# Team:Evry/Auxin diffusion

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Overview

Overview

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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. 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.

# 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:
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

## References

References:

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