Team:Grenoble-EMSE-LSU/Project/Modelling/Density

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<div id="template" style="text-align: center; font-weight: bold; font-size: large; color: #f6f6f6; padding: 5px;">
 
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ATTENTION, RABBIE JACOB IL VA DANSER !!
 
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<div id="instructions" style="text-align: center; font-weight: normal; font-size: small; color: #f6f6f6; padding: 5px;">
 
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<strong>DO NOT EDIT THIS PAGE!!!</strong> </br>
 
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<h1>Density Control</h1>
<h1>Density Control</h1>
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<p>The final aim is to control our bacterial population with only one data available : the global fluorescence. But in the experiments, in order to prove that our system works, besides the fluorescence we measured the $OD_{600}$ and the amount of living cells.</p>
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<p>The aim of our project is to control the density of a living bacterial population with light-transmitted signals:</p>
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<p> At first, we wanted to have a constant population of living bacteria, therefore an $OD_{600}$ going up in a linear way. The model was necessary to find what would be the best illumination to provide to have as quickly as possible a stable level of the amount of living bacteria.</p>
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<p>$\bullet$From the cells to computer: fluorescence and backscattered light as measurable signals from the cell suspension.</p>
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<p>$\bullet$From the computer to the cells: white illumination to kill cells expressing KillerRed proteins. Its intensity needs to be adjusted.</p>
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<p>The density of living cells cannot be measured instantaneously: it is determined by plating the cells on LB-agar plates and counting the colonies.
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<p> In order to validate our system, we first tried to have a constant population of living bacteria. This situation is characterized by an $OD_{600}$ growing up in a <a href="https://2013.igem.org/Team:Grenoble-EMSE-LSU/Project/Modelling/Building#linGrowth" >linear way</a> (see <a href=#ODprev>below</a>). Our model was necessary to find the best time profile of the illumination $I(t)$ to reach as quickly as possible a stable level of the amount of living bacteria</p>
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<h2 id="RbL">Regulation by Light</h2>
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<h2 id="RbL">Stabilization of the living cell density by light</h2>
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<p>We have shown in the <a href="https://2013.igem.org/Team:Grenoble-EMSE-LSU/Project/Modelling/Building#AnaSol">construction of the model</a> that it was theorically possible to stabilize the amount of living bacteria with a constant light. With the complete model, this is still true. On simulations, the light intensity is a very sensitive parameters. Here the predictions of 3 kinetics of 16 hours, illuminated with a power of $1 UF$ for the first (red), $0,327 UF$ for the second (blue) and $0.25 UF$ for the last one (green).</p>
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<p>In our<a href="https://2013.igem.org/Team:Grenoble-EMSE-LSU/Project/Modelling/Building">initial model</a> we showed that it was theoretically possible to stabilize the amount of living bacteria with a <a href="https://2013.igem.org/Team:Grenoble-EMSE-LSU/Project/Modelling/Building#AnaSol">constant light intensity</a>. With the complete model, this still holds true, as shown by the following simulations. In addition, numerical  simulation shows that the light intensity is a very sensitive parameter. Below are displayed three simulated 16-hours long kinetics, for a cell suspension illuminated at different percentage of the maximal illumination:</p>
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<p> 1. $25\%$ for the green one.</p>
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<p> 2. $30\%$ for the cyan one.</p>
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<p> 3. $32.7\%$ for the blue one.</p>
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<p> 4. $35\%$ for the magenta one.</p>
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<p> 5. $100\%$ for the red one.</p>
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<p>These simulations were conducted with the following values of the parameters: </p>
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<p> $r=8,3.10^{-3} min^{-1}$, or $R=83 min$ (time of division)</p>
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<p> $a=130 RFU.OD^{-1}.min^{-1}$</p>
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<p> $b=0,9.10^{-2}RFU.UL^{-1}.min^{-1}$</p>
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<p> $k=0,9.10^{-7}OD.RFU^{-1}.UL^{-1}.min^{-1}$</p>
