Abstract
Motor proteins drive persistent motion and self-organization of cytoskeletal filaments. However, state-of-the-art microscopy techniques and continuum modeling approaches focus on large length and time scales. Here, we perform component-based computer simulations of polar filaments and molecular motors linking microscopic interactions and activity to self-organization and dynamics from the filament level up to the mesoscopic domain level. Dynamic filament cross-linking and sliding and excluded-volume interactions promote formation of bundles at small densities and of active polar nematics at high densities. A buckling-type instability sets the size of polar domains and the density of topological defects. We predict a universal scaling of the active diffusion coefficient and the domain size with activity, and its dependence on parameters like motor concentration and filament persistence length. Our results provide a microscopic understanding of cytoplasmic streaming in cells and help to develop design strategies for novel engineered active materials.
INTRODUCTION
Active systems are driven by nonthermal, energy-consuming processes that lead to very rich collective behavior (1–4). Examples are found from macroscopic length scales (fish and birds) to microscopic length scales (bacteria and algae) to even subcellular structures like the cell’s cytoskeleton. The cytoskeleton is composed of filaments that are dynamically interconnected by passive and active cross-linkers. It provides mechanical stability to biological cells, acts as a force-generating element, and serves as a track network for active intracellular transport (5, 6). Moreover, its dynamics generates internal motion of the cytoplasm that secures nutrient availability and the distribution of organelles (7). Fundamental knowledge about the relationship between cytoskeleton structure and dynamics and about the molecular driving forces helps to obtain a deeper understanding of cellular function and dysfunction in vivo and to design active gel materials, e.g., artificial cells (8) in vitro. Experimental studies of purified cell extracts containing cytoskeletal filaments and molecular motors show a plethora of dynamical phenomena. The long, thin, and stiff filaments self-organize into various liquid crystalline structures depending on filament concentration, motor type, and presence of passive cross-linkers. In vitro experiments on microtubule-kinesin and actin-myosin mixtures, where polymer-induced depletion interaction drives bundling of filaments and confines them at an oil-water interface, have shown exciting nonequilibrium behaviors, such as persistent spontaneous flows, turbulent-like motion, and the formation of highly mobile +1/2 topological defects, a characteristic signature of active nematics (9–13).
Various theoretical modeling approaches (14–22) have been used to reproduce and explain the nonequilibrium features of cytoskeletal filament-motor mixtures. Continuum descriptions based on active gel theory have been amazingly successful to capture essential aspects on the mesoscopic scale, particularly the formation and dynamics of topological defects (14, 15, 17–20). In this approach, activity is incorporated into a model of passive nematics by an active current J = ζ∇ · Q, where Q is the nematic tensor and ζ is the activity strength. This active current originates from the active force dipoles, which generate a particle flux along or against the nematic curvature (23, 24). For coarse-grained descriptions of active nematics at the filament level, nematic symmetry and activity are difficult to reconcile. Models of rods expanding in length and breaking in periodic intervals represent growing filament bundles (25) and have been used, as well as models of filaments that intermittently move forward and backward (26). At the microscopic level of filaments and motor proteins, the system is not nematic but polar. Studies of concentrated systems modeling polar filaments and molecular motors show the formation of polar bands and persistent motion generated at the boundary between the bands, where the filaments are antialigned (16, 27–30). The generation of extensile stresses occurs predominantly at the interfaces between polar domains and can drive the instability of a global nematic phase. This has been shown by a multiscale modeling approach that uses filament-based computer simulations of rods to provide parameters for a continuum polar-fluid model (16, 28, 29).
An important question arising from all the previous studies is the role of hydrodynamic interactions. It has been shown that systems with extensile force dipoles (corresponding to pusher-type hydrodynamics) destabilize the nematic order, whereas contractile force dipoles (corresponding to puller-type hydrodynamics) stabilize it (1, 19, 23, 24). These hydrodynamic stresses are at the origin of the active current in active nematic theory. However, in the absence of hydrodynamic interactions, active forces can also render the nematic phase unstable, like in experiments (31) and simulations (32) of granular ellipsoids and in a phenomenological continuum model for overdamped active nematics (33). These different approaches raise several pertinent questions: What is the relevance of polarity in active gels? Are the individual filaments the appropriate fundamental units of modeling, or should it rather be the filament bundles? Are the predictions of active gel theory and of polar active filament descriptions consistent? Which advantages can a more microscopic approach provide? What is the role of filament flexibility?
