We consider the problem of inverting the artifacts associated with scanning a page from an open book, i.e. "xeroxing." The process typically leads to a non-uniform combination of distortion, blurring and darkening owing to the fact that the page is bound to a stiff spine that causes the sheet of paper to be bent inhomogeneously. Complementing purely data-driven approaches, we use knowledge about the geometry and elasticity of the curved sheet to pose and solve a minimal physically consistent inverse problem to reconstruct the image. Our results rely on 3 dimensionless parameters, all of which can be measured for a scanner, and show that we can improve on the data-driven approaches. More broadly, our results might serve as a "textbook" example and a tutorial of how knowledge of generative mechanisms can speed up the solution of inverse problems.

Evolutionary adaptations associated with the formation of a folded cortex in many mammalian brains are thought to be a critical specialization associated with higher cognitive function. The dramatic surface expansion and highly convoluted folding of the cortex during early development is a theme with variations that suggest the need for a comparative study of cortical gyrification. Here, we use a combination of physical experiments using gels, computational morphogenesis, and geometric morphometrics to study the folding of brains across different species. Starting with magnetic resonance images of brains of a newborn ferret, a fetal macaque, and a fetal human, we construct two-layer physical gel brain models that swell superficially in a solvent, leading to folding patterns similar to those seen in vivo. We then adopt a three-dimensional continuum model based on differential growth to simulate cortical folding in silico. Finally, we deploy a comparative morphometric analysis of the in vivo, in vitro, and in silico surface buckling patterns across species. Our study shows that a simple mechanical instability driven by differential growth suffices to explain cortical folding and suggests that variations in the tangential growth and different initial geometries are sufficient to explain the differences in cortical folding across species.

A mechanistic understanding of neurodevelopment requires us to follow the multiscale processes that connect molecular genetic processes to macroscopic cerebral cortical formations and thence to neurological function. Using magnetic resonance imaging of the brain of the ferret, a model organism for studying cortical morphogenesis, we create in vitro physical gel models and in silico numerical simulations of normal brain gyrification. Using observations of genetically manipulated animal models, we identify cerebral cortical thickness and cortical expansion rate as the primary drivers of dysmorphogenesis and demonstrate that in silico models allow us to examine the causes of aberrations in morphology and developmental processes at various stages of cortical ontogenesis. Finally, we explain analogous cortical malformations in human brains, with comparisons with human phenotypes induced by the same genetic defects, providing a unified perspective on brain morphogenesis that is driven proximally by genetic causes and affected mechanically via variations in the geometry of the brain and differential growth of the cortex.

To investigate real-time decision-making in a simplified context, we analyze the trade-offs individuals make using a game of Roulette. In this game, players observe a ball that gradually decelerates as it moves around a circular track, and the participants must predict where the ball will ultimately stop. Players are rewarded for making rapid, accurate predictions, while slower but accurate predictions, or fast yet inaccurate ones, result in lower rewards. In this speed-accuracy trade-off setup we can calculate the optimal timing for placing a bet based on the ball's initial speed and the deceleration rate, given the capacity of an individual's tracking abilities. We find that the participants improve their performance by accruing more reward with each trial and saturates close to the optimal performance, indicating that individuals learn the parameters of the physical model as well as a representation of the form of the reward under constrained time. Looking at the correlation between the bet time of the participant and the optimal betting time, the response delay and reward accrued, we find that the participant can be classified on a novice-expert spectrum. Our study offers ways to quantify human adaptation in competitive and dynamic environments such as sports, which may be helpful in enhancing participant performance.

Random packings of stiff rods are self-supporting mechanical structures stabilized by long-range interactions induced by contacts. To understand the geometrical and topological complexity of the packings, we first deploy X-ray computerized tomography to unveil the structure of the packing. This allows us to directly visualize the spatial variations in density, orientational order, and the entanglement, a mesoscopic field that we define in terms of a local average crossing number, a measure of the topological complexity of the packing. We find that increasing the aspect ratio of the constituent rods in a packing leads to a proliferation of regions of strong entanglement that eventually percolate through the system and correlated with a sharp transition in the mechanical stability of the packing. To corroborate our experimental findings, we use numerical simulations of contacting elastic rods and characterize their stability to …

Inspired by recent experimental observations of a harmonically excited elastic foil hovering near a wall while supporting substantial weight, we develop a theoretical framework that describes the underlying physical effects. Using elastohydrodynamic lubrication theory, we quantify how the dynamic deformation of the soft foil couples to the viscous fluid flow in the intervening gap. Our analysis shows that the soft foil rectifies the reversible forcing, breaking time-reversal symmetry; the relative spatial support of the forcing determines whether the sheet is attracted to or repelled from the wall. A simple scaling law predicts the time-averaged equilibrium hovering height and the maximum weight the sheet can sustain before detaching from the surface. Numerical simulations of the governing equation corroborate our theoretical predictions, are in qualitative agreement with experiments, and might explain the behavior of organisms while providing design principles for soft robotics.

The vertebrate bauplan is primarily established via the formation of embryonic tissues in a head-to-tail progression. The mechanics of this elongation, which requires the presomitic mesoderm (PSM), remain poorly understood. Here, we find that avian PSM explants can elongate autonomously when physically confined in vitro, producing a pushing force promoting posterior elongation of the embryo. This tissue elongation is caused by volumetric expansion, which results from an increase in the extracellular fraction accompanied by graded cellular motility. We show that fibroblast growth factor (FGF) signaling promotes glycolysis-dependent production of hyaluronic acid (HA), which is required for expansion of the posterior PSM. Our findings link body axis elongation to tissue expansion through the metabolic control of extracellular matrix production downstream of FGF signaling.

