Pattern-induced local symmetry breaking in active-matter systems

Surface Topography Induces and Orients Nematic Swarms of Active Filaments: Considerations for Lab-On-A-Chip Devices

Surface-bound molecular motors can drive the collective motion of cytoskeletal filaments in the form of nematic bands and polar flocks in reconstituted gliding assays. Although these "swarming transitions" are an emergent property of active filament collisions, they can be controlled and guided by tuning the surface chemistry or topography of the substrate. To date, the impact of surface topography on collective motion in active nematics is only partially understood, with most experimental studies focusing on the escape of a single filament from etched channels. Since the late 1990s, significant progress has been made to utilize the nonequilibrium properties of active filaments and create a range of functional nanodevices relevant to biosensing and parallel computation; however, the complexity of these swarming transitions presents a challenge when attempting to increase filament surface concentrations. In this work, we etch shallow, linear trenches into glass substrates to induce the formation of swarming nematic bands and investigate the mechanisms by which surface topography regulates the two-dimensional (2D) collective motion of driven filamentous actin (F-actin). We demonstrate that nematic swarms only appear at intermediate trench spacings and vanish if the trenches are made too narrow, wide, or tortuous. To rationalize these results, we simulate the F-actin as self-propelled, semiflexible chains subject to a soft, spatially modulated potential that encodes the energetic cost of bending a filament along the edge of a trench. In our model, we hypothesize that an individual filament experiences a penalty when its projected end-to-end distance is smaller than the trench spacing ("bending and turning"). However, chains that span the channel width glide above the trenches in a force- and torque-free manner ("crowd-surfing"). Our simulations demonstrate that collections of filaments form nematic bands only at intermediate trench spacings, consistent with our experimental findings.

Coarsening and wavelength selection far from equilibrium: A unifying framework based on singular perturbation theory

Intracellular protein patterns are described by (nearly) mass-conserving reaction-diffusion systems. While these patterns initially form out of a homogeneous steady state due to the well-understood Turing instability, no general theory exists for the dynamics of fully nonlinear patterns. We develop a unifying theory for nonlinear wavelength-selection dynamics in (nearly) mass-conserving two-component reaction-diffusion systems independent of the specific mathematical model chosen. Previous work has shown that these systems support an extremely broad band of stable wavelengths, but the mechanism by which a specific wavelength is selected has remained unclear. We show that an interrupted coarsening process selects the wavelength at the threshold to stability. Based on the physical intuition that coarsening is driven by competition for mass and interrupted by weak source terms that break strict mass conservation, we develop a singular perturbation theory for the stability of stationary patterns. The resulting closed-form analytical expressions enable us to quantitatively predict the coarsening dynamics and the final pattern wavelength. We find excellent agreement with numerical results throughout the diffusion- and reaction-limited regimes of the dynamics, including the crossover region. Further, we show how, in these limits, the two-component reaction-diffusion systems map to generalized Cahn-Hilliard and conserved Allen-Cahn dynamics, therefore providing a link to these two fundamental scalar field theories. The systematic understanding of the length-scale dynamics of fully nonlinear patterns in two-component systems provided here builds the basis to reveal the mechanisms underlying wavelength selection in multicomponent systems with potentially several conservation laws.

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.

Dynamic phase transition induced by active molecules in a supercooled liquid

The purpose of this work is to use active particles to investigate the effect of facilitation on supercooled liquids. To this end we examine the behavior of a model supercooled liquid that is doped with a mixture of active particles and slowed particles. To simulate the facilitation mechanism, the activated particles are subjected to a force that follows the mobility of their most mobile neighboring molecule, while the slowed particles experience a friction force. Upon activation, we observe a fluidization of the entire medium along with a significant increase in dynamic heterogeneity. This effect is reminiscent of the fluidization observed experimentally when introducing molecular motors into soft materials. Interestingly, when the characteristic time τ_{μ}, used to define the mobility in the facilitation mechanism, matches the physical time t^{*} that characterizes the spontaneous cooperativity of the material, we observe a phase transition accompanied by structural aggregation of the active molecules. This transition is characterized by a sharp increase in fluidization and dynamic heterogeneity.

