Topological defects and density fluctuations in collectively moving systems

Abstract

Ensembles of collectively moving particles like flocks of birds, bacteria, or filamentous polymers show a broad range of intriguing phenomena, yet seem to obey very similar physical principles. These generic principles have been predicted to lead to characteristic density fluctuations, which are in sharp contrast to normal fluctuations determining the properties of ordered systems in thermal equilibrium. Using high-density motility assays of driven filaments, we characterize here the origin and nature of giant fluctuations that emerge in this class of systems. By showing that these unique statistical properties result from the coupling between particle density and the topology of the velocity field of the particles, we provide insight in the physics of collective motion.

The composition of active systems that show collective motion is quite generic: They consist of a sufficiently high density of “particles” like birds (1), insects (2), or fish (3), vibrated granules (4, 5), bacteria and cells (6–10), or filamentous proteins (11–17) that are either self-propelled or actively driven. However, the dynamics that results from mostly local interactions among the driven constituents (18) is anything but simple: Apart from collective motion and nematic or polar order, these systems can show such intriguing phenomena as swirling motion (5, 6, 12, 15), spontaneous and collective changes in the direction of motion (1, 3, 11), and persistent density inhomogeneities (7, 11, 12, 16) and can generate macroscopic fluid flow (17). The theoretical description of these systems proves exceedingly difficult. To elucidate the underlying physical principles and to assess whether these systems rely on unifying organizing principles, numerous theoretical studies have been devoted to model (self-)propelled particle systems. Pioneered by the work of Vicsek et al. (19), they approach the problem on all levels of description ranging from agent-based simulations (20–24) and mesoscopic models coarse-graining microscopic interaction rules (25–29) to mean field models in the hydrodynamic limit (10, 30–32).

The dynamic properties of all these models are strikingly similar to the phenomena observed in this broad class of systems. However, due to the lack of adequate and well-controlled experimental systems, a quantitative comparison with experimental results so far proved difficult. Of particular interest are certain key properties of collective motion that were first identified by Toner and Tu (30). These hallmarks of collective motility include the occurrence of abnormally large fluctuations in the particle density, the so-called giant number fluctuations (4, 7, 16, 24, 30, 31, 33) and correlations that are anisotropic with respect to the direction of collective motion (7, 20, 30, 34).

Despite the utmost importance of these unifying observables for a sound understanding of the physics of collective motion, most experimental systems studied to date defy their unambiguous measurement. On the one hand, this can be attributed to the limited system size of most experimental setups, whereas theoretic predictions are based on infinitely large system sizes in the mean field limit. On the other hand, many experimental systems are prone to rely on higher ordering principles like the exchange of signals or even cognitive processes.

To overcome these limitations, we use high-density motility assay experiments where highly concentrated actin filaments are propelled by heavy meromyosin (HMM) motor proteins in a quasi-2D geometry (11, 12, 17) (Fig. 1A). The molecular nature of the system allows us to access and control the mean field properties of collectively moving driven filaments with only a few and easily adjustable key parameters and at the same time permits large system sizes and high particle densities.

Fig. 1.

Collectively moving filaments show anomalously huge fluctuations in the particle density. A schematically shows the motility assay experiments where highly concentrated actin filaments are propelled by HMM motor proteins. At low F-actin concentrations, the filaments move independently from one another and no collective motion develops (B). Above a critical density, a phase of collectively moving filaments emerges (C) that is characterized by anomalously high fluctuations in the particle density. These fluctuations can be quantified by recording the temporal fluctuations of the fluorescence intensity ΔI (for details, see Materials and Methods). D shows ΔI as a function of the mean intensity <I>. For systems obeying the central limit theorem, these fluctuations should scale according to ΔI α I^0.5—and in the absence of collective motion, this is the case (D, gray data points). In the presence of phase boundaries between dense and dilute regions, like it is the case in the density wave regime, the fluctuations scale like ΔI α I^1.0 (D, black data points). Already slightly above the critical density, where the individual filaments move collectively, but the overall density is still homogeneous, the fluctuations are anomalously large with the fluctuations scaling according to ΔI α I^0.8 (D, blue, red, and green data points). These fluctuations are long-lived, as can be seen from the autocorrelation C(t) = <δI(T), δI (T + t)> of the local density fluctuations δI (E). In the absence of collective motion, density fluctuations decorrelate diffusively according to C(t) ∼ t⁻¹ (E, red data points); in the ordered phase, an additional exponential contribution for long times is found (E, blue data points). Parameters: gray: filament density: ρ = 3 ± 1 µm⁻²; filament length: l = 6.8 µm; red: ρ = 12 ± 2 µm⁻²; l = 6.8 µm; blue: ρ = 16 ± 2 µm⁻²; l = 6.8 µm; green: ρ = 30 ± 2 µm⁻²; l = 1.1 µm; black: ρ = 24 ± 2 µm⁻²; l = 6.8 µm. (Scale bars: 50 µm.)

