Team:TU-Delft/Modeling/Diffusion

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One of the main objectives of the project was to synthesize a practical device, the '''Snifferometer''' for tuberculosis detection. As a first step towards achieving this goal, we built a temporal model of the system using '''PDE's''' which was simulated in matlab. A 2D reaction-diffusion system was then implemented in COMSOL multiphysics using the knowledge obtained from single cell pathway model,ombining the behaviours of the which helped us get a better understanding of how such a device could be implemented and the response times involved in such a process.  
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One of the main objectives of the project was to synthesize a practical device, the '''Snifferometer''' for tuberculosis detection. As a first step towards this, we built a spatio temporal 2D reaction-diffusion system which was implemented in COMSOL multiphysics. Combining the behaviours of the diffusion and the pathway model, helped us get a better understanding of how such a device could be implemented and the response times involved in such a process.  
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= Diffusion Model =
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= The device set up =
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== Setup of diffusion model ==
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The device at the bottom has a membrance preventing the yeast coming in direct contact with [[File:SnifferometerCAD.png|200px|left|thumb|'''Figure 5''': Computer aided design of the Snifferometer]]external substances, above which is a layer of agar in which yeast is placed,the nutrients for it's growth is fed in through a channel along the sides of the tube using which the yeast cells can also be replaced as and when needed. A micro-optrode is then used to sense the photons emitted by the fluoroscent proteins. A key question was the time it took for the ligand to diffuse into the layer of yeast. To answer this question, the System was modeled as a 1D diffusion problem using PDE’s.
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[[File:Diffusion Model1.png|320px|middle|thumb|'''Figure 1''': Full structure of diffusion device.]]
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The values for various dimensions of the model were chosen using an iterative approach of design and analysis which would yield an optimal response. The model dimensions can be obtained from the code in the codes section. Once the structure of the device was designed, a simple reaction model assuming linear kinetics was implemented in the agar layer using the results of the single cell pathway model([https://2012.igem.org/Team:TU-Delft/Modeling/SingleCellModel#Results '''Results'''])
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= Modeling Approach =
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== PDEs ==
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The diffusion phenomenon is modelled by '''Fick's second law'''.  
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https://static.igem.org/mediawiki/igem.org/8/83/DiffusionFormula1.PNG
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where C represents the ratio of the pressure between gasphase and atmosphere, x is the distance from the surface of petridish to the agar layer, and D is the diffusion coefficient, the value of which is calculated from [1].
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In order to solve PDEs, numerical methods are used as approximation.
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On the left side of the equation, the Euler forward method is taken:
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https://static.igem.org/mediawiki/igem.org/1/1a/DiffusionFormula2.PNG
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On the right side of the equation, the central differential method is used:
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https://static.igem.org/mediawiki/igem.org/b/b6/DiffusionFormula3.PNG
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It can be noticed that the equations both to the right and left of the equality are the same.
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https://static.igem.org/mediawiki/igem.org/3/35/DiffusionFormula4.PNG
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Therefore, the PDEs can be numerically replaced by the equation:
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The diffusion setup contains two parts: petridish of smelling liquid/solid in the bottom and agar of olfactory yeast on the top.
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https://static.igem.org/mediawiki/igem.org/4/4c/DiffusionFormula5.PNG
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Modeling of this setup was especially done to evaluate the speed of diffusion from the ligand phase to the yeast cells themselves and see if this behavior is slow or quick with respect to the biochemical behavior of the yeast.  
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where https://static.igem.org/mediawiki/igem.org/b/b8/DiffusionFormula6.PNG. In the implementation by MATLAB, dx is taken as the interval of x.
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Partial Differential Equations (PDE) are used to solve this problem.
 
