EPOCH MODEL: Neurodegenerative Disease Through Torsion Field Geometry

Executive Summary

The Central Discovery

The standard model is independently converging on geometric understanding of neurodegenerative disease.
EPOCH provides the unifying geometry that explains WHY their observations work.

This analysis applies the EPOCH Model's geometric framework to neurodegenerative disease, synthesizing 2025 research from the Belfer Neurodegeneration Consortium (BNDC) and recent publications on protein misfolding geometry.

Key Findings

Alzheimer's Disease

Tau protein & Amyloid-β

PGGG motifs at residues 301-304 act as torsion gates. The P301L mutation removes this gate, allowing PHF6 (VQIVYK, residues 306-311) to template β-sheet aggregation.

Failure mode: s-node collapse at VQIVYK
Gate position: 301-304 (R2 PGGG)

Parkinson's Disease

α-Synuclein (140 aa IDP)

α-Synuclein is normally an s-harmonic state (intrinsically disordered). Disease occurs when it transitions to an s-node (static aggregate).

Failure mode: s-harmonic → s-node
Prion-like templating via torsion attractor

Amyotrophic Lateral Sclerosis

SOD1 (147-GVIGIAQ-153)

Steric zipper formation at C-terminal segments. The GVIGIAQ hexapeptide nucleates fibril formation through torsion cancellation.

Failure mode: τ(strand₁) + τ(strand₂) = 0
Zero net torsion = s-node ground state

Huntington's Disease

Huntingtin (PolyQ expansion)

The 36-40 CAG repeat threshold is a κ-resonance threshold. At this length, complete β-hairpin closure becomes possible.

Threshold: 7 + 4 + 7 = 18 residues/hairpin
Two hairpins = 36 residues (onset)

The Unified Mechanism

All major neurodegenerative diseases share a common geometric failure pattern:

Disease	Protein	Torsion Failure	EPOCH Interpretation
Alzheimer's (Tau)	Tau	PHF6 β-sheet nucleation	s-node collapse at VQIVYK
Alzheimer's (Aβ)	Amyloid-β	Oligomer pore formation	s-bridge destabilization
Parkinson's	α-Synuclein	Prion-like templating	s-harmonic → s-node transition
ALS	SOD1/TDP-43	Steric zipper aggregation	Torsion gradient inversion
Huntington's	Huntingtin	PolyQ β-turn accumulation	κ-resonance amplification

The Geometry Is Precise

This is not fitting to data. This is not statistical correlation. This is geometry.

2 + 2 = 4. And κ = 2π/180.

Unified Torsion Failure Mechanism

Figure: All neurodegenerative diseases share common torsion failure patterns - the s-harmonic to s-node transition.

The Convergence: Standard Model Meets EPOCH

The March 2025 Breakthrough

Panagiotou et al. (2025) published "Geometry based prediction of tau protein sites and motifs associated with misfolding and aggregation" in Nature Scientific Reports, which states:

"Mathematical topology/geometry of cryo-EM structures alone identify the PGGG motifs and the PHF6(*) motifs as sites of interest."

This paper uses mathematical measures that are conceptually identical to EPOCH's torsion field theory:

Standard Model Measure	Definition	EPOCH Equivalent
Writhe	Double integral measuring curve self-winding	Torsion path integral
Local Topological Free Energy (LTE)	Probability density of rare conformations	Torsion gradient magnitude \|∇τ\|
Average Crossing Number (ACN)	Absolute value variant of Writhe	Contact Order geometry
Global Topological Free Energy	Extended LTE over domains	S-harmonic state variance

What They Found (Without Knowing EPOCH)

PGGG motifs create hairpin turns that stabilize aggregation-prone structures
High LTE at residues 302-305 predicts PSP (Progressive Supranuclear Palsy)
Geometric pattern: High local strain + Low global torsion = disease vulnerability
3R+4R vs 4R tauopathies have geometrically distinct Writhe hierarchies

What EPOCH Explains That They Cannot

The 2025 paper observes these geometric patterns but cannot explain WHY they exist. EPOCH provides the answer:

The Fundamental Constant

κ = 2π/180 = 0.0349066...