With this setup, we show that the dynamical, statistical, and morphological properties of collectively moving propelled particles result from a close coupling between the density field and the velocity field of the propelled particles. We show that collective motion is always accompanied by anomalously large and long-lived fluctuations in the particle density and relate their emergence to the topology of the velocity field of the particles: Fluctuations in the particles’ velocity field lead to nonvanishing local divergences in the velocity that produce mass flow. As the directional fluctuations are large, this results in large fluctuations in the local particle density and the destabilization of any large scale polar and homogeneous state. The directional fluctuations themselves are anisotropic and decorrelate faster longitudinal to the direction of motion than perpendicular to it. As the density field is coupled to the velocity field, this results in anisotropic density fluctuations that also become manifest in the morphology of the collectively moving structures.

Results and Discussion

The temporal fluctuations of the filament density are quantified by evaluating the temporal fluctuations ΔI of the fluorescence intensity I of the filaments within subsystems of varying sizes (for details, see Materials and Methods and Fig. S1). At low densities, the filaments move quasiindependent from each other without any directional preference (Fig. 1B). In this disordered phase, the filaments are homogenously distributed and show only normal random fluctuations, with ΔI scaling according to I^0.5 (Fig. 1D and Fig. S2).

Above a critical filament density of ρ_c ∼ 5 µm⁻², the driven filaments form an ordered phase of collectively moving filaments (11, 12). The ordered phase is characterized by persistent density inhomogeneities as the filaments self-organize to form coherently moving structures such as homogenous clusters and density waves that coexist with a disordered background (12). Trivially, the phase boundaries of these inhomogeneities give rise to large density fluctuations, which scale with I^1.0 (Fig. 1D). However, within the phase boundaries of a large cluster (diameter > 1 mm), the filament density is homogeneous, yet persistent and coherent currents of collectively moving filaments are observable (Fig. 1C and Movie S1). To enable the quantitative comparison with existing theories of collective motion, it is mandatory to observe the local density fluctuations in such a phase where the overall density can still be considered to be homogenous (Fig. S3), yet still coherently moving filaments are observable. In these homogenous ordered phases, the fluctuations of the fluorescence intensity ΔI grow far more rapidly than expected for systems in thermal equilibrium and are found to be proportional to I^0.8. The observed scaling exponent of 0.8 is a generic feature of the system and highly conserved upon variations of the filament density and the filament lengths (Fig. 1D).

The occurrence of these anomalously large fluctuations in the ordered phase is intimately related to the lifetime of local density fluctuations δI that can be quantified by their autocorrelation function C(t) = <δI(T + t) δI(T)>_r. In the homogenous ordered phase, C(t) decays according C(t) ∼ t⁻¹ on short timescales. For longer timescales, however, C(t) shows a pronounced exponential contribution (Fig. 1E). It is this exponential contribution that discriminates the ordered state from the disordered state, where C(t) decays according to C(t) ∼ t⁻¹ for all times. Thus, the fluctuations in the ordered phase are not only anomalously large in magnitude, they are also very persistent. Although the exact functional dependence is different, the occurrence of long time tails is in agreement with the theoretic predictions (30).

