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Team:Evry/auxin pde
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<h3>Issues arising from over-modelling</h3> | <h3>Issues arising from over-modelling</h3> | ||
| - | With precise geometry and a powerful PDE integration algorithm, it would 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. | + | 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 | + | 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 <strong>a good model is non necessarily one extremely close to reality</strong>. <br/> |
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. | 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. | ||
Revision as of 16:14, 22 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
Blood compartment
Kidney compartment
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 non 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. Nevertheless, this idealistic diffusion model is more or less what we considered as a basis for our simplifications.
