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.

Assumptions

There are the different hypothesis we were constrained to make in order to model the system:
1. 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:
1. Skin
2. Blood vessels or muscles (depending on the model)
3. Notochord
4. Spinal cord
5. Aorta
6. 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:
1. blood <-> notochord
2. 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:
• P_{x <-> 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

Calibration

Results

Conclusion

References

References:

1. Atlas of xenopus development,G. Bernardini, Springer, 1999.
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