids, one must also take horizontal genes transfer into account. </p>
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| - | A Probabilistic Model is useful because easily undertstandable, but requires some assumptions. </p>
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| - | </section>
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| - | <section style="text-align: justify; margin: -35px"></section>
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| - | <section style="text-align: justify; margin: 50px">
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| - | <h3>$1.2)$ $By$ $Logistic$ $\small\&\normalsize$ $Lotka$ $Equations$</h3>
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| - | The Logistic Equation was initially introduced during the beginning of the XIXth Century, by the belgian mathematician P.F. Verhulst. Now, this equation is mainly used in Population Dynamics Models, especially in Biological Sciences. Mathematicians currently finish Ph.D thesis using it, and the analytical Lotka-Volterra model is directly associated with the Verhulst theory. Another interesting model is obtained from Euler-Lotka equation to the Leslie matrix coefficients. </p>
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| - | Other models exist, f.e. by Monod equation, but these ideas are less consistent with our global and partial systems.
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| - | </section>
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| - | </p>-</p>
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| - | <h1>Toxin-Antitoxin Systems</h1>
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| - | <!-- Hypertxt -->
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| - | <section style="margin: -70px"></section>
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| - | <section style="text-align: right; margin: 70px">
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| - | <a href="https://2014.igem.org/Team:ULB-Brussels/Modelling/TA-System"><b> ~TA~ </b></a> </section>
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| - | <section style="margin: -70px"></section>
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| - | <p>Two type II TA systems are investigating in our $\MyColi$ project.</p>
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| - | The $\hspace{0.12cm}\small\mathtt{1}\normalsize^{st}$ consists of ccdB (the toxin, T) and ccdA (the antitoxin, A) and for the $\hspace{0.12cm}\small\mathtt{2}\normalsize^{nd}$ these are Kid (T) and Kis (A):</p>
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| - | <!--LaTex Raccourcis-->
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| - | $\newcommand{\AA}{\mathbb{A}}
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| - | \newcommand{\CC}{\mathbb{C}}
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| - | \newcommand{\TT}{\mathbb{T}}
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| - | \newcommand{\GG}{\mathbb{G}}
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| - | \newcommand{\KK}{\mathcal{K}}$
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| - | <section style="text-align: justify; margin: 50px">
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| - | <h3>$2.1)$ $CcdBA$</h3>
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| - | <p>One of the most studied and characterized TA systems, CcdBA involves two principal components : ccdB and its antidote ccdA.
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| - | As we explained in the introduction page of our project, ccdB is an inhibitor of the DNA gyrase, so it binds the subunit A of the DNA gyrase complex when it's bound to DNA.
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| - | When DNA double strand is broken, there is activation of emergency signals (SOS system blocks cellular division in bacteria). If the DNA gyrase is not protected by a mutation (such events are possible, but excessively rare) or if the antidote is degraded (very frequent because ccdA is unstable in comparison with ccdA), the death of a bacterium in unavoidable. </p>
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| - |
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| - | <!-- \begin{equation} \hspace{-0.08cm}\wp\hspace{0.02cm}\equiv[\small GFP \normalsize]\\ -->
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| - | <!--
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| - | \begin{align*}
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| - | \AA\hspace{0.02cm}&\equiv[ccdA] &
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| - | \CC\hspace{0.02cm}&\equiv[ccdBA\small - \normalsize complex]\\[0.1cm]
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| - | \TT\hspace{0.02cm}&\equiv[ccdB] &
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| - | \GG\hspace{0.012cm}&\equiv[\small GFP \normalsize]\\[-0.3cm]
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| - | a\hspace{0.03cm}&\equiv[\mathrm{ara}] &
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| - | \mathrm{g}\hspace{0.05cm}&\equiv[glu] \hspace{1.45cm}
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| - | \mathring{x}=\dfrac{dx}{dt}
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| - | \end{align*}
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| - |
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| - | In presence of arabinose, AraC activates the transcription of RNA$\hspace{0.01cm}_{\textbf{m}}$ (catalysed by RNA$\hspace{0.02cm}_{\textbf{poly}}$) :
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| - | \begin{array}.
