Section: Application Domains

Protein polymerisation

Self-assembly of proteins into amyloid aggregates is an important biological phenomenon associated with various human neurodegenerative diseases such as Alzheimer's, Parkinson's, Prion (in particular variant Creutzfeldt-Jakob disease, epidemically linked to bovine spongiform encephalopathy, or so-called “mad cow”, disease), Huntington's disease. Amyloid fibrils also have potential applications in nano-engineering of biomaterials.

However, the mechanisms of polymerisation are far from being quantitatively understood by biologists. They can be modelled with the help of coagulation-fragmentation equations, a field of expertise of MAMBA [67] , [65] , or with stochastic models. One difficulty of this application is that the reactions imply both very small and very large scales for the sizes of polymers, experimental data giving only access to the time evolution of size-averaged quantities. Moreover, there exists an intrinsic variability among experiments, which has to be distinguished from a lack of reproducibility [44] .

The European starting grant SKIPPER${}^{AD},$ which follows the ANR project TOPPAZ, came up very naturally from a cooperation established with Human Rezaei, a biologist expert in amyloid diseases at INRA Jouy-en-Josas. It allowed us to further develop new collaborations, in particular with W.F. Xue's team in Canterbury, who is one of the rare biophysicists in this area who is able to measure not only size-averaged quantities, as for instance the time-evolution of the total polymerised mass, but also size distribution of polymers (at least over a certain threshold). Such measurements allow us to use much more powerful inverse problems methods, linked to the ones previously developed for bacteria [13] .

Moreover, this field of applications to human neurogenerative diseases brings us new questions, which is a stimulation for our mathematical research and at the same time allows us to provide biologists with a new and efficient tool.