Team:Evry/auxin pde

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Revision as of 17:21, 23 September 2012

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"

    Issues arising from over-modelling

    With precise geometry and a powerful PDE integration algorithm, it would be possible to consider making simulations. But this ideal model also introduces many parameters which would in practice be very difficult to set. Although most of them could be set with very extensive experimentation, such a rigorous PDE approach is definitely excessive. Moreover, although already very difficult to deal with, this model still requires very strong assumptions like tissues homogeneity and the possibility to apply diffusion equations straightforwardly. Therefore, its realistic feature is only relative. This model has been introduced to explicitly show the idea that a good model is not necessarily one extremely close to reality.
    Indeed, modelling is sometimes presented as a way to "summarize data" in a more concise way. But the question whether a model can really represent reality is a philosophical dead end. In our opinion a model should focus on explaining a precise mechanism and demonstrate if the modelled version of the mechanism is sufficient to explain data.

    Although easy to criticize, this idealistic diffusion model is more or less what we considered as a basis for our simplifications, assuming that this hypothetical model would capture all the phenomena of interest.

    Diffusion model goals

    As stated, a good model is designed to answer specific questions. In our case the goal is to understand what is the global dynamic of Auxin transportation. More specifically, in our view two aspects are to be considered :
    • Mean Auxin concentration in each of the 3 compartments when the system is "triggered" and in steady state
    • Spatial and time pattern that could arise when considering not only mean, but also local Auxin concentration

    We can summarize this as capturing Qualitatively and Quantitatively Auxin's transport. In order to investigate this question we came up with two models :
    • A PDE model based on a simplified but realistic geometry to access the qualitative possible behaviour
    • A global ODE model to model the quantitative dynamic of Auxin transportation between the 3 compartments

    From PDE to ODEs

    Our global ODE based model captures the global fluxes and concentration in and between compartments. The corresponding equations come from adapting the previously stated equations and "removing" space considerations. But in order to be able to link our sub-models to each other we need to formally state this transition.

    Simplified model equations

    Before describing the simplification from the general PDE model, we remind you the simplified model equations :
    ODE system

    In this system, the J terms represent the fluxes between the different compartments. We made them depend on the concentrations of both the in and out compartments as explained in hypothesis 2. Their mathematical formulation is the following:

    mathematical expression of fluxes

    Where:
    • S in m2, represents the area of the exchange surface between the two compartments.
    • P in m2, represents the permeability of the membrane between the specified compartments.
    • C in [quantity] / m3, represents the concentration of auxin in the specified compartment

    Constructing the mean concentration of Auxin

    Mean concentration definition

    Defining the creation term Dborn from plasmid repartition model and Auxin creation model

    Mean concentration definition

    Dealing with diffusion terms and constructing Ddie

    In the way we wrote the diffusion equations, we have included transfer terms between compartments. Formally, these terms are equal to the global flux of Auxin crossing the surface separating two compartments. Therefore, these terms are more associated to boundary conditions. Nevertheless, as in heat transfer, this creates many problems and it is more convenient to write a specific term for exchange and use more simple boundary conditions. This being equivalent by superposition.
    Mean concentration definition

    Defining the transfer terms

    Mean concentration definition