The transformation of a two-dimensional epithelial sheet into various three-dimensional structures is a critical process in generating the diversity of animal forms. Previous studies of epithelial folding have revealed diverse mechanisms driven by epithelium-intrinsic or -extrinsic forces. Yet little is known about the biomechanical basis of epithelial splitting, which involves extreme folding and eventually a topological transition breaking the epithelial tube. Here, we leverage tracheal-esophageal separation (TES), a critical and highly conserved morphogenetic event during tetrapod embryogenesis, as a model system for interrogating epithelial tube splitting both in vivo and ex vivo. Comparing TES in chick and mouse embryos, we identified an evolutionarily conserved, compressive force exerted by the mesenchyme surrounding the epithelium, as being necessary to drive epithelial constriction and splitting. The compressive force is mediated by localized convergent flow of mesenchymal cells towards the epithelium. We further found that Sonic Hedgehog (SHH) secreted by the epithelium functions as an attractive cue for mesenchymal cells. Removal of the mesenchyme, inhibition of cell migration, or loss of SHH signaling all abrogate TES, which can be rescued by externally applied pressure. These results unveil the biomechanical basis of epithelial splitting and suggest a mesenchymal origin of tracheal-esophageal birth defects.

The archetypal instability of a structure is associated with the eponymous Euler beam, modeled as an inextensible curve which exhibits a supercritical bifurcation at a critical compressive load. In contrast, a soft compressible beam is capable of a subcritical instability, a problem that is far less studied, even though it is increasingly relevant in the context of soft materials and structures. Here, we study the stability of a soft extensible elastic beam on an elastic foundation under the action of a compressive axial force, using the Lyapunov-Schmidt reduction method which we corroborate with numerical calculations. Our calculated bifurcation diagram differs from those associated with the classical Euler-Bernoulli beam, and shows two critical loads, , for each buckling mode . The beam undergoes a supercritical pitchfork bifurcation at for all and slenderness. Due to the elastic foundation, the lower order modes at exhibit subcritical pitchfork bifurcations, and perhaps surprisingly, the first supercritical pitchfork bifurcation point occurs at a higher critical load. The presence of the foundation makes it harder to buckle the elastic beam, but when it does so, it tends to buckle into a more undulated shape. Overall, we find that an elastic support can lead to a myraid of buckled shapes for the classical elastica and one can tune the substrate stiffness to control desired buckled modes -- an experimentally testable prediction.

The dynamics of many macromolecular machines are characterized by chemically mediated structural changes that achieve large-scale functional deployment through local rearrangements of constitutive protein subunits. Motivated by recent high-resolution structural microscopy of a particular class of such machines, contractile injection systems (CISs), we construct a coarse-grained semianalytical model that recapitulates the geometry and bistability of CISs in terms of a minimal set of measurable physical parameters. We use this model to predict the size, shape, and speed of a dynamical actuation front that underlies contraction. Scaling laws for the velocity and physical extension of the contraction front are consistent with our numerical simulations and may be generally applicable to related systems.

We use Brownian dynamics simulations and theory to study the over-damped spatiotemporal dynamics and pattern formation in a fluid-permeated array of equally spaced, active, elastic filaments that are pinned at one end and free at the other. The filaments are modeled as connected colloidal chains with activity incorporated via compressive follower forces acting along the filament backbone. The length of the chains is smaller than the thermal persistence length. For a range of filament separation and activity values, we find that the filament array eventually self-assembles into a series of regularly spaced, kinetically arrested, compact clusters. Filament activity, geometry, elasticity, and grafting density are each seen to crucially influence the size, shape, and spacing of emergent clusters. Furthermore, cluster shapes for different grafting densities can be rescaled into self-similar forms with activity-dependent scaling exponents. We derive theoretical expressions that relate the number of filaments in a cluster and the spacing between clusters, to filament activity, filament elasticity, and grafting density. Our results provide insight into the physical mechanisms involved in the initiation of clustering and suggest that steric contact forces and friction balance active forces and filament elasticity to stabilize the clusters. Our simulations suggest design principles to realize filament-based clusters and similar self-assembling biomimetic materials using active colloids or synthetic microtubule-motor systems.

Posture and its control are fundamental aspects of animal behavior that capture the complex interplay between sensorimotor activity driven by muscular forces and mediated by environmental feedback. An extreme example of this is seen in brown tree snakes and juvenile pythons, which can stand almost upright, with 70% of their body length in the air. We quantify experimental observations of this behavior and present a minimal theoretical framework for postural stability by modeling the snake as an active elastic filament whose shape is controlled by muscular forces. We explore two approaches to characterize the musculature needed to achieve a specific posture: proprioceptive feedback (whereby the snake senses and reacts to its own shape) and a control-theoretic optimization approach (whereby the snake minimizes the expended energy to stand up), and also analyze the dynamic stability of the snake in its upright pose. Our results lead to a three-dimensional postural stability diagram in terms of muscle extent and strength, and gravity, consistent with experimental observations. In addition to general predictions about posture control in animals, our study suggests design principles for robotic mimics.

