How a Soft Rod Wraps around a Rotating Cylinder.

The unique characteristics of helical coils are utilized in nature, manufacturing processes, and daily life. These coils are also pivotal in the development of soft machines, such as artificial muscles and soft grippers. The stability of these helical coils is generally dependent on the mechanical properties of the rods and geometry of the supporting objects. In this Letter, the shapes formed by a flexible, heavy rod wrapping around a slowly rotating rigid cylinder are investigated through a combination of experimental and theoretical approaches. Three distinct morphologies-tight coiling, helical wrapping, and no wrapping-are identified experimentally. These findings are rationalized by numerical simulations and a geometrically nonlinear Kirchhoff rod theory. Despite the frictional contact present, the local shape of the rod is explained by the interplay between bending elasticity, gravity, and the geometry of the system. Our Letter provides a comprehensive physical understanding of the ordered morphology of soft threads and rods. Implications of this understanding are significant for a wide range of phenomena, from the recently discovered wrapping motility mode of bacterial flagella to the design of an octopus-inspired soft gripper.

Erratum: Twist-Induced Snapping in a Bent Elastic Rod and Ribbon [Phys. Rev. Lett. 122, 114301 (2019)].

This corrects the article DOI: 10.1103/PhysRevLett.122.114301.

Tree of motility - A proposed history of motility systems in the tree of life.

Motility often plays a decisive role in the survival of species. Five systems of motility have been studied in depth: those propelled by bacterial flagella, eukaryotic actin polymerization and the eukaryotic motor proteins myosin, kinesin and dynein. However, many organisms exhibit surprisingly diverse motilities, and advances in genomics, molecular biology and imaging have showed that those motilities have inherently independent mechanisms. This makes defining the breadth of motility nontrivial, because novel motilities may be driven by unknown mechanisms. Here, we classify the known motilities based on the unique classes of movement-producing protein architectures. Based on this criterion, the current total of independent motility systems stands at 18 types. In this perspective, we discuss these modes of motility relative to the latest phylogenetic Tree of Life and propose a history of motility. During the ~4 billion years since the emergence of life, motility arose in Bacteria with flagella and pili, and in Archaea with archaella. Newer modes of motility became possible in Eukarya with changes to the cell envelope. Presence or absence of a peptidoglycan layer, the acquisition of robust membrane dynamics, the enlargement of cells and environmental opportunities likely provided the context for the (co)evolution of novel types of motility.

Mechanics of a Snap Fit.

Snap fits are versatile mechanical designs in industrial products that enable the repeated assembling and disassembling of two solid parts. This important property is attributed to a fine balance between geometry, friction, and bending elasticity. In this Letter, we combine theory, simulation, and experiment to reveal the fundamental physical principles of snap-fit functions in the simplest possible setup consisting of a rigid cylinder and a thin elastic shell. We construct a phase diagram using geometric parameters and identify four distinct mechanical phases. We develop analytical predictions based on the linear elasticity theory combined with the law of static friction and rationalize the numerical and experimental results. The study reveals how an operational asymmetry of snap fits (i.e., easy to assemble but difficult to disassemble) emerges from an exquisite combination of geometry, elasticity, and friction and suggests optimization of the tunable functionalities for a range of mechanical designs.

Twist-Induced Snapping in a Bent Elastic Rod and Ribbon.

Snapping of a slender structure is utilized in a wide range of natural and manmade systems, mostly to achieve rapid movement without relying on musclelike elements. Although several mechanisms for elastic energy storage and rapid release have been studied in detail, a general understanding of the approach to design such a kinetic system is a key challenge in mechanics. Here we study a twist-driven buckling and fast flip dynamics of a geometrically constrained ribbon by combining experiments, numerical simulations, and an analytical theory. We identify two distinct types of shape transitions: A narrow ribbon snaps, and a wide ribbon forms a pair of localized helices. We construct a phase diagram and explain the origin of the boundary, which is determined largely by the geometry. We quantify the effects of gravity and clarify the timescale dictating the rapid flipping. Our study reveals the unique role of geometric twist-bend coupling in the fast dynamics of a thin constrained structure, which has implications for a wide range of biophysical and applied physical problems.

Slip Morphology of Elastic Strips on Frictional Rigid Substrates.

The morphology of an elastic strip subject to vertical compressive stress on a frictional rigid substrate is investigated by a combination of theory and experiment. We find a rich variety of morphologies, which-when the bending elasticity dominates over the effect of gravity-are classified into three distinct types of states: pinned, partially slipped, and completely slipped, depending on the magnitude of the vertical strain and the coefficient of static friction. We develop a theory of elastica under mixed clamped-hinged boundary conditions combined with the Coulomb-Amontons friction law and find excellent quantitative agreement with simulations and controlled physical experiments. We also discuss the effect of gravity in order to bridge the difference in the qualitative behaviors of stiff strips and flexible strings or ropes. Our study thus complements recent work on elastic rope coiling and takes a significant step towards establishing a unified understanding of how a thin elastic object interacts vertically with a solid surface.

