We use experiments and numerical simulations to study the rapid buckling of thinwalled cones as they impact a solid surface at high velocities. The buildup of air pressure inside the cone localizes the deformations to the impacting interface with the solid surface, leading to the hierarchical formation of an ordered pattern of small rhomboidal cells. In contrast, when the inner air pressure is not allowed to develop, the ordered pattern is destabilized and the cone collapses in a highly disordered state on long length scales. Numerical simulations confirm that the transition between ordered and disordered crumpling is governed by the competition between the elastic deformation energy of the shells and the work required to pressurize the air. Our results show how dynamic stabilization via tensioning suppresses long wavelength subcritical instabilities in shells and leads to the localization and propagation of short wavelength patterns.

Recent progress in additive manufacturing and materials engineering has led to a surge of interest in shape-changing plate and shell-like structures. Such structures are typically printed in a planar configuration and, when exposed to an ambient stimulus such as heat or humidity, swell into a desired three-dimensional geometry. Viewed through the lens of differential geometry and elasticity, the application of the physical stimulus can be understood as a local change in the metric of a two dimensional surface embedded in three dimensions. To relieve the resulting elastic frustration, the structure will generally bend and buckle out-of-plane. Here, we propose a numerical approach to convert the discrete geometry of filament bilayers, associated with print paths of inks with given material properties, into continuous plates with inhomogeneous growth patterns and thicknesses. When subject to prescribed growth anisotropies, we can then follow the evolution of the shapes into their final form. We show that our results provide a good correspondence between experiments and simulations, and lead to a framework for the prediction and design of shape-changing structures.

We consider meniscus instabilities in thin elastic layers perfectly adhered to, and confined between, much stiffer bodies. When the free boundary associated with the meniscus of the elastic layer recedes into the layer, for example by pulling the stiffer bodies apart or injecting air between them, then the meniscus will eventually undergo a purely elastic instability in which fingers of air invade the layer. Here we show that the form of this instability is identical in a range of different loading conditions, provided only that the thickness of the meniscus, a, is small compared to the in-plane dimensions and to two emergent in-plane length scales that arise if the substrate is soft or if the layer is compressible. In all such situations, we predict that the instability will occur when the meniscus has receded by approximately 1.27a, and that the instability will have wavelength l E 2.75a. We illustrate this by also calculating the threshold for fingering in a thin wedge of elastic material bonded to two rigid plates that are pried apart, and the threshold for fingering when a flexible plate is peeled from an elastic layer that glues the plate to a rigid substrate

Inspired by the question of quantifying wing shape, we propose a computational approach for analysing planar shapes. We first establish a correspondence between the boundaries of two planar shapes with boundary landmarks using geometric functional data analysis and then compute a landmark-matching curvature-guided Teichmüller mapping with uniform quasi-conformal distortion in the bulk. This allows us to analyse the pair-wise difference between the planar shapes and construct a similarity matrix on which we deploy methods from network analysis to cluster shapes. We deploy our method to study a variety of Drosophila wings across species to highlight the phenotypic variation between them, and Lepidoptera wings over time to study the developmental progression of wings. Our approach of combining complex analysis, computation and statistics to quantify, compare and classify planar shapes may be usefully deployed in other biological and physical systems.

Honeybee Apis mellifera swarms form large congested treehanging clusters made solely of bees attached to each other 1 .How these structures are maintained under the influence of dynamic mechanical forcing is unknown. To address this, we created pendant clusters and subject them to dynamic loads of varying orientation, amplitude, frequency and duration. We find that horizontally shaken clusters adapt by spreading out to form wider, flatter cones that recover their original shape when unloaded. Measuring the response of a cluster to an impulsive pendular excitation shows that flattened cones deform less and relax faster than the elongated ones (that is, they are more stable). Particle-based simulations of a passive assemblage suggest a behavioural hypothesis: individual bees respond to local variations in strain by moving up the strain gradient, which is qualitatively consistent with our observations of individual bee movement during dynamic loading. The simulations also suggest that vertical shaking will not lead to significant differential strains and thus no shape adaptation, which we confirmed experimentally. Together, our findings highlight how a super-organismal structure responds to dynamic loading by actively changing its morphology to improve the collective stability of the cluster at the expense of increasing the average mechanical burden of an individual.

The identification of relationships in complex networks is critical in a variety of scientific contexts. This includes the identification of globally central nodes and analysing the importance of pairwise relationships between nodes. In this paper, we consider the concept of topological proximity (or ‘closeness’) between nodes in a weighted network using the generalized Erdo´´s numbers (GENs). This measure satisfies a number of desirable properties for networks with nodes that share a finite resource. These include: (i) real-valuedness, (ii) non-locality and (iii) asymmetry. We show that they can be used to define a personalized measure of the importance of nodes in a network with a natural interpretation that leads to new methods to measure centrality. We show that the square of the leading eigenvector of an importance matrix defined using the GENs is strongly correlated with well-known measures such as PageRank, and define a personalized measure of centrality that is also well correlated with other existing measures. The utility of this measure of topological proximity is demonstrated by showing the asymmetries in both the dynamics of random walks and the mean infection time in epidemic spreading are better predicted by the topological definition of closeness provided by the GENs than they are by other measures.