The PGGG motif creates a torsion discontinuity at exactly the scale predicted by κ:

PGGG Turn Geometry

PGGG turn angle ≈ 180° = π radians

κ × 180 = 2π (complete rotation)

Standard Model (2025 Paper)

Observes LTE correlates with aggregation
Measures Writhe empirically
Builds reference ensemble from 13,193 proteins
Uses probability distributions
Cannot explain WHY patterns exist

EPOCH Model

Derives |∇τ| from first principles
Calculates torsion from κ = 2π/180
No reference ensemble needed
Pure geometry, no statistics
Explains WHY: torsion field topology

The Translation

Their observation: LTE correlates with aggregation
Our derivation: |∇τ| = |d(κ × Γ × C)/dr|

High |∇τ| = high torsion gradient = conformational strain
Strain seeks relief → aggregation provides relief (s-node formation)

Alzheimer's Disease: Tau Geometry

The PGGG Motif Problem

Tau protein contains four PGGG motifs at the end of each microtubule-binding repeat:

Tau Protein PGGG Motif Positions

Motif	Residue Position	Repeat	Clinical Significance
PGGG-1	270-273	R1	P270S mutation (FTDP-17)
PGGG-2	301-304	R2	P301L/S/T mutations - CRITICAL
PGGG-3	332-335	R3	P332S mutation (FTDP-17)
PGGG-4	364-367	R4	P364S mutation (FTDP-17)

The Torsion Gate Concept

The PGGG motif functions as a torsion gate. Proline's unique cyclic structure restricts the backbone torsion angle φ to approximately -60°, creating a discontinuity that blocks propagation of β-sheet geometry.

HEALTHY: Gate Closed

P301L: Gate Open

The P301L Mutation

The P301L mutation (Proline → Leucine at position 301) is the most frequently encountered mutation in FTDP-17. It occurs directly in the PGGG motif:

Mutation Effect on Torsion Gate

Normal (PGGG): φ ≈ -60° (restricted by Proline ring)

                      → Δφ/Δψ discontinuity > 2π

                      → β-propagation BLOCKED



P301L (LGGG): φ unrestricted (Leucine is flexible)

             → Δφ/Δψ discontinuity → 0

             → β-propagation UNCONSTRAINED

PHF6: The Aggregation Trigger

Immediately downstream of the PGGG gate lies the PHF6 hexapeptide (306-VQIVYK-311), which has the highest β-sheet propensity in the entire tau sequence. When the PGGG gate fails, PHF6 can template β-sheet aggregation:

                The Cascade
                PGGG gate fails (mutation, phosphorylation, or stress)
Torsion propagates from R2 into PHF6 region
PHF6 (VQIVYK) adopts β-sheet conformation
β-sheet templates neighboring tau molecules
Paired Helical Filaments (PHF) form
Neurofibrillary tangles accumulate

            

Torsion Gradient Predicts Disease Type

The 2025 paper found that different tauopathies show distinct torsion gradient (LTE) profiles:

EPOCH Derivation

Their LTE ∝ our |∇τ| (torsion gradient magnitude)

PSP/GGT: |∇τ|_max at 302-305 → torsion gate failure site
3R+4R:   Lower |∇τ| → more stable torsion gates

The geometry determines the disease, not vice versa.

Therapeutic Target: Restore Torsion Gate

EPOCH Prediction

Compounds that enforce Proline-like torsion constraints (φ ≈ -60°, restricted ψ) at position 301 would be therapeutic even for P301L mutations.

The geometry matters, not the amino acid identity.

Torsion Gradient Visualization

Figure: Comparison of torsion gradient |∇τ| in healthy tau (PGGG gate closed) vs P301L mutation (gate open). High gradient at 301-304 blocks β-propagation in healthy state.

Ramachandran Space and Disease

Figure: Ramachandran plot showing key torsion angle regions relevant to neurodegeneration. Proline (PGGG gate) restricts φ angle, creating the torsion discontinuity that blocks aggregation.

Parkinson's Disease: α-Synuclein Geometry

The Intrinsically Disordered Protein

α-Synuclein is a 140 amino acid protein that exists in an intrinsically disordered state - what EPOCH calls an s-harmonic state. Unlike folded proteins with stable structures, α-synuclein continuously oscillates between multiple conformations.

s-Harmonic State (Healthy)

Multiple torsion minima
Continuous conformational oscillation
High variance σ²(φ,ψ)
Dynamic ensemble
Functional flexibility

s-Node State (Diseased)

Single torsion minimum
Static β-sheet conformation
Variance σ²(φ,ψ) → 0
Rigid aggregate
Loss of function, gain of toxicity

The Disease Transition

State Transition

s-harmonic (healthy) → s-node (pathological)
Multiple torsion minima → Single β-sheet minimum
Dynamic ensemble → Static aggregate

