Polar Pattern Formation in Driven Filament Systems Require Non-Binary Particle Collisions

Breakdown of Boltzmann-type models for the alignment of self-propelled rods

Studies in the collective motility of organisms use a range of analytical approaches to formulate continuous kinetic models of collective dynamics from rules or equations describing agent interactions. However, the derivation of these kinetic models often relies on Boltzmann's "molecular chaos" hypothesis, which assumes that correlations between individuals are short-lived. While this assumption is often the simplest way to derive tractable models, it is often not valid in practice due to the high levels of cooperation and self-organization present in biological systems. In this work, we illustrated this point by considering a general Boltzmann-type kinetic model for the alignment of self-propelled rods where rod reorientation occurs upon binary collisions. We examine the accuracy of the kinetic model by comparing numerical solutions of the continuous equations to an agent-based model that implements the underlying rules governing microscopic alignment. Even for the simplest case considered, our comparison demonstrates that the kinetic model fails to replicate the discrete dynamics due to the formation of rod clusters that violate statistical independence. Additionally, we show that introducing noise to limit cluster formation helps improve the agreement between the analytical model and agent simulations but does not restore the agreement completely. These results highlight the need to both develop and disseminate improved moment-closure methods for modeling biological and active matter systems.

Actin-membrane linkers: Insights from synthetic reconstituted systems

At the cell surface, the actin cytoskeleton and the plasma membrane interact reciprocally in a variety of processes related to the remodeling of the cell surface. The actin cytoskeleton has been known to modulate membrane organization and reshape the membrane. To this end, actin-membrane linking molecules play a major role in regulating actin assembly and spatially direct the interaction between the actin cytoskeleton and the membrane. While studies in cells have provided a wealth of knowledge on the molecular composition and interactions of the actin-membrane interface, the complex molecular interactions make it challenging to elucidate the precise actions of the actin-membrane linkers at the interface. Synthetic reconstituted systems, consisting of model membranes and purified proteins, have been a powerful approach to elucidate how actin-membrane linkers direct actin assembly to drive membrane shape changes. In this review, we will focus only on several actin-membrane linkers that have been studied by using reconstitution systems. We will discuss the design principles of these reconstitution systems and how they have contributed to the understanding of the cellular functions of actin-membrane linkers. Finally, we will provide a perspective on future research directions in understanding the intricate actin-membrane interaction.

Hierarchical defect-induced condensation in active nematics

Topological defects play a central role in the formation and organization of various biological systems. Historically, such nonequilibrium defects have been mainly studied in the context of homogeneous active nematics. Phase-separated systems, in turn, are known to form dense and dynamic nematic bands, but typically lack topological defects. In this paper, we use agent-based simulations of weakly aligning, self-propelled polymers and demonstrate that contrary to the existing paradigm phase-separated active nematics form -1/2 defects. Moreover, these defects, emerging due to interactions among dense nematic bands, constitute a novel second-order collective state. We investigate the morphology of defects in detail and find that their cores correspond to a strong increase in density, associated with a condensation of nematic fluxes. Unlike their analogs in homogeneous systems, such condensed defects form and decay in a different way and do not involve positively charged partners. We additionally observe and characterize lateral arc-like structures that separate from a band's bulk and move in transverse direction. We show that the key control parameters defining the route from stable bands to the coexistence of dynamic lanes and defects are the total density of particles and their path persistence length. We introduce a hydrodynamic theory that qualitatively recapitulates all the main features of the agent-based model, and use it to show that the emergence of both defects and arcs can be attributed to the same anisotropic active fluxes. Finally, we present a way to artificially engineer and position defects, and speculate about experimental verification of the provided model.

