| This research models the development of prion populations replicating by nucleated polymerization. Prions are infectious, modified forms of normal cellular proteins, and are associated with many varieties of transmissible spongiform encephalopathies. Examples include scrapie in sheep, bovine spongiform encephalopathy (“Mad Cow Disease”) in cows, and variant Creutzfeld-Jakob disease in humans. This association remains controversial, as prions contain no DNA or RNA, yet must somehow replicate if they are the infectious agent in question. A key mechanism currently under study is that of nucleated polymerization, through which prions multiply by converting uninfected proteins; hence no replication in the standard sense needs to occur.; We have chosen to model polymer length as a continuous structure variable, measurable in thousands of monomer units. The result is a nonlinear coupled system of ordinary and partial differential equations describing the dynamics of the normal protein monomers and the infectious polymers. We use this system of equations to determine several parameter changes which would result in extinction of the disease; to numerically simulate progress of the disease based on initial conditions and parameter input; and to determine existence and uniqueness of solutions in a special case. |
