Quaternionic response geometry for proteins: toward a noncommutative theory of ordered deformations

Abstract 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 quaternionic variables, with infinitesimal rotation encoded by $$ \\Omega (\\ell )=2\\,q(\\ell )^{-1}\\partial _\\ell q(\\ell ) $$ . 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 $$\\alpha $$ -helix shows how localized pitch and bending perturbations can yield similar endpoint descriptors while producing a nonzero endpoint-derived ordered-transport discrepancy. At the formal level, the framework separates an order-sensitive transport-memory sector, lost under a commutative shadow, from a spectral-response sector that remains visible.