Emergence of diverse patterns driven by molecular motors in the motility assay

Actomyosin contractility originating from interactions between F-actin and myosin motors in the actin cytoskeleton generates mechanical forces and drives a wide range of cellular processes including cell migration and cytokinesis. To probe the interactions between F-actin and myosin motors, the myosin motility assay has been popularly employed, which consists of myosin heads attached to a glass surface and F-actins gliding on the surface via interactions with the heads. Several experiments have shown that F-actins move in a collective fashion due to volume-exclusion effects between neighboring F-actins. Furthermore, Computational models have shown how changes in key parameters lead to diverse pattern formation in motility assay. However, in most of the computational models, myosin motors were implicitly considered by applying a constant propulsion force to filaments to reduce computational cost. This simplification limits the physiological relevance of the insights provided by the models and potentially leads to artifacts. In this study, we employed an agent-based computational model for the motility assay with explicit immobile motors interacting with filaments. We rigorously account for the kinetics of myosin motors including the force-velocity relationship for walking and the binding and unbinding behaviors. We probed the effects of the length, rigidity, and concentration of filaments and repulsive strength on collective movements and pattern formation. It was found that four distinct types of structures-homogeneous networks, flocks, bands, and rings-emerged as a result of collisions between gliding filaments. We further analyzed the frequency and morphology of these structures and the curvature, alignment, and rotational motions of filaments. Our study provides better insights into the origin and properties of patterns formed by gliding filaments beyond what was shown before.

Active Ising Models of flocking: a field-theoretic approach

Using an approach based on Doi-Peliti field theory, we study several different Active Ising Models (AIMs), in each of which collective motion (flocking) of self-propelled particles arises from the spontaneous breaking of a discrete symmetry. We test the predictive power of our field theories by deriving the hydrodynamic equations for the different microscopic choices of aligning processes that define our various models. At deterministic level, the resulting equations largely confirm known results, but our approach has the advantage of allowing systematic generalization to include noise terms. Study of the resulting hydrodynamics allows us to confirm that the various AIMs share the same phenomenology of a first-order transition from isotropic to flocked states whenever the self-propulsion speed is nonzero, with an important exception for the case where particles align only pairwise locally. Remarkably, this variant fails entirely to give flocking-an outcome that was foreseen in previous work, but is confirmed here and explained in terms of the scalings of various terms in the hydrodynamic limit. Finally, we discuss our AIMs in the limit of zero self-propulsion where the ordering transition is continuous. In this limit, each model is still out of equilibrium because the dynamical rules continue to break detailed balance, yet it has been argued that an equilibrium universality class (Model C) prevails. We study field-theoretically the connection between our AIMs and Model C, arguing that these particular models (though not AIMs in general) lie outside the Model C class. We link this to the fact that in our AIMs without self-propulsion, detailed balance is not merely still broken, but replaced by a different dynamical symmetry in which the dynamics of the particle density is independent of the spin state. .

Signatures of irreversibility in microscopic models of flocking

Flocking in d=2 is a genuine nonequilibrium phenomenon for which irreversibility is an essential ingredient. We study a class of minimal flocking models whose only source of irreversibility is self-propulsion and use the entropy production rate (EPR) to quantify the departure from equilibrium across their phase diagrams. The EPR is maximal in the vicinity of the order-disorder transition, where reshuffling of the interaction network is fast. We show that signatures of irreversibility come in the form of asymmetries in the steady-state distribution of the flock's microstates. These asymmetries occur as consequences of the time-reversal symmetry breaking in the considered self-propelled systems, independently of the interaction details. In the case of metric pairwise forces, they reduce to local asymmetries in the distribution of pairs of particles. This study suggests a possible use of pair asymmetries both to quantify the departure from equilibrium and to learn relevant information about aligning interaction potentials from data.

Snowdrift game induces pattern formation in systems of self-propelled particles

Evolutionary games between species are known to lead to intriguing spatiotemporal patterns in systems of diffusing agents. However, the role of interspecies interactions is hardly studied when agents are (self-)propelled, as is the case in many biological systems. Here, we combine aspects from active matter and evolutionary game theory and study a system of two species whose individuals are (self-)propelled and interact through a snowdrift game. We derive hydrodynamic equations for the density and velocity fields of both species from which we identify parameter regimes in which one or both species form macroscopic orientational order as well as regimes of propagating wave patterns. Interestingly, we find simultaneous wave patterns in both species that result from the interplay between alignment and snowdrift interactions-a feedback mechanism that we call game-induced pattern formation. We test these results in agent-based simulations and confirm the different regimes of order and spatiotemporal patterns as well as game-induced pattern formation.

