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

1. Problem Statement

Neurodegenerative diseases, such as Alzheimer's, are often driven by the prion-like propagation of misfolded proteins (e.g., tau, amyloid-$\beta$). Previous work by the author introduced the HeMiTo framework, a class of heterodimer models describing these dynamics through three phases:

• He-phase (Healthy): Healthy protein concentration increases; toxic protein remains constant.
• Mi-phase (Mixed): Onset of disease; toxic protein increases at the expense of healthy protein.
• To-phase (Toxic): System evolves toward a toxic steady state.

While the He-phase was previously characterized analytically, the Mi-phase was only described via numerical approximations, and the To-phase was inferred from linear stability arguments. The lack of rigorous analytical characterizations for the Mi- and To-phases limited the ability to derive explicit conditions for disease onset and saturation, hindering predictive modeling of biomarker trajectories.

2. Methodology

The paper employs singular perturbation analysis on a dimensionless class of heterodimer models. The core system of Ordinary Differential Equations (ODEs) is:
$$\frac{du}{d\tau} = c_1 - c_2u - \varepsilon uvf(v)$$
$$\frac{dv}{d\tau} = \varepsilon v (uf(v) - 1)$$
Where:

• $u(\tau)$: Healthy species concentration.
• $v(\tau)$: Toxic species concentration.
• $f(v)$: A general conversion function defining the specific prion-like mechanism.
• $\varepsilon \in (0, 1)$: A small perturbation parameter.

Analytical Approach:

1. Perturbation Expansion: The solutions are expanded as series in $\varepsilon$: $u = u_{He} + \varepsilon u_{Mi} + O(\varepsilon^2)$ and $v = v_{He} + \varepsilon v_{Mi} + O(\varepsilon^2)$.
2. Outer Solutions ($O(1)$): Describe the He-phase (already derived in prior work).
3. Inner Solutions ($O(\varepsilon)$): Derived in this work to describe the Mi-phase dynamics.
4. Asymptotic Matching: The inner solutions are matched with the outer solutions to ensure continuity and determine the maximum of the healthy species.
5. Quasi-Steady-State (QSS) Reduction: Applied near the Toxic Steady State (TSS) to analyze the To-phase, reducing the coupled system to a single scalar equation.

3. Key Contributions and Results

A. Analytical Characterization of the Mi-Phase (Mixed Phase)

• Exact Inner Solutions: The author derived exact analytical expressions for the inner solutions $u_{Mi}(\tau)$ and $v_{Mi}(\tau)$ (Theorem 1). These solutions govern the transition from the healthy to the toxic state.
• Concavity of Healthy Species: By matching the inner and outer solutions, the paper proves that the healthy species trajectory is concave-like. It increases during the He-phase, reaches a maximum exactly at the Healthy Steady State (HSS) value ($c_1/c_2$), and then decreases during the Mi-phase.
• Exponential Growth of Toxic Species: The paper establishes that the toxic species exhibits exponential growth during the early Mi-phase under a specific condition:
  • Condition: The initial healthy concentration must be sufficiently high: $u_0 > 1/f(v_0)$.
  • Growth Rate: The growth rate is $\lambda = f(v_0)u_0 - 1$. This provides a mechanistic interpretation where the rate depends on the initial healthy load and the specific conversion mechanism $f(v)$.

B. Analytical Characterization of the To-Phase (Toxic Phase)

• Logistic Saturation: By applying a quasi-steady-state approximation (assuming $du/d\tau \approx 0$) near the Toxic Steady State (TSS), the coupled system reduces to a single scalar ODE for the toxic species.
• Logistic Growth Law: Theorem 3 demonstrates that the dynamics in the To-phase reduce to a logistic growth equation:
  $$\frac{dv}{d\tau} = \gamma v \left(1 - \frac{v}{\kappa}\right)$$
  Where $\gamma$ is the growth rate and $\kappa$ is the carrying capacity.
• Mechanistic Link: The carrying capacity $\kappa$ corresponds to the toxic species concentration at the TSS ($v^\star_2$). This formally links the exponential growth observed in the Mi-phase to the saturation observed in the To-phase, showing they are complementary regimes of the same underlying dynamics.

4. Significance and Implications

• Unified Framework: The work provides a complete, unified analytical description of the entire HeMiTo trajectory, moving from qualitative numerical observations to rigorous mathematical proofs.
• Biomarker Prediction: The results offer a mechanistic foundation for the widely used sigmoidal trajectory model of neurodegenerative disease biomarkers (proposed by Jack et al.). The paper explains why biomarkers follow a sigmoidal curve: an initial exponential take-off (Mi-phase) driven by prion-like conversion, transitioning naturally into logistic saturation (To-phase) as the system approaches the toxic steady state.
• Predictive Modeling: Unlike purely statistical models, these mechanistic models provide interpretable parameters (e.g., conversion rates, degradation rates) grounded in biological laws (mass action). This facilitates the calibration of models to patient-specific longitudinal data (e.g., plasma p-tau217) for forecasting individual disease progression.
• Generalizability: The framework applies to a broad class of heterodimer models with general conversion functions $f(v)$, making it applicable to various prion-like proteins (tau, amyloid-$\beta$, yeast prions).

Conclusion

This paper closes the analytical gap in the HeMiTo dynamics framework. It proves that the transition from health to toxicity is governed by a switch from exponential growth (driven by initial conditions and conversion efficiency) to logistic saturation (driven by system capacity). These findings transform the HeMiTo framework from a qualitative description into a quantitative, predictive tool for understanding and modeling neurodegenerative disease progression.