Stability analysis of a steady state of a model describing Alzheimer’s disease and interactions with prion proteins

Abstract

Alzheimer’s disease (AD) is a neuro-degenerative disease affecting more than 46 million people worldwide in 2015. AD is in part caused by the accumulation of Aβ peptides inside the brain. These can aggregate to form insoluble oligomers or fibrils. Oligomers have the capacity to interact with neurons via membrane receptors such as prion proteins (PrP C ). This interaction leads PrP C to be misfolded in oligomeric prion proteins (PrP ol ), transmitting a death signal to neurons. In the present work, we aim to describe the dynamics of Aβ assemblies and the accumulation of toxic oligomeric species in the brain, by bringing together the fibrillation pathway of Aβ peptides in one hand, and in the other hand Aβ oligomerization process and their interaction with cellular prions, which has been reported to be involved in a cell-death signal transduction. The model is based on Becker–Döring equations for the polymerization process, with delayed differential equations accounting for structural rearrangement of the different reactants. We analyse the well-posedness of the model and show existence, uniqueness and non-negativity of solutions. Moreover, we demonstrate that this model admits a non-trivial steady state, which is found to be globally stable thanks to a Lyapunov function. We finally present numerical simulations and discuss the impact of model parameters on the whole dynamics, which could constitute the main targets for pharmaceutical industry.