---
abstract: 'In active matter systems, self-propelled particles can self-organize to undergo collective motion, leading to persistent dynamical behavior out of equilibrium. In cells, cytoskeletal filaments and motor proteins exhibit activity and self-organization into complex structures important for cell mechanics, motility, and division. Collective dynamics of cytoskeletal systems can be reconstituted using filament gliding experiments, in which cytoskeletal filaments are propelled by surface-bound motor proteins. These experiments have observed diverse behavior, including flocks, polar streams, swirling vortices, and single filament spirals. Recent experiments with microtubules and kinesin motor proteins found that the effective repulsive interaction between filaments can be tuned by crowding agents in solution, altering the collective behavior. Adding a crowder reduced filament crossing, promoted alignment, and led to a transition from active, isotropically oriented filaments to locally aligned polar streams. These results suggest that tunable soft repulsion can control active phase behavior, but how altering steric interactions and filament stiffness alter collective motion is not fully understood. Here we use simulations of driven filaments with tunable soft repulsion and rigidity in order to better understand how the interplay between filament flexibility and steric effects can lead to different active steady states. We identify swirling flocks, polar streams, buckling bands, and spirals, and describe the physics that govern transitions between these states. In addition to repulsion, tuning filament stiffness can promote collective behavior, and controls the transition between active isotropic filaments, locally aligned flocks, and polar streams.'
author:
- 'Jeffrey M. Moore'
- 'Tyler N. Thompson'
- 'Matthew A. Glaser'
- 'Meredith D. Betterton'
bibliography:
- 'collective\_motion\_driven\_semiflexible\_filaments.bib'
title: '**[Collective motion of driven semiflexible filaments tuned by soft repulsion and stiffness]{}**'
---

Active particles exhibit complex and dynamical order at length scales much larger than the scale of a single particle. Living systems with collective dynamics include swimming bacteria, schools of fish, flocks of birds, and crowds of people [@vicsek12; @czirok96; @dombrowski04; @parrish02; @bajec09; @silverberg13a]. The study of active biopolymers is motivated by the activity of cells, because intracellular organization and dynamics are largely governed by the cytoskeleton. Cytoskeletal filaments and motor proteins generate active forces that control the assembly of critical cellular structures and long-range patterns [@hyman96; @nedelec97; @surrey01], or active nematics [@sanchez12; @m.lemma19]. Collective behavior can occur in filament gliding experiments, wherein cytoskeletal filaments are propelled by motor proteins bound to a surface. Previous work on filament gliding has reported several nonequilibrium steady states [@butt10; @schaller10; @schaller11; @liu11; @sumino12; @huber18]. However, our understanding of the physics that controls these phases is incomplete.

Systems of microtubules propelled by kinesin motors often do not interact and align sufficiently for the emergence of collective motion [@liu11; @sumino12; @inoue15]. Recent work showed that adding a molecular crowding agent can reduce filament crossing and cause microtubules to sterically interact and locally align [@inoue15; @saito17; @kim18]. Similar behavior occurs in actomyosin motility, where the presence of a crowding agent increased filament alignment [@huber18]. This transition has been proposed to occur due to a depletion force between filaments, which promotes alignment [@inoue15; @saito17; @kim18; @huber18; @m.lemma19]. As a result, the collective dynamics of gliding filaments can be tuned by depletant concentration. However, the underlying physics contributing to the collective motion observed in these experiments is not well understood. Previous theoretical work investigating collective behavior of active polar particles includes Vicsek-type models [@vicsek95; @levine00; @gregoire04; @aldana07; @chate08], driven rigid rods [@kraikivski06; @ginelli10; @peruani12; @abkenar13; @kuan15] and flexible filaments with fixed interaction potential [@duman18a; @vliegenthart19]. However, previous models did not explore how the phase behavior changes with varying rigidity and tunable interaction strength.

In this paper, we study the non-equilibrium phase behavior of driven semiflexible filaments that interact with a tunable soft repulsive potential. The steady-state behavior depends both on the repulsive potential and rigidity. Increasing the repulsion leads to a transition between active isotropic filaments and aligned polar streams, in agreement with experiment. However, we also find that increasing filament rigidity promotes collective motion, which is counterintuitive because higher rigidity lowers the probability that two intersecting filaments align [@supp]. Therefore, we propose that while filament alignment by collisions is important, the degree of filament orientational persistence (controlled by stiffness) is an additional key physical effect for collective motion. Our results suggest that collisional alignment is not solely responsible for the emergence of collective behavior in active filament systems.

