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M. Marchetti, J. Joanny, S. Ramaswamy, T. Liverpool, J. Prost, M. Rao, R. Simha (2013)
Hydrodynamics of soft active matterReviews of Modern Physics, 85
Samantha Stam, Simon Freedman, Shiladitya Banerjee, Kimberly Weirich, A. Dinner, M. Gardel (2017)
Filament rigidity and connectivity tune the deformation modes of active biopolymer networksProceedings of the National Academy of Sciences, 114
T. Sanchez, Daniel Chen, Stephen DeCamp, M. Heymann, Z. Dogic (2012)
Spontaneous motion in hierarchically assembled active matterNature, 491
(2008)
National Institutes of Mental Health
J. Fukuda, H. Yokoyama, M. Yoneya, H. Stark (2005)
Interaction between Particles in a Nematic Liquid Crystal: Numerical Study Using the Landau-de Gennes Continuum TheoryMolecular Crystals and Liquid Crystals, 435
Maureen Cetera, G. Juan, Patrick Oakes, Lindsay Lewellyn, M. Fairchild, Guy Tanentzapf, M. Gardel, S. Horne-Badovinac (2014)
Epithelial rotation promotes the global alignment of contractile actin bundles during Drosophila egg chamber elongationNature communications, 5
C. Denniston, D. Marenduzzo, E. Orlandini, J. Yeomans (2003)
Lattice Boltzmann algorithm for three–dimensional liquid–crystal hydrodynamicsPhilosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 362
S. Margossian, S. Lowey (1982)
[7] Preparation of myosin and its subfragments from rabbit skeletal muscleMethods in Enzymology, 85
Pau Guillamat, J. Ignés-Mullol, F. Sagués (2016)
Control of active liquid crystals with a magnetic fieldProceedings of the National Academy of Sciences, 113
L. Giomi, M. Bowick, Xu Ma, M. Marchetti (2013)
Defect annihilation and proliferation in active nematics.Physical review letters, 110 22
J. Spudich, S. Watt (1971)
The regulation of rabbit skeletal muscle contraction. I. Biochemical studies of the interaction of the tropomyosin-troponin complex with actin and the proteolytic fragments of myosin.The Journal of biological chemistry, 246 15
Rui Zhang, Nitin Kumar, J. Ross, M. Gardel, J. Pablo (2017)
Interplay of structure, elasticity, and dynamics in actin-based nematic materialsProceedings of the National Academy of Sciences, 115
L. Giomi, M. Bowick, Prashant Mishra, R. Sknepnek, M. Marchetti (2014)
Defect dynamics in active nematicsPhilosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 372
A. Beris, B. Edwards (1994)
Thermodynamics of flowing systems : with internal microstructure
Zhaoli Guo, C. Zheng, B. Shi (2002)
Discrete lattice effects on the forcing term in the lattice Boltzmann method.Physical review. E, Statistical, nonlinear, and soft matter physics, 65 4 Pt 2B
P. Ellis, D. Pearce, Ya-Wen Chang, G. Goldsztein, L. Giomi, A. Fernández-Nieves (2017)
Curvature-induced defect unbinding and dynamics in active nematic toroidsNature Physics, 14
Xingzhou Tang, Jonathan Selinger (2017)
Orientation of topological defects in 2D nematic liquid crystals.Soft matter, 13 32
Zhaoli Guo, C. Shu (2013)
Lattice Boltzmann Method and Its Applications in Engineering
E. Hemingway, Prashant Mishra, M. Marchetti, M. Suzanne, Fielding (2016)
Correlation lengths in hydrodynamic models of active nematics.Soft matter, 12 38
C. Dombrowski, L. Cisneros, S. Chatkaew, R. Goldstein, J. Kessler (2004)
Self-concentration and large-scale coherence in bacterial dynamics.Physical review letters, 93 9
W. Paxton, Kevin Kistler, Christine Olmeda, Ayusman Sen, S. Angelo, Yanyan Cao, T. Mallouk, P. Lammert, V. Crespi (2004)
Catalytic nanomotors: autonomous movement of striped nanorods.Journal of the American Chemical Society, 126 41
M. Genkin, A. Sokolov, O. Lavrentovich, I. Aranson (2017)
Topological Defects in a Living Nematic Ensnare Swimming BacteriaPhysical Review X, 7
Nitin Kumar, Harsh Soni, S. Ramaswamy, A. Sood (2014)
Flocking at a distance in active granular matterNature Communications, 5
A. Verkhovsky, G. Borisy (1993)
Non-sarcomeric mode of myosin II organization in the fibroblast lamellumThe Journal of Cell Biology, 123
V. Narayan, S. Ramaswamy, Narayanan Menon (2006)
Long-Lived Giant Number Fluctuations in a Swarming Granular NematicScience, 317
A. Cavagna, I. Giardina (2014)
Bird Flocks as Condensed MatterAnnual Review of Condensed Matter Physics, 5
Sandra Palmgren, P. Ojala, M. Wear, J. Cooper, P. Lappalainen (2001)
Interactions with PIP2, ADP-actin monomers, and capping protein regulate the activity and localization of yeast twinfilinThe Journal of Cell Biology, 155
S. Gueron, S. Levin, D. Rubenstein (1996)
The Dynamics of Herds: From Individuals to AggregationsJournal of Theoretical Biology, 182
