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The Greer, Pujo-Menjouet andWebb model [Greer et al., J. Theoret. Biol., 242(2006), 598-606] for prion dynamics was found to be in good agreement withexperimental observations under no-flow conditions. The objective of this workis to generalize the problem to the framework of generalpolymerization-fragmentation under flow motion, motivated by the fact thatlaboratory work often involves prion dynamics under flow conditions in order toobserve faster processes. Moreover, understanding and modelling themicrostructure influence of macroscopically monitored non-Newtonian behaviouris crucial for sensor design, with the goal to provide practical informationabout ongoing molecular evolution. This paper's results can then be consideredas one step in the mathematical understanding of such models, namely the proofof positivity and existence of solutions in suitable functional spaces. To thatpurpose, we introduce a new model based on the rigid-rod polymer theory toaccount for the polymer dynamics under flow conditions. As expected, whenapplied to the prion problem, in the absence of motion it reduces to that inGreer et al. (2006). At the heart of any polymer kinetical theory there is aconfigurational probability diffusion partial differential equation (PDE) ofFokker-Planck-Smoluchowski type. The main mathematical result of this paper isthe proof of existence of positive solutions to the aforementioned PDE for aclass of flows of practical interest, taking into account the flow inducedsplitting/lengthening of polymers in general, and prions in particular.

Fragmentation and monomer lengthening of rod-like polymers, a relevant model for prion proliferation