Team:Evry/auxin pde


Revision as of 15:52, 22 September 2012 by Arty (Talk | contribs)

From realistic to simplified auxin diffusion model

The main goal of this section is to clearly present our though process in modelling the diffusion and transportation of Auxin between Xenopus' tissues.

An ideal model

Ideally, modelling Auxin's diffusion in tissues and its transportation through blood would require a 4D (3D space + time) PDE representation.
Assuming a concentration can be defined, and considering steady state, a powerful representation would use the general Reaction-Diffusion equation from which the famous Fisher-KPP equation is derived. Using this formalism, we propose to consider the 3 compartments of interest : emitter - blood - receiver and to write one PDE for each.
We model Auxin flux according to Fick's law which is an adaptation of Fourier's law for heat transport.

The according equations, using the Nabla operator and using skin as emitter and kidney as receiver are therefore :

Skin compartment

Skin compartment's equation

Blood compartment

Blood compartment's equation

Kidney compartment

Kidney compartment's equation

Limit conditions and geometry

In order to be able to integrate these equations it is mandatory to specify boundary conditions and the geometry of the 3 comparments. In this ideal case, we would use the true geometry of Xenopus, this can be retrieved from histological cuts, various 3D imaging techniques or sometimes by using various plane images. Permeability experiments of the various interfaces are also required to classify them as permeable or not. Boundary conditions being therefore a composition of Dirichlet and Neumann conditions according to the permeability.
A last requirement is the condition at t=0. Our goal being to have Auxin synthesis induced by an external stimuli (for instance, a pollutant detection) the concentration would be equal to the residual Auxin concentration measured when the biosensor is "off"