Our simulation model expands our previous work on self-propelled filaments [@kuan15; @gao15; @moore19] by adding semiflexibility and tunable repulsion. Each filament is composed of finite rigid segments (Fig. \[fig:model\]A, [@supp]). Neighboring segments on a chain are coupled by a bending potential, giving a filament persistence length $L_p$. Each segment of the filament experiences random forces so that its dynamics obey the fluctuation-dissipation theorem for slender filaments at thermal equilibrium [@doi88]. We implement the algorithm of Montesi, Morse, and Pasquali for a semiflexible chain with anisotropic friction [@montesi05]. The activity of motor proteins is modeled by a polar driving force per unit length $f_{dr}$ parallel to each filament segment. This choice reflects experimental observations that filament velocity is constant, even during filament crossing events [@liu11], suggesting that stochastic effects due to the motors (e.g., binding and unbinding, variation in motor stepping) occur at short enough time scales so as to not significantly alter large-scale behavior.

![Model schematic and simulation snapshot. A) Schematic of self-propelled, semiflexible, inextensible, filaments, which contain rigid segments of length $a$ with a bending potential between adjacent segments $U(\theta)$. Each segment experiences a parallel driving force $F_{\text{dr}}$. B) Plot of the repulsive potential $U(r)$ centered on a filament of diameter $\sigma$. C) Snapshot of a simulation with packing fraction $\phi = 0.2$, filament rigidity $\kappa = 100$, and repulsion $\epsilon^* = 5.6$.[]{data-label="fig:model"}](model){width="50.00000%"}

Interactions between filaments are repulsive but soft, defined by the generalized exponential potential (GEM-8), $U(r) = \epsilon e^{-(r/\sigma)^8} $, with cutoff $U(r>\sqrt{2}\sigma) = 0$. Here $r$ is the minimum distance between neighboring segments and $\sigma$ the diameter of a filament (Fig. \[fig:model\]B). The maximum potential value $\epsilon$ represents the energy required for two segments to overlap. This potential is steep near the edge of a filament. Interactions occur between each filament segment, except for nearest-neighbor segments of the same filament. While filaments experience local drag, we neglect long-range hydrodynamic interactions because previous experiments found these forces to be negligible [@saito17].

Filaments are inserted at a packing fraction $\phi = A_\text{fil}/A_\text{sys}$, where $A_{\text{sys}}$ is the area of the simulation box and $A_{\text{fil}}=N(L\sigma + \pi\sigma^2)$ is the total area occupied by $N$ spherocylindrical filaments of length $L$ and diameter $\sigma$. The characteristic timescale is the time for a filament to move the distance of its contour length, $\tau_A = L/v$, with the velocity depending on the total driving force and the coefficient of friction acting parallel to the filament, $v = \xi_{\parallel}^{-1} F_{\text{dr}}$. The filament driving force per unit length is set by the Péclet number, which is the ratio of active and diffusive transport time scales, $Pe = \tau_D/\tau_A = f_{\text{dr}} L^2/k_B T.$ We set $Pe = 10^5$, based on calculations of active forces in experiments [@liu11]. Simulations were run for $10^3-10^4 \tau_A$.

Our results depend on five dimensionless parameters: rigidity $\kappa=L_p/L$, interaction energy $\epsilon^* = \epsilon/\epsilon_{\text{dr}}$, packing fraction $\phi = A_{\text{fil}}/A_{\text{sys}}$, aspect ratio $l=L/\sigma=60$, system size $l_{sys}=L_{sys}/L=20$, and Péclet number. The interaction energy has been rescaled by $\epsilon_{\text{dr}}$, which is the energy required for the potential to exert a maximum force equal to the driving force of a particle with length $\sigma$. We varied $\epsilon^*$ from 0.93–9.3, $\kappa=20$, $50$, and $100$, and $\phi = 0.04$, $0.1$, $0.2$, and $0.4$. Although the filament aspect ratio is fixed, varying the filament rigidity is analogous to varying the length of a filament with a fixed persistence length.