Pau Guillamat, J. Ignés-Mullol, Suraj Shankar, M. Marchetti, F. Sagués (2016)
Probing the shear viscosity of an active nematic film.Physical review. E, 94 6-1
Rui Zhang, Ye Zhou, M. Rahimi, J. Pablo (2016)
Dynamic structure of active nematic shellsNature Communications, 7
R. Simha, S. Ramaswamy (2001)
Hydrodynamic fluctuations and instabilities in ordered suspensions of self-propelled particles.Physical review letters, 89 5
M. Murrell, M. Gardel (2012)
F-actin buckling coordinates contractility and severing in a biomimetic actomyosin cortexProceedings of the National Academy of Sciences, 109
G. Toth, C. Denniston, J. Yeomans (2002)
Hydrodynamics of topological defects in nematic liquid crystals.Physical review letters, 88 10
wrote the paper. Competing interests: The authors declare that they have no competing interests. Data and materials availability: All data
Shuang Zhou, A. Sokolov, O. Lavrentovich, I. Aranson (2013)
Living liquid crystalsProceedings of the National Academy of Sciences, 111
Colin Denniston, Enzo Orlandini, J. Yeomans (2000)
Lattice Boltzmann simulations of liquid crystal hydrodynamics.Physical review. E, Statistical, nonlinear, and soft matter physics, 63 5 Pt 2
D. Marenduzzo, E. Orlandini, M. Cates, J. Yeomans (2007)
Steady-state hydrodynamic instabilities of active liquid crystals: hybrid lattice Boltzmann simulations.Physical review. E, Statistical, nonlinear, and soft matter physics, 76 3 Pt 1
F. Keber, E. Loiseau, T. Sanchez, Stephen DeCamp, L. Giomi, L. Giomi, M. Bowick, M. Marchetti, Z. Dogic, Z. Dogic, A. Bausch (2014)
Topology and dynamics of active nematic vesiclesScience, 345
T. Saw, A. Doostmohammadi, V. Nier, L. Kocgozlu, S. Thampi, Y. Toyama, P. Marcq, C. Lim, J. Yeomans, B. Ladoux (2017)
Topological defects in epithelia govern cell death and extrusionNature, 544
Stephen DeCamp, Gabriel Redner, A. Baskaran, M. Hagan, Z. Dogic (2015)
Orientational order of motile defects in active nematics.Nature materials, 14 11
Shuang Zhou, S. Shiyanovskii, Heung‐Shik Park, O. Lavrentovich (2017)
Fine structure of the topological defect cores studied for disclinations in lyotropic chromonic liquid crystalsNature Communications, 8
Kyogo Kawaguchi, R. Kageyama, M. Sano (2016)
Topological defects control collective dynamics in neural progenitor cell culturesNature, 545
Rui Zhang, T. Roberts, I. Aranson, J. Pablo (2016)
Lattice Boltzmann simulation of asymmetric flow in nematic liquid crystals with finite anchoring.The Journal of chemical physics, 144 8
P. Gennes, Richard Alben (1974)
The physics of liquid crystals
D. Frederiksen, L. Cunningham (1982)
Structural and contractile proteins. Part B. The contractile apparatus and the cytoskeleton.Methods in enzymology, 85 Pt B
C. Peng, T. Turiv, Yubing Guo, Q. Wei, O. Lavrentovich (2016)
Command of active matter by topological defects and patternsScience, 354
Hudson, Thomas (1989)
Frank elastic-constant anisotropy measured from transmission-electron-microscope images of disclinations.Physical review letters, 62 17
D. Needleman, Z. Dogic, Z. Dogic (2017)
Active matter at the interface between materials science and cell biologyNature Reviews Materials, 2
J. Fournier, P. Galatola (2005)
Modeling planar degenerate wetting and anchoring in nematic liquid crystalsEPL (Europhysics Letters), 72
J. Alicea, L. Balents, Matthew Fisher, A. Paramekanti, L. Radzihovsky (2004)
Transition to zero resistance in a two-dimensional electron gas driven with microwavesPhysical Review B, 71
A. Vromans, L. Giomi (2015)
Orientational properties of nematic disclinations.Soft matter, 12 30
Y. Hatwalne, S. Ramaswamy, Madan Rao, Madan Rao, R. Simha (2003)
Rheology of active-particle suspensions.Physical review letters, 92 11
SCIENCE ADVANCES RESEARCH ARTICLE MATERIALS SCIENCE Copyright © 2018 The Authors, some rights reserved; exclusive licensee American Association 1,2 3 3,4† 1,2,5† Nitin Kumar *, Rui Zhang *, Juan J. de Pablo , Margaret L. Gardel for the Advancement of Science. No claim to Active materials are capable of converting free energy into directional motion, giving rise to notable dynamical original U.S. Government phenomena. Developing a general understanding of their structure in relation to the underlying nonequilibrium Works. Distributed physics would provide a route toward control of their dynamic behavior and pave the way for potential applica- under a Creative tions. The active system considered here consists of a quasi–two-dimensional sheet of short (≈1 mm) actin fila- Commons Attribution ments driven by myosin II motors. By adopting