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Emergence of coexisting ordered states in active matter systems

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Science  20 Jul 2018:
Vol. 361, Issue 6399, pp. 255-258
DOI: 10.1126/science.aao5434

A balance between motion and cooperation

In active matter systems, the infusion of energy and motion drives ordering processes. Huber et al. present a combination of experiments and numerical simulations on an active matter system consisting of actin filaments propelled across a surface by surface-attached myosin motor proteins. Adding a depletion agent—polymer chains that weakened interactions between the actin filaments—drove the system between ferromagnetic (polar) and nematic (liquid crystal) ordering.

Science, this issue p. 255

Abstract

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.

The distinctive feature of active matter is the local supply of energy that is transduced into mechanical motion. Examples include assemblies of self-propelled colloidal particles (15), self-organizing systems composed of biopolymers and molecular motors (69), and layers of migrating cells (10, 11). These systems exhibit a rich phenomenology of collective phenomena and emergent properties, with features absent in passive equilibrium systems. Self-propelled colloidal particles interacting solely by steric repulsion have been predicted (12, 13) to show phase separation into an ordered, solid-like phase with a disordered gas-like phase, similar to experimental observations (24). Active systems composed of rod-shaped particles, cytoskeletal filaments, or colloidal particles with velocity alignment interactions show an even broader range of collective behavior, including polar clusters (1, 57), nematic lanes (9), and vortex patterns (8, 14), which, in all cases, phase-separate with a dilute isotropic, disordered background. Theoretical studies have shown that, in principle, alignment interactions can explain how these different types of orientational order and the transitions between them emerge on the basis of either agent-based (1521) or mean-field models (2028). All these studies tacitly assume that, as in systems in thermal equilibrium, the symmetry of the observed macroscopic order is largely dictated by the symmetry of local alignment interactions. But to what degree is the symmetry of the macroscopic order constrained by the symmetry of the microscopic interactions? More broadly, can active systems depart from these constraints and express a multitude of different ordering simultaneously, as is the case for living systems such as actin stress fibers and filopodia (29, 30)?

To study these fundamental questions, we use the high-density actomyosin motility assay (Fig. 1A), which is ideally suited to address the microscopic processes that underlie pattern formation in active systems (6, 7, 3134). By sensitively tuning the interactions between the myosin-driven filaments with a depletion agent, we can observe the emergence of a phase in which nematic and polar order stably coexist. The complete phase diagram is recovered from agent-based simulations of self-propelled filaments, in which weak alignment interactions quantitatively reproduce the experimentally determined microscopic collision statistics. We show that sufficiently weak interactions generically lead to dynamic coexistence of three phases (isotropic, nematic, and polar).

Fig. 1 Interactions in the actomyosin assay.

(A) Schematic of the actomyosin motility assay. PEG acts as a depletion agent. (B) Illustration of different filament collision geometries with an incoming angle θin and (C) corresponding binary collision curves. Whereas strong polar or nematic collision rules lead to full alignment or anti-alignment, weak collisions cause a gradual change of orientation and may exhibit both polar and nematic features (purple line). The dashed line depicts neutral collisions (θout = θin). (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 normalized filament velocities as a function of motor density. v0.1 is the velocity at 0.1 mg/ml nonprocessive heavy meromyosin. The inset shows absolute filament velocities.

In the actomyosin motility assay, hydrolysis of adenosine triphosphate (ATP) enables actin filaments to actively glide over a lawn of nonprocessive heavy meromyosin motor proteins (31, 32). Previous studies have shown that increasing the filament density beyond a critical value results in the emergence of polar clusters and waves (6, 7) (Fig. 2A). These patterns are produced by collisions in which filaments may align in a polar or nematic fashion. The degree and symmetry of the alignment depends on the change in the relative orientation of the interacting filaments, Δ = θout – θin, where θin and θout are the angles before and after a collision event, respectively (Fig. 1B). In theoretical studies (1528), these collisions have been idealized by assuming that filaments either align in a strictly polar or strictly nematic fashion upon colliding (Fig. 1C). However, in actual experimental active matter systems (8, 9, 34, 35), the degree of alignment caused by a single collision event is weak, that is, the relative change in filament orientation is small, Embedded Image (Fig. 1D). Moreover, the resulting alignment exhibits neither perfectly nematic nor perfectly polar symmetry. Instead, depending on the collision angle θin, in the motility assay, there is a weak tendency to favor either alignment or anti-alignment of the filaments (Fig. 1, C and D). How, then, can such weak interactions without a clear alignment symmetry on a local scale lead to collective order at the system level, and what features of the local interactions determine the global symmetry of the macroscopic state?

