Team:Grenoble-EMSE-LSU/Project/Modelling

From 2013.igem.org

(Difference between revisions)
 
(38 intermediate revisions not shown)
Line 26: Line 26:
<ul class="texte">
<ul class="texte">
<li id="titre">
<li id="titre">
-
<h1>Modeling</h1>
+
<h1>Modelling</h1>
-
<p>Modeling took a large place in the project; it was not only used for the characterization of KillerRed and the Voigt plasmids, it was needed for the control of the bacteria’s population. With our device, we cannot control a population of living cells with a simple closed-loop transfer function. First, this is because optical measurements (OD 600 nm or fluorescence) originate from all cells, whether they are alive or not. This fluorescence intensity gives clues about living cell activity and therefore its temporal evolution permits to find the number of living cells, but there is no simple relation between them. Second, there is a large delay between an action and its effect: there are about one or two hours between the onset of illumination and the deceleration of fluorescence. In those conditions, a simple closed-loop transfer function is predictably unstable, and a model predictive control is needed to stabilize the population of living cells.</p>
+
<p>Modelling took a large place in the project. We used modelling for the characterization of KillerRed and the Voigt plasmids and also needed it for the control of the bacterial population. With our device, we cannot control a population of living cells with a simple closed-loop transfer function. The first reason is that optical measurements (OD at 600 nm or fluorescence) originate from all cells, whether they are alive or not. As a consequence, it is not easy to reconstruct the size of a population of living cells from a fluorescent intensity or an $OD_{600}$ reading. Second, there is a large delay between an action and its effect: there are about one or two hours between the onset of illumination and a decrease in fluorescence growth rate. In those conditions, a simple closed-loop transfer function is predictably unstable, and a predictive model is needed to stabilize the population of living cells.</p>
-
</li>
+
                                </li>
-
+
-
<li>
+
-
<h1>Building the Model</h1>
+
-
<h2>Initial Model</h2>
+
-
<h3>The equation</h3>
 
-
<p>    Our system is made of bacterial cells and ‘KillerRed’ proteins. Bacteria divide and produce KillerRed proteins, and KillerRed proteins respond to light: they fluoresce, degrade (photobleaching) and produce Radical Oxygen Species or ROS (phototoxicity). These reactions are exhibited by all fluorescent proteins, but <a href="/Team:Grenoble-EMSE-LSU/Project/Biology">the 3D structure of KillerRed</a> makes its degradation quicker and its high concentration allows ROS to reach proteins, DNA and membrane within the bacteria and damage its vital functions. )</p>
 
-
<br>
 
-
<p>$\bullet$ $C$ the amount of living bacteria per milliter of cell suspension.</p>
 
-
<p>$\bullet$ $K$ the amount of KillerRed inside the living bacteria per milliter of cell suspension.</p>
 
-
<p>$\bullet$ $I$ the amount of incident (white) light.</p>
 
-
<br>
 
-
<p>The evolution of C and K is linked to I by the set of equations : </p>
 
-
<center style="font-size:150%;">
+
                                <li>
 +
<a href="/Team:Grenoble-EMSE-LSU/Project/Modelling/Building" title="BuildingLink"><h3>Building the Model</h3></a>
 +
<a href="/Team:Grenoble-EMSE-LSU/Project/Modelling/Building" title="BuildingLink2">
 +
<img src="https://static.igem.org/mediawiki/2013/5/5b/Gre_Mod_Lego.png" style="float:left"></a>
 +
<p style="height:100px;float:none;padding-left:125px;padding-right:150px"> Come, sit, and listen to the story of the model's construction! You will hear about the journey of this model, from its genesis to its completion ! Going through the reasons that drove it to consider the time of <a href="https://2013.igem.org/Team:Grenoble-EMSE-LSU/Project/Modelling/Building#MatTime">maturation of KillerRed</a> or the <a href="https://2013.igem.org/Team:Grenoble-EMSE-LSU/Project/Modelling/Building#AccDam">resilience of bacteria</a>.</p>
-
$
 
-
\left\{
 
-
  \begin{array}{l l}
 
-
    \frac{dC}{dt}=rC-kIK \\
 
-
    \frac{dK}{dt}=aC-bIK-kI\frac{K^2}{C} \\
 
-
  \end{array}
 
-
\right.
 
