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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
  • 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.

  • 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.

  • 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°.

  • 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°.

Supplementary Materials

  • Emergence of coexisting ordered states in active matter systems

    L. Huber, R. Suzuki, T. Krüger, E. Frey, A. R. Bausch

    Materials/Methods, Supplementary Text, Tables, Figures, and/or References

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    • Materials and Methods
    • Figs. S1 to S7
    • Table S1
    • Captions for Movie S1 to S9
    • References

    Images, Video, and Other Media

    Movie S1
    Large network of nematic lanes.
    In this movie actin filaments form nematic high-density structures which constitute branches of a large network, surrounded by disordered regions. This movie is associated with figure 2B and has dimensions 605 μm times 461μm. Visible filaments are labelled with a GFP tracer and represent a fraction of 1:60 of all filaments.
    Movie S2
    Close-up of a nematic lane.
    This movie shows the dynamics of single actin filaments, which are exchanged between nematic lanes and the disordered environment. Within a lane, reversal of filament orientations occurs. The movie is associated with figure 2D and has dimensions 212 μm times 161 μm (labelling ratio 1:60).
    Movie S3
    Nematic lanes merging.
    Movie of close-by nematic lanes that merge and form a new branch, indicated by white arrows. This happens on the time scale of minutes. The movie dimensions are 328 μm times 246 μm (labelling ratio 1:30).
    Movie S4
    Nematic lanes shrinking.
    In this movie, nematic lanes indicated by white arrows become depleted on the time scale of minutes. The movie dimensions are 295 μm times 222 μm (labelling ratio 1:50).
    Movie S5
    Simulation of polar density waves.
    This movie (associated with figure 3C) shows a large system that produces polar density waves which coarsen over time. Left panel: density field. Upper right panel: local nematic order. Lower right panel: local polar order. Parameters: α = 3, ϕp = 2.1°, box size 650.2L (periodic boundaries, random initial conditions), ρ0 = 1.29/L2, simulation time 0-3175.
    Movie S6
    Simulation of nematic high-density lanes.
    This movie (associated with figure 3D) shows the evolution of a large system, forming a network of nematic lanes. Left panel: density field. Upper right panel: local nematic order. Lower right panel: local polar order. Parameters: α = 6.25, ϕp = 2.1°, box size 650.2L (periodic boundaries, random initial conditions), ρ0 = 1.29/L2, simulation time 0-3175.
    Movie S7
    Simulation with simultaneously emerging polar and nematic structures.
    This movie shows the emergence of both nematic and polar structures, which interact and coarsen over time. Left panel: density field. Upper right panel: local nematic order. Lower right panel: local polar order. Parameters: α =4.25, ϕp =2.1°, box size 650.2L (periodic boundaries, random initial conditions), ρ0 = 1.29/L2, simulation time 0-3175.
    Movie S8
    Coexisting polar and nematic structures.
    Polar clusters are embedded between nematic lanes, which get spatially rearranged by interacting with the clusters. The movie has dimensions 605 μm times 461μm and is associated with figure 4D (labelling ratio 1:30).
    Movie S9
    Filament exchange processes between structures of different symmetry.
    This movie shows exchange processes between polar and nematic structures within a single experiment in the coexistence regime. Left panel: polar clusters “eat up” nematic lanes by crossing them. Right panel: polar cluster “leaves” a nematic lane. The movie is associated with figures S6A,B (labelling ratio 1:30).

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