Abstract
From the selforganization of the cytoskeleton to the synchronous motion of bird flocks, living matter has the extraordinary ability to behave in a concerted manner^{1,2,3,4}. The Boltzmann equation for selfpropelled particles is frequently used in silico to link a system’s meso or macroscopic behaviour to the microscopic dynamics of its constituents^{5,6,7,8,9,10}. But so far such studies have relied on an assumption of simplified binary collisions owing to a lack of experimental data suggesting otherwise. We report here experimentally determined binarycollision statistics by studying a recently introduced molecular system, the highdensity actomyosin motility assay^{11,12,13}. We demonstrate that the alignment induced by binary collisions is too weak to account for the observed ordering transition. The transition density for polar pattern formation decreases quadratically with filament length, indicating that multifilament collisions drive the observed ordering phenomenon and that a gaslike picture cannot explain the transition of the system to polar order. Our findings demonstrate that the unique properties of biological activematter systems require a description that goes well beyond that developed in the framework of kinetic theories.
Main
Unlike animals that possess interactions such as spatial cognition or hierarchical dispositions^{14,15,16}, the interactions of reconstituted^{11,12,13,17,18} or synthetic^{19,20,21,22} model systems stem purely from physical interactions among the constituents. Although weak alignment forces have been proposed to be sufficient for the polar ordering transitions^{11}, no experimental data are available that quantify the interaction rules between the constituents. Owing to the lack of experimental data for such interaction rules, microscopic studies assume either an average rule over the particles in the neighbourhood^{23,24}, or that all binary interactions lead to perfect polar alignment^{5}.
Here we provide the angleresolved binarycollision statistics for a paradigmatic experimental active system; the actomyosin motility assay. The experimental system consists of only two main components: actin filaments and nonprocessive motor proteins heavy meromyosin (HMM; refs 11, 12, 13, 25). Actin filaments move on a lawn of HMM by consumption of adenosine triphosphate (ATP) at a constant speed of approximately 4 μm s^{−1}, which is ensured in the experiment by using high ATP concentrations. The orientation of a single filament motion is determined solely by the head, and the motion itself exhibits no longterm directionality, where the randomness of the system is generated by the pushing of the motors.
To obtain the binarycollision statistics, the experiments are conducted under dilute conditions with filament densities between 0.005ρ_{c} and 0.06ρ_{c}, where ρ_{c} ≈ 5 filaments μm^{−2} is the critical density for the disorder–order transition^{11}. Only short filaments with L = 2–5 μm are considered, to ensure that collisions are binary and also that the collision angles can be unambiguously defined, which becomes difficult for longer filaments owing to the intrinsic bending of the filaments. When two filaments encounter each other, for the majority of the collision events the filaments readily cross each other. Despite the frequent crossing events, we observe various tendencies in the binary collisions: polar alignment, antipolar alignment and events where the filament orientation hardly changes (Fig. 1a–c). To quantify these tendencies we measure the incoming angle θ_{12} = θ_{1}–θ_{2} along with the outgoing angle θ_{12}′ = θ_{1}′–θ_{2}′ (Fig. 1d; see Methods). Binary collisions alone are extracted and all nonbinary collisions are neglected. All filament densities in the dilute regime demonstrate similar quantitative behaviour for the experimentally obtained binarycollision statistics θ_{12}′(θ_{12}) (Fig. 2a), where unaffected collisions, θ_{12}′ = θ_{12}, are represented by a diagonal line in the diagram. The similarity in the behaviours indicates that there are no significant spatial correlations between the filaments for the dilute densities considered.