Laboratory studies of the heart use cell and tissue cultures to dissect heart function yet rely on animal models to measure pressure and volume dynamics. Here, we report tissue-engineered scale models of the human left ventricle, made of nanofibrous scaffolds that promote native-like anisotropic myocardial tissue genesis and chamber-level contractile function. Incorporating neonatal rat ventricular myocytes or cardiomyocytes derived from human induced pluripotent stem cells, the tissue-engineered ventricles have a diastolic chamber volume of ~500 µl (comparable to that of the native rat ventricle and approximately 1/250 the size of the human ventricle), and ejection fractions and contractile work 50–250 times smaller and 104–108 times smaller than the corresponding values for rodent and human ventricles, respectively. We also measured tissue coverage and alignment, calcium-transient propagation and pressure–volume loops in the presence or absence of test compounds. Moreover, we describe an instrumented bioreactor with ventricular-assist capabilities, and provide a proof-of-concept disease model of structural arrhythmia. The model ventricles can be evaluated with the same assays used in animal models and in clinical settings.

Developing adaptive materials with geometries that change in response to external stimuli provides fundamental insights into the links between the physical forces involved and the resultant morphologies and creates a foundation for technologically relevant dynamic systems1,2 . In particular, reconfigurable surface topography as a means to control interfacial properties3 has recently been explored using responsive gels4 , shape-memory polymers5 , liquid crystals6–8 and hybrid composites9–14, including magnetically active slippery surfaces12–14. However, these designs exhibit a limited range of topographical changes and thus a restricted scope of function. Here we introduce a hierarchical magnetoresponsive composite surface, made by infiltrating a ferrofluid into a microstructured matrix (termed ferrofluid-containing liquid-infused porous surfaces, or FLIPS). We demonstrate various topographical reconfigurations at multiple length scales and a broad range of associated emergent behaviours. An applied magneticfield gradient induces the movement of magnetic nanoparticles suspended in the ferrofluid, which leads to microscale flow of the ferrofluid first above and then within the microstructured surface. This redistribution changes the initially smooth surface of the ferrofluid (which is immobilized by the porous matrix through capillary forces) into various multiscale hierarchical topographies shaped by the size, arrangement and orientation of the confining microstructures in the magnetic field. We analyse the spatial and temporal dynamics of these reconfigurations theoretically and experimentally as a function of the balance between capillary and magnetic pressures15–19 and of the geometric anisotropy of the FLIPS system. Several interesting functions at three different length scales are demonstrated: self-assembly of colloidal particles at the micrometre scale; regulated flow of liquid droplets at the millimetre scale; and switchable adhesion and friction, liquid pumping and removal of biofilms at the centimetre scale. We envision that FLIPS could be used as part of integrated control systems for the manipulation and transport of matter, thermal management, microfluidics and fouling-release materials.

We describe a minimal realization of reversibly programmable matter in the form of a featureless smooth elastic plate that has the capacity to store information in a Braille-like format as a sequence of stable discrete dimples. Simple experiments with cylindrical and spherical shells show that we can control the number, location, and the temporal order of these dimples, which can be written and erased at will. Theoretical analysis of the governing equations in a specialized setting and numerical simulations of the complete equations allow us to characterize the phase diagram for the formation of these localized elastic states, elastic bits (e-bits), consistent with our observations. Given that the inherent bistability and hysteresis in these low-dimensional systems arise exclusively due to the geometrical-scale separation, independent of material properties or absolute scale, our results might serve as alternate approaches to small-scale mechanical memories.

Active stresses can cause instabilities in contractile gels and living tissues. Here we provide a generic hydrodynamic theory that treats these systems as a mixture of two phases of varying activity and different mechanical properties. We find that differential activity between the phases causes a uniform mixture to undergo a demixing instability. We follow the nonlinear evolution of the instability and characterize a phase diagram of the resulting patterns. Our study complements other instability mechanisms in mixtures driven by differential adhesion, differential diffusion, differential growth, and differential motion.

Inside cells, complex metabolic reactions are distributed across the modular compartments of organelles1,2. Reactions in organelles have been recapitulated in vitro by reconstituting functional protein machineries into membrane systems3–5. However, maintaining and controlling these reactions is challenging. Here we designed, built, and tested a switchable, light-harvesting organelle that provides both a sustainable energy source and a means of directing intravesicular reactions. An ATP (ATP) synthase and two photoconverters (plant-derived photosystem II and bacteriaderived proteorhodopsin) enable ATP synthesis. Independent optical activation of the two photoconverters allows dynamic control of ATP synthesis: red light facilitates and green light impedes ATP synthesis. We encapsulated the photosynthetic organelles in a giant vesicle to form a protocellular system and demonstrated optical control of two ATP-dependent reactions, carbon fixation and actin polymerization, with the latter altering outer vesicle morphology. Switchable photosynthetic organelles may enable the development of biomimetic vesicle systems with regulatory networks that exhibit homeostasis and complex cellular behaviors.