Pollen tubes, root hairs and fungal hyphae elongate by adding material at their tips. The region in the immediate vicinity of the tip is dynamically pliable and allows for the addition of new wall material even as it flows in response to the turgor that drives it. Any imbalance between the rate of material addition and its consequent flow will cause a tip growing cell to either burst because its wall thins or stop growing because its wall thickens. We use an experimentally-inspired feedback law that couples vesicle exocytosis to wall mechanics via the local strain rate to construct a minimal theory for the dynamics of tip growing cells. Our theory characterizes the parameter regime where stable steady tip growth is possible in terms of two dimensionless parameters: a scaled turgor pressure, and a ratio of length scales describing gradients of wall extensibility and vesicle composition near the tip. Our analysis also explains the experimentally observed shape response of tip growing cells observed when external turgor is dynamically varied. All together our formalism provides a general framework for the coupled dynamics of internal vesicular transport and wall mechanics in tip-growing cells.

Self-organized branching structures can emerge spontaneously as interfacial instabilities in both simple and complex fluids, driven by the interplay between bulk material rheology, boundary constraints, and interfacial forcing. In our experiments, injecting dye between a source and a sink in a Hele-Shaw cell filled with a yield-stress fluid reveals an abrupt transition in morphologies as a function of injection rate. Slow injection leads to a direct path connecting the source to the sink, while fast injection leads to a rapid branching morphology that eventually converges to the sink. This shift from an exploitative (direct) to an exploratory (branched) strategy resembles search strategies in living systems; however, here it emerges in a simple physical system from a combination of global constraints (fluid conservation) and a switch-like local material response. We show that the amount of fluid needed to achieve breakthrough is minimal at the transition, and that there is a trade-off between speed and accuracy in these arborization patterns. Altogether, our study provides an embodied paradigm for fluidic computation driven by a combination of local material response (body) and global boundary conditions (environment).

Synthetic active collectives, composed of many nonliving individuals capable of cooperative changes in group shape and dynamics, hold promise for practical applications and for the elucidation of guiding principles of natural collectives. However, the design of collective robotic systems that operate effectively without intelligence or complex control at either the individual or group level is challenging. We investigate how simple steric interaction constraints between active individuals produce a versatile active system with promising functionality. Here we introduce the link-bot: a V-shape-based, single-stranded chain composed of active bots whose dynamics are defined by its geometric link constraints, allowing it to possess scale- and processing-free programmable collective behaviors. A variety of emergent properties arise from this dynamic system, including locomotion, navigation, transportation, and competitive or cooperative interactions. Through the control of a few link parameters, link-bots show rich usefulness by performing a variety of divergent tasks, including traversing or obstructing narrow spaces, passing by or enclosing objects, and propelling loads in both forward and backward directions. The reconfigurable nature of the link-bot suggests that our approach may significantly contribute to the development of programmable soft robotic systems with minimal information and materials at any scale.

Extracellular matrix (ECM) viscoelasticity broadly regulates cell behavior. While hydrogels can approximate the viscoelasticity of native ECM, it remains challenging to recapitulate the rapid stress relaxation observed in many tissues without limiting the mechanical stability of the hydrogel. Here, we develop macroporous alginate hydrogels that have an order of magnitude increase in the rate of stress relaxation as compared to bulk hydrogels. The increased rate of stress relaxation occurs across a wide range of polymer molecular weights (MWs), which enables the use of high MW polymer for improved mechanical stability of the hydrogel. The rate of stress relaxation in macroporous hydrogels depends on the volume fraction of pores and the concentration of bovine serum albumin, which is added to the hydrogels to stabilize the macroporous structure during gelation. Relative to cell spheroids encapsulated in bulk …

We introduce an additive approach for the design of a class of transformable structures based on two-bar linkages ("scissor mechanisms") joined at vertices to form a two dimensional lattice. Our discussion traces an underlying mathematical similarity between linkage mechanisms, origami, and kirigami and inspires our name for these structures: karigami. We show how to design karigami which unfold from a one dimensional collapsed state to two-dimensional surfaces of single and double curvature. Our algorithm for growing karigami structures is provably complete in providing the ability to explore the full space of possible mechanisms, and we use it to computationally design and physically realize a series of examples of varying complexity.

Patterns arise spontaneously in a range of systems spanning the sciences, and their study typically focuses on mechanisms to understand their evolution in space-time. Increasingly, there has been a transition towards controlling these patterns in various functional settings, with implications for engineering. Here, we combine our knowledge of a general class of dynamical laws for pattern formation in non-equilibrium systems, and the power of stochastic optimal control approaches to present a framework that allows us to control patterns at multiple scales, which we dub the "Hamiltonian bridge". We use a mapping between stochastic many-body Lagrangian physics and deterministic Eulerian pattern forming PDEs to leverage our recent approach utilizing the Feynman-Kac-based adjoint path integral formulation for the control of interacting particles and generalize this to the active control of patterning fields. We demonstrate the applicability of our computational framework via numerical experiments on the control of phase separation with and without a conserved order parameter, self-assembly of fluid droplets, coupled reaction-diffusion equations and finally a phenomenological model for spatio-temporal tissue differentiation. We interpret our numerical experiments in terms of a theoretical understanding of how the underlying physics shapes the geometry of the pattern manifold, altering the transport paths of patterns and the nature of pattern interpolation. We finally conclude by showing how optimal control can be utilized to generate complex patterns via an iterative control protocol over pattern forming pdes which can be casted as gradient flows …

Mechanical metamaterials -- structures with unusual properties that emerge from their internal architecture -- that are designed to undergo large deformations typically exploit large internal rotations, and therefore, necessitate the incorporation of flexible hinges. In the mechanism limit, these metamaterials consist of rigid bodies connected by ideal hinges that deform at zero energy cost. However, fabrication of structures in this limit has remained elusive. Here, we demonstrate that the fabrication and integration of textile hinges provides a scalable platform for creating large structured metamaterials with mechanism-like behaviors. Further, leveraging recently introduced kinematic optimization tools, we demonstrate that textile hinges enable extreme shape-morphing responses, paving the way for the development of the next generation of mechanism-based metamaterials.

