some detail, along with other important factors, such as private patronage from wealthy figures who were unhealthily obsessed with the hypothetical phenom-enon of antigravity. The renaissance of general relativity rovides a rich case study for thinking about fundamental problems of scien-tific change. How are theories used and not used? What is the difference between healthy and withering research pro-grams? Can the resurgence of a heory be explained by new observational discov-eries? Technological advances? Individ-ual personalities? Social and political de-velopments? The book is a great cross section of those different approaches, which will make it valuable to scholars in history and sociology of physics—although it might set a fairly high methodological barrier to entry for non-specialist readers. Nevertheless, The Re - naissance of General Relativity in Context will ickly become a classic in the his-tory of the field, and it will perhaps spur new research programs of its own.

Inspired by the allure of additive fabrication, we pose the problem of origami design from a different perspective: How can we grow a folded surface in three dimensions from a seed so that it is guaranteed to be isometric to the plane? We solve this problem in two steps: by first identifying the geometric conditions for the compatible completion of two separate folds into a single developable fourfold vertex, and then showing how this foundation allows us to grow a geometrically compatible front at the boundary of a given folded seed. This yields a complete marching, or additive, algorithm for the inverse design of the complete space of developable quad origami patterns that can be folded from flat sheets. We illustrate the flexibility of our approach by growing ordered, disordered, straight, and curved-folded origami and fitting surfaces of given curvature with folded approximants. Overall, our simple shift in perspective from a global search to a local rule has the potential to transform origami-based metastructure design.

Pattern formation driven by differential strain in constrained elastic systems is a common motif in many technological and biological systems. Here we introduce a biologically motivated case of elastic patterning that allows us to explore the conditions for the existence of local puckering and global wrinkling patterns: a soft growing composite ring adhered elastically to a constraining rigid ring. We explore how differential growth of the soft ring and the elastic resistance to shear and stretching deformations induced by soft adherence lead to a range of phenomena that include uniform aperture-like modes, localized puckers that are Nambu–Goldstone-like modes and global wrinkles in the system. Our analysis combines computer simulations of a discrete rod model with a nonlinear stability analysis of the differential equations in the continuum limit. We provide phase diagrams and scaling relations that reveal the nature and extent of the deformation patterns. Overall, our study reveals how geometry and mechanics conspire to yield a rich phenomenology that could serve as a guide to the design of programmable localized elastic deformations while being relevant for the mechanical basis of biological morphogenesis.

Interactions with animal pollinators have helped shape the stunning diversity of flower morphologies across the angiosperms. A common evolutionary consequence of these interactions is that some flowers have converged on suites of traits, or pollination syndromes, that attract and reward specific pollinator groups. Determining the genetic basis of these floral pollination syndromes can help us understand the processes that contributed to the diversification of the angiosperms. Here, we characterize the genetic architecture of a bee-to-hummingbird pollination shift in Aquilegia (columbine) using QTL mapping of 17 floral traits encompassing color, nectar composition, and organ morphology. In this system, we find that the genetic architectures underlying differences in floral color are quite complex, and we identify several likely candidate genes involved in anthocyanin and carotenoid floral pigmentation. Most morphological and nectar traits also have complex genetic underpinnings; however, one of the key floral morphological phenotypes, nectar spur curvature, is shaped by a single locus of large effect.

Adhesive interactions between elastic structures such as graphene sheets, carbon nanotubes, and microtubules have been shown to exhibit hysteresis due to irrecoverable energy loss associated with bond breakage, even in static (rate-independent) experiments. To understand this phenomenon, we start with a minimal theory for the peeling of a thin sheet from a substrate, coupling the local event of bond breaking to the nonlocal elastic relaxation of the sheet and show that this can drive static adhesion hysteresis over a bonding/debonding cycle. Using this model we quantify hysteresis in terms of the adhesion and elasticity parameters of the system. This allows us to derive a scaling relation that preserves hysteresis at different levels of granularity while resolving a seeming paradox of lattice trapping in the continuum limit of a discrete fracture process. Finally, to verify our theory, we use new experiments to demonstrate and measure adhesion hysteresis in bundled microtubules.

Locomotion of an organism interacting with an environment is the consequence of a symmetry-breaking action in space-time. Here we show a minimal instantiation of this principle using a thin cir- cular sheet, actuated symmetrically by a pneumatic source, using pressure to change shape nonlinearly via a spontaneous buckling instability. This leads to a polarized, bilaterally symmetric cone that can walk on land and swim in water. In either mode of loco- motion, the emergence of shape asymmetry in the sheet leads to an asymmetric interaction with the environment that generates movement––via anisotropic friction on land, and via directed iner- tial forces in water. Scaling laws for the speed of the sheet of the actuator as a function of its size, shape, and the frequency of ac- tuation are consistent with our observations. The presence of eas- ily controllable reversible modes of buckling deformation further allows for a change in the direction of locomotion in open arenas and the ability to squeeze through confined environments––both of which we demonstrate using simple experiments. Our simple approach of harnessing elastic instabilities in soft structures to drive locomotion enables the design of novel shape-changing ro- bots and other bioinspired machines at multiple scales.

