Team:INSA Toulouse/contenu/project/modelling

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<h1 class="title1">Modelling</h1>
<h1 class="title1">Modelling</h1>
<p class="texte">The full adder was tailored taking into account the diffusion of the carry from the bit n to the bit n+1. Evidently, the molecule should reach the n+1 colony prior the calculation step! Therefore, we have modeled the diffusion of AHL through the agar plate. The model would then help us determine the spacing between the different <i>E. calculus</i> colonies and the time necessary between two bits counting.  
<p class="texte">The full adder was tailored taking into account the diffusion of the carry from the bit n to the bit n+1. Evidently, the molecule should reach the n+1 colony prior the calculation step! Therefore, we have modeled the diffusion of AHL through the agar plate. The model would then help us determine the spacing between the different <i>E. calculus</i> colonies and the time necessary between two bits counting.  
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<br>For the modelling, we used a strain of Chromobacterium violaceum deleted in the gene producing AHL. This strai can then only react to the externalm presence of AHL, coloring nicely with violacein, a violet(!!) pigment.</p>
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<br>For the modelling, we used a strain of Chromobacterium violaceum deleted in the gene producing AHL. This strain can then only react to the external presence of AHL, coloring nicely with violacein, a violet pigment.</p>
<h2 class="title2"><i>N</i>-acetyl Homoserine lactone diffusion in agar medium</h2>
<h2 class="title2"><i>N</i>-acetyl Homoserine lactone diffusion in agar medium</h2>
<p class="texte"><i>N</i>-acyl Homoserine Lactone (AHL), 3-oxohexanoyl-homoserine lactone was chosen as the biological messenger in our system.
<p class="texte"><i>N</i>-acyl Homoserine Lactone (AHL), 3-oxohexanoyl-homoserine lactone was chosen as the biological messenger in our system.
<!--/* Different models can be devised to implement the expression and diffusion of the AHL. */-->
<!--/* Different models can be devised to implement the expression and diffusion of the AHL. */-->
Let's imagine a simple system. A petri dish containing colonies equidistant from each other. The lights provide the information for the addition to perform, and the expression and diffusion of AHL from one colony to another allows the carry propagation.<br><br>
Let's imagine a simple system. A petri dish containing colonies equidistant from each other. The lights provide the information for the addition to perform, and the expression and diffusion of AHL from one colony to another allows the carry propagation.<br><br>
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This system seems quite simple but nevertheless raises a certain number of problems:
+
This system seems quite simple but nevertheless raises a certain number of questions:
    
    
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<center><p class="textecaption"><i>Figure 2: Bacterial full adder system in wells.</i></p></center>
<center><p class="textecaption"><i>Figure 2: Bacterial full adder system in wells.</i></p></center>
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<h3 class="title3">Inoculation of cultures</h3>
<h3 class="title3">Inoculation of cultures</h3>
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   <p class="texte">A gradual inoculation of wells during the addition process allows, first to avoid the direct interference between the wells.
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   <p class="texte">A gradual inoculation of wells during the addition process avoids the direct interference between the wells.
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   Furthermore, progressive innoculation would lower the problem cells dying on the plate that could not respond anymore to the AHL messenger.  
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   Furthermore, progressive inoculation would lower the problem of cells dying on the plate that could not respond anymore to the AHL messenger.  
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  <center><p class="textecaption"><i>But is this a problem as dead cells would not respond either to the lights?????</i></p></center>
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</p>
</p>
 +
<h3 class="title3">The ideal distance between wells</h3>
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<h3 class="title3">The ideal distance between well</h3>
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   <p class="texte">In order to find the ideal distance between two colonies, we searched a model that would calculate how does AHL diffuse into the medium and how long does the diffusion process take place to pass from one colony to another.
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+
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   <p class="texte">In order to find the ideal distance between two colonies we searched a model that would calculate how does AHL diffuse into the medium and how long does the diffusion process takes place to pass from one colony to another.
+
</p>
</p>
    
    
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<h2 class="title2">Modelling steps</h2>
 