Our simulations generated five primary phases: active isotropic, flocking, banding, swirling, and spooling (Fig. \[fig:examples\]). To quantify the dynamical phases, we used six order parameters: the global polar order $P$, global nematic order $Q$, contact number $c$, local polar order $p$, positional order $S(q_0)$, and spiral number $s$ [@supp]. In the active isotropic phase, filaments cross each other in all directions, resembling filament gliding experiments that do not exhibit collective motion (Fig. \[fig:examples\]A,  [@supp]). The flocking phase is characterized by polar domains of aligned filaments (Fig. \[fig:examples\]B), and can be stable as one large flock (giant flock phase) or with multiple coexisting flocks (flocking phase). Flocks are dynamic, with filaments continuously joining and leaving the flocking state. The banding phase occurs when most filaments join a stable band of filaments that spans the system length (Fig. \[fig:examples\]C), and can be either one band (polar band phase) or many counter-propagating bands (nematic band phase). In the spooling phase, filament are flexible enough to self-interact, and many filaments form spirals (Fig. \[fig:examples\]D). The swirling phase contains large, swarming flocks that collide, self-interact, and form transient vortices (Fig. \[fig:examples\]E).

![image](diagram){width="90.00000%"}

Collective motion emerges with increasing repulsion or rigidity. When both filament repulsion and rigidity are small, we find the active isotropic phase at all densities (Fig. \[fig:examples\]F). Self-organization and collective motion can occur by increasing either repulsion or stiffness; increasing $\epsilon^*$ tends to increase collective behavior, while increasing $\kappa$ or $\phi$ tends to increase long-range order. Remarkably, there are regions of phase space where increasing stiffness drives the emergence of collective motion, and regions where increasing repulsion abolishes collective behavior by lowering long-range organization. This was surprising because previous experimental work found that the transition from an active isotropic phase to polar streams was driven by an increase in alignment events between pairs of colliding filaments [@sumino12; @inoue15; @saito17; @kim18]. In our simulations, increasing $\epsilon^*$ increases $P_{\text{align}}$, in agreement with experiments [@inoue15; @saito17; @kim18]. In contrast, increasing the rigidity $\kappa$ decreases $P_{\text{align}}$, yet increases filament collective motion (Fig. \[fig:palign\]).

The counter-intuitive result that increasing stiffness lowers collisional alignment but increases collective motion reflects a trade-off between filament crossing and the orientational persistence of filament trajectories. We measured the time autocorrelation of filament orientation. Longer correlation time occurs for larger rigidity because filaments resist deflection [@supp]. Flocking filaments that move ballistically have longer-lived alignment, resulting in more filaments joining the flock than leaving it. In other words, more flexible filaments reorient more easily, so fewer filaments can join the flock, requiring a larger $\epsilon^*$ to cross the active isotropic–flocking boundary. Therefore, our results suggest that both filament alignment and persistent filament orientation contribute to collective motion and stable flocks (Fig. \[fig:palign\]).

In the active isotropic phase, filaments rarely align and do not stay aligned, leading to random orientation on average (Fig. \[fig:behavior\]A). The alignment lifetime–alignment probability interplay controls whether collective motion occurs. Increasing rigidity at low interaction potential strength ($\epsilon^* = 1.4$–$4.7$) decreases the absolute number of filament alignment events, but increases the alignment lifetime, allowing flock formation (Fig. \[fig:behavior\]B).

At steady state, the flocking phase may consist of multiple polar domains of aligned filaments, or all flocks may merge. At moderate density ($\phi = 0.1, 0.2$), flock coalescence results in a giant flock that spans the length of the system ($L_{\text{flock}} > l_{\text{sys}}$), forming a stable, polar band. At high density ($\phi=0.4$), multiple independent bands can form simultaneously, resulting in the nematic band phase. We note that high filament density paired with high $\kappa$ results in more collective motion even at very low $\epsilon^*$. It appears that high local density further increases the alignment lifetime of flocks by decreasing rotation of gliding filaments.