Stretching an Elastic Loop: Crease, Helicoid, and Pop Out.

Under geometric constraints, a thin structure can respond to an external loading in an unexpected way. A paper strip that is looped and pulled can be used for simple experimentation of such a process. Here, we study this seemingly very simple phenomenon in detail by combing experiments and theory. We identify the three types of shape transitions, i.e., crease, helicoid, and pop out, from a stretched loop, and classify them in terms of parameters characterizing a ribbon geometry. We establish a transition-type diagram by compiling our extensive experimental data. Numerical simulations based on the Kirchhoff rod theory and scaling argument reveal that the pop-out transition is governed by a single characteristic length ξ∼b^{2}/h, where b and h are the ribbon's width and thickness, respectively. We also reveal the key roles of other physical effects such as the anisotropy of the bending elasticity and plastic deformations upon the shape selection mechanisms of a constraint ribbon.

Structural mechanics and helical geometry of thin elastic composites.

Helices are ubiquitous in nature, and helical shape transition is often observed in residually stressed bodies, such as composites, wherein materials with different mechanical properties are glued firmly together to form a whole body. Inspired by a variety of biological examples, the basic physical mechanism responsible for the emergence of twisting and bending in such thin composite structures has been extensively studied. Here, we propose a simplified analytical model wherein a slender membrane tube undergoes a helical transition driven by the contraction of an elastic ribbon bound to the membrane surface. We analytically predict the curvature and twist of an emergent helix as functions of differential strains and elastic moduli, which are confirmed by our numerical simulations. Our results may help understand shapes observed in different biological systems, such as spiral bacteria, and could be applied to novel designs of soft machines and robots.

Helical flow of surface protein required for bacterial gliding motility.

Cells of Flavobacterium johnsoniae and of many other members of the phylum Bacteroidetes exhibit rapid gliding motility over surfaces by a unique mechanism. These cells do not have flagella or pili; instead, they rely on a novel motility apparatus composed of Gld and Spr proteins. SprB, a 669-kDa cell-surface adhesin, is required for efficient gliding. SprB was visualized by electron microscopy as thin 150-nm-long filaments extending from the cell surface. Fluorescence microscopy revealed movement of SprB proteins toward the poles of the cell at ∼2 µm/s. The fluorescent signals appeared to migrate around the pole and continue at the same speed toward the opposite pole along an apparent left-handed helical closed loop. Movement of SprB, and of cells, was rapidly and reversibly blocked by the addition of carbonyl cyanide m-chlorophenylhydrazone, which dissipates the proton gradient across the cytoplasmic membrane. In a gliding cell, some of the SprB protein appeared to attach to the substratum. The cell body moved forward and rotated with respect to this point of attachment. Upon reaching the rear of the cell, the attached SprB often was released from the substratum, and apparently recirculated to the front of the cell along a helical path. The results suggest a model for Flavobacterium gliding, supported by mathematical analysis, in which adhesins such as SprB are propelled along a closed helical loop track, generating rotation and translation of the cell body.

Indentation of an elastic arch on a frictional substrate: Pinning, unfolding, and snapping.

In this study, we investigate the morphology and mechanics of a naturally curved elastic arch loaded at its center and frictionally supported at both ends on a flat, rigid substrate. Through systematic numerical simulations, we classify the observed behaviors of the arch into three configurations in terms of the arch geometry and the coefficient of static friction with the substrate. A linear theory is developed based on a planar elastica model combined with Amontons-Coulomb's frictional law, which quantitatively explains the numerically constructed phase diagram. The snapping transition of a loaded arch in a sufficiently large indentation regime, which involves a discontinuous force jump, is numerically observed. The proposed model problem enables a fully analytical investigation and demonstrates a rich variety of mechanical behaviors owing to the interplay among elasticity, geometry, and friction. This study provides a basis for understanding more common but complex systems, such as a cylindrical shell subjected to a concentrated load and simultaneously supported by frictional contact with surrounding objects.

Bidirectional bacterial gliding motility powered by the collective transport of cell surface proteins.

