Hierarchical self-assembly of polydisperse colloidal bananas into a two-dimensional vortex phase

Self-assembly of microscopic building blocks into highly ordered and functional structures is ubiquitous in nature and found at all length scales. Hierarchical structures formed by colloidal building blocks are typically assembled from monodisperse particles interacting via engineered directional interactions. Here, we show that polydisperse colloidal bananas self-assemble into a complex and hierarchical quasi–two-dimensional structure, called the vortex phase, only due to excluded volume interactions and polydispersity in the particle curvature. Using confocal microscopy, we uncover the remarkable formation mechanism of the vortex phase and characterize its exotic structure and dynamics at the single-particle level. These results demonstrate that hierarchical self-assembly of complex materials can be solely driven by entropy and shape polydispersity of the constituting particles.

Self-assembly of microscopic building blocks is a powerful route for preparing materials with predesigned structure and engineered properties (1–7). Nature provides a fascinating range of self-assembled architectures offering insight into how structural organization can emerge at different length scales (8–13). In the biological world, for instance, tobacco mosaic virus coat proteins self-organize into sophisticated capsids around viral RNA strands (11, 14). In molecular systems, lipid molecules, such as fatty acids, form a range of self-assembled structures as relevant as cell membranes and vesicles (15, 16). At the colloidal scale, a rich variety of crystals with remarkable optical properties, such as opal and other gemstones, also assembles from a range of colloidal constituents (12, 17–20). The structural complexity of self-assembled materials is typically dictated by the combination of the type of interactions between the constituent building blocks and their shape (2, 3, 5, 6). Colloids are ideal systems to independently study the role of these key parameters, as their shape and interactions can be systematically tuned and rationally designed (5, 18, 21–23).In colloidal systems interacting solely via excluded volume interactions, the shape of the particles can already lead to the assembly of complex structures (24–28). For instance, binary colloidal crystals (25) are obtained from spherical particles, complex dodecagonal quasicrystals are formed by tetrahedrons (26), and exotic banana-shaped liquid crystals are assembled from colloidal bananas (28). Introducing complex interactions between the colloidal building blocks—on the top of their shape—leads to their assembly into hierarchical materials with structural order at multiple length scales (3, 29–31). Examples include colloidal diamond structures assembled by patchy tetrahedrons functionalized with DNA strands (20) and superlattice structures formed by octapod-like particles functionalized with hydrophobic molecules (32). The successful hierarchical self-assembly of these structures relies not only on the directionality of the particle interactions but also, on the uniformity in size of the constituent building blocks, as polydispersity typically disrupts ordering via the formation of defects (33, 34).In this work, however, we show that a colloidal suspension of polydisperse banana-shaped particles interacting only via simple excluded volume interactions (28) self-assembles into remarkably ordered concentric structures, which we term colloidal vortices. At high packing fractions, these structures form a quasi–two-dimensional (quasi-2D) hierarchical material, which we term the vortex phase. Using confocal microscopy, we uncover the formation mechanism of this tightly packed phase and characterize its exotic structure and dynamics at the single-particle level.     

Chemotactic self-caging in active emulsions

A common feature of biological self-organization is how active agents communicate with each other or their environment via chemical signaling. Such communications, mediated by self-generated chemical gradients, have consequences for both individual motility strategies and collective migration patterns. Here, in a purely physicochemical system, we use self-propelling droplets as a model for chemically active particles that modify their environment by leaving chemical footprints, which act as chemorepulsive signals to other droplets. We analyze this communication mechanism quantitatively both on the scale of individual agent–trail collisions as well as on the collective scale where droplets actively remodel their environment while adapting their dynamics to that evolving chemical landscape. We show in experiment and simulation how these interactions cause a transient dynamical arrest in active emulsions where swimmers are caged between each other’s trails of secreted chemicals. Our findings provide insight into the collective dynamics of chemically active particles and yield principles for predicting how negative autochemotaxis shapes their navigation strategy.

Motile microorganisms have evolved to sense their environment and react to external chemical or physical cues via taxis. Specifically, organisms respond to a gradient in the concentration field of a chemical species by chemotaxis (1) or autochemotaxis when the gradient is generated by the organisms themselves (2). In microorganisms, chemotaxis and autochemotaxis guide many collective processes, such as colony migration (3, 4), aggregation (5, 6), or biofilm formation (7), where the emergent complex behavior is governed by the interplay of physical effects and biological processes. Many aggregatory, quorum-sensing (8) behaviors are based on attractive signaling (i.e., positive autochemotaxis). Repulsive signaling (negative autochemotaxis) is of practical importance to efficient space exploration (e.g., when ant colonies forage using mutual avoidance) (9).Complex collective behavior can result from intricate biological mechanisms but also, can be solely caused by nonequilibrium dynamics (refs. 10–14 have such examples), such that there is a need to untangle physics and biology. To this end, current research in artificial active matter aims to design and develop synthetic microswimmers that can mimic strategies like chemotaxis by purely physicochemical means (15). Self-phoretic particles, which propel via a self-generated local chemical gradient (16–18), are widely studied in theory and experiment. Suspensions of these particles exhibit nontrivial dynamics influenced by autochemotaxis (19–24). Specifically, self-propelling droplets (25) provide an experimental model for repulsive chemical signaling (12, 26–28). Along their way, the droplets shed a persistent trail of depleted fuel, which acts as a chemorepellent to other droplets. Hence, the motion of such a droplet is affected by the previous passage of another droplet.In this study, we show that in an active emulsion, droplets remodel their chemical environment while adapting their dynamics to that evolving resource landscape. In spirit, this resembles Pseudomonas aeruginosa organizing their interactions by shedding attractive trails (7, 29), with the difference that droplet trails are chemorepulsive. We start by a quantitative analysis of individual “delayed collision” events in quasi–two-dimensional (quasi-2D) confinement; we directly visualize and map the chemical footprints of droplets and measure the diffusion coefficient of the constituent chemicals. We use these results to fit a generic analytical model in 2D based on time delay, angle of impact, and chemical coupling strength. We show that these parameters determine whether a droplet crosses a chemical trail or rebounds from it. We then proceed to the collective dynamics, comparing experimental data with simulations using our single-event fits. We demonstrate how such individual binary collisions cause autochemotactic arrest in ensembles of chemically active droplets, a kind of “history caging,” where swimmers are transiently trapped in an evolving network of repulsive trails. We finally address the question of whether this type of caging is possible in three dimensions (3D) via experiments in density-matched bulk media.     

Active topolectrical circuits

The transfer of topological concepts from the quantum world to classical mechanical and electronic systems has opened fundamentally different approaches to protected information transmission and wave guidance. A particularly promising emergent technology is based on recently discovered topolectrical circuits that achieve robust electric signal transduction by mimicking edge currents in quantum Hall systems. In parallel, modern active matter research has shown how autonomous units driven by internal energy reservoirs can spontaneously self-organize into collective coherent dynamics. Here, we unify key ideas from these two previously disparate fields to develop design principles for active topolectrical circuits (ATCs) that can self-excite topologically protected global signal patterns. Realizing autonomous active units through nonlinear Chua diode circuits, we theoretically predict and experimentally confirm the emergence of self-organized protected edge oscillations in one- and two-dimensional ATCs. The close agreement between theory, simulations, and experiments implies that nonlinear ATCs provide a robust and versatile platform for developing high-dimensional autonomous electrical circuits with topologically protected functionalities.

Information transfer and storage in natural and man-made active systems, from sensory organs (1–3) to the internet, rely on the robust exchange of electrical signals between a large number of autonomous units that balance local energy uptake and dissipation (4, 5). Major advances in the understanding of photonic (6–9), acoustic (10–12), and mechanical (13–16) metamaterials have shown that topological protection (17–24) enables the stabilization and localization of signal propagation in passive and active (25–27) dynamical systems that violate time-reversal and/or other symmetries. Recent studies have successfully applied these ideas to realize topolectrical circuits (28) in the passive linear (29–34) and passive nonlinear (35, 36) regimes. However, despite substantial progress in the development of topological wave guides (37), lasers (38, 39), and transmission lines (40–43), the transfer of these concepts to active (44, 45) nonlinear circuits made from autonomously acting units still poses an unsolved challenge. From a broader perspective, not only does harnessing topological protection in nonlinear active circuits promise a new generation of autonomous devices, but also understanding their design and self-organization principles may offer insights into information storage and processing mechanisms in living systems, which integrate cellular activity, electrical signaling, and nonlinear feedback to coordinate essential biological functions (46, 47).Exploiting a mathematical analogy with active Brownian particle systems (26), we theoretically develop and experimentally demonstrate general design principles for active topolectrical circuits (ATCs) that achieve self-organized, self-sustained, topologically protected electric current patterns. The main building blocks of ATCs are nonlinear dissipative elements that exhibit an effectively negative resistance over a certain voltage range. Negative resistances can be realized using van der Pol (vdP) circuits (48), tunnel diodes, unijunction transistors, solid-state thyristors (49), or operational amplifiers set as negative impedance converters through current inversion (50), and the design principles described below are applicable to all these systems. Indeed, we expect them to apply to an even broader class of nonlinear systems, as similar dynamics also describe electromagnetic resonators with Kerr-type nonlinearities (51–53).     