Soft slender structures are ubiquitous in natural and artificial systems, in active and passive settings and across scales, from polymers and flagella, to snakes and space tethers. In this paper, we demonstrate the use of a simple and practical numerical implementation based on the Cosserat rod model to simulate the dynamics of filaments that can bend, twist, stretch and shear while interacting with complex environments via muscular activity, surface contact, friction and hydrodynamics. We validate our simulations by solving a number of forward problems involving the mechanics of passive filaments and comparing them with known analytical results, and extend them to study instabilities in stretched and twisted filaments that form solenoidal and plectonemic structures. We then study active filaments such as snakes and other slender organisms by solving inverse problems to identify optimal gaits for limbless locomotion on solid surfaces and in bulk liquids.

Thin adhesive pads used to attach objects to each other often fail catastrophically. Here we consider the nature of failure of such a pad under loading parallel to the adhesive substrate. To determine the modes of failure of the pad and to understand what limits its load bearing capacity, we conduct experiments with finite pads composed of a soft adhesive layer with a stiff backing and load them parallel to the surface of adhesion. We find that two different peeling mechanisms emerge as a function of the slenderness of the adhesive pad: an interfacial peeling mechanism that starts close to the pulling end for very long pads, and an unstable curling mechanism that starts at the opposite end for relatively short pads. A minimal theoretical framework allows us to explain our observations and reveals the adhesive bond stiffness as a dominant parameter in defining the peeling mode. A phase diagram that delineates the different regimes of peeling modes brings our experiments and theory together. Our results suggest that unstable peeling by curling may be more common than previously thought, and could perhaps occur naturally in such examples as the gecko foot.

Reconstructive surgeries often use topological manipulation of tissue to minimize postoperative scarring. The most common version of this, Z-plasty, involves modifying a straight line cut into a Z shape, followed by a rotational transposition of the resulting triangular pedicle flaps, and a final restitching of the wound. This locally reorients the anisotropic stress field and reduces the potential for scarring. We analyze the planar geometry and mechanics of the Z-plasty to quantify the rotation of the overall stress field and the local forces on the restitched cut using theory, simulations, and simple physical Z-plasty experiments with foam sheets that corroborate each other. Our study rationalizes the most typical surgical choice of this angle, and opens the way for a range of surgical decisions by characterizing the stresses along the cut.

The worm-like chain model is a simple continuum model for the statistical mechanics of a flexible polymer subject to an external force. We offer a tutorial introduction to it using three approaches. First, we use a mesoscopic view, treating a long polymer (in two dimensions) as though it were made of many groups of correlated links or “clinks,” allowing us to calculate its average extension as a function of the external force via scaling arguments. We then provide a standard statistical mechanics approach, obtaining the average extension by two different means: the equipartition theorem and the partition function. Finally, we work in a probabilistic framework, taking advantage of the Gaussian properties of the chain in the large-force limit to improve upon the previous calculations of the average extension

Hollow vesicular tissues of various sizes and shapes arise in biological organs such as ears, guts, hearts, brains and even entire organisms. Regulating their size and shape is crucial for their function. Although chemical signaling has been thought to play a role in the regulation of cellular processes that feed into larger scales, it is increasingly recognized that mechanical forces are involved in the modulation of size and shape at larger length scales. Motivated by a variety of examples of tissue cyst formation and size control that show simultaneous growth and size oscillations, we create a minimal theoretical framework for the growth and dynamics of a soft, fluidpermeable, spherical shell. We show that these shells can relieve internal pressure by bursting intermittently, shrinking and re-growing, providing a simple mechanism by which hydraulically gated oscillations can regulate size. To test our theory, we develop an in vitro experimental set-up to monitor the growth and oscillations of a hollow tissue spheroid growing freely or when confined. A simple generalization of our theory to account for irreversible deformations allows us to explain the time scales and the amplitudes of oscillations in terms of the geometry and mechanical properties of the tissue shells. Taken together, our theory and experimental observations show how soft hydraulics can regulate the size of growing tissue shells.

Morphogenesis, the study of how forms arise in biology, has attracted scientists for aeons. A century ago, D’Arcy Wentworth Thompson crystallized this question in his opus On Growth and Form (Thompson, 1917) using a series of biological examples and geometric and physical analogies to ask how biological forms arise during development and across evolution. In light of the advances in molecular and cellular biology since then, a succinct modern view of the question states: how do genes encode geometry?

Fluid-resistance limited transport of vesicles through narrow constrictions is a recurring theme in many biological and engineering applications. Inspired by the motor-driven movement of soft membrane-bound vesicles into closed neuronal dendritic spines, here we study this problem using a combination of passive three-dimensional simulations and a simplified semianalytical theory for the active transport of vesicles forced through constrictions by molecular motors. We show that the motion of these objects is characterized by two dimensionless quantities related to the geometry and to the strength of forcing relative to the vesicle elasticity. We use numerical simulations to characterize the transit time for a vesicle forced by fluid pressure through a constriction in a channel and find that relative to an open channel, transport into a blind end leads to the formation of a smaller forward-flowing lubrication layer that strongly impedes motion. When the fluid pressure forcing is complemented by forces due to molecular motors that are responsible for vesicle trafficking into dendritic spines, we find that the competition between motor forcing and fluid drag results in multistable dynamics reminiscent of the real system. Our study highlights the role of nonlocal hydrodynamic effects in determining the kinetics of vesicular transport in constricted geometries.

