Theoretical Framework for Prion Aggregation Kinetics: From Nucleation to Spatial Propagation

Prion diseases are characterized by the templated conversion of native proteins into misfolded amyloid aggregates exhibiting sigmoidal growth kinetics with distinct lag and growth phases. We present a unified theoretical framework integrating four mathematical models: the Nucleated Polymerization Model (NPM) using moment equations, the Smoluchowski coagulation equation with size-resolved kinetics, the heterodimer reaction-diffusion system for spatial propagation, and the Fisher-Kolmogorov equation for traveling wave analysis. Theoretical analysis of the NPM reveals a lag phase of $39\pm5$ hours followed by exponential growth and monomer depletion to steady state. Size-resolved analysis predicts bimodal aggregate distributions with mean size increasing from 5 to 31 monomers while maintaining mass conservation. Spatial propagation analysis yields a propagation speed of $0.051$ mm/hr, consistent with the analytical prediction $v=2\sqrt{Dk}$ within 14\% error. Stochastic analysis demonstrates significant heterogeneity in lag times (CV=0.35). Parameter sensitivity analysis identifies the critical nucleus size $n_0$ as the dominant control of aggregation onset, with larger nuclei dramatically increasing lag times due to combinatorial penalties. The theoretical framework reproduces experimental observations including concentration-dependent scaling ($\tau \propto c^{-0.15}$) and seeding effects. This theoretical synthesis provides a mathematical foundation for quantitative analysis of protein aggregation mechanisms, validating classical theoretical predictions while generating new insights into kinetic control points for therapeutic intervention.