Kirigami, the art of paper cutting, has become a paradigm for mechanical metamaterials in recent years. The basic building blocks of any kirigami structures are repetitive deployable patterns that derive inspiration from geometric art forms and simple planar tilings. Here, we complement these approaches by directly linking kirigami patterns to the symmetry associated with the set of 17 repeating patterns that fully characterize the space of periodic tilings of the plane. We start by showing how to construct deployable kirigami patterns using any of the wallpaper groups, and then design symmetry-preserving cut patterns to achieve arbitrary size changes via deployment. We further prove that different symmetry changes can be achieved by controlling the shape and connectivity of the tiles and connect these results to the underlying kirigami-based lattice structures. All together, our work provides a systematic approach for creating a broad range of kirigami-based deployable structures with any prescribed size and symmetry properties.

The termite nest is one of the architectural wonders of the liv- ing world, built by the collective action of workers in a colony. Each nest has several characteristic structural motifs that allow for efficient ventilation, cooling, and traversal. We use tomography to quantify the nest architecture of the African termite Apicotermes lamani, consisting of regularly spaced floors connected by scattered linear and helicoidal ramps. To understand how these elaborate structures are built and arranged, we formulate a minimal model for the spatiotemporal evolution of three hydrodynamic fields—mud, termites, and pheromones— linking environmental physics to collective building behavior using simple local rules based on experimental observations. We find that floors and ramps emerge as solutions of the governing equations, with statistics consistent with observations of A. lamani nests. Our study demonstrates how a local self-reinforcing biotectonic scheme is capable of generating an architecture that is simultaneously adaptable and functional, and likely to be relevant for a range of other animal-built structures.

Walking is a common bipedal and quadrupedal gait and associated with terrestrial and aquatic organisms. Inspired by recent evidence of the neural underpinnings of primitive aquatic walking in the little skate Leucoraja erinacea, we introduce a theoretical model of aquatic walking that reveals robust and efficient gaits with modest requirements for body morphology and control. The model predicts undulatory behaviour of the system body with a regular foot placement pattern, which is also observed in the animal, and additionally predicts the existence of gait bistability between two states, one with a large energetic cost for locomotion and another associated with almost no energetic cost. We show that these can be discovered using a simple reinforcement learning (RL) scheme. To test these theoretical frameworks, we built a bipedal robot and show that its behaviours are similar to those of our minimal model: its gait is also periodic and exhibits bistability, with a low efficiency gait separated from a high efficiency gait by a ‘jump’ transition. Overall, our study highlights the physical constraints on the evolution of walking and provides a guide for the design of efficient biomimetic robots.

How can we manipulate the topological connectivity of a three-dimensional prismatic assembly to control the number of internal degrees of freedom and the number of connected components in it? To answer this question in a deterministic setting, we use ideas from elementary number theory to provide a hierarchical deterministic protocol for the control of rigidity and connectivity. We then show that it is possible to also use a stochastic protocol to achieve the same results via a percolation transition. Together, these approaches provide scale-independent algorithms for the cutting or gluing of three-dimensional prismatic assemblies to control their overall connectivity and rigidity.

Tissues undergoing morphogenesis impose mechanical effects on one another. How developmental programs adapt to or take advantage of these effects remains poorly explored. Here, using a combination of live imaging, modeling, and microsurgical perturbations, we show that the axial and paraxial tissues in the forming avian embryonic body coordinate their rates of elongation through mechanical interactions. First, a cell motility gradient drives paraxial presomitic mesoderm (PSM) expansion, resulting in compression of the axial neural tube and notochord; second, elongation of axial tissues driven by PSM compression and polarized cell intercalation pushes the caudal progenitor domain posteriorly; finally, the axial push drives the lateral movement of midline PSM cells to maintain PSM growth and cell motility. These interactions form an engine-like positive feedback loop, which sustains a shared elongation rate for coupled tissues. Our results demonstrate a key role of inter-tissue forces in coordinating distinct body axis tissues during their co-elongation.

Channel formation and branching is widely seen in physical systems where movement of fluid through a porous structure causes the spatiotemporal evolution of the medium. We provide a simple theoretical framework that embodies this feedback mechanism in a multiphase model for flow through a frangible porous medium with a dynamic permeability. Numerical simulations of the model show the emergence of branched networks whose topology is determined by the geometry of external flow forcing. This allows us to delineate the conditions under which splitting and/or coalescing branched network formation is favored, with potential implications for both understanding and controlling branching in soft frangible media.

Rectilinear crawling locomotion is a primitive and common mode of loco- motion in slender soft-bodied animals. It requires coordinated contractions that propagate along a body that interacts frictionally with its environment. We propose a simple approach to understand how this coordination arises in a neuromechanical model of a segmented, soft-bodied crawler via an itera- tive process that might have both biological antecedents and technological relevance. Using a simple reinforcement learning algorithm, we show that an initial all-to-all neural coupling converges to a simple nearest-neighbour neural wiring that allows the crawler to move forward using a localized wave of contraction that is qualitatively similar to what is observed in Drosophila melanogaster larvae and used in many biomimetic solutions. The resulting solution is a function of how we weight gait regularization in the reward, with a trade-off between speed and robustness to proprioceptive noise. Overall, our results, which embed the brain–body–environment triad in a learning scheme, have relevance for soft robotics while shedding light on the evolution and development of locomotion.

