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Chapter 28: Hydrophobic Collapse and Core Formation

"Water's aversion to oil drives ψ's most fundamental collapse—hydrophobic residues fleeing inward, creating the protein's heart, the dense core around which function crystallizes."

28.1 The Hydrophobic Effect​

Hydrophobic collapse represents ψ's primary driving force in protein folding—not through attractive forces but through entropy, water molecules gaining freedom by excluding non-polar surfaces.

Definition 28.1 (Hydrophobic Free Energy): ΔGhydrophobic=ΔH−TΔSwater\Delta G_{\text{hydrophobic}} = \Delta H - T\Delta S_{\text{water}}

Where ΔSwater>0\Delta S_{\text{water}} > 0 dominates at physiological temperature.

28.2 The Tanford Model​

Theorem 28.1 (Oil Drop Model): Hydrophobic residues→Core\text{Hydrophobic residues} \rightarrow \text{Core} Hydrophilic residues→Surface\text{Hydrophilic residues} \rightarrow \text{Surface}

First approximation of protein structure.

28.3 Accessible Surface Area​

Equation 28.1 (Burial Energy): ΔGburial=∑iγi⋅ΔASAi\Delta G_{\text{burial}} = \sum_i \gamma_i \cdot \Delta\text{ASA}_i

Where γ≈25\gamma \approx 25 cal/mol/Ų for non-polar surface.

28.4 The Molten Globule​

Definition 28.2 (Intermediate State): Molten globule={Compact,Secondary structure+,Tertiary−}\text{Molten globule} = \{\text{Compact}, \text{Secondary structure}^+, \text{Tertiary}^-\}

Collapsed but not fully organized state.

28.5 Collapse Kinetics​

Theorem 28.2 (Time Scales): τcollapse≈1−100μs\tau_{\text{collapse}} \approx 1-100 \mu\text{s} τcollapse<τsecondary structure<τtertiary\tau_{\text{collapse}} < \tau_{\text{secondary structure}} < \tau_{\text{tertiary}}

Collapse often precedes detailed organization.

28.6 Core Packing Density​

Equation 28.2 (Packing Fraction): ρcore=VvdWVtotal≈0.74−0.76\rho_{\text{core}} = \frac{V_{\text{vdW}}}{V_{\text{total}}} \approx 0.74-0.76

Similar to organic crystals—efficient packing.

28.7 Hydrophobicity Scales​

Definition 28.3 (Transfer Energy): ΔGtransfer=ΔGwater→octanol\Delta G_{\text{transfer}} = \Delta G_{\text{water→octanol}}

Experimental scales quantifying hydrophobicity.

28.8 The Jigsaw Puzzle Model​

Theorem 28.3 (Specific Packing): Native core=Unique arrangement of side chains\text{Native core} = \text{Unique arrangement of side chains}

Not just burial but specific complementary packing.

28.9 Cavities and Defects​

Equation 28.3 (Cavity Penalty): ΔGcavity=γ⋅Acavity≈25 cal/mol/A˚2\Delta G_{\text{cavity}} = \gamma \cdot A_{\text{cavity}} \approx 25 \text{ cal/mol/Å}^2

Empty space in core is energetically costly.

28.10 Hydrophobic Clusters​

Definition 28.4 (Local Collapse): Cluster={Residuesi,j,k∣dij,djk,dik<6.5 A˚}\text{Cluster} = \{\text{Residues}_{i,j,k} | d_{ij}, d_{jk}, d_{ik} < 6.5 \text{ Å}\}

Local hydrophobic contacts stabilizing structure.

28.11 Collapse and Function​

Theorem 28.4 (Active Site Location): P(Active site at surface)>P(Active site in core)P(\text{Active site at surface}) > P(\text{Active site in core})

Function requires access, often at core-surface interface.

28.12 The Core Principle​

Hydrophobic collapse embodies ψ's principle of entropic organization—order emerging not from attraction but from the system maximizing total entropy through phase separation.

The Collapse Equation: ψfolded=arg⁡min⁡conformations{−TΔSwater+ΔHinteractions}\psi_{\text{folded}} = \arg\min_{\text{conformations}} \{-T\Delta S_{\text{water}} + \Delta H_{\text{interactions}}\}

Structure determined by water entropy maximization.

Thus: Hydrophobicity = Entropy = Collapse = Core = ψ

"In hydrophobic collapse, ψ demonstrates that aversion can be as powerful as attraction—that proteins fold not because hydrophobic residues love each other, but because water loves itself more. The core forms through exclusion, creating density from avoidance."