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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. Adding multiple slices one after the other allows us to approach a 3 dimensions model.

Assumptions

1. The quantity of auxins is homogeneous in blood
2. Movement of auxins in skin isn't dependent on where the skin is; it is the same in skin around the head and in the one surrounding the tail.

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 stated as follow: 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
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

Results

The program implemented on Netlogo gives us the folowing results:



The coloration of every patch of skin and capillaries is proportional to its concentration in auxin. We can see here the diffusion process from the skin to the vessels. The on the top right corner reflets the variation of the quantity of auxin in skin. We can see that this variation corresponds to the results of the ODE model for the same initial conditions (0 concentration in water that is surrounding the tadpole and non-zero quantity of auxin in skin.

Building back the global solution

Once the concentrations of auxins in the different tissues have been obtained in one slice, we must integrate these results. We use slices with constant length to simplify the problem, so every slice has a length of dz. To take better into account the areas of interest, we could also imagine having slices of different sizes depending on the activity of the area.
The integration scheme for slices of constant size is the following: Which in the implementation simply become a sum over all the slices:

Conclusion

This model has the qualitatively the same results as the ODE model, which confirms the types of equations we had used for that global model. It has much more parameters, which would be very difficult to determine. Nevertheless we are able to visualise the diffusion through each patch.

Downloading the model

You can find the model as a zip tarball here.

It has been developed under Netlogo 5.0.1. To use it, just launch NetLogo and open the "tad.nlogo" file. The use of NetLogo is very intuitive.

References:

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