a concerted theoretical and experimental strategy, new insights NonCommercial are gained into the nonequilibrium properties of active nematics over a wide range of internal activity levels. In License 4.0 (CC BY-NC). particular, it is shown that topological defect interactions can be led to transition from attractive to repulsive as a function of initial defect separation and relative orientation. Furthermore, by examining the +1/2 defect morphol- ogy as a function of activity, we found that the apparent elastic properties of the system (the ratio of bend-to-splay elastic moduli) are altered considerably by increased activity, leading to an effectively lower bend elasticity. At high levels of activity, the topological defects that decorate the material exhibit a liquid-like structure and adopt preferred orientations depending on their topological charge. Together, these results suggest that it should be possible to tune internal stresses in active nematic systems with the goal of designing out-of-equilibrium structures with engineered dynamic responses. INTRODUCTION be exploited to probe the LC over a range of active stresses. We mea- Materials that contain mechanochemically active constituents are broad- sure changes in the LC’s orientational and velocity correlation lengths ly referred to as active matter and are ubiquitous in natural (1, 2), as a function of motor density and find that these are consistent with biological (3), and physical (4–6) systems. The internal stresses that theoretical calculations of nematic LCs with varying levels of internal activity generates result in materials that can spontaneously flow and stress. We then use the morphology of +1/2 defects to show, in both deform over macroscopic length scales (7). A fundamental question experiments and simulations, that increased activity reduces the LC’s in active matter physics is how internal energy affects the structure, bend elasticity relative to its splay elasticity. Thus, the degree to which mechanics, and dynamics of a material that is out of thermodynamic an LC with known mechanics is driven out of equilibrium can be as- equilibrium. certained by the +1/2 defect morphology. We further demonstrate Structured fluids are a particularly rich system in which to explore that varying internal activity can completely alter defect interaction, these questions. On the one hand, nematic liquid crystals (LCs) can turning it from attractive to effectively repulsive. The activity at which be used to manipulate active matter (8–10). On the other hand, activ- this transition occurs is found to be a function of a defect pair’sinitial ity may destroy orientational order of LCs and lead to generation of separation and its relative orientation. To accurately capture these dy- defect pairs and spontaneous flows (11). This behavior has been ex- namics in simulations, contributions of both bend and splay elasticity perimentally realized in vibrated granular matter (12), dense micro- in LC mechanics must be accounted for. We also analyze pair posi- tubule solutions driven by kinesin motors (13), bacterial suspensions tional and orientational correlations of defects. Our calculations, (14), and cell colonies (15, 16). In the microtubule-kinesin active ne- which are confirmed by experimental observations and measure- matics of relevance to this work, recent efforts have sought to alter the ments, demonstrate that defects in active nematics exhibit liquid- defect structure using confinement (17) or surface fields (18)and like behavior. Last, we show that two preferable configurations for have sought to characterize transport properties such as viscosity ±1/2 defect pairs exist, in contrast to like-charge defect pairs, which (19), elasticity, and active stresses (20). Despite this increased interest, have single preferable configuration. These results demonstrate how foundational questions regarding the role of activity on characteristic internal stresses can be used to systematically change the mechanics length scales of active flows, or the nature of defect pair interactions and dynamics of LCs to enable structured liquids with tunable trans- far from equilibrium, remain unanswered. Addressing these questions port properties. might enable design and engineering of new classes of active and adaptive materials. Here, we introduce a nematic LC composed of short actin fila- RESULTS ments driven into an active state by myosin II motors. We first dem- Actin-based active nematics with varying activity onstrate that the long-time clustering dynamics