-
$
 
-
</center>
 
-
 
-
<p>$\diamond$ $rC$ describes bacterial growth.</p>
 
-
<p>$\diamond$ $kIK=kI\frac{K}{C}C$ the amount of bacteria killed by KillerRed and light.</p>
 
-
<p>$\diamond$ $aC$ the production of KillerRed.</p>
 
-
<p>$\diamond$ $bIK$ the amount of KillerRed photobleached.</p>
 
-
<p>$\diamond$ $kIK\frac{K}{C}$ the amount of KillerRed in the bacteria killed in the last step of time.</p>
 
<br>
<br>
-
<p>    Unfortunately, $C$ and $K$ are not measurable variables. The only thing we can quickly and easily measure are the optical density (OD) bound to the amount of bacteria dead AND alive, and the global fluoresence bound to the amount of KillerRed in the bacteria dead AND alive. To be able to compare our model to experimental results, we need two other variables : </p>
 
<br>
<br>
-
<p>$\bullet$ $D$ the amount of dead bacteria </p>
 
-
<p>$\bullet$ $K_D$ the amount of KillerRed inside the dead bacteria</p>
 
-
<center style="font-size:150%;">
+
<a href="/Team:Grenoble-EMSE-LSU/Project/Modelling/Parameters" title="ParamLink"><h3>Finding Parameters</h3></a>
-
 
+
<a href="/Team:Grenoble-EMSE-LSU/Project/Modelling/Parameters" title="ParamLink2">
-
$
+
<img src="https://static.igem.org/mediawiki/2013/d/d4/Gre_Mod_Tron.png" style="float:left"></a>
-
\left\{
+
<p style="height:100px;float:none;padding-left:125px;padding-right:150px"> Here, you traveler, you will read about the way the parameters were chosen to best fit the experiments, and thus make the model properly predict the evolution of the bacterial concentration. You will see how a <a href="https://2013.igem.org/Team:Grenoble-EMSE-LSU/Project/Modelling/Parameters#AlgoGen">genetic algorithm</a> can be used to understand a genetic network. And of course you will appreciate the <a href="https://2013.igem.org/Team:Grenoble-EMSE-LSU/Project/Modelling/Parameters#Results">final results</a>.</p>
-
  \begin{array}{l l}
+
-
    \frac{dD}{dt}=kIK \\
+
-
    \frac{dK_D}{dt}=kI\frac{K^2}{C}-bIK_D\\
+
-
  \end{array}
+
-
\right.
+
-
$
+
-
</center>
+
<br>
<br>
-
<br>
 
-
<p> The simplest possible units were used, that correspond to the measurable quantities : </p>
 
-
<p> $C$ and $D$ are in '$OD_600nm$' units.</p>
 
-
<p> $K$ and $K_D$ are in 'units of fluorescence' : UF. Bacterial auto-fluorescence is considered as negligible compared to KillerRed fluorescence. </p>
 
<br>
<br>
-
 
-
<h3>Analytical Solution</h3>
 
-
 
-
<p> This simple model can be partially solved, for $C(t)$ or $I(t)$ constant for example : </p>
 
-
<br>
 
-
<p>If we $C$ is constant, $\forall t, C(t)=C_0$, we have :</p>
 
-
<p>
 
-
$
 
-
\left\{
 
-
  \begin{array}{l l}
 
-
    rC_0=kIK \\
 
-
    \frac{dK}{dt}=aC_0-bIK-kI\frac{K^2}{C_0}\\
 
-
  \end{array}
 
-
\right.
 
-
$
 
-
</p>
 
-
<br>
 
-
<p> and so : $\frac{dK}{dt}=\left(a-\frac{br}{k}\right)C_0-rK$</p>
 
-
<p> which gives : $K=\left(\frac{a}{r}-\frac{b}{k}\right)C_0+Be^{-rt}$</p>
 
-
<br>
 
-
<p> then $I(t)=\frac{rC_0}{kK(t)}$ should give a constant concentration of living cell.</p>
 
-
<p> For time long enough, the light intensity that stabilizes the concentration of living cells is  $I_0=\frac{r^2}{ak-rb}$.</p>
 
-
<br>
 
-
<br>
 
-
<p> But if we assume that $I$ is constant, $\forall t, I(t)=I_0$, we need another variable to solve easily our equation : </p>
 
-
<p> We define : $Y=\frac{K}{C}$ the amount of KillerRed per bacteria.</p>
 
-
<p> $\frac{dY}{dt}=\frac{d}{dt}\left(\frac{K}{C}\right)=a-(bI_0+r)Y$</p>
 
-
<p> which gives : $Y=\frac{a}{bI_0+r}+Be{-(bI_0+r)t} $</p>
 
-
<br>
 
-
<p> $Y$ tends toward a steady state value, $\frac{a}{bI_0+r}$. Let's see how C </p>
 
