Hydro-Responsive Curling of the Resurrection Plant Selaginella lepidophylla

The spirally arranged stems of the spikemoss Selaginella lepidophylla, an ancient resurrection plant, compactly curl into a nest-ball shape upon dehydration. Due to its spiral phyllotaxy, older outer stems on the plant interlace and envelope the younger inner stems forming the plant centre. Stem curling is a morphological mechanism that limits photoinhibitory and thermal damages the plant might experience in arid environments. Here, we investigate the distinct conformational changes of outer and inner stems of S. lepidophylla triggered by dehydration. Outer stems bend into circular rings in a relatively short period of desiccation, whereas inner stems curl slowly into spirals due to hydro-actuated strain gradient along their length. This arrangement eases both the tight packing of the plant during desiccation and its fast opening upon rehydration. The insights gained from this work shed light on the hydro-responsive movements in plants and might contribute to the development of deployable structures with remarkable shape transformations in response to environmental stimuli

Global Energy Matching Method for Atomistic-to-Continuum Modeling of Self-Assembling Biopolymer Aggregates

This paper studies mathematical models of biopolymer supramolecular aggregates that are formed by the self-assembly of single monomers. We develop a new multiscale numerical approach to model the structural properties of such aggregates. This theoretical approach establishes micro-macro relations between the geometrical and mechanical properties of the monomers and supramolecular aggregates. Most atomistic-to-continuum methods are constrained by a crystalline order or a periodic setting and therefore cannot be directly applied to modeling of soft matter. By contrast, the energy matching method developed in this paper does not require crystalline order and, therefore, can be applied to general microstructures with strongly variable spatial correlations. In this paper we use this method to compute the shape and the bending stiffness of their supramolecular aggregates from known chiral and amphiphilic properties of the short chain peptide monomers. Numerical implementation of our approach demonstrates consistency with results obtained by molecular dynamics simulations

Discrete Curvature and Torsion from Cross-Ratios

Motivated by a M\"obius invariant subdivision scheme for polygons, we study a curvature notion for discrete curves where the cross-ratio plays an important role in all our key definitions. Using a particular M\"obius invariant point-insertion-rule, comparable to the classical four-point-scheme, we construct circles along discrete curves. Asymptotic analysis shows that these circles defined on a sampled curve converge to the smooth curvature circles as the sampling density increases. We express our discrete torsion for space curves, which is not a M\"obius invariant notion, using the cross-ratio and show its asymptotic behavior in analogy to the curvature

Asymptotic Analysis of Discrete Normals and Curvatures of Polylines

Accurate estimations of geometric properties of a smooth curve from its discrete approximation are important for many computer graphics and computer vision applications. To assess and improve the quality of such an approximation, we assume that the curve is known in general form. Then we can represent the curve by a Taylor series expansion and compare its geometric properties with the corresponding discrete approximations. In turn we can either prove convergence of these approximations towards the true properties as the edge lengths tend to zero, or we can get hints on how to eliminate the error. In this paper, we propose and study discrete schemes for estimating tangent and normal vectors as well as for estimating curvature and torsion of a smooth 3D curve approximated by a polyline. Thereby we make some interesting findings about connections between (smooth) classical curves and certain estimation schemes for polylines