Kirigami is the art of cutting paper to make it articulated and deployable, allowing for it to be shaped into complex two and three-dimensional geometries. The mechanical response of a kirigami sheet when it is pulled at its ends is enabled and limited by the presence of cuts that serve to guide the possible non-planar deformations. Inspired by the geometry of this art form, we ask two questions: (i) What is the shortest path between points at which forces are applied? (ii) What is the nature of the ultimate shape of the sheet when it is strongly stretched? Mathematically, these questions are related to the nature and form of geodesics in the Euclidean plane with linear obstructions (cuts), and the nature and form of isometric immersions of the sheet with cuts when it can be folded on itself, and are related to a class of questions associated with isometric immersions in differential geometry and geodesics in discrete and …

Safe, all-solid-state lithium metal batteries enable high energy density applications, but suffer from instabilities during operation that lead to rough interfaces between the metal and electrolyte and subsequently cause void formation and dendrite growth that degrades performance and safety. Inspired by the morphogenetic control of thin lamina such as tree leaves that robustly grow into flat shapes -- we propose a range of approaches to control lithium metal stripping and plating. To guide discovery of materials that will implement these feedback mechanisms, we develop a reduced order model that captures couplings between mechanics, interface growth, temperature, and electrochemical variables. We find that long-range feedback is required to achieve true interface stability, while approaches based on local feedback always eventually grow into rough interfaces. All together, our study provides the beginning of a practical framework for analyzing and designing stable electrochemical interfaces in terms of the mechanical properties and the physical chemistry that underlie their dynamics.

When placed on an inclined plane, a perfect 2D disk or 3D sphere simply rolls down in a straight line under gravity. But how is the rolling affected if these shapes are irregular or random? Treating the terminal rolling speed as an order parameter, we show that phase transitions arise as a function of the dimension of the state space and inertia. We calculate the scaling exponents and the macroscopic lag time associated with the presence of first and second order transitions, and describe the regimes of co-existence of stable states and the accompanying hysteresis. Experiments with rolling cylinders corroborate our theoretical results on the scaling of the lag time. Experiments with spheres reveal closed orbits and their period-doubling in the overdamped and inertial limits respectively, providing visible manifestations of the hairy ball theorem and the doubly-connected nature of SO(3), the space of 3-dimensional rotations. Going beyond simple curiosity, our study might be relevant in a number of natural and artificial systems that involve the rolling of irregular objects, in systems ranging from nanoscale cellular transport to robotics.

The synthesis of life from non-living matter has captivated and divided scientists for centuries. This bold goal aims at unraveling the fundamental principles of life and leveraging its unique features, such as its resilience, sustainability, and ability to evolve. Synthetic life represents more than an academic milestone—it has the potential to revolutionize biotechnology, medicine, and materials science. Although the fields of synthetic biology, systems chemistry, and biophysics have made great strides toward synthetic life, progress has been hindered by social, philosophical, and technical challenges, such as vague goals, misaligned interdisciplinary efforts, and incompletely addressing public and ethical concerns. Our perspective offers a roadmap toward the synthesis of life based on discussions during a 2-week workshop with scientists from around the globe.

Temporal imaging of biological epithelial structures yields shape data at discrete time points, leading to a natural question: how can we reconstruct the most likely path of growth patterns consistent with these discrete observations? We present a physically plausible framework to solve this inverse problem by creating a framework that generalizes quasiconformal maps to quasiconformal flows. By allowing for the spatio-temporal variation of the shear and dilatation fields during the growth process, subject to regulatory mechanisms, we are led to a type of generalized Ricci flow. When guided by observational data associated with surface shape as a function of time, this leads to a constrained optimization problem. Deploying our data-driven algorithmic approach to the shape of insect wings, leaves and even sculpted faces, we show how optimal quasiconformal flows allow us to characterize the morphogenesis of a range of surfaces.

Hox transcription factors play crucial roles in organizing developmental patterning across metazoa, but how these factors trigger regional morphogenesis has largely remained a mystery. In the developing gut, Hox genes help demarcate identities of intestinal subregions early in embryogenesis, which ultimately leads to their specialization in both form and function. Although the midgut forms villi, the hindgut develops sulci that resolve into heterogeneous outgrowths. Combining mechanical measurements of the embryonic chick intestine and mathematical modeling, we demonstrate that the posterior Hox gene HOXD13 regulates biophysical phenomena that shape the hindgut lumen. We further show that HOXD13 acts through the transforming growth factor β (TGF-β) pathway to thicken, stiffen, and promote isotropic growth of the subepithelial mesenchyme—together, these features lead to hindgut-specific surface buckling. TGF-β, in turn, promotes collagen deposition to affect mesenchymal geometry and growth. We thus identify a cascade of events downstream of positional identity that direct posterior intestinal morphogenesis.

