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 spatio temporal 2D reaction-diffusion system was then implemented in COMSOL multiphysics using the knowledge obtained from single cell pathway model, combining 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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= Initial model of the device =
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= The device set up =
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== Setup of the 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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The initial setup of the device consisted of two parts: a petridish with the ligand which forms the bottom layer and agar with the olfactory yeast which forms the top layer. The ligand molecules diffuse through air from the petridish into the agar.
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== Modeling Approach ==
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= Modeling Approach =
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=== PDEs ===
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== PDEs ==
The diffusion phenomenon is modelled by '''Fick's second law'''.  
The diffusion phenomenon is modelled by '''Fick's second law'''.  
https://static.igem.org/mediawiki/igem.org/8/83/DiffusionFormula1.PNG
https://static.igem.org/mediawiki/igem.org/8/83/DiffusionFormula1.PNG
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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.
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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=== Boundary conditions ===
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== 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.
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.
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The model was the simulated in matlab, the results of which are below
The model was the simulated in matlab, the results of which are below
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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 pressure ratio)]]
 
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[[File:output of diffusion2.png|800px|left|thumb|'''Figure 3''': Simulation of diffusion model at x=0 and x=0.01m)]]
 
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Figures 2 and 3 depict the results of the simulation with distance dimension 0 - 0.01 meter and the time dimension 0 - 100 seconds.
 
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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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= Results =
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== conclusion ==
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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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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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* The results suggest that at approximately two minutes, there is steady state concentration at the agar layer
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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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* 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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= Final Device Model - The Snifferometer =
 
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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 4''': Snifferometer - Device with the modified yeast cells for sensing]]
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Different models were analyzed for their suitability before arriving at the final design in Figure 4. 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 Spatio-Temporal Model =
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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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https://www.circuitlab.com/circuit/2xaw7c/screenshot/1024x768/
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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'''])[[File:NoConvection.png|300px|left|thumb|'''Figure 5''': Device Modeled without convection]]
 
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We wanted to build a system that is able to provide a rapid response. Towards achieving this goal we simulated the model for two different scenarios.[[File:WithConvection.png|300px|Right|thumb|'''Figure 6''': Device Modelled with convection. The flow direction is from left to the right]]
 
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* Without Convection (which simulates the case of a stationary ligand like a sample of saliva).
 
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* With Convection (which simulates a continuous flow of Ligand for eg: A spirometer)
 
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== Simulation Results ==
 
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The results of the simulation over 45 mins duration is shown in the figures below. Please click on the link for the complete simulation.[[File:DiffusionModelSimulation.swf]]
 
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{{:Team:TU-Delft/DiffusionSimulation}}
 
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= Conclusion =
= Conclusion =
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From the simulations and analysis that were done, the following things could be concluded
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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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*''' The convection model responds much faster when compared to the convectionless model.'''
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*''' It takes around 40 mins for the system to generate quantifiable output.'''
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*''' The rate limiting factor was found not to be the diffusion but the GFP production from the pathway.'''
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= Limitations =
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The time scale of the project did not allow us to conduct experiments to validate the findings of the model.
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= Future Work =
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As the model is in it's initial stages of conception, it leaves room open for further research into the characteristics of the device. Some of the points we are looking at working in the future are
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* Conduct experiments to validate the results of the model.
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* Re-estimating the parameters if needed.
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* Building the actual device
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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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