of myosin motors can We construct an active LC formed by the semiflexible biopolymer, F-actin, and the molecular motor, myosin II. A dilute suspension of monomeric actin (2 mM) is polymerized in the presence of a capping 1 2 James Franck Institute, The University of Chicago, Chicago, IL 60637, USA. De- protein (CP) (21) to construct filaments with a mean length of l ≈ partment of Physics, The University of Chicago, Chicago, IL 60637, USA. Institute 1 mm. Filaments are crowded onto an oil-water interface by adding for Molecular Engineering, The University of Chicago, Chicago, IL 60637, USA. In- methylcellulose as a depletion agent (Fig. 1A), resulting in a dense film stitute for Molecular Engineering, Argonne National Laboratory, Lemont, IL 60439, USA. Institute for Biophysical Dynamics, The University of Chicago, Chicago, IL of filaments that form a two-dimensional (2D) nematic LC with an 60637, USA. abundance of ±1/2 topological defects (movie S1) (22). Myosin II *These authors contributed equally to this work. assembles into bipolar filaments of several hundred motor heads that †Corresponding author. Email: depablo@uchicago.edu (J.J.d.P.); gardel@uchicago. edu (M.L.G.) appear as near diffraction–limited puncta in fluorescence microscopy Kumar et al., Sci. Adv. 2018; 4 : eaat7779 12 October 2018 1of12 | SCIENCE ADVANCES RESEARCH ARTICLE Fig. 1. F-actin–based active nematic LC driven by myosin II motors. (A) Schematic of the experimental setup. Short actin filaments (black) are crowded to an oil- water interface supported by a layer of surfactant molecules (magenta) to form a 2D nematic LC. After formation of the passive LC, myosin motors (green) are added. (B) Time sequence of fluorescence images of actin filaments (gray scale) and myosin motors (green) showing the generation of a ±1/2 defect pair (blue and red arrows). −2 Motor concentration, c = 0.019 mm . Filament length, l =1 mm. (C) Schematic of two actin filaments with antiparallel polarities sliding relative to each other due to the myosin II motor activity. (D) Schematic of ±1/2 defects. (E) Fluorescence images of actin LC (l =1 mm) at different motor densities c. The director field (cyan lines) and ±1/2 defects (blue and red, respectively) are overlaid. (F) Simulation snapshots of LC at different activity levels a. Short black lines depict the local director field, and −1/2 curves show streamlines (color indicates speed), with warmer colors indicating higher speeds. (G) Mean defect spacing l as a function of c. Inset: l plotted against c . d d −1/2 (H) Orientational and velocity-velocity correlation lengths (x and x , respectively) plotted against c .(I) l as a function of a in simulation. Inset: l plotted against q v d d −1/2 −1/2 a .(J) x and x plotted against a in simulation. q v (Fig. 1B and movie S2). Myosin II filaments generate stress on anti- to the defects (fig. S3), indicating that motor clustering does not affect parallel actin filament pairs (Fig. 1C) to drive changes in the LC struc- the LC structure. To explore the LC structure as the myosin puncta den- ture and dynamics, including the formation, transport, and annihilation sity, c, changes, we extract the nematic director field (24)and identify of defect pairs (movies S2 to S4). Creation of new ±1/2 defects occurs the ±1/2 defects over time as the puncta density decreases from 0.02 to −2 over tens of seconds, with defects moving apart at a rate of ~0.8 mm/s 0.0016 mm (Fig. 1E). The fast relaxation time of the underlying actin (Fig. 1B). The direction of +1/2 defect motion indicates that the acto- LC relative to therateofchangeofthe motordensity allows onetocon- myosin generates extensile stresses (23), consistent with previous active sider the system to be in a quasisteady state (see the “Analysis” section nematics formed with microtubule-kinesin mixtures (13). This leads to in Methods and fig. S4). the manifestation of active nematics, which, to the best of our knowl- edge, has not been reported in actin-based systems. Effect of activity on correlation lengths: Myosin Over the course of 50 min, motors cluster into larger aggregates, re- concentration acts as an activity parameter sulting in a decrease in the myosin puncta number density (movie S2 To understand how internal stress drives nematic activity, we turn to a and fig. S1). Motor clustering occurs concomitantly with a gradual de- hydrodynamic model of active nematics (25). In the model, a phenom- crease of the instantaneous velocity of the nematic (fig. S2), suggesting enological free energy is written in terms