Understanding the morphology of self-assembled fibrillar bundles and aggregates is relevant to a range of problems in molecular biology, supramolecular chemistry and materials science. Here, we propose a coarse-grained approach that averages over specific molecular details and yields an effective mechanical theory for the spatial complexity of self-assembling fibrillar structures that arises due to the competing effects of (the bending and twisting) elasticity of individual filaments and the adhesive interactions between them. We show that our theoretical framework accounting for this allows us to capture a number of diverse fibril morphologies observed in natural and synthetic systems, ranging from Filopodia to multi-walled carbon nanotubes, and leads to a phase diagram of possible fibril shapes. We also show how the extreme sensitivity of these morphologies can lead to spatially chaotic structures. Together, these results suggest a common mechanical basis for mesoscale fibril morphology as a function of the nanoscale mechanical properties of its filamentous constituents.

Many complex systems experience damage accumulation, which leads to aging, manifest as an increasing probability of system collapse with time. This naturally raises the question of how to maximize health and longevity in an aging system at minimal cost of maintenance and intervention. Here, we pose this question in the context of a simple interdependent network model of aging in complex systems and show that it exhibits cascading failures. We then use both optimal control theory and reinforcement learning alongside a combination of analysis and simulation to determine optimal maintenance protocols. These protocols may motivate the rational design of strategies for promoting longevity in aging complex systems with potential applications in therapeutic schedules and engineered system maintenance.

Animal hearts are soft shells that actively pump blood to oxygenate tissues. Here, we propose an allometric scaling law for the heart rate based on the idea of elastohydrodynamic resonance of a fluid-loaded soft active elastic shell that buckles and contracts axially when twisted periodically. We show that this picture is consistent with numerical simulations of soft cylindrical shells that twist-buckle while pumping a viscous fluid, yielding optimum ejection fractions of 35%–40% when driven resonantly. Our scaling law is consistent with experimental measurements of heart rates over 2 orders of magnitude, and provides a mechanistic basis for how metabolism scales with organism size. In addition to providing a physical rationale for the heart rate and metabolism of an organism, our results suggest a simple design principle for soft fluidic pumps.

When a volatile droplet is deposited on a floating swellable sheet, it becomes asymmetric, lobed and mobile. We describe and quantify this phenomena that involves nonequilibrium swelling, evaporation and motion, working together to realize a self-excitable spatially extended oscillator. Solvent penetration causes the film to swell locally and eventually buckle, changing its shape and the drop responds by moving. Simultaneously, solvent evaporation from the swollen film causes it to regain its shape once the droplet has moved away. The process repeats and leads to complex pulsatile spinning and/or sliding movements. We use a one-dimensional experiment to highlight the slow swelling of and evaporation from the film and the fast motion of the drop, a characteristic of excitable systems. Finally, we provide a phase diagram for droplet excitability as a function of drop size and film thickness and scaling laws for the motion of the droplet.

Inspired by the jamming in leaky systems that arises in many physiological and industrial settings, we study the propagation of clogs in a leaky microfluidic channel. By driving a colloidal suspension through such a channel with a fluid-permeable wall adjoining a gutter, we follow the formation and propagation of jams and show that they move at a steady speed, in contrast with jams in channels that have impermeable walls. Furthermore, by varying the ratio of the resistance from the leaky wall and that of the gutter, we show that it is possible to control the shape of the propagating jam, which is typically wedge shaped. We complement our experiments with numerical simulations, where we implement an Euler-Lagrangian framework for the simultaneous evolution of both immersed colloidal particles and the carrier fluid. Finally, we show that the particle ordering in the clog can be tuned by adjusting the geometry of the leaky wall. Altogether, the leaky channel serves both as a filter and a shunt with the potential for a range of uses.

The ability to infer intention lies at the basis of many social interactions played out via motor actions. We consider a simple paradigm of this ability in humans using data from experiments simulating an antagonistic game between an Attacker and a Blocker. Evidence shows early inference of an Attacker move by as much as 100ms but the nature of the informational cues signaling the impending move remains unknown. We show that the transition to action has the hallmark of a critical transition that is accompanied by early warning signals. These early warning signals occur as much as 130 ms before motion ensues—showing a sharp rise in motion autocorrelation at lag-1 and a sharp rise in the autocorrelation decay time. The early warning signals further correlate strongly with Blocker response times. We analyze the variance of the motion near the point of transition and find that it diverges in a manner consistent with the dynamics of a fold-transition. To test if humans can recognize and act upon these early warning signals, we simulate the dynamics of fold-transition events and ask people to recognize the onset of directional motion: participants react faster to fold-transition dynamics than to its uncorrelated counterpart. Together, our findings suggest that people can recognize the intent and onset of motion by inferring its early warning signals.