Inspired by the differential-growth-driven morphogenesis of leaves, flowers, and other tissues, there is increasing interest in artificial analogs of these shape-shifting thin sheets made of active materials that respond to environmental stimuli such as heat, light, and humidity. But how can we determine the growth patterns to achieve a given shape from another shape? We solve this geometric inverse problem of determining the growth factors and directions (the metric tensors) for a given isotropic elastic bilayer to grow into a target shape by posing and solving an elastic energy minimization problem. A mathematical equivalence between bilayers and curved monolayers simplifies the inverse problem considerably by providing algebraic expressions for the growth metric tensors in terms of those of the final shape. This approach also allows us to prove that we can grow any target surface from any reference surface using orthotropically growing bilayers. We demonstrate this by numerically simulating the growth of a flat sheet into a face, a cylindrical sheet into a flower, and a flat sheet into a complex canyon-like structure.

In vitro system supporting stable segmentation clock oscillations d A cell-density effect controls the onset of oscillations in vitro d Yap-dependent mechanical signal acts as a control parameter for oscillations d Segmentation clock has properties of an excitable system

How termite mounds function to facilitate climate control is still only partially understood. Recent experimental evidence in the mounds of a single species, the south Asian termite Odontotermes obesus, suggests that the daily oscillations of radiant heating associated with diurnal insolation patterns drive convective flow within them. How general this mechanism is remains unknown. To probe this, we consider the mounds of the African termite Macrotermes michaelseni, which thrives in a very different environment. By directly measuring air velocities and temperatures within the mound, we see that the overall mechanisms and patterns involved are similar to that in the south Asian species. However, there are also some notable differences between the physiology of these mounds associated with the temporal variations in radiant heating patterns and CO2 dynamics. Because of the difference between direct radiant heating driven by the position of the sun in African conditions, and the more shaded south Asian environments, we see changes in the convective flows in the two types of mounds. Furthermore, we also see that the south Asian mounds show a significant overturning of stratified gases, once a day, while the African mounds have a relatively uniform concentration of CO2. Overall, our observations show that despite these differences, termite architectures can harness periodic solar heating to drive ventilation inside them in very different environments, functioning as an external lung, with clear implications for human engineering.

Birdsong is the product of the controlled generation of sound embodied in a neuromotor system. From a biophysical perspective, a natural question is that of the difficulty of producing birdsong. To address this, we built a biomimetic syrinx consisting of a stretched simple rubber tube through which air is blown, subject to localized mechanical squeezing with a linear actuator. A large static tension on the tube and small dynamic variations in the localized squeezing allow us to control transitions between three states: a quiescent state, a periodic state and a solitary wave state. The static load brings the system close to threshold for spontaneous oscillations, while small dynamic loads allow for rapid transitions between the states. We use this to mimic a variety of birdsongs via the slow– fast modulated nonlinear dynamics of the physical substrate, the syrinx, regulated by a simple controller. Finally, a minimal mathematical model of the system inspired by our observations allows us to address the problem of song mimicry in an excitable oscillator for tonal songs

Motivated by the redrawing of hot glass into thin sheets, we investigate the shape and stability of a thin viscous sheet that is inhomogeneously stretched in an imposed nonuniform temperature field. We first determine the associated base flow by solving the long-time-scale stretching flow of a flat sheet as a function of two dimensionless parameters: the normalized stretching velocity α and a dimensionless width of the heating zone β. This allows us to determine the conditions for the onset of an out-of-plane wrinkling instability stated in terms of an eigenvalue problem for a linear partial differential equation governing the displacement of the midsurface of the sheet. We show that the sheet can become unstable in two regions that are upstream and downstream of the heating zone where the minimum in-plane stress is negative. This yields the shape and growth rates of the most unstable buckling mode in both regions for various values of the stretching velocity and heating zone width. A transition from stationary to oscillatory unstable modes is found in the upstream region with increasing β, while the downstream region is always stationary. We show that the wrinkling instability can be entirely suppressed when the surface tension is large enough relative to the magnitude of the in-plane stress. Finally, we present an operating diagram that indicates regions of the parameter space that result in a required outlet sheet thickness upon stretching while simultaneously minimizing or suppressing the out-of-plane buckling, a result that is relevant for the glass redraw method used to create ultrathin glass sheets.