Pattern formation and polarity sorting of driven actin filaments on lipid membranes

Pattern formation processes in active systems give rise to a plethora of collective structures. Predicting how the emergent structures depend on the microscopic interactions between the moving agents remains a challenge. By introducing a high-density actin gliding assay on a fluid membrane, we demonstrate the emergence of polar structures in a regime of nematic binary interactions dominated by steric repulsion. The transition from a microscopic nematic symmetry to a macroscopic polar structure is linked to microscopic polarity sorting mechanisms, including accumulation in wedge-like topological defects. Our results should be instrumental for a better understanding of pattern formation and polarity sorting processes in active matter.

Collective motion of active matter is ubiquitously observed, ranging from propelled colloids to flocks of bird, and often features the formation of complex structures composed of agents moving coherently. However, it remains extremely challenging to predict emergent patterns from the binary interaction between agents, especially as only a limited number of interaction regimes have been experimentally observed so far. Here, we introduce an actin gliding assay coupled to a supported lipid bilayer, whose fluidity forces the interaction between self-propelled filaments to be dominated by steric repulsion. This results in filaments stopping upon binary collisions and eventually aligning nematically. Such a binary interaction rule results at high densities in the emergence of dynamic collectively moving structures including clusters, vortices, and streams of filaments. Despite the microscopic interaction having a nematic symmetry, the emergent structures are found to be polar, with filaments collectively moving in the same direction. This is due to polar biases introduced by the stopping upon collision, both on the individual filaments scale as well as on the scale of collective structures. In this context, positive half-charged topological defects turn out to be a most efficient trapping and polarity sorting conformation.

Field-theoretic model for chemotaxis in run and tumble particles

In this paper, we develop a field-theoretic description for run and tumble chemotaxis, based on a density-functional description of crystalline materials modified to capture orientational ordering. We show that this framework, with its in-built multiparticle interactions, soft-core repulsion, and elasticity, is ideal for describing continuum collective phases with particle resolution, but on diffusive timescales. We show that our model exhibits particle aggregation in an externally imposed constant attractant field, as is observed for phototactic or thermotactic agents. We also show that this model captures particle aggregation through self-chemotaxis, an important mechanism that aids quorum-dependent cellular interactions.

Non-reciprocal phase transitions

Out of equilibrium, a lack of reciprocity is the rule rather than the exception. Non-reciprocity occurs, for instance, in active matter^1-6, non-equilibrium systems^7-9, networks of neurons^10,11, social groups with conformist and contrarian members¹², directional interface growth phenomena^13-15 and metamaterials^16-20. Although wave propagation in non-reciprocal media has recently been closely studied^1,16-20, less is known about the consequences of non-reciprocity on the collective behaviour of many-body systems. Here we show that non-reciprocity leads to time-dependent phases in which spontaneously broken continuous symmetries are dynamically restored. We illustrate this mechanism with simple robotic demonstrations. The resulting phase transitions are controlled by spectral singularities called exceptional points²¹. We describe the emergence of these phases using insights from bifurcation theory^22,23 and non-Hermitian quantum mechanics^24,25. Our approach captures non-reciprocal generalizations of three archetypal classes of self-organization out of equilibrium: synchronization, flocking and pattern formation. Collective phenomena in these systems range from active time-(quasi)crystals to exceptional-point-enforced pattern formation and hysteresis. Our work lays the foundation for a general theory of critical phenomena in systems whose dynamics is not governed by an optimization principle.

Polar pattern formation induced by contact following locomotion in a multicellular system

Biophysical mechanisms underlying collective cell migration of eukaryotic cells have been studied extensively in recent years. One mechanism that induces cells to correlate their motions is contact inhibition of locomotion, by which cells migrating away from the contact site. Here, we report that tail-following behavior at the contact site, termed contact following locomotion (CFL), can induce a non-trivial collective behavior in migrating cells. We show the emergence of a traveling band showing polar order in a mutant Dictyostelium cell that lacks chemotactic activity. We find that CFL is the cell-cell interaction underlying this phenomenon, enabling a theoretical description of how this traveling band forms. We further show that the polar order phase consists of subpopulations that exhibit characteristic transversal motions with respect to the direction of band propagation. These findings describe a novel mechanism of collective cell migration involving cell-cell interactions capable of inducing traveling band with polar order.