A traveling-wave solution for bacterial chemotaxis with growth

Bacterial cells navigate their environment by directing their movement along chemical gradients. This process, known as chemotaxis, can promote the rapid expansion of bacterial populations into previously unoccupied territories. However, despite numerous experimental and theoretical studies on this classical topic, chemotaxis-driven population expansion is not understood in quantitative terms. Building on recent experimental progress, we here present a detailed analytical study that provides a quantitative understanding of how chemotaxis and cell growth lead to rapid and stable expansion of bacterial populations. We provide analytical relations that accurately describe the dependence of the expansion speed and density profile of the expanding population on important molecular, cellular, and environmental parameters. In particular, expansion speeds can be boosted by orders of magnitude when the environmental availability of chemicals relative to the cellular limits of chemical sensing is high. Analytical understanding of such complex spatiotemporal dynamic processes is rare. Our analytical results and the methods employed to attain them provide a mathematical framework for investigations of the roles of taxis in diverse ecological contexts across broad parameter regimes.

As a fundamental part of their life cycle, bacteria spread by dispersing into and colonizing new habitats. Many species of bacteria navigate in these new habitats by sensing gradients of certain chemicals and biasing their flagellum-based swimming to move themselves along these gradients (1, 2). This process, known as chemotaxis, is among the most extensively investigated topics in molecular biology (1, 3–7) and was observed in diverse microbial habitats such as the gut (8); the soil (9); leaves (10, 11); and marine environments such as the phycosphere, sinking marine particles, and coral reefs (2, 12–14). Further, chemotaxis is employed by many eukaryotic cells such as the free-living Dictyostelium (15) and is an important element of many tissue-forming processes involved in embryogenesis (16), neuronal patterning (17), wound healing (18), and tumor metastasis (19).Beyond promoting the movements by individual cells, chemotaxis also drives the collective movement of cells leading to emergent patterns and behaviors at the population level (20, 21). Such collective dynamics have been best studied with bacteria in culture plates and microfluidic devices. For example, when Escherichia coli cells are inoculated at the center of a soft agar plate replete with nutrients, consumption of preferred chemicals (referred to as attractants) results in collective cell movement up self-generated attractant gradients (22), leading to the emergence of striking migrating bands that propagate radially outward from the inoculation site (23–25). These migrating bands typically comprise one or two peaks in population density, which stand in contrast to the predictions of canonical models of front propagation and population expansion (26–28); they also expand at much faster speeds than predicted by canonical models. These population-level changes can strongly shape fitness and ecological interactions as recent laboratory studies have shown (29–32).The first attempt to understand these migrating bands mathematically was made by Keller and Segel, who recovered a traveling-wave solution using a pair of reaction–diffusion–convection equations to describe the density of bacterial populations and the concentration of the attractant they consume (33). While being highly influential, the Keller–Segel (KS) model neglected cell growth, a substantial factor in the expansion process. It further required unrealistic assumptions on attractant sensing without which the migrating bands lose stability (34). Subsequent modeling efforts including cell growth managed to recover the stability of the bands, but their predictions did not match major experimental observations such as the sharply peaked density profiles and their rapid migration speeds (31, 35–38).Recent work by Cremer et al. (39) demonstrated that the major features of the migrating bands for E. coli in soft agar can be accurately captured using a model in which bacterial growth is independent of the attractant. Numerical solutions to their growth-expansion (GE) model quantitatively described not only the boosted speed of the migrating band but also the signature spatial profile of the bands and their dependence on molecular parameters (39). Their results established the role of attractants as an environmental cue exploited by bacteria independent of possible nutritional values to promote rapid expansion.The success of the GE model in describing E. coli in soft agar raises the possibility that the phenotype of rapid expansion and distinct density bands might also occur for chemotactic systems in the wild, in situations where growth, diffusion, and chemotaxis dominate. However, from the numerical work of Cremer et al. (39), it is not clear what aspects of their results are generalizable given that both bacterial and environmental characteristics can be vastly different in the wild. For example, bacteria living near sulphidic sediments move more than 30 times faster than E. coli (39, 40), while bacterial motility is significantly reduced by high viscosity in the gut (41, 42). Addressing the generalizability of the GE model requires a detailed mathematical analysis of the interplay of 1) growth, 2) diffusion, and 3) chemotaxis, preferably with analytical solutions. While growth and diffusion have been studied together in the canonical models of front propagation (26–28), as have diffusion and chemotaxis in the KS model (33, 36, 43–45), a sufficient understanding of the interplay of all three is still lacking.Toward obtaining such an understanding, we describe here a detailed analytical study of the GE model. Through a heuristic analysis, we derive analytic relations that describe the dependence of the expansion speed and density profile on important molecular, cellular, and environmental parameters, including the rate of cell growth, the diffusivity and availability of the attractants, the motility and sensitivity of the bacteria, carrying capacity, and the limit of attractant sensing. Our analysis reveals the key condition for the population to attain rapid expansion speed and suggests a very broad parameter regime for which rapid expansion can be expected.     

Living liquid crystals

Collective motion of self-propelled organisms or synthetic particles, often termed “active fluid,” has attracted enormous attention in the broad scientific community because of its fundamentally nonequilibrium nature. Energy input and interactions among the moving units and the medium lead to complex dynamics. Here, we introduce a class of active matter––living liquid crystals (LLCs)––that combines living swimming bacteria with a lyotropic liquid crystal. The physical properties of LLCs can be controlled by the amount of oxygen available to bacteria, by concentration of ingredients, or by temperature. Our studies reveal a wealth of intriguing dynamic phenomena, caused by the coupling between the activity-triggered flow and long-range orientational order of the medium. Among these are (i) nonlinear trajectories of bacterial motion guided by nonuniform director, (ii) local melting of the liquid crystal caused by the bacteria-produced shear flows, (iii) activity-triggered transition from a nonflowing uniform state into a flowing one-dimensional periodic pattern and its evolution into a turbulent array of topological defects, and (iv) birefringence-enabled visualization of microflow generated by the nanometers-thick bacterial flagella. Unlike their isotropic counterpart, the LLCs show collective dynamic effects at very low volume fraction of bacteria, on the order of 0.2%. Our work suggests an unorthodox design concept to control and manipulate the dynamic behavior of soft active matter and opens the door for potential biosensing and biomedical applications.Active matter has recently emerged as an important physical model of living systems that can be described by the methods of nonequilibrium statistical mechanics and hydrodynamics (1–3). Active matter is driven by the internal sources of energy, associated with the self-propelled particles such as bacteria or synthetic swimmers. The interaction of these active particles among themselves and with the medium produces a rich variety of dynamic effects and patterns. Most of the studies deal with active particles embedded into a Newtonian isotropic fluid. In this case the interactions among particles are caused by long-range hydrodynamic and short-range excluded volume effects (4–13). In this work, we conceive a general class of active fluids, termed living liquid crystals (LLCs). The suspending medium is a nontoxic liquid crystal (LC) that supports the activity of self-propelled particles, namely bacteria. At the very same time, the medium imposes long-range anisotropic interactions onto bacteria, thanks to the intrinsic orientational order that persists even when the bacteria are not active. The importance of this system is twofold. Firstly, the bacterial activity modifies the orientational order of the system, by producing well-defined and reproducible patterns with or without topological defects. Secondly, the orientational order of the suspending medium reveals facets of bacterial behavior, allowing one to control trajectories of individual bacteria and to visualize rotation of flagella through birefringence of the host. The LLCs represent an example of a biomechanical system, capable of controlled transduction of stored energy into a systematic movement, which is of critical importance in a variety of applications, from bioinspired micromachines to self-assembled microrobots (14,15). The study of bacterial motion in LCs and non-Newtonian fluids takes us a step closer to realizing in vitro environments that more closely resemble conditions in vivo (16,17).     

Evidence for biosurfactant-induced flow in corners and bacterial spreading in unsaturated porous media

The spread of pathogenic bacteria in unsaturated porous media, where air and liquid coexist in pore spaces, is the major cause of soil contamination by pathogens, soft rot in plants, food spoilage, and many pulmonary diseases. However, visualization and fundamental understanding of bacterial transport in unsaturated porous media are currently lacking, limiting the ability to address the above contamination- and disease-related issues. Here, we demonstrate a previously unreported mechanism by which bacterial cells are transported in unsaturated porous media. We discover that surfactant-producing bacteria can generate flows along corners through surfactant production that changes the wettability of the solid surface. The corner flow velocity is on the order of several millimeters per hour, which is the same order of magnitude as bacterial swarming, one of the fastest known modes of bacterial surface translocation. We successfully predict the critical corner angle for bacterial corner flow to occur based on the biosurfactant-induced change in the contact angle of the bacterial solution on the solid surface. Furthermore, we demonstrate that bacteria can indeed spread by producing biosurfactants in a model soil, which consists of packed angular grains. In addition, we demonstrate that bacterial corner flow is controlled by quorum sensing, the cell–cell communication process that regulates biosurfactant production. Understanding this previously unappreciated bacterial transport mechanism will enable more accurate predictions of bacterial spreading in soil and other unsaturated porous media.

