ACTIVE MATTER

Emergence of coexisting orderedstates in active matter systemsL. Huber1*, R. Suzuki2,3,4*†, T. Krüger1*, E. Frey1‡, A. R. Bausch2‡

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

The distinctive feature of active matter isthe local supply of energy that is transducedinto mechanical motion. Examples includeassemblies of self-propelled colloidal par-ticles (1–5), self-organizing systems com-

posed of biopolymers and molecular motors(6–9), and layers of migrating cells (10, 11). Thesesystems exhibit a rich phenomenology of collec-tive phenomena and emergent properties, withfeatures absent in passive equilibrium systems.Self-propelled colloidal particles interacting sole-ly by steric repulsion have been predicted (12, 13)to show phase separation into an ordered, solid-like phase with a disordered gas-like phase, sim-ilar to experimental observations (2–4). Activesystems composed of rod-shaped particles, cyto-skeletal filaments, or colloidal particles withvelocity alignment interactions show an evenbroader range of collective behavior, includingpolar clusters (1, 5–7), nematic lanes (9), andvortex patterns (8, 14), which, in all cases, phase-separate with a dilute isotropic, disordered back-ground. Theoretical studies have shown that,in principle, alignment interactions can explainhow these different types of orientational orderand the transitions between them emerge onthe basis of either agent-based (15–21) or mean-field models (20–28). All these studies tacitlyassume that, as in systems in thermal equilib-rium, the symmetry of the observed macroscopicorder is largely dictated by the symmetry of localalignment interactions. But to what degree is

the symmetry of the macroscopic order con-strained by the symmetry of the microscopicinteractions? More broadly, can active systemsdepart from these constraints and express amultitude of different ordering simultaneously,as is the case for living systems such as actinstress fibers and filopodia (29, 30)?To study these fundamental questions, we use

thehigh-density actomyosinmotility assay (Fig. 1A),which is ideally suited to address the microscopicprocesses that underlie pattern formation in activesystems (6, 7, 31–34). By sensitively tuning the in-teractions between the myosin-driven filamentswith a depletion agent, we can observe the emer-gence of a phase inwhich nematic and polar orderstably coexist. The complete phase diagram is re-covered from agent-based simulations of self-propelled filaments, in which weak alignmentinteractions quantitatively reproduce the experi-mentally determined microscopic collision statis-tics. We show that sufficiently weak interactionsgenerically lead to dynamic coexistence of threephases (isotropic, nematic, and polar).In the actomyosin motility assay, hydrolysis

of adenosine triphosphate (ATP) enables actinfilaments to actively glide over a lawn of non-processive heavy meromyosin motor proteins(31, 32). Previous studies have shown that in-creasing the filament density beyond a criticalvalue results in the emergence of polar clustersand waves (6, 7) (Fig. 2A). These patterns areproduced by collisions in which filaments mayalign in a polar or nematic fashion. The degreeand symmetry of the alignment depends on thechange in the relative orientation of the inter-acting filaments, D = qout – qin, where qin andqout are the angles before and after a collisionevent, respectively (Fig. 1B). In theoretical studies(15–28), these collisions have been idealized byassuming that filaments either align in a strictlypolar or strictly nematic fashion upon colliding(Fig. 1C). However, in actual experimental activematter systems (8, 9, 34, 35), the degree ofalignment caused by a single collision event isweak, that is, the relative change in filament

orientation is small, jqout � qinj≪p (Fig. 1D).Moreover, the resulting alignment exhibitsneither perfectly nematic nor perfectly polarsymmetry. Instead, depending on the collisionangle qin, in the motility assay, there is a weaktendency to favor either alignment or anti-alignment of the filaments (Fig. 1, C and D).How, then, can such weak interactions withouta clear alignment symmetry on a local scale leadto collective order at the system level, and whatfeatures of the local interactions determine theglobal symmetry of the macroscopic state?To answer these questions, we tuned the local

