Efficient navigation in swarms often relies on the emergence of decentralized approaches that minimize traversal time or energy. Stigmergy, where agents modify a shared environment that then modifies their behavior, is a classic mechanism that can encode this strategy. We develop a theoretical framework for stigmergic transport by casting it as a stochastic optimal control problem: agents (collectively) lay and (individually) follow trails while minimizing expected traversal time. Simulations and analysis reveal two emergent behaviors: path straightening in homogeneous environments and path refraction at material interfaces, both consistent with experimental observations of insect trails. While reminiscent of Fermat's principle, our results show how local, noisy agent+field interactions can give rise to geodesic trajectories in heterogeneous environments, without centralized coordination or global knowledge, relying instead on an embodied slow fast dynamical mechanism.

Patterns in non-equilibrium systems—from phase separation to tissue morphogenesis—arise from microscopic stochastic dynamics but are most easily specified and measured at macroscopic scales. This work introduces a unified physics-based framework for generative control of such patterns that bridges particle (Lagrangian) and field (Eulerian) descriptions. Starting from overdamped Langevin dynamics for interacting particles with internal states, this work defines smoothed mesoscopic fields and a free-energy functional that governs macroscopic evolution. The resulting flows are projected back to particles via smoothed-particle hydrodynamics, closing a micro-macro modeling loop. Control objectives are posed at the field level but actuated at the particle level, enabling scale-dependent design. To solve the resulting high-dimensional stochastic control problem, an adjoint-based path integral control (APIC) is introduced, which marries an exact Feynman-Kac linearization of the Hamilton-Jacobi-Bellman (HJB) equation with continuous-time backpropagation and automatic differentiation to compute exact gradients of the value functional from sampled trajectories. Instead of solving the full HJB equation, APIC samples uncontrolled trajectories forward and propagates adjoint co-states backward to compute control gradients efficiently. The controlled dynamics define a Hamiltonian bridge: a physically constrained interpolation between initial and target patterns that respects conservation laws and energetic structure. The framework is demonstrated across canonical systems—navigation on complex landscapes, Allen-Cahn (non-conserved) and Cahn-Hilliard (conserved) phase separation, thin-film droplet self-assembly, reaction-diffusion patterning, and a cell-fate/migration model—showing scalable control and interpretable transport geometries. Conceptually, this links generative modeling and optimal control with variational physics, offering a route to programmable pattern formation in soft matter, active media, and developmental systems and suggesting a path toward inference-in-the-loop and experimental actuation.

Effective wound healing relies on thousands of cells collectively migrating to close the gap. It is increasingly clear that part of this migration is driven by endogenous electrochemical fields which point towards the center of skin wounds and guide collective cell migration through “electrotaxis”. Mounting evidence suggests exogenously applied electric fields can accelerate healing, but progress in this field has been limited by the use of brute-force, global stimulation strategies that are wound agnostic and reflect neither the collective nature of the healing process nor the dynamic nature of the wound geometry. Here, we develop an experimental system to study how monolayer mouse skin responds to spatiotemporally patterned electric fields. We first show that local electrical stimulation can produce near global cell migration responses arising from cell-cell mechanical adhesion. We apply this strategy to 2D circular wounds using a local ring electric field near the wound edge and reveal how to tune the timing of local stimulation to avoid cellular jamming. Finally, we integrate our findings with a biophysics-informed optimal control strategy to tune both when and where the electric field should be applied in time, resulting in dramatic improvements to healing.

We explore the dynamical response of the free surface of an ultra-soft solid driven by a localized moving pressure disturbance. Experiments reveal a steady V-shaped wake analogous to a surface Mach wedge. A simple geometric argument provides a qualitative explanation consistent with observations. A theoretical framework combining elastodynamic, capillary, and gravitational effects yields a generalized dispersion relation that smoothly interpolates between Kelvin's theory of liquid interface wakes and Rayleigh's theory of elastic surface waves. Together, our experiments and theory reveal the existence of a soft wake regime that bridges fluid and solid surface wave physics, offering new routes for probing the dynamics of soft surfaces.