Gliding filament system giving both global orientational order and clusters in collective motion

Emergence and collapse of coherent motions of self-propelled particles are affected more by particle motions and interactions than by their material or biological details. In the reconstructed systems of biofilaments and molecular motors, several types of collective motion including a global-order pattern emerge due to the alignment interaction. Meanwhile, earlier studies show that the alignment interaction of a binary collision of biofilaments is too weak to form the global order. The multiple collision is revealed to be important to achieve global order, but it is still unclear what kind of multifilament collision is actually involved. In this study, we demonstrate that not only alignment but also crossing of two filaments is essential to produce an effective multiple-particle interaction and the global order. We design the reconstructed system of biofilaments and molecular motors to vary a probability of the crossing of biofilaments on a collision and thus control the effect of volume exclusion. In this system, biofilaments glide along their polar strands on the turf of molecular motors and can align themselves nematically when they collide with each other. Our experiments show the counterintuitive result, in which the global order is achieved only when the crossing is allowed. When the crossing is prohibited, the cluster pattern emerges instead. We also investigate the numerical model in which we can change the strength of the volume exclusion effect and find that the global orientational order and clusters emerge with weak and strong volume exclusion effects, respectively. With those results and simple theory, we conclude that not only alignment but also finite crossing probability are necessary for the effective multiple-particles interaction forming the global order. Additionally, we describe the chiral symmetry breaking of a microtubule motion which causes a rotation of global alignment.

Collective and contractile filament motions in the myosin motility assay

Cells require mechanical forces for their physiological functions. The forces are generated mainly from molecular interactions between actin filaments, cross-linking proteins, and myosin motors in the actin cytoskeleton. To better understand the molecular interactions, many studies employed myosin motility assays with actin filaments propelled by myosin heads fixed on a surface. Various interesting behaviors of actin filaments have been observed in the motility assay experiments. Despite the popularity of the motility assays, there were only a few computational models designed for simulating the motility assay systems. Most of the previous models have limitations which precluded full understanding of molecular origins for behaviors of actin filaments. In this study, we used an agent-based computational model based on Brownian dynamics for simulating the motility assay system. Our model rigorously describes the mechanics, dynamics, and interactions of actin filaments, cross-linking proteins, and molecular motors. Using the model, we first investigated how properties of actin filaments and motors affect gliding motions of actin filaments without volume-exclusion effects as a base study. We found that actin filaments can continuously glide at relative fast speed only when they are sufficiently longer than the average spacing between neighboring motors and that the gliding speed of F-actins shows a biphasic dependence on processivity of motors. Then, we showed that volume-exclusion effects between actin filaments can induce diverse collective movements and alignment of actin filaments, thus creating thick bundles and ring-like structures in the absence of cross-linking proteins. Lastly, we demonstrated that cross-linking proteins can lead to distinct contractile behaviors of actin networks depending on the density and kinetics of the cross-linking proteins. Results from our study show the ability of our model to simulate the motility assay system under various conditions and provide insights into understanding of different behaviors of actin filaments.

Three-body interactions drive the transition to polar order in a simple flocking model

A large class of mesoscopic or macroscopic flocking theories are coarse-grained from microscopic models that feature binary interactions as the chief aligning mechanism. However, while such theories seemingly predict the existence of polar order with just binary interactions, actomyosin motility assay experiments show that binary interactions are insufficient to obtain polar order, especially at high densities. To resolve this paradox, here we introduce a solvable one-dimensional flocking model and derive its stochastic hydrodynamics. We show that two-body interactions are insufficient to generate polar order unless the noise is non-Gaussian. We show that noisy three-body interactions in the microscopic theory allow us to capture all essential dynamical features of the flocking transition, in systems that achieve orientational order above a critical density.