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| - | \hspace{0.02cm}\mathring{\AA} &=& v_{s_{1}} \dfrac{a}{a + \small\KK\normalsize_{\mathtt{1}}} - \hspace{0.05cm}v_{d_{1}} \hspace{0.01cm}\AA\hspace{0.02cm} - \hspace{0.05cm}v_{a} \hspace{0.02cm}(\AA\TT)\hspace{0.02cm} + \hspace{0.05cm}v_{d_{3}}\hspace{0.02cm}\CC\hspace{0.03cm} \\[0.1cm]
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| - | %
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| - | \hspace{0.02cm}\mathring{\TT} &=& v_{s_{2}} \dfrac{a}{a + \small\KK\normalsize_{2}} - \hspace{0.05cm}v_{d_{2}} \hspace{0.03cm}\TT\hspace{0.02cm} - \hspace{0.05cm}v_{a} \hspace{0.02cm}(\AA\TT)\hspace{0.02cm} + \hspace{0.05cm}v_{d_{4}} \hspace{0.02cm}\CC\hspace{0.03cm} \\[0.1cm]
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| - | %
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| - | \hspace{0.02cm}\mathring{\CC} &=& v_{a} \hspace{0.02cm}(\AA\TT)\hspace{0.02cm} - \hspace{0.02cm}(v_{d_{3}}+v_{d_{4}}) \hspace{0.05cm}\CC\\[0.25cm]
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| - | %
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| - | \hspace{0.02cm}\mathring{\GG} &=& v_{s_{3}} \dfrac{a}{a + \small\KK\normalsize_{3}} - \hspace{0.05cm}v_{d_{5}} \hspace{0.04cm}\GG
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| - | \end{array}
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| - |
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| - | In presence of glucose, AraC becomes a repressor of the promotor pBAD, so the three first differential equations (for
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| - | $\hspace{0.1cm}\AA\hspace{0.04cm}$, $\hspace{0.04cm}\TT\hspace{0.04cm}$ $\small\&\normalsize$ $\hspace{0.04cm}\GG\hspace{0.1cm}$)
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| - | are changed by :
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| - |
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| - | \begin{equation}
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| - | \left( v_{s_{j}} \dfrac{a}{a + \small\KK\normalsize_{j}} \right) \rightarrow \left( v_{s_{j}} \dfrac{\small\KK\normalsize_{j}}{\mathrm{g} + \small\KK\normalsize_{j}} \right).
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| - | \end{equation}
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| - |
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| - | NB:$\hspace{0.06cm}$ We have chosen a Michaelis-Menten kinetics, maybe a higher Hill coefficient would be desirable.</p>
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| - | Because we will preserve some fragment of the population, it is necessary to controle its level. In practical, different parameters are introduced in the mathematical model to describe all the configurations of the biological system (in the equations above, the parameters are the constants $\hspace{0.04cm}\small\mathcal{K}\normalsize_{j}\hspace{0.02cm}$ and the velocities $\hspace{0.04cm}v_{j}\hspace{0.06cm}$). These parameters influence the global dynamics of the TA system, with or without an additional proline with p2A.</p>
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| - | -->
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| - | This TA system will be implemented for $\EColi$ bacteria.</p>
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| - | </section>
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| - | <section style="text-align: justify; margin: -35px"></section>
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| - | <section style="text-align: justify; margin: 50px">
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| - | <h3>$2.2)$ $Kis/Kid$</h3>
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| - | The same is relevant about the second TA system studied. In this case, the two principal components are Kid and its antidote Kis and the parameters are chosen a little bit different than in the first system.</p>
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| - | This TA system will be implemented for $\SCere$ yeasts.
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| - | By modelling and by comparison with experiments, we would obtain finally a model that correctly describes our TA system.</p>
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| - | </section>
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| - |
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| - | </p>-</p>
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| - | <h1>Bioreactor and 2A Peptide</h1>
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| - |
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| - | <!-- Hypertxt, without really "Modelling the 2A Peptide"-->
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| - | <section style="margin: -70px"></section>
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| - | <section style="text-align: right; margin: 70px">
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| - | <a href="https://2014.igem.org/Team:ULB-Brussels/Modelling/2A-Peptid"><b> ~2A~ </b></a> </section>
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| - |
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| - | <section style="margin: -70px"></section>
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| - | <p>The p2A peptide is an important building block in our system. </p>
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| - | We think that its use can enhance the production of bioreactors. </p>
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| - |
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| - | <section style="text-align: justify; margin: 50px">
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| - | In this perspective, we will compare the classical $\Stabi$ system with our $\MyColi$ system
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| - | and identify when Mighty Coli should be applied in a bioreactor. We will also investigate the effective gain in production of a protein with the Mighty Coli system. </p>
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| - |
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| - | We would like to mention that different 2A peptids exists(f.e. F2A and P2A), but these peptids are not
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| - | completely characterized. A future continuation of our project could be a study of this aspect
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| - | to refine our current model and to build new Mighty Coli Biobricks that use different 2A peptids.
