We studied the mechanical process of seed pods opening in Bauhinia variegate and found a chirality-creating mechanism, which turns an initially flat pod valve into a helix. We studied configurations of strips cut from pod valve tissue and from composite elastic materials that mimic its structure. The experiments reveal various helical configurations with sharp morphological transitions between them. Using the mathematical framework of "incompatible elasticity," we modeled the pod as a thin strip with a flat intrinsic metric and a saddle-like intrinsic curvature. Our theoretical analysis quantitatively predicts all observed configurations, thus linking the pod's microscopic structure and macroscopic conformation. We suggest that this type of incompatible strip is likely to play a role in the self-assembly of chiral macromolecules and could be used for the engineering of synthetic self-shaping devices.

We provide a geometric-mechanical model for calculating equilibrium configurations of chemical systems that self-assemble into chiral ribbon structures. The model is based on incompatible elasticity and uses dimensionless parameters to determine the equilibrium configurations. As such, it provides universal curves for the shape and energy of self-assembled ribbons. We provide quantitative predictions for the twisted-to-helical transition, which was observed experimentally in many systems, and demonstrate it with synthetic ribbons made of responsive gels. In addition, we predict the bi-stability of wide ribbons and also show how geometrical frustration can cause arrest of ribbon widening. Finally, we show that the model's predictions provide explanations for experimental observations in different chemical systems.

Author contributions: S.A. and M.P. designed research; S.A., M.S.B., A.A.-D., and M.P. performed research; S.A., M.S.B., and A.A.-D. analyzed data; and S.A. and M.P. wrote the paper.The authors declare no conflict of interest.

Differentiation into well-defined patterns and tissue growth are recognized as key processes in organismal development. However, it is unclear whether patterns are passively, homogeneously dilated by growth or whether they remodel during tissue expansion. Leaf vascular networks are well-fitted to investigate this issue, since leaves are approximately two-dimensional and grow manyfold in size. Here we study experimentally and computationally how vein patterns affect growth. We first model the growing vasculature as a network of viscoelastic rods and consider its response to external mechanical stress. We use the so-called texture tensor to quantify the local network geometry and reveal that growth is heterogeneous, resembling non-affine deformations in composite materials. We then apply mechanical forces to growing leaves after veins have differentiated, which respond by anisotropic growth and reorientation of the network in the direction of external stress. External mechanical stress appears to make growth more homogeneous, in contrast with the model with viscoelastic rods. However, we reconcile the model with experimental data by incorporating randomness in rod thickness and a threshold in the rod growth law, making the rods viscoelastoplastic. Altogether, we show that the higher stiffness of veins leads to their reorientation along external forces, along with a reduction in growth heterogeneity. This process may lead to the reinforcement of leaves against mechanical stress. More generally, our work contributes to a framework whereby growth and patterns are coordinated through the differences in mechanical properties between cell types.

epithelium of T. adhaerens and the "active cohesion" 1 hypothesis. 2One Sentence Summary: We report the fastest epithelial cell contractions known to date, show they fit 3 the kinematics arising from current cytoskeletal models, and suggest the extreme tissue dynamics is a means 4 to actively avoid rupture. Abstract: 10By definition of multi-cellularity, all animals need to keep their cells attached and intact, despite internal 11 and external forces. Cohesion between epithelial cells provides this key feature. In order to better 12 understand fundamental limits of this cohesion, we study the epithelium mechanics of an ultra-thin (~25 13 um) primitive marine animal Trichoplax adhaerens, composed essentially of two flat epithelial layers. With 14 no known extra-cellular-matrix and no nerves or muscles, T. adhaerens was claimed the "simplest known 15 living animal", yet is still capable of coordinated locomotion and behavior. Here we report the discovery 16 of the fastest epithelial cellular contractions to date to be found in T. adhaerens dorsal epithelium (50% 17 shrinkage of apical cell area within one second, at least an order of magnitude faster than known examples). 18Live imaging reveals emergent contractile patterns that are mostly sporadic single-cell events, but also 19 include propagating contraction waves across the tissue. We show that cell contraction speed can be 20 explained by current models of non-muscle actin-myosin bundles without load, while the tissue architecture 21 and unique mechanical properties are softening the tissue, minimizing the load on a contracting cell. We 22 propose a hypothesis, in which the physiological role of the contraction dynamics is to avoid tissue rupture 23 ("active cohesion"), a novel concept that can be further applied to engineering of active materials. 24 . CC-BY-NC-ND 4.0 International license peer-reviewed) is the author/funder. It is made available under a The copyright holder for this preprint (which was not . http://dx.doi.org/10.1101/258103 doi: bioRxiv preprint first posted online Feb. 16, 2018; Page 2 of 53 Main Text: 25Epithelial apical contractions are mostly known to occur during embryonic developmental stages (5)(6)(7)(8). 26These contractions are slow (each contraction lasting minutes to hours) and precisely patterned in both 27 space and time. They play a crucial role in the morphogenesis of the embryo-and then desist. The molecular 28 and mechanical mechanism of contraction in these non-muscle cells, as well as their tissue level control (9-29 11), are under intensive investigation (9, 10,(12)(13)(14)(15)(16)(17). Recently, in vitro spreading assays of adult epithelial 30 monolayers showed similarly-slow cellular contractions, though not as canonically patterned (18)(19)(20)(21)(22)(23). The 31 triggering and patterning mechanisms of these contractions in somatic tissues are still unknown. 32In an evolutionary context, cellular contractions have been suggested to play a role in cohesion and 33 coordination in early animals lacking neurons or ...

SummaryThree-dimensional geometry of leaf margins is an important shape characteristic to distinguish different leaf phenotypes. Novel geometrical methods were defined, measured, and used to quantify waviness and lobiness of leaves.

Growth pattern dynamics lie at the heart of morphogenesis. Here, we investigate the growth of plant leaves. We compute the conformal transformation that maps the contour of a leaf at a given stage onto the contour of the same leaf at a later stage. Based on the mapping we predict the local displacement field in the leaf blade and find it to agree with the experimentally measured displacement field to 92%. This approach is applicable to any two-dimensional system with locally isotropic growth, enabling the deduction of the whole growth field just from observation of the tissue contour.

A growing leaf is a prototypical active solid, as its active units, the cells, locally deform during the out-of-equilibrium process of growth. During this local growth, leaves increase their area by orders of magnitude, yet maintain a proper shape, usually flat. How this is achieved in the lack of a central control, is unknown. Here we measure the in-plane growth tensor of Tobacco leaves and study the statistics of growth-rate, isotropy and directionality. We show that growth strongly fluctuates in time and position, and include multiple shrinkage events. We identify the characteristic scales of the fluctuations. We show that the area-growth distribution is broad and non-Gaussian, and use multiscale statistical methods to show how growth homogenizes at larger/longer scales. In contrast, we show that growth isotropy does not homogenize in time. Mechanical analysis shows that with such growth statistics, a leaf can stay flat only if the fluctuations are regulated/correlated.