Long-time behaviour of a model for p62-ubiquitin aggregation in cellular autophagy

TL;DR

This paper analyzes a mathematical model describing the long-term behavior of protein aggregate formation relevant to cellular autophagy, combining analytical and numerical methods to understand stability and growth dynamics.

Contribution

It provides a rigorous analysis of the bifurcation behavior and stability of solutions in a model for p62-ubiquitin aggregation, including proofs of certain conjectures.

Findings

Stability of the zero state is characterized using blow-up techniques.

Existence of polynomially growing solutions is established via Poincaré compactification.

Bifurcation behavior of the model is conjectured and partially proven.

Abstract

The qualitative behavior of a recently formulated ODE model for the dynamics of heterogenous aggregates is analyzed. Aggregates contain two types of particles, oligomers and cross-linkers. The motivation is a preparatory step of cellular autophagy, the aggregation of oligomers of the protein p62 in the presence of ubiquitin cross-linkers. A combination of explicit computations, formal asymptotics, and numerical simulations has led to conjectures on the bifurcation behavior, certain aspects of which are proven rigorously in this work. In particular, the stability of the zero state, where the model has a smoothness deficit is analyzed by a combination of regularizing transformations and blow-up techniques. On the other hand, in a different parameter regime, the existence of polynomially growing solutions is shown by Poincar\'e compactification, combined with a singular perturbation analysis .