During dynamic instability, self-assembling microtubules (MTs) stochastically alternate between phases of growth and shrinkage. This process is driven by the presence of two distinct states of MT subunits, GTP- and GDP-bound tubulin dimers, that have different structural properties. Here, we use a combination of analysis and computer simulations to study the mechanical and kinetic regulation of dynamic instability in three-dimensional (3D) self-assembling MTs. Our model quantifies how the 3D structure and kinetics of the distinct states of tubulin dimers determine the mechanical stability of MTs. We further show that dynamic instability is influenced by the presence of quenched disorder in the state of the tubulin subunit as reflected in the fraction of non-hydrolysed tubulin. Our results connect the 3D geometry, kinetics and statistical mechanics of these tubular assemblies within a single framework, and may be applicable to other self-assembled systems where these same processes are at play.

A submerged finite cylinder moving under its own weight along a soft incline lifts off and slides at a steady velocity while also spinning. Here, we experimentally quantify the steady spinning of the cylinder and show theoretically that it is due to a combination of an elastohydrodynamic torque generated by flow in the variable gap, and the viscous friction on the edges of the finite-length cylinder. The relative influence of the latter depends on the aspect ratio of the cylinder, the angle of the incline, and the deformability of the substrate, which we express in terms of a single scaled compliance parameter. By independently varying these quantities, we show that our experimental results are consistent with a transition from an edge-effect dominated regime for short cylinders to a gap-dominated elastohydrodynamic regime when the cylinder is very long.

Kirigami, the creative art of paper cutting, is a promising paradigm for mechanical metamaterials. However, to make kirigami-inspired structures a reality requires controlling the topology of kirigami to achieve connectivity and rigidity. We address this question by deriving the maximum number of cuts (minimum number of links) that still allow us to preserve global rigidity and connectivity of the kirigami. A deterministic hierarchical construction method yields an efficient topological way to control both the number of connected pieces and the total degrees of freedom. A statistical approach to the control of rigidity and connectivity in kirigami with random cuts complements the deterministic pathway, and shows that both the number of connected pieces and the degrees of freedom show percolation transitions as a function of the density of cuts (links). Together, this provides a general framework for the control of rigidity and connectivity in planar kirigami.

The Poisson’s ratio and residual strain of ultra-thin films (<100 nm) are characterized using the phenomenon of transverse wrinkling in stretched bridges. The test methodology utilizes residual stress driven structures and easy to replicate clean-room fabrication and metrology techniques that can be seamlessly incorporated into a thin-film production assembly line. Freestanding rectangular ultra-thin film bridges are fabricated using dimensions that generate repeatable transverse wrinkling patterns. Numerical modeling based on the non-linear Koiter plate and shell energy formulation is conducted to correlate the Poisson’s ratio and residual strain to the measured wrinkling deformation. Poisson’s ratio affects the peak amplitudes without significantly changing the wavelength of the wrinkles. By contrast, the strain affects both the wavelength and amplitude. The proof of concept is demonstrated using 65 nm thick copper films. A Poisson’s ratio of 0.34 ± 0.05 and a tensile residual strain of

Inspired by the patterns of multicellularity in choanoflagellates, the closest living relatives of animals, we quantify the biophysical processes underlying the morphogenesis of rosette colonies in the choanoflagellate Salpingoeca rosetta. We find that rosettes reproducibly transition from an early stage of 2-dimensional (2D) growth to a later stage of 3D growth, despite the underlying variability of the cell lineages. Our perturbative experiments demonstrate the fundamental importance of a basally secreted extracellular matrix (ECM) for rosette morphogenesis and show that the interaction of the ECM with cells in the colony physically constrains the packing of proliferating cells and, thus, controls colony shape. Simulations of a biophysically inspired model that accounts for the size and shape of the individual cells, the fraction of ECM, and its stiffness relative to that of the cells suffices to explain our observations and yields a morphospace consistent with observations across a range of multicellular choanoflagellate colonies. Overall, our biophysical perspective on rosette development complements previous genetic perspectives and, thus, helps illuminate the interplay between cell biology and physics in regulating morphogenesis.

Morphogenetic flows in developmental biology are characterized by the coordinated motion of thousands of cells that organize into tissues, naturally raising the question of how this collective organization arises. Using only the kinematics of tissue deformation, which naturally integrates local and global mechanisms along cell paths, we identify the dynamic morphoskeletons behind morphogenesis, i.e., the evolving centerpieces of multicellular trajectory patterns. These features are model- and parameter-free, frame-invariant, and robust to measurement errors and can be computed from unfiltered cell-velocity data. We reveal the spatial attractors and repellers of the embryo by quantifying its Lagrangian deformation, information that is inaccessible to simple trajectory inspection or Eulerian methods that are local and typically frame-dependent. Computing these dynamic morphoskeletons in wild-type and mutant chick and fly embryos, we find that they capture the early footprint of known morphogenetic features, reveal new ones, and quantitatively distinguish between different phenotypes.