Here, we study the emergent structures, persistent motion, and instability of the aligned nematic phase in mixtures of polar semiflexible filaments and molecular motors. Two-dimensional Langevin dynamics simulations are designed to mimic experimental systems of filament-motor suspensions confined at oil-water interfaces (9, 12, 13). Our modeling approach bridges length scales from nanometer-sized molecular motors to micrometer-long semiflexible filaments. The results presented here show that the average motor-induced force on antiparallel filaments is a robust measure for the activity in the system and that hydrodynamic interactions are not necessary to describe the system dynamics. In an initially globally nematic filament suspension, the generated active stresses induce filament sorting in polar bands, a buckling-type instability, the formation of nematic-type topological defects, and—at steady state—the emergence of complex flow patterns. The dynamics of individual filaments is characterized by active Brownian particle–like motion. The active-force dependence of several inherent length scales, like the domain size, the length of quasi-ballistic motion, and the effective persistence length of single filaments, all follow a universal inverse square root dependence. Furthermore, we show how active nematic behavior emerges with increasing filament concentration. Where applicable, the predictions from our “polar active nematics” description agree well with those of active nematic theory.
The simulations presented here are based on a model of semiflexible filaments of contour length L and persistence length ℓp in two dimensions. The filaments consist of ns beads of diameter σ connected by stiff harmonic springs with rest length a0 = L/(ns − 1), where a0 = σ. The filament area fraction
RESULTS
Activity drives the formation of polar domains
The microscopic origin of filament motion is the dynamic cross-linking of nf polar filaments by nm molecular motors. The motors walk in the direction of polarity p on the filaments and induce sliding forces in direction −p. If two filaments are polar aligned and two consecutive motor steps occur on the two different filaments, the motors induce no net filament motion (see Fig. 1A). However, if the filaments are antialigned, the motors get stretched and net filament motion results (see Fig. 1B). The nm motors are classified as
(A and B) Sketches of two consecutive motor steps when filaments are oriented (A) parallel and (B) antiparallel. The initially relaxed gray motor steps toward the filament polar end, it relaxes, and it makes a second step on the other filament and relaxes again. This results in net filament motion only for antiparallel motors. (C and D) Simulation snapshots for an initially disordered nematic system. The thin black box indicates the central simulation cell. (C) Short-time band formation and (D) long-time disordered structures. Filament colors indicate their orientation, illustrated by the color axis. (E) Number of antiparallel motors as a function of the persistence length
When motors are added to an initially globally nematic suspension of filaments with random left-right filament orientation, the motor-induced sliding forces first lead to a rapid sorting of filaments into narrow polar bands with antiparallel alignment at the domain boundaries (see Fig. 1C). In time, these polar bands coarsen. When the activity is large enough, a buckling instability leads to disordered configurations of polar domains with topological defects (see Fig. 1D). If the activity is too small or the filaments are too stiff, the steady state consists of stable parallel bands (see fig. S1). Movie S1 illustrates the polarity-sorting process in more detail. Movie S2 shows the entire dynamical evolution from the initial nematic state to the stationary state. Because in this system both nematic alignment and polar sorting play an important role, we call it an “active polar nematic.”
