I am attempting a project on modelling branching morphogenesis, but am getting very confused looking at the literature.
On the one hand, the structure formation itself is clearly best described by a branching and annihilating random walk (perhaps with some bias as to the angle of branching, and the rate of branching given how recently the last branches were formed. But this would be non-Markovian an add a lot more complexity!)
However I have seen models using instead 'mean field theory' and reaction-diffusion type models. As well as epidemic type models. In all honesty, I have no idea what the differences are between these and why we would use one over the other.
My current understanding- and I would very much appreciate if someone could correct and add to this- is that
- We would like to model this as a branching and annihilating random walk. Perhaps this is doable on the computer (perhaps it is computationally demanding? I'm not sure).
- I think we use reaction-diffusion type model considering species active/proliferating tip and inactive tip or duct as our 'mean field'. The thing is, I only see this as being able to give us paramters like the evoltion of active/inactive tip density within a region, and not allow us to create simulations if the branching itself. I see the branching/annihilating random walk model as allowing this, but certainly not using this course/grain model for densty. I am certainly mistunderstanding the papers I have been reading when they say that they used this model to create the branching simulations. Maybe instead the idea was to use both branching-annihilating random walk and compute the average densities this results in, and reaction-diffusion equation, and see if the results wcompare?
- I don't see the difference between using reaction/diffusion and epidemic-type models. They both involve empty/uninfected cells, infected/active proliferating cells, and inactive/immune cells...
Additionally, I am slightly confused as to the significance of adding perturbations about the mean field.
I appreciate that my query is regarding a rather specific topic, however I am hoping that someone familiar wiith statistics, and that has an understanding of why different models may be useful- particualry for simulations- and what we might get out of them- might be able to give some insights. At the moment, I feel quite lost with all of the terminology and models!
Some relevant papers I have been looking at:
A unified theory of branching morphogenesis, Hannezo et al
Field theory and branching and annihilating random walks, Cardy and Tauber
NOTE: I was unsure whether it would be more appropriate to put this on Math stack exchange or physics stack exchange under statistical physics. The modelling seems to have relevance to condensed matter physics.
EDIT: I forgot also percolation theory seems to be relavnt. Another model/theory that is getting mixed with all of the others!