What question did this study set out to answer?

This research aims to explore the relationship between geometric descriptors and protein topological order, focusing on allosteric classification and oncogenic mutations.

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 Oncogenic Mutation Detection

Key Points

This research aims to explore the relationship between geometric descriptors and protein topological order, focusing on allosteric classification and oncogenic mutations.
Population sample of 569 allosteric proteins analyzed for sector occupancy.
Geometric proof establishes a mathematical foundation for nodal sets in protein networks.
Sample of 72 wild-type/mutant pairs assessed for oncogenic mutation impact on sector occupancy.
Allosteric proteins enriched in Golden sectors (p=0.0018), depleted in Danger sectors (p=0.0003).
Oncogenic mutations increase Danger zone occupancy by 8.43% (p=0.0005, Cohen d=+0.431) and decrease Golden zone occupancy by 6.25% (p=0.019).
Notable mutations include EGFR T790M (increase of 96.4% in Danger zone) and KRAS G12C (75.8% increase).

Abstract

We report a geometric isomorphism between the classical Chinese Twenty-Four Mountain (Er-Shi-Si Shan) geomantic compass and the Fiedler eigenvector phase space of protein contact networks, validated across three complementary experimental layers: Layer 1 (Population, n=569 AlloBench): Allosteric proteins show significant enrichment in Golden (antinodal) sectors (p=0.0018) and depletion from Danger (nodal) sectors (p=0.0003), with N-to-C Life-Axis anchoring. Non-allosteric proteins show no deviation from uniform (both ns). Layer 2 (Geometric proof): The six Danger sectors constitute an exact centrosymmetric nodal set of the 45-degree rotational lattice: all six centers at 45-degree lattice points +7.5 degrees (offset std = 0.000 deg), vector sum = (0.00000000, 0.00000000). This is a mathematical theorem identifying Danger sectors as nodal lines and Golden sectors as antinodal regions of the Fiedler vibrational mode. A three-level isomorphism hierarchy is established: Lo-Shu (9 nodes, 40 deg) -- Twenty-Four Mountain (24 sectors, 15 deg) -- Fiedler phase space (continuous), with GCD = 5 deg (72-sector common grid). Layer 3 (Mutation, n=72 WT/MUT pairs, 30 cancer-driver genes): Oncogenic mutations significantly increase Danger zone occupancy (delta-Danger = +8.43%, p=0.0005***, Cohen d=+0.431) and decrease Golden zone occupancy (delta-Golden = -6.25%, p=0.019*). Notable cases: EGFR T790M (+96.4%), KRAS G12C (+75.8%), TP53 G245S (+61.8%), PIK3CA triple hotspots (+29-30% each). Feature ablation confirms that Twenty-Four Mountain descriptors add no incremental AUC (delta-AUC = -0.007, ns) because lambda2 (exact eigenvalue) subsumes the discrete angular approximation -- confirming isomorphism rather than indicating failure. Related: Genesis v8 (doi:10.5281/zenodo.20252223); Paper K Kan-Li axis (doi:10.5281/zenodo.20228189); Route 2 Spatial Validation (doi:10.5281/zenodo.20251581).

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Twenty-Four Mountain Celestial Compass as a Multi-Scale Geometric Descriptor of Protein Topological Order: From Allosteric Classification to Oncogenic Mutation Detection

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Abstract

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