The gliding motility of Flavobacterium johnsoniae is driven by moving surface adhesive proteins. Recently, these motility components were observed to travel along a closed loop on the cell surface. The mechanism by which such moving surface adhesins give rise to cell motion remains unknown. On the basis of the unique motility properties of F. johnsoniae, we present a generic model for bidirectional motion of rigidly coupled adhesins, which are propelled in opposite directions. Using analytical and numerical methods, we demonstrate that, for a sufficiently large adhesin speed, bidirectional motion arises from spontaneous symmetry breaking. The model also predicts that, close to the bifurcation point, a weak asymmetry in the binding dynamics is sufficient to facilitate directed motility, indicating that the direction of motion could be sensitively regulated internally in response to inhomogeneity of the environment.

Hierarchical helical order in the twisted growth of plant organs.

The molecular and cellular basis of left-right asymmetry in plant morphogenesis is a fundamental issue in biology. A rapidly elongating root or hypocotyl of twisting mutants of Arabidopsis thaliana exhibits a helical growth with a handedness opposite to that of the underlying cortical microtubule arrays in epidermal cells. However, how such a hierarchical helical order emerges is currently unknown. We propose a model for investigating macroscopic chiral asymmetry in Arabidopsis mutants. Our elastic model suggests that the helical pattern observed is a direct consequence of the simultaneous presence of anisotropic growth and tilting of cortical microtubule arrays. We predict that the root helical pitch angle is a function of the microtubule helical angle and elastic moduli of the tissues. The proposed model is versatile and is potentially important for other biological systems ranging from protein fibrous structures to tree trunks.

Hydrodynamics of helical-shaped bacterial motility.

To reveal the underlying hydrodynamic mechanism for the directed propulsion of the bacterium Spiroplasma, we formulate a coarse-grained elastic polymer model with domains of alternating helicities along the contour. Using hydrodynamic simulations and analytic arguments, we show that the propagation of helical domain walls leads to the directed propulsion of the cell body opposite to the domain-wall traveling direction. Several key features of Spiroplasma motility are reproduced by our model. We in particular show that the helical pitch angle observed for Spiroplasma meliferum, psi=35 degrees , is optimized for maximal swimming speed and energy-conversion efficiency. Our analytic theory based on the slender-body hydrodynamic approximation agrees very well with our numerical data demonstrating how the chirality switch propagating along the helical cell body is converted to a translational thrust for the cell body itself. We in detail consider thermal effects on the propulsion efficiency in the form of orientational fluctuations and conformational fluctuations of the helix shape. The body length dependence of the cell motility is studied numerically and compared to our approximate analytic theory. For fixed pitch angle psi=35 degrees , the swimming speed is maximized at a ratio of cell-body length to domain length of about 2-3, which are typical values for real cells. We also propose simple analytic arguments for an enhancement of the swimming velocity with increasing solution viscosity by taking into account the effects of transient confinement of a helical cell body in a polymeric meshwork. Comparison with a generalized theory for the swimming speed of flagellated bacteria in polymeric meshworks shows that the presence of a finite-sized bacterial head gives rise to a maximal swimming speed at a finite solution viscosity, whereas in the absence of a head the swimming speed monotonically increases with increasing viscosity.

Structure, design, and mechanics of a paper spring.

A paper spring is a simple paper craft popular with children. It can be constructed by interfolding and gluing two long strips of paper of equal sizes, with the simplest possible crease patterns. In addition to its curious springy response, this origami-based composite exhibits a twist deformation during its extension. Although its interlocking structure is expected to underly the strong stretch-twist coupling, a detailed understanding of it remains elusive. Here we quantify the kinematics and mechanics of a paper spring during its extensional actuation by combining experimental, numerical, and analytical approaches. We directly link the nonlinear mechanics of a paper spring with its structural design and the sheet elasticity. We show that the unique interlocking provides an enhanced structural rigidity because the thin sheets suffer from geometric frustrations and must locally bend and stretch during extension. This structural design allows for a reversible transformation between the rotatory and linear motions solely by controlling forces and moments applied at the ends of the structure. Such deployment kinematics could provide a unique avenue of the mode conversion for potential applications and will broaden the possibilities of future designs of origami-based springs with tunable functionalities.

Snap-buckling in asymmetrically constrained elastic strips.

When a flat elastic strip is compressed along its axis, it is bent in one of two possible directions via spontaneous symmetry breaking, forming a cylindrical arc. This is a phenomenon well known as Euler buckling. When this cylindrical section is pushed in the other direction, the bending direction can suddenly reverse. This instability is called "snap-through buckling" and is one of the elementary shape transitions in a prestressed thin structure. Combining experiments and theory, we study snap-buckling of an elastic strip with one end hinged and the other end clamped. These asymmetric boundary constraints break the intrinsic symmetry of the strip, generating mechanical behaviors, including largely hysteretic but reproducible force responses and switchlike discontinuous shape changes. We establish the set of exact analytical solutions to fully explain all our major experimental and numerical findings. Asymmetric boundary conditions arise naturally in diverse situations when a thin object is in contact with a solid surface at one end. The introduction of asymmetry through boundary conditions yields new insight into complex and programmable functionalities in material and industrial design.