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<h3 class="title3">Diffusion reminder</h3>
<h3 class="title3">Diffusion reminder</h3>
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<div class="clear"></div>
<div class="clear"></div>
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<p class="texte">The graph above represents cylinder containing bacterias. Bacterias can produce AHL to send a message to another well. Here we can imagine that AHL diffuse into the medium. In order to introduce the theory of diffusion we can realize a simple model, with stationary state condition. In fact, we can establish a mass balance on AHL over a thickness of Δr :
+
<p class="texte">The graph above represents cylinder containing bacterias. Bacterias can produce AHL to send a message to another well. Here we can imagine that AHL diffuse into the medium. In order to introduce the theory of diffusion we can realize a simple model, with stationary state conditions.
</p>
</p>
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<div class="clear"></div>
<div class="clear"></div>
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<p class="texte">In fact, we can establish a mass balance on AHL over a thickness of Δr :
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<p class="texte">In fact, we can establish a mass balance on AHL over a thickness of delta(r) :
</p>
</p>
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<div class="clear"></div>
<div class="clear"></div>
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<img style="width:340px" src="https://static.igem.org/mediawiki/2013/b/b5/Carry4.png" class="imgcontentleft"/>
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<p class="texte">Here we can introduce the Fick’s law for diffusion.
 +
</p>
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<img style="width:340px" src="https://static.igem.org/mediawiki/2013/8/85/Carry5.png" class="imgcontentrightcaption"/>
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<img style="width:340px" src="https://static.igem.org/mediawiki/2013/e/ea/Integration.png" class="imgcontentleft"/>
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<img style="width:340px" src="https://static.igem.org/mediawiki/2013/7/73/Concentration_profile.PNG" class="imgcontentrightcaption"/>
<center><p class="textecaptionright"><i>Figure 3: Evolution of AHL concentration versus distance.</i></p></center>
<center><p class="textecaptionright"><i>Figure 3: Evolution of AHL concentration versus distance.</i></p></center>
<div class="clear"></div>
<div class="clear"></div>
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   <p class="texte">Figure 3 shows that the largest amount of AHL is between 2.5 and 20 mm. This information is important but we need to develop a model based on time.
+
   <p class="texte">With this equation we can establish the concentration profile (figure 3) of the diffusion of AHL. This equation is only valid in our case of well geometry. However, this model does not depend on time, that’s why we must try to develop a more complex system in order to model the diffusion of AHL into LBagar medium with time.
</p>
</p>
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<h1 class="title1">What modelling has tought us.</h1>
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<p class="texte">
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<h3 class="title3">Experimentation</h3>
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If you are not afraid of maths, you can go to the <a href="https://2013.igem.org/Team:INSA_Toulouse/contenu/lab_practice/results/carry" class="texte" style="display : list-item; list-style-image : url(https://static.igem.org/mediawiki/2013/8/84/Insa-toulouse2013-listpuce1.png);"> Modelling results  </a>
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Otherwise here is a brief summary of our main findings:
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<br> - the best model for diffusion is a system where the supply of AHL is not infinite and basically that will be the case in <i>E. calculus</i>. Two elements supports this theory : i) production of AHL is linked to growing. When cells growing will stop, production of AHL will stop too; ii )production of AHL is also linked to presence of the general inducer in the medium (which is not permanent)
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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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<br> - in order to avoid that AHL goes to n+1 and n+2 (and many other) colonies, AHL must be taken up and destroyed by the n+1 colony. That should also be the case as the n+1 colony will pump up AHL and destroy it. <br>
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Figure 4 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.  
+
So basically, the design would be OK provided that we could have extracted the right parameters for AHL diffusion starting from an <i>E. calculus</i> strain instead of pure AHL.
 +
<br>Now that you have had the conclusions, you can go to the modelling results, there are very nice animated pictures at the end!!!
</p>
</p>
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<center><img style="width:700px" src="https://static.igem.org/mediawiki/2013/9/90/Tri_boite.png" class="imgcontentcaption"/></center>
 
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<center><p class="textecaption"><i>Figure 4: Photographs at 25h of petri dishes containing each 8 Chromobacterium violaceum colonies from 5 mm to 40 mm.</i></p></center>
 
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<p class="texte">Figure 5 represent the evolution of AHL diffusion versus time.
 
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</p>
 
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<img style="width:500px" src="https://static.igem.org/mediawiki/2013/6/6b/Carry7.png" class="imgcontentcaption"/>
 
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<p class="textecaption"><i>Figure 5: Evolution of AHL diffusion into petri dish.</i></p>
 
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<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.
 
