It is surely possible to assign percolation thresholds to any kind of structures , a very interesting case is reported in the attached paper, where we see how the percolation threshold of water molecules network palys a crucial role in life.

dear Dr.Giuliani i read the article and it was interesting to know the relation between optimum biological acting temperature and percolation threshold of covering water clusters. do you think cluster formation of water molecules ( in protein networks) may be a function of the most robust lattice structure for these type of molecules? as you know water molecules can form different lattice networks in different physico chemical conditions

Well, we demonstrated protein architecture is more similar to that of a sponge than to a dense globule,

http://pubs.acs.org/doi/abs/10.1021/ci2005127

this implies the 'water network' not only stays on the surface but enters protein cavities, if you are interested in pursuing this avenue of research I sugegst you to try and study the hydration dynamics of a suitable protein molecule (e.g.lysozime) and try to define percolation thresholds at different microenvironmental conditions.

Another intriguing possibility is to apply percolation thresholds to bones trabeculas (the architectural support of the bones) in presence of disease or aging with respect to normal bones, I'm convinced you could find many interesting facts, see for example:

http://onlinelibrary.wiley.com/doi/10.1002/(SICI)1096-9896(199601)178:1%3C100::AID-PATH429%3E3.0.CO;2-K/full

http://onlinelibrary.wiley.com/doi/10.1111/j.1469-7580.2012.01556.x/full

I deeply believe every structure and every network with its functionality and communication to other networks of a unified system(like different networks in a cell or an organ) can be logically defined by mathematical models.Fractals are one of these models that may be useful to get more clear perspective toward the formation and development of biological systems (tissues and organs).

Fractals have amazing properties. they have infinite self repetitive modules and due to this character fractals and structures like that are robust and show stability against perturbations (internal and external). protein networks (like metabolic networks) and biological structures like bone or cytoplasmic organization in cells (and many other examples) indicate fractal pattern and acquire fractal dimensions, but here comes another more important question and that is: how we can define and understand communications between different kind of networks in our system (e.g. a cellular unit)? is it possible to generalize and extent fractal pattern and modeling to network-network communication? does this kind of interactions follow the same pattern?

many researches have done about the single elements or single networks but i think to step forward its vital to look for models which cover several distinct networks to construct a live unit like cell.

It seems that a molecular network in a cell is so spatially constrained that the mere existence of the components of a network readily leads to the assembly of a robust network. If a percolation threshold exists, it will be very low. Also the connections in a molecular network is a small number, which should further reduce the percolation threshold, if it is still meaningful to define it in a cell or a cellular compartment.

Dear Dr.Yang we have calculated robustness of metabolic networks in different organisms for 14 in silico models based on a method using bond percolation theory. these models include the latest existing genetic and metabolic data about organisms.

for example results shows that percolation threshold of arabidopsis thaliana ( as i see you have published papers working on arabidopsis) is Pc=0.71 and it means that if you randomly delete 71 percent of reactions in metabolic network of arabidopsis (in average) the metabolic network of this organism will collapse with no more functionality. as you see mathematical analysis helps us to have more detailed view toward biological networks and function of living organisms.

Dr.Yang if you like you can check out more explanation and comparative data in my published paper

Thank you

So the above means that you can delete less than 71% of the reactions with no significant effect?

I think so. Or in other words, if you "put in" just >29 percent of the reactions randomly in an "empty" cell, the reactions will self-organize into a network or networks. Perhaps in a real cell, the threshold (percentage) is even less.

Yes Matteo. in Arabidopsis thaliana (of course these analysis was done for in silico model of this organism) you can delete less than 71% of metabolic reaction (in average) with no fatal effect on whole system. 71% is the threshold of system collapse. for another organism like Aspergilus nidulans percolation threshold is 0.93% (in average) and it means that you can delete 93 percent of reactions in metabolic network to reach the point that network has no functionality at all. to get clearer view you can see the paper in my profile if you like. it provides comparative data for different organism with different life style and different necessities of growth media(e.g. bacteria, yeast,mycobacteria,pathogens, ...) indeed causing different percolation thresholds.

I'm wondering Havva, did it make a difference which parts of the metabolic network was removed? Are there 'hubs', essential parts of pathways, that when removed let the network collapse at lower percentages?