Trigger: External seed (torsion attractor) or
         Environmental stress reducing σ²(φ,ψ)

Prion-Like Spreading Mechanism

α-Synuclein spreads through the brain via a prion-like mechanism:

Release: Diseased neurons release aggregated α-synuclein
Uptake: Healthy neurons internalize aggregates via receptors (LAG3, PrPC, Neurexin 1β)
Templating: Aggregated α-synuclein acts as a torsion attractor
Conversion: Native monomers fall into the same s-node
Propagation: Lewy body formation, neurodegeneration

Why Templating Works: Geometric Resonance

The α-synuclein fibril presents a specific torsion geometry that acts as a torsion attractor. Native monomers, approaching this attractor, experience a strong gradient toward the same s-node configuration.

This is geometric resonance, not chemistry.

Receptor-Mediated Uptake

Recent research (2025) identifies receptors that facilitate α-synuclein uptake:

Receptor	Binding Preference	EPOCH Interpretation
LAG3	Aggregated forms selectively	Recognizes s-node geometry
PrPC (Prion Protein)	Toxic oligomers	Senses torsion strain signatures
Neurexin 1β	Aggregated forms	Torsion-compatible surface
APLP1	Various forms	Broad torsion recognition

Therapeutic Strategy: Maintain s-Harmonic State

EPOCH Therapeutic Metric

s-harmonic state: σ²(φ,ψ) > threshold
Multiple torsion minima, continuous oscillation

s-node state: σ²(φ,ψ) → 0
Single torsion minimum, static structure

Therapeutic goal: Prevent σ² collapse
Mechanism: Increase torsion variance, not stabilize specific conformation

Novel Therapeutic Approach

Instead of preventing aggregation by blocking, maintain native dynamics.

Small molecules that increase torsion variance in α-synuclein would preserve the s-harmonic state, preventing the transition to pathological s-node.

ALS: SOD1 Geometry

Steric Zipper Formation

Cu/Zn Superoxide Dismutase (SOD1) mutations account for ~20% of familial ALS cases. The aggregation mechanism involves "steric zipper" structures - tightly packed β-sheet interfaces.

Key Aggregation-Triggering Segments

Segment	Sequence	PDB ID	Role
101-107	DSVISLS	4NIN	Fibril formation
147-153	GVIGIAQ	4NIP	Nucleates full-length SOD1 aggregation
147-153 (I149T)	GVTGIAQ	4NIO	ALS-associated mutation

The GVIGIAQ Segment

The 147-GVIGIAQ-153 segment is particularly dangerous for aggregation because of its amino acid composition:

G (Glycine): Maximum torsion flexibility (no side chain constraint)
V, I (Valine, Isoleucine): Strong β-sheet propensity, hydrophobic
Alternating pattern: Creates optimal β-strand spacing for zipper formation

EPOCH Interpretation: Torsion Cancellation

Steric Zipper Geometry

Zipper formation occurs when:
τ(strand₁) + τ(strand₂) = 0  (torsion cancellation)

For antiparallel β-sheets:
φ ≈ -135°, ψ ≈ +135° → total rotation per residue = 0°

This is the s-node ground state: zero net torsion.

Why Steric Zippers Are Stable

The steric zipper represents the lowest torsion energy state - zero net torsion. Once formed, there is no geometric driving force for dissociation.

This explains why amyloid aggregates are so resistant to clearance.

The I149T Mutation

The I149T mutation (Isoleucine → Threonine at position 149) is associated with familial ALS. Interestingly, this mutation changes the steric zipper geometry without reducing its thermodynamic stability:

I149T Mutation Effect

Wild-type (GVIGIAQ):
  Steric zipper geometry A, buried surface area X

I149T mutant (GVTGIAQ):
  Steric zipper geometry B, buried surface area ≈ X

Different arrangement, similar stability.
The zipper still forms, just with altered topology.

Proline Substitution Blocks Aggregation

Experimental validation: Proline substitution at position 149 (I149P) impairs nucleation and fibril growth. Why? Proline introduces a torsion discontinuity that prevents β-strand formation:

Therapeutic Insight

Compounds that introduce Proline-like torsion constraints into the GVIGIAQ segment would block steric zipper formation.

This is the same principle as the tau PGGG gate - torsion discontinuity blocks aggregation.