Local polar order controls mechanical stress and triggers layer formation in developing Myxococcus xanthus colonies

Colonies of the social bacterium Myxococcus xanthus go through a morphological transition from a thin colony of cells to three-dimensional droplet-like fruiting bodies as a strategy to survive starvation. The biological pathways that control the decision to form a fruiting body have been studied extensively. However, the mechanical events that trigger the creation of multiple cell layers and give rise to droplet formation remain poorly understood. By measuring cell orientation, velocity, polarity, and force with cell-scale resolution, we reveal a stochastic local polar order in addition to the more obvious nematic order. Average cell velocity and active force at topological defects agree with predictions from active nematic theory, but their fluctuations are anomalously large due to polar active forces generated by the self-propelled rod-shaped cells. We find that M. xanthus cells adjust their reversal frequency to tune the magnitude of this local polar order, which in turn controls the mechanical stresses and triggers layer formation in the colonies.

Bridging scales in a multiscale pattern-forming system

Self-organized pattern formation is vital for many biological processes. Reaction-diffusion models have advanced our understanding of how biological systems develop spatial structures, starting from homogeneity. However, biological processes inherently involve multiple spatial and temporal scales and transition from one pattern to another over time, rather than progressing from homogeneity to a pattern. To deal with such multiscale systems, coarse-graining methods are needed that allow the dynamics to be reduced to the relevant degrees of freedom at large scales, but without losing information about the patterns at small scales. Here, we present a semiphenomenological approach which exploits mass conservation in pattern formation, and enables reconstruction of information about patterns from the large-scale dynamics. The basic idea is to partition the domain into distinct regions (coarse grain) and determine instantaneous dispersion relations in each region, which ultimately inform about local pattern-forming instabilities. We illustrate our approach by studying the Min system, a paradigmatic model for protein pattern formation. By performing simulations, we first show that the Min system produces multiscale patterns in a spatially heterogeneous geometry. This prediction is confirmed experimentally by in vitro reconstitution of the Min system. Using a recently developed theoretical framework for mass-conserving reaction-diffusion systems, we show that the spatiotemporal evolution of the total protein densities on large scales reliably predicts the pattern-forming dynamics. Our approach provides an alternative and versatile theoretical framework for complex systems where analytical coarse-graining methods are not applicable, and can, in principle, be applied to a wide range of systems with an underlying conservation law.

Impact of dipole-dipole interactions on motility-induced phase separation

We present a hydrodynamic theory for systems of dipolar active Brownian particles which, in the regime of weak dipolar coupling, predicts the onset of motility-induced phase separation (MIPS), consistent with Brownian dynamics (BD) simulations. The hydrodynamic equations are derived by explicitly coarse-graining the microscopic Langevin dynamics, thus allowing for a mapping of the coarse-grained model and particle-resolved simulations. Performing BD simulations at fixed density, we find that dipolar interactions tend to hinder MIPS, as first reported in [Liao et al., Soft Matter, 2020, 16, 2208]. Here we demonstrate that the theoretical approach indeed captures the suppression of MIPS. Moreover, the analysis of the numerically obtained, angle-dependent correlation functions sheds light into the underlying microscopic mechanisms leading to the destabilization of the homogeneous phase.

Active nematic order and dynamic lane formation of microtubules driven by membrane-bound diffusing motors

Active nematics are ordered liquid crystalline fluids that exhibit spontaneous persistent flows and collective dynamics. The development of motile biopolymer systems inspired by nature has recently attracted considerable attention to out-of-equilibrium soft materials. We report the formation of an active nematic in which microtubules are propelled by kinesin motors coupled to a lipid membrane substrate. The system exhibits apolar order in a globally aligned nematic phase and locally ordered dynamic lanes. Use of a fluid substrate represents a significant advance for active matter as it allows for spatial re-organization of motors, which generate force, in response to the dynamics of the aligning microtubules. This self-organized feedback mechanism may have implications in vivo and for engineering efficient dynamic and reconfigurable materials.

Dynamic lane formation and long-range active nematic alignment are reported using a geometry in which kinesin motors are directly coupled to a lipid bilayer, allowing for in-plane motor diffusion during microtubule gliding. We use fluorescence microscopy to image protein distributions in and below the dense two-dimensional microtubule layer, revealing evidence of diffusion-enabled kinesin restructuring within the fluid membrane substrate as microtubules collectively glide above. We find that the lipid membrane acts to promote filament–filament alignment within the gliding layer, enhancing the formation of a globally aligned active nematic state. We also report the emergence of an intermediate, locally ordered state in which apolar dynamic lanes of nematically aligned microtubules migrate across the substrate. To understand this emergent behavior, we implement a continuum model obtained from coarse graining a collection of self-propelled rods, with propulsion set by the local motor kinetics. Tuning the microtubule and kinesin concentrations as well as active propulsion in these simulations reveals that increasing motor activity promotes dynamic nematic lane formation. Simulations and experiments show that, following fluid bilayer substrate mediated spatial motor restructuring, the total motor concentration becomes enriched below the microtubule lanes that they drive, with the feedback leading to more dynamic lanes. Our results have implications for membrane-coupled active nematics in vivo as well as for engineering dynamic and reconfigurable materials where the structural elements and power sources can dynamically colocalize, enabling efficient mechanical work.

