Date of Award
Spring 2026
Document Type
Open Access Thesis
Department
Mathematics
First Advisor
Mitchel Colebank
Abstract
Alzheimerโs disease is characterized by the presence of beta-amyloid (A๐ฝ) plaques in the brain, caused by an accumulation of misfolded A๐ฝ proteins. Research suggests that hallmark proteins involved in neurodegenerative disorders such as Alzheimerโs disease, Parkinsonโs disease, and Amyotrophic Lateral Sclerosis (ALS) exhibit prion- like behavior. This means that the misfolded, pathogenic proteins influence non- pathogenic proteins to misfold. There is also a small rate of spontaneous conversion which causes A๐ฝ to misfold without the presence of pathogenic proteins. Additionally, events that limit blood flow to the brain, such as strokes or direct trauma, are known to increase A๐ฝ in the bloodstream.
Prusinerโs Heterodimer model is a system of differential equations that depicts the behavior of prion diseases and has been adapted to simulate the concentration of A๐ฝ in the bloodstream. The model relates concentrations of both non-pathogenic and misfolded A๐ฝ to their associated production, conversion, and clearance rates. Equilibria of the system have been identified in both disease-free and diseased cases. Cerebral blood flow impacts the production and clearance of proteins and molecules in the blood stream, which is explicitly represented in a further modification of the Heterodimer model. Pathogenic A๐ฝ causes an increase in the presence of reactive oxygen species (๐ ๐๐). ๐ ๐๐ encourages the production of endothelin-1, which causes blood vessels to constrict. The modified Heterodimer model assumes that changes in cerebral blood flow, ๐ ๐๐ and endothelin-1 are instantaneous. However, it is important to document the interactions between ๐ ๐๐, endothelin-1, and vessel radius to better understand the prion-like behavior and clearance of A๐ฝ.
In this thesis, we will analyze the existing Heterodimer model by examining its stability through linearization of the nonlinear system at its equilibrium points. Fol- lowing our analysis, updates to the existing model will be introduced to include separate equations for flow-dependent dynamics. These equations will model the presence of ๐ ๐๐ and endothelin-1, as well as blood vessel radius. We will then ana- lyze the equilibria of a simplified version of the new model using the same methods applied to the Heterodimer model. Numerical simulations will be used to confirm the model analysis and depict the modelโs behavior. In more complex cases where flow is radius-dependent, we will use simulations to anticipate the true behavior of the system.
Rights
ยฉ 2026, Caroline Lloyd
Recommended Citation
Lloyd, C.(2026). A Dynamical Systems Analysis of Amyloid-Beta Production. (Master's thesis). Retrieved from https://scholarcommons.sc.edu/etd/8792