Team:INSA Toulouse/contenu/lab practice/results/carry

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<div class="maincontent" style="width: 720px; margin: 25px 0 25px 0; float: right;">
<div class="maincontent" style="width: 720px; margin: 25px 0 25px 0; float: right;">
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   <h1 class="title1">Modelling Results</h1>
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   <h1 class="title1">Results: Modelling </h1>
    
    
   <h3 class="title3">Experimentation</h3>
   <h3 class="title3">Experimentation</h3>
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  <p class="texte">We realized many experiences in order to measure how AHL diffuse into agar medium. We used a particular bacteria, Chromobacterium violaceum, which is capable to detect presence of AHL by producing a purple pigment, the violacein.<br><br>
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  <p class="texte">We realized different experiments in order to measure how AHL diffuses into thez agar medium. We used a mutant strain <i>Chromobacterium violaceum</i> that cannot produce AHL. However, in the presence of AHL, the mutated strain starts  producing a purple pigment: violacein.<br><br>
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Figure 1 shows petri dishes containing eight colonies of bacteria placed at 5, 10, 15, 20, 25, 30, 35 and 40 mm from the center of the box. With this experience we can track the diffusion of the AHL. Also we can see that the colonies are in a spiral disposition. This typical disposition allow us to avoid the problem of utilisation of AHL of the n-1 colony to the n colony.  
+
Figure 1 shows a petri dish containing eight colonies of bacteria placed at 5, 10, 15, 20, 25, 30, 35 and 40 mm from the center of the box. With this experiment, we can study the diffusion of AHL. We also can see that colonies are disposed in spiral. This geometric disposition allows us to avoid the problem of utilisation of AHL of the n-1 colony to the n colony.  
</p>
</p>
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  <p class="texte">Figure 2 represent the evolution of AHL diffusion versus time.
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  <p class="texte">Figure 2 represents the evolution of AHL diffusion versus time.
</p>
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<p class="textecaption"><i>Figure 2: Evolution of AHL diffusion into petri dish.</i></p>
<p class="textecaption"><i>Figure 2: Evolution of AHL diffusion into petri dish.</i></p>
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  <p class="texte">With this experience we can represent on figure 3 the surface of AHL diffusion (surface is a disk) versus time in order to find the coefficient of diffusion of AHL into LBagar. The directing coefficient of the regression line give us the order of magnitude of coefficient of diffusion, we find an order of 10-8 m²/s. This coefficient is an important value to develop analytical and numerical modelling.
+
  <p class="texte">With this experiment, we can represent the surface of AHL diffusion on figure 3(surface is a disk) versus time in order to find the diffusion coefficient of AHL into LB agar. The regression coefficient of the straight line gives us an order of magnitude for the diffusion coefficient at 10<SUP>-8</SUP> m²/s. This coefficient is an important value to develop analytical and numerical modelling.
</p>
</p>
<img style="width:500px" src="https://static.igem.org/mediawiki/2013/0/04/Coefficient_of_diffusivity.PNG" class="imgcontentcaption"/>
<img style="width:500px" src="https://static.igem.org/mediawiki/2013/0/04/Coefficient_of_diffusivity.PNG" class="imgcontentcaption"/>
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<p class="textecaption"><i>Figure 3: Determination of coefficient of diffusion of AHL into LBagar medium.</i></p>
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<p class="textecaption"><i>Figure 3: Determination of diffusion coefficient of AHL into LB agar medium.</i></p>
<h3 class="title3">Analytical Model</h3>
<h3 class="title3">Analytical Model</h3>
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  <p class="texte">After the creation of our first model above which allowed us to find relevant answer for our system, we thought that it term of modeling we weren’t really satisfied. That is why we developed a new model based on equation of continuity in cylindrical coordinates. The general equation in our case is the following.
+
  <p class="texte">After the creation of our first model, which allowed us to find some relevant answers for our system, we thought that in terms of modelling we were not really satisfied. That is why we developed a new model based on equations of continuity in cylindrical coordinates. The general equation in our case is the following:
Equation of continuity:
Equation of continuity:
</p>
</p>
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<p class="texte">In our case we have:  
<p class="texte">In our case we have:  
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         No reaction