Bacteria are widely present in unsaturated porous media, where air and liquid coexist in pore spaces, such as in natural soils (1), plant tissues (2), food storage and packaging (3), and the lungs (4). The spread of pathogenic bacteria in these unsaturated porous media is the major cause of soil contamination by pathogens (5, 6), soft rot in plants (7), food spoilage (3, 8), and many pulmonary diseases (9). Fundamental understanding of the transport of bacterial cells in unsaturated porous media is key to addressing the above contamination- and disease-related issues. However, most current studies focus on bacterial transport in bulk liquids, driven by fluid advection, diffusion, and bacterial swimming and gliding motilities (10–13), and in liquid films on flat surfaces, due to solid-surface locomotion, biofilm expansion (14–18), and the production of a spatial gradients of biosurfactants that drive Marangoni flows, which are due to surface tension gradients (19, 20). The transport of bacterial cells in unsaturated porous media, to our knowledge, has not been directly visualized and remains to be characterized.Current understanding of bacterial transport in unsaturated porous media is primarily inferred from macroscopic observations and statistical analyses. For example, in soil science, researchers estimate the bacterial transport rate by injecting a bacterial solution into a sand column or aquifer for a short duration and then measure the concentration of bacterial cells at different distances, for example, a few meters away from the injection site as a function of time (21–23). Many of these studies show that the transport of bacterial cells in soil increased after adding surfactants (23–25). In plant pathology, investigation of the invasion of bacterial mutants revealed that surfactant-producing bacteria cause more severe disease in host tissues or soft rot (7). In addition, the development of many lung diseases is closely related to the functionality of some surfactant-producing genes (9, 26). The above observations highlight the importance of surfactants in bacterial transport in unsaturated porous media. However, the mechanisms by which surfactants affect bacterial transport in these contexts are not clear.Here, we demonstrate a previously unreported role of biosurfactants in inducing bacterial spreading in unsaturated porous media by generating corner flow. We discover that surfactant-producing bacteria generate flows along corners by producing surfactants that coat the solid surface and change its wettability. It has been known for decades that when a wetting liquid is placed in the corner region of a container or an angular pore the air–water interface curves at the corner to maintain a constant contact angle at the solid surface, whose value is determined by the properties of the liquid and the solid surface (27–33). The triangular geometry of the corner requires the air–water interface to be concave when the sum of the contact angle and half the corner angle is less than π/2 (27, 28). Classic corner flow theory suggests that when a wetting liquid forms a concave interface at the corner, a pressure gradient will build up along the corner due to the surface tension acting at the interface (27–31). This pressure gradient generates a flow along the corner, which has been shown to play an important role in the transport of wetting liquids in soil (29, 33). However, the transport of initially nonwetting liquids and bacteria, along corners and in soil, due to the production of biosurfactants that changes the contact angle, has not been reported previously.In this work, we first visualize bacterial flows in transparent, triangular prism-shaped chambers with different corner angles to mimic angular pores of unsaturated porous media. The material of the chamber, transparent polydimethylsiloxane (PDMS), is a compound with a silicon-oxygen main chain and hydrocarbon side chains, which acts as a surrogate for natural hydrophobic hydrocarbon-covered soils. We grow Pseudomonas aeruginosa, a typical biosurfactant-producing soil bacterium and major human pathogen, in an M9 solution (see Materials and Methods for details) that initially does not wet the chamber, and we discover that this biosurfactant-producing bacterium can self-generate flows along corners, a phenomenon that, to our knowledge, has not been observed or suggested previously. Second, we demonstrate that the corner flow is induced by bacteria-produced biosurfactants and the corner geometry rather than bacterial motility or the Marangoni effect. We successfully predict the critical corner angle for the bacterial corner flow to occur based on classic corner flow theory developed for pure uniform wetting liquids lacking bacteria. Third, we demonstrate biosurfactant-producing bacteria can indeed spread in a model soil, which consists of irregular PDMS grains packed in a confined space, while surfactant-deficient bacteria cannot. Finally, we demonstrate that the bacterial corner flow is controlled by quorum sensing, the cell–cell communication process that regulates biosurfactant production.     

Morphological instability and roughening of growing 3D bacterial colonies

How do growing bacterial colonies get their shapes? While colony morphogenesis is well studied in two dimensions, many bacteria grow as large colonies in three-dimensional (3D) environments, such as gels and tissues in the body or subsurface soils and sediments. Here, we describe the morphodynamics of large colonies of bacteria growing in three dimensions. Using experiments in transparent 3D granular hydrogel matrices, we show that dense colonies of four different species of bacteria generically become morphologically unstable and roughen as they consume nutrients and grow beyond a critical size—eventually adopting a characteristic branched, broccoli-like morphology independent of variations in the cell type and environmental conditions. This behavior reflects a key difference between two-dimensional (2D) and 3D colonies; while a 2D colony may access the nutrients needed for growth from the third dimension, a 3D colony inevitably becomes nutrient limited in its interior, driving a transition to unstable growth at its surface. We elucidate the onset of the instability using linear stability analysis and numerical simulations of a continuum model that treats the colony as an “active fluid” whose dynamics are driven by nutrient-dependent cellular growth. We find that when all dimensions of the colony substantially exceed the nutrient penetration length, nutrient-limited growth drives a 3D morphological instability that recapitulates essential features of the experimental observations. Our work thus provides a framework to predict and control the organization of growing colonies—as well as other forms of growing active matter, such as tumors and engineered living materials—in 3D environments.

Bacteria are known to thrive in diverse ecosystems and habitats (1–4). In nature, bacteria can be found growing on surfaces, which in addition to the ease of visualization in two dimensions, have led laboratory studies to typically focus on colony growth on two-dimensional (2D) planar surfaces. However, in many cases in nature, bacteria grow in three-dimensional (3D) habitats, such as gels and tissues inside of hosts (5–7), soils and other subsurface media (8, 9), wastewater treatment devices, and naturally occurring bodies of water (10–12). Nonetheless, despite their prevalence, the morphodynamics of bacterial colonies growing in such 3D environments remains largely unknown. Here, we ask: what determines the shape of a bacterial colony growing in three dimensions? And are there general characteristics and universal principles that span across species and specific environmental conditions?Studies of bacteria growing on 2D planar surfaces have revealed a variety of growth patterns, ranging from circular-shaped colonies (13–19) to herringbone patterns (20) and ramified, rough interfaces (13–18, 21). Some of these patterns become 3D as the colony can grow and deform into the third dimension. These morphologies are now understood to arise from friction between the growing colony and the surface and differential access to nutrients, which may also be available from the third dimension (13–18, 20, 22–26). The emergent patterns have been rationalized by incorporating these key ingredients into reaction–diffusion equations (14–16, 18, 22, 27–36), active continuum theories (19–21, 28, 31, 37–53), and agent-based models (28, 31, 38, 40, 45, 46, 54–57). Moreover, it has been suggested that these growth patterns may, in turn, influence the global function and physiology of bacterial colonies (58–60), including resistance to antibiotics (61–63) and parasites (64), resilience to environmental changes (65, 66), and genetic diversity (56, 58, 59, 67–73). However, in stark contrast to the considerable scientific effort devoted to studying 2D growth, little is known about the collective processes, the resulting morphologies, or their functional consequences for bacteria growing in three dimensions.From a physics standpoint, growing in three dimensions is fundamentally different from growing in two dimensions in terms of both nutrient access and the ability to grow and deform into an additional dimension. Consequently, we expect colony morphodynamics—the way a colony’s overall shape changes over time—to also be different. Some recent studies hint that this is indeed the case, showing how specific mechanical interactions imposed by a 3D environment can influence the morphology of growing biofilms. For instance, external fluid flows are now known to trigger the formation of streamers that stem from an initially surface-attached colony (10, 74). Also, under quiescent conditions, small (at most tens of cells across) biofilm colonies constrained in cross-linked gels adopt internally ordered structures as they grow and push outward (75), mediated by elastic stresses arising at the interface between the colony and its stiff environment. However, the behavior of larger bacterial colonies growing freely in quiescent 3D environments remains underexplored, despite the fact that they represent a fundamental building block of more complex natural colonies.Here, we combine experiments, theoretical modeling, and numerical simulations to unravel the morphodynamics of large colonies growing in 3D environments. By performing experiments with four different species of bacteria growing in transparent and easily deformed granular hydrogel matrices, we find that dense colonies growing in three dimensions undergo a morphological instability and roughen when they become nutrient limited in their interior. Independent of variations in cell type and environmental conditions, growing colonies eventually adopt a generic highly branched, broccoli-like morphology with a characteristic roughness exponent and power spectrum. Employing a continuum “active fluid” model that incorporates the coupling between nutrient diffusion, consumption, and cell growth, we trace the origin of the instability to an interplay of competition for nutrients with growth pressure–driven colony expansion. In particular, we find that these dense 3D colonies inevitably become nutrient limited in their interior, which eventually drives the periphery of the colony to become unstable and roughen. Our results thus help establish a framework to predict and control the organization of growing colonies in 3D habitats. These principles could also extend to other morphogenesis processes driven by growth in 3D environments, such as developmental processes (76, 77), tumor growth (78–82), and the expansion of engineered soft living materials (83, 84).     

