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

<title>Abstract</title> <p>Protein function may depend not only on endpoint conformations but also on the ordered deformation histories through 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 visible structures while retaining distinct internal transport histories. Current state-centered or endpoint-centered representations do not always preserve this order-sensitive information. The practical motivation is therefore to provide a foundation for future descriptors of protein deformation trajectories that can 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 quater-nionic variables, with infinitesimal rotation encoded by Ω(ℓ) = 2 q(ℓ) −1 ∂ ℓ q(ℓ). Ordered concatenation of admissible deformation paths generates a noncommu-tative 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. MSC Classification: Primary 46L87 , 58J52; Secondary 81Q10 , 53C80 , 92C40</p>