The distribution P(θ_{12}) of the incoming angle exhibits an asymmetric shape which is consistent with the Boltzmann scattering cylinder for rods^{8,9} when considering the assumption of slender rods (Supplementary Information), which gives P(θ_{12}) ∝ Γ(θ_{12}) = 2Lv_{0}sin(θ_{12}/2) ⋅ sinθ_{12} (see red line in Fig. 2b). Here, L denotes the filament length and v_{0} is the filament speed. The limiting case of slender rods is justified as actin filaments have a length of L = 2–5 μm and a diameter of d ≈ 8 nm. The Boltzmann scattering cylinder Γ describes the frequency of collisions between propelled particles of constant speed as a function of the relative angle θ_{12}, which is derived by geometric considerations. Owing to the large aspect ratio, incoming angles close to 0° and 180° are less likely for slender rods, and the corresponding distribution is asymmetric.
Averaging the experimentally obtained binarycollision statistics θ_{12}′(θ_{12}), we see that although most events for incoming angles θ_{12} > 80° are indeed largely unaffected by the collision, at highly acute θ_{12} a clear polar bias can be recognized: θ_{12}′ is smaller than θ_{12} (Fig. 2c). The distribution of θ_{12}′(θ_{12}) is significantly skewed towards polar outcomes for highly acute θ_{12} and antipolar outcomes for θ_{12} close to 180° (Fig. 2d). Because of this strong skewness, all data points of the experimental binarycollision statistics, and not just the average and standard deviation, are required to make a proper prediction on the existence of transition to polar order. Using the average of the collision statistics can lead to the loss of important information concerning the collision events, and ultimately cause a wrong prediction on the existence of polar order^{7}.
To connect the microscopic dynamics to the meso or macroscopic pattern formation and ordering transition in active systems, the Boltzmann equation for propelled particles has been successfully applied^{5,6,7,8,9,10}. The Boltzmann equation describes the mesoscopic particle motion and maps the precollision states onto the postcollision states, which enables the analytical prediction of the onset to polar order. It determines the time evolution of the oneparticle distribution function f(r, θ, t), where r are the coordinates, θ is the particle orientation and t is time:
The streaming term, , accounts for the movement of particles with velocity , where v_{0} denotes the constant speed and . Angular fluctuations are described as D_{θ}∂_{θ}^{2}f (see refs 20, 26), with D_{θ} denoting the angular diffusion constant. The collision integral, , captures the effect of binary filament collisions and the filament geometry. It consists a twoparticle density f^{(2)}(r, θ_{1}, θ_{2}, t), which can be written in the absence of correlations as f^{(2)}(r, θ_{1}, θ_{2}, t) = f(r, θ_{1}, t)f(r, θ_{2}, t). The collision integral plays a key role as the experimentally obtained data enter this term, which determines whether a transition to polar order exists or not. can be divided into a loss (−) and a gain (+), , (ref. 7):
where we omitted time and space dependencies for brevity. The gain and loss contributions are derived from integrating over all possible precollision orientations of the particles. A collision occurs with a frequency given by the Boltzmann collision cylinder for rods, Γ(θ_{12}), and changes the orientation according to θ_{j} → θ_{j} + η_{j}(θ_{12}) = θ_{j}′, j ∈ {1,2}, where η_{j} denotes the angular change of the jth particle orientation. For a given relative precollision angle θ_{12}, the orientation of particle j changes by η_{j}(θ_{12}) with probability p_{j}(η_{j}θ_{12})dη_{j}. The experimental scattering statistics (Fig. 2a) is equivalently expressed in terms of p_{j}(η_{j}θ_{12}) (Fig. 3a, b; Supplementary Information). If the filaments are indistinguishable, two important symmetry properties of p_{j} need to be satisfied: particle exchange symmetry p_{1}(η_{1}θ_{12}) = p_{2}(η_{2}−θ_{12}) and mirror symmetry p_{1}(−η_{1}θ_{12}) = p_{1}(η_{1}−θ_{12}), where the same argument applies for p_{2}. Indeed both are obeyed in the experimentally obtained binarycollision statistics of the actomyosin motility assay system (Fig. 3c, d and Supplementary Information). This indicates that experimental uncertainties, such as the narrow filament length distribution, do not influence or bias the binarycollision statistics.