Emergent ultra–long-range interactions between active particles in hybrid active–inactive systems

Particle–particle interactions determine the state of a system. Control over the range of such interactions as well as their magnitude has been an active area of research for decades due to the fundamental challenges it poses in science and technology. Very recently, effective interactions between active particles have gathered much attention as they can lead to out-of-equilibrium cooperative states such as flocking. Inspired by nature, where active living cells coexist with lifeless objects and structures, here we study the effective interactions that appear in systems composed of active and passive mixtures of colloids. Our systems are 2D colloidal monolayers composed primarily of passive (inactive) colloids, and a very small fraction of active (spinning) ferromagnetic colloids. We find an emergent ultra–long-range attractive interaction induced by the activity of the spinning particles and mediated by the elasticity of the passive medium. Interestingly, the appearance of such interaction depends on the spinning protocol and has a minimum actuation timescale below which no attraction is observed. Overall, these results clearly show that, in the presence of elastic components, active particles can interact across very long distances without any chemical modification of the environment. Such a mechanism might potentially be important for some biological systems and can be harnessed for newer developments in synthetic active soft materials.Active-matter systems have received much interest due to their emergent nonequilibrium phase and collective dynamical behavior. This interest is well founded as some of the most ubiquitous and important biological systems or processes can exhibit such emergent nonequilibrium behavior, which is perhaps most recognizable in macroscopic examples ranging from schools of aquatic organisms like rays or fish (1–3), herds of livestock (4), flocks of birds (5–7), and even a mosh pit at a heavy metal concert (8). Active-matter systems are additionally unique in that such phenomena spans multiple length scales from meter to nanometer. At these smaller length scales, it is clear that such dynamical adaptation and phase separation are necessary to perform many vital biological processes. Dense crowds of cells move collectively through tissue during development and in many of the immune response processes, i.e., wound healing (9). Sea urchin sperm cells have been found to phase separate and organize into arrays of vortices when the density of spermatozoa is large enough (10). In fact, a myriad of biological systems, and experimental systems with biological components, have reported swarming (11–13), flocking (6, 14, 15), spiraling (13, 16), and many more nonequilibrium steady states (17, 18). It is clear that, in all these systems, a combination of the activity, shape of the active agents, and the environment lead to effective out-of-equilibrium interactions that determine their steady states. These active-matter systems have been particularly attractive for theoretical studies as there are relatively few artificial systems that can mimic these active-matter systems, in addition to a wide range of parameter space that can be explored. Simulations have been able to capture a wide range of collective dynamical behavior and spontaneous phase separation in these nonequilibrium systems (19–22).Although most experimental and simulation studies have focused on systems composed of purely active components (23), here we investigate a hybrid system composed of active and passive components. This hybrid active–passive system serves as an excellent model for collective motion in complex, crowded biological environments where active units must move through or past nonmotile cells or tissues. Such a model system could potentially help to explain collective motion and phase separation observed in bacterial biofilms (24) and cells migrating through tissue interfaces or within tissues and cell sheets (25–29). In the past, there have been very few studies investigating hybrid active–passive systems and almost all of them are simulation or theoretical studies. Interestingly, these studies have reported nonequilibrium phase segregation in systems of motile and nonmotile rods (21), passive spheres and active rods (22), active and passive agents (15, 29), and active and passive hard spheres (30). Attempts to observe similar behavior in experimental artificial model systems have proved to be a more difficult task, presumably due to the fact that other interactions are present in such systems compared with the relatively simple models that theory and simulation have considered. Despite this, there are many systems, both experimental and theoretical, that show an emergent attraction between active agents in mixtures of active and passive populations and subsequent phase separation. Nevertheless, the origin of this nonequilibrium phase segregation is not well understood. Here, we clearly show that two active spinning particles in a dense monolayer of passive colloids attract, and such attraction can be felt at extremely long distances. This range is much longer than the magnetic dipole–dipole interaction and tunable via the activity and kinetics of the system. The origin of such long-range attraction is due to the elasticity of the monolayer. Our results, thus, provide much-needed understanding of the emergent interactions in synthetic mixed active systems and can help distinguish what biological interactions can be due to purely physical phenomena and which interactions require presumably physical and biological/biochemical stimuli.     

Encapsulated bacteria deform lipid vesicles into flagellated swimmers

We study a synthetic system of motile Escherichia coli bacteria encapsulated inside giant lipid vesicles. Forces exerted by the bacteria on the inner side of the membrane are sufficient to extrude membrane tubes filled with one or several bacteria. We show that a physical coupling between the membrane tube and the flagella of the enclosed cells transforms the tube into an effective helical flagellum propelling the vesicle. We develop a simple theoretical model to estimate the propulsive force from the speed of the vesicles and demonstrate the good efficiency of this coupling mechanism. Together, these results point to design principles for conferring motility to synthetic cells.

The interaction of active and passive matter lies at the heart of biology. Active matter (1) consists of collections of entities that consume energy from their environment to generate mechanical forces, which often result in motion. Thus, the cell membrane, whose essential component is a passive lipid bilayer, can actively remodel during cell growth; motility; and, ultimately, cell evolution, under the forces exerted by a host of active agents. For example, continuous polymerization–depolymerization of actin in the cytoskeleton deforms the eukaryotic cell membrane into two- and one-dimensional protrusions (lamellipodia and filopodia) that can move the whole cell (2, 3). Similar actin-supported membrane protrusions are thought to have facilitated the accidental engulfment of bacteria that led to the emergence of eukaryotic cells (4). The biophysics of active membranes has therefore been subjected to interdisciplinary scrutiny (5). More recently, learning how to create active membranes systems that deform, divide, and propel has become a priority area in the drive to synthesize life ab initio (6).Lipid vesicles enclosing natural or artificial microswimmers are becoming a model system for studying active membranes in vitro (7–15). Such composites have also direct biological relevance. For instance, from inside their eukaryotic hosts, bacterial pathogens such as Rickettsia rickettsii or Listeria monocytogenes (16, 17) continue their life cycles by hijacking the actin polymerization–depolymerization apparatus of their hosts and pushing out a tube-like protuberance from the plasma membrane. The pathogens then contact other host cells or escape into the surrounding medium by means of these membrane tubes (18).To date, research in coupling swimmers with membranes has mostly been theoretical and numerical. Such models have predicted a range of interfacial morphological changes and, in some cases, net motion of the interface (7–13). The experimental realization of these systems was only recently achieved by encapsulating swimming Bacillus subtilis bacteria (14) and synthetic Janus particles (15) in giant lipid vesicles. Both experiments reported nonequilibrium membrane fluctuations and vesicle deformations, ranging from tubular protrusions to dendritic shapes. However, net motion of the vesicles was not observed in either case.Here we present a similar experimental design but with markedly different outcome. Escherichia coli, another common motile bacterium, also extrudes membrane tubes but in addition sets the whole vesicle into motion. We demonstrate that such motion is due to a physical coupling between the flagella bundle of the enclosed cells and the tubes. The tube–flagella composite functions as a helical propeller for the entire vesicle.In biology, the specificity of interactions between bacteria and the membranes of eukaryotic hosts underlies the plethora of parasitic and symbiotic relations that have emerged between cells (18–20). Likewise, our observations illustrate the importance of small details in the design of active matter systems (21). Encapsulated bacteria propelled by a single bundle of helical flagella can generate net motion of the vesicles, whereas encapsulated swimmers propelled at similar speeds by phoresis fail to do so (15). These observations illustrate the fact that it is dangerous to proceed from coarse-grained simulations or theory that neglect such details to predict the behavior of particular systems. At the same time, our results point to a design principle for conferring motility to artificial cell models.     

Conditions for metachronal coordination in arrays of model cilia

On surfaces with many motile cilia, beats of the individual cilia coordinate to form metachronal waves. We present a theoretical framework that connects the dynamics of an individual cilium to the collective dynamics of a ciliary carpet via systematic coarse graining. We uncover the criteria that control the selection of frequency and wave vector of stable metachronal waves of the cilia and examine how they depend on the geometric and dynamical characteristics of a single cilium, as well as the geometric properties of the array. We perform agent-based numerical simulations of arrays of cilia with hydrodynamic interactions and find quantitative agreement with the predictions of the analytical framework. Our work sheds light on the question of how the collective properties of beating cilia can be determined using information about the individual units and, as such, exemplifies a bottom-up study of a rich active matter system.