Soft elastic filaments that can be stretched, bent, and twisted exhibit a range of topologically and geometrically complex morphologies. Recently, a number of experiments have shown how to use these building blocks to create filament-based artificial muscles that use the conversion of writhe to extension or contraction, exposing the connection between topology, geometry, and mechanics. Here, we combine numerical simulations of soft elastic filaments that account for geometric nonlinearities and self-contact to map out the basic structures underlying artificial muscle fibers in a phase diagram that is a function of the extension and twist density. We then use ideas from computational topology to track the interconversion of link, twist, and writhe in these geometrically complex physical structures to explain the physical principles underlying artificial muscle fibers and provide guidelines for their design.

The size, shape and structure of insect wings are intimately linked to their ability to fly. However, there are few systematic studies of the variability of the natural patterns in wing morphology across insects. We have assembled a dataset of 789 insect wings with representatives from 25 families and performed a comprehensive computational analysis of their morphology using topological and geometric notions in terms of (i) wing size and contour shape, (ii) vein topology, and (iii) shape and distribution of wing membrane domains. These morphospaces are complementary to existing methods for quantitatively characterizing wing morphology and are likely to be useful for investigating wing function and evolution. This Methods and Techniques paper is accompanied by a set of computational tools for open use.

Thin shape-shifting structures are often limited in their ability to morph into complex and doubly curved shapes. Such transformations require both large in-plane expansion or contraction gradients and control over extrinsic curvature, which are hard to achieve with single materials arranged in simple architectures. We solve this problem by 4-dimensional printing of multiple materials in heterogeneous lattice designs. Our material system provides a platform that achieves in-plane growth and out-of-plane curvature control for 4-material bilayer ribs. The lattice design converts this into large growth gradients, which lead to complex, predictable 3-dimensional (3D) shape changes. We demonstrate this approach with a hemispherical antenna that shifts resonant frequency as it changes shape and a flat lattice that transforms into a 3D human face.

Animals make organs of precise size, shape, and symmetry but how developing embryos do this is largely unknown. Here, we combine quantitative imaging, physical theory, and physiological measurement of hydrostatic pressure and fluid transport in zebrafish to study size control of the developing inner ear. We find that fluid accumulation creates hydrostatic pressure in the lumen leading to stress in the epithelium and expansion of the otic vesicle. Pressure, in turn, inhibits fluid transport into the lumen. This negative feedback loop between pressure and transport allows the otic vesicle to change growth rate to control natural or experimentally-induced size variation. Spatiotemporal patterning of contractility modulates pressure-driven strain for regional tissue thinning. Our work connects molecular-driven mechanisms, such as osmotic pressure driven strain and actomyosin tension, to the regulation of tissue morphogenesis via hydraulic feedback to ensure robust control of organ size.

Controlled gliding is one of the most energetically efficient modes of transportation for natural and human powered fliers. Here we demonstrate that gliding and landing strategies with different optimality criteria can be identified through deep-reinforcement-learning without explicit knowledge of the underlying physics. We combine a two-dimensional model of a controlled elliptical body with deep-reinforcement-learning (D-RL) to achieve gliding with either minimum energy expenditure, or fastest time of arrival, at a predetermined location. In both cases the gliding trajectories are smooth, although energy/time optimal strategies are distinguished by small/high frequency actuations. We examine the effects of the ellipse's shape and weight on the optimal policies for controlled gliding. We find that the model-free reinforcement learning leads to more robust gliding than model-based optimal control strategies with a modest additional computational cost. We also demonstrate that the gliders with D-RL can generalize their strategies to reach the target location from previously unseen starting positions. The model-free character and robustness of D-RL suggests a promising framework for developing robotic devices capable of exploiting complex flow environments.

The gastrointestinal tract is enveloped by concentric and orthogonally aligned layers of smooth muscle; however, an understanding of the mechanisms by which these muscles become patterned and aligned in the embryo has been lacking. We find that Hedgehog acts through Bmp to delineate the position of the circumferentially oriented inner muscle layer, whereas localized Bmp inhibition is critical for allowing formation of the later-forming, longitudinally oriented outer layer. Because the layers form at different developmental stages, the muscle cells are exposed to unique mechanical stimuli that direct their alignments. Differential growth within the early gut tube generates residual strains that orient the first layer circumferentially, and when formed, the spontaneous contractions of this layer align the second layer longitudinally. Our data link morphogen-based patterning to mechanically controlled smooth muscle cell alignment and provide a mechanistic context for potentially understanding smooth muscle organization in a wide variety of tubular organs.

Inspired by observations of beads packed on a thin string in such systems as sea-grapes and dental plaque, we study the random sequential adsorption of spheres on a cylinder. We determine the asymptotic fractional coverage of the cylinder as a function of the sole parameter in the problem, the ratio of the sphere radius to the cylinder radius (for a very long cylinder) using a combination of analysis and numerical simulations. Examining the asymptotic structures, we find weak chiral ordering on sufficiently small spatial scales. Experiments involving colloidal microspheres that can attach irreversibly to a silica wire via electrostatic forces or DNA hybridization allow us to verify our predictions for the asymptotic coverage.