Ducks display a unique and dramatic sexual dimorphism in their vocal organ, the syrinx. Males have a left-sided bulla that is not present in females and that has been long hypothesized to play a role in courtship vocalizations, though this connection has never been tested. The large, hollow morphology of the bulla and its proximity to the sound-producing vocal folds introduce the possibility that it may work as a Helmholtz resonator, which makes it possible to predict the resonance frequencies enhanced by this structure. We found that during early ontogeny, the distribution of energy across the harmonic spectrum of contact calls is not different between males and females. We then used microcomputed tomography (µCT) scans of duck syringes to estimate resonance frequencies of the bullae and compared these with spectral features of their vocalizations. This comparison overall supports the idea that the bulla resonance may specifically enhance aspects of courtship vocalizations, especially in species that have a tonal courtship whistle. This was further supported when we tested the frequencies produced when air was blown through 3D printed bullae. We also saw potential influence of the bulla in non-courtship vocalizations, which could be explored further with a greater understanding of the input of other vocal tract features that influence vocalization. We observed that, in general, excepting the common eider, bulla size shows a weak positive correlation with male bird body mass. This study provides support for the long-held hypothesis that the adult male duck bulla influences resonance frequencies, in particular in courtship vocalizations.

Brains likely evolved from diffuse nerve nets in the Precambrian, but we do not know what the first brains looked like or how they were organized. Acoel worms, the sister lineage to all other animals with brains, offer a window into this transition. We studied the three-banded panther worm Hofstenia miamia, whose brain is diffuse and unlike any previously described: it shows little anatomical or functional regionalization or stereotypy. Worms forage successfully even after large portions of the brain are removed, suggesting most regions can perform most computations. Neural cell type markers are also distributed across the brain with little regionalization. High-resolution studies of hunting reveal that more brain tissue improves performance, but no specific brain region is required. These results lead us to propose that H. miamia’s brain is built from computationally pluripotent ‘tiles’, whose interactions generate coherent behavior. This architecture suggests that early brains arose by condensation of diffuse nerve nets into unregionalized brains, with regionalization evolving secondarily.

Soft and frangible materials that remodel under flow can give rise to branched patterns shaped by material properties, boundary conditions, and the time scales of forcing. We present a general theoretical framework for emergent branching in these frangible (or threshold) materials that switch abruptly from resisting flow to permitting flow once local stresses exceed a threshold, relevant for examples as varied as dielectric breakdown of insulators and the erosion of soft materials. Simulations in 2D and 3D show that branching is adaptive and tunable via boundary conditions and domain geometry, offering a foundation for self-organized engineering of functional transport architectures.

Epithelial cortinoids, fluid filled shells formed from induced pluripotent stem cells (iPSCs), must accommodate large deformations during growth and morphogenesis. Using inflation–deflation assays and high-resolution imaging, we find that these fluid-filled shells are weakly-pressurized and achieve extreme deformability through reversible soft modes of deformation accommodated by the cytoskeleton. We show that cytoskeletal elements such as actin localized along lateral cell edges undergo tilt and bend instabilities that buffer mechanical load by decoupling apico–basal stretching from lateral extension. These reversible instabilities act as elastic safety valves, permitting large shape changes without loss of epithelial hydraulic and topological integrity. A minimal theoretical and computational model demonstrates how tilt and bend reduce effective resistance to radial thinning and explains the observed pressure–strain softening. Thus, iPSC shells exploit reversible cytoskeletal instabilities as mechanical buffers, enabling robust tolerance of large deformations in developing epithelia.

Analyzing kinetic experiments on protein aggregation using integrated rate laws has led to numerous advances in our understanding of the fundamental chemical mechanisms behind amyloidogenic disorders such as Alzheimer's and Parkinson's diseases. However, the description of biologically relevant processes may require rate equations that are too complex to solve using existing methods, hindering mechanistic insights into these processes. An example of significance is coaggregation in environments containing multiple amyloid-beta (Aβ) peptide alloforms, which may play a crucial role in the biochemistry of Alzheimer's disease but whose mechanism is still poorly understood. Here, we use the mathematics of Lie symmetry to derive a general integrated rate law valid for most plausible linear self-assembly reactions. We use it in conjunction with experimental data to determine the mechanism of …