Motile cilia are hair-like organelles that beat with a whip-like stroke that breaks time-reversal symmetry to create fluid flow or propel swimming microorganisms under low Reynolds number conditions (1–3). The beat is actuated by many dynein motors, which generate forces between microtubules that cause the cilium to bend in a robust cyclic manner with moderate fluctuations (4, 5). On surfaces with many cilia, the actuating organelles can coordinate with each other and collectively beat in the form of metachronal waves, where neighboring cilia beat sequentially (i.e., with a phase lag) rather than synchronously (6). The flows created from this coordinated beating are important for breaking symmetry in embryonic development (7, 8), creation of complex and dynamic flow patterns for the cerebrospinal fluid in the brain (9, 10), and providing access to nutrients (11). In microorganisms such as Paramecium and Volvox, the metachronal beating of cilia provides propulsion strategies in viscous environments (12, 13). It has been shown that depending on the parameters, beating ciliary carpets can exhibit globally ordered and turbulent flow patterns (14), which can be stable even with a moderate amount of quenched disorder (15), and that metachronal coordination optimizes the efficiency of fluid pumping (16, 17). Natural cilia have inspired various designs of artificial cilia (18–22), which may be used for pumping fluid (23, 24) and mixing (25), or fabrication of microswimmers (26).Hydrodynamic interactions have been shown to play a key role in coordinated beating of cilia (27, 28) and mediating cell polarity control (29). To achieve synchronization between two cilia via hydrodynamic interactions, it is necessary to break the permutation symmetry between them [e.g., by exploiting the dependence of the drag coefficient on the distance from a surface (30), flexibility of the anchoring of the cilia (31), nonuniform beat patterns (32, 33), or any combination of these effects (34)]. In addition to the hydrodynamic interactions, the basal coupling between cilia can also facilitate the coordination (35–37).How can we predict the collective behavior of arrays of many cilia coordinated by hydrodynamic interactions, and in particular, the properties of the emerging metachronal waves, from the single-cilium characteristics? Extensive numerical simulations using explicitly resolved beating filaments (16, 17, 27, 38–40) and simplified spherical rotors (13, 14, 41, 42) have demonstrated that metachronal coordination emerges from hydrodynamic interactions. However, insight into this complex many-body dynamical system at the level that has been achieved in studies of two cilia is still lacking. Here, we propose a theoretical framework for understanding the physical conditions for coordination of many independently beating cilia, which are arranged on a substrate in the form of a two-dimensional (2D) array immersed in a three-dimensional (3D) fluid. We uncover the physical conditions for the emergence of stable metachronal waves and predict the properties of the wave in terms of single-cilium geometric and dynamic characteristics.     

Intermittent collective dynamics emerge from conflicting imperatives in sheep herds

Among the many fascinating examples of collective behavior exhibited by animal groups, some species are known to alternate slow group dispersion in space with rapid aggregation phenomena induced by a sudden behavioral shift at the individual level. We study this phenomenon quantitatively in large groups of grazing Merino sheep under controlled experimental conditions. Our analysis reveals strongly intermittent collective dynamics consisting of fast, avalanche-like regrouping events distributed on all experimentally accessible scales. As a proof of principle, we introduce an agent-based model with individual behavioral shifts, which we show to account faithfully for all collective properties observed. This offers, in turn, an insight on the individual stimulus/response functions that can generate such intermittent behavior. In particular, the intensity of sheep allelomimetic behavior plays a key role in the group’s ability to increase the per capita grazing surface while minimizing the time needed to regroup into a tightly packed configuration. We conclude that the emergent behavior reported probably arises from the necessity to balance two conflicting imperatives: (i) the exploration of foraging space by individuals and (ii) the protection from predators offered by being part of large, cohesive groups. We discuss our results in the context of the current debate about criticality in biology.The social interactions and behavioral mechanisms involved in the coordination of collective movements in animal groups largely determine the animals’ ability to display adapted responses when they face challenges, such as finding, efficiently, food sources (1–4) or safe resting places (5–7) or avoiding predators (8–13). Thus, the diversity of collective motion patterns observed in group-living species reflects the multiple forms of interactions individuals use for coordinating their behavioral actions (14, 15). Deciphering these interactions, their relation with the patterns emerging at the collective level, and their connections with the physiological and ecological constraints peculiar to each group-living species is crucial to understanding the evolution of collective phenomena in biological systems (16–18). So far, only a handful of quantitative datasets have been gathered for large animal groups (19–21). Most of them have focused on elementary cases where the prevailing biological imperative seems to be group cohesion, either to gain protection from potential predators, such as for the spontaneous collective motion exhibited by starling flocks (19, 22) and some fish schools (23–25), or for reproductive purposes, as in swarms of midges (21, 26).One important and, so far, often neglected aspect of collective motion is the existence of individual-level behavioral shifts, which, in turn, may trigger a transition at the collective level. For instance, in many species of fish, groups regularly alternate between a swarming state, in which fish simply aggregate with a low level of polarization, and a schooling state, in which individuals are aligned and move in the same direction (27, 28). This transition is elicited by a sudden change in the velocity of a single or a few individuals that propagates to the whole group. In many cases, the behavioral shift occurs without any perceived threat in the neighborhood, resulting in a spontaneous transition at the collective level that can be interpreted as a consequence of random individual decisions. Such alternating behavioral phases at the collective level have also been reported in refs. 29, 30.Here, we report a quantitative study of the collective behavior of large groups of Merino sheep (Ovis Aries), a highly gregarious domestic breed (31), under controlled experimental conditions. Our analysis reveals an intermittent collective dynamics where long dispersion phases—during which grazing sheep slowly spread out, exploring the foraging field—are punctuated by fast packing events, triggered by an individual-level behavioral shift. We find that these events are distributed on all experimentally accessible scales. To gain insight on the sheep individual stimulus/response function, we introduce an agent-based model that explicitly includes behavioral shifts and strong allelomimetic effects. Our model results suggest that the observed collective behavior can be generated when parameters quantifying allelomimetic behavior are sufficiently large. In this parameter range, sheep regrouping time is minimized, and a large per capita grazing surface is at sheep disposal during dispersion phases.     

Two-dimensional electronic–vibrational sum frequency spectroscopy for interactions of electronic and nuclear motions at interfaces

Interactions of electronic and vibrational degrees of freedom are essential for understanding excited-states relaxation pathways of molecular systems at interfaces and surfaces. Here, we present the development of interface-specific two-dimensional electronic–vibrational sum frequency generation (2D-EVSFG) spectroscopy for electronic–vibrational couplings for excited states at interfaces and surfaces. We demonstrate this 2D-EVSFG technique by investigating photoexcited interface-active (E)-4-((4-(dihexylamino) phenyl)diazinyl)-1-methylpyridin-1- lum (AP3) molecules at the air–water interface as an example. Our 2D-EVSFG experiments show strong vibronic couplings of interfacial AP3 molecules upon photoexcitation and subsequent relaxation of a locally excited (LE) state. Time-dependent 2D-EVSFG experiments indicate that the relaxation of the LE state, S₂, is strongly coupled with two high-frequency modes of 1,529.1 and 1,568.1 cm⁻¹. Quantum chemistry calculations further verify that the strong vibronic couplings of the two vibrations promote the transition from the S₂ state to the lower excited state S₁. We believe that this development of 2D-EVSFG opens up an avenue of understanding excited-state dynamics related to interfaces and surfaces.

Electronic and vibrational degrees of freedom are the most important physical quantities in molecular systems at interfaces and surfaces. Knowledge of interactions between electronic and vibrational motions, namely electronic–vibrational couplings, is essential to understanding excited-states relaxation pathways of molecular systems at interfaces and surfaces. Many excited-states relaxation processes occur at interfaces and surfaces, including charge transfer, energy transfer, proton transfer, proton-coupled electron transfer, configurational dynamics, and so on (1–11). These relaxation processes are intimately related to the electronic–vibrational couplings at interfaces and surfaces. Strong electronic–vibrational couplings could promote nonadiabatic evolution of excited potential energy and thus, facilitate chemical reactions or intramolecular structural changes of interfacial molecules (10, 12, 13). Furthermore, these interactions of electronic and vibrational degrees of freedom are subject to solvent environments (e.g., interfaces/surfaces with a restricted environment of unique physical and chemical properties) (9, 14, 15). Despite the importance of interactions of electronic and vibrational motions, little is known about excited-state electronic–vibrational couplings at interfaces and surfaces.Interface-specific electronic and vibrational spectroscopies enable us to characterize the electronic and vibrational structures separately. As interface-specific tools, second-order electronic sum frequency generation (ESFG) and vibrational sum frequency generation (VSFG) spectroscopies have been utilized for investigating molecular structure, orientational configurations, chemical reactions, chirality, static potential, environmental issues, and biological systems at interfaces and surfaces (16–52). Recently, structural dynamics at interfaces and surfaces have been explored using time-resolved ESFG and time-resolved VSFG with a visible pump or an infrared (IR) pump thanks to the development of ultrafast lasers (6–9, 13–15, 49, 53–61). Doubly resonant sum frequency generation (SFG) has been demonstrated to probe both electronic and vibration transitions of interfacial molecular monolayer (15, 62–71). This frequency-domain two-dimensional (2D) interface/surface spectroscopy could provide information regarding electronic–vibrational coupling of interfacial molecules. However, contributions from excited states are too weak to be probed due to large damping rates of vibrational states in excited states (62, 63). As such, the frequency-domain doubly resonant SFG is used only for electronic–vibrational coupling of electronic ground states. Ultrafast interface-specific electronic–vibrational spectroscopy could allow us to gain insights into how specific nuclear motions drive the relaxation of electronic excited states. Therefore, development of interface-specific electronic–vibrational spectroscopy for excited states is needed.In this work, we integrate the specificity of interfaces and surfaces into the capabilities of ultrafast 2D spectroscopy for dynamical electronic–vibrational couplings in excited states of molecules; 2D interface-specific spectroscopies are analogous to those 2D spectra in bulk that spread the information contained in a pump−probe spectrum over two frequency axes. Thus, one can better interpret congested one-dimensional signals. Two-dimensional vibrational sum frequency generation (2D-VSFG) spectroscopy was demonstrated a few year ago (72–74). Furthermore, heterodyne 2D-VSFG spectroscopy using middle infrared (mid-IR) pulse shaping and noncollinear geometry 2D-VSFG experiments have also been developed to study vibrational structures and dynamics at interfaces (31, 75–78). Recently, two-dimensional electronic sum frequency generation (2D-ESFG) spectroscopy has also been demonstrated for surfaces and interfaces (79). On the other hand, bulk two-dimensional electronic–vibrational (2D-EV) spectroscopy has been extensively used to investigate the electronic relaxation and energy transfer dynamics of molecules, biological systems, and nanomaterials (80–90). The 2D-EV technique not only provides electronic and vibrational interactions between excitons or different excited electronic states of systems but also, identifies fast nonradiative transitions through nuclear motions in molecules, aggregations, and nanomaterials. However, an interface-specific technique for two-dimensional electronic–vibrational sum frequency generation (2D-EVSFG) spectroscopy has yet to be developed.Here, we present the development of 2D-EVSFG spectroscopy for the couplings of electronic and nucleic motions at interfaces and surfaces. The purpose of developing 2D-EVSFG spectroscopy is to bridge the gap between the visible and IR regions to reveal how structural dynamics for photoexcited electronic states are coupled with vibrations at interfaces and surfaces. As an example, we applied this 2D-EVSFG experimental method to time evolution of electronic–vibrational couplings at excited states of interface-active molecules at the air–water interface.     

