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This paper introduces a mathematical framework using quaternionic geometry to track the ordered sequence of structural deformations in proteins, not just their final configurations. The authors demonstrate that different sequences of perturbations can lead to similar endpoint structures while retaining distinct internal transformation histories, which is important for understanding protein allostery, conformational switching, and mutation effects. A proof-of-concept application to an idealized alpha-helix shows the framework can detect differences in deformation pathways that conventional endpoint-centered methods miss.
Why it matters
The framework could improve understanding of how proteins function through their dynamic conformational changes rather than static structures alone. This may have implications for predicting mutation effects, designing allosteric drugs, and understanding epistatic interactions where the order of genetic changes affects the outcome.
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⚠️ Preprint – Noch nicht peer-reviewed
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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.
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.
Source: Quaternionic Response Geometry for Proteins: Toward a Noncommutative Theory of Ordered Deformations