However, what exactly is responsible for these intriguing statistical properties that sharply discriminate the ordered from the disordered phase? Plainly, the main difference between the two phases is the emergence of coherent currents of collectively moving filaments (Fig. 2A). To investigate this coupling between order and density, we calculate the coarse-grained velocity field v(r,t) of the collectively moving particles with a particle image velocimetry scheme (6, 12). Indeed, the dynamic properties of the velocity field and the density field are closely related: qualitatively, at points where the velocity field converges and has a sink, material is accumulated and the density increases—and at points where the velocity field has a source, material is transported away and the density locally decreases (Fig. 2 B and C and Movie S1).

Fig. 2.

Fluctuations in the particle density arise from the coupling between density and velocity field. Local density fluctuations of collectively moving filaments (A) are quantified by the local deviation δI from the mean fluorescence intensity (B). At sources, where material moves away from [div(v) < 0], the local density is reduced; at sinks [div(v) > 0] the local density is increased (C). The filament density was adjusted to ρ = 12 ± 2 µm⁻², and the average filament length was 6.8 µm.

The change in density as a response to local defects is only one part of the coupling between the density and the velocity field. The sinks and sources in the velocity field are not stationary either. They not only constantly dissolve and emerge, but generically propagate through the system and interact with one another. Defects of the same topological charge, two sources for instance, can bind and even merge to form a bigger and stronger defect (Fig. 3A and Fig. S4). However, defects of opposite topological charge can readily annihilate one another (Fig. 3A), reminiscent to the dynamics in nematic liquid crystals (35). Sinks and sources propagate independently through the system (Movie S1). Remarkably, their propagation speeds are up to one order of magnitude higher than the velocity of individual filaments v = 4.8 ± 0.3 µm/s (Fig. 3B)—much higher than the maximal velocity attainable if the defects were only advected with the fluid.

Fig. 3.

Defect dynamics in the ordered state. The image in A shows a merging event of two sink defects. The yellow isosurface corresponds to a defect strength of 0.3 (definition of the defect strength: Fig. S4). During the merging event, the strength of the sink slightly rises. Over time, more and more material accumulates. As the particle density is conserved and as this particular sink barely moves, the local rise in density has to be compensated by the emergence of a source in the vicinity of the sink (T = 8 s; light blue isosurface corresponds to a defect strength of 0.3). In general, however, defects are not stationary in space as they are dynamically propagate through the system. This is illustrated in B, where a time overlay of the spatiotemporal evolution of a source is shown. The defect propagates through the system and constantly changes its shape and its strength. Statistically, sinks and sources are equally likely to develop. This can be seen in C, where the size distribution of the 0.3-isosurfaces for sinks and sources is plotted. Both distributions are approximately exponentially distributed; sinks and sources have comparable sizes. The filament density was adjusted to ρ = 12 ± 2 µm⁻², and the average filament length was 6.8 µm.

The dynamics of the defects is closely related to the local filament density: Defects emerge predominately at spots with a low filament density and are transported from high to low filament densities. Physically, this happens because the filaments are less ordered and thus exhibit stronger directional fluctuations at spots with low filament densities. Sinks and sources are equally likely to develop. Both sinks and sources are not only approximately exponentially distributed, they also have similar occurrence probabilities and are on average of the same size (Fig. 3C). Their lifetime, of the order of 10 s (Fig. 3C), is yet again determined by the local filament density: Defects only persist as long as a further reduction or accumulation of material is not limited by a too low or too high local material density, respectively. Thus, the density not only follows the flow field, but the dynamic transport of defects within the flow field is also tightly coupled to the filament density, which is finite.

This bidirectional coupling between density and velocity can even lead to persistent local oscillations of the filament density and the velocity field (Fig. 4A): Conceptually, a sink [div(v) > 0] in the velocity field locally causes a rise in density until the local density is so high that no more filaments can be accumulated. At this point, the sink either has to dissolve or move to lower densities. This in turn causes a density outflux in the respective region and a source develops [div(v) < 0]. The decreased density leads to stronger fluctuations in the velocity until the density rises again. A sink emerges and the cycle starts over. The region of these local oscillations can extend up to 200 µm in diameter with a corresponding oscillation period of up to 30 s (Fig. 4A).