Tissue buckling is an increasingly appreciated mode of morphogenesis in the embryo, but it is often unclear how geometric and material parameters are molecularly determined in native developmental contexts to generate diverse functional patterns. Here, we study the link between differential mechanical properties and the morphogenesis of distinct anteroposterior compartments in the intestinal tract—the esophagus, small intestine, and large intestine. These regions originate from a simple, common tube but adopt unique forms. Using measured data from the developing chick gut coupled with a minimal theory and simulations of differential growth, we investigate divergent lumen morphologies along the entire early gut and demonstrate that spatiotemporal geometries, moduli, and growth rates control the segment-specific patterns of mucosal buckling. Primary buckling into wrinkles, folds, and creases along the gut, as well as secondary buckling phenomena, including period-doubling in the foregut and multiscale creasing-wrinkling in the hindgut, are captured and well explained by mechanical models. This study advances our existing knowledge of how identity leads to form in these regions, laying the foundation for future work uncovering the relationship between molecules and mechanics in gut morphological regionalization.

Muscle is a complex, hierarchically organized, soft contractile engine. To understand the limits on the rate of contraction and muscle energetics, we construct a coarse-grained multiscale model that describes muscle as an active sponge. Our analysis of existing experiments across species and muscle types highlights the importance of spatially heterogeneous strains and local volumetric deformations. Our minimal theoretical model shows how contractions induce intracellular fluid flow and power active hydraulic oscillations, yielding the limits of ultrafast muscular contractions. We further demonstrate that the viscoelastic response of muscle is naturally non-reciprocal—or odd—owing to its active and anisotropic nature. This enables an alternate mode of muscular power generation from periodic cycles in spatial strain alone, contrasting with previous descriptions based on temporal cycles. Our work suggests a revised view of muscle dynamics that emphasizes the multiscale spatiotemporal origins of soft hydraulic power, with potential implications for physiology, biomechanics and locomotion.

Organs have a distinctive yet often overlooked spatial arrangement in the body1,2,3,4,5. We propose that there is a logic to the shape of an organ and its proximity to its neighbours. Here, by using volumetric scans of many Drosophila melanogaster flies, we develop methods to quantify three-dimensional features of organ shape, position and interindividual variability. We find that both the shapes of organs and their relative arrangement are consistent yet differ between the sexes, and identify unexpected interorgan adjacencies and left–right organ asymmetries. Focusing on the intestine, which traverses the entire body, we investigate how sex differences in three-dimensional organ geometry arise. The configuration of the adult intestine is only partially determined by physical constraints imposed by adjacent organs; its sex-specific shape is actively maintained by mechanochemical crosstalk between gut muscles and vascular-like trachea. Indeed, sex-biased expression of a muscle-derived fibroblast growth factor-like ligand renders trachea sexually dimorphic. In turn, tracheal branches hold gut loops together into a male or female shape, with physiological consequences. Interorgan geometry represents a previously unrecognized level of biological complexity which might enable or confine communication across organs and could help explain sex or species differences in organ function.

Stem cell-derived kidney organoids contain nephron segments that recapitulate morphological and functional aspects of the human kidney. However, directed differentiation protocols for kidney organoids are largely conducted using biochemical signals to control differentiation. Here, the hypothesis that mechanical signals regulate nephrogenesis is investigated in 3D culture by encapsulating kidney organoids within viscoelastic alginate hydrogels with varying rates of stress relaxation. Tubular nephron segments are significantly more convoluted in kidney organoids differentiated in encapsulating hydrogels when compared with those in suspension culture. Hydrogel viscoelasticity regulates the spatial distribution of nephron segments within the differentiating kidney organoids. Consistent with these observations, a particle-based computational model predicts that the extent of deformation of the hydrogel–organoid interface regulates the morphology of nephron segments. Elevated extracellular calcium levels in the culture medium, which can be impacted by the hydrogels, decrease the glomerulus-to-tubule ratio of nephron segments. These findings reveal that hydrogel encapsulation regulates nephron patterning and morphology and suggest that the mechanical microenvironment is an important design variable for kidney regenerative medicine.

During vertebrate gastrulation, an embryo transforms from a layer of epithelial cells into a multilayered gastrula. This process requires the coordinated movements of hundreds to tens of thousands of cells, depending on the organism. In the chick embryo, patterns of actomyosin cables spanning several cells drive coordinated tissue flows. Here, we derive a minimal theoretical framework that couples actomyosin activity to global tissue flows. Our model predicts the onset and development of gastrulation flows in normal and experimentally perturbed chick embryos, mimicking different gastrulation modes as an active stress instability. Varying initial conditions and a parameter associated with active cell ingression, our model recapitulates distinct vertebrate gastrulation morphologies, consistent with recently published experiments in the chick embryo. Altogether, our results show how changes in the patterning of critical cell behaviors associated with different force-generating mechanisms contribute to distinct vertebrate gastrulation modes via a self-organizing mechanochemical process.