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         <br>No reaction
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         No evolution in z and θ direction
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         <br>No evolution in z and ? directions
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         dwA/dr=0 because of the mass equation (/Dt=0) with ρ=cste
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         <br>dwA/dr=0 because of the mass equation (D?/Dt=0) with ?=cste
After simplification:
After simplification:
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<div class="clear"></div>
<div class="clear"></div>
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<p class="texte">To solve this equation we first thought to use the variables separation but we faced different issues, but we found after some more research a new solution which fit better with our system. This solution is corresponding to a Dirac impulsion. This is not the perfect solution but this is the closest analytical solution to our system. The only difference with our system is that the boundary condition for C(0;0), is not exactly the same. For the Dirac impulsion C(0;0)->,  but this approximation can be acceptable for our system considering that our AHL concentration is high at the beginning of the experiment and the loading is really small.This is our analytical solution:
+
<p class="texte">To solve this equation we first thought to use the separation of variables method but found a new solution which fits better with our system. This solution corresponds to a Dirac impulsion. This is not the perfect solution but this is the closest analytical solution to our system. The only difference with our system is that the boundary condition for C(0;0), is not exactly the same. For the Dirac impulsion C(0;0)->8,  but this approximation can be acceptable for our system considering that our AHL concentration is high at the beginning of the experiment and the loading surface is really small. This is our analytical solution:
</p>
</p>
<center><img src="https://static.igem.org/mediawiki/2013/c/c0/Equation_of_continuity_analitycal_solution.PNG" class="imgcontent"/></center>
<center><img src="https://static.igem.org/mediawiki/2013/c/c0/Equation_of_continuity_analitycal_solution.PNG" class="imgcontent"/></center>
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<p class="texte">In this case as we consider our system as a dirac impulsion system, we cannot be focused on the concentration value (because C(0;0)-> ). That is why here, we are working with normalized concentration value. The point of interest with this model is the evolution of the concentration in time and space and also the determination of the diffusion coefficient D.  
+
<p class="texte">In this case as we consider our system as a dirac impulsion, we cannot focus on the concentration values (because C(0;0)-> 8). We are thus working with normalized concentration values. The interest of this model is the evolution of the concentration over time and space and also the determination of the diffusion coefficient D.  
-
We tested different diffusivity coefficient to find which one allow the model to fit better with the experimentation. We found that coefficient of diffusivity of AHL into LBagar medium is around: D = 1.10-8 m²/s
+
We tested different diffusivity coefficients to find the one which allows the model with the best fit to our experimenations . We found that the coefficient of AHL diffusivity into LBagar medium is around: D = 1.10-8 m²/s
-
On this 1st chart, we can see for a given distance from the petri dish center the evolution of the concentration. Close to the center the concentration increase really fast and goes down more slowly. More the radius is important more the concentration increase is weak.
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On this first chart, we can see the evolution of the concentration for a given distance from the petri dish center. Close to the center the concentration increases really fast and goes down more slowly.  
</p>
</p>
<img style="width:500px" src="https://static.igem.org/mediawiki/2013/2/28/Analytical_fig_1.PNG" class="imgcontentcaption"/>
<img style="width:500px" src="https://static.igem.org/mediawiki/2013/2/28/Analytical_fig_1.PNG" class="imgcontentcaption"/>
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<p class="textecaption"><i>Figure 4: Evolution of concentration with time for analytic solution.</i></p>
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<p class="textecaption"><i>Figure 4: Evolution of concentration in time for analytic solution.</i></p>
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<p class="texte">On this second chart we can see at different time the concentration profile along the petri dish. This graph is relevant to compare at different time how the concentration profile is. For instance at 70min the concentration is still high at r=0 but AHL reached 25 mm, whereas at 20min at r=0 the concentration is 2 times higher but the AHL only reached 10mm.
+