Localized deformation patterns are a common motif in morphogenesis and are increasingly finding applications in materials science and engineering, in such instances as mechanical memories. Here, we describe the emergence of spatially localized deformations in a minimal mechanical system by exploring the impact of growth and shear on the conformation of a semi-flexible filament connected to a pliable shearable substrate. We combine numerical simulations of a discrete rod model with theoretical analysis of the differential equations recovered in the continuum limit to quantify (in the form of scaling laws) how geometry, mechanics and growth act together to give rise to such localized structures in this system. We find that spatially localized deformations along the filament emerge for intermediate shear modulus and increasing growth. Finally, we use experiments on a 3D-printed multi-material model system to demonstrate that external control of the amount of shear and growth may be used to regulate the spatial extent of the localized strain texture.

Kirigami tessellations, regular planar patterns formed by partially cutting flat, thin sheets, allow compact shapes to morph into open structures with rich geometries and unusual material properties. However, geometric and topological constraints make the design of such structures challenging. Here we pose and solve the inverse problem of determining the number, size and orientation of cuts that enables the deployment of a closed, compact regular kirigami tessellation to conform approximately to any prescribed target shape in two or three dimensions. We first identify the constraints on the lengths and angles of generalized kirigami tessellations that guarantee that their reconfigured face geometries can be contracted from a non-trivial deployed shape to a compact, non-overlapping planar cut pattern. We then encode these conditions into a flexible constrained optimization framework to obtain generalized kirigami patterns derived from various periodic tesselations of the plane that can be deployed into a wide variety of prescribed shapes. A simple mechanical analysis of the resulting structure allows us to determine and control the stability of the deployed state and control the deployment path. Finally, we fabricate physical models that deploy in two and three dimensions to validate this inverse design approach. Altogether, our approach, combining geometry, topology and optimization, highlights the potential for generalized kirigami tessellations as building blocks for shape-morphing mechanical metamaterials.

Moving along a straight path is a surprisingly difficult task. This is because, with each ensuing step, noise is generated in the motor and sensory systems, causing the animal to deviate from its intended route. When relying solely on internal sensory information to correct for this noise, the directional error generated with each stride accumulates, ultimately leading to a curved path. In contrast, external compass cues effectively allow the animal to correct for errors in its bearing. Here, we studied straight-line orientation in two different sized dung beetles. This allowed us to characterize and model the size of the directional error generated with each step, in the absence of external visual compass cues (motor error) as well as in the presence of these cues (compass and motor errors). In addition, we model how dung beetles balance the influence of internal and external orientation cues as they orient along straight paths under the open sky. We conclude that the directional error that unavoidably accumulates as the beetle travels is inversely proportional to the step size of the insect, and that both beetle species weigh the two sources of directional information in a similar fashion.

Diversity in the shapes of avian eggs has intrigued biologists for centuries, and recent studies at a range of taxonomic scales suggest that egg shape can be a powerful lens through which to view morphological adaptation. At a broad taxonomic level, we previously examined egg shape in 1400 species (Stoddard et al. 2017). Our study, which contained a detailed analysis of the egg shape morphospace and a new biophysical model of egg shape formation, included a broad-scale phylogenetic comparative analysis of egg shape across more than 1200 species representing 34 orders and 143 families. We found that, at this global scale, most variation in egg shape is correlated with phylogenetic history, an adult bird's body mass and egg size. We also found a significant correlation between egg shape and hand-wing index, a measure of wing shape that provides a general estimate of flight ability, prompting us to consider the possibility that adaptations for flight might be important drivers of egg shape variation.

The embryonic gut tube is a cylindrical structure from which the respiratory and gastrointestinal tracts develop1. Although the early emergence of the endoderm as an epithelial sheet2,3 and later morphogenesis of the definitive digestive and respiratory organs4,5,6 have been investigated, the intervening process of gut tube formation remains relatively understudied7,8. Here we investigate the molecular control of macroscopic forces underlying early morphogenesis of the gut tube in the chick embryo. The gut tube has been described as forming from two endodermal invaginations—the anterior intestinal portal (AIP) towards the rostral end of the embryo and the caudal intestinal portal (CIP) at the caudal end—that migrate towards one another, internalizing the endoderm until they meet at the yolk stalk (umbilicus in mammals)1,6. Migration of the AIP to form foregut has been descriptively characterized8,9, but the hindgut is likely to form by a distinct mechanism that has not been fully explained10. We find that the hindgut is formed by collective cell movements through a stationary CIP, rather than by movement of the CIP itself. Further, combining in vivo imaging, biophysics and mathematical modelling with molecular and embryological approaches, we identify a contractile force gradient that drives cell movements in the hindgut-forming endoderm, enabling tissue-scale posterior extension of the forming hindgut tube. The force gradient, in turn, is established in response to a morphogenic gradient of fibroblast growth factor signalling. As a result, we propose that an important positive feedback arises, whereby contracting cells draw passive cells from low to high fibroblast growth factor levels, recruiting them to contract and pull more cells into the elongating hindgut. In addition to providing insight into the early gut development, these findings illustrate how large-scale tissue level forces can be traced to developmental signals during vertebrate morphogenesis.

