Team:Evry/Auxin diffusion

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<h2>Overview</h2>
<h2>Overview</h2>
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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.
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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.
<h2>Assumptions</h2>
<h2>Assumptions</h2>

Revision as of 02:13, 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:
    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:
    diffusion equation
    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
    tadpole geometry
    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:
    neumann ext
    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.
    neumann int
    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

    Calibration

    Results

    Conclusion

    References

    References:

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

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