Fig. 4.

The local divergences in the velocity field div(v) correlate with variations in the density δI. In A, the local temporal evolution of δI and div(v) is shown. Starting at T = 0 s, a sink in the velocity field develops and the local density rises until the packing of the filaments is maximal and no more material can accumulate (T ∼ 12 s). A source develops that reduces the material density again (T ∼ 12 s to T ∼ 32 s). Low densities are not stable either: due to the sparse packing, individual filaments fluctuate more, thereby generating new defects. A sink emerges and the cycle starts over. The coupling between div(v) and δI is characterized by a delay time τ, as can be seen from the temporal correlation function (B). Light gray curves: sample correlation functions for different spots; black curve: correlation for the spot shown in D; blue curve: ensemble average over 768 independent spots. The filament density was adjusted to ρ = 12 ± 2 µm⁻², and the average filament length was 6.8 µm. (Scale bars: 50 µm; the local spot was set to 10 µm × 10 µm.)

The bidirectional coupling between local defects in the velocity field and the density can further be quantified by introducing the averaged local temporal correlation function G(t) = <δI(T) div(v)(T + t)>_r between the density fluctuations δI and the divergence of the velocity field div(v) measuring the strength of the sinks and sources in the velocity field (Fig. 4B). Remarkably, the correlation between density fluctuations and the divergence of the velocity field is maximal for a finite time interval τ = 2.4 s, before it decays. The maximum at τ indicates that the coupling between density and velocity is characterized by a delay time of the order of τ. This implies that the density follows distortions in the flow field only with a certain retardation and that density changes result from local defects in the velocity field and vice versa. By contrast, in the absence of local defects, for a perfectly homogenous and steady current with div(v) = 0 no anomalously large fluctuations do emerge (Fig. S5).

The directional fluctuations that underlie the formation of defects in the velocity field themselves are strongly anisotropic with respect to the direction of motion, as can be seen from the angle-dependent two-point velocity correlation function P(r,α) ∼ <v(R + r,α) v(R)>_t,R, with α being the angle between v(R) and r (Fig. 5A, Inset). While the velocity field is highly correlated transversal to the direction of movement (α= 90°, 270°), P(r,α) decays faster along the direction of motion (α= 0°) (Fig. 5A). Both longitudinal and transversal to the direction of motion velocity correlations can be described by stretched exponential according to P(r,α) ∼ exp[−(Γ/x)]^β with a stretching exponent of β = 1.2 and a correlation length of Γ = 197.4 µm transversal (averaged over α= 90° and α= 270°) and Γ = 149.7 µm longitudinal (α= 0°) to the direction of motion, respectively (Fig. 5B). The deviations from the stretched exponential that become visible at high correlation lengths are due to finite size effects.

Fig. 5.

The correlation of the flow profile is anisotropic. A shows the correlation map of the velocity field of coherently moving filaments. To quantify the angle-dependent correlation in the system, the correlation function is evaluated in a coordinate system that corotates with the local direction of motion (Inset). The resulting correlation of the velocity field is anisotropic with respect to the direction of motion and shows a pronounced asymmetry in the transversal direction (B). In all cases, the correlation functions can be described by stretched exponentials according to exp[−(Γ/x)^β] with a stretching exponent of β = 1.2 (dashed lines). Both the anisotropy and the correlation length Γ are a function of average filament length (C). In general, shorter average filament lengths lead to smaller correlation lengths with a concomitantly decreasing anisotropy. The filament concentration was adjusted to a monomeric concentration of 18 µM.

