A mathematical analysis of the dynamics of prion proliferation.

How do the normal prion protein (PrP(C)) and infectious prion protein (PrP(Sc)) populations interact in an infected host? To answer this question, we analyse the behavior of the two populations by studying a system of differential equations. The system is constructed under the assumption that PrP(Sc) proliferates using the mechanism of nucleated polymerization. We prove that with parameter input consistent with experimentally determined values, we obtain the persistence of PrP(Sc). We also prove local stability results for the disease steady state, and a global stability result for the disease free steady state. Finally, we give numerical simulations, which are confirmed by experimental data.

[1]  B. Chesebro,et al.  Prion protein biosynthesis in scrapie-infected and uninfected neuroblastoma cells , 1989, Journal of virology.

[2]  G. D. Hunter,et al.  Nature of the Scrapie Agent , 1967, Nature.

[3]  S. Prusiner,et al.  Scrapie and cellular prion proteins differ in their kinetics of synthesis and topology in cultured cells , 1990, The Journal of cell biology.

[4]  Statistical mechanics of prion diseases. , 2001, Physical review letters.

[5]  S. Prusiner,et al.  Scrapie agent contains a hydrophobic protein. , 1981, Proceedings of the National Academy of Sciences of the United States of America.

[6]  A. Aguzzi,et al.  Mammalian Prion Biology One Century of Evolving Concepts , 2004, Cell.

[7]  Adam Douglass,et al.  Mechanism of Prion Propagation: Amyloid Growth Occurs by Monomer Addition , 2004, PLoS biology.

[8]  Stanley B. Prusiner,et al.  Nobel Lecture: Prions , 1998 .

[9]  Martin A. Nowak,et al.  Prion infection dynamics , 1998 .

[10]  M A Nowak,et al.  Quantifying the kinetic parameters of prion replication. , 1999, Biophysical chemistry.

[11]  Arthur L Horwich,et al.  Deadly Conformations—Protein Misfolding in Prion Disease , 1997, Cell.

[12]  J. Hale,et al.  Ordinary Differential Equations , 2019, Fundamentals of Numerical Mathematics for Physicists and Engineers.

[13]  L. Hood,et al.  Purification and structural studies of a major scrapie prion protein , 1984, Cell.

[14]  B. Ghetti,et al.  Bax deletion prevents neuronal loss but not neurological symptoms in a transgenic model of inherited prion disease. , 2005, Proceedings of the National Academy of Sciences of the United States of America.

[15]  D. Mobley,et al.  Simulations of oligomeric intermediates in prion diseases. , 2003, Biophysical journal.

[16]  P. Lansbury,et al.  The chemistry of scrapie infection: implications of the 'ice 9' metaphor. , 1995, Chemistry & biology.

[17]  A. Aguzzi,et al.  Efficient inhibition of prion replication by PrP-Fc(2) suggests that the prion is a PrP(Sc) oligomer. , 2005, Journal of molecular biology.

[18]  Christoph Walker,et al.  On the solvability of a mathematical model for prion proliferation , 2006 .

[19]  X. Roucou,et al.  Neuroprotective functions of prion protein , 2004, Journal of neuroscience research.

[20]  David A Agard,et al.  Structural studies of the scrapie prion protein by electron crystallography , 2002, Proceedings of the National Academy of Sciences of the United States of America.

[21]  J. C. Cressoni,et al.  Stochastic modeling approach to the incubation time of prionic diseases. , 2003, Physical review letters.

[22]  S. Prusiner.  Novel proteinaceous infectious particles cause scrapie. , 1982, Science.

[23]  Robert H. Martin,et al.  Nonlinear operators and differential equations in Banach spaces , 1976 .

[24]  Ruedi Aebersold,et al.  A cellular gene encodes scrapie PrP 27-30 protein , 1985, Cell.

[25]  Thorsten Pöschel,et al.  Kinetics of prion growth. , 2003, Biophysical journal.

[26]  R J Fletterick,et al.  Structural clues to prion replication. , 1994, Science.

[27]  A. LeBlanc,et al.  Prion Protein Protects Human Neurons against Bax-mediated Apoptosis* , 2001, The Journal of Biological Chemistry.

[28]  F. Cohen,et al.  Evidence for assembly of prions with left-handed beta-helices into trimers. , 2004, Proceedings of the National Academy of Sciences of the United States of America.

[29]  R J Fletterick,et al.  Conversion of alpha-helices into beta-sheets features in the formation of the scrapie prion proteins. , 1993, Proceedings of the National Academy of Sciences of the United States of America.

[30]  J. Griffith,et al.  Nature of the Scrapie Agent: Self-replication and Scrapie , 1967, Nature.

[31]  E. J. Field.  The scrapie agent. , 1967, Lancet.

[32]  S. Lindquist,et al.  Bidirectional amyloid fiber growth for a yeast prion determinant , 2001, Current Biology.

[33]  Laurent Pujo-Menjouet,et al.  Analysis of a model for the dynamics of prions , 2005 .

[34]  Glenn F. Webb,et al.  Analysis of a model for the dynamics of prions II , 2005 .

[35]  D. Harris,et al.  Mammalian Prion Protein Suppresses Bax-induced Cell Death in Yeast* , 2005, Journal of Biological Chemistry.

[36]  M. Eigen,et al.  Prionics or the kinetic basis of prion diseases. , 1996, Biophysical chemistry.

[37]  A. Aguzzi,et al.  Normal host prion protein necessary for scrapie-induced neurotoxicity , 1996, Nature.

[38]  Hyoung-Gon Lee,et al.  The cellular prion protein (PrPC) prevents apoptotic neuronal cell death and mitochondrial dysfunction induced by serum deprivation. , 2004, Brain research. Molecular brain research.

[39]  David R. Brown,et al.  Lack of prion protein expression results in a neuronal phenotype sensitive to stress , 2002, Journal of neuroscience research.

[40]  F. Cohen,et al.  Proposed three-dimensional structure for the cellular prion protein. , 1994, Proceedings of the National Academy of Sciences of the United States of America.

[41]  M. Laurent.  Autocatalytic processes in cooperative mechanisms of prion diseases , 1997, FEBS letters.

[42]  P. Lansbury,et al.  Models of amyloid seeding in Alzheimer's disease and scrapie: mechanistic truths and physiological consequences of the time-dependent solubility of amyloid proteins. , 1997, Annual review of biochemistry.

[43]  X. Roucou,et al.  Cytosolic Prion Protein Is Not Toxic and Protects against Bax-mediated Cell Death in Human Primary Neurons* , 2003, Journal of Biological Chemistry.

[44]  P. Lansbury,et al.  Seeding “one-dimensional crystallization” of amyloid: A pathogenic mechanism in Alzheimer's disease and scrapie? , 1993, Cell.

[45]  R. Carp,et al.  Scrapie-infected spleens: analysis of infectivity, scrapie-associated fibrils, and protease-resistant proteins. , 1991, The Journal of infectious diseases.

[46]  S. Prusiner,et al.  Molecular biology of prion diseases , 1991, Science.

[47]  R. Kulkarni,et al.  Theoretical modeling of prion disease incubation. , 2001, Biophysical journal.

[48]  E. Masliah,et al.  Anchorless Prion Protein Results in Infectious Amyloid Disease Without Clinical Scrapie , 2005, Science.