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| - |
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| - | <!-- Bacteria versus yeasts and types of 2A
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| - | <section style="text-align: justify; margin: 50px">
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| - | <h3>$3.1)$ $F2A$</h3> <h3>$3.2)$ $P2A$</h3> </section>
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| - | -->
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| - | <br><br>
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| - | <!-- Nice Biblio effect with: </table> -->
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| - | <table style="background-color:rgb(204,214,234)">
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| - | <section style="background-color:#CCD6EA; margin: 25px">
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| - | <section style="text-align: justify; margin: 25px">
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| - | <h3> Bibliography </h3>
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| - | <ul>
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| - | <li>[9] prob refer </li>
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| - | <li>[10] A.V. Hill, (1910). <i>The possible effects of the Aggregation of the molecules of hemoglobin on its Dissociation curves</i>, J. Physiol, No.40, iv-vii. </li>
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| - | <!-- Intro on Hill functions with hemoglobin-->
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| - | <li>[11] T. Ogura, S. Hiraga, (1983). <i>Mini-F plasmids genes that couple Host cell division to Plasmid proliferation</i>, Proc. Natl. Acad. Sci. USA, 80, 4784-4788. </li>
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| - | <!--
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| - | (Abstract): Stable Maintenance plasmids, ccd region dissected into ccdB (cell division) and ccdA (inhibition), plasmid proloferation.
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| - | (p4785): oriC plasmids, EcoR1, replication
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| - | (p4787): Dissection of the cdd region + culture, kinetics and growing conditions.
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| - | (p4788): Mutants of F plasmid
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| - | -->
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| - | <li>[12] M. Santillán, (2008). <i>On the use of the Hill functions in Mathematical models od Gene regulatory networks</i>, Math. Model. Nat. Phenom., Vol.3, No.2, 85-97. </li>
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| - | <!--
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| - | (p86): Cooperative binding sequences: The Hill coefficient is appropriately described as an interaction coefficient reflecting cooperativity.
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| - | (p94): If P is an activator(/repressor), the regulatory function R([P]) comes out to be monotocally increasing(/decreasing). Transcription rate, probability of gene copy.
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| - | (p95): Interestingly, the most extensively studied gene regulatory systems (cf lactose operon of E.coli) make use of cooperativity to increase the sigmoidicity of the regulatory functions.
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| - | (p96): Significance of the parameters by modelling.
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| - | -->
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| - | <li>[13] L. Gelens, L. Hill, A. Vandervelde, J. Danckaert, R. Loris, (2013). <i>A general model for Toxin-antitoxin module dynamics can explain Persister cell formation in E.coli</i>, PLOS Computational Biology, Vol.9, Iss.8, e1003190. </li>
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| - | <!--
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| - | (p1): Among the elements involved in bacterial stress response are the type TT toxine-antitoxin modules. CcdB and ParE family mambers inhibit gyrase, although via different molecular mechanisms.
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| - | (p2): Persisters are subpopulations of bacteria wich are tolerant to biological stresses such antibiotics because they are in a dormant, non-dividing state.
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| - | (p11): Fig @ simulations: Large toxin spikes -> route to persister cell formation through growth rate suppression.
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| - | (p14): Kinds of TA, TAT Decay complexes.
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| - | -->
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| - | <li>[14] N. Goeders, L. Van Melderen, (2014). <i>Toxin-antitoxin systems as Multilevel interaction systems</i>, Toxins, 6, 304-324, ISSN 2072-6651. </li>
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| - | <!--
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| - | (p305): Fig: ccdA-ccdB
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| - | (p306): Evolution of TA systems + bioinfo approaches.
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| - | (p313): Effect on DNA-gyrase.
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| - | -->
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| - | <li>[15] 2A refer 1 </li>
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| - | <li>[16] 2A refer 2 </li>
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| - | </ul>
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| - | <br>
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| - | </section></section>
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| - | <section style="margin: -25px"></section>
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| - | </tr>
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| | <!-- Previous and Next pages --> | | <!-- Previous and Next pages --> |
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| | <section style="text-align: right"> | | <section style="text-align: right"> |
| | <a href="https://2014.igem.org/Team:ULB-Brussels/Modelling/Population-Dynamics"><b> Population Dynamics > </b></a> | | <a href="https://2014.igem.org/Team:ULB-Brussels/Modelling/Population-Dynamics"><b> Population Dynamics > </b></a> |
| - | </section> | + | </section></tr> |
| - | </tr> | + | |
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| | <tr><td><br/><br/></td></tr> | | <tr><td><br/><br/></td></tr> |
| - | </table></th></tr>
| + | </th></tr> |
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| | </div> | | </div> |
$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
\newcommand{\MyColi}{{\small Mighty\hspace{0.12cm}Coli}}
\newcommand{\Stabi}{\small Stabi}$
$\newcommand{\EColi}{\small E.coli}
\newcommand{\SCere}{\small S.cerevisae}\\[0cm]
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
\newcommand{\PI}{\small PI}$
$\newcommand{\Igo}{\Large\mathcal{I}}
\newcommand{\Tgo}{\Large\mathcal{T}}
\newcommand{\Ogo}{\Large\mathcal{O}}
~$
"
- Université Libre de Bruxelles -