Apart from the pronounced anisotropy, the correlation function shows a pronounced asymmetry along the direction of motion (Fig. 5B). Although the correlation length ahead of the direction of motion (α= 0°) averages to Γ = 149.7 µm, it is considerably enhanced behind the direction of motion (α= 180°) where Γ evaluates to Γ = 168.6. This asymmetry can be attributed to a lateral memory effect within the collectively moving flock: Succeeding filaments that lie in an angular sector of around α= 180° cannot immediately loose the orientational bias of the leading filaments, naturally enhancing the correlation in this direction.

Microscopically, the anisotropy that characterizes the collective motion of driven filaments stems from the extreme aspect ratio of the individual filaments. With a length in the order of 10 µm and a thickness of 7.5 nm, the filaments have an aspect ratio of around 1,000. It is this extreme aspect ratio that allows for an efficient alignment and a swift information transfer transversal to the direction of motion. Conceptually, any change in this aspect ratio should become manifest in both the overall correlation length and the anisotropy of the velocity field. This is indeed the case: Although the shortening of the filaments only barely affects the phase behavior of the system, the correlation length of the velocity field decreases (Fig. 5C). With a correlation length in the order of 100 µm, velocity correlations decay twofold faster for short, on average 0.8-µm–long, filaments than for filaments with an average length of 6.8 µm. The decrease in the overall correlation is accompanied by a concomitant decrease in the anisotropy of the velocity field (Fig. 5C), illustrating that shorter filaments provide a less effective information transfer transversal to the direction of motion.

According to the identified coupling between density and velocity, this anisotropy should also become manifest in the dynamical properties of the density field. As can be inferred from the correlation map of the density fluctuations P(r,α) ∼ <δI(R + r,α) δI(R)>_t,R, this is indeed the case (Fig. 6A). Just like the velocity, the density fluctuations are anisotropic and decay faster along the direction of motion than perpendicular to it (Fig. 6B and Fig. S6) in accordance with theoretic predictions (30).

Fig. 6.

The anisotropy in the correlation of the density fluctuations affects the morphology of clusters of collectively moving filaments. As in the case of the velocity (Fig. 5), correlations P(r,α) among intensity fluctuations δI are anisotropic (A). Compared with correlations of the velocity field, the correlation length is approximately a factor of two smaller and the anisotropy is less pronounced (B). The dashed lines denote the corresponding velocity correlations. The anisotropy of the density correlations with respect to the direction of motion also becomes manifest in the morphology of the collectively moving clusters. C shows the distribution of the main axes a and b of a moving cluster (C, Inset). b denotes the axes length perpendicular to the direction of motion (red arrow); a is the length of the main axes alongside the mean direction of motion. Whereas a ratio of b/a = 1 is equivalent with a spherical cluster, b/a > 1 denotes an oblate cluster that is stretched out perpendicular to the direction of motion, and b/a < 1 results from an elongated cluster. As can be seen from the distribution of b/a values in C, clusters are predominantly oblate, once again exemplifying the close coupling between density, velocity, and morphology of the structures. The filament density was adjusted to ρ = 12 ± 2 µm⁻² for A and B and to ρ = 7 ± 2 μm⁽⁻⁾² for C, and the average filament length was 6.8 µm.

The observed anisotropy in both the density fluctuations and the velocity field has severe consequences for the dynamic structure formation of the driven filaments. Practically, the enhanced correlation transversal to the direction of motion means that density accumulations transversal to the direction of motion are more stable than along the direction of motion and by consequence they get amplified.

In addition, the anisotropy in both the velocity and density correlations also becomes manifest in the cluster phase at lower filament concentrations. Although the cluster sizes are a function of the filament density (Figs. S7 and S8), the morphology of the clusters is determined by the collective dynamics of the coherently moving filaments: On average clusters are predominately oblate with their long axis pointing perpendicular to the direction of motion—independent of the cluster sizes (Fig. 6C).

