Rigorous analytical characterization of Mi- and To-phase dynamics in the HeMiTo heterodimer models

Establish a rigorous analytical characterisation of the mixed (Mi) phase and a corresponding analysis of the toxic (To) phase dynamics for the dimensionless class of heterodimer models defined by du/dτ = c1 − c2 u − ε u v f(v) and dv/dτ = ε v (u f(v) − 1) with initial conditions u(0) = u0 and v(0) = v0, where ε ∈ (0,1) and f is an analytic conversion function, in order to provide an analytical description beyond the previously derived He-phase outer solutions.

Background

The HeMiTo framework partitions prion-like protein dynamics into three phases: healthy (He), mixed (Mi), and toxic (To). Prior work derived exact outer solutions for the He-phase but only provided numerical approximations for Mi-phase inner solutions and heuristic or linear-stability-based insights for the To-phase.

The class of heterodimer models considered is given by coupled ODEs for healthy (u) and toxic (v) species, with a conversion function f(v) and a small parameter ε enabling perturbation analysis. The open problem highlighted the lack of a rigorous analytical treatment of the Mi-phase and a corresponding analysis of the To-phase dynamics within this general framework.