The rapid evolution of three-dimensional (3D) printing technology has enabled new manufacturing capabilities1,2 and allowed for the creation of complex patterns via the additive deposition of complex materials through a computer-controlled nozzle. However, the ultimate limits on print quality are imposed by the rheodynamics of the printed material that is a function of material rheology and the ability to design robust print paths for the nozzle. Defects in printing typically arise in non-uniformly extruded/deposited material and instabilities such as folding and coiling of fluid jets.3 A simple protocol to prevent these defects follows by forcing the nozzle trajectory to exactly mimic the target print pattern from a very small height offset.

In biology, epithelial morphogenesis of thin sheet-like structures in plants and animals is responsible for the vast majority of functional structures that make up organs and organisms. These may be modeled effectively as two-dimensional surfaces whose geometry is driven by active processes that are intimately connected to the presence of orientational order in the tangent plane that modifies the embedding and in turn is modified by it. The nature of in-plane order is akin to that of polar molecules, liquid crystals, etc., or more generally to p-fold rotational order, denoted as ‘‘p-atics’’.1 There is a growing body of evidence suggesting that topological defects, singular disruptions of the rotational order, play a crucial role in guiding or controlling morphogenesis, as seen in experimental observations of cell extrusion and apoptosis,2 mound formation,3,4 layer formation,5 and body shaping using bulges, pits and tentacles.6 These observations suggest a natural question: how might one construct a theoretical framework that accounts for the dynamics of topological defects on non-uniform surfaces which themselves deform in response to the forces induced by the defects?

Adaptive avian radiations associated with the diversification of bird beaks into a multitude of forms enabling different functions are exemplified by Darwin’s finches and Hawaiian honeycreepers. To elucidate the nature of these radiations, we quantified beak shape and skull shape using a variety of geo- metric measures that allowed us to collapse the variability of beak shape into a minimal set of geometric parameters. Furthermore, we find that just two measures of beak shape—the ratio of the width to length and the normal- ized sharpening rate (increase in the transverse beak curvature near the tip relative to that at the base of the beak)—are strongly correlated with diet. Finally, by considering how transverse sections to the beak centreline evolve with distance from the tip, we show that a simple geometry-driven growth law termed ‘modified mean curvature flow’ captures the beak shapes of Darwin’s finches and Hawaiian honeycreepers. A surprising consequence of the simple growth law is that beak shapes that are not allowed based on the developmental programme of the beak are also not observed in nature, suggesting a link between evolutionary morphology and development in terms of growth-driven developmental constraints.

Experiments over many decades are suggestive that the combination of cellular contractility and active phase separation in cell-matrix composites can enable spatiotemporal patterning in multicellular tissues across scales. To characterize these phenomena, we provide a general theory that incorporates active cellular contractility into the classical Cahn-Hilliard-Larché model for phase separation in passive viscoelastic solids. Within this framework, we show how a homogeneous cell-matrix mixture can be destabilized by activity via either a pitchfork or Hopf bifurcation, resulting in stable phase separation and/or traveling waves. Numerical simulations of the full equations allow us to track the evolution of the resulting self-organized patterns in periodic and mechanically constrained domains, and in different geometries. Altogether, our study underscores the importance of integrating both cellular activity and mechanical phase separation in understanding patterning in soft, active biosolids in both in vivo and in vitro settings.

When a hyperelastic hydrogel confined between two parallel glass plates begins to dry from a lateral boundary, the volume lost by evaporation is accommodated by an inward displacement of the air-hydrogel interface that induces an elastic deformation of the hydrogel. Once a critical front displacement is reached, we observe intermittent fracture events initiated by a geometric instability resulting in localized bursts at the interface. These bursts relax the stresses and irreversibly form air cavities that lead to cellular networks. We show that the spatial extent of the strain field prior to a burst, influenced by the air-hydrogel interfacial tension and the confinement of the gel, determines the characteristic size of the cavities.

Effective multicellularity requires both cooperation and competition between constituent cells. Cooperation involves sacrificing individual fitness in favour of that of the community, but excessive cooperation makes the community susceptible to senescence and ageing. Competition eliminates unfit senescent cells via natural selection and thus slows down ageing, but excessive competition makes the community susceptible to cheaters, as exemplified by cancer and cancer-like phenomena. These observations suggest that an optimal level of intercellular competition in a multicellular organism maximizes organismal vitality by delaying the inevitability of ageing. We quantify this idea using a statistical mechanical framework that leads to a generalized replicator dynamical system for the population of cells that changes their vitality and cooperation due to somatic mutations that make them susceptible to ageing and/or cancer. By accounting for the cost of cooperation and strength of competition in a minimal setting, we show that our model predicts an optimal value of competition that maximizes vitality and delays the inevitability of senescence or cancer. The results have implications for the design of strategies aimed at delaying ageing in biological, technical and social systems that exhibit similar processes.

Growing a flat lamina such as a leaf is almost impossible without some feedback to stabilize long wavelength modes that are easy to trigger since they are energetically cheap. Here we combine the physics of thin elastic plates with feedback control theory to explore how a leaf can remain flat while growing. We investigate both in-plane (metric) and out-of-plane (curvature) growth variation and account for both local and nonlocal feedback laws. We show that a linearized feedback theory that accounts for both spatially nonlocal and temporally delayed effects suffices to suppress long wavelength fluctuations effectively and explains recently observed statistical features of growth in tobacco leaves. Our work provides a framework for understanding the regulation of the shape of leaves and other leaflike laminar objects.