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Equation of continuity:
 
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</p>
 
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<p class="texte">In our case we have:
 
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        No reaction
 
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        No evolution in z and θ direction
 
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        dwA/dr=0 because of the mass equation (Dρ/Dt=0) with ρ=cste
 
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</p>
 
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<p class="texte">After simplification
 
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</p>
 
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<p class="texte">After simplification and changing the mass in concentration we obtained:
 
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</p>
 
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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:
 
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</p>
 
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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.
 
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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
 
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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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</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.
 
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</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.
 
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</p>
 
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<h3 class="title3">Numerical Model</h3>
 
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<p class="texte">The last step of our modeling was to solve on a numerical way our diffusion problem.
 
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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.
 
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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.
 
-
 
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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:
 
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C(0,0)=cste
 
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C(r=R,t->∞)= cste
 
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</p>
 
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<p class="texte">On these second and third animations we tried to simulate what could be our final system with AHL diffusion from a reaction term.
 
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</p>
 
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</div>
</div>

Latest revision as of 03:16, 5 October 2013

logo


Modelling

The full adder was tailored taking into account the diffusion of the carry from the bit n to the bit n+1. Evidently, the molecule should reach the n+1 colony prior the calculation step! Therefore, we have modeled the diffusion of AHL through the agar plate. The model would then help us determine the spacing between the different E. calculus colonies and the time necessary between two bits counting.
For the modelling, we used a strain of Chromobacterium violaceum deleted in the gene producing AHL. This strain can then only react to the external presence of AHL, coloring nicely with violacein, a violet pigment.

N-acetyl Homoserine lactone diffusion in agar medium

N-acyl Homoserine Lactone (AHL), 3-oxohexanoyl-homoserine lactone was chosen as the biological messenger in our system. Let's imagine a simple system. A petri dish containing colonies equidistant from each other. The lights provide the information for the addition to perform, and the expression and diffusion of AHL from one colony to another allows the carry propagation.

This system seems quite simple but nevertheless raises a certain number of questions:

  1. Can a colony produce enough AHL to induce a response on the n+1 colony?
  2. Do the colonies have to be inoculated all at once or progressively during the calculation step?
  3. What is the ideal distance between the colonies? How can we avoid excessive AHL diffusion that would reach the colony n+1 but also n+2, n+3 etc.

Figure 1: Diffusion of AHL through colonies.

Production of AHL

To overcome the problem of the amount of AHL required for a rapid diffusion of the messenger, we also imagined a system in which liquid precultures may be deposited. A higher cell density would be obtained as well as a greater production of AHL.

Figure 2: Bacterial full adder system in wells.

Inoculation of cultures

A gradual inoculation of wells during the addition process avoids the direct interference between the wells. Furthermore, progressive inoculation would lower the problem of cells dying on the plate that could not respond anymore to the AHL messenger.

The ideal distance between wells

In order to find the ideal distance between two colonies, we searched a model that would calculate how does AHL diffuse into the medium and how long does the diffusion process take place to pass from one colony to another.

Diffusion reminder

The graph above represents cylinder containing bacterias. Bacterias can produce AHL to send a message to another well. Here we can imagine that AHL diffuse into the medium. In order to introduce the theory of diffusion we can realize a simple model, with stationary state conditions.

In fact, we can establish a mass balance on AHL over a thickness of delta(r) :

Here we can introduce the Fick’s law for diffusion.

Figure 3: Evolution of AHL concentration versus distance.

With this equation we can establish the concentration profile (figure 3) of the diffusion of AHL. This equation is only valid in our case of well geometry. However, this model does not depend on time, that’s why we must try to develop a more complex system in order to model the diffusion of AHL into LBagar medium with time.

What modelling has tought us.

If you are not afraid of maths, you can go to the Modelling results Otherwise here is a brief summary of our main findings:
- the best model for diffusion is a system where the supply of AHL is not infinite and basically that will be the case in E. calculus. Two elements supports this theory : i) production of AHL is linked to growing. When cells growing will stop, production of AHL will stop too; ii )production of AHL is also linked to presence of the general inducer in the medium (which is not permanent)
- in order to avoid that AHL goes to n+1 and n+2 (and many other) colonies, AHL must be taken up and destroyed by the n+1 colony. That should also be the case as the n+1 colony will pump up AHL and destroy it.
So basically, the design would be OK provided that we could have extracted the right parameters for AHL diffusion starting from an E. calculus strain instead of pure AHL.
Now that you have had the conclusions, you can go to the modelling results, there are very nice animated pictures at the end!!!

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