-
<p>$\frac{dC}{dt}=C(r-kI_0Y)$</p>
 
-
<p>$\frac{dC}{dt}=C\left(r-\frac{kI_0a}{bI_0+r}+Be^{-(bI_0+r)t}\right)$</p>
 
-
<p> Thus, in the specific case where $I_0=\frac{r^2}{ak-rb}$, we have : $\lim_{t\to\infty}\frac{dC}{dt}(t)=0$</p>
 
-
<br>
 
-
<p> The resolution of this equation have shown the possibility to stabilize the system.</p> 
 
-
<br>
 
-
<p>The analytical solution of this set of equations clearly shows the possibility to stabilize the system thanks to a suitable (constant) light intensity.</p>
 
-
 
-
<h3>Comparison with experiments</h3>
 
-
 
-
<p> This first model is very interesting to understand which parameters govern the evolution of the living cell population et to show that conditions exist stabilize it. But this set of equation is insufficient to explain the experiments : </p>
 
-
 
-
 
-
 
 +
<a href="/Team:Grenoble-EMSE-LSU/Project/Modelling/Density" title="ParamLink"><h3>Density Control</h3></a>
 +
<a href="/Team:Grenoble-EMSE-LSU/Project/Modelling/Density" title="ControlLink">
 +
<img src="https://static.igem.org/mediawiki/2013/d/d7/Gre_Mod_Manette.png" style="float:left"></a>
 +
<p style="height:100px;float:none;padding-left:125px;padding-right:150px""> At last, you will discover <a href="https://2013.igem.org/Team:Grenoble-EMSE-LSU/Project/Modelling/Density#RbL">the power and the impact</a> that light has on our system. Then you will be introduced to a <a href="https://2013.igem.org/Team:Grenoble-EMSE-LSU/Project/Modelling/Density#MPC">crafty way</a> to master it and control a bacterial population with its help. You will be shown <a href="https://2013.igem.org/Team:Grenoble-EMSE-LSU/Project/Modelling/Density#MCBP">how it is supposed to be done</a>.</p>
<br>
<br>
<br>
<br>
-
<p> Whereas we observe a lag between the onset of light and the decrease of fluorescence, the first model predicts an immediate decrease. This discrepancy requires another phenomena to be introduced to explain the lag between the stimulus (the light) and the reaction (the decrease of fluorescence and the OD stabilization). Of course this explanation should be borne out by biological facts. </p>
 
-
      </li>
 
-
 
-
<li>
 
-
<h2>Maturation Time</h2>
 
-
<h3>The maturation of fluorescent proteins</h3>
 
-
<p>After traduction and spontaneous polypeptide folding, a fluorescent protein still has to maturate before becoming fluorescent. Fluorescent proteins mature after an oxidation reaction where three amino acids rearrange to form the fluorophore. For GFP, this time is typically 30 minutes [1]. The maturation time of KillerRed is significant for our hands. </p>
 
-
<br>
 
-
<p>[1]<a ref=http://www.chem.ufl.edu/~fanucci/courses/BiochemistryJournalClub/Spring2007/ChromophoreFormationinGFP_biochemistry_1997.pdf">REID Brian G., FLYNN Gregiry C. Chromophore Formation in Green Fluorescent Protein. Biochemistry, 1997, 36, p 6786-6791</a>.</p>
 
-
<h3>Second model </h3>
 
-
<p> We consider maturation to be a simple chemical reaction, and the concentration on immature protein $[K_im]$ is governed by the equation : $[K_im]'=-m[K_im]$. It is a realistic hypothesis since maturation is an oxydation. A new variable is needed : </p>
 
-
 
-
<br>
 
-
<p>$\bullet$ $K_m$ the amount of mature KillerRed inside the living bacteria</p>
 
-
<p>$\bullet$ $K_i$ the amount of immature KillerRed inside the living bacteria</p>
 
-
 
-
<center style="font-size:150%;">
 
-
$
 
-
\left\{
 
-
  \begin{array}{l l}
 
-
    \frac{dC}{dt}=rC-kIK_m \\
 
-
    \frac{dK_i}{dt}=aC-kI\frac{K_i^2}{C}-mK_i\\
 
-
    \frac{dK_m}{dt}=-kI\frac{K_m^2}{C}-bIK_m+mK_i\\
 
-
  \end{array}
 
-
\right.
 