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<p> $l=0.087$</p>
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<p> $m=6,3.10^{-3} min^{-1}$, or $M=110 min$ (half-time of maturation)</p>
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<p> And the initial contitions were:</p>
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<p> $OD_{600}=0.015$</p>
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<p> $fluorescence=0RFU$</p>
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<p>Here the predictions of the amount of living cells (in $OD_{600}$). If the light intensity is too strong, all the bacteria die, if it is too weak, they grow. And there is a value of light intensity in the middle that permits to get a constant value of living cells. </p>
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<p id="AliveCells">In this first figure, the density of living cells $C(t)$ is displayed in $OD_{600}$ units. When the light intensity is too strong, all bacteria die, when it is too weak, they grow exponentially. A particular value of the light intensity allows to get a constant density of living cells. Note that this light intensity $I^*$ should be precisely regulated, since a 2.5% increase or decrease around this particular light intensity will result in a 30% decrease or increase of the living cell density after 10 hours. </p>
<center><img src="https://static.igem.org/mediawiki/2013/2/2c/Control_3_cells.png"></center>
<center><img src="https://static.igem.org/mediawiki/2013/2/2c/Control_3_cells.png"></center>
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<p>Here is the prediction of the global $OD_{600}$, with both living and dead bacteria. When the light intensity is too strong, the $OD_{600}$ turns constant, when light is too weak, it growths exponentially. But on the right value of light, it growths in a linear way.</p>
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<p id="ODprev">The evolution of $OD_{600}$ is due to both living and dead bacteria. When the light intensity is too strong, the OD600 tends to a constant, when the light intensity is too weak, it growths exponentially. At the light intensity $I^*$, $OD_{600}$ increases linearly.</p>
<center><img src="https://static.igem.org/mediawiki/2013/b/ba/Control_3_OD600.png"></center>
<center><img src="https://static.igem.org/mediawiki/2013/b/ba/Control_3_OD600.png"></center>
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<p></p>
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<p>The evolution of KillerRed fluorescence is also very sensitive to the light intensity. It decreases when the illumination is too strong, increases rapidly when it is too weak, and increases more slowly at the stabilizing light intensity $I^*$. </p>
<center><img src="https://static.igem.org/mediawiki/2013/a/af/Control_3_fluo.png"></center>
<center><img src="https://static.igem.org/mediawiki/2013/a/af/Control_3_fluo.png"></center>
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<p>Since even small deviations from the stabilizing light intensity $I^*$ results in large variations of cell density, an iterative process was designed to determine it with high precision.</p> 
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<h2 id="MPC">Model Predictive Control</h2>
<h2 id="MPC">Model Predictive Control</h2>
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<p>A Model Predictive Control is a Process Control able to deal with complex systems like ours. As our system cannot be stabilized with a simple closed-loop control, it is therefore required to a control more advanced like this one. The aim is to drive our system to a setpoint by predicting the evolution of the system, and adapting the answer to these predictions. To do so, the control will use different kinds of variables:</p>
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<p>Since our system cannot be stabilized with a simple closed-loop control, we used a more advanced control method: <a href="http://en.wikipedia.org/wiki/Model_predictive_control">Model Predictive Control</a>. This algorithm is used to control dynamical systems. It requires a mathematical model of the behavior of the system, and the history of the past control actions. The aim is to drive our system to a setpoint by predicting the evolution of the system, and adapting the control parameter (the light intensity here) to these predictions. To do so, the control will use different kinds of variables:</p>
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<p>$\bullet$ Some variables defines the setpoint, here, we want to drive our living bacteria concentration to a determinate value, called $C_{target}.</p>
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<p>$\bullet$ Some variables defines the setpoint, here, we want to drive our living bacteria concentration to a determinate value, called $C_{target}$.</p>
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<p>$\bullet$ Some variables are measurable durong the experiment. Here, the fluorescence. They are the only ones that will enable us to see if the system behaves according to plans.</p>