of a second-order, symmet- decreased active stress. We observe only a small, local distortion of the ric, and traceless Q-tensor: Q = q(nn − I/3) under uniaxial condition, nematic field around large motor clusters. These clusters do not localize where q is the nematic scalar order parameter, n is the director field, Kumar et al., Sci. Adv. 2018; 4 : eaat7779 12 October 2018 2of12 | SCIENCE ADVANCES RESEARCH ARTICLE −2 and I is the identity tensor (see supplementary text). The active stress In the presence of activity (c =0.0015 mm ), the defect morphol- P caused by the presence of motors is written as ogy changes from V-shaped to “U-shaped” (Fig. 2A and movie S5). This is reflected in the q(f) plot, from which we calculate k′ =0.72 P ¼aQ (Fig. 2B). By analyzing defects over a wide range of c,we find that k decreases linearly with c (Fig. 2C). Thus, increased activity results in a where the activity parameter a has units of N/m and is related to the lower value of effective bend modulus relative to the splay modulus, magnitude of the force dipole that gives rise to local active, extensile consistent with the fact that extensile active nematics are unstable to stress. Physically, the competition of active stress and elastic stress bend distortion (23, 29). That is, activity reduces the elastic penalty of leads to the generation of new defects, a feature that is characteristic nematic bend modes. The net effect is thus a reduction of the effective of active nematics. If we introduce an elastic constant K for the nematic bend modulus of the LC as activity increases. material, a/K bears the same units as c in the experiment. This is The change of the defect morphology induced by activity can also consistent with our intuition that motor number density is related to be explained in terms of hydrodynamic effects. We show in Fig. 2D the the activity of the system. One can therefore construct a natural length flow pattern obtained from our simulations that are associated with pﬃﬃﬃﬃﬃﬃﬃﬃﬃ scale K=a, and as discussed later, it should dictate the characteristic the motion of a +1/2 defect as a is increased from 0 (left) to 0.003 lengths that arise in our active nematics. (right). There are shear flows on the two sides of the symmetry axis In Fig. 1F, we illustrate the hydrodynamic flows obtained from si- of the defect, with which the director field of a nematic LC tends to mulations with different levels of activity a.As a is increased, the av- align, assuming a flow-aligning nematic (30). With this flow-aligning erage speed increases, indicated by the warmer color of the stream effect, the surrounding director field becomes more horizontal. This lines. The +1/2 defects are always associated with high-velocity re- leads to the defect morphology becoming more U-shaped and a lower gions, whereas the −1/2 defects are stagnation points. We also observe apparent bend modulus as the internal stress increases (Fig. 2E, circles). the formation of eddies, induced by the motion of defect pairs, and Further analysis, detailed in fig. S5, shows that although this effect is find that the average eddy size decreases with increased activity. The amplified by hydrodynamic flow alignment (31), activity-promoted simulations also show that the distance between defect pairs in active bending is present even when the coupling to flow is turned off. In nematics is susceptible to large fluctuations, caused by the competi- the simulations, we can change the sign of the active stress from exten- tion between elastic forces and active stresses. The mean defect spacing sile to contractile. These calculations predict that contractile stresses decreases as a function of active stress (Fig. 1I) and agrees qualita- lower the effective splay elasticity, resulting in a tendency for the defect tively with that observed experimentally (Fig. 1G). We observe that to become more V-shaped (Fig. 2E, crosses). These observations help pﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃ l º1= cº K=a(Fig. 1, G and I, insets), consistent with theoretical establish that the defect morphology provides a direct reflection of the expectations (26). extent to which the LC is driven out of equilibrium. The nature of the To further characterize the LC structure and dynamics, we measure defects’ morphological change also provides a simple visual marker to the orientational and velocity-velocity correlation lengths, x and x ,re- differentiate between the extensile or contractile nature of active stresses. q v spectively (see the “Analysis” section in