Modeling and analysis of prion dynamics in the presence of a chaperone

Prions are infectious agents and are polymers called PrP Sc —Prion protein scrapies, of a normal protein, a monomer called PrP c —Prion protein cellular. These PrP Sc s cause TSEs—transmissible spongiform encephalopathies such as bovine spongiform encephalopathy (BSE) in cattle, scrapies in sheep, Kuru and Creutzfeld–Jacob diseases in humans. Cellular molecular chaperones, which are ubiquitous, stress-induced proteins, and newly found chemical and pharmacological chaperones have been found to be effective in preventing misfolding of different disease-causing proteins, essentially reducing the severity of several neurodegenerative disorders and many other protein-misfolding diseases. In this work, we propose a model for the replication of prions by nucleated polymerization in the presence of a chaperone. According to this model, the biological processes of coagulation, splitting and the inhibitory effects of the chaperone can be described by a coupled system consisting of ordinary differential equations and a partial differential equation. The model is converted into a system of ordinary differential equations and the equilibrium points are computed and their stability is studied. We give a numerical simulation of the model and we find that a disease free state can be achieved in the presence of a chaperone. The duration of the disease free state is found to increase with the amount of chaperone and this amount of chaperone can be computed from the model.