Conclusion

The unique properties and accessibility of our experimental systems allow the demonstration and verification of the existence of two key signatures of collective motion: The anisotropy of the collective motion and the emergence of anomalously large fluctuations in the particle density. Both properties agree quantitatively with generic theories (30) and agent-based simulations (24). The anomalous properties arise from fluctuations in the velocity field with div(v) ≠ 0 that produce mass flow. Because the fluctuations in the velocity field are large, this results in large fluctuations in the particle density as well (31). Although this qualitative argument in principle is straightforward to understand, the systematic and quantitative investigation of this principle still proved difficult. On the theoretical site, although defects are explicitly allowed in generic theories, their role for the dynamic structure formation and for the unique statistical properties of collectively moving living matter remains to be systematically addressed in the full nonlinear regime. The importance of the interplay between divergences in the velocity field and the emergence of anomalously large fluctuations has been addressed indirectly in a recent combined experimental and theoretical study on collectively moving bacteria (10). There, it was shown that incompressible systems where div(v) = 0 strictly applies, do not show anomalously large fluctuations, further substantiating the coupling between divergences in the velocity field and fluctuations in the particle density we identify here. The further investigation of this interplay will not only contribute to a better understanding of the unique statistical properties of active systems but will pave the way for a further advance in our general understanding of the complex physics of flocking, swarming, and collective motion.

Materials and Methods

Protein Preparation.

G-actin was obtained from rabbit skeletal muscle following a standardized protocol (36, 37). Actin polymerization was initiated by adding 1/10th of the sample volume of a 10-fold concentrated F-buffer (20 mM Tris, 20 mM MgCl₂, 2 mM CaCl₂, 2 mM DTT, and 1 M KCl). For fluorescence microscopy, fluorescently labeled reporter filaments are used at a ratio of labeled to unlabeled filaments ranging from 1:4 to 1:50. They were stabilized with Alexa Fluor 488-phalloidin (Invitrogen) at a ratio of 1:2. Unlabeled filaments are stabilized with phalloidin (Sigma), likewise at a ratio of 1:2. Once polymerized, actin was used within 2 d. HMM is prepared from myosin II obtained from rabbit skeletal muscle following a standardized protocol (38).

Sample Preparation.

Flow chambers were prepared with precleaned microscope slides (Carl Roth) and coverslips (Carl Roth; 20 × 20 mm; no. 1). Coverslips were coated with a 0.1% nitrocellulose solution diluted from a 2% (vol/vol) solution (Electron Microscopy Sciences) in amyl acetate (Carl Roth). The coverslips were fixed to the microscope slides using parafilm, yielding an overall chamber volume of ∼30 µL. Before each experiment, a 30-µL actin dilution (1–25 µM monomeric actin) was prepared by gently mixing labeled and unlabeled actin filaments with Assay Buffer (25 mM imidazole hydrochloride, pH 7.4; 25 mM KCl; 4 mM MgCl₂; 1 mM EGTA; 1 mM DTT). The flow chamber was incubated with HMM diluted in Assay Buffer. Before the insertion of the respective actin dilutions, surfaces are passified with a BSA solution [1 mg/mL BSA (Sigma) dissolved in Assay Buffer]. To start the experiment, 2 mM ATP dissolved in Assay Buffer (50 µM) is flushed into the flow chamber. Oxidation of the fluorophore was prevented by adding a standard antioxidant buffer supplement GOC (2 mg of glucose oxidase, Sigma; 0.5 mg of catalase, Fluka). To verify that the observed patterns are not affected by the use of GOC, control assays without the antioxidant supplement were performed. After addition of the Motility Buffer, flow chambers were sealed with vacuum grease (Bayer Silicones).

Image Acquisition and Calculation of the Density Fluctuations.

All data are acquired on a Leica DMI 2000 inverted microscope with a 40× oil objective (numerical aperture, 1.35). Images (resolution, 1,344 × 1,024 pixels; color depth, 16 bit) were captured with a charge-coupled device camera (C4880-80; Hamamatsu) attached via a 0.35 camera mount. Image acquisition and storage are carried out with the image-processing software “OpenBox” (39).

