{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,23]],"date-time":"2026-10-23T19:22:55Z","timestamp":1792783375477,"version":"4.5.0"},"reference-count":31,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2026,9,21]],"date-time":"2026-09-21T00:00:00Z","timestamp":1789948800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2026,9,21]],"date-time":"2026-09-21T00:00:00Z","timestamp":1789948800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"name":"R\u00c3\u0192\u00c2\u00a9gion R\u00c3\u0192\u00c2\u00a9union and European Union","award":["DESVE\/20251283; Ref. D2026\/3446"],"award-info":[{"award-number":["DESVE\/20251283; Ref. 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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\n                    <jats:inline-formula>\n                      <jats:tex-math>$$ \\Omega (\\ell )=2\\,q(\\ell )^{-1}\\partial _\\ell q(\\ell ) $$<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    . Ordered concatenation of admissible deformation paths generates a noncommutative transport algebra, recording that deformation\n                    <jats:inline-formula>\n                      <jats:tex-math>$$A$$<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    followed by\n                    <jats:inline-formula>\n                      <jats:tex-math>$$B$$<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    need not be equivalent to\n                    <jats:inline-formula>\n                      <jats:tex-math>$$B$$<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    followed by\n                    <jats:inline-formula>\n                      <jats:tex-math>$$A$$<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    . 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\n                    <jats:inline-formula>\n                      <jats:tex-math>$$\\alpha $$<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    -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.\n                  <\/jats:p>","DOI":"10.1007\/s12064-026-00494-7","type":"journal-article","created":{"date-parts":[[2026,9,21]],"date-time":"2026-09-21T18:29:17Z","timestamp":1790015357000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Quaternionic response geometry for proteins: toward a noncommutative theory of ordered deformations"],"prefix":"10.1007","volume":"145","author":[{"given":"Xiaoting","family":"Chen","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Chon-Fai","family":"Kam","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yu","family":"Li","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"David","family":"Medina-Ortiz","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Cedric","family":"Damour","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jean Pierre","family":"Chabriat","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Alain","family":"Miranville","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Miloud","family":"Bessafi","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Frederic","family":"Cadet","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2026,9,21]]},"reference":[{"key":"494_CR1","doi-asserted-by":"publisher","first-page":"493","DOI":"10.1038\/s41586-024-07487-w","volume":"630","author":"J Abramson","year":"2024","unstructured":"Abramson J, Adler J, Dunger J et al (2024) Accurate structure prediction of biomolecular interactions with AlphaFold 3. 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All scientific content, methodological decisions, software development, analyses, and conclusions were conceived, implemented, verified, and validated by the authors.","order":1,"name":"Ethics","label":"Artificial intelligence usage","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare no conflict of interest.","order":2,"name":"Ethics","label":"Conflict of interest","group":{"name":"EthicsHeading","label":"Declarations"}}],"article-number":"45"}}