Fig. 2 Experimental phenomenology.

(A) Polar actin clusters formed in the absence of PEG, moving in the same direction as the filaments. (The fraction of fluorescently labeled filaments is 1/50, and the monomeric actin concentration is 10 μΜ.) (B) A large network of high-density nematic lanes formed at a PEG concentration of 3% and 5 μM actin. The image is an overlay covering a period of 100 s to demonstrate that the structure is frozen and stable. Filaments move along the lane contours in opposite directions. (The labeled filament fraction is 1/60.) (C) Probability density P(vx, vy) of instantaneous velocities shows the preferred bidirectional motion of filaments within a lane. (D) Single filaments move inside lanes (bright region). Two representative trajectories are shown (turquoise and orange) at 10 μM actin and 2% PEG. The inset shows an overlay covering a period of 50 s. Polar (A) and nematic (B) motion are depicted by uni- and bidirectional arrows, respectively. Scale bars, 100 μm; a.u., arbitrary units.

To answer these questions, we tuned the local interactions between the filaments by adding polyethylene glycol (PEG, 35 kDa), a depletion agent, at concentrations of up to 3% (w/v) to the assay (Fig. 1D and fig. S1). The observed change in the binary collision statistics can be attributed to the excluded-volume effect of the PEG molecules, which forces the filaments closer to the bottom surface covered with motors, enabling each to interact with more motors on average, with a concomitant increase in motor processivity (Fig. 1E). This reduces the incidence of collisions where filaments just pass over each other (9) and increases the likelihood that filaments will repel each other sterically, thus enhancing the tendency to align nematically [Fig. 1D, (36)]. This technique enabled us to continuously modulate the symmetry of alignment interactions at the microscopic level and probe the robustness of pattern formation in the gliding assay at high filament densities. Despite the rather minute changes in interaction characteristics caused by adding PEG at a concentration of 3% (Fig. 1D), we found that polar flocks no longer form. Instead, the moving filaments quickly, within a few minutes, self-organize into a network of “ant trails” (Fig. 2B and movie S1). In contrast to the unidirectional filament motion found within polar clusters, the filaments that form these “lanes” move bidirectionally, as do many colonial ant species (37). Because the filaments move along these tracks in either direction with equal probability (Fig. 2C and fig. S2), the overall order is nematic, not polar, and stable; this is quantified by the local nematic order (fig. S2A) and the autocorrelation function of the filament orientations (fig. S2, D and E). Moreover, whereas polar clusters propagate through the system at uniform speed, nematic lanes form static networks with branches spanning up to several hundred micrometers in length (Fig. 2B). Filaments are also seen to continuously leave and enter the trails (Fig. 2D and movie S2), such that these branches remain fixed in orientation and slowly grow and shrink at their ends (fig. S2F). These processes, operating on a time scale of minutes, lead to a slow reorganization of network architecture, with new branches forming (movie S3) while others contract (movie S4). Note that these networks are isotropically oriented and that no notable actin bundling was observed below 3% PEG.

This fundamental qualitative change in macroscopic order, from propagating waves of polar order to branched networks of stable lanes within which filaments move bidirectionally, induced by relatively minor changes in interaction characteristics at the microscopic scale, is puzzling. To reveal the underlying mechanism, we developed an agent-based computational model that goes beyond simple collision rules and faithfully reproduces the experimentally observed (microscopic) binary collision statistics and used it to predict 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 a constant speed v with the body of the filament following the tip. The direction of motion changes upon interaction with other filaments, as well as through interaction with molecular motors. When the leading segment of a given filament collides with a segment of another filament at a relative orientation θ, an alignment potential U(θ) acts on the tip. This potential is assumed to be the sum of terms with polar (p) and nematic (n) symmetry, U(θ) ∝ ϕϕp cos θ + ϕϕn cos 2θ, where ϕϕp and ϕϕn represent the respective mean change in orientation during a collision. We adjusted ϕϕp and ϕϕn such that the binary collision statistics of the computational model (Fig. 3B) closely resemble those observed experimentally (Fig. 1D).

Fig. 3 Simulation model and phenomenology.