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<p>$\bullet$ Some variables are measurable during the experiment. Here, the fluorescence and the optical density. They are the only observable quantities enabling us to see whether the system behaves according to plans or not.</p>
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<p>$\bullet$ Other cannont be measured and have to be estimated with the model. Precisely, the living bacteria concentration cannot be measured. This is a reason why the model has to be precise : we will not be able to measure our setpoint, it shall be estimated.
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<p>$\bullet$ Other variables cannot be measured and have to be estimated from the model. In our case, the concentration of living bacteria cannot be measured in real time, so it will be estimated from the measurable quantities. The model thus needs to be precise.
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<p>$\bullet$ And variables are the ones used to act on the system. The light intensity will affect the system by decreasing the amount of living bacteria and the concentration of KillerRed.</p>
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<p>$\bullet$ And some variables are the ones used to act on the system. In our case, this is the light intensity which will affect the system by decreasing the amount of living bacteria and the concentration of KillerRed.</p>
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<p>This is how it works:</p>
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<p>$1$. For the first point, we have all the datas : the fluorescence $I(0)$and the amount of living cells $C(0)$(no bacteria has died, so $C(0)=OD_{600}$).</p>
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<p>$2$. A illumination $I_1(t)$ is created, it is supposed, according to the model, drive $C(t)$ to its setpoint $C_{target}$. The fluorescence $F_1(t)$ and the amount of cells $C_1(t)$ are also estimated.</p>
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<p>$3$. For a determinate time $\tau$, around 10 minutes to have a start of effect, the experiment will be run with the illumination $I_1(t)$</p>
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<p>Here are the outlines of the algorithm:</p>
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<p>$1$. To start with, cells are grown in the dark before the onset of the experiment. We thus have full information on the system because the fluorescence $K(0)$ and the $OD_{600}$ of $C(0)$ correspond to living cells only. </p>
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<p>$2$. An illumination $I_1(t)$ is calculated, which according to the model is supposed to drive $C(t)$ to its setpoint $C_{target}$. The total fluorescence $F_1(t)$ and the living cell $C_1(t)$ kinetics are also computed.</p>
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<p>$3$. For a certain amount of time $\tau$, more than 10 minutes to see the effect of the illumination, light is applied to the cell suspension at intensity $I_1(t)$</p>
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<p>$4$. At time $t=\tau$, the real fluorescence, $F(\tau)$, is measured and compared to the estimated one, $F_1(\tau)$. </p>
<p>$4$. At time $t=\tau$, the real fluorescence, $F(\tau)$, is measured and compared to the estimated one, $F_1(\tau)$. </p>
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<p>$5$. The others parameters like $C(\tau)$ are estimated according to the difference between $F(\tau)$ and $F_1(\tau)$. If $F(\tau)< F_1(\tau)$, it means that we had overestimated the growth of cells, and so now : $C_{real}(\tau)< C_1(\tau)$. </p>
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<p>$6$. From these estimated and measured values, it goes back to $2$ and $I_2(t)$, $F_2(t)$ and $C_2(t)$ are created.</p>
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<p>$5$. Others hidden variables as $C(\tau)$ are estimated using to the difference between $F(\tau)$ and $F_1(\tau)$. If $F(\tau)< F_1(\tau)$, it means hat we overestimated cell growth, and thus $C_{real}(\tau)< C_1(\tau)$. </p>
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<p>$6$. From the estimated and measured values of $C$, we recalculate the value of the illumination : $I_2(t)$, $F_2(t)$ and $C_2(t)$ are created and the algorithm loops to step $2$ .</p>
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<p>It will not drive perfectly $C(t)$ to its setpoint $C_{target}$, the imperfections of the model will create a gap between them. But we have shown the gap is not too big compared to the value of $C_{target}$.</p>
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<p>This algorithm will not perfectly drive $C(t)$ to its setpoint $C_{target}$. Imperfections in the model will create deviations. But our <a href="https://2013.igem.org/Team:Grenoble-EMSE-LSU/Project/Validation">proof of concept</a> has shown that these are not too big compared to the value of $C_{target}$.</p>
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<h2 id=MCBP">Method to Control a Bacterial Population</h2>