Methods). These correlation lengths, along with l , have been shown to scale with activity x º Activity as a means to switch the interaction between d q −1/2 x º a (26), consistent with our simulation results (Fig. 1J). When ±1/2 defects these correlation lengths are extracted from the experimental data, we Beyond the LC structure, activity also influences dynamics. In the ab- −1/2 −2 observe that they are proportional to c (Fig. 1H). These findings sence of active stress (c =0 mm ), defects of opposite charge experi- serve to establish that the number density of myosin puncta is a good ence an attractive interaction, as the elastic energy is reduced through measure of internal active stress in the actin-based nematic. Further, the annihilation of +1/2 and −1/2 defects (Fig. 3A, top, and movie S6). We time-dependent motor clustering provides a tool to directly observe the quantify defect annihilation by tracking the distance between defect effects of varying activity on nematic structure and dynamics. pairs, Dr, over time (Fig. 3C). Defect annihilation occurs slowly, at a rate of 2 mm/min (Fig. 3C, green squares), a phenomenon that has Change of the +1/2 defect morphology indicates lowering of been studied previously (22, 32). In contrast, at high motor density −2 effective bend-to-splay modulus ratio (c > 0.0015 mm ), we observe that +1/2 and −1/2 defects effectively Next, we explore how changes in active stresses affect the relative bal- “repel” each other (Fig. 3A, bottom, and movie S7) such that the defect ance of bend and splay energies, which is manifested in the morphol- spacing increases at a rate of ≥10 mm/min (Fig. 3C, red triangles and ogy of +1/2 defects (22). Figure 2A shows a fluorescence image of a blue circles). A similar phenomenon, namely, the “unbinding of de- −2 passive (c =0 mm ) LC with average filament length, l =2 mm. For fects,” has been reported in microtubule-kinesin–based systems (13) clarity, the region around a +1/2 defect has been enlarged, and the and 2D hydrodynamic simulations (33). Here, we examine this effect corresponding director field is shown. In a 2D nematic system, the using our 3D simulations. Becauseof symmetrybreakingin the only relevant elastic modes are splay (K ) and bend (K ), and their surrounding director field, a +1/2 defect moves along its orientation 11 33 ratio, k = K /K ,dictates the morphology of +1/2 defects (27, 28). (indicated by an arrow in Fig. 1D), activated by extensile stresses. In 33 11 Qualitatively, the “V-shaped” defect morphology can be understood the absence of any far-field flows and elastic forces, simulations indi- by the relative dominance of the bend elasticity (K ) to the splay elas- cate that +1/2 defects are mobile, while −1/2 defects remain relatively ticity (K ). We quantify the defect morphology by circumnavigating immobile. The transition from attractive to repulsive interaction be- the defect and plotting the angle the director field subtends with the tween defects of opposite topological charge is also observed in the tangent, q, as a function of the angular coordinate f (Fig. 2, A and B) simulations in the range from a = 0 to 0.001 (Fig. 3, B and D) and can averaged over a radial distance from the core where it remains rela- be qualitatively understood as activity generating propulsive stresses tively constant (22). These results are then fitted with a theoretical ex- within the nematic field that are sufficiently strong to overcome elas- pression to extract a value of k =2.19(22). tic stresses. Kumar et al., Sci. Adv. 2018; 4 : eaat7779 12 October 2018 3of12 | SCIENCE ADVANCES RESEARCH ARTICLE −2 −2 Fig. 2. Effect of activity on defect structure and effective elasticity. (A) Fluorescence actin images of a passive (c =0 mm ) and active (c = 0.0015 mm )LC(l =2 mm). The region enclosed by the box is enlarged below, and the director field (cyan lines) and defect morphology (red dashed lines) are indicated. The ratio of bend (K ) to splay (K ) elasticity calculated from the defect morphology are indicated in the bottom right. (B) Plot of q versus f corresponding to experimental images of (A) for the passive ′ ′ (red circles) and active (green diamonds) LC. (C) Apparent elasticityk′ ¼ K =K as a function of c for experimental data. Dashed line highlights the linear scaling. 