European honey bees (Apis mellifera L.) live in large congested nest cavities with a single opening that limits passive ventilation. When the local air temperature exceeds a threshold, the nests are actively ventilated by bees fanning their wings at the nest entrance. Here, we show that colonies with relatively large nest entrances use an emergent ventilation strategy where fanning bees self-organize to form groups, separating regions of continuous inflow and outflow. The observed spatio-temporal patterns correlate the air velocity and air temperature along the entrances to the distribution of fanning bees. A mathematical model that couples these variables to known fanning behaviour of individuals recapitulates their collective dynamics. Additionally, the model makes predictions about the temporal stability of the fanning group as a function of the temperature difference between the environment and the nest. Consistent with these predictions, we observe that the fanning groups drift, cling to the entrance boundaries, break-up and reform as the ambient temperature varies over a period of days. Overall, our study shows how honeybees use flow-mediated communication to self-organize into a steady state in fluctuating environments.

Several species of millimetric-sized termites across Africa, Asia, Australia, and South America collectively construct large, metersized, porous mound structures that serve to regulate mound temperature, humidity, and gas concentrations. These mounds display varied yet distinctive morphologies that range widely in size and shape. To explain this morphological diversity, we introduce a mathematical model that couples environmental physics to insect behavior: The advection and diffusion of heat and pheromones through a porous medium are modified by the mound geometry and, in turn, modify that geometry through a minimal characterization of termite behavior. Our model captures the range of naturally observed mound shapes in terms of a minimal set of dimensionless parameters and makes testable hypotheses for the response of mound morphology to external temperature oscillations and internal odors. Our approach also suggests mechanisms by which evolutionary changes in odor production rate and construction behavior coupled to simple physical laws can alter the characteristic mound morphology of termites.

Origami structures with a large number of excess folds are capable of storing distinguishable geometric states that are energetically equivalent. As the number of excess folds is reduced, the system has fewer equivalent states and can eventually become rigid. We quantify this transition from a floppy to a rigid state as a function of the presence of folding constraints in a classic origami tessellation, Miura-ori. We show that in a fully triangulated Miura-ori that is maximally floppy, adding constraints via the elimination of diagonal folds in the quads decreases the number of degrees of freedom in the system, first linearly and then nonlinearly. In the nonlinear regime, mechanical cooperativity sets in via a redundancy in the assignment of constraints, and the degrees of freedom depend on constraint density in a scale-invariant manner. A percolation transition in the redundancy in the constraints as a function of constraint density suggests how excess folds in an origami structure can be used to store geometric information in a scale-invariant way.

Bacterial swarming and biofilm formation are collective multicellular phenomena through which diverse microbial species colonize and spread over water-permeable tissue. During both modes of surface translocation, fluid uptake and transport play a key role in shaping the overall morphology and spreading dynamics. Here we develop a generalized two-phase thin-film model that couples bacterial growth, extracellular matrix swelling, fluid flow, and nutrient transport to describe the expansion of both highly motile bacterial swarms, and sessile bacterial biofilms. We show that swarm expansion corresponds to steady-state solutions in a nutrient-rich, capillarity dominated regime. In contrast, biofilm colony growth is described by transient solutions associated with a nutrient-limited, extracellular polymer stress driven limit. We apply our unified framework to explain a range of recent experimental observations of steady and unsteady expansion of microbial swarms and biofilms. Our results demonstrate how the physics of flow and transport in slender geometries serve to constrain biological organization in microbial communities.

Liquid cellular compartments form in the cyto- or nucleoplasm and can regulate aberrant protein aggregation. Yet, the mechanisms by which these compartments affect protein aggregation remain unknown. Here, we combine kinetic theory of protein aggregation and liquid-liquid phase separation to study the spatial control of irreversible protein aggregation in the presence of liquid compartments. We find that even for weak interactions aggregates strongly partition into the liquid compartment. Aggregate partitioning is caused by a positive feedback mechanism of aggregate nucleation and growth driven by a flux maintaining the phase equilibrium between the compartment and its surrounding. Our model establishes a link between specific aggregating systems and the physical conditions maximizing aggregate partitioning into the compartment. The underlying mechanism of aggregate partitioning could be used to confine cytotoxic protein aggregates inside droplet-like compartments but may also represent a common mechanism to spatially control irreversible chemical reactions in general.

Tropisms, growth-driven responses to environmental stimuli, cause plant organs to respond in space and time and reorient themselves. Classical experiments from nearly a century ago reveal that plant shoots respond to the integrated history of light and gravity stimuli rather than just responding instantaneously. We introduce a temporally non-local response function for the dynamics of shoot growth formulated as an integro-differential equation whose solution allows us to qualitatively reproduce experimental observations associated with intermittent and unsteady stimuli. Furthermore, an analytic solution for the case of a pulse stimulus expresses the response function as a function of experimentally tractable variables, which we calculate for the case of the phototropic response of Arabidopsis hypocotyls. All together, our model enables us to predict tropic responses to time-varying stimuli, manifested in temporal integration phenomena, and sets the stage for the incorporation of additional effects such as multiple stimuli, gravitational sagging, etc.