The fluctuations in the filament density were calculated by recording and evaluating the fluorescence intensity of 2,500 consecutive images (corresponding to 4.88 min) for five different spots per sample. To quantify the intensity fluctuations, the raw images are first subjected to a background subtraction. Background images were recorded separately with homogeneous samples that contain the same amount of fluorophores as the actual experiments. The subtraction of these homogeneous background images proved to be the best way to correct for the slight but inevitable inhomogeneities in the fluorescence illumination that are particularly pronounced at the margins of the field of view. To further correct for these inhomogeneities, a region of interest was defined that leaves out the boundary regions of the field of view, thereby containing only 60% (1,024 × 768) of the original image.

To rule out systematic intensity fluctuations that occur due to bleaching of the sample, the averaged decay (averaged over the entire region of interest) in the temporal evolution of the decay in the fluoresence intensity was fitted with a single exponential according to exp(−t/I_d). This was used to correct the image sequence for bleaching by adding exp(t/I_d) to each pixel. The temporal intensity fluctuations ΔI are then evaluated for rectangular subsystems of varying size and plotted against the mean intensity I in the subsystem—following the approach introduced in ref. 4.

With the image-processing described here, the fluorescence intensity provides a good measure for the particle density in the field of view. To illustrate this, we performed experiments at a lower labeling ratio of 1 labeled filament to 200 unlabeled filaments and observed the collectively moving filaments at a higher magnification with a 100× objective and using a 1.0 camera mount. The sparse labeling at high magnifications thereby allows us to simultaneously record the fluorescence intensity and to track individual filaments. As can be seen from Fig. S2A, the number of filaments N is indeed directly proportional to the fluorescence intensity I. As a consequence, the temporal fluctuations of both quantities are highly correlated and coincide in magnitude and duration (Fig. S2B). These temporal fluctuations average up to anomalously large fluctuations with comparable exponents for both quantities (Fig. S2 C and D): the particle fluctuations ΔN scale with 0.73 ± 0.03 and the fluctuations in the fluorescence intensity ΔI scale with 0.76 ± 0.05 for the given image sequence. This exemplifies that the evaluation of the density fluctuations can readily be calculated via the fluorescence intensity.

However, using the fluorescence intensity as readout has the advantage that much larger regions of interest that comprise millions of individual filaments can be analyzed, which is mandatory for any comparison with macroscopic theories. Moreover, using the fluorescence intensity to quantify the density fluctuations, we can use higher labeling ratios, minimizing labeling artifacts and at the same time allowing for the precise evaluation of the corresponding macroscopic velocity profiles.

Particle Image Velocimetry.

The velocity profiles were calculated with a customized particle image velocity (PIV) algorithm written in MATLAB. The PIV code estimates the most probable displacement of small quadratic blocks by evaluating the cross-correlation function of two consecutive images (40). For all image sequences, a block size of 32 pixels was used.

Velocity and Density Correlations.

The angle resolved correlation functions for the velocity field are defined as follows:

whereby <•>_t denotes a temporal average over 2,500 frames corresponding to ∼5 min. The angle α is the angle between the local direction of motion and the vector r connecting the two PIV blocks (Fig. 5A, Inset). Mathematically, this can be conceived as a transformation of the velocity field into a coordinate system that corotates with the local direction of motion. The output is a fully angle-resolved correlation map that allows extracting the correlation in any given direction relative to the local polarity (7). The statistics for the correlation map is determined by the number of time points and the spatial average. Each point in the correlation map corresponds to a statistical average over 2,500 individual data points, at least. For the fit presented in Figs. 5 and 6, the statistical weight is explicitly taken into account. The correlation map for the density fluctuations is calculated similarly.

Cluster Morphology and Cluster Size Distributions.

To quantify the morphology of the clusters and to extract the cluster size distributions from the microscope images (Fig. S7A), we first smooth the raw images by using a Gaussian blur filter (radius, 2 pixels; algorithm, ImageJ) (Fig. S7B). We then apply a cutoff in the fluorescence intensity that defines the margin of the cluster (Fig. S7 C–E). At low filament densities, the cluster sizes are approximately exponentially distributed (Fig. S8A, light blue). At higher filament densities, the probability for finding large clusters increases compared with the exponential distribution (Fig. S8A, dark blue).