Spiral geometries commonly occur in natural and engineered systems and are fundamentally described by curvature and torsion. In deformation-dominated systems, these variables evolve dynamically, requiring a continuum mechanical framework to link geometry and deformation. This study focused on refractile bodies (R-bodies), protein supramolecular assemblies that undergo reversible roll-spiral transformations in response to stimuli such as pH changes. Although multiple R-body types with distinct morphologies and unrolling behaviours were experimentally identified, their deformation mechanisms lack quantitative theoretical descriptions. We proposed a deformation-gradient-tensor-based continuum model incorporating geometrical mapping from the rolled to spiral state within a unified framework. The model successfully reconstructed macroscopic deformation behaviours of types 51, 7, and Pa R-bodies by capturing differences in unrolling behaviours, tapered geometry, and spatio-temporal evolution. The analysis revealed that deformation proceeds through a coupled process in which the curvature decreases via straightening, while the torsion increases by twisting. Importantly, the framework connected the macroscopic morphology with microscopic lattice deformation, enabling quantitative inference of lattice intervals and angles. The proposed comprehensive geometric model of the R-body roll-spiral transformation offers a general mathematical foundation for understanding deformation-driven spiral transformations in soft matter systems.
Tsugawa, S., Kikuchi, K., Date, K., Nonoyama, T., Kang, Z., Ueno, T.
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