33 11 (D) Director field from the simulation for both passive (a = 0) and active (a = 0.003) LC. Red arrows around the defect represent the shear flow caused by the velocity field shown in the background. (E) Apparent elasticity k′ as a function of a obtained from simulations for extensile (black circles) and contractile (red crosses) stresses. Our simulations also indicate that a critical activity (a*)exists for how activity can alter the nature of defect interactions over varying which the propulsive stresses are perfectly balanced by the elasticity of length scales will be an exciting topic for future research. the LC, leading to the “stalling” of defect pairs where their separation To generalize the above findings, we also consider arbitrary relative stays constant for several hundred seconds (Fig. 3, B and D). We also orientations of a defect pair, as illustrated in Fig. 3G and fig. S6. The observe this defect stalling experimentally at a critical motor density c* angle Q between the +1/2 defect orientation and the line connecting (Fig. 3, A and C, and movie S8). Both experiments and simulations the two defect cores has a profound effect on defect dynamics (34, 35). show that although the interdefect distance remains constant over Using simulations (see supplementary text for details), we explore the course of several hundred seconds, their positions shift over time, how defect pair interactions are affected by changes to Q and activity possibly due to uncontrolled background flows. This demonstrates a (Fig. 3G). When Q is small, as the +1/2 defect faces the −1/2 defect, that propulsive stresses from activity can be used to qualitatively alter their interaction is always attractive; when Q is large, as the +1/2 defect the defect dynamics. points away from the −1/2 defect, there is a transition activity a*(Q) To quantify the change from attractive to repulsive behavior, we (as a function of Q) above which defects become repulsive. Our simu- plot the relative speeds between paired defects (Dv = v − v )as lations also show that when defect separation is closer, the phase +1/2 −1/2 afunctionof c for our experimental data (Fig. 3E). This shows that the boundary shifts to higher a, a feature consistent with experimental ob- transition from attractive to repulsive interactions for defects with an servations (Fig. 3, E and F). Thus, internal stresses can qualitatively −2 initial separation Dr =30 mmoccurs around c =0.003 mm ,and the change the interactions between defect pairs in LCs. relative velocity is linearly controlled by motor concentration. Last, we find that, for a constant activity, Dv also scales linearly with the initial Defect density in an extensile active nematic is mainly defect separation, Dr (Fig.3F),suchthat wecan define alengthscale at determined by bend modulus which the transition between attractive and repulsive interactions oc- The inherent elasticity of a nematic LC can be viewed as a measure of cur. We see evidence that this length scale increases from 20 to 30 mmas the restoring force acting against spatial distortions (30). In two dimen- −2 the motor density decreases from 0.005 to 0.0015 mm .Understanding sions, a nematic LC opposes splay (K ) and bend (K )deformations, 11 33 Kumar et al., Sci. Adv. 2018; 4 : eaat7779 12 October 2018 4of12 | SCIENCE ADVANCES RESEARCH ARTICLE Fig. 3. Regulation of defect interactions by internal stress. (A) Fluorescence images of actin LC showing dynamics of ±1/2 defect pair (blue and red arrows, respectively) for varying levels of c showing annihilation (top), stalling (middle), and repulsion (bottom). (B) Director field obtained around a ±1/2 defect pair from the simulations at varying levels of active stress showing annihilation (top), stalling (middle), and repulsion (bottom). (C) Defect separation, Dr,as a function of time −2 at different values of c obtained from experimental data for defects with an initial separation of 30 mm. At c* = 0.0015 mm , the defect spacing remains constant. (D) Defect separation, Dr, as a function of time as a is increased from 0 to 0.001 obtained from simulation data. (E) Relative velocity of defect separation, Dv,as a −2 function of c obtained from experiments; the red asterisk corresponds to c*. Dashed line is the linear fit to the data. (F) Dv as a function of Dr for c = 0.005 and 0.0015 mm (inset). Solid black lines show linear fits. Red dashed lines indicate the length scale where Dv is zero. Data correspond to initial defect spacing, Dr =30 mm. (G)Phase diagram of defect pair dynamics in terms of activity a and initial relative orientation Q. Dashed lines indicate that phase boundary moves when the defect separation becomes smaller. but existing models of active nematics have been generally assumed K /K =0.5 (22). As described earlier, the