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Team:Evry/Auxin diffusion
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Revision as of 03:28, 27 September 2012
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 in concentration in each compartment.We are then able to estimate delays between arrival in one end of a compartement and exit from the other end. 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]. Adding multiple slices one after the other allows us to approach a 3 dimensions model.
Assumptions
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
null Neumann condition
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:
This condition is also found at these other boundaries:
- blood <-> notochord
- blood <-> spinal cord
Exchange between different tissues
For the exchanges between other compartments, we use the same Neumann condition but modify the value of the right hand term.
This values is now computed depending on the neighbors that do not belong to the same tissue.
where:
- Px <-> y is the permeability between compartments x and y
- neighbors is a function computing the 4-connexity neighborhood of a point
Parameters
Description
Symbol
Type
Values
Permeabilities
P
calculated
here
Diffusion constants
D
calculated
here
Volumes
V
calculated
here
Degradation rate
Ddie
estimated
unknown
Creation rate
Dborn
computed
plasmid repartition model
Results
Conclusion
References
References:
- Atlas of xenopus development,G. Bernardini, Springer, 1999.
Assumptions
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
null Neumann condition
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:
- blood <-> notochord
- blood <-> spinal cord
Exchange between different tissues
For the exchanges between other compartments, we use the same Neumann condition but modify the value of the right hand term. This values is now computed depending on the neighbors that do not belong to the same tissue.
- Px <-> y is the permeability between compartments x and y
- neighbors is a function computing the 4-connexity neighborhood of a point
Parameters
| Description | Symbol | Type | Values |
|---|---|---|---|
| Permeabilities | P | calculated | here |
| Diffusion constants | D | calculated | here |
| Volumes | V | calculated | here |
| Degradation rate | Ddie | estimated | unknown |
| Creation rate | Dborn | computed | plasmid repartition model |
Results
Conclusion
References
References:
- Atlas of xenopus development,G. Bernardini, Springer, 1999.