Equations (1)–(3) are now analysed in terms of Fourier modes, where all Fourier coefficients are determined by the comprehensive binarycollision statistics p_{j} (ref. 7). The Fourier representation is a good starting point to derive the homogeneous and linearized equations for the coarsegrained momentum g, ∂_{t}g = ν g. The calculation also yields the coefficients ν in terms of p_{j}. The equations for the momentum become unstable for ν > 0, thus the transition density of polar order corresponds to ν = 0 (refs 5, 26). Specifically, the coefficient , where contains the comprehensive binarycollision statistics p_{j} (Fig. 3a, b). See Supplementary Information for the detailed mathematical expressions. Importantly, for the onset of polar order (ν > 0), must be positive.
Using the experimental data (Figs 2b and 3a, b), we find for binning of Δθ_{12} = 15° and Δη = 15° (see Supplementary Information for details on the computation of using experimental data). The negative sign of computed from the experimental data is robust against binning of η and θ (Supplementary Information). This implies that the binarycollision description is insufficient to explain the ordering transition in the experimental system for any density ρ, and the isotropic state is linearly stable. This statement is independent of the value of the rotational diffusion constant D_{θ} as it opposes the formation of polar order, expressed by a negative sign in ν. We test the consistency of our results with analytic predictions for a wellestablished theoretical model system where rods interact by halfangle alignment (η_{j} = θ_{i}/2 − θ_{j}/2; refs 5, 6), which results in (Supplementary Information), agreeing well with refs 6, 7.
Although we show that polar order cannot be reached when neglecting angular correlations (molecular chaos), small locally aligned filament clusters could still trigger the onset of polar order by binary collisions^{7,27}. To model such weak orientational correlations we write f^{(2)}(r, θ_{1}, θ_{2}, t) = χ(θ_{12})f(r, θ_{1}, t)f(r, θ_{2}, t), where χ(θ_{12}) characterizes the precursor angular correlations; for χ = 1, angular correlations are absent, leading to the assumption of molecular chaos. Led by recent studies of angular correlations in a selfpropelled particle system^{7}, we emulate angular correlations as χ(θ_{12}) = 1 + A/θ_{12}, where A is a free parameter that determines the strength of orientational correlations. For A = 0, correlations are absent, whereas large A values correspond to small, locally aligned filament clusters. We find that using different strengths of angular paircorrelations A does not change the negative sign of the kinetic coefficient (Supplementary Information). This implies that even a collection of preformed weakly aligned filaments, which collide accordingly to the experimental data p_{j} (Fig. 3a, b), cannot cause the system to order. It effectively augments the dealigning contributions, causing to decrease with increasing magnitude of χ. We conclude that, independent of the assumptions of zero noise and molecular chaos, the pattern formation observed in the motility assay cannot be explained solely by a succession of binary filament collisions.
The transition to polar order seen in the experiment can, however, be understood by multiparticle collisions: Already for densities one order of magnitude below the transition density, binary collisions are rare, characterized by the experimentally determined ratio R = n_{binary}/n_{all} of binary collisions n_{binary} to all collisions n_{all} (Fig. 4a). At ρ = 0.06ρ_{c} less than 10% of all collisions are binary. Thus, for densities close to the transition density ρ_{c}, filaments predominantly encounter multiparticle collisions.
More importantly, multifilament collisions are essential for the transition to polar order, which is manifest in the dependence of the transition density on the filament length. At a given filament density ρ ≈ 16 filaments μm^{−2}, long filaments with length L_{long} = 4–7 μm are able to form polar structures (Fig. 4b), whereas an assay of short filaments with length L_{short} = 0.5–3 μm remain in the isotropic state (Fig. 4c). Yet, at a higher concentration ρ ≈ 30 filaments μm^{−2}, short filaments are able to create clusters. At this higher concentration of filaments, the longfilament system is already in the densitywave regime^{11}.