Inspired by recent experiments demonstrating that vibrating elastic sheets can function as seemingly contactless suction cups, we investigate the elastohydrodynamic hovering of a thin elastic sheet vibrating near a rigid substrate. Previous theoretical work suggests that the hovering height results from a balance between the active forcing that triggers the vibrations, the bending forces associated with the sheet's deformation, the viscous lubrication flow between the sheet and the substrate, and the sheet's weight. Here, we extend this analysis beyond the asymptotic regime of weak forcing and explore the regime of strong forcing through numerical simulations. We further quantify the influence of fluid inertia and compressibility on the equilibrium hovering height and the maximum load that can be supported. Both effects are found to introduce repulsive contributions to the net force on the sheet, which can significantly reduce its adhesive strength. Beyond providing insights into soft contactless grippers and swimming near surfaces, our analysis is relevant to the elastohydrodynamics of squeeze films and near-field acoustic levitation.

Collective cell migration lies at the intersection of developmental biology and non-equilibrium physics, where active processes give rise to emergent patterns that are biologically relevant. Here, we investigate dilatational modes--cycles of expansion and contraction--in epithelial monolayers, and show that the divergence of the velocity field exhibits robust, large-scale temporal oscillations. These oscillatory patterns, reminiscent of excitable media and their biological analogs, emerge spontaneously from the coupled dynamics of actively pulsing cells. We find that the temporal persistence of these oscillations varies non-monotonically with cell density: synchrony initially increases with density, reaches a maximum at intermediate densities and is lost at higher values. This trend mirrors changes in the spatial correlation length of cell-cell interactions, and the density of topological defects in the system, suggesting a shared physical origin. We develop a continuum model in which a complex-valued Ginzburg-Landau-type field that governs the amplitude and phase of oscillations is coupled to local cell density. Simulations reproduce the observed behavior, revealing that local density adapts to phase patterns, reinforcing temporal coherence up to a critical density, and variations in the density of topological defects as a function of cell density. Extending our analysis to breast cancer cell lines with increasing invasiveness, we find that malignant cells exhibit longer phase persistence and fewer topological defects, suggesting a mechanistic link between temporal coherence and metastatic potential. Together, these results highlight the role of density-dependent synchrony dynamics as a fundamental, quantifiable mode of collective behavior in active epithelial matter, with implications for morphogenesis, cancer progression, and tissue diagnostics.

When navigating complex environments, animals often combine multiple strategies to mitigate the effects of external disturbances. These modalities often correspond to different sources of information, leading to speed – accuracy trade-offs. Inspired by the intermittent reorientation strategy seen in the behaviour of the dung beetle, we consider the problem of the navigation strategy of a correlated random walker moving in two dimensions. We assume that the heading of the walker can be reoriented to the preferred direction by paying a fixed cost as it tries to maximize its total displacement in a fixed direction. Using optimal control theory, we derive analytically and confirm numerically the strategy that maximizes the walker’s speed, and show that the average time between reorientations scales inversely with the magnitude of the environmental noise. We then extend our framework to describe execution errors and …

All-solid-state lithium metal batteries enable high-energy-density applications, such as electric aviation, but suffer from instabilities during operation that lead to rough interfaces between the metal and electrolyte. These 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 via a range of feedback mechanisms. A minimal model that captures the coupling between interface motion, thermodynamics, electrochemistry and mechanics shows that local feedback cannot stop the formation of rough interfaces, while long-range feedback allows us to stabilize the interface and keep it flat. Our theoretical study suggests various approaches to achieve this, and provides the beginning of a practical framework for …

Perturbation theory plays a central role in the approximate solution of nonlinear differential equations. However, its naïve application often yields divergent series solutions. While these can be made convergent using singular perturbation methods of various types, the procedures used can be subtle owing to the lack of globally applicable algorithms. Inspired by the fact that all exact solutions of differential equations are consequences of their (Lie) symmetries, we reformulate perturbation theory for differential equations as a series expansion of their solutions’ symmetries. This is a change in perspective from the usual method of obtaining series expansions of the solutions themselves. We show that these approximate symmetries are straightforward to calculate and are never singular; their integration is therefore an easier way of constructing uniformly convergent solutions. This geometric viewpoint naturally …