Steady-state filament dynamics emerges from a complex interplay between various filament and motor properties. For example, the number of antiparallel motors is an important factor driving the filament dynamics and the suspension structure. We quantify the activity by the motor force per filament
Single-filament motion and active diffusion
We first characterize single-filament motion in dense systems in the active steady states by calculating the filament orientational autocorrelation functions and filament center-of-mass mean squared displacements (MSDs). The latter can be measured experimentally by fluorescently labeled tracer filaments or by using tracer particles that can be tracked. For nonzero
(A) Filament MSD divided by the lag time for different motor concentrations
Buckling polar bands
The strength of the active force does determine not only the average filament velocity but also structure, size, and stability of the domains. Polar bands are stable for weak active forces and large persistence lengths, whereas they become unstable and buckle at a particular wavelength λ for sufficiently large active forces. The time evolution of the motor-filament mixture in Fig. 3 (A to C) shows first polarity-sorted bands and then progressive bending and breaking of bands (see also movie S3). The wavelength λ of the instability displays a square root dependence on the filament persistence length ℓp (see Fig. 3D). The assumption that the active force in all these systems evolves independent of the persistence length (see fig. S5) suggests an Euler buckling–type instability with buckling force Fb ∝ E/λ2, where E ∝ ℓp is the (effective) elastic modulus (35). This relation applies under the assumption that the effective elasticity E is only weakly affected by the activity. Together with
(A to C) Snapshots of the time evolution of an initially nematic system with
Dynamics of topological defects
The system of polar filaments and motor proteins shares features with both active nematics and polar active fluids, although it is clearly distinguishable from both of them. The high densities favor nematic ordering for which the polarity is not important, whereas the two-filament pair interaction mediated by the motors has a pronounced polar character. The latter leads to local polar order in the domains. However, the characteristic topological defects that appear in polar active fluids, +1 or −1 defects (14), are never observed here. Instead, the defect structures in the dynamic disordered phase are
(A) Snapshot of a pair of
The f motion of the
From dilute to dense filament systems
In passive lyotropic liquid-crystalline systems, particle density and shape determine both structural and dynamical properties. The isotropic-nematic (IN) transition in a system of hard rods with aspect ratio L/σ = 20 at ϕ ≈ 0.2 (36) is a lower bound for the IN transition for our semiflexible filaments. To investigate the effect of filament density, we show in Fig. 5 (A to D) the simulation snapshots for various filament densities, below and above the IN transition in the passive systems, and different
(A to D) Snapshots for systems at various densities, ϕ = (0.15, 0.29, 0.44, 0.66), where nm is varied to result in similar
Last, we study the effect of activity and concentration on the effective filament stiffness or effective persistence length
A simple argument for this behavior can be extracted from the properties of single tangentially driven self-propelled filaments (37). Under the assumption that filament motion is predominantly along their contour (“railway motion”), the rotational diffusion DR of the end-to-end tangent vector (the change of the contour) can be calculated explicitly and only depends on the active velocity and the persistence length,
DISCUSSION
Our microscopic model of active gels—based on semiflexible polar filaments cross-linked by molecular motors—shows that for large enough motor concentration, an initially nematic arrangement of filaments is unstable and evolves via polarity sorting, band formation, and buckling into a stationary but highly dynamic polar-domain structure with persistent defect formation and annihilation. We find universal scaling of the domain sizes with the active force determined by the number of antiparallel motors and their extension. The predictions of our model are in good qualitative agreement with recent experiments. While polarity has received little attention for a long time, a recent in vitro experiment provides evidence for the existence of polar bands (38). Furthermore, the instability of the polar bands in our model is characterized by a unique length scale, in agreement with recent in vitro observations (39). In the disordered phase with polar domains, filaments display a short-time active ballistic motion followed by active diffusion at later times with a diffusion constant that increases with activity, in agreement with results of a model of growing and breaking filaments (9). For polar bands, similar active ballistic motion has been found in simulations of stiff rods (28).
Although polarity is important in driving the filament dynamics, the formation of
Our model can readily be extended to three-dimensional systems, to study mixtures of stiff and (semi)flexible filaments, as well as of models with an increased level of complexity like the inclusion of passive cross-linkers or hydrodynamic interactions. Because of the simplicity of the underlying mechanisms, we hope that our results can also help in the design of novel engineered active biomimetic materials. The Supplementary Material accompanies this paper at www.scienceadvances.org/.
MATERIALS AND METHODS
Suspensions of nf semiflexible filaments and nm motors are studied using Langevin dynamics (40) in two dimensions with periodic boundary conditions. Semiflexible filaments of length contour L are modeled as discrete chains of ns beads of mass m with position vectors
The Langevin equation of motion for each bead i on filament q
The coupling of filaments by molecular motors leads to sliding and binding forces that depend on the relative orientation of the filaments (see Fig. 1, A and B), where antiparallel motors (coupling two antiparallel filaments) exert larger forces than parallel motors (coupling two parallel filaments). Moreover, the activity induced by dimeric motors (one motor arm is grafted) tetrameric motors (both motor arms move simultaneously) is larger (30). The motor model here is a hybrid of the two, and both motor arms can move with equal probability but only one at a time.
Rotational diffusion is measured through the filament orientation correlation function, where the orientation is defined as the eigenvector corresponding to the largest eigenvalue of the moment of inertia tensor. Defects are traced using the method outlined in (25).
SUPPLEMENTARY MATERIALS
Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/6/30/eaaw9975/DC1
This is an open-access article distributed under the terms of the Creative Commons Attribution-NonCommercial license, which permits use, distribution, and reproduction in any medium, so long as the resultant use is not for commercial advantage and provided the original work is properly cited.
REFERENCES AND NOTES
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