It is known that an object translating parallel to a soft wall in a viscous fluid produces hydrodynamic stresses that deform the wall, which in turn results in a lift force on the object. Recent experiments with cylinders sliding under gravity near a soft incline, which confirmed theoretical arguments for the lift force, also reported an unexplained steady-state rotation of the cylinders [B. Saintyves et al., Proc. Natl. Acad. Sci. USA 113, 5847 (2016)]. Motivated by these observations, we show, in the lubrication limit, that an infinite cylinder that translates in a viscous fluid parallel to a soft wall at constant speed and separation distance must also rotate in order to remain free of torque. Using the Lorentz reciprocal theorem, we show analytically that for small deformations of the elastic layer, the angular velocity of the cylinder scales with the cube of the sliding speed. These predictions are confirmed numerically. We then apply the theory to the gravity-driven motion of a cylinder near a soft incline and find qualitative agreement with the experimental observa

components that are required to assemble chromosomes in a test tube (9). Three components were found to be essential: histones, together with assembly factors that they require to load onto DNA; condensin; and topoisomerase II. The latter is an enzyme that allows DNA strands to pass each other, thereby preventing DNA from getting hopelessly tangled up.

Growing plant stems and shoots exhibit a variety of shapes that embody growth in response to various stimuli. Building on experimental observations, we provide a quantitative biophysical theory for these shapes by accounting for the inherent observed passive and active effects: (i) the active controllable growth response of the shoot in response to its orientation relative to gravity, (ii) proprioception, the shoot’s growth response to its own observable current shape, and (iii) the passive elastic deflection of the shoot due to its own weight, which determines the current shape of the shoot. Our theory separates the sensed and actuated variables in a growing shoot and results in a morphospace diagram in terms of two dimensionless parameters representing a scaled local active gravitropic sensitivity, and a scaled passive elastic sag. Our computational results allow us to explain the variety of observed transient and steady morphologies with effective positive, negative and even oscillatory gravitropic behaviours, without the need for ad hoc complex spatio-temporal control strategies in terms of these parameters. More broadly, our theory is applicable to the growth of soft, floppy organs where sensing and actuation are dynamically coupled through growth processes via shape.

Growing plant stems and shoots exhibit a variety of shapes that embody growth in response to various stimuli. Building on experimental observations, we provide a quantitative biophysical theory for these shapes by accounting for the inherent observed passive and active effects: (i) the active controllable growth response of the shoot in response to its orientation relative to gravity, (ii) proprioception, the shoot’s growth response to its own observable current shape, and (iii) the passive elastic deflection of the shoot due to its own weight, which determines the current shape of the shoot. Our theory separates the sensed and actuated variables in a growing shoot and results in a morphospace diagram in terms of two dimensionless parameters representing a scaled local active gravitropic sensitivity, and a scaled passive elastic sag. Our computational results allow us to explain the variety of observed transient and steady morphologies with effective positive, negative and even oscillatory gravitropic behaviours, without the need for ad hoc complex spatio-temporal control strategies in terms of these parameters. More broadly, our theory is applicable to the growth of soft, floppy organs where sensing and actuation are dynamically coupled through growth processes via shape.

Controlled self-assembly of three-dimensional shapes holds great potential for fabrication of functional materials. Their practical realization requires a theoretical framework to quantify and guide the dynamic sculpting of the curved structures that often arise in accretive mineralization. Motivated by a variety of bioinspired coprecipitation patterns of carbonate and silica, we develop a geometrical theory for the kinetics of the growth front that leaves behind thin-walled complex structures. Our theory explains the range of previously observed experimental patterns and, in addition, predicts unexplored assembly pathways. This allows us to design a number of functional base shapes of optical microstructures, which we synthesize to demonstrate their light-guiding capabilities. Overall, our framework provides a way to understand and control the growth and form of functional precipitating microsculptures.

Looping of the initially straight embryonic gut tube is an essential aspect of intestinal morphogenesis, permitting proper placement of the lengthy small intestine within the confines of the body cavity. The formation of intestinal loops is highly stereotyped within a given species and results from differential-growth–driven mechanical buckling of the gut tube as it elongates against the constraint of a thin, elastic membranous tissue, the dorsal mesentery. Although the physics of this process has been studied, the underlying biology has not. Here, we show that BMP signaling plays a critical role in looping morphogenesis of the avian small intestine. We first exploited differences between chicken and zebra finch gut morphology to identify the BMP pathway as a promising candidate to regulate differential growth in the gut. Next, focusing on the developing chick small intestine, we determined that Bmp2 expressed in the dorsal mesentery establishes differential elongation rates between the gut tube and mesentery, thereby regulating the compressive forces that buckle the gut tube into loops. Consequently, the number and tightness of loops in the chick small intestine can be increased or decreased directly by modulation of BMP activity in the small intestine. In addition to providing insight into the molecular mechanisms underlying intestinal development, our findings provide an example of how biochemical signals act on tissue-level mechanics to drive organogenesis, and suggest a possible mechanism by which they can be modulated to achieve distinct morphologies through evolution.

The interaction of a robotic manipulator with unknown soft objects represents a significant challenge for traditional robotic platforms because of the difficulty in controlling the grasping force between a soft object and a stiff manipulator. Soft robotic actuators inspired by elephant trunks, octopus limbs and muscular hydrostats are suggestive of ways to overcome this fundamental difficulty. In particular, the large intrinsic compliance of soft manipulators such as ‘pneu-nets’—pneumatically actuated elastomeric structures—makes them ideal for applications that require interactions with an uncertain mechanical and geometrical environment. Using a simple theoretical model, we show how the geometric and material nonlinearities inherent in the passive mechanical response of such devices can be used to grasp soft objects using force control, and stiff objects using position control, without any need for active sensing or feedback control. Our study is suggestive of a general principle for designing actuators with autonomous intrinsic impedance control.

