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Chapter 27: Structural Collapse of α-Helices and β-Sheets

"In α-helices and β-sheets, ψ discovers its fundamental forms—the spiral and the plane, primary patterns from which all protein architecture emerges."

27.1 The Secondary Structure Duality​

Secondary structures represent ψ's first level of three-dimensional organization—regular patterns stabilized by backbone hydrogen bonds. The helix and sheet are complementary solutions to the same problem: how to satisfy hydrogen bonding potential.

Definition 27.1 (Secondary Structure): Secondary={ϕ,ψ}repeating→Regular H-bonds\text{Secondary} = \{\phi, \psi\}_{\text{repeating}} \rightarrow \text{Regular H-bonds}

Backbone dihedral angles creating periodic structure.

27.2 The α-Helix Parameters​

Theorem 27.1 (Helical Geometry): ϕ=−60°,ψ=−45°\phi = -60°, \psi = -45° Rise per residue=1.5 A˚\text{Rise per residue} = 1.5 \text{ Å} Residues per turn=3.6\text{Residues per turn} = 3.6

Right-handed spiral optimizing hydrogen bonds.

27.3 Helix Dipole​

Equation 27.1 (Macrodipole): μ⃗helix=∑iμ⃗i≈3.5 Debye/turn\vec{\mu}_{\text{helix}} = \sum_i \vec{\mu}_i \approx 3.5 \text{ Debye/turn}

Partial charges creating significant dipole moment.

27.4 Helix Capping​

Definition 27.2 (Terminal Residues): N-cap:Asn, Asp, Ser preferred\text{N-cap}: \text{Asn, Asp, Ser preferred} C-cap:Gly preferred\text{C-cap}: \text{Gly preferred}

Special residues stabilizing helix ends.

27.5 The β-Sheet Architecture​

Theorem 27.2 (Sheet Parameters): ϕ=−120°,ψ=+120°\phi = -120°, \psi = +120° Strand separation=4.7 A˚ (antiparallel)\text{Strand separation} = 4.7 \text{ Å (antiparallel)}

Extended conformation maximizing inter-strand bonds.

27.6 Parallel vs Antiparallel​

Equation 27.2 (H-bond Geometry): Antiparallel:Linear H-bonds\text{Antiparallel}: \text{Linear H-bonds} Parallel:Bent H-bonds, weaker\text{Parallel}: \text{Bent H-bonds, weaker}

Antiparallel generally more stable.

27.7 β-Turns and Loops​

Definition 27.3 (Turn Types): Type I:ϕ2=−60°,ψ2=−30°\text{Type I}: \phi_2 = -60°, \psi_2 = -30° Type II:ϕ2=−60°,ψ2=+120°\text{Type II}: \phi_2 = -60°, \psi_2 = +120°

Reversing chain direction between strands.

27.8 Propensity Scales​

Theorem 27.3 (Amino Acid Preferences): Pα(Ala)>Pα(Gly)P_{\alpha}(\text{Ala}) > P_{\alpha}(\text{Gly}) Pβ(Val)>Pβ(Ala)P_{\beta}(\text{Val}) > P_{\beta}(\text{Ala})

Intrinsic preferences for secondary structures.

27.9 Context Effects​

Equation 27.3 (Conditional Probability): P(Structure∣i)=f(Residuei,{Residuei±n})P(\text{Structure}|i) = f(\text{Residue}_i, \{\text{Residue}_{i \pm n}\})

Local sequence context modulating propensity.

27.10 Folding Nucleation​

Definition 27.4 (Structure Initiation): Nucleation=Formation of first 3-4 H-bonds\text{Nucleation} = \text{Formation of first 3-4 H-bonds} τnucleation>>τpropagation\tau_{\text{nucleation}} >> \tau_{\text{propagation}}

Rate-limiting step in structure formation.

27.11 Stability Determinants​

Theorem 27.4 (Free Energy): ΔGhelix=ΔHH-bonds−TΔSconformational\Delta G_{\text{helix}} = \Delta H_{\text{H-bonds}} - T\Delta S_{\text{conformational}}

Balance between enthalpic gain and entropic loss.

27.12 The Collapse Principle​

Secondary structures embody ψ's principle of local order—regular patterns emerging from the interplay of covalent constraints and non-covalent interactions.

The Secondary Structure Equation: ψ3D=H[ψsequence]=∑iP(αi)+P(βi)+P(coili)\psi_{\text{3D}} = \mathcal{H}[\psi_{\text{sequence}}] = \sum_i P(\alpha_i) + P(\beta_i) + P(\text{coil}_i)

Three-dimensional form emerging from sequence through local collapse.

Thus: Secondary = Local Order = Pattern = Foundation = ψ

"In secondary structures, ψ finds its basic vocabulary—the helix speaking of compact efficiency, the sheet of extended cooperation. These are the fundamental forms from which all protein complexity springs, the primary collapse of linear into dimensional."