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Team:Evry/Auxin diffusion
From 2012.igem.org
Model using Partial Differential Equations(PDE)
Overview
Using PDE instead of ODE allows one to take into account the space dimensions. Where in the ODE model, the concentration of auxin in a compartment was considered homogeneous, here we can represent the variations internal to the compartment. When taking into account the space dimensions, new problems arise: finding a coherent geometry, 2D vs 3D model, precision, etc. We chose to model a slice of a tadpole's tail based on images from [ADD BOOK REF]. Putting multiple such slices one after the other allows us to approximate a 3 dimensions model.
Hypothesis
There are the different hypothesis we were constrained to make in order to model the system:
- The quantity of auxins is homogeneous in the blood
Model description
The PDE model is similar to the ODE one except that it takes into account the geometry. This allows us to model more complex phenomenon such as diffusion and transport.
Equations
The diffusion equation is used to model the repartition of auxin's molecule when not subject to any flow (in the skin for example).
The equation is states as follows:
where:
- x = (x1, x2) is a 2 dimensional vector
- c is a function representing the auxin concentration
- D is the diffusion constant
- Δ is the Laplacian operator
Geometry
To keep the complexity of the numerical simulations low, we had to simplify the considered geometry. Hence, the slide of tadpole only contains the following elements:
- Skin
- Blood vessels or muscles (depending on the model)
- Notochord
- Spinal cord
- Aorta
- Veins: caudal and dorsal
As seen on the picture above, the geometry is composed of very elementary shapes: circles and ellipses. Note that in the model, the shapes and positions of the different areas can be modified. In fact their are no really fixed geometry because it varies from tadpoles to tadpoles.
Limit conditions
We consider in this model the tadpole as a closed system: no exchanges are allowed between the external medium and the skin. This hypothesis is modelized by using the neumann boundary condition with a value of 0:
Calibration
Results
Conclusion
References
References:
- Atlas of xenopus development,G. Bernardini, Springer, 1999.
Hypothesis
There are the different hypothesis we were constrained to make in order to model the system:- The quantity of auxins is homogeneous in the blood
Model description
The PDE model is similar to the ODE one except that it takes into account the geometry. This allows us to model more complex phenomenon such as diffusion and transport.Equations
The diffusion equation is used to model the repartition of auxin's molecule when not subject to any flow (in the skin for example). The equation is states as follows:
- x = (x1, x2) is a 2 dimensional vector
- c is a function representing the auxin concentration
- D is the diffusion constant
- Δ is the Laplacian operator
Geometry
To keep the complexity of the numerical simulations low, we had to simplify the considered geometry. Hence, the slide of tadpole only contains the following elements:- Skin
- Blood vessels or muscles (depending on the model)
- Notochord
- Spinal cord
- Aorta
- Veins: caudal and dorsal
Limit conditions
We consider in this model the tadpole as a closed system: no exchanges are allowed between the external medium and the skin. This hypothesis is modelized by using the neumann boundary condition with a value of 0:
Calibration
Results
Conclusion
References
References:
- Atlas of xenopus development,G. Bernardini, Springer, 1999.
