Modeling and analysis of prion dynamics in the presence of a chaperone Rajiv Kumar, Padma Murali * Mathematics Group, Birla Institute of Technology and Science, Pilani, Rajasthan 333031, India Received 21 August 2007; received in revised form 14 January 2008; accepted 8 February 2008 Available online 20 February 2008 Abstract 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 mis- folding of different disease-causing proteins, essentially reducing the severity of several neurodegenerative disorders and many other pro- tein-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 numer- ical 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. Ó 2008 Elsevier Inc. All rights reserved. Keywords: Prions; Chaperones; Nucleated polymerization 1. Introduction Prions are pathogens responsible for a variety of animal and human neurodegenerative diseases, such as bovine spongiform encephalopathy (BSE), scrapie of sheep, Cre- utzfeldt–Jacob and Gerstmann–Straussler–Scheinker dis- eases of humans. Bewilderingly, all these diseases can be sporadic, genetic and infectious, thus making the identifica- tion of the disease mechanism a challenging task. For many years, the prion diseases were thought to be caused by slow-acting viruses. These diseases were often referred to as slow virus diseases, transmissible spongiform encepha- lopathies, or unconventional viral diseases. Considerable effort was expended searching for the scrapie virus; yet none was found either with respect to the discovery of a virus-like particle or a genome composed of RNA or DNA [1]. The unusual properties of the infectious agent became the focus of attention beginning in the 1960s, and in the early 1980s Stanley Prusiner, building upon earlier sugges- tions proposed the prion hypothesis [2]. This stated that the infectious agent in human and animal spongiform enceph- alopathies was composed exclusively of a single kind of protein molecule designated PrP Sc without any encoding nucleic acid. Based on foregoing findings, the term prion was introduced to distinguish the proteinaceous infectious particles that cause scrapie from both viroids and viruses. Perhaps, the best current working definition of a prion is a proteinaceous infectious particle that lacks nucleic acid [3]. This protein can appear in two forms that differ only in their conformation. One form is the mainly a-helical form, called cellular prion protein (or PrP c ). This is the native form of the protein which naturally appears in many tis- sues, however with a notable abundance in the brain, 0025-5564/$ - see front matter Ó 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.mbs.2008.02.002 * Corresponding author. Tel.: +91 1596 245073x260; fax: +91 1596 244183. E-mail addresses: padmab@bits-pilani.ac.in, padmamurali_b@yahoo. com (P. Murali). www.elsevier.com/locate/mbs Available online at www.sciencedirect.com Mathematical Biosciences 213 (2008) 50–55