Bursts of activity in collective cell migration

Dense monolayers of living cells display intriguing relaxation dynamics, reminiscent of soft and glassy materials close to the jamming transition, and migrate collectively when space is available, as in wound healing or in cancer invasion. Here we show that collective cell migration occurs in bursts that are similar to those recorded in the propagation of cracks, fluid fronts in porous media, and ferromagnetic domain walls. In analogy with these systems, the distribution of activity bursts displays scaling laws that are universal in different cell types and for cells moving on different substrates. The main features of the invasion dynamics are quantitatively captured by a model of interacting active particles moving in a disordered landscape. Our results illustrate that collective motion of living cells is analogous to the corresponding dynamics in driven, but inanimate, systems.Collective cell movement depends on intracellular biological mechanisms as well as environmental cues due to the extracellular matrix (1–5), mainly composed of collagen which is organized in hierarchical structures, such as fibrils and fibers. The mechanical properties of collagen fibril networks are essential to offer little resistance and high sensitivity to small deformations, allowing easy local remodeling and strong strain stiffening needed to ensure cell and tissue integrity (6). Wound healing is a typical biological assay to study collective migration of cells under controlled conditions in vitro and is a prototypical experimental method to study active matter (7–10). Experiments performed on soluble collagen (11) or other gels (12), micropatterned (13, 14) and deformable substrates (1) show that cell migration is guided by the substrate structure and stiffness (5, 15, 16).It has been argued that collective migration properties arise from stresses transmitted between neighboring cells (1) giving rise to long-ranged stress waves in the monolayer (17, 18). Hence the dynamics of an invading cell sheet is ruled by a combination of long-range internal stresses and interactions with the substrate, suggesting an analogy with driven elastic systems moving in a disordered medium such as cracks lines (19, 20), imbibition fronts (21), or ferromagnetic domain walls (22). The scaling laws in these systems are usually associated with a depinning critical point that has been widely studied by simple models for interface dynamics. Thanks to a combination of numerical simulations (23, 24) and renormalization group theory (23, 25–27), we now have a detailed picture of the nonequilibrium phase transitions and universality classes in these systems. Here we substantiate the analogy between collective cell migration and depinning by revealing and characterizing widely distributed bursts of activity in the collective migration of different types of cells (human cancer cells and epithelial cells, mouse endothelial cells) over different substrates (plastic, soluble, and fibrillar collagen) and experimental conditions [vascular endothelial (VE)-cadherin knockdown] and compare the experiments with simulations of a computational model of active particles (10). We find that in all these cases the statistical properties of the bursts follow universal scaling laws that are quantitatively similar to those observed in driven disordered systems (28).     

Mechanically-driven phase separation in a growing bacterial colony

Secretion of extracellular polymeric substances (EPSs) by growing bacteria is an integral part of forming biofilm-like structures. In such dense systems, mechanical interactions among the structural components can be expected to significantly contribute to morphological properties. Here, we use a particle-based modeling approach to study the self-organization of nonmotile rod-shaped bacterial cells growing on a solid substrate in the presence of self-produced EPSs. In our simulation, all of the components interact mechanically via repulsive forces, occurring as the bacterial cells grow and divide (via consuming diffusing nutrient) and produce EPSs. Based on our simulation, we show that mechanical interactions control the collective behavior of the system. In particular, we find that the presence of nonadsorbing EPSs can lead to spontaneous aggregation of bacterial cells by a depletion attraction and thereby generates phase separated patterns in the nonequilibrium growing colony. Both repulsive interactions between cell and EPSs and the overall concentration of EPSs are important factors in the self-organization in a nonequilibrium growing colony. Furthermore, we investigate the interplay of mechanics with the nutrient diffusion and consumption by bacterial cells and observe that suppression of branch formation occurs due to EPSs compared with the case where no EPS is produced.A common underlying theme of biophysics of living matter is the quest to understand how local interactions of individual components lead to collective behavior and formation of highly self-organized systems (1–6). In this regard, bacterial systems are an especially interesting example. Bacteria are known to self-organize into multicellular communities, commonly known as biofilms, in which microbial cells live in close association with a solid surface or liquid–air interface and are embedded in a self-produced extracellular matrix. Extracellular polymeric substances (EPSs) play an important role in determining the structural and mechanical architecture of a biofilm (7–12). Generally, the collective dynamics of bacterial colony involves a complex interplay of various physical, chemical, and biological mechanisms, such as growth and differentiation of cells, production of EPSs, the collective movement of cells determined by interacting physical forces and chemical cues, e.g., chemotaxis, motility, cell–cell signaling, adhesion, and gene regulation (13–19). At low density, communication among cells occurs mainly through chemical signals (20). However, at a higher density, as bacteria aggregate and form dense communities, direct mechanical interaction becomes increasingly relevant in the self-organization (21–23). In this regard, microfluidics-based experiments coupled with continuum modeling of cellular dynamics by Volfson et al. (24) was a pioneering study emphasizing the role of cell–cell mechanical interaction in the growth of a highly organized bacterial colony. As a significant extension, Farrell and coworkers (25, 26) have recently proposed a theoretical model for bacterial cells, the colony of which grows purely based on cell–cell mechanical interaction; they have studied both circular and branched colonies. However, none of the previous studies has explicitly considered the presence of extracellular matrix; the potentially important role of direct mechanical interaction of EPS with bacterial cells has remained unexplored.In this article, we investigate the role of direct mechanical interaction of EPSs with bacterial cells in a dense bacterial colony on a solid substrate. These EPSs could contribute to patterning the system either by bridging (27) or by a depletion attraction (28, 29), as observed in colloid–polymer mixtures. Bridging is a common phenomenon where polymers adsorb or adhere to multiple colloidal particles and pull them closer. On the other hand, in the latter case of depletion, which is the central focus of the present study, the polymer coils are nonadsorbing/nonadhering and the center of the coil cannot approach the surface of colloidal particles by a distance less than the size of its own radius (e.g., the radius of gyration). Due to this depletion effect, large objects experience mutual effective attraction via an entropy-driven mechanism (for an explanation, see Fig. 2), possibly leading to aggregation and phase separation in the system. This depletion effect is well appreciated as one of the driving forces of structure formation in biological systems (30–34). For instance, it has been suggested to play an important role in the compression of chromatin fibers and formation of nuclear bodies such a nucleoli and adhesion of red blood cells, etc. (30–34). Recently, a number of studies revealed that crowding effects by extracellular substances can operate in bacterial cultures as well (35–40). However, these earlier works study the role of depletion attraction operating in an equilibrium mixture of fixed bacterial cells and polymer concentration. Our focus instead is on how a nonequilibrium growing bacterial colony undergoes spontaneous aggregation of bacterial cells in the presence of self-produced EPSs.Open in a separate windowFig. 2.Schematic representation of how depletion attraction operates in a colony of bacteria producing exopolysaccharide. Bacterial cells are large spherocylinders (magenta) and coiled polymers (EPSs) are assumed as small spheres (yellow). The dashed lines around each bacteria are zones of excluded volume as shown in A and B, which is not accessible to the center of mass of a given EPS after it has been secreted into the medium. In A, as bacterial cells are separated from each other, the area of excluded volume is large, an area that could be partially accessible for the EPS particles if the cells come closer and form aggregates, as is shown in B.To investigate this question, we construct a particle-based model to study the growth of a colony of nonmotile bacterial cells in two dimensions, which consume nutrient from the solid substrate to grow and divide and produce EPSs that are secreted into the medium. We use a coarse-grained approach to treat surrounding polymeric EPS coils as small spherical particles. Via computer simulation, we show that it is the cell–EPS mechanical interaction that drives the nature of the morphology of the growing assembly: whereas at a low cell–EPS mechanical interaction, the bacterial colony remains homogeneous and isotropic, a high cell–EPS mechanical interaction drives the system into a patterned colony. In addition, our results show that the secretion rate of EPS also plays an important role; the resultant pattern depends on the ratio of EPS to cell density and on the growth velocity as determined by the level of nutrient.     