interactions between the filaments by addingpolyethylene glycol (PEG, 35 kDa), a depletionagent, at concentrations of up to 3% (w/v) tothe assay (Fig. 1D and fig. S1). The observedchange in the binary collision statistics can beattributed to the excluded-volume effect of thePEG molecules, which forces the filaments closerto the bottom surface covered with motors, en-abling each to interact with more motors onaverage, with a concomitant increase in motorprocessivity (Fig. 1E). This reduces the incidenceof collisions where filaments just pass over eachother (9) and increases the likelihood that fil-aments will repel each other sterically, thusenhancing the tendency to align nematically[Fig. 1D, (36)]. This technique enabled us tocontinuously modulate the symmetry of align-ment interactions at the microscopic level andprobe the robustness of pattern formation inthe gliding assay at high filament densities.Despite the rather minute changes in interac-tion characteristics caused by adding PEG at aconcentration of 3% (Fig. 1D), we found thatpolar flocks no longer form. Instead, the movingfilaments quickly, within a few minutes, self-organize into a network of “ant trails” (Fig. 2Band movie S1). In contrast to the unidirectionalfilament motion found within polar clusters,the filaments that form these “lanes” move bi-directionally, as do many colonial ant species(37). Because the filaments move along thesetracks in either direction with equal probability(Fig. 2C and fig. S2), the overall order is nematic,not polar, and stable; this is quantified by thelocal nematic order (fig. S2A) and the auto-correlation function of the filament orientations(fig. S2, D and E). Moreover, whereas polar clus-ters propagate through the system at uniformspeed, nematic lanes form static networks withbranches spanning up to several hundred micro-meters in length (Fig. 2B). Filaments are alsoseen to continuously leave and enter the trails(Fig. 2D and movie S2), such that these branchesremain fixed in orientation and slowly grow andshrink at their ends (fig. S2F). These processes,operating on a time scale of minutes, lead to aslow reorganization of network architecture,with new branches forming (movie S3) whileothers contract (movie S4). Note that thesenetworks are isotropically oriented and thatno notable actin bundling was observed below3% PEG.This fundamental qualitative change in macro-

scopic order, from propagating waves of polar

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1Arnold Sommerfeld Center for Theoretical Physics andCenter for NanoScience, Department of Physics, Ludwig-Maximilians-Universität München, D-80333 Munich,Theresienstrasse 37, Germany. 2Lehrstuhl für Biophysik(E27), Technische Universität München, James-Franck-Strasse 1, D-85748 Garching, Germany. 3Graduate School ofMedicine, Kyoto University, 606-8501 Kyoto, Japan.4Institute for Integrated Cell-Material Sciences, KyotoUniversity, 606-8501 Kyoto, Japan.*These authors contributed equally to this work.†Present address: Center for Integrative Medicine and Physics,Institute for Advanced Study, Kyoto University, 606-8501 Kyoto, Japan.‡Corresponding author. Email: frey@lmu.de (E.F.); abausch@mytum.de (A.R.B.)

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order to branched networks of stable lanes with-in which filaments move bidirectionally, inducedby relatively minor changes in interaction char-acteristics at the microscopic scale, is puzzling.To reveal the underlying mechanism, we devel-oped an agent-based computational model thatgoes beyond simple collision rules and faithfullyreproduces the experimentally observed (micro-scopic) binary collision statistics and used it topredict the collective dynamics at large scales.Propelled actin filaments are modeled as discrete,slender chains of length L [Fig. 3A and fig. S3,(36)]. Each filament is assumed to move at aconstant speed v with the body of the filamentfollowing the tip. The direction of motion changesupon interaction with other filaments, as well asthrough interaction with molecular motors. Whenthe leading segment of a given filament collideswith a segment of another filament at a relativeorientation q, an alignment potential U(q) actson the tip. This potential is assumed to be thesum of terms with polar (p) and nematic (n) sym-metry, U(q) º φp cos q + φn cos 2q, where φpand φn represent the respective mean change

in orientation during a collision. We adjustedφp and φn such that the binary collision statisticsof the computational model (Fig. 3B) closely re-semble those observed experimentally (Fig. 1D).Having validated the computational model

at the microscopic level, we asked whether itcaptures the collective dynamics of the high-density actomyosin motility assay. We firstperformed large-scale simulations for model pa-rameters corresponding to the absence of PEG.Starting from a random uniform distribution offilaments, we observed that high-density wavefronts of polar-ordered filaments rapidly form,surrounded by disordered, low-density regions(Fig. 3C and movie S5). This matches the phe-nomenology observed in the motility assay. Next,we performed simulations in a parameter regimecorresponding to 3% PEG. Again, in agreementwith our experiments, we found networks ofhigh-density nematic lanes surrounded by dis-ordered, low-density regions (Fig. 3D), reminis-cent of chaotic structures that were predicted foractive nematics (21). The overall network archi-tecture changed slowly, with trails extending

