The Geometry of Allostery: Quantifying Pathway Organization in Protein Networks.

A central challenge in dynamic allostery is understanding how local ligand-binding events selectively route and couple distant communication pathways. Here, we develop the Laplacian minor hierarchy, a parameter-free mathematical framework that extracts higher-order geometric invariants directly from protein contact networks. To incorporate explicit ligand interactions without computational truncation, we develop a block-matrix Schur complement reduction scheme that projects pathways introduced by bound ligands directly onto an effective protein Laplacian. We apply this framework across all eight benchmark structures of the PSD95-PDZ3 domain to map its evolutionary adaptation from Class I (CRIPT) to Class II (T-2F) ligand specificity via the G330T and H372A mutations. Evaluating doublet, triplet, and quadruplet pathway couplings shows that allosteric rewiring is governed by network-level epistasis, where the surface-loop mutation G330T opens communication routes that the active-site mutation H372A subsequently exploits. Quadruplet correlation profiles collapse into two distinct quantitative regimes governed entirely by the pivot residue Thr332. These results indicate the potential of the Laplacian minor hierarchy as a rigorous, physically grounded framework for analyzing path organization, long-range communication, and the topological basis of allosteric epistasis in proteins.