Anomalous Topological Active Matter

Active systems exhibit spontaneous flows induced by self-propulsion of microscopic constituents and can reach a nonequilibrium steady state without an external drive. Constructing the analogy between the quantum anomalous Hall insulators and active matter with spontaneous flows, we show that topologically protected sound modes can arise in a steady-state active system in continuum space. We point out that the net vorticity of the steady-state flow, which acts as a counterpart of the gauge field in condensed-matter settings, must vanish under realistic conditions for active systems. The quantum anomalous Hall effect thus provides design principles for realizing topological metamaterials. We propose and analyze the concrete minimal model and numerically calculate its band structure and eigenvectors, demonstrating the emergence of nonzero bulk topological invariants with the corresponding edge sound modes. This new type of topological active systems can potentially expand possibilities for their experimental realizations and may have broad applications to practical active metamaterials. Possible realization of non-Hermitian topological phenomena in active systems is also discussed.

Scaling behavior of nonequilibrium measures in internally driven elastic assemblies

Detecting and quantifying nonequilibrium activity is essential for studying internally driven assemblies, including synthetic active matter and complex living systems such as cells or tissue. We discuss a noninvasive approach of measuring nonequilibrium behavior based on the breaking of detailed balance. We focus on "cycling frequencies"-the average frequency with which the trajectories of pairs of degrees of freedom revolve in phase space-and explain their connection with other nonequilibrium measures, including the area enclosing rate and the entropy production rate. We test our approach on simple toy models composed of elastic networks immersed in a viscous fluid with site-dependent internal driving. We prove both numerically and analytically that the cycling frequencies obey a power law as a function of distance between the tracked degrees of freedom. Importantly, the behavior of the cycling frequencies contains information about the dimensionality of the system and the amplitude of active noise. The mapping we use in our analytical approach thus offers a convenient framework for predicting the behavior of two-point nonequilibrium measures for a given activity distribution in the network.

Advances in Biological Liquid Crystals

Biological liquid crystals, a rich set of soft materials with rod-like structures widely existing in nature, possess typical lyotropic liquid crystalline phase properties both in vitro (e.g., cellulose, peptides, and protein assemblies) and in vivo (e.g., cellular lipid membrane, packed DNA in bacteria, and aligned fibroblasts). Given the ability to undergo phase transition in response to various stimuli, numerous practices are exercised to spatially arrange biological liquid crystals. Here, a fundamental understanding of interactions between rod-shaped biological building blocks and their orientational ordering across multiple length scales is addressed. Discussions are made with regard to the dependence of physical properties of nonmotile objects on the first-order phase transition and the coexistence of multi-phases in passive liquid crystalline systems. This work also focuses on how the applied physical stimuli drives the reorganization of constituent passive particles for a new steady-state alignment. A number of recent progresses in the dynamics behaviors of active liquid crystals are presented, and particular attention is given to those self-propelled animate elements, like the formation of motile topological defects, active turbulence, correlation of orientational ordering, and cellular functions. Finally, future implications and potential applications of the biological liquid crystalline materials are discussed.

Mobility of Molecular Motors Regulates Contractile Behaviors of Actin Networks

Cells generate mechanical forces primarily from interactions between F-actin, cross-linking proteins, myosin motors, and other actin-binding proteins in the cytoskeleton. To understand how molecular interactions between the cytoskeletal elements generate forces, a number of in vitro experiments have been performed but are limited in their ability to accurately reproduce the diversity of motor mobility. In myosin motility assays, myosin heads are fixed on a surface and glide F-actin. By contrast, in reconstituted gels, the motion of both myosin and F-actin is unrestricted. Because only these two extreme conditions have been used, the importance of mobility of motors for network behaviors has remained unclear. In this study, to illuminate the impacts of motor mobility on the contractile behaviors of the actin cytoskeleton, we employed an agent-based computational model based on Brownian dynamics. We find that if motors can bind to only one F-actin like myosin I, networks are most contractile at intermediate mobility. In this case, less motor mobility helps motors stably pull F-actins to generate tensile forces, whereas higher motor mobility allows F-actins to aggregate into larger clustering structures. The optimal intermediate motor mobility depends on the stall force and affinity of motors that are regulated by mechanochemical rates. In addition, we find that the role of motor mobility can vary drastically if motors can bind to a pair of F-actins. A network can exhibit large contraction with high motor mobility because motors bound to antiparallel pairs of F-actins can exert similar forces regardless of their mobility. Results from this study imply that the mobility of molecular motors may critically regulate contractile behaviors of actin networks in cells.