Hydrodynamic flow and concentration gradients in the gut enhance neutral bacterial diversity

The gut microbiota features important genetic diversity, and the specific spatial features of the gut may shape evolution within this environment. We investigate the fixation probability of neutral bacterial mutants within a minimal model of the gut that includes hydrodynamic flow and resulting gradients of food and bacterial concentrations. We find that this fixation probability is substantially increased, compared with an equivalent well-mixed system, in the regime where the profiles of food and bacterial concentration are strongly spatially dependent. Fixation probability then becomes independent of total population size. We show that our results can be rationalized by introducing an active population, which consists of those bacteria that are actively consuming food and dividing. The active population size yields an effective population size for neutral mutant fixation probability in the gut.

In the human body, bacteria are approximately as numerous as human cells, and about 99% of these bacteria are located in the digestive tract (1). The gut microbiota is very diverse and collectively harbors more genes than there are human genes (2). One source of this genetic diversity is evolution occurring within the gut, which is the natural environment of these bacteria. Such evolution can have important public health implications, as the gut can constitute a reservoir of antibiotic resistance both in humans and in farm animals (3). How does the environment in the gut affect the evolution of bacteria? A crucial feature of the gut is the flow of its contents along its main axis and the associated gradients of concentration of food and bacteria. Going downstream along this axis, food is first ingested; then, simple nutrients are absorbed by the body; next more complex molecules are broken down by bacteria; and eventually, what remains of the food exits the system together with many bacteria, which make up from a quarter to half of fecal mass (4). These features yield a very particular spatial structure that can impact the evolution of bacteria.Evolutionary models that investigate population spatial structure generally consider discrete patches of population with migrations between them and the same environment in each of them (5–13). Complex spatial structures are investigated through models on graphs where each individual (14–17) or each patch of population (18–21) occupies a node of the graph. Population structure can impact the rapidity of adaption (22–28) because local competition can allow the maintenance of larger genetic diversity. In simple population structures where migration is symmetric between patches (5, 6), the fixation probability of a mutant is unaffected by population structure (7, 8), unless extinctions of patches occur (11). However, more complex population structures with asymmetric migrations can impact the fixation probabilities of beneficial and deleterious mutants (13, 14, 21). In the case of the gut, the flow can be viewed as yielding asymmetric migrations, but the system is continuous. In large-scale turbulent systems, hydrodynamic flow has been shown to strongly impact fixation probabilities and fixation times (29–31). In addition, environmental gradients (e.g., of antibiotic concentration) can strongly impact evolution (32–35). How do population structure, hydrodynamic flow, and gradients shape the evolution of bacteria in the gut microbiota?Here, we propose a minimal model of evolution of bacteria in the gut. Because most bacteria in the human digestive tract are located in the bulk of the colon lumen (1, 36), we focus on this compartment. Since most bacteria in the digestive tract have no self-motility (37, 38), we consider that they are carried passively with the digesta. The motion of the digesta is complex, but it was shown in refs. 39 and 40 that it can be approximated as a one-dimensional flow with net velocity and effective diffusion representing mixing. Within this model of the gut that includes hydrodynamic flow and resulting gradients of food and bacterial concentrations, we ask how the fixation probability of a neutral mutant compares with that in an equivalent well-mixed chemostat. We find that the structure of the gut can increase this fixation probability, specifically in the regime where the profiles of food and bacterial concentration are strongly spatially dependent. In this regime, fixation probability becomes independent of total population size, in stark contrast with a well-mixed population, where fixation probability is inversely proportional to total population size (41, 42). We show that this behavior can be understood by introducing the notion of active population, which corresponds to the fraction of the bacterial population that is actively consuming food and dividing.     

Mammalian sperm hyperactivation regulates navigation via physical boundaries and promotes pseudo-chemotaxis

Mammalian sperm migration within the complex and dynamic environment of the female reproductive tract toward the fertilization site requires navigational mechanisms, through which sperm respond to the tract environment and maintain the appropriate swimming behavior. In the oviduct (fallopian tube), sperm undergo a process called “hyperactivation,” which involves switching from a nearly symmetrical, low-amplitude, and flagellar beating pattern to an asymmetrical, high-amplitude beating pattern that is required for fertilization in vivo. Here, exploring bovine sperm motion in high–aspect ratio microfluidic reservoirs as well as theoretical and computational modeling, we demonstrate that sperm hyperactivation, in response to pharmacological agonists, modulates sperm–sidewall interactions and thus navigation via physical boundaries. Prior to hyperactivation, sperm remained swimming along the sidewalls of the reservoirs; however, once hyperactivation caused the intrinsic curvature of sperm to exceed a critical value, swimming along the sidewalls was reduced. We further studied the effect of noise in the intrinsic curvature near the critical value and found that these nonthermal fluctuations yielded an interesting “Run–Stop” motion on the sidewall. Finally, we observed that hyperactivation produced a “pseudo-chemotaxis” behavior, in that sperm stayed longer within microfluidic chambers containing higher concentrations of hyperactivation agonists.

The navigational mechanisms that regulate sperm migration through the complex and dynamic physical and chemical environments of the female reproductive tract to the site of fertilization are poorly understood (1, 2). Over many years, studies have revealed that the biophysical navigational cues for sperm in the female tract include fluid flow (3–7), wall architecture (8–12), ambient rheological properties such as fluid viscoelasticity (13), and possible temperature gradients (14, 15). There is also evidence that biochemical cues from the female tract serve to modulate sperm migration (16). These may include chemoattractants (17, 18), molecular triggers that change sperm flagellar beating patterns (19–21), and sperm receptors on the epithelium of the tract that anchor sperm (22–24).The in vivo biochemical factors that transform sperm flagellar beating patterns are not precisely known (16), but in vitro exposure to certain chemical stimuli results in similar transformation of the flagellar beating pattern (25). Specifically, exposure to certain chemical stimuli results in the rise of cytoplasmic Ca²⁺ in sperm (26, 27) through either activation of CATSPER membrane ion channels and flux of exogenous Ca²⁺ ions into the flagellum (28, 29) or by mobilization of intracellular Ca²⁺ stores (30–32) or both. In turn, this rise of cytoplasmic Ca²⁺ concentration results in an increase of asymmetry in the flagellar beat cycle. This process is called “hyperactivation,” and it is required for fertilization (33, 34), as it enhances the ability of sperm to penetrate the matrix of the oocyte’s cumulus oophorus and zona pellucida to reach the plasma membrane of the oocyte (21). Furthermore, there is evidence that hyperactivation assists sperm swimming through viscoelastic substances in the female reproductive tract (35) and plays a role in detaching sperm from epithelial cells in the oviduct (36). It has been observed that hyperactivation is stimulated via a concentration-dependent mechanism (21).Although past findings have revealed roles that hyperactivation plays in supporting the success of fertilization, it remains elusive whether this functional state of motility is directly involved in sperm navigation within the female reproductive tract. Accordingly, we hypothesized that hyperactivation influences sperm–sidewall interactions and thus regulates sperm navigation via nearby wall architecture. Furthermore, we expected this regulatory mechanism to be dependent on the concentration of hyperactivation agonists.Here, using microfluidic experimentation with bovine sperm as well as theoretical and computational modeling, we investigated the effect of hyperactivation on hydrodynamic sperm–sidewall interactions in both standard and viscoelastic media. We used two established pharmacological agents to trigger hyperactivation in sperm: caffeine (25) and 4-aminopyridine (4-AP) (37). We demonstrated that hyperactivation directly regulates sperm interactions with sidewalls of our microfluidic reservoirs and thus navigation via physical boundaries. As a result of this concentration-dependent regulatory mechanism, we observed a “pseudo-chemotaxis” phenomenon in which sperm accumulated within reservoirs with higher concentrations of hyperactivation agonists. Our results revealed a potential role of hyperactivation in the navigational response of sperm to biochemical cues within the female reproductive tract.     

