Course: Physics of Molecular Diseases
Topic: Protein Folding Diseases II — Protein Quality Control (PQC)
Subunit: Deterministic model of protease dynamics in Parkinson's disease
Question: "In the presented version, protease production was assumed constant. How do the results change if you include the fact that protease is induced by (i) proto-fibrils s, (ii) oligomers ααα, (iii) fibrils F?"
In the base Sneppen model (2009), proteasome production σ is a constant. But cells do not passively accept protein misfolding — they actively upregulate their protein quality control (PQC) machinery in response to aggregation stress. This is a cellular stress response: a negative feedback loop that competes directly with the positive feedback (fibrils sequestering proteasomes) that drives disease.
The three inducers differ in where they sit along the αSN aggregation pathway:
α-synuclein monomers (α)
↓ nucleation (slow)
protofilaments (s) ← (i) EARLIEST signal — upstream, fast
↓ maturation
mature fibrils (F) ← (iii) DOWNSTREAM — slow accumulation
↓ off-pathway seeds (ααα)
oligomers (ααα) ← (ii) TOXIC ALARM — responds to low P
↓
cell death
The central question is: does the induced proteasome create enough negative feedback to counteract the positive feedback loop (F sequesters P)? And which inducer is most effective at raising the disease threshold m_T?
Protofilaments are at quasi-steady-state:
Physics: this creates a direct, instantaneous negative feedback on P depletion. When P drops → s rises → σ_eff rises → P is restored. The response is on the same timescale as the P oscillations themselves.
From the paper's Appendix (eqs. A.9–A.10), the oligomer concentration in quasi-steady-state is proportional to
Physics: when P drops → oligomers rise sharply → σ_eff rises. The parameter ε prevents divergence as P → 0 (biological saturation of induction). This is also a fast feedback, but slightly attenuated by ε at very low P.
Physics: when F accumulates → σ_eff rises → P rises → F is cleared. But F accumulates slowly — it is a time-integrated variable reflecting the history of fibril production. By the time F is large enough to significantly induce σ, the proteasome may already have been largely sequestered into complexes C.
| Parameter | Value | Meaning |
|---|---|---|
ALPHA_S |
2.0 | Protofilament-induction strength |
K_S |
5.0 | Half-saturation for s-induction |
ALPHA_O |
1.0 | Oligomer-induction strength |
EPS |
0.1 | Saturation at low P (prevents σ → ∞) |
ALPHA_F |
2.0 | Fibril-induction strength |
K_F |
20.0 | Half-saturation for F-induction |
GAMMA = NU = SIGMA0 |
1.0 | Base Sneppen reference values |
All three induction forms raise the disease threshold m_T compared to the base model:
Base (constant σ): m_T ≈ 8 (analytical: (1+ν)(1+σ)² = 8)
(i) Induced by s: m_T ≈ 25 ← STRONGEST rescue
(ii) Induced by ααα: m_T ≈ 19 ← moderate rescue
(iii)Induced by F: m_T ≈ 16 ← weakest rescue
| Model | P_mean | P_min | F_mean | ⟨1/P⟩ | Oscillates? |
|---|---|---|---|---|---|
| Base (σ = const) | 1.16 | 0.13 | 78.6 | 4.07 | YES ★ |
| (i) Induced by s | 2.22 | 2.08 | 3.5 | 0.45 | no (SS) |
| (ii) Induced by ααα | 1.76 | 0.70 | 10.1 | 0.84 | no (SS) |
| (iii) Induced by F | 1.70 | 0.46 | 17.1 | 1.24 | YES ★ |
At m = 25, only induction by s and ααα successfully rescue the system to a steady state. Induction by F is insufficient at this m value — the system still oscillates, though with reduced severity.
Why is (i) s-induction the strongest?
Because s = m/(1+P) responds instantaneously and directly to P depletion — it is the same variable that drives fibril production. The feedback loop is:
P↓ → s = m/(1+P) rises immediately → σ_eff↑ → P↑
This is a fast, tight negative feedback on the same timescale as the P oscillations. There is no delay between P dropping and s rising.