<p class="texte">On this second chart we can see at different times the concentration profile along the petri dish. With this graph, it is relevant to compare how the concentration profile evolves in time. For instance at 70 min the concentration is still high at r=0 and AHL reached 25 mm, whereas at 20 min at r=0 the concentration is 2 times higher but the AHL only reached 10 mm.
</p>
</p>
<img style="width:500px" src="https://static.igem.org/mediawiki/2013/3/38/Analytical_fig_2.PNG" class="imgcontentcaption"/>
<img style="width:500px" src="https://static.igem.org/mediawiki/2013/3/38/Analytical_fig_2.PNG" class="imgcontentcaption"/>
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<p class="textecaption"><i>Figure 5: Evolution of concentration with radius for analytic solution.</i></p>
<p class="textecaption"><i>Figure 5: Evolution of concentration with radius for analytic solution.</i></p>
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<p class="texte">This model fit well in term of evolution with our experimentation. Thanks to this model we can approximately predict the diffusion of AHL into the LB agar. But we have to be careful here because we don’t have the real concentration values and for high time the analytical model isn’t corresponding to the reality.
+
<p class="texte">This model fits well in terms of evolution with our experimentation. Thanks to this model we can approximately predict the diffusion of AHL into the LB agar. But we have to be careful here because we don’t have the real concentration values and for longer times the analytical model does not correspond to the reality.
</p>
</p>
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<p class="texte">The last step of our modeling was to solve on a numerical way our diffusion problem.
<p class="texte">The last step of our modeling was to solve on a numerical way our diffusion problem.
The aim was to solve the same differential equation presented in the analytical solution part. But here as we solved it with a computer we could use the right boundary condition, and also added a reaction term.
The aim was to solve the same differential equation presented in the analytical solution part. But here as we solved it with a computer we could use the right boundary condition, and also added a reaction term.
-
Therefore we developed a numerical solution of AHL diffusion into LB agar medium with the software Comsol. We just needed to enter all the physical parameter of our petri dish and the AHL diffusivity.
+
Therefore we developed a numerical solution of AHL diffusion into LB agar medium with the software Comsol. We just needed to enter all the physical parameters of our petri dish and the AHL diffusivity.
-
We can see on this first animation the diffusion of AHL into a petri dish with colonies but without reaction term. We note that the concentration of AHL becomes progressively homogeneous in the medium. This model is closest to the reality than the analytical one because here the boundary conditions are the experimental boundary condition:
+
On this first animation, we can see  the diffusion of AHL into a petri dish with colonies but without reaction term. We note that the concentration of AHL becomes progressively homogeneous in the medium. This model is closest to the reality than the analytical one because here the boundary conditions are equivalent to the experimental boundary conditions:
C(0,0)=cste
C(0,0)=cste
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C(r=R,t->)= cste
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C(r=R,t->8)= cste
</p>
</p>
<img src="https://static.igem.org/mediawiki/2013/c/c8/Ahl_diffusion3.gif" class="imgcontentcaption"/>
<img src="https://static.igem.org/mediawiki/2013/c/c8/Ahl_diffusion3.gif" class="imgcontentcaption"/>
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<center><p class="textecaption"><i>Figure 8: Reception of AHL in well 2 and 3.</i></p></center>
<center><p class="textecaption"><i>Figure 8: Reception of AHL in well 2 and 3.</i></p></center>
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<p class="texte">We can note that the AHL easily reached the well 2, but also 3 after some time. The passage of the AHL from a well n to well n+1 without reaching the point n+2 is really a key point. It is certainly difficult to implement such a system. In addition, it is imperative that the production of AHL is not continuous, because the concentration in the petri reach all the wells. We need a short pulse in time for transferring the AHL only in well 2. It also requires that the AHL is consumed in well 2 to limit the diffusion in well 3.
+
<p class="texte">We can note that the AHL easily reached the well 2, but also 3 after some time. The passage of the AHL from a well n to well n+1 without reaching the point n+2 is really a key point. It is certainly difficult to implement such a system. In addition, it is imperative that the production of AHL is not continuous, because the concentration in the petri reaches all the wells. We need a short pulse in time for transferring the AHL only in well 2. It also requires that the AHL is consumed in well 2 to limit the diffusion in well 3.
</p>
</p>

