Team:Evry/Auxin diffusion

From 2012.igem.org

(Difference between revisions)
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<h3>Limit conditions</h3>
<h3>Limit conditions</h3>
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<h4>null Neumann condition</h4>
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:
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:
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<img src="https://static.igem.org/mediawiki/2012/1/13/Neumann-ext.png" alt="neumann ext" />
<img src="https://static.igem.org/mediawiki/2012/1/13/Neumann-ext.png" alt="neumann ext" />
</center>
</center>
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This condition is also found at these other boundaries:
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<ol>
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  <li>blood <-> notochord</li>
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  <li>blood <-> spinal cord</li>
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</ol>
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<h4>Exchange between different tissues</h4>
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For the exchanges between other compartments, we use the same Neumann condition but modify the value of the right hand term:
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<center>
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<img src="https://static.igem.org/mediawiki/2012/e/ef/Neumann-int.png" alt="neumann int" />
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</center>
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This values is now computed depending on the neighbors (with a 4-connexity) that do not belong to the same tissue.
<h2>Calibration</h2>
<h2>Calibration</h2>

Revision as of 23:37, 26 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 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:
    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:
    neumann int
    This values is now computed depending on the neighbors (with a 4-connexity) that do not belong to the same tissue.

    Calibration

    Results

    Conclusion

    References

    References:

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

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