C. elegans collectively forms dynamical networks

Understanding physical rules underlying collective motions requires perturbation of controllable parameters in self-propelled particles. However, controlling parameters in animals is generally not easy, which makes collective behaviours of animals elusive. Here, we report an experimental system in which a conventional model animal, Caenorhabditis elegans, collectively forms dynamical networks of bundle-shaped aggregates. We investigate the dependence of our experimental system on various extrinsic parameters (material of substrate, ambient humidity and density of worms). Taking advantage of well-established C. elegans genetics, we also control intrinsic parameters (genetically determined motility) by mutations and by forced neural activation via optogenetics. Furthermore, we develop a minimal agent-based model that reproduces the dynamical network formation and its dependence on the parameters, suggesting that the key factors are alignment of worms after collision and smooth turning. Our findings imply that the concepts of active matter physics may help us to understand biological functions of animal groups.

Understanding collective motions in a group of interacting animal is a challenge owing to the lack of control over, for example, real fish schools. Here, the authors study the aggregation of C. elegans at controllable conditions and reproduce the experimental observations using a minimal model.

From cytoskeletal assemblies to living materials

Many subcellular structures contain large numbers of cytoskeletal filaments. Such assemblies underlie much of cell division, motility, signaling, metabolism, and growth. Thus, understanding cell biology requires understanding the properties of networks of cytoskeletal filaments. While there are well established disciplines in biology dedicated to studying isolated proteins - their structure (Structural Biology) and behaviors (Biochemistry) - it is much less clear how to investigate, or even just describe, the structure and behaviors of collections of cytoskeletal filaments. One approach is to use methodologies from Mechanics and Soft Condensed Matter Physics, which have been phenomenally successful in the domains where they have been traditionally applied. From this perspective, collections of cytoskeletal filaments are viewed as materials, albeit very complex, 'active' materials, composed of molecules which use chemical energy to perform mechanical work. A major challenge is to relate these material level properties to the behaviors of the molecular constituents. Here we discuss this materials perspective and review recent work bridging molecular and network scale properties of the cytoskeleton, focusing on the organization of microtubules by dynein as an illustrative example.

Emergence of coexisting ordered states in active matter systems

Active systems can produce a far greater variety of ordered patterns than conventional equilibrium systems. In particular, transitions between disorder and either polar- or nematically ordered phases have been predicted and observed in two-dimensional active systems. However, coexistence between phases of different types of order has not been reported. We demonstrate the emergence of dynamic coexistence of ordered states with fluctuating nematic and polar symmetry in an actomyosin motility assay. Combining experiments with agent-based simulations, we identify sufficiently weak interactions that lack a clear alignment symmetry as a prerequisite for coexistence. Thus, the symmetry of macroscopic order becomes an emergent and dynamic property of the active system. These results provide a pathway by which living systems can express different types of order by using identical building blocks.

Active colloids with collective mobility status and research opportunities

The collective mobility of active matter (self-propelled objects that transduce energy into mechanical work to drive their motion, most commonly through fluids) constitutes a new frontier in science and achievable technology. This review surveys the current status of the research field, what kinds of new scientific problems can be tackled in the short term, and what long-term directions are envisioned. We focus on: (1) attempts to formulate design principles to tailor active particles; (2) attempts to design principles according to which active particles interact under circumstances where particle-particle interactions of traditional colloid science are augmented by a family of nonequilibrium effects discussed here; (3) attempts to design intended patterns of collective behavior and dynamic assembly; (4) speculative links to equilibrium thermodynamics. In each aspect, we assess achievements, limitations, and research opportunities.

Do hydrodynamically assisted binary collisions lead to orientational ordering of microswimmers?

We have investigated the onset of collective motion in systems of model microswimmers, by performing a comprehensive analysis of the binary collision dynamics using three-dimensional direct numerical simulations (DNS) with hydrodynamic interactions. From this data, we have constructed a simplified binary collision model (BCM) which accurately reproduces the collective behavior obtained from the DNS for most cases. Thus, we show that global alignment can mostly arise solely from binary collisions. Although the agreement between both models (DNS and BCM) is not perfect, the parameter range in which notable differences appear is also that for which strong density fluctuations are present in the system (where pseudo-sound mound can be observed (N. Oyama et al., Phys. Rev. E 93, 043114 (2016))).

