Analysis for a model incorporating γ−secretase inhibitor to explore the mechanism of ‘Amyloid-β rise’

The abnormal accumulation of amyloid-β (Aβ) is a key factor in the advancement of Alzheimer’s disease (AD), prompting the development of numerous strategies to reduce Aβ levels in the brain. Aβ is produced through the hydrolysis of amyloid precursor protein (APP), beginning with its cleavage by β-secretase to yield a C-terminal fragment, C99, which is further processed by γ−secretase to release Aβ. The crucial role of γ−secretase in Aβ synthesis has led to the creation of various γ−secretase inhibitors aimed at regulating Aβ concentrations. Nonetheless, experimental observations have reported a paradoxical ‘Aβ rise’ phenomenon, where lower concentrations of γ−secretase inhibitor paradoxically lead to increased Aβ levels, thereby complicating the development of these inhibitors. To elucidate the underlying mechanisms of ‘Aβ rise’, we constructed and analyzed a mathematical model incorporating γ−secretase inhibitor. Our analysis reveals bistable behavior driven by saddle–node bifurcation within the model. Through comprehensive analysis, we identified the conditions precipitating ‘Aβ rise’ and discovered that varying concentrations of γ−secretase inhibitor result in three distinct Aβ trends: (i) a consistent decrease, (ii) an initial increase followed by a decrease (‘Aβ rise’), and (iii) a decrease, subsequent increase, and final decrease. These trends are consistent with empirical findings. Our study further reveals that the emergence of ‘Aβ rise’ is primarily attributed to both lower η cleaving rate for APP and degradation rate of C99. Therefore, inhibiting ‘Aβ rise’ can be achieved by enhancing the η cleaving rate for APP and degradation rate of C99. These findings give new insights into the underlying mechanism of ‘Aβ rise’ and potential therapeutic strategies for treating AD.