Size control is fundamental in tissue development and homeostasis1,2. Although the role of cell proliferation in these processes has been widely studied, the mechanisms that control embryo size—and how these mechanisms affect cell fate—remain unknown. Here we use the mouse blastocyst as a model to unravel a key role of fluid-filled lumen in the control of embryo size and specification of cell fate. We find that there is a twofold increase in lumenal pressure during blastocyst development, which translates into a concomitant increase in cell cortical tension and tissue stiffness of the trophectoderm that lines the lumen. Increased cortical tension leads to vinculin mechanosensing and maturation of functional tight junctions, which establishes a positive feedback loop to accommodate lumen growth. When the cortical tension reaches a critical threshold, cell–cell adhesion cannot be sustained during mitotic entry, which leads to trophectoderm rupture and blastocyst collapse. A simple theory of hydraulically gated oscillations recapitulates the observed dynamics of size oscillations, and predicts the scaling of embryo size with tissue volume. This theory further predicts that disrupted tight junctions or increased tissue stiffness lead to a smaller embryo size, which we verified by biophysical, embryological, pharmacological and genetic perturbations. Changes in lumenal pressure and size can influence the cell division pattern of the trophectoderm, and thereby affect cell allocation and fate. Our study reveals how lumenal pressure and tissue mechanics control embryo size at the tissue scale, which is coupled to cell position and fate at the cellular scale.

Protein aggregation has been implicated in many medical disorders, including Alzheimer’s and Parkinson’s diseases. Potential therapeutic strategies for these diseases propose the use of drugs to inhibit specific molecular events during the aggregation process. However, viable treatment protocols require balancing the efficacy of the drug with its toxicity, while accounting for the underlying events of aggregation and inhibition at the molecular level. To address this key problem, we combine here protein aggregation kinetics and control theory to determine optimal protocols that prevent protein aggregation via specific reaction pathways. We find that the optimal inhibition of primary and fibril-dependent secondary nucleation require fundamentally different drug administration protocols. We test the efficacy of our approach on experimental data for the aggregation of the amyloid-(1-42) peptide of Alzheimer’s disease in the model organism Caenorhabditis elegans. Our results pose and answer the question of the link between the molecular basis of protein aggregation and optimal strategies for inhibiting it, opening up avenues for the design of rational therapies to control pathological protein aggregation.

The growth, form, and division of prebiotic vesicles, membraneous bags of fluid of varying components and shapes is hypothesized to have served as the substrate for the origin of life. The dynamics of these out-of-equilibrium structures is controlled by physicochemical processes that include the intercalation of amphiphiles into the membrane, fluid flow across the membrane, and elastic deformations of the membrane. To understand prebiotic vesicular forms and their dynamics, we construct a minimal model that couples membrane growth, deformation, and fluid permeation, ultimately couched in terms of two dimensionless parameters that characterize the relative rate of membrane growth and the membrane permeability. Numerical simulations show that our model captures the morphological diversity seen in extant precursor mimics of cellular life, and might provide simple guidelines for the synthesis of these complex shapes from simple ingredients.

Geometric reasoning has an inherent dissonance: its abstract axioms and propositions refer to perfect, idealized entities, whereas its use in the physical world relies on dynamic perception of objects. How do abstract Euclidean concepts, dynamics, and statistics come together to support our intuitive geometric reasoning? Here, we address this question using a simple geometric task – planar triangle completion. An analysis of the distribution of participants’ errors in localizing a fragmented triangle’s missing corner reveals scale-dependent deviations from a deterministic Euclidean representation of planar triangles. By considering the statistical physics of the process characterized via a correlated random walk with a natural length scale, we explain these results and further predict participants’ estimates of the missing angle, measured in a second task. Our model also predicts the results of a categorical reasoning task about changes in the triangle size and shape even when such completion strategies need not be invoked. Taken together, our fndings suggest a critical role for noisy physical processes in our reasoning about elementary Euclidean geometry.

We use optical trapping to continuously bend an isolated microtubule while simultaneously measuring the applied force and the resulting filament strain, thus allowing us to determine its elastic properties over a wide range of applied strains. We find that, while in the lowstrain regime, microtubules may be quantitatively described in terms of the classical Euler-Bernoulli elastic filament, above a critical strain they deviate from this simple elastic model, showing a softening response with increasingdeformations. A three-dimensional thin-shell model, in which the increased mechanical compliance is caused by flattening and eventual buckling of the filament cross-section, captures this softening effect in the high strain regime and yields quantitative values of the effective mechanical properties of microtubules. Our results demonstrate that properties of microtubules are highly dependent on the magnitude of the applied strain and offer a new interpretation for the large variety in microtubule mechanical data measured by different methods.

The release of nanoscale structures from microcapsules, triggered by changes in the capsule in response to external stimuli, has significant potential for active component delivery. Here, we describe an orthogonal strategy for controlling molecular species’ release across oil/water interfaces by modulating their intrinsic self-assembly state. We show that although the soluble peptide Boc-FF can be stably encapsulated for days, its self-assembly into nanostructures triggers jet-like release within seconds. Moreover, we exploit this self-assembly-mediated release to deliver other molecular species that are transported as cargo. These results demonstrate the role of self-assembly in modulating the transport of peptides across interfaces.