or retracting from their ends, and some lanesmerging on longer time scales (fig. S4A andmovie S6).The model was then used to predict the de-

pendence of nematic versus polar order on thefilament density r0 and the ratio of nematic topolar alignment strength, a = φn/φp. To fa-cilitate simulations over a broad parameter range,we considered smaller systems with a box sizeof 81.3L. We monitored the (global) polar andnematic order parameters P ¼ jhexpðiqÞij andN ¼ jhexpð2iqÞij, respectively, measured over allfilaments after the dynamics had become sta-tionary (fig. S4B). In initial parameter sweeps,we observed that, within certain intervals of a,simulations starting from different realizationsof randomly distributed filaments, but with iden-tical parameter sets, sometimes resulted in polarand sometimes in nematic patterns (Fig. 3E, bot-tom panel). Similar observations were made in aVicsek-type model, but only if strong additionalmemory in the particle movement is included(17). The patterns in our simulation were stablewithin the simulation times, and no switching

Huber et al., Science 361, 255–258 (2018) 20 July 2018 2 of 4

Fig. 2. Experimental phenomenology.(A) Polar actin clusters formed in the absenceof PEG, moving in the same direction as thefilaments. (The fraction of fluorescently labeledfilaments is 1/50, and the monomeric actinconcentration is 10 mM.) (B) A large networkof high-density nematic lanes formed at aPEG concentration of 3% and 5 mM actin. Theimage is an overlay covering a period of 100 sto demonstrate that the structure is frozenand stable. Filaments move along the lanecontours in opposite directions. (The labeledfilament fraction is 1/60.) (C) Probability densityP(vx, vy) of instantaneous velocities shows thepreferred bidirectional motion of filaments within a lane. (D) Single filamentsmove inside lanes (bright region).Two representative trajectories are shown(turquoise and orange) at 10 mM actin and 2% PEG. The inset shows an

overlay covering a period of 50 s. Polar (A) and nematic (B) motion aredepicted by uni- and bidirectional arrows, respectively. Scale bars, 100 mm;a.u., arbitrary units.

Fig. 1. Interactions in the actomyosin assay.(A) Schematic of the actomyosin motility assay.PEG acts as a depletion agent. (B) Illustrationof different filament collision geometrieswith an incoming angle qin and (C) correspondingbinary collision curves. Whereas strong polaror nematic collision rules lead to full alignmentor anti-alignment, weak collisions cause a gradualchange of orientation and may exhibit both polarand nematic features (purple line). The dashedline depicts neutral collisions (qout = qin).(D) Binary collision statistics. Blue squares,PEG 3% (389 collisions); red circles, no PEG[1113 collisions; data from (34)]. Error bars, ±SD.(E) Processivity increases with PEG concentration,as indicated by the earlier saturation of normalizedfilament velocities as a function of motor density.v0.1 is the velocity at 0.1 mg/ml nonprocessiveheavy meromyosin. The inset shows absolutefilament velocities.

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between them was observed, suggesting the ex-istence of a regime of interaction strengths inwhich the dynamics exhibit multistability. Toprobe these initial observations further, wechecked for hysteresis effects in the collectivedynamics [fig. S5, (36)]. To this end, we initiatedour simulations in a parameter regime in whichthe system shows polar waves only (a = 2.75,Fig. 3E, top-left panel), waited until the dynamicsbecame stationary, and then quasi-statically in-creased the value of a (i.e., giving the systemsufficient time to equilibrate between successiveadjustments of a) and monitored both nematicand polar order parameters (Fig. 3F, closedsymbols). After reaching a regime in which thesystem gave rise to nematic lanes only (a = 6,Fig. 3E, top-right panel), we reduced the valueof a quasi-statically (Fig. 3F, open symbols). Al-though the nematic order parameter remainedessentially unchanged, we observed a hysteresisloop in the polar order parameter P. As the rel-ative strength of nematic-to-polar alignment isincreased, the degree of polar order (P+) grad-ually declines until it reaches zero at some crit-ical value a+. Conversely, in the reverse direction,polar order (P−) remains negligible up to a dif-ferent critical value a− and then suddenly jumpsto a rather large value. The phase diagram inFig. 4A was obtained using dP = P+ − P− toquantify the degree of multistability, where dPis the difference between the degree of polarorder for the forward (increasing a) and back-ward (decreasing a) processes.To test these predictions, we performed ex-

periments over a broad range of actin and PEGconcentrations and obtained a phase diagram(Fig. 4B) whose topology closely resembles thatobtained from the computational model (Fig. 4A).