![A) Plot of the probability for two filaments to align upon collision. Increasing repulsion $\epsilon^*$ increases $P_{\text{align}}$ while increasing rigidity $\kappa$ decreases it. B) Simulation snapshots illustrating how increasing filament rigidity increases collective motion. All images show simulations with $\epsilon^* = 4.7$. []{data-label="fig:palign"}](align_prob){width="50.00000%"}

While increasing $\epsilon^*$ typically increases alignment events, promoting collective motion, for flexible filaments (low $\kappa$), filaments can bend and self-interact via collisions to form a spiral (Fig. \[fig:behavior\]C). Spirals are often stable until another collision deforms the filament enough to release it. The system therefore forms a spooling phase, related to the frozen, active steady states found in previous experimental work [@schaller11]. The principle mechanism of spool formation has been of significant interest [@liu11; @luria11; @t.lam14; @isele-holder15; @vandelinder16a; @duman18a; @mokhtari19], with suggestions ranging from defects in the motor lattice to thermal activation. Previous experiments with microtubules and dynein have observed stable filament vortices attributed to filament curvature induced by motors [@sumino12], which is absent in our simulations. The spooling phase demonstrated here is governed entirely by steric self-interactions and collisions, as found in previous modeling work [@isele-holder15; @duman18a]. Therefore while spool formation may be affected by defects in the protein lattice, pinning of filaments by dead motors, or intrinsic curvature, it does not require these mechanisms.

Our phase diagram shows a narrow region of stability for long-range collective motion in the banding and giant flocking phases. This occurs because while increasing $\kappa$ promotes long-range order, increasing $\epsilon^*$ too high can break it. At high $\kappa$ and high $\epsilon^*$, collisions lead to large deformations of flocking and banding filaments, causing buckling. In the simulation shown in Fig. \[fig:examples\]C, the filaments repeatedly form a polar band, which buckles, shears, falls apart, and reforms. Similar dynamic phases have been observed in systems of self-propelled rods [@ginelli10] and semiflexible filaments driven by motors [@vliegenthart19]. Therefore, intermediate values of $\epsilon^*$ facilitate stable collective motion.

![Diagram depicting the phase behavior of active filaments with varying rigidity $\kappa = 20, 100$ and repulsion $\epsilon^*=2.1, 9.3$ at $\phi = 0.2$. Increasing repulsion decreases the probability of filament crossing. Increasing filament rigidity increases polar order of flocks and increases resistance to filament bending in collisions. []{data-label="fig:behavior"}](transitions){width="50.00000%"}

When repulsion and stiffness are both high, even transient polar bands can no longer form, and the system enters the swirling phase. Collisions cause large deformation of flocks. Sharp bends can result in flock self-interaction, which may form a large, transient vortex of filaments (Fig. \[fig:behavior\]D). Near the banding-swirling transition, flocking filaments may coalesce into one large flock similar to the polar band phase, but the high degree of bending results in shorter end-to-end flock lengths, inhibiting the ability to form stable polar bands.

Our simulations do not display symmetry breaking of the system chirality as observed in some previous work [@schaller11; @sumino12; @kim18], due to zero preferred filament curvature in our model. When an intrinsic curvature is added with simulation parameters that otherwise form stable polar bands, we observe a rotation in the polar order vector [@supp], similar to previous simulations of gliding filaments with intrinsic curvature [@kim18]. Future work will likely explore the additional phases arising from chiral symmetry breaking due to intrinsic filament curvature.

The phase diagram presented here makes predictions that could guide future experiments seeking to observe collective behavior. Systems with tunable filament repulsion and rigidity can exhibit a variety of active steady-states. The ability to control the transition between these behaviors may have applications for drugs targeting cortical cytoskeletal filaments or nanodevices that use cytoskeletal filaments as molecular shuttles [@nitta06; @hess11].

We would like to thank Michael Stefferson for helpful discussions. This work was funded by the Soft Materials Research Center under NSF MRSEC Grant No. DMR-1420736, and the National Science Foundation under NSF GRFP Award No. DGE-1144083 and NSF Grant No. DMR-1725065. This work utilized the RMACC Summit supercomputer, which is supported by the National Science Foundation (awards ACI-1532235 and ACI-1532236), the University of Colorado Boulder, and Colorado State University. The Summit supercomputer is a joint effort of the University of Colorado Boulder and Colorado State University.