(A) Illustration of the simulation model: Filaments (green) are propelled along their contours (solid black arrows). Upon collision, the orientations of tips (gray arrows) are redirected in proportion to the polar and nematic alignment strengths (red and blue arrows). (B) Binary collision data from simulations for two selected curves with different α. Error bars, 1 SD. (C and D) Emergence of (C) polar waves (α = 3) and (D) a network of nematic lanes (α = 6.25) in large-scale systems. The insets show filaments within a single pixel with local density ρ and local (C) polar or (D) nematic order. In both panels, 544,000 filaments were simulated in a box of length 650.2L, with a homogeneous density ρ0 = 1.29/L2. Scale bars, 100L. Uni- and bidirectional arrows denote local polar and nematic filament motion, respectively. (E) Different steady states for small simulation boxes, with ρ0 = 1.29/L2: Whereas α = 2.75 always produces polar waves and α = 6 always nematic lanes, at α = 4, either waves or lanes can be obtained in different realizations. Scale bars, 10L. (F) Global order parameters during a hysteresis loop in α. Black arrows denote the direction of the loop. Regions of nonzero δP (shaded in green) exhibit multistable behavior. For (B) to (F), ϕϕp = 2.1°.

Having validated the computational model at the microscopic level, we asked whether it captures the collective dynamics of the high-density actomyosin motility assay. We first performed large-scale simulations for model parameters corresponding to the absence of PEG. Starting from a random uniform distribution of filaments, we observed that high-density wave fronts of polar-ordered filaments rapidly form, surrounded by disordered, low-density regions (Fig. 3C and movie S5). This matches the phenomenology observed in the motility assay. Next, we performed simulations in a parameter regime corresponding to 3% PEG. Again, in agreement with our experiments, we found networks of high-density nematic lanes surrounded by disordered, low-density regions (Fig. 3D), reminiscent of chaotic structures that were predicted for active nematics (21). The overall network architecture changed slowly, with trails extending or retracting from their ends, and some lanes merging on longer time scales (fig. S4A and movie S6).

The model was then used to predict the dependence of nematic versus polar order on the filament density ρ0 and the ratio of nematic to polar alignment strength, α = ϕϕn/ϕϕp. To facilitate simulations over a broad parameter range, we considered smaller systems with a box size of 81.3L. We monitored the (global) polar and nematic order parameters Embedded Image and Embedded Image, respectively, measured over all filaments after the dynamics had become stationary (fig. S4B). In initial parameter sweeps, we observed that, within certain intervals of α, simulations starting from different realizations of randomly distributed filaments, but with identical parameter sets, sometimes resulted in polar and sometimes in nematic patterns (Fig. 3E, bottom panel). Similar observations were made in a Vicsek-type model, but only if strong additional memory in the particle movement is included (17). The patterns in our simulation were stable within the simulation times, and no switching between them was observed, suggesting the existence of a regime of interaction strengths in which the dynamics exhibit multistability. To probe these initial observations further, we checked for hysteresis effects in the collective dynamics [fig. S5, (36)]. To this end, we initiated our simulations in a parameter regime in which the system shows polar waves only (α = 2.75, Fig. 3E, top-left panel), waited until the dynamics became stationary, and then quasi-statically increased the value of α (i.e., giving the system sufficient time to equilibrate between successive adjustments of α) and monitored both nematic and polar order parameters (Fig. 3F, closed symbols). After reaching a regime in which the system gave rise to nematic lanes only (α = 6, Fig. 3E, top-right panel), we reduced the value of α quasi-statically (Fig. 3F, open symbols). Although the nematic order parameter remained essentially unchanged, we observed a hysteresis loop in the polar order parameter P. As the relative strength of nematic-to-polar alignment is increased, the degree of polar order (P+) gradually declines until it reaches zero at some critical value α+. Conversely, in the reverse direction, polar order (P) remains negligible up to a different critical value α and then suddenly jumps to a rather large value. The phase diagram in Fig. 4A was obtained using δP = P+P to quantify the degree of multistability, where δP is the difference between the degree of polar order for the forward (increasing α) and backward (decreasing α) processes.

Fig. 4 Phase diagrams and coexisting symmetries in experiment and simulation.

(A) Simulation phase diagrams for different filament densities ρ0 and relative alignment strengths α. (B) Experimental phase diagram of emergent patterns for varying monomeric actin and PEG concentrations. Gray crosses, disorder; red triangles, polar clusters; blue squares, nematic lanes; green diamonds, coexisting polar and nematic structures. Actin concentrations were normalized with respect to the estimated critical concentration in the absence of PEG (see supplementary materials for details). (C) Emergence of both polar waves and nematic lanes in large-scale simulations (scale bar, 100L) for α = 4 and a homogeneous density ρ0 = 1.29/L2. (D) Coexistence of polar clusters and nematic lanes in the motility assay at 2% PEG and 5 μM actin. Scale bar, 100 μm. (E) Phase diagrams for different polar alignment strengths ϕϕp and ρ0 = 1.29/L2. The total strength of alignment increases with both ϕp and α. The shape of the phase diagram only slightly changes for larger system sizes (see fig. S7A). (F) Scaling analysis of time scales at two different parameter sets (orange data: ϕϕp = 2.1°, α = 4.17; purple data: ϕϕp = 3.3°, α = 3.13). The average coexistence lifetime tfix (solid lines) grows roughly linear with system size, whereas the average initial order time t0 (dashed lines) remains small and constant. Averages taken over 25 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 diagrams were obtained by hysteresis analysis in α, and white dashed lines depict the domain boundaries of the observed steady states. For (A) and (C), ϕϕp = 2.1°.