In particular, upon varying the strength of in-teraction between the filaments by changing thePEG level and thus a, we find a broad regime ofnonequilibrium steady states where polar wavesand nematic lanes coexist simultaneously. More-over, both simulations of large systems (Fig. 4Cand movie S7) and experiments (Fig. 4D andmovie S8) consistently show that the equilib-rium is highly dynamic. Polar waves may invaderegions containing nematic trails and therebydisrupt their network structure (fig. S6A). Afterthe passage of these waves, nematic-lane networksare observed to reform locally, often close to theiroriginal positions. The formation of nematic laneswas also observed at the left and right edges ofpolar waves (fig. S6B and movie S9). Whereasin experiments this coexistence remained stableduring the full experiment duration (fig. S6C),in simulations we performed a scaling analy-sis to probe the lifetime of coexistence tfix as afunction of the finite system size, at differentpoints in the multistable parameter region. Wefound that this lifetime grows linearly with thesystem size, whereas the time of initial patternformation t0 remains small and constant [Fig. 4Fand fig. S7, (36)], implying a diverging time-scaleseparation and stable coexistence in the thermo-dynamic limit.These observations fromexperiment and theory

imply that polar waves and nematic lanes areboth intrinsically stable structures, suggestingthat the nonequilibrium steady state representsa dynamic equilibrium between different pat-terns, which, although they have conflictingpolar and nematic symmetries, coexist in a dilute,disordered background. We attribute their co-existence to the weak interaction between theactive particles, which determines macroscopic

order not at the microscopic level but insteadrenders the symmetry of collective order itselfto become an emergent property, which is dy-namic in space and time. If this picture is valid,then an increase in the alignment strength atthe binary level should eliminate the ambigu-ity in symmetry and prevent the emergence ofcoexistence. To test this hypothesis, we performedextensive numerical simulations by varying a andφp (Fig. 4E) and looking formultistability. Indeed,we find that as the total degree of alignment, thatis, both φn and φp, is increased, the multistableregion contracts and eventually vanishes com-pletely. In this limit, there appears to be a sharptransition between a polar and nematic phase,similar to previous findings in a Vicsek-type toymodel (18). We therefore conclude that the co-existence of patterns with mutual polar andnematic symmetries depends on sufficientlyweak alignment interactions between individ-ual filaments. Furthermore, it seems to be crucialthat the computational model includes arbitrarypairwise interactions and spatiotemporal corre-lations without relying on any ad hoc trunca-tion. This allows for coarsening dynamics, wheremany different mesoscale filament configura-tions are explored until they take the form ofeither polar clusters or nematic lanes. Thesepatterns become local attractors of the dynam-ics, such that, despite their conflicting symme-tries, they can exist in juxtaposition within thesame system. This indicates that the celebratedGibbs phase rule—which states that, in thermal-equilibrium one-component systems, a three-phase coexistence only occurs at a singular pointin parameter space—is invalid in active systems.Overcoming this thermodynamic constraint maybe an essential and simple prerequisite for

Huber et al., Science 361, 255–258 (2018) 20 July 2018 3 of 4

Fig. 3. Simulation model and phenomenology.(A) Illustration of the simulation model: Filaments(green) are propelled along their contours (solidblack arrows). Upon collision, the orientations oftips (gray arrows) are redirected in proportion to thepolar and nematic alignment strengths (red andblue arrows). (B) Binary collision data from simu-lations for two selected curves with different a.Error bars, 1 SD. (C andD) Emergence of (C) polarwaves (a = 3) and (D) a network of nematiclanes (a = 6.25) in large-scale systems.The insetsshow filaments within a single pixel with localdensity r and local (C) polar or (D) nematic order.In both panels, 544,000 filaments were simulatedin a box of length 650.2L, with a homogeneousdensity r0 = 1.29/L2. Scale bars, 100L. Uni- andbidirectional arrows denote local polar and nematicfilament motion, respectively. (E) Different steadystates for small simulation boxes, with r0 = 1.29/L2:Whereas a = 2.75 always produces polar wavesand a = 6 always nematic lanes, at a = 4, eitherwaves or lanes can be obtained in different realiza-tions. Scale bars, 10L. (F) Global order parametersduring a hysteresis loop in a. Black arrows denotethe direction of the loop. Regions of nonzero dP(shaded in green) exhibit multistable behavior. For(B) to (F), φp = 2.1°.