Cell-body rocking is a dominant mechanism for flagellar synchronization in a swimming alga

The unicellular green alga Chlamydomonas swims with two flagella that can synchronize their beat. Synchronized beating is required to swim both fast and straight. A long-standing hypothesis proposes that synchronization of flagella results from hydrodynamic coupling, but the details are not understood. Here, we present realistic hydrodynamic computations and high-speed tracking experiments of swimming cells that show how a perturbation from the synchronized state causes rotational motion of the cell body. This rotation feeds back on the flagellar dynamics via hydrodynamic friction forces and rapidly restores the synchronized state in our theory. We calculate that this “cell-body rocking” provides the dominant contribution to synchronization in swimming cells, whereas direct hydrodynamic interactions between the flagella contribute negligibly. We experimentally confirmed the two-way coupling between flagellar beating and cell-body rocking predicted by our theory.Eukaryotic cilia and flagella are long, slender cell appendages that can bend rhythmically and thus present a prime example of a biological oscillator (1). The flagellar beat is driven by the collective action of dynein molecular motors, which are distributed along the length of the flagellum. The beat of flagella, with typical frequencies ranging from 20–60 Hz, pumps fluids, for example, mucus in mammalian airways (2), and propels unicellular microswimmers such as Paramecia, spermatozoa, and algae (3). The coordinated beating of collections of flagella is important for efficient fluid transport (2, 4, 5) and fast swimming (6). This coordinated beating represents a striking example for the synchronization of oscillators, prompting the question of how flagella couple their beat. Identifying the specific mechanism of synchronization can be difficult because synchronization may occur even for weak coupling (7). Further, the effect of the coupling is difficult to detect once the synchronized state has been reached.Hydrodynamic forces were suggested to play a significant role for flagellar synchronization already in 1951 by Taylor (8). Since then, direct hydrodynamic interactions between flagella were studied theoretically as a possible mechanism for flagellar synchronization (9–12). Another synchronization mechanism that is independent of hydrodynamic interactions was recently described in the context of a minimal model swimmer (13–15). This mechanism crucially relies on the interplay of swimming motion and flagellar beating.Here, we address the hydrodynamic coupling between the two flagella in a model organism for flagellar coordination (16–19), the unicellular green alga Chlamydomonas reinhardtii. Chlamydomonas propels its ellipsoidal cell body, which has typical diameter of 10 μm, using a pair of flagella, whose lengths are about 10 μm (16). The two flagella beat approximately in a common plane, which is collinear with the long axis of the cell body. In that plane, the two beat patterns are nearly mirror-symmetric with respect to this long axis. The beating of the two flagella of Chlamydomonas can synchronize, that is, adopt a common beat frequency and a fixed phase relationship (16–19). In-phase synchronization of the two flagella is required for swimming along a straight path (19). The specific mechanism leading to flagellar synchrony is unclear.Here, we use a combination of realistic hydrodynamic computations and high-speed tracking experiments to reveal the nature of the hydrodynamic coupling between the two flagella of free-swimming Chlamydomonas cells. Previous hydrodynamic computations for Chlamydomonas used either resistive force theory (20, 21), which does not account for hydrodynamic interactions between the two flagella, or computationally intensive finite element methods (22). We employ an alternative approach and represent the geometry of a Chlamydomonas cell by spherical shape primitives, which provides a computationally convenient method that fully accounts for hydrodynamic interactions between different parts of the cell. Our theory characterizes flagellar swimming and synchronization by a minimal set of effective degrees of freedom. The corresponding equation of motion follows naturally from the framework of Lagrangian mechanics, which was used previously to describe synchronization in a minimal model swimmer (13, 15). These equations of motion embody the key assumption that the flagellar beat speeds up or slows down according to the hydrodynamic friction forces acting on the flagellum, that is, if there is more friction and therefore higher hydrodynamic load, then the beat will slow down. This assumption is supported by previous experiments that showed that the flagellar beat frequency decreases when the viscosity of the surrounding fluid is increased (23, 24). The simple force–velocity relationship for the flagellar beat employed by us coarse-grains the behavior of thousands of dynein molecular motors that collectively drive the beat. Similar force–velocity properties have been described for individual molecular motors (25) and reflect a typical behavior of active force generating systems.Our theory predicts that any perturbation of synchronized beating results in a significant yawing motion of the cell, reminiscent of rocking of the cell body. This rotational motion imparts different hydrodynamic forces on the two flagella, causing one of them to beat faster and the other to slow down. This interplay between flagellar beating and cell-body rocking rapidly restores flagellar synchrony after a perturbation. Using the framework provided by our theory, we analyze high-speed tracking experiments of swimming cells, confirming the proposed two-way coupling between flagellar beating and cell-body rocking.Previous experiments restrained Chlamydomonas cells from swimming, holding their cell body in a micropipette (17–19). Remarkably, flagellar synchronization was observed also for these constrained cells. This observation seems to argue against a synchronization mechanism that relies on swimming motion. However, the rate of synchronization observed in these experiments was faster by an order of magnitude than the rate we predict for synchronization by direct hydrodynamic interactions between the two flagella in the absence of any motion. In contrast, we show that rotational motion with a small amplitude of a few degrees only, which may result from either a residual rotational compliance of the clamped cell or an elastic anchorage of the flagellar pair, provides a possible mechanism for rapid synchronization, which is analogous to synchronization by cell-body rocking in free-swimming cells.     

The spontaneous emergence of conventions: An experimental study of cultural evolution

How do shared conventions emerge in complex decentralized social systems? This question engages fields as diverse as linguistics, sociology, and cognitive science. Previous empirical attempts to solve this puzzle all presuppose that formal or informal institutions, such as incentives for global agreement, coordinated leadership, or aggregated information about the population, are needed to facilitate a solution. Evolutionary theories of social conventions, by contrast, hypothesize that such institutions are not necessary in order for social conventions to form. However, empirical tests of this hypothesis have been hindered by the difficulties of evaluating the real-time creation of new collective behaviors in large decentralized populations. Here, we present experimental results—replicated at several scales—that demonstrate the spontaneous creation of universally adopted social conventions and show how simple changes in a population’s network structure can direct the dynamics of norm formation, driving human populations with no ambition for large scale coordination to rapidly evolve shared social conventions.Social conventions are the foundation for social and economic life (1–7), However, it remains a central question in the social, behavioral, and cognitive sciences to understand how these patterns of collective behavior can emerge from seemingly arbitrary initial conditions (2–4, 8, 9). Large populations frequently manage to coordinate on shared conventions despite a continuously evolving stream of alternatives to choose from and no a priori differences in the expected value of the options (1, 3, 4, 10). For instance, populations are able to produce linguistic conventions on accepted names for children and pets (11), on common names for colors (12), and on popular terms for novel cultural artifacts, such as referring to junk email as “SPAM” (13, 14). Similarly, economic conventions, such as bartering systems (2), beliefs about fairness (3), and consensus regarding the exchangeability of goods and services (15), emerge with clear and widespread agreement within economic communities yet vary broadly across them (3, 16).Prominent theories of social conventions suggest that institutional mechanisms—such as centralized authority (14), incentives for collective agreement (15), social leadership (16), or aggregated information (17)—can explain global coordination. However, these theories do not explain whether, or how, it is possible for conventions to emerge when social institutions are not already in place to guide the process. A compelling alternative approach comes from theories of social evolution (2, 18–20). Social evolutionary theories maintain that networks of locally interacting individuals can spontaneously self-organize to produce global coordination (21, 22). Although there is widespread interest in this approach to social norms (6, 7, 14, 18, 23–26), the complexity of the social process has prevented systematic empirical insight into the thesis that these local dynamics are sufficient to explain universally adopted conventions (27, 28).Several difficulties have limited prior empirical research in this area. The most notable of these limitations is scale. Although compelling experiments have successfully shown the creation of new social conventions in dyadic and small group interactions (29–31), the results in small group settings can be qualitatively different from the dynamics in larger groups (Model), indicating that small group experiments are insufficient for demonstrating whether or how new conventions endogenously form in larger populations (32, 33). Important progress on this issue has been made using network-based laboratory experiments on larger groups (15, 24). However, this research has been restricted to studying coordination among players presented with two or three options with known payoffs. Natural convention formation, by contrast, is significantly complicated by the capacity of individuals to continuously innovate, which endogenously expands the “ecology” of alternatives under evaluation (23, 29, 31). Moreover, prior experimental studies have typically assumed the existence of either an explicit reward for universal coordination (15) or a mechanism that aggregates and reports the collective state of the population (17, 24), which has made it impossible to evaluate the hypothesis that global coordination is the result of purely local incentives.More recently, data science approaches to studying norms have addressed many of these issues by analyzing behavior change in large online networks (34). However, these observational studies are limited by familiar problems of identification that arise from the inability to eliminate the confounding influences of institutional mechanisms. As a result, previous empirical research has been unable to identify the collective dynamics through which social conventions can spontaneously emerge (8, 34–36).We addressed these issues by adopting a web-based experimental approach. We studied the effects of social network structure on the spontaneous evolution of social conventions in populations without any resources to facilitate global coordination (9, 37). Participants in our study were rewarded for coordinating locally, however they had neither incentives nor information for achieving large scale agreement. Further, to eliminate any preexisting bias in the evolutionary process, we studied the emergence of arbitrary linguistic conventions, in which none of the options had any a priori value or advantage over the others (3, 23). In particular, we considered the prototypical problem of whether purely local interactions can trigger the emergence of a universal naming convention (38, 39).