To test these predictions, we performed experiments over a broad range of actin and PEG concentrations and obtained a phase diagram (Fig. 4B) whose topology closely resembles that obtained from the computational model (Fig. 4A). In particular, upon varying the strength of interaction between the filaments by changing the PEG level and thus α, we find a broad regime of nonequilibrium steady states where polar waves and nematic lanes coexist simultaneously. Moreover, both simulations of large systems (Fig. 4C and movie S7) and experiments (Fig. 4D and movie S8) consistently show that the equilibrium is highly dynamic. Polar waves may invade regions containing nematic trails and thereby disrupt their network structure (fig. S6A). After the passage of these waves, nematic-lane networks are observed to reform locally, often close to their original positions. The formation of nematic lanes was also observed at the left and right edges of polar waves (fig. S6B and movie S9). Whereas in experiments this coexistence remained stable during the full experiment duration (fig. S6C), in simulations we performed a scaling analysis to probe the lifetime of coexistence tfix as a function of the finite system size, at different points in the multistable parameter region. We found that this lifetime grows linearly with the system size, whereas the time of initial pattern formation t0 remains small and constant [Fig. 4F and fig. S7, (36)], implying a diverging time-scale separation and stable coexistence in the thermodynamic limit.

These observations from experiment and theory imply that polar waves and nematic lanes are both intrinsically stable structures, suggesting that the nonequilibrium steady state represents a dynamic equilibrium between different patterns, which, although they have conflicting polar and nematic symmetries, coexist in a dilute, disordered background. We attribute their coexistence to the weak interaction between the active particles, which determines macroscopic order not at the microscopic level but instead renders the symmetry of collective order itself to become an emergent property, which is dynamic in space and time. If this picture is valid, then an increase in the alignment strength at the binary level should eliminate the ambiguity in symmetry and prevent the emergence of coexistence. To test this hypothesis, we performed extensive numerical simulations by varying α and ϕϕp (Fig. 4E) and looking for multistability. Indeed, we find that as the total degree of alignment, that is, both ϕϕn and ϕϕp, is increased, the multistable region contracts and eventually vanishes completely. In this limit, there appears to be a sharp transition between a polar and nematic phase, similar to previous findings in a Vicsek-type toy model (18). We therefore conclude that the coexistence of patterns with mutual polar and nematic symmetries depends on sufficiently weak alignment interactions between individual filaments. Furthermore, it seems to be crucial that the computational model includes arbitrary pairwise interactions and spatiotemporal correlations without relying on any ad hoc truncation. This allows for coarsening dynamics, where many different mesoscale filament configurations are explored until they take the form of either polar clusters or nematic lanes. These patterns become local attractors of the dynamics, such that, despite their conflicting symmetries, they can exist in juxtaposition within the same system. This indicates that the celebrated Gibbs phase rule—which states that, in thermal-equilibrium one-component systems, a three-phase coexistence only occurs at a singular point in parameter space—is invalid in active systems. Overcoming this thermodynamic constraint may be an essential and simple prerequisite for biological systems to produce heterogeneous, multitasking structures out of a single set of constituents, as is the case for the cellular actin network (29, 30) and migrating cell layers (10, 11).

Supplementary Materials

www.sciencemag.org/content/361/6399/255/suppl/DC1

Materials and Methods

Figs. S1 to S7

Table S1

References (3848)

Movies S1 to S9

References and Notes

  1. Materials and methods are available as supplementary materials.
Acknowledgments: Funding: E.F. acknowledges support from the DFG via project B02 within SFB 863. A.R.B. acknowledges support from the European Research Council in the framework of the Advanced Grant 289714-SelfOrg and partial support from the DFG via project B01 within the SFB 863. All authors acknowledge the continuous support of the German Excellence Initiatives via the NanoSystems Initiative Munich (NIM). Author contributions: L.H., R.S., T.K., and A.R.B. performed and designed the experiments. L.H., T.K., and E.F. performed and designed the simulations. All authors participated in interpreting the experimental and theoretical results and in writing the manuscript. Competing interests: The authors declare no competing financial interests. Data and materials availability: All relevant data are provided in the paper and supplementary materials or online at https://github.com/LorenzHuber/ActivePolymerDataRepository/.
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