In crowded environments, individuals must navigate around other occupants to reach their destinations. Understanding and controlling traffic flows in these spaces is relevant to coordinating robot swarms and designing infrastructure for dense populations. Here, we combine simulations, theory, and robotic experiments to study how noisy motion can disrupt traffic jams and enable flow as agents travel to individual goals. Above a critical noise level, large jams do not persist. From this observation, we analytically approximate the goal attainment rate as a function of the noise level, then solve for the optimal agent density and noise level that maximize the swarm's goal attainment rate. We perform robotic experiments to corroborate our simulated and theoretical results. Finally, we compare simple, local navigation approaches with a sophisticated but computationally costly central planner. A simple reactive scheme performs well up to moderate densities and is far more computationally efficient than a planner, suggesting lessons for real-world problems.

Starting from a three-dimensional description of an active nematic layer, we employ an asymptotic theory to derive a series of low-dimensional continuum models that capture the coupled dynamics of flat and curved films, including variations in film thickness, shape deformations, internal velocity fields, and the dynamics of orientational order. Using this asymptotic theory, we investigate instabilities driven by activity in both the nematic and isotropic phases for cylindrical and flat films. In the flat case, we demonstrate that incorporating shape and thickness variations fundamentally alters the bend and splay nature of instabilities compared to conventional two dimensional nematic instabilities. In the isotropic phase, we find that both extensile and contractile activity can induce nematic order, in contrast with active nematics on fixed surfaces, where only extensile activity leads to ordering. For the case of curved geometries such as a cylindrical film, we reveal that thickness and shape instabilities are inherently coupled. In the isotropic phase, the emergence of nematic order triggers both thickness and shape instabilities. In the nematic phase, contractile activity induces thickness instabilities, which in turn drive geometric deformations. Our results highlight the crucial interplay between activity, thickness variations, and curvature, providing new insights into the behavior of active nematic films beyond the conventional two dimensional paradigm that has been studied to date.

Soft valves serve to modulate and rectify flows in complex vasculatures across the tree of life, e.g. in the heart of every human reading this. Here we consider a minimal physical model of the heart mitral valve modeled as a flexible conical shell capable of flow rectification via collapse and coaptation in an impinging (reverse) flow. Our experiments show that the complex elastohydrodynamics of closure features a noise-activated rectification mechanism. A minimal theoretical model allows us to rationalize our observations while illuminating a dynamical bifurcation driven by stochastic hydrodynamic forces. Our theory also suggests a way to trigger the coaptation of soft valves on demand, which we corroborate using experiments, suggesting a design principle for their efficient operation.

In vivo and in vitro systems of cells and extra-cellular matrix (ECM) systems are well known to form ordered patterns of orientationally aligned fibers. Here, we interpret them as active analogs of the (disordered) isotropic to the (ordered) nematic phase transition seen in passive liquid crystalline elastomers. A minimal theoretical framework that couples cellular activity (embodied as mechanical stress) and the finite deformation elasticity of liquid crystal elastomers sets the stage to explain these patterns. Linear stability analysis of the governing equations about simple homogeneous isotropic base states shows how the onset of periodic morphologies depends on the activity, elasticity, and applied strain, provides an expression for the wavelength of the instability, and is qualitatively consistent with observations of cell-ECM experiments. Finite element simulations of the nonlinear problem corroborate the results of linear analysis. These results provide quantitative insights into the onset and evolution of nematic order in cell-matrix composites.

Snakes exhibit a wide variety of gaits, including gliding in air and sidewinding on land, which is particularly notable for its out-of-plane motion. Here we report the observation of another non-planar gait used as an escape strategy from threatening situations by juvenile anacondas (Eunectes notaeus), which we refer to as the S-start due to its shape. In this transient mode of locomotion, the snake writhes and bends out of the plane while rolling forward about its midsection without slippage. To quantify our observations, we develop a model for an active non-planar filament that interacts anisotropically with a frictional substrate. We demonstrate that locomotion is due to a propagating localized pulse of a topological quantity—the link density. A two-dimensional phase space characterized by scaled body weight and muscular torque shows that relatively light juveniles are capable of S-starts, whereas heavy adults are not, consistent with our experiments. We also show that a periodic sequence of S-starts naturally leads to a sidewinding gait.

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.