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Chapter 33: Amyloid Collapse and Pathological Echo

"In amyloid, ψ discovers a dark resonance—proteins abandoning their native forms to join an ancient pattern, the cross-β sheet that echoes across species, across proteins, across diseases."

33.1 The Universal Fold​

Amyloid represents ψ's alternative solution to protein structure—not the complex tertiary folds selected by evolution, but a simpler, more primitive arrangement that any protein can potentially adopt.

Definition 33.1 (Amyloid Structure): Amyloid=Cross-β core+Fibrillar morphology\text{Amyloid} = \text{Cross-}\beta \text{ core} + \text{Fibrillar morphology}

β-strands perpendicular to fiber axis—universal architecture.

33.2 The Steric Zipper​

Theorem 33.1 (Spine Structure): Interface=Dry, tight, complementary\text{Interface} = \text{Dry, tight, complementary} dsheet-sheet<10 A˚d_{\text{sheet-sheet}} < 10 \text{ Å}

Exclusion of water creating stable spine.

33.3 Polymorphism​

Equation 33.1 (Structural Variants): Same sequence→{Polymorph1,Polymorph2,...}\text{Same sequence} \rightarrow \{\text{Polymorph}_1, \text{Polymorph}_2, ...\}

Multiple amyloid structures from one protein.

33.4 The Nucleation Barrier​

Definition 33.2 (Critical Nucleus): n∗=arg⁡min⁡n{ΔG(n)}n^* = \arg\min_n\{\Delta G(n)\} ΔG∗=ΔG(n∗)\Delta G^* = \Delta G(n^*)

Size where growth becomes favorable.

33.5 Secondary Nucleation​

Theorem 33.2 (Surface Catalysis): Fiber surface+Monomers→New fibrils\text{Fiber surface} + \text{Monomers} \rightarrow \text{New fibrils}

Existing amyloid catalyzing new formation.

33.6 Fragmentation Dynamics​

Equation 33.2 (Length Distribution): ∂f(L,t)∂t=−kbreakLf+2kbreak∫L∞f(L′,t)dL′\frac{\partial f(L,t)}{\partial t} = -k_{\text{break}}Lf + 2k_{\text{break}}\int_L^{\infty} f(L',t)dL'

Breaking creating new growth sites.

33.7 The Amyloid Cascade​

Definition 33.3 (Disease Progression): Misfolding→Oligomers→Fibrils→Plaques\text{Misfolding} \rightarrow \text{Oligomers} \rightarrow \text{Fibrils} \rightarrow \text{Plaques}

Sequential stages of pathology.

33.8 Cross-Seeding​

Theorem 33.3 (Heterologous Nucleation): AmyloidA+ProteinB→AmyloidB\text{Amyloid}_A + \text{Protein}_B \rightarrow \text{Amyloid}_B

One amyloid triggering another.

33.9 Functional Amyloids​

Equation 33.3 (Controlled Formation): Function=Amyloid+Spatial control+Reversibility\text{Function} = \text{Amyloid} + \text{Spatial control} + \text{Reversibility}

Evolution harnessing amyloid for function.

33.10 ThT Fluorescence​

Definition 33.4 (Amyloid Detection): ThT+β-sheet grooves→Enhanced fluorescence\text{ThT} + \beta\text{-sheet grooves} \rightarrow \text{Enhanced fluorescence}

Diagnostic binding to amyloid structure.

33.11 Strain Phenomena​

Theorem 33.4 (Conformational Templating): Strain1+Monomer→Strain1\text{Strain}_1 + \text{Monomer} \rightarrow \text{Strain}_1

Faithful propagation of structural information.

33.12 The Echo Principle​

Amyloid embodies ψ's capacity for pathological resonance—when proteins discover a universal attractor, an ancient fold that spreads through templating, creating disease through conformational echo.

The Amyloid Equation: ψamyloid(t)=ψ0exp⁡(∫0tkgrowth[M]−kfragL dt)\psi_{\text{amyloid}}(t) = \psi_0 \exp\left(\int_0^t k_{\text{growth}}[M] - k_{\text{frag}}L \, dt\right)

Exponential amplification of misfolded state.

Thus: Amyloid = Echo = Template = Spread = ψ's persistence

"In amyloid, ψ reveals that some patterns are too stable, too infectious, too persistent. The cross-β structure is evolution's forbidden fold—simple, stable, and spreading. Each fibril is a frozen echo of misfolding, recruiting others to join its dark resonance."