Quaternionic Response Geometry for Proteins: Toward a Noncommutative Theory of Ordered Deformations

Abstract

Protein function may depend on both endpoint conformations and the ordered deformation histories by which they are reached. This distinction is relevant to allostery, conformational switching, mutation-induced rearrangements, and epistatic effects, where different perturbation sequences may produce similar structures while retaining distinct internal transport histories. Current state- or endpoint-centered representations may not preserve this order-sensitive information. We therefore provide a foundation for descriptors of protein deformation trajectories that distinguish ordered histories even when endpoint conformations are similar. Such descriptors could support analyses of allosteric switching, mutation-order effects, conformational memory, and path-dependent response in molecular-dynamics trajectories, NMR ensembles, structural families, and outputs of geometric generative models. We propose a deformation-first geometric framework based on quaternionic frame transport along the protein backbone. Local backbone frames are lifted to quaternionic variables, with infinitesimal rotation encoded by \(Ω(\ell)=2\,q(\ell)^{-1}\partial_\ell q(\ell)\). Ordered concatenation of admissible deformation paths generates a noncommutative transport algebra, recording that deformation A followed by B need not be equivalent to B followed by A. From this ordered transport layer, we construct a spectral-response layer comprising a global Dirac-type operator, local spectral germs, a renormalized spectral density, and a mixed response form. A minimal realization on an idealized \(α\)-helix shows how localized pitch and bending perturbations can yield similar endpoint descriptors while preserving a nonzero order-memory signal. The framework separates an order-sensitive transport-memory sector, lost under a commutative shadow, from a spectral-response sector that remains visible.

43 pages, including 10 pages of appendices and 2 pages of references.5 figures. Submitted to Theory in Biosciences