Force percolation of contractile active gels

Living systems provide a paradigmatic example of active soft matter. Cells and tissues comprise viscoelastic materials that exert forces and can actively change shape. This strikingly autonomous behavior is powered by the cytoskeleton, an active gel of semiflexible filaments, crosslinks, and molecular motors inside cells. Although individual motors are only a few nm in size and exert minute forces of a few pN, cells spatially integrate the activity of an ensemble of motors to produce larger contractile forces (∼nN and greater) on cellular, tissue, and organismal length scales. Here we review experimental and theoretical studies on contractile active gels composed of actin filaments and myosin motors. Unlike other active soft matter systems, which tend to form ordered patterns, actin-myosin systems exhibit a generic tendency to contract. Experimental studies of reconstituted actin-myosin model systems have long suggested that a mechanical interplay between motor activity and the network's connectivity governs this contractile behavior. Recent theoretical models indicate that this interplay can be understood in terms of percolation models, extended to include effects of motor activity on the network connectivity. Based on concepts from percolation theory, we propose a state diagram that unites a large body of experimental observations. This framework provides valuable insights into the mechanisms that drive cellular shape changes and also provides design principles for synthetic active materials.

The emergence and transient behaviour of collective motion in active filament systems

Most living systems, ranging from animal flocks, self-motile microorganisms to the cytoskeleton rely on self-organization processes to perform their own specific function. Despite its importance, the general understanding of how individual active constituents initiate the intriguing pattern formation phenomena on all these different length scales still remains elusive. Here, using a high density actomyosin motility assay system, we show that the observed collective motion arises from a seeding process driven by enhanced acute angle collisions. Once a critical size is reached, the clusters coarsen into high and low density phases each with fixed filament concentrations. The steady state is defined by a balance of collision induced randomization and alignment effects of the filaments by multi-filament collisions within ordered clusters.

Self-organization is observed in cytoskeletal systems but emergence of order from disorder is poorly understood. Using a high density actomyosin system, the authors capture the transition from disorder to order, which is driven by enhanced alignment effects caused by increase in multi-filament collisions.

Long-range nematic order and anomalous fluctuations in suspensions of swimming filamentous bacteria

We study the collective dynamics of elongated swimmers in a very thin fluid layer by devising long filamentous nontumbling bacteria. The strong confinement induces weak nematic alignment upon collision, which, for large enough density of cells, gives rise to global nematic order. This homogeneous but fluctuating phase, observed on the largest experimentally accessible scale of millimeters, exhibits the properties predicted by standard models for flocking, such as the Vicsek-style model of polar particles with nematic alignment: true long-range nematic order and nontrivial giant number fluctuations.

Non-Boltzmann stationary distributions and nonequilibrium relations in active baths

Most natural and engineered processes, such as biomolecular reactions, protein folding, and population dynamics, occur far from equilibrium and therefore cannot be treated within the framework of classical equilibrium thermodynamics. Here we experimentally study how some fundamental thermodynamic quantities and relations are affected by the presence of the nonequilibrium fluctuations associated with an active bath. We show in particular that, as the confinement of the particle increases, the stationary probability distribution of a Brownian particle confined within a harmonic potential becomes non-Boltzmann, featuring a transition from a Gaussian distribution to a heavy-tailed distribution. Because of this, nonequilibrium relations (e.g., the Jarzynski equality and Crooks fluctuation theorem) cannot be applied. We show that these relations can be restored by using the effective potential associated with the stationary probability distribution. We corroborate our experimental findings with theoretical arguments.

Active Curved Polymers Form Vortex Patterns on Membranes

Recent in vitro experiments with FtsZ polymers show self-organization into different dynamic patterns, including structures reminiscent of the bacterial Z ring. We model FtsZ polymers as active particles moving along chiral, circular paths by Brownian dynamics simulations and a Boltzmann approach. Our two conceptually different methods point to a generic phase behavior. At intermediate particle densities, we find self-organization into vortex structures including closed rings. Moreover, we show that the dynamics at the onset of pattern formation is described by a generalized complex Ginzburg-Landau equation.