The body of vertebrate embryos forms by posterior elongation from a terminal growth zone called the Tail Bud (TB). The TB produces highly motile cells forming the presomitic mesoderm (PSM), a tissue playing an important role in elongation movements. PSM cells establish an anterior-posterior cell motility gradient which parallels the degradation of a specific cellular signal (Fgf8) known to be implicated in cell motility. Here, we combine electroporation of fluorescent reporters in the PSM to time-lapse imaging in the chicken embryo to quantify cell diffusive movements along the motility gradient. We show that simple microscopic and macroscopic mechano-chemical models for tissue extension that couple Fgf activity, cell motility and tissue rheology at both the cellular and continuum levels suffice to capture the speed and extent of elongation. These observations explain how the continuous addition of cells that exhibit a gradual reduction in motility combined with lateral confinement can be converted into an oriented movement that drives body elongation. The results of the models compare well with our experimental results, with implications for other elongation processes in the embryo.

Wet starch cracks when it dries inhomogeneously, while hot glass cracks when it cools non-uniformly. In both cases, differential shrinkage induced by drying/cooling from the surface causes superficial cracks to grow perpendicular to the surface in different patterns. In contrast with these observations of bulk cracking in brittle materials, when a soft and homogeneously swollen polymer gel dries, differential strains lead to the peeling of a thin layer that spontaneously tears away from the bulk. Continued drying leads to the process repeating itself, forming a peeled-layered structure. The emergent thickness of the exfoliated layer is a function of both the geometry of the original gel and the physical parameters associated with the drying rate and external temperature. We characterize the experimental conditions under which layer peeling can arise, and use simulations to corroborate these observations. Finally, a minimal theory explains the scaling of the peel thickness, consistent with our experiments

Flat paper stained with water buckles and wrinkles as it swells and deforms out of the original plane. Here we quantify the geometry and mechanics of a strip of paper that swells when it imbibes water from a narrow capillary. Characterizing the hygroexpansive nature of paper shows that thickness-wise swelling is much faster than in-plane water imbibition, leading to a simple picture for the process by which the strip of paper bends out of the plane. We model the out-of-plane deformation using a quasi-static theory and show that our results are consistent with quantitative experiments. C

We consider the dynamics of an elastic sheet as it starts to adhere to a wall, a process that is limited by the viscous squeeze flow of the intervening liquid. Elastohydrodynamic lubrication theory allows us to derive a partial di↵erential equation coupling the elastic deformation of the sheet, the microscopic van der Waals adhesion, and viscous thin film flow. We use a combination of numerical simulations of the governing equation and a scaling analysis to describe the self-similar touchdown of the sheet as it approaches the wall. An analysis of the equation in terms of similarity variables in the vicinity of the touchdown event shows that only the fundamental similarity solution is observed in the time-dependent numerical simulations, consistent with the fact that it alone is stable. Our analysis generalizes similar approaches for rupture in capillary thin film hydrodynamics and suggests experimentally verifiable predictions for a new class of singular flows linking elasticity, hydrodynamics, and adhesion

Relative motion between soft wet solids arises in a number of applications in natural and artificial settings, and invariably couples elastic deformation fluid flow. We explore this in a minimal setting by considering a fluid-immersed negatively buoyant cylinder moving along a soft inclined wall. Our experiments show that there is an emergent robust steady-state sliding regime of the cylinder with an effective friction that is significantly reduced relative to that of rigid fluid-lubricated contacts. A simple scaling approach that couples the cylinder-induced flow to substrate deformation allows us to explain the elastohydrodynamic lift that underlies the self-sustained lubricated motion of the cylinder, consistent with recent theoretical predictions. Our results suggest an explanation for a range of effects such as reduced wear in animal joints and long-runout landslides, and can be couched as a design principle for low-friction interfaces.

Animals use a combination of egocentric navigation driven by the internal integration of environmental cues, interspersed with geocentric course correction and reorientation. These processes are accompanied by uncertainty in sensory acquisition of information, planning and execution. Inspired by observations of dung beetle navigational strategies that show switching between geocentric and egocentric strategies, we consider the question of optimal reorientation rates for the navigation of an agent moving along a preferred direction in the presence of multiple sources of noise. We address this using a model that takes the form of a correlated random walk at short time scales that is punctuated by reorientation events leading to a biased random walks at long time scales. This allows us to identify optimal alternation schemes and characterize their robustness in the context of noisy sensory acquisition as well as performance errors linked with variations in environmental conditions and agent–environment interactions.

