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This research aims to explore the geometric relationships of protein structures using a traditional compass framework and assess their implications for allosteric mechanisms.

May 19, 2026Open Access

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Twenty-Four Mountain Celestial Compass as a Multi-Scale Geometric Descriptor of Protein Topological Order: From Allosteric Classification to Proteogenomic Backbone Geometry — A Five-Layer Investigation with Two-Scale Operator Framework

Key Points

This research aims to explore the geometric relationships of protein structures using a traditional compass framework and assess their implications for allosteric mechanisms.
Utilized the Twenty-Four Mountain compass framework to analyze protein topological order across five experimental layers.
Conducted statistical analyses, including permutation tests and odds ratios for allosteric classification.
Introduced the Two-Scale Operator Framework to evaluate protein geometric properties in various contexts.
Allosteric proteins showed significant enrichment in Golden sectors (p=0.0018) and a trend towards Danger depletion.
Kan-type proteins were found to be 4.5 times more likely to be allosteric (OR=4.50, p<0.001).
Distinct dihedral distributions in backbone geometry were observed in EGFR, BRAF, and ABL1, with statistically significant findings for multiple structures.

Abstract

The classical Chinese Twenty-Four Mountain (Er-Shi-Si Shan) compass partitions 360 degrees into 24 sectors: Golden (7 sectors, 29. 2%), Danger (6 sectors, 25. 0%), and Neutral (11 sectors, 45. 8%). We report a geometric isomorphism between this system and protein contact network spectral geometry, investigated across five experimental layers. Layer 1 (n=569): Allosteric proteins show Golden enrichment (p=0. 0018**; protein-level permutation pₚerm=0. 003**) and Danger depletion trend (p=0. 0003***; pₚerm=0. 14, attenuates after spatial autocorrelation correction). Layer 2: Algebraic theorem — Danger sectors are the exact nodal set of the 45-degree-lattice Fiedler mode (offset std=0. 000 degrees). Layer 3 (n=336): Kan-type proteins are 4. 5x more likely allosteric (OR=4. 50, p<0. 001). Layer 4 (n=72 pairs): Residue-level shows Danger increase (p=0. 0005***) but gene-level aggregation yields p=0. 869 (ns), indicating pseudoreplication; additionally, EGFR WT structure 2GS7 exhibits near-zero lambda2 (degenerate Fiedler space). Layer 4 is downgraded to exploratory. Layer 5 NEW (4 kinases, 18 PDB structures): Anchor-free Ramachandran phi/psi analysis demonstrates that exon boundaries correspond to distinct backbone dihedral distributions in EGFR (KW-phi p=0. 000143***, 7 structures), BRAF (p<0. 0001***, 3 structures), and ABL1 (p<0. 0001***, 4 structures) ; KRAS shows partial significance (psi only, p=0. 042). The Two-Scale Operator Framework is introduced as a formal boundary condition: Life-Axis anchoring is valid as a global protein-level operator (Layers 1-3) but invalid on local exon subspaces; Ramachandran phi/psi provides the appropriate local alternative. This version (v4. 1) incorporates three methodological corrections: (1) protein-level permutation test for L1, revealing Golden enrichment is robust but Danger depletion is trend-level; (2) explicit documentation of 2GS7 PCN degeneracy and its effect on L4; (3) extended L5 to 4 kinase families with clarified common-residue methodology.

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Twenty-Four Mountain Celestial Compass as a Multi-Scale Geometric Descriptor of Protein Topological Order: From Allosteric Classification to Proteogenomic Backbone Geometry — A Five-Layer Investigation with Two-Scale Operator Framework

Key Points

Abstract

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