== Boundary conditions ==
== Boundary conditions ==
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Two boundary conditions are set for these two surfaces of petridish-gas and gas-agar.
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In order to solve this equation, boundary conditions are required. Two boundary conditions are set for these two surfaces of petridish-air and air-agar.
*At x=0, the Dirichlet boundary condition was placed:  
*At x=0, the Dirichlet boundary condition was placed:  
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https://static.igem.org/mediawiki/igem.org/b/b8/DirichletBC.PNG
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https://static.igem.org/mediawiki/igem.org/archive/b/b8/20120926220011%21DirichletBC.PNG
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with Psat being the saturation pressure of the specific compound being IsoAmylAcetate as an example in our model. For IsoAmylAcetate this value is 533.3 Pa at room temperature.  
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with Csat being the pressure ratio with respect to the saturation pressure of the specific compound being IsoAmylAcetate as an example in our model. For IsoAmylAcetate this value is 533.3 Pa at room temperature. Thus Csat is equal to 0.05333.
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*One can take Neumann boundary condition at x=l, where l is the distant between petridish and agar and assumed to be 0.01 meter.
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*One can take Neumann boundary condition at x=l, where l is the distant between petridish and agar and assumed to be 0.01 meter in the model. The reason we can use Neumann boundary is because the molecules bounce back into the gasphase against the agar surface.
https://static.igem.org/mediawiki/igem.org/e/ea/NeumannBC.PNG
https://static.igem.org/mediawiki/igem.org/e/ea/NeumannBC.PNG
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In this part of the system the molecules bounce against the agar surface back into the gasphase.
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The model was the simulated in matlab, the results of which are below
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A van Neumann boundary is the way to give a mathematical workable structure to that.
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For the problem definition to be complete and be able to start programming in MATLAB
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we only need one more value, being the diffusion coefficient. For this an empirical relation from the [1] was used.
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== PDE ==
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</div>
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Now we have a complete problem that we will solve using a numerical approach.
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Put this in the pressure domain.
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= Results =
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[[File:output of diffusionNew.png|320px|middle|thumb|'''Figure 2''': Simulation of diffusion model.(in 3D Cartesian coordinate, X axis is time, Y axis is the distance from the petridish, Z axis is the pressure ratio)]]Figures 2 and 3 depict the results of the simulation with distance dimension 0 - 0.01 meter and the time dimension 0 - 100 seconds. Two key observations were drawn from this model
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*The results suggest that at approximately two minutes, there is steady state concentration at the agar layer
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*The input to the single cell pathway model can be considered a step function, if we assume that the Ligand is not degraded by yeast.
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And you probably guessed but my value for h is ok in the model and in my code this translates to the line:
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*Matlab script for diffusion model can be found [https://static.igem.org/mediawiki/igem.org/c/c2/IGEM_Model_Pertidish.txt'''here'''].
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dcdt(i,1)=  a*x(i-1,1)+a*x(i+1,1) -2*a*x(i,1);
 
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and YES a=h here and YES that IS confusing considering the other above statements.
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What I basically did is grab a bunch of Ordinary differential equations and couples them
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So that it becomes a PDE ( partial differential equation ).  H*time step should also not be bigger than 0.5 and that does not happen, so the solution is stable.
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Analytical solutions for this problem appeared a little more troublesome than expected.
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= Comsol Spatio-Temporal Model =
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There are solutions ( in Sum row form ), but I was anticipating a nice function.
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A spatio temporal 2D reaction-diffusion system which was implemented in COMSOL multiphysics. Combining the behaviours of the diffusion and the pathway model.The simulation of which is shown below.
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I will get a book at the library to see if there are any better ones, but for now this works out fine.
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I can also use a Cranck-Nicolson approach to the problem to have another numerical solution for verification. And last but not least. The results:
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[[File:output of diffusion.png|320px|middle|thumb|'''Figure 2''': Simulation of diffusion model.(in 3D Cartesian coordinate, X axis is time, Y axis is the distance from the petridish, Z axis is the concentration)]]
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Figure 2 shows the simulation result with distance dimension 0 - 0.01 meter and the time dimension 0 - 100 seconds. From Figure2 it can be seen after 100 seconds it approximately reaches steady state at the x=0.01 boundary.
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<iframe width="560" height="315" src="http://www.youtube.com/embed/qoObgkd4tsY" frameborder="0" allowfullscreen></iframe>
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<h6>In case the place seems to be blank, zoom into the page by using Ctrl + Mouse-wheel and hit F5.</h6>
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<div id="contentbox" style="text-align:justify;">
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= Snifferometer =
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= Conclusion =
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From the simulations and analysis it can be concluded that the response of the system can be improved by improving the kinetics of the signalling pathway and the diffusion has a limited influence on the system response.
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One of the other goals of the diffusion modeling was to model the device which we intended to build for the project. We made use of the finite element analysis simulator Comsol Multiphysics[[Team:TU-Delft/Modeling/Diffusion#Ref3|[3]]]
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for developing this model.[[File:Snifferometer.png|300px|Right|thumb|'''Figure 1''': Snifferometer - Device with the modified yeast cells for sensing]]
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Different models were analyzed for their suitability before coming to the final design in Figure. The device at the bottom has a membrance preventing the yeast coming in direct contact with external substances, above which is a layer of agar in which yeast is placed, the nutrients for it's growth is fed in through a channel along the sides of the tube using which the yeast cells can also be replaced as and when needed. A micro-optrode is then used to sense the photons emitted by the fluoroscent proteins.
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= Comsol Code =
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=== Final Model ===
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[[File:TUD-Download.png|50px|link=http://thenewview.eu/igem/feromone.mph|left]]
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== Saliva Model ==
 