There is growing need for hybrid curricula that integrate constructivist methods from Science and Technology Studies (STS) into both engineering and policy courses at the undergraduate and graduate levels. However, institutional and disciplinary barriers have made implementing such curricula difficult at many institutions. While several programs have recently been launched that mix technical training with consideration of “societal” or “ethical issues,” these programs often lack a constructivist element, leaving newly-minted practitioners entering practical fields ill-equipped to unpack the politics of knowledge and technology or engage with skeptical publics. This paper presents a novel format for designing interdisciplinary coursework that combines conceptual content from STS with training in engineering and policy. Courses following this format would ideally be team taught by instructors with advanced training in diverse fields, and hence co-learning between instructors and disciplines is a key element of the format. Several instruments for facilitating both student and instructor collaborative learning are introduced. The format is also designed for versatility: in addition to being adaptable to both technical and policy training environments, topics are modularized around a conceptual core so that issues ranging from biotech to nuclear security can be incorporated to fit programmatic needs and resources.

We present an additive approach for the inverse design of kirigami-based mechanical metamaterials by focusing on the empty (negative) spaces instead of the solid tiles. By considering each negative space as a four-bar linkage, we identify a simple recursive relationship between adjacent linkages, yielding an efficient method for creating kirigami patterns. This allows us to solve the kirigami design problem using elementary linear algebra, with compatibility, reconfigurability and rigid-deployability encoded into an iterative procedure involving simple matrix multiplications. The resulting linear design strategy circumvents the solution of a non-convex global optimization problem and allows us to control the degrees of freedom in the deployment angle field, linkage offsets and boundary conditions. We demonstrate this by creating a large variety of rigid-deployable, compact, reconfigurable kirigami patterns. We then realize our kirigami designs physically using two simple but effective fabrication strategies with very different materials. Altogether, our additive approaches present routes for efficient mechanical metamaterial design and fabrication based on ori/kirigami art forms.

The presence of incomplete cuts in a thin planar sheet can dramatically alter its mechanical and geometrical response to loading, as the cuts allow the sheet to deform strongly in the third dimension, most beautifully demonstrated in kirigami art-forms. We use numerical experiments to characterize the geometric mechanics of kirigamized sheets as a function of the number, size and orientation of cuts. We show that the geometry of mechanically loaded sheets can be approximated as a composition of simple developable units: flats, cylinders, cones and compressed Elasticae. This geometric construction yields scaling laws for the mechanical response of the sheet in both the weak and strongly deformed limit. In the ultimately stretched limit, this further leads to a theorem on the nature and form of geodesics in an arbitrary kirigami pattern, consistent with observations and simulations. Finally, we show that by varying the shape and size of the geodesic in a kirigamized sheet, we can control the deployment trajectory of the sheet, and thence its functional properties as an exemplar of a tunable structure that can serve as a robotic gripper, a soft light window or the basis for a physically unclonable device. Overall our study of disordered kirigami sets the stage for controlling the shape and shielding the stresses in thin sheets using cuts.

Drawing on elementary invariance principles, we propose that a statistical geometric object, the probability distribution of the normalized contour curvatures (NCC) in the intensity field of a planar image has the potential to categorize objects. We show that NCC is sufficient for discriminating between cognitive categories such as animacy, size and type, and demonstrate the robustness of this metric to variation in illumination and viewpoint, consistent with psychological experiments. A generative model for producing artificial images with the observed NCC distributions highlights the key features that our metric captures, and those that it does not. More broadly, our study points to the need for statistical geometric approaches to cognition that build in both the statistics and the natural invariances of the visual world.

Active fluids, such as cytoskeletal filaments, bacterial colonies and epithelial cell layers, exhibit distinctive orientational coherence, often characterized by nematic order and its breakdown, defined by the presence of topological defects. In contrast, little is known about positional coherence, that is, whether there is an organization in the underlying fluid motion—despite this being both a prominent and an experimentally accessible feature. Here we characterize the organization of fluid motion in active nematics using the notion of Lagrangian coherent structures by analyzing experimental data of two-dimensional mixtures of microtubules and kinesin, as well as numerical data obtained from the simulation of the active nematodynamic equations. Coherent structures consist of moving attractors and repellers, which orchestrate complex motion. To understand the interaction of positional and orientational coherence, we analyse experiments and simulations and find that +1/2 defects move and deform the attractors, functioning as control centres for collective motion. Additionally, we find that regions around isolated +1/2 defects undergo high bending and low stretching/shearing deformations, consistent with the local stress distribution. The stress is the minimum at the defect, whereas high differential stress along the defect orientation induces folding. Our work offers a new perspective to describe and control self-organization in active fluids, with potential applications to multicellular systems.

The steelpan is a pitched percussion instrument that takes the form of a concave bowl with several localized dimpled regions of varying curvature. Each of these localized zones, called notes, can vibrate independently when struck, and produce a sustained tone of a well-defined pitch. While the association of the localized zones with individual notes has long been known and exploited, the relationship between the shell geometry and the strength of the mode confinement remains unclear. Here, we explore the spectral properties of the steelpan modelled as a vibrating elastic shell. To characterize the resulting eigenvalue problem, we generalize a recently developed theory of localization landscapes for scalar elliptic operators to the vector-valued case, and predict the location of confined eigenmodes by solving a Poisson problem. A finite-element discretization of the shell shows that the localization strength is determined by the difference in curvature between the note and the surrounding bowl. In addition to providing an explanation for how a steelpan operates as a two-dimensional xylophone, our study provides a geometric principle for designing localized modes in elastic shells.