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biological systems to produce heterogeneous,multitasking structures out of a single set ofconstituents, as is the case for the cellular actinnetwork (29, 30) and migrating cell layers (10, 11).

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ACKNOWLEDGMENTS

Funding: E.F. acknowledges support from the DFG viaproject B02 within SFB 863. A.R.B. acknowledges support fromthe European Research Council in the framework of theAdvanced Grant 289714-SelfOrg and partial support fromthe DFG via project B01 within the SFB 863. All authorsacknowledge the continuous support of the German ExcellenceInitiatives via the NanoSystems Initiative Munich (NIM).Author contributions: L.H., R.S., T.K., and A.R.B. performedand designed the experiments. L.H., T.K., and E.F. performedand designed the simulations. All authors participated ininterpreting the experimental and theoretical results and inwriting the manuscript. Competing interests: The authorsdeclare no competing financial interests. Data and materialsavailability: All relevant data are provided in the paper andsupplementary materials or online at https://github.com/LorenzHuber/ActivePolymerDataRepository/.

SUPPLEMENTARY MATERIALS

www.sciencemag.org/content/361/6399/255/suppl/DC1Materials and MethodsFigs. S1 to S7Table S1References (38–48)Movies S1 to S9

7 August 2017; resubmitted 22 January 2018Accepted 16 May 2018Published online 28 June 201810.1126/science.aao5434

Huber et al., Science 361, 255–258 (2018) 20 July 2018 4 of 4

Fig. 4. Phase diagrams and coexistingsymmetries in experiment and simulation.(A) Simulation phase diagrams for differentfilament densities r0 and relative alignmentstrengths a. (B) Experimental phase diagram ofemergent patterns for varying monomericactin and PEG concentrations. Gray crosses,disorder; red triangles, polar clusters; bluesquares, nematic lanes; green diamonds,coexisting polar and nematic structures. Actinconcentrations were normalized with respectto the estimated critical concentration in theabsence of PEG (see supplementary materialsfor details). (C) Emergence of both polar wavesand nematic lanes in large-scale simulations(scale bar, 100L) for a = 4 and a homogeneousdensity r0 = 1.29/L2. (D) Coexistence of polarclusters and nematic lanes in the motility assay at2% PEG and 5 mM actin. Scale bar, 100 mm.(E) Phase diagrams for different polar alignmentstrengths φp and r0 = 1.29/L2. The total strengthof alignment increases with both fp and a.The shape of the phase diagram only slightlychanges for larger system sizes (see fig. S7A).(F) Scaling analysis of time scales at two differentparameter sets (orange data: φp = 2.1°, a = 4.17;purple data: φp = 3.3°, a = 3.13). The averagecoexistence lifetime tfix (solid lines) grows roughlylinear with system size, whereas the average initial order time t0(dashed lines) remains small and constant. Averages taken over25 simulations per size; error bars represent 15th and 85th percentiles(see supplementary materials and fig. S7 for details). The triangle is

a slope of 1 (linear) to guide the eye. For (A) and (E), phase diagramswere obtained by hysteresis analysis in a, and white dashed linesdepict the domain boundaries of the observed steady states. For (A)and (C), φp = 2.1°.

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Emergence of coexisting ordered states in active matter systemsL. Huber, R. Suzuki, T. Krüger, E. Frey and A. R. Bausch

originally published online June 28, 2018DOI: 10.1126/science.aao5434 (6399), 255-258.361Science 

, this issue p. 255Scienceordering.

drove the system between ferromagnetic (polar) and nematic (liquid crystal)−−interactions between the actin filamentspolymer chains that weakened−−across a surface by surface-attached myosin motor proteins. Adding a depletion agent

combination of experiments and numerical simulations on an active matter system consisting of actin filaments propelled present a et al.In active matter systems, the infusion of energy and motion drives ordering processes. Huber

A balance between motion and cooperation

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