As the exact nature of the length dependence on the transition density ρ_{c} indicates the microscopic mechanism of the transition, we systematically vary the average filament lengths from 2–10 μm and determine ρ_{c} (Fig. 4d). For a gas of propelled particles, where binary collisions are dominant (equations (1)–(3)), the transition density is defined by ν = 0; hence, ρ_{c} is expected to scale with 1/L (see Supplementary Equation 8). Thus, in this slenderrod limit, the collision probability is determined only by the length of the collision partner. Yet, we observe here an approximately quadratic behaviour for ρ_{c}(L) (Fig. 4d), ruling out the gaslike picture for this system. At the transition density ρ_{c}, the total number of simultaneous collisions for a filament of length L is constant: ρ_{c}L^{2} ≈ 200–300 (Fig. 4d inset). Importantly, this implies that it is the total number of simultaneous collisions and not the filament length itself which defines the order transition.
The presented results challenge our current understanding on the polar ordering transition in active systems. Such polar ordering transition is not determined purely by the number of particles in the system, but rather by the degree of multiparticle collisions. Systems governed by such features could well be described by kinetic models that incorporate collisions of arbitrary numbers of partners^{28,29} or go beyond Boltzmann’s meanfield assumption of molecular chaos^{30}. The uniqueness of the actomyosin motility assay, which enables access to all microscopic interactions and parameters, sets a quantitative basis for further development of our understanding of ordering phenomena in this diverse class of materials.
Methods
Assay preparation.
We used standard protocols to prepare the actin filaments and heavy meromyosin (HMM) motor proteins. Fluorescently labelled filaments stabilized with Alexa Fluor 488 phalloidin were used to visualize filaments with a fluorescence microscope. Flow chambers built from nitrocellulosecoated coverslips were incubated with HMM (0.05 μg ml^{−1}). Bovine serum albumin was used to passivate the surfaces inside the chamber after the incubation with HMM, then a dilute solution of actin filaments (binary collision: approximately 10–100 nM, filament length dependency: approximately 3–10 μM) was introduced. We added 2 mM ATP to enable the HMM to drive the filaments and a standard antioxidant buffer supplement was used to prevent oxidation of the fluorophore. Filament length was adjusted by shearing. For details of protein and assay chamber preparations, refer to Supplementary Information.
Imaging.
A Leica DMI 6000B inverted microscope was used to acquire data. A ×100 oil objective (NA: 1.4) was used for the binarycollision experiments and a ×40 oil objective (NA: 1.25) for the filament length dependency experiment. Images of resolution 1,344 × 1,024 pixels at a time resolution of 0.13 s were captured with a chargecoupled device (CCD) camera (C474295, Hamamatsu) attached to a ×1 (for binary collisions) and ×0.35 (for filament length dependency) camera mount.
Data analysis.
For the binarycollision experiments, filaments are identified by labelling connected components in the binary images and then are skeletonized, using Matlab. A cubic spline fit is applied to the skeletonized filaments to obtain coordinates for the filament contour. The coordinates for the filament head are used to determine θ_{1,2} and θ_{1,2}′. A collision is defined from when two filaments touch each other, until they are separated. Any collision events involving three or more filaments are classified as nonbinary. See Supplementary Information for more details.
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Acknowledgements
This research was supported by the European Research Council in the framework of the Advanced Grant 289714SelfOrg, Deutsche Forschungsgemeinschaft via project No. B2 within the SFB No. 863, and the German Excellence Initiative via the programme ‘NanoSystems Initiative Munich’ (NIM).
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R.S., C.A.W., E.F. and A.R.B. designed the project. R.S. and A.R.B. performed and designed all experiments. C.A.W. and E.F. theoretically analysed the experimental data. All authors participated in interpreting the experimental and theoretical results and in writing the manuscript.
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Suzuki, R., Weber, C., Frey, E. et al. Polar pattern formation in driven filament systems requires nonbinary particle collisions. Nature Phys 11, 839–843 (2015). https://doi.org/10.1038/nphys3423
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