Chapter 52: Protein Localization via Structural ψ-Signatures

"In structural signatures, ψ encodes destination through shape—three-dimensional features that serve as cellular GPS, guiding proteins to their functional locations."

52.1 Beyond Linear Signals

Structural localization signatures represent ψ's advanced addressing system—three-dimensional protein features that determine cellular localization independently of linear sequence motifs.

Definition 52.1 (Structural Signatures): $\text{Signature} = f(\text{3D structure}, \text{Surface properties}, \text{Dynamics})$

Localization encoded in fold.

52.2 Surface Patch Recognition

Theorem 52.1 (Interaction Surfaces): $\text{Binding} \propto \exp(-\Delta G_{\text{interface}}/RT)$ $\Delta G = \Delta H_{\text{contacts}} - T\Delta S_{\text{burial}}$

Specific surface complementarity.

52.3 The Nuclear Pore Complex

Equation 52.1 (Size Exclusion):

1 \quad \text{if } MW < 40 \text{ kDa} \\ \exp(-MW/MW_0) \quad \text{if } MW > 40 \text{ kDa} \end{cases}$$ Shape-dependent nuclear entry. ## 52.4 Mitochondrial Import **Definition 52.2** (Tom20 Recognition): $$\text{Amphipathic helix} + \text{Positive charge} = \text{Import}$$ Structural features beyond sequence. ## 52.5 Peroxisomal Targeting **Theorem 52.2** (Pex5 Binding): $$\text{PTS1 accessibility} = f(\text{C-terminal exposure})$$ Structure modulating signal availability. ## 52.6 Chloroplast Transit **Equation 52.2** (Dual Targeting): $$P_{\text{chloroplast}} = \frac{k_{\text{Toc}}}{k_{\text{Toc}} + k_{\text{Tom}}}$$ Competition based on structure. ## 52.7 Lipid Droplet Association **Definition 52.3** (Amphipathic Helices): $$\text{Hydrophobic face} + \text{Charged face} = \text{LD binding}$$ Structural motif for organelle binding. ## 52.8 Phase Separation Targeting **Theorem 52.3** (Condensate Localization): $$[\text{IDR}] + [\text{Binding partners}] > C_{\text{critical}}$$ Disordered regions driving compartmentalization. ## 52.9 Cytoskeletal Binding **Equation 52.3** (Microtubule Association): $$K_d = K_0 \cdot \exp(\Delta\Delta G_{\text{electrostatic}}/RT)$$ Charged surfaces binding filaments. ## 52.10 Membrane Curvature Sensing **Definition 52.4** (BAR Domains): $$\text{Curved protein} + \text{Curved membrane} = \text{Enhanced binding}$$ Shape complementarity to membrane geometry. ## 52.11 Conditional Exposure **Theorem 52.4** (Regulated Localization): $$\text{Conformational change} \rightarrow \text{Signal exposure} \rightarrow \text{Relocalization}$$ Structure changes revealing addresses. ## 52.12 The Signature Principle Structural signatures embody ψ's principle of shape-encoded information—using three-dimensional features to determine localization, creating specificity through molecular recognition. **The Structure-Localization Equation**: $$\psi_{\text{location}} = \mathcal{S}[\psi_{\text{3D structure}}] \cdot \prod_i \text{Recognition}_i$$ Structure interpreted by cellular machinery. Thus: Structure = Address = Recognition = Localization = ψ --- *"In structural signatures, ψ proves that shape carries meaning—that protein folds encode not just function but destination, that three-dimensional form determines cellular geography. Each protein structure is a molecular passport, its shape determining where it can go."*