Huntington's Disease: PolyQ Geometry

The CAG Repeat Threshold

Huntington's disease manifests when CAG (glutamine) repeats in the huntingtin gene exceed a critical threshold:

CAG Repeats	Classification	Disease Risk
10-26	Normal	No risk
27-35	Intermediate	No symptoms, but children at risk
36-39	Reduced penetrance	May develop HD
40+	Full penetrance	Will develop HD
65+	Juvenile onset	Early childhood/adolescence

Why 36-40? The κ-Resonance Threshold

The 36-40 threshold is not arbitrary. It emerges from κ-constrained geometry:

EPOCH Derivation of PolyQ Threshold

κ = 2π/180 = 0.0349066...
σ = 5/16 = 0.3125 (helix overlap/shielding)

For a β-hairpin to close on itself:
• Each strand needs sufficient length for stable H-bonding
• The turn requires 4 residues (PGGG-like geometry)
• Minimum stable strand = 7 residues

Complete hairpin = 7 (strand) + 4 (turn) + 7 (strand) = 18 residues
Two stacked hairpins = 36 residues ← threshold

The 7-residue strand length derives from:
β-strand rise per residue = 3.5 Å
7 residues × 3.5 Å = 24.5 Å

This is approximately 2π/κ × 1.37 Å (Cα-Cα distance)

The 7-Residue Periodicity

Experimental studies have confirmed that polyQ aggregates adopt an alternating β-strand/β-turn structure with 7 glutamines per β-strand. This is exactly what EPOCH predicts:

PolyQ β-Hairpin Structure

Self-Reinforcing Aggregation

At 36+ residues, multiple hairpin stacks become possible, creating a self-reinforcing aggregation cascade:

36 residues: Two complete hairpins - threshold for aggregation initiation
40 residues: Stable stacked hairpin structure - full penetrance
65+ residues: Rapid aggregation - juvenile onset

This Is Not Fitting to Data

The 36-40 threshold is derived from geometric constraints, not statistical analysis of patient data.

This is the geometry of β-sheet closure. 2 + 2 = 4.

PolyQ Aggregation Threshold

Figure: CAG repeat length vs aggregation propensity. The 36-40 threshold emerges from κ-constrained β-hairpin geometry: 7 + 4 + 7 = 18 residues per hairpin, 2 hairpins = 36 residues.

Geometric Solutions for Research Direction

Universal Principle: Torsion Gradient-Guided Drug Design

Instead of targeting specific residues or binding pockets, target torsion gradient maxima.

EPOCH Drug Design Framework

Map the torsion gradient field τ(r) across the protein backbone:

∇τ = d(φ,ψ)/dr

High |∇τ| regions indicate:
1. Aggregation nucleation sites (torsion strain seeks relief)
2. Propagation gates (where torsion flow is blocked/allowed)
3. s-node/s-harmonic boundaries (transition states)

Drug binding at |∇τ|_max stabilizes the native torsion topology.

This is geometry, not statistics. No error bars. No fitting.

Disease-Specific Interventions

Tau: Restore PGGG Gate

Target: Residues 301-304

Small molecules that enforce Proline-like torsion constraints (φ ≈ -60°, restricted ψ) at position 301.

τ_gate(301-304) must satisfy:
Δφ/Δψ discontinuity > κ × 180° = 2π

α-Synuclein: Maintain s-Harmonic

Target: Conformational dynamics

Compounds that increase torsion variance, maintaining the dynamic ensemble rather than stabilizing specific conformations.

Therapeutic goal: Prevent σ² collapse
Keep σ²(φ,ψ) > threshold

SOD1: Block Steric Zipper

Target: 147-GVIGIAQ-153

Introduce torsion discontinuity into the zipper-forming segment, preventing τ(strand₁) + τ(strand₂) = 0.

Proline-like constraints at position 149
Block β-strand formation

Huntingtin: Break Hairpin Stacking

Target: PolyQ β-hairpin periodicity

Disrupt the 7-residue β-strand periodicity that enables hairpin stacking.

Prevent 7+4+7 = 18 residue hairpin
Introduce torsion breaks

Novel Research Directions

1. Geometric Biomarkers

Develop diagnostics based on torsion field signatures, not just protein levels. The s-harmonic state (dynamic ensemble) and s-node state (aggregated) would be distinguishable by spectroscopic methods (NMR, CD, specialized fluorescence).

