Instability-Driven Torsion Sign Flips Revealed by the Laplacian Structure of Curvature Support Fields in Protein Backbones

Torsion sign flips are fundamental discrete events in protein backbone dynamics, yet their geometric triggers remain poorly understood. While traditionally associated with singular critical points, we show that these transitions are primarily driven by local geometric instability, quantified by the Laplacian of a curvature-derived support field, ΔS. Through a large-scale analysis of over 1.3 million residue positions from 3,000 PDB structures, we demonstrate a robust physical law: flip events are significantly depleted in Laplacian-zero (locally balanced) regions and exhibit progressively higher enrichment as the instability magnitude |ΔS| increases. This supports an instability-associated statistical interpretation in which geometric imbalance is correlated with an increased propensity for torsion inversions in the analyzed static structural data set. We further identify a "topological locking" effect, where looplike constraints systematically suppress flips, particularly in high-instability regimes. Multivariate modeling confirms that ΔS provides predictive information inaccessible to standard local descriptors (curvature and torsion). Our results establish a physically interpretable framework linking discrete differential geometry to stochastic backbone dynamics, revealing how local geometric imbalance and global structural constraints are jointly associated with torsion sign-flip propensity in experimentally determined static protein structures.