Latest revision as of 23:05, 4 October 2013

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Results: Modelling

Experimentation

We realized different experiments in order to measure how AHL diffuses into thez agar medium. We used a mutant strain Chromobacterium violaceum that cannot produce AHL. However, in the presence of AHL, the mutated strain starts producing a purple pigment: violacein.

Figure 1 shows a petri dish containing eight colonies of bacteria placed at 5, 10, 15, 20, 25, 30, 35 and 40 mm from the center of the box. With this experiment, we can study the diffusion of AHL. We also can see that colonies are disposed in spiral. This geometric disposition allows us to avoid the problem of utilisation of AHL of the n-1 colony to the n colony.

Figure 1: Photographs at 25h of petri dishes containing each 8 Chromobacterium violaceum colonies from 5 mm to 40 mm.

Figure 2 represents the evolution of AHL diffusion versus time.

Figure 2: Evolution of AHL diffusion into petri dish.

With this experiment, we can represent the surface of AHL diffusion on figure 3(surface is a disk) versus time in order to find the diffusion coefficient of AHL into LB agar. The regression coefficient of the straight line gives us an order of magnitude for the diffusion coefficient at 10-8 m²/s. This coefficient is an important value to develop analytical and numerical modelling.

Figure 3: Determination of diffusion coefficient of AHL into LB agar medium.

Analytical Model

After the creation of our first model, which allowed us to find some relevant answers for our system, we thought that in terms of modelling we were not really satisfied. That is why we developed a new model based on equations of continuity in cylindrical coordinates. The general equation in our case is the following: Equation of continuity:

Bird, R. B., Stewart, W. E., & Lightfoot, E. N. (1960). Transport phenomena (Vol. 2). New York: Wiley.

In our case we have:
No reaction
No evolution in z and ? directions
dwA/dr=0 because of the mass equation (D?/Dt=0) with ?=cste After simplification:

After simplification and changing the mass in concentration we obtained:

To solve this equation we first thought to use the separation of variables method but found a new solution which fits better with our system. This solution corresponds to a Dirac impulsion. This is not the perfect solution but this is the closest analytical solution to our system. The only difference with our system is that the boundary condition for C(0;0), is not exactly the same. For the Dirac impulsion C(0;0)->8, but this approximation can be acceptable for our system considering that our AHL concentration is high at the beginning of the experiment and the loading surface is really small. This is our analytical solution:

In this case as we consider our system as a dirac impulsion, we cannot focus on the concentration values (because C(0;0)-> 8). We are thus working with normalized concentration values. The interest of this model is the evolution of the concentration over time and space and also the determination of the diffusion coefficient D. We tested different diffusivity coefficients to find the one which allows the model with the best fit to our experimenations . We found that the coefficient of AHL diffusivity into LBagar medium is around: D = 1.10-8 m²/s On this first chart, we can see the evolution of the concentration for a given distance from the petri dish center. Close to the center the concentration increases really fast and goes down more slowly.

Figure 4: Evolution of concentration in time for analytic solution.

On this second chart we can see at different times the concentration profile along the petri dish. With this graph, it is relevant to compare how the concentration profile evolves in time. For instance at 70 min the concentration is still high at r=0 and AHL reached 25 mm, whereas at 20 min at r=0 the concentration is 2 times higher but the AHL only reached 10 mm.

Figure 5: Evolution of concentration with radius for analytic solution.

This model fits well in terms of evolution with our experimentation. Thanks to this model we can approximately predict the diffusion of AHL into the LB agar. But we have to be careful here because we don’t have the real concentration values and for longer times the analytical model does not correspond to the reality.

Numerical Model

The last step of our modeling was to solve on a numerical way our diffusion problem. The aim was to solve the same differential equation presented in the analytical solution part. But here as we solved it with a computer we could use the right boundary condition, and also added a reaction term. Therefore we developed a numerical solution of AHL diffusion into LB agar medium with the software Comsol. We just needed to enter all the physical parameters of our petri dish and the AHL diffusivity. On this first animation, we can see the diffusion of AHL into a petri dish with colonies but without reaction term. We note that the concentration of AHL becomes progressively homogeneous in the medium. This model is closest to the reality than the analytical one because here the boundary conditions are equivalent to the experimental boundary conditions: C(0,0)=cste C(r=R,t->8)= cste

Figure 6: Diffusion of AHL in LBagar medium with numerical solution.

On these second and third animations we tried to simulate what could be our final system with AHL diffusion from a well 1 to other wells. These following two animations are complementary. The first shows the diffusion of the AHL from well 1, the second allows us to visualize the reach of the AHL in wells 2 and 3.

Figure 7: Diffusion of AHL in LBagar medium with numerical solution from well 1.

Figure 8: Reception of AHL in well 2 and 3.

We can note that the AHL easily reached the well 2, but also 3 after some time. The passage of the AHL from a well n to well n+1 without reaching the point n+2 is really a key point. It is certainly difficult to implement such a system. In addition, it is imperative that the production of AHL is not continuous, because the concentration in the petri reaches all the wells. We need a short pulse in time for transferring the AHL only in well 2. It also requires that the AHL is consumed in well 2 to limit the diffusion in well 3.