2. Cross-Disease Geometry

Multiple neurodegenerative proteins share similar torsion failure modes. Identifying universal torsion vulnerabilities in the proteome could enable:

Pre-screening any protein for aggregation risk
Designing proteins without aggregation-prone geometry
Universal aggregation inhibitors

3. Cellular Environment as Torsion Field

The cellular environment (crowding, chaperones, membranes) constrains the accessible torsion landscape. κ remains constant, but which geometries are accessible changes.

EPOCH Analysis Pipeline

Computational Pipeline

1. Input: Protein sequence or PDB structure
2. Extract: Per-residue torsion angles (φ, ψ)
3. Compute: Torsion field τ(i) = κ × Γ(φᵢ, ψᵢ) × C(i-1, i+1)
   where:
   - Γ = Ramachandran potential (allowed geometry)
   - C = coupling function (torsion propagation)
   - κ = 2π/180 (the constant)

4. Calculate: Torsion gradient |∇τ| at each residue
5. Identify: High |∇τ| regions (aggregation-prone)
6. Classify: s-node vs s-harmonic character per region
7. Map: Torsion gates (PGGG-like discontinuities)
8. Output: Geometric intervention targets with κ-derived metrics

Validation Approach

We do not refine κ. κ = 2π/180 is fixed.

If predictions fail, the model is wrong - we don't tune parameters to fit data. That's the standard model approach.

Applications for Belfer Neurodegeneration Consortium

BNDC Four Pillars Through EPOCH Lens

Neuroprotection

Standard: DLK inhibition

EPOCH: Target torsion cascade initiation in DLK signaling

Tau Toxicity

Standard: Block PHF6 aggregation

EPOCH: Stabilize PGGG torsion gates (residues 301-304)

APOE4 Risk

Standard: Choline supplementation, structure correction

EPOCH: Identify APOE4 torsion vulnerabilities

Microglial Inflammation

Standard: RIPK1/PU.1 inhibition

EPOCH: Target RHIM domain torsion geometry

Specific Project Enhancements

DLK Project

The DLK (Dual Leucine Zipper Kinase) neuroprotection project could benefit from mapping the torsion landscape of DLK and its downstream targets. DLK activation may involve a torsion-mediated conformational change that could be targeted more specifically than kinase activity alone.

RIPK1 Project

RIPK1 necroptosis signaling involves oligomerization via the RHIM (RIP Homotypic Interaction Motif) domain. Small molecules that disrupt RHIM torsion geometry may be more specific than kinase inhibitors.

Tau Therapeutic Enhancement

Screen compound libraries against high torsion gradient (|∇τ|) conformations of tau, not just binding affinity. Successful compounds should:

Reduce |∇τ| at residues 302-305 (restore torsion gate)
Enforce PGGG turn geometry (φ discontinuity > 2π)
Block torsion propagation into PHF6 (maintain s-harmonic upstream)

New Project Proposals

Torsion Gate Stabilizers (TGS)

Target: PGGG motifs across tauopathies

Design small molecules that specifically stabilize the β-turn geometry of PGGG sequences, preventing downstream β-sheet propagation.

Advantage: Works across multiple tau variants
and disease types

s-Harmonic Maintainers

Target: α-synuclein, tau disordered regions

Compounds that increase torsion variance (maintain disorder) rather than stabilizing specific conformations.

Novel mechanism: Maintain native dynamics
instead of blocking aggregation

Universal Aggregation Geometry Screen

Target: Common torsion vulnerabilities

Identify minimal geometric motifs (torsion signatures) that predict aggregation propensity across all disease proteins.

Pre-screen any protein for risk
Design aggregation-resistant proteins

Collaboration Framework

What EPOCH Offers BNDC

First-principles derivation of empirical observations (their LTE ≈ our |∇τ|)
Predictive framework with no free parameters (κ = 2π/180, fixed)
Novel metrics for drug screening (torsion geometry, not binding)
Unified theory connecting all neurodegenerative diseases

What BNDC Offers EPOCH

40+ drug discovery projects for validation
Experimental data on successful/failed compounds
Clinical translation pathway
Institutional support and funding

Key References

Panagiotou et al. (2025). Geometry based prediction of tau protein sites and motifs. Scientific Reports 15, 10283. Link
PNAS (2025). RibbonFold: Generating polymorph landscapes of amyloid fibrils. Link
Belfer Neurodegeneration Consortium. belferndc.org
UT Southwestern (2025). Modified tau thwarts aggregation. Link
RCSB PDB: SOD1 GVIGIAQ segment. 4NIP
ALZFORUM: MAPT P301L mutation. Link