Locomotion is necessary for survival in most animal species. However, injuries to the appendages mediating locomotion are common. We assess the recovery of walking in Drosophila melanogaster following leg amputation. Whereas flies preamputation explore open arenas in a symmetric fashion on average, foreleg amputation induces a strong turning bias away from the side of the amputation. However, we find that unbiased walking behavior returns over time in wild-type flies, while recovery is significantly impaired in proprioceptive mutants. To identify the biomechanical basis of this locomotor impairment and recovery, we then examine individual leg motion (gait) at a fine scale. A minimal mathematical model that links neurodynamics to body mechanics during walking shows that redistributing leg forces between the right and left side enables the observed recovery. Altogether, our study suggests that proprioceptive input from the intact limbs plays a crucial role in the behavioral plasticity associated with locomotor recovery after injury.

Locomotion in an organism is a consequence of the coupled interaction between brain, body and environment. Motivated by qualitative observations and quantitative perturbations of crawling in Drosophila melanogaster larvae, we construct a minimal integrative mathematical model for its locomotion. Our model couples the excitation-inhibition circuits in the nervous system to force production in the muscles and body movement in a frictional environment, thence linking neural dynamics to body mechanics via sensory feedback in a heterogeneous environment. Our results explain the basic observed phenomenology of crawling with and without proprioception, and elucidate the stabilizing role that proprioception plays in producing a robust crawling phenotype in the presence of biological perturbations. More generally, our approach allows us to make testable predictions on the effect of changing body-environment interactions on crawling, and serves as a step in the development of hierarchical models linking cellular processes to behavior.

Inspired by the relatively simple morphological blueprint provided by batoid fish such as stingrays and skates, we created a biohybrid system that enables an artificial animal—a tissue-engineered ray—to swim and phototactically follow a light cue. By patterning dissociated rat cardiomyocytes on an elastomeric body enclosing a microfabricated gold skeleton, we replicated fish morphology at 1=10 scale and captured basic fin deflection patterns of batoid fish. Optogenetics allows for phototactic guidance, steering, and turning maneuvers. Optical stimulation induced sequential muscle activation via serpentinepatterned muscle circuits, leading to coordinated undulatory swimming. The speed and direction of the ray was controlled by modulating light frequency and by independently eliciting right and left fins, allowing the biohybrid machine to maneuver through an obstacle course.

In nature, fast, high-power-density actuation can be achieved through the release of stored elastic energy by exploiting mechanical instabilities in systems including the closure of the Venus flytrap1 and the dispersal of plant or fungal spores2. Here, we use droplet microfluidics to tailor the geometry of a nanoscale self-assembling supra-molecular polymer to create a mechanical instability. We show that this strategy allows the build-up of elastic energy as a result of peptide selfassembly, and its release within milliseconds when the buckled geometry of the nanotube confined within microdroplets becomes unstable with respect to the straight form. These results overcome the inherent limitations of self-assembly for generating large-scale actuation on the sub-second timescale and illuminate the possibilities and performance limits of irreversible actuation by supra-molecular polymers.

The rapid growth of the human cortex during development is accompanied by the folding of the brain into a highly convoluted structure1,2,3. Recent studies have focused on the genetic and cellular regulation of cortical growth4,5,6,7,8, but understanding the formation of the gyral and sulcal convolutions also requires consideration of the geometry and physical shaping of the growing brain9,10,11,12,13,14,15. To study this, we use magnetic resonance images to build a 3D-printed layered gel mimic of the developing smooth fetal brain; when immersed in a solvent, the outer layer swells relative to the core, mimicking cortical growth. This relative growth puts the outer layer into mechanical compression and leads to sulci and gyri similar to those in fetal brains. Starting with the same initial geometry, we also build numerical simulations of the brain modelled as a soft tissue with a growing cortex, and show that this also produces the characteristic patterns of convolutions over a realistic developmental course. All together, our results show that although many molecular determinants control the tangential expansion of the cortex, the size, shape, placement and orientation of the folds arise through iterations and variations of an elementary mechanical instability modulated by early fetal brain geometry.

Origami describes rules for creating folded structures from patterns on a flat sheet, but does not prescribe how patterns can be designed to fit target shapes. Here, starting from the simplest periodic origami pattern that yields one-degree-of-freedom collapsible structures—we show that scale-independent elementary geometric constructions and constrained optimization algorithms can be used to determine spatially modulated patterns that yield approximations to given surfaces of constant or varying curvature. Paper models confirm the feasibility of our calculations. We also assess the diculty of realizing these geometric structures by quantifying the energetic barrier that separates the metastable flat and folded states. Moreover, we characterize the trade-o between the accuracy to which the pattern conforms to the target surface, and the eort associated with creating finer folds. Our approach enables the tailoring of origami patterns to drape complex surfaces independent of absolute scale, as well as the quantification of the energetic and material cost of doing so.

Shape-morphing systems can be found in many areas, including smart textiles1 , autonomous robotics2 , biomedical devices3 , drug delivery4 and tissue engineering5 . The natural analogues of such systems are exemplified by nastic plant motions, where a variety of organs such as tendrils, bracts, leaves and flowers respond to environmental stimuli (such as humidity, light or touch) by varying internal turgor, which leads to dynamic conformations governed by the tissue composition and microstructural anisotropy of cell walls6–10. Inspired by these botanical systems, we printed composite hydrogel architectures that are encoded with localized, anisotropic swelling behaviour controlled by the alignment of cellulose fibrils along prescribed four-dimensional printing pathways. When combined with a minimal theoretical framework that allows us to solve the inverse problem of designing the alignment patterns for prescribed target shapes, we can programmably fabricate plant-inspired architectures that change shape on immersion in water, yielding complex three-dimensional morphologies.

