Sub-residue sharpness of protein helix-coil transitions reveals a spatial-spectral uncertainty limit

Abstract

The boundaries of cooperative helix--coil transitions directly affect protein allostery and conformational dynamics, yet the physical origin of the persistent one-to-two-residue assignment ambiguity at these structural interfaces remains unresolved. We apply the discrete Hasimoto map to translate three-dimensional protein backbone geometry into a one-dimensional discrete nonlinear Schrödinger effective potential and analyze its spatial-frequency fluctuations. Helical segments display near-integrable, low-entropy soliton-like states, while coil regions exhibit broadband conformational noise. Statistical analysis of over 19,000 boundaries across 1,986 proteins reveals a median geometric transition width of only 0.145 residues, providing an independent kinematic counterpart to the high thermodynamic cooperativity of the Zimm--Bragg model. This sub-residue spatial narrowness indicates an intrinsic observational constraint governed by the Gabor uncertainty principle, whereby any macroscopic spectral probe tends to blur the microscopic phase boundary, suggesting that the boundary ambiguity in structural biology is not merely algorithmic but reflects a physical resolution limit inherent to the biopolymer lattice.