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[[File:Test1.swf|300px|Right|thumb|'''Figure 1''': Snifferometer - Device with the modified yeast cells for sensing]]
 
= References =  
= References =  
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Chen, N. H. (1962). New Generalized Equation for Gas Diffusion Coefficient. J. Chem. Eng. Data, 37–41.
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<h6>
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1. Chen, N. H. (1962). New Generalized Equation for Gas Diffusion Coefficient. J. Chem. Eng. Data, 37–41.
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</h6>
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<h6>
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2. Comsol Multiphysics.
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</h6>
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Latest revision as of 01:46, 27 October 2012

Menu

Diffusion Model

One of the main objectives of the project was to synthesize a practical device, the Snifferometer for tuberculosis detection. As a first step towards this, we built a spatio temporal 2D reaction-diffusion system which was implemented in COMSOL multiphysics. Combining the behaviours of the diffusion and the pathway model, helped us get a better understanding of how such a device could be implemented and the response times involved in such a process.

Contents

The device set up

The device at the bottom has a membrance preventing the yeast coming in direct contact with
Figure 5: Computer aided design of the Snifferometer
external substances, above which is a layer of agar in which yeast is placed,the nutrients for it's growth is fed in through a channel along the sides of the tube using which the yeast cells can also be replaced as and when needed. A micro-optrode is then used to sense the photons emitted by the fluoroscent proteins. A key question was the time it took for the ligand to diffuse into the layer of yeast. To answer this question, the System was modeled as a 1D diffusion problem using PDE’s.

The values for various dimensions of the model were chosen using an iterative approach of design and analysis which would yield an optimal response. The model dimensions can be obtained from the code in the codes section. Once the structure of the device was designed, a simple reaction model assuming linear kinetics was implemented in the agar layer using the results of the single cell pathway model(Results)

Modeling Approach

PDEs

The diffusion phenomenon is modelled by Fick's second law. DiffusionFormula1.PNG

where C represents the ratio of the pressure between gasphase and atmosphere, x is the distance from the surface of petridish to the agar layer, and D is the diffusion coefficient, the value of which is calculated from [1].

In order to solve PDEs, numerical methods are used as approximation.

On the left side of the equation, the Euler forward method is taken:

DiffusionFormula2.PNG

On the right side of the equation, the central differential method is used:

DiffusionFormula3.PNG

It can be noticed that the equations both to the right and left of the equality are the same.

DiffusionFormula4.PNG

Therefore, the PDEs can be numerically replaced by the equation:

DiffusionFormula5.PNG

where DiffusionFormula6.PNG. In the implementation by MATLAB, dx is taken as the interval of x.

Boundary conditions

In order to solve this equation, boundary conditions are required. Two boundary conditions are set for these two surfaces of petridish-air and air-agar.

  • At x=0, the Dirichlet boundary condition was placed:

20120926220011%21DirichletBC.PNG

with Csat being the pressure ratio with respect to the saturation pressure of the specific compound being IsoAmylAcetate as an example in our model. For IsoAmylAcetate this value is 533.3 Pa at room temperature. Thus Csat is equal to 0.05333.

  • One can take Neumann boundary condition at x=l, where l is the distant between petridish and agar and assumed to be 0.01 meter in the model. The reason we can use Neumann boundary is because the molecules bounce back into the gasphase against the agar surface.

NeumannBC.PNG

The model was the simulated in matlab, the results of which are below

Results

Figure 2: Simulation of diffusion model.(in 3D Cartesian coordinate, X axis is time, Y axis is the distance from the petridish, Z axis is the pressure ratio)
Figures 2 and 3 depict the results of the simulation with distance dimension 0 - 0.01 meter and the time dimension 0 - 100 seconds. Two key observations were drawn from this model
  • The results suggest that at approximately two minutes, there is steady state concentration at the agar layer
  • The input to the single cell pathway model can be considered a step function, if we assume that the Ligand is not degraded by yeast.


  • Matlab script for diffusion model can be found here.


Comsol Spatio-Temporal Model

A spatio temporal 2D reaction-diffusion system which was implemented in COMSOL multiphysics. Combining the behaviours of the diffusion and the pathway model.The simulation of which is shown below.

In case the place seems to be blank, zoom into the page by using Ctrl + Mouse-wheel and hit F5.

Conclusion

From the simulations and analysis it can be concluded that the response of the system can be improved by improving the kinetics of the signalling pathway and the diffusion has a limited influence on the system response.

Comsol Code

Final Model

TUD-Download.png


References

1. Chen, N. H. (1962). New Generalized Equation for Gas Diffusion Coefficient. J. Chem. Eng. Data, 37–41.
2. Comsol Multiphysics.

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