Why is (iii) F-induction the weakest?
Because F is a slow, integrating variable. F accumulates gradually as the integral of past fibril production — it does not spike when P drops. By the time F is large enough to strongly induce σ, most of the proteasome has already been sequestered in complexes C = [P:F]. The feedback arrives too late to prevent the oscillatory collapse.
Why does (ii) ααα-induction sit between them?
Oligomers respond quickly to P drops (ααα ∝ 1/P, like s), giving a fast feedback. However, the saturation parameter ε limits the σ_eff at very low P, slightly reducing the rescue efficiency compared to s-induction. The response is fast but bounded.
Common finding across all three:
Induction creates negative feedback that competes with the positive feedback (F sequesters P). This raises m_T significantly but does not eliminate the threshold — at sufficiently large m, the system always enters the disease state regardless of induction strength.
| Figure | Description |
|---|---|
| Fig 1 | Trajectories of F and P for all 4 models at m = 25. Black dashed line shows σ_eff(t) for induction models. Oscillatory vs steady-state behavior visible at a glance. |
| Fig 2 | Threshold scan: oligomer accumulation ⟨1/P⟩ and Lewy body burden ∫F dt vs m (log scale). Shows how each induction type raises m_T and by how much. |
| Fig 3 | σ_eff(t) dynamics for each induction type, computed both on the base trajectory and the induced trajectory. Illustrates the different response speeds and amplitudes. |
| Fig 4 | Rescue efficiency scan: ⟨1/P⟩ vs induction strength α for each inducer at m = 25. Points coloured red (still oscillating) vs blue/purple/orange (rescued to SS). |
| Fig 5 | Phase portraits F vs P for all 4 models. Fixed point (health) vs limit cycle (disease) clearly visible in trajectory topology. |
Requirements: Python 3, NumPy, SciPy (solve_ivp, find_peaks), Matplotlib.
NumPy compatibility: works on both NumPy < 2.0 and ≥ 2.0 (trapz/trapezoid auto-detected).
| Concept | Implementation |
|---|---|
| s quasi-SS from paper Appendix eq. A.3 | s = m/(1+P) in all induced models |
| Oligomer proxy from paper eq. A.9–A.10 | ααα ∝ 1/(ε+P) in model (ii) |
| Negative feedback competing with positive | σ_eff(X) rising when X rises |
| Threshold condition m_T = (1+ν)(1+σ)² | Shifts up because effective σ > σ₀ |
| σ/m ratio as key disease parameter | All inductions increase σ_eff, raising σ_eff/m |
| Sneppen fig. 2: F and P mutually exclusive | Fig 1 and 5 reproduce this for all 4 models |
Biological analogues of the three inducers:
-
(i) s-induction corresponds to the early UPS stress response — cells detect early aggregation intermediates (e.g. through Hsp70/Hsp40 chaperone systems) and rapidly upregulate the ubiquitin-proteasome pathway. This is the fastest and most effective response precisely because it acts on a signal that anti-correlates with P.
-
(ii) ααα-induction corresponds to apoptotic and stress signalling triggered by toxic oligomers damaging membranes. The cell upregulates degradation machinery as an emergency response to the most toxic species. Effective but slightly delayed relative to s.
-
(iii) F-induction corresponds to mature fibril stress response — e.g. the heat shock response (HSF1 activation) triggered by large insoluble aggregates. Biologically important but therapeutically less attractive because by the time mature fibrils accumulate, the proteasome may already be irreversibly occupied.
- Sneppen, K., Lizana, L., Jensen, M.H., Pigolotti, S. & Otzen, D. (2009). Modeling proteasome dynamics in Parkinson's disease. Physical Biology, 6, 036005.
- Lecture slides: Physics of Molecular Diseases – Week 2, Protein Quality Control (Prof. Ala Trusina, Niels Bohr Institute, 2020).
- Axelsen, J.B. & Sneppen, K. (2004). Quantifying the benefits of translation regulation in the unfolded protein response. Physical Biology, 1, 159 — for negative feedback in stress response systems.