Two-dimensional numerical model of Marangoni surfers: From single swimmer to crystallization

We numerically study the dynamics of an ensemble of Marangoni surfers in a two-dimensional and unconfined space. The swimmers are modeled as Gaussian sources of surfactant generating surface tension gradients and are shown to follow the Marangoni flow filtered at their spatial scale in the lubrication regime, an unstable situation leading to spontaneous motion as soon as the Marangoni effect is intense enough. As the system is fully unconstrained, it is possible to study the various dynamical regimes from single swimmer, two-body interaction, to the many-particles case characterized by an efficient particle dispersion. We show that, although the present model is very simple, it reproduces the experimentally observed transition between a regime of dispersion by random agitation when the number of swimmers is moderate to the regime of crystallization with imperfect hexagonal lattice at high density.

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.

Long-Range Nematic Order in Two-Dimensional Active Matter

Working in two space dimensions, we show that the orientational order emerging from self-propelled polar particles aligning nematically is quasi-long-ranged beyond ℓ_{r}, the scale associated to induced velocity reversals, which is typically extremely large and often cannot even be measured. Below ℓ_{r}, nematic order is long-range. We construct and study a hydrodynamic theory for this de facto phase and show that its structure and symmetries differ from conventional descriptions of active nematics. We check numerically our theoretical predictions, in particular the presence of π-symmetric propagative sound modes, and provide estimates of all scaling exponents governing long-range space-time correlations.

Discrete and periodic complex Ginzburg-Landau equation for a hydrodynamic active lattice

A discrete and periodic complex Ginzburg-Landau equation, coupled to a mean equation, is systematically derived from a driven and dissipative lattice oscillator model, close to the onset of a supercritical Andronov-Hopf bifurcation. The oscillator model is inspired by recent experiments exploring active vibrations of quasi-one-dimensional lattices of self-propelled millimetric droplets bouncing on a vertically vibrating fluid bath. Our systematic derivation provides a direct link between the constitutive properties of the lattice system and the coefficients of the resultant amplitude equations, paving the way to compare the emergent nonlinear dynamics-namely, the onset and formation of discrete dark solitons, breathers, and traveling waves-against experiments. The framework presented herein is expected to be applicable to a wider class of oscillators characterized by the presence of a dynamic coupling potential between particles. More broadly, our results point to deeper connections between nonlinear oscillators and the physics of active and driven matter.

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.

Design of nematic liquid crystals to control microscale dynamics

The dynamics of small particles, both living such as swimming bacteria and inanimate, such as colloidal spheres, has fascinated scientists for centuries. If one could learn how to control and streamline their chaotic motion, that would open technological opportunities in the transformation of stored or environmental energy into systematic motion, with applications in micro-robotics, transport of matter, guided morphogenesis. This review presents an approach to command microscale dynamics by replacing an isotropic medium with a liquid crystal. Orientational order and associated properties, such as elasticity, surface anchoring, and bulk anisotropy, enable new dynamic effects, ranging from the appearance and propagation of particle-like solitary waves to self-locomotion of an active droplet. By using photoalignment, the liquid crystal can be patterned into predesigned structures. In the presence of the electric field, these patterns enable the transport of solid and fluid particles through nonlinear electrokinetics rooted in anisotropy of conductivity and permittivity. Director patterns command the dynamics of swimming bacteria, guiding their trajectories, polarity of swimming, and distribution in space. This guidance is of a higher level of complexity than a simple following of the director by rod-like microorganisms. Namely, the director gradients mediate hydrodynamic interactions of bacteria to produce an active force and collective polar modes of swimming. The patterned director could also be engraved in a liquid crystal elastomer. When an elastomer coating is activated by heat or light, these patterns produce a deterministic surface topography. The director gradients define an activation force that shapes the elastomer in a manner similar to the active stresses triggering flows in active nematics. The patterned elastomer substrates could be used to define the orientation of cells in living tissues. The liquid-crystal guidance holds a major promise in achieving the goal of commanding microscale active flows.