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

Johannes G. Borgqvist

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.

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

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.

Paper Structure

This paper contains 8 sections, 4 theorems, 54 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Mathematical preliminaries
  3. Results
  4. Mi-phase dynamics are governed by inner solutions
  5. Asymptotic matching in the Mi-phase yields the healthy steady state maximum
  6. Mi-phase dynamics yield exponential growth of the toxic species
  7. To-phase dynamics through quasi steady-state reduction yield logistic growth
  8. Discussion

Key Result

Theorem 1

Consider the perturbation ansätze for the healthy species in Eq. eq:u_approx and the toxic species in Eq. eq:v_approx for the solutions of the class of heterodimer models in Eqs. eq:ODE_u to eq:IC_u_and_v. Furthermore, assume that the initial conditions in Eq. eq:IC_u_and_v satisfy $u_{0},v_{0}=\mat which satisfy the initial conditions in Eqs. eq:u_Mi_IC and eq:v_Mi_IC, respectively. $\blacktriang

Figures (3)

  • Figure 1: Infectious conversion of prion-like proteins. When the toxic and infectious form $v(\tau)$ of the prion-like protein interacts with the healthy form $u(\tau)$, the latter gets converted into a toxic particle. This figure is adapted from Fig. 20 in thompson2020protein.
  • Figure 2: Asymptotic matching of the inner and outer solutions. The inner and outer solutions for the healthy species denoted by $u_{\mathrm{He}}$ and $u_{\mathrm{Mi}}$ are illustrated in blue curves and bounded by the maximum value $c_{1}/c_{2}$, while the inner and outer solutions for the toxic species denoted by $v_{\mathrm{He}}$ and $v_{\mathrm{Mi}}$ are illustrated in the green curves. The parameters and initial conditions for the illustrated curves are $c_{1}=1.75$, $c_{2}=0.70$, $u_{0}=1.00$, $v_{0}=0.05$ and $f(v_{0})=1.80$.
  • Figure 3: Exponential approximation of the toxic species during the Mi-phase. The inner solution $v_{\mathrm{Mi}}(\tau)$ in Eq. \ref{['eq:v_Mi_thm_2']}, the solid line, is compared to the exponential approximation $v_{0}\exp\left((u_{0}f(v_{0})-1)\tau\right)$ in Eq. \ref{['eq:exp_approx']}, the dashed line. The exponential approximation is accurate for early time points when $\tau\in[0,0.5]$. The parameters and initial conditions for the illustrated curves are $c_{1}=1.75$, $c_{2}=0.70$, $u_{0}=1.00$, $v_{0}=0.05$ and $f(v_{0})=1.80$.

Theorems & Definitions (8)

  • Theorem 1: Inner solutions $u_{\mathrm{Mi}}(\tau)$ and $v_{\mathrm{Mi}}(\tau)$
  • proof
  • Theorem 2: Exponential approximation of the toxic species
  • proof
  • Lemma 1: Quasi steady-state near the toxic steady state
  • proof
  • Theorem 3: Local logistic growth of the toxic species near the TSS
  • proof