Analytical characterisation of the Mi- and To-phases in HeMiTo dynamics: exponential growth and logistic saturation of toxic prion-like proteins

TL;DR

This paper provides a comprehensive analytical framework for understanding the different phases of prion-like protein dynamics in neurodegenerative diseases, including exponential growth and logistic saturation.

Contribution

It offers the first complete analytical characterization of the Mi- and To-phases in the HeMiTo model, linking growth and saturation mechanisms.

Findings

Derived exact solutions for the Mi-phase dynamics.

Established conditions for exponential toxic species growth.

Reduced the To-phase dynamics to a logistic growth model.

Abstract

Prion-like propagation of misfolded proteins is a key mechanism underlying the progression of neurodegenerative diseases such as Alzheimer's disease. In previous work, we introduced the HeMiTo framework, describing these prion-like dynamics for a class of heterodimer models in terms of three phases: the healthy (He), mixed (Mi), and toxic (To) phases. While the He-phase was characterised analytically, the Mi-phase was described numerically and the To-phase was inferred from linear stability arguments. In this work, we provide a complete analytical characterisation of the Mi- and To-phases for our class of heterodimer models. We derive exact inner solutions governing the Mi-phase and match them with outer solutions from the He-phase, explaining the concave-like behaviour of the healthy species and establishing explicit conditions for exponential growth of the toxic species with a mechanistically interpretable growth rate. Furthermore, we formalise a quasi steady-state reduction near the toxic steady state and show that the dynamics reduce to a logistic growth equation, linking exponential growth to saturation. Together, these results provide a unified and mechanistic description of prion-like dynamics across all phases of disease progression and establish a foundation for predictive modelling of biomarker trajectories.