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<h2 id="MCBP">Heuristic method to Control the density of living cells in a bacterial population</h2>
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<p>When the model's parameters were <a href="https://2013.igem.org/Team:Grenoble-EMSE-LSU/Project/Modelling/Parameters#Results">calculated</a>, it appeared that some of them ($b$, $k$ and $l$) were very variable, their value depended a lot on the preparation of the experiment. Sadly, it is important that they are the more accurate possible, the quality of the predictions depend greatly on them. That is the reason why those 3 parameters shall be slightly modified for each experiment.</p>
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<p>We previously <a href="https://2013.igem.org/Team:Grenoble-EMSE-LSU/Project/Modelling/Parameters#Results">observed</a> that some of the parameters of our model ($b$, $k$ and $l$) were quite variable, from experiment to experiment. Since the quality of the predictions greatly depends on them, it is important to determine them accurately. We therefore designed a procedure to obtain them during each experiment.</p>
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<p>$1$ A first period of approximately 100 minutes without illumination to let the bacteria grow and produce KillerRed.<p>
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<p>$2$ A second one with an illumination at maximal intensity. As the effects of light appear, it becomes possible to improve the fit of the volatile parameters ($b$, $k$ and $l$). The length of this period depends on the precision wanted for the parameters: the more it lasts, the more the effects of ligth are obvious, and the more the improvement of the fit is efficient.</p>
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<p>$1$ Cells are first incubated in the dark during a first period of approximately 100 minutes without illumination to let the bacteria grow and produce KillerRed.<p>
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<p> These two first steps are also essential, according to the model, to accelerate the apparition of the level of the amount of living bacteria. If the light of stabilization is switched on from the beginning, the level will appear after 8 hours. But if bacteria are exposed to an important intensity, they will stabilized quicker.</p>
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<p>$3$ Once the parameters are chosen, the model is used to determinate the light intensity that will stabilised our system.</p>
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<p>$2$ During a second period, the cell suspension was illuminated at maximal intensity ($I=1$). This makes it possible to precisely determine the capricious parameters ($b$, $k$ and $l$). The length of this period depends on the precision wanted for the parameters: the longer it lasts, the largest the effects of light are, and more efficiently is the fit improved.</p>
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<p>$4$ The third step passed, the model and its parameters are fixed. But the light intensity still can be modified to control the system if its measured values are drifting away from the trajectory calculated.</p>
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<p>$3$ Once the parameters are chosen, the model is used to determinate the light intensity predicted to stabilize our system.</p>
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<p>$4$ Each time the measured values of fluorescence or absorbance are too far from the predicted ones, parameters are re-estimated, and so is the light intensity that is supposed to stabilize the system. This stabilization could be automatically performed by the model predictive control. We instead run our model to determine changes to apply to the light intensity to reach the desired cell density value</p>
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<p>It is not already the model predictive control, it is the first way to stabilize safely and quickly the population of living bacteria. This experimement is the one used as a <a href="https://2013.igem.org/Team:Grenoble-EMSE-LSU/Project/Validation">proof of concept</a>.</p>
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<p>Another fact should be taken into consideration. We want to stabilize the living cell density in the minimum time, because otherwise the cell culture may become contaminated or run out of nutrients.  We empirically observed that the two first steps of our procedure are essential to speed up the stabilization of the living bacteria population. If the light is set at the stabilizing value $I^*$ from the beginning, the living cell density stabilizes after 8 hours (see the <a href=#AliveCells">predition of living cells</a>). Exposing the bacteria to a large intensity, then reducing the light, makes the living cell population to stabilize more quickly.</p>

Latest revision as of 01:27, 5 October 2013

Grenoble-EMSE-LSU, iGEM


Grenoble-EMSE-LSU, iGEM

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