-
$
 
-
</center>
 
-
 
-
 
-
<p>$\diamond$ $mK_i$ is the term traducing the maturation of KillerRed </p>
 
-
 
-
<br>
 
-
<p> Here too, other equations are required to find the measureable variables : </p>
 
-
 
-
<center style="font-size:150%;">
 
-
$
 
-
\left\{
 
-
  \begin{array}{l l}
 
-
    \frac{dD}{dt}=kIK_m \\
 
-
    \frac{dK_{Di}}{dt}=kI\frac{K_i^2}{C}-mK_{Di}\\
 
-
    \frac{dK_{Dm}}{dt}=kI\frac{K_m^2}{C}-bIK_{Dm}+mK_{Di}\\
 
-
  \end{array}
 
-
\right.
 
-
$
 
-
</center>
 
-
 
-
<br>
 
-
<h3> Numeric results</h3>
 
-
<p> This model draws curves that properly follow the tendancy of the experiments : the lag of the reaction, the peak of fluorescence and then the brutal decrease are there. </p>
 
-
<br>
 
-
<br>
 
-
<p>Nonetheless, it is impossible to get a good fit between the prediction of the model and the experiment. The integration of maturation doesn't explain why the production of KillerRed is so hight even two hours after the start of illumination and the decrease of fluorescence is so brutal four hours after the illumination.</p>
 
-
 
</li>
</li>
-
 
-
<li>
 
-
<h2>Damage Accumulation</h2>
 
-
<p> Until now, in the equations, KillerRed and light produced ROS, those reacted in the instant, killed or not the bacteria, and disappeared. Now, we consider that bacteria are unable to instantly repear all the damages caused by ROS, and with damages mounting up, they are more and more fragile and close to death. Considering this accumulation allows us to shift the effect of illumination, and so to have a model more accurate.</p>
 
-
<br>
 
-
<p> Writen in its numeric form, the evolution of C was : </p>
 
-
<center style="font-size:150%;">
 
-
$
 
-
C(t+1)-C(t)=rC(t)-kI(t)K(t)
 
-
$
 
-
</center>
 
-
<p> It is now : </p>
 
-
<center style="font-size:150%;">
 
-
$
 
-
\left\{
 
-
  \begin{array}{l l}
 
-
    C(t+1)-C(t)=rC(t)-\mbox{tox}(t) \\
 
-
    \mbox{tox}(t+1)=l.\mbox{tox}(t)+k'I(t)K(t)\\
 
-
  \end{array}
 
-
\right.
 
-
$
 
-
</center>
 
-
<center>
 
-
with $l\in[0,1]$
 
-
</center>
 
-
<p>
 
-
The variable 'tox' is representative of the amount of damages inflicted to bacteria, and so their probability to die. With each step of time (for us, a minute), bacteria cures a part of their injuries ($l<1$) and suffers other damages ($k'I(t)K(t)$). At first, few bacteria die : $\mbox{tox}(t)\cong k'I(t)K(t)$, then, 'tox' increases until it reach $\mbox{tox}(t)\cong\frac{k'}{1-l}I(t)K(t)$
 
-
</p>
 
-
<br>
 
-
<p> This new equation can also be writen in an analytic form :</p>
 
-
<center style="font-size:150%;">
 
-
$
 
-
\frac{dC}{dt}(t)=rC(t)-\int_{u=0}^t k'I(u)K(u)e^{\ln(l)(t-u)}du
 
-
$
 
-
</center>
 
-
<p> But it is unsolvable and brings only complexity.</p>
 
-
 
-
<br>
 
-
 
-
<p> With this model, we can explain properly the look of our curves : </p>
 
-
 
-
 
-
</li>
 
-
<li id="next"><a href="/Team:Grenoble-EMSE-LSU/Project/Modeling/Parameters">Next Page</a></li>
+
<li id="next"><a href="/Team:Grenoble-EMSE-LSU/Project/Modelling/Building">Next Page</a></li>
</ul>
</ul>
</div>
</div>
</html>
</html>

Latest revision as of 01:55, 5 October 2013

Grenoble-EMSE-LSU, iGEM


Grenoble-EMSE-LSU, iGEM

Retrieved from "http://2013.igem.org/Team:Grenoble-EMSE-LSU/Project/Modelling"