Plant-inspired pipettes.

The female sex organ of the liverwort (Marchantia polymorpha) has a characteristic parasol-like form highly suitable for collecting water droplets containing sperm for fertilization. Motivated by this observation and using three-dimensional printing techniques, we develop a parasol-like rigid object that can grab, transport and release water droplets of a maximum size of about 1 cm. By combining experiments and scaling theory, we quantify the object's fundamental wetting and fluid dynamical properties. We construct a stability phase diagram and suggest that it is largely insensitive to properties of liquids such as surface tension and viscosity. A simple scaling argument is developed to explain the phase boundary. Our study provides basic design rules of a simple pipette-like device with bubble-free capture and drop of liquids, which can be used in laboratory settings and has applications within soft robotics. Through systematic experimental investigations, we suggest the optimal design criteria of the liverwort-inspired object to achieve maximal pipetting performance. We also provide, based on our scalable model experiments, a biological implication for the mechanistic advantage of this structure in liverwort reproduction.

Emergence of a new valence-ordered structure and collapse of the magnetic order under high pressure in EuPtP.

The layered hexagonal EuPtP is a rare substance that exhibits two successive valence transitions occurring simultaneously with valence ordering transitions and an antiferromagnetic order. Anticipating that the application of pressure to this sample would induce a new valence-ordered structure and/or a new phenomenon associated with valence fluctuation, we examined the electrical resistivity ρ, the Eu L3-edge x-ray absorption spectroscopy, and the powder x-ray diffraction under high pressure. We found a new valence transition at around P = 2.5 GPa. After the transition, a new valence-ordered structure is realized at the lowest temperature. The valence-ordered structure is inferred to be stacking of [Formula: see text] (2+: Eu2+ layer, 3+: Eu3+ layer) along the c-axis. Upon further increases in pressure, the valence-ordered structure is suppressed and another valance-ordered phase is realized up to P = 6 GPa. The antiferromagnetic order collapses in the pressure range between 6 GPa and 8 GPa.

Nonlinear elasticity of single collapsed polyelectrolytes.

Nonlinear elastic responses of short and stiff polyelectrolytes are investigated by dynamic simulations on a single-molecule level. When a polyelectrolyte condensate undergoes a mechanical unfolding, two types of force-extension curves--i.e., a force plateau and a stick-release pattern--are observed depending on the strength of the electrostatic interaction. We provide a physical interpretation of such force-extension behavior in terms of intramolecular structures of the condensates. We also describe charge distributions of counterions condensed onto a polyelectrolyte, which clarify formation of one-dimensional strongly correlated liquid at large Coulomb coupling regime. These findings may provide significant insights into the relationship between a molecular elasticity and a molecular mechanism of like-charge attractions observed in a wide range of charged biopolymer systems.

Combination of MD Simulations with Two-State Kinetic Rate Modeling Elucidates the Chain Melting Transition of Phospholipid Bilayers for Different Hydration Levels.

The phase behavior of membrane lipids plays an important role in the formation of functional domains in biological membranes and crucially affects molecular transport through lipid layers, for instance, in the skin. We investigate the thermotropic chain melting transition from the ordered Lß phase to the disordered Lα phase in membranes composed of dipalmitoylphosphatidylcholine (DPPC) by atomistic molecular dynamics simulations in which the membranes are subject to variable heating rates. We find that the transition is initiated by a localized nucleus and followed by the propagation of the phase boundary. A two-state kinetic rate model allows characterizing the transition state in terms of thermodynamic quantities such as transition state enthalpy and entropy. The extrapolated equilibrium melting temperature increases with reduced membrane hydration and thus in tendency reproduces the experimentally observed dependence on dehydrating osmotic stress.

Shear-induced quench of long-range correlations in a liquid mixture.

A static correlation function of concentration fluctuations in a (dilute) binary liquid mixture subjected to both a concentration gradient and uniform shear flow is investigated within the framework of fluctuating hydrodynamics. It is shown that a well-known |c|(2)/k(4) long-range correlation at large wave numbers k crosses over to a weaker divergent one at wave numbers satisfying k<(gamma;/D)(1/2), while an asymptotic shear-controlled power-law dependence is found at much smaller wave numbers given by k<<(gamma;/nu)(1/2), where c, gamma;, D, and nu are the mass concentration, the rate of the shear, the mass diffusivity, and the kinematic viscosity of the mixture, respectively. The result will provide the possibility to observe the shear-induced suppression of a long-range correlation experimentally by using, for example, a low-angle light scattering technique.