Developmental morphogenesis is driven by tissue stresses acting on tissue rheology. Direct measurements of forces in small tissues (100 µm-1 mm) in situ, such as in early embryos, require high spatial precision and minimal invasiveness. Here, we introduce a control-based approach, tissue force microscopy (TiFM), that integrates a mechanical cantilever probe and live imaging with closed-loop feedback control of mechanical loading in early chicken embryos. By testing previously qualitatively characterized force-producing tissues in the elongating body axis, we show that TiFM quantitatively captures stress dynamics with high sensitivity. TiFM also provides the means to apply stable, minimally invasive and physiologically relevant loads to drive tissue deformation and to follow the resulting morphogenetic progression associated with large-scale cell movements. Together, TiFM allows us to control tissue force measurement and manipulation in small developing embryos, and promises to contribute to the quantitative understanding of complex multi-tissue mechanics during development.

Controlling the connectivity and rigidity of kirigami, i.e. the process of cutting paper to deploy it into an articulated system, is critical in the manifestations of kirigami in art, science and technology, as it provides the resulting metamaterial with a range of mechanical and geometric properties. Here, we combine deterministic and stochastic approaches for the control of rigidity in kirigami using the power of k choices, an approach borrowed from the statistical mechanics of explosive percolation transitions. We show that several methods for rigidifying a kirigami system by incrementally changing either the connectivity or the rigidity of individual components allow us to control the nature of the explosive transition by a choice of selection rules. Our results suggest simple lessons for the design of mechanical metamaterials.

The solution of complex problems by the collective action of simple agents in both biologically evolved and synthetically engineered systems involves cooperative action. Understanding the resulting emergent solutions requires integrating across the organismal behavior of many individuals. Here, we investigate an ecologically relevant collective task in black carpenter ants Camponotus pennsylvanicus: excavation of a soft, erodible confining corral. These ants show a transition from individual exploratory excavation at random locations to spatially localized collective exploitative excavation and escape from the corral. Agent-based simulations and a minimal continuum theory that coarse-grains over individual actions and considers their integrated influence on the environment leads to the emergence of an effective phase space of behaviors, characterized in terms of excavation strength and cooperation intensity. To test the theory over the range of both observed and predicted behaviors, we use custom-built robots (RAnts) that respond to stimuli to characterize the phase space of emergence (and failure) of cooperative excavation. Tuning the amount of cooperation between RAnts, allows us to vary the efficiency of excavation and synthetically generate the entire range of macroscopic phases predicted by our theory. Overall, our approach shows how the cooperative completion of tasks can arise from simple rules that involve the interaction of agents with a dynamically changing environment that serves as both an enabler and a modulator of behavior.

The persistence of imperfect mimicry in nature presents a challenge to mimicry theory. Some hypotheses for the existence of imperfect mimicry make differing predictions depending on how mimetic fidelity is measured. Here, we measure mimetic fidelity in a brood parasite–host system using both trait-based and response-based measures of mimetic fidelity. Cuckoo finches Anomalospiza imberbis lay imperfectly mimetic eggs that lack the fine scribbling characteristic of eggs of the tawny-flanked prinia Prinia subflava, a common host species. A trait-based discriminant analysis based on Minkowski functionals—that use geometric and topological morphometric methods related to egg pattern shape and coverage—reflects this consistent difference between host and parasite eggs. These methods could be applied to quantify other phenotypes including stripes and waved patterns. Furthermore, by painting scribbles onto cuckoo finch eggs and testing their rate of rejection compared to control eggs (i.e. a response-based approach to quantify mimetic fidelity), we show that prinias do not discriminate between eggs based on the absence of scribbles. Overall, our results support relaxed selection on cuckoo finches to mimic scribbles, since prinias do not respond differently to eggs with and without scribbles, despite the existence of this consistent trait difference.

The physics of signal propagation in a collection of organisms that communicate with each other both enables and limits how active excitations at the individual level reach, recruit and lead to collective patterning. Inspired by the spatio-temporal patterns in a planar swarm of bees that use pheromones and fanning flows to recruit additional bees, we develop a theoretical framework for patterning via active flow-based recruitment. Our model generalizes the well-known Patlak–Keller–Segel model of diffusion dominated aggregation and leads to an enhanced phase space of patterns spanned by two dimensionless parameters that measure the scaled stimulus/activity and the scaled chemotactic response. Together these determine the efficacy of signal communication via fluid flow (which we dub rheomergy) that leads to a variety of migration and aggregation patterns, consistent with observations.

The morphology of gastrulation driving the internalization of the mesoderm and endoderm differs markedly among vertebrate species. It ranges from involution of epithelial sheets of cells through a circular blastopore in amphibians to ingression of mesenchymal cells through a primitive streak in amniotes. By targeting signaling pathways controlling critical cell behaviors in the chick embryo, we generated crescent- and ring-shaped mesendoderm territories in which cells can or cannot ingress. These alterations subvert the formation of the chick primitive streak into the gastrulation modes seen in amphibians, reptiles, and teleost fish. Our experimental manipulations are supported by a theoretical framework linking cellular behaviors to self-organized multicellular flows outlined in detail in the accompanying paper. Together, this suggests that the evolution of gastrulation movements is largely determined by changes in a few critical cell behaviors in the mesendoderm territory across different species and controlled by a relatively small number of signaling pathways.