Chemotaxis, the directional migration of cells in a chemical gradient, is robust to fluctuations associated with low chemical concentrations and dynamically changing gradients as well as high saturating chemical concentrations. Although a number of reports have identified cellular behavior consistent with a directional memory that could account for behavior in these complex environments, the quantitative and molecular details of such a memory process remain unknown. Using microfluidics to confine cellular motion to a 1D channel and control chemoattractant exposure, we observed directional memory in chemotactic neutrophil-like cells. We modeled this directional memory as a long-lived intracellular asymmetry that decays slower than observed membrane phospholipid signaling. Measurements of intracellular dynamics revealed that moesin at the cell rear is a longlived element that when inhibited, results in a reduction of memory. Inhibition of ROCK (Rho-associated protein kinase), downstream of RhoA (Ras homolog gene family, member A), stabilized moesin and directional memory while depolymerization of microtubules (MTs) disoriented moesin deposition and also reduced directional memory. Our study reveals that long-lived polarized cytoskeletal structures, specifically moesin, actomyosin, and MTs, provide a directional memory in neutrophil-like cells even as they respond on short time scales to external chemical cues.

When red blood cells (RBCs) move through narrow capillaries in the microcirculation, they deform as they flow. In pathophysiological processes such as sickle cell disease and malaria, RBC motion and flow are severely restricted. To understand this threshold of occlusion, we use a combination of experiment and theory to study the motion of a single swollen RBC through a narrow glass capillary of varying inner diameter. By tracking the movement of the squeezed cell as it is driven by a controlled pressure drop, we measure the RBC velocity as a function of the pressure gradient as well as the local capillary diameter, and find that the effective blood viscosity in this regime increases with both decreasing RBC velocity and tube radius by following a power-law that depends upon the length of the confined cell. Our observations are consistent with a simple elasto-hydrodynamic model and highlight the role of lateral confinement in the occluded pressure-driven slow flow of soft confined objects.

Joseph Bishop Keller was an applied mathematician of world renown, whose research interests spanned a wide range of topics, including wave propagation, semi-classical mechanics, geophysical fluid dynamics, operations research, finance, biomechanics, epidemiology, biostatistics, and the mathematics of sports. His work combined a love of physics, mathematics, and natural phenomena with an irrepressible curiosity to pursue explanations of practical and often playful enigmas.

The main analytical ingredients of the first part of this paper are two independent results: a theorem on approximation of W2,2 solutions of the Monge–Ampère equation by smooth solutions, and a theorem on the matching (in other words, continuation) of second order isometries to exact isometric embeddings of 2d surface in R3. In the second part, we rigorously derive the -limit of 3-dimensional nonlinear elastic energy of a shallow shell of thickness h, where the depth of the shell scales like hα and the applied forces scale like hα+2, in the limit when h → 0. We offer a full analysis of the problem in the parameter range α ∈ (1/2, 1). We also complete the analysis in some specific cases for the full range α ∈ (0, 1), applying the results of the first part of the paper. © 2015 Elsevier Masson SAS. All rights reserved.

We propose a minimal mathematical model for the physical basis of membrane protein patterning in the immunological synapse (IS), which encompass membrane mechanics, protein binding kinetics and motion, and fluid flow in the synaptic cleft. Our theory leads to simple predictions for the spatial and temporal scales of protein cluster formation, growth and arrest as a function of membrane stiffness, rigidity and kinetics of the adhesive proteins, and the fluid flow in the synaptic cleft. Numerical simulations complement these scaling laws by quantifying the nucleation, growth and stabilization of proteins domains on the size of the cell. Direct comparison with experiment shows that passive elastohydrodynamics and kinetics of protein binding in the synaptic cleft can describe the short-time formation and organization of protein clusters, without evoking any active processes in the cytoskeleton. Despite the apparent complexity of the process, our analysis shows that just two dimensionless parameters characterize the spatial and temporal evolution of the protein pattern: a ratio of membrane elasticity to protein stiffness, and the ratio of a hydrodynamic time scale for fluid flow relative to the protein binding rate. A simple phase diagram encompasses the variety of patterns that can arise

A rigid cylinder placed on a soft gel deforms its surface. When multiple cylinders are placed on the surface, they interact with each other via the topography of the deformed gel which serves as an energy landscape; as they move, the landscape changes which in turn changes their interaction. We use a combination of experiments, simple scaling estimates and numerical simulations to study the self-assembly of cylinders in this elastic analog of the “Cheerios Effect”, which describes capillary interactions on a fluid interface. Our results show that the effective two-body interaction can be well described by an exponential attraction potential as a result of which the dynamics also show an exponential behavior with respect to the separation distance. When many cylinders are placed on the gel, the cylinders cluster together if they are not too far apart; otherwise their motion gets elastically arrested.