Supplemental Materials from A criterion characterizing accumulated neurotoxicity of Aβ oligomers in Alzheimer's disease

The paper develops a criterion to quantify the accumulated neurotoxicity of Aβ oligomers in Alzheimer's disease (AD). Accumulated neurotoxicity is determined by integrating the concentration of Aβ oligomers within the control volume over time. In the scenario of a low rate of free Aβ oligomer deposition into senile plaques and dysfunctional degradation machinery, resulting in an infinitely long half-life of Aβ monomers and aggregates, the obtained analytical solution reveals a quadratic relationship between accumulated neurotoxicity and time. This suggests that initially, neurotoxicity increases slowly but accelerates as time progresses. This could help to understand the prolonged delay in the onset of AD symptoms. Furthermore, as the model indicates that accumulated neurotoxicity increases with the duration of the aggregation process, it implies that if the protein degradation system is compromised, the onset of AD becomes unavoidable. Eventually, neuronal death is only a question of time. The only way to prevent this outcome is to ensure that the degradation machinery for Aβ peptides and their aggregates remains functional. A threshold value of accumulated neurotoxicity is suggested. The developed theory suggests that if this value is exceeded, nearby neurons will die. The progression of accumulated neurotoxicity over time is analysed. An S-shaped growth pattern as the half-deposition time of Aβ aggregates into senile plaques increases is revealed. Additionally, the sensitivity of accumulated neurotoxicity to different parameter values is examined.