addition of motors drives 33 11 K = K (7). Our results in Fig. 2 suggest that this may be insufficient LC dynamics, as shown in the series of optical images in Fig. 4A. We 11 33 to faithfully capture active LC mechanical response and, thus, their design two simulation systems, one with k = 0.5 and another with k =1, dynamics. To explore this, we construct an LC composed of actin fil- for which the initial director field is directly taken from the experiments ament length l=1 mm and use the +1/2 defect morphology to calculate at time t = 0 s (see the “Numerical details” section in Methods). The Kumar et al., Sci. Adv. 2018; 4 : eaat7779 12 October 2018 5of12 | SCIENCE ADVANCES RESEARCH ARTICLE Fig. 4. LC mechanics is essential for predicting active-state dynamics and structure. (A) Time-lapse fluorescence images of an active actin-based LC (k = 0.5). ±1/2 defects are indicated by blue arcs and red triangles, respectively. The director field obtained from simulations of an active LC with the same instantaneous activity level as in the experiments evolved over time. Simulation is initiated with the director field from the experiments in (A) at t = 0. The mechanics of the LC are k = 0.5 (K = 0.5K, K = K with K = 1 pN) in (B) and k =1 (K = K = 0.75K)in(C). The black circle highlights a defect pair that undergoes an annihilation event in (C) but not in (A) or 11 33 11 (B). (D) Defect density as a function of k.< n > decreases as a function of k when K decreases while keeping K + K constant for several different activity levels defect 33 11 33 (black, green, and blue symbols). In contrast, it merely changes when K is varied while keeping K constant (red symbols). 11 33 dynamics obtained from these simulations are shown in Fig. 4 (B and Topological defects exhibit liquid-like structure and C). We find that for k = 0.5, locations and trajectories of defects in Fig. preferred orientations 4B exhibit good agreement with experiments, whereas agreement is To further understand the combined effects of activity and elasticity on poor for k = 1. In particular, we find that the encircled defect pair under- the microstructure of active nematics, and gain insights into the seem- goes annihilation for k = 1, an event that is not observed in the exper- ingly chaotic behavior of topological defects, we rely on measures of imental data or in the simulations with accurate mechanical properties. order that have been particularly useful in the context of simple liquids, This shows that the defect dynamics at mesoscopic length and time namely, radial distribution functions [g(r)]. Note that the correlation scales strongly depends on the choice of splay and bend elasticity in length calculations presented in Fig. 1 neglect the presence of defects. the model. To isolate the roles of K and K ,werun simulationson As a complimentary analysis tool, we introduce g(r) between defects 11 33 a larger system size with variable elasticity. We find that the defect den- and measure it as a function of activity. In this view, the active nematic sity, < n >, defined as the total number of defects per unit area, de- system can be regarded as a binary system of positive and negative par- defect creases with k with constant K + K at all activity values, as shown in ticles (defects; movie S9). In the first step, we ignore defect orientations 11 33 Fig. 4D. Furthermore, by keeping K constant and varying K alone, and focus only on their spatial distribution. The radial distribution 33 11 we find that the defect density merely changes over a wide range of k. functions corresponding to defect cores in our active nematic system Thus, for extensile active nematics, defect density in the active state is are akin to those observed in liquids, with a first peak corresponding to mainly controlled by K , by regulating the propensity of defect pairs (+) and (−) defect pairs, and higher-order peaks arising from longer- to annihilate. range correlations. The predictions of simulations (Fig. 5A) are in Kumar et al., Sci. Adv. 2018; 4 : eaat7779 12 October 2018 6of12 | SCIENCE ADVANCES RESEARCH ARTICLE Fig. 5. Radial distribution function of defect structure. (A) Radial distribution function g(r) for +1/2 and −1/2 defects from simulations for z = 0.03 and k = 1.0. (B) g(r) of defects of specific charge (+1/2 and −1/2) from simulation. (C) Zoom-in of (B) reveals higher-order peaks in g(r). (D) Radial distribution function g(r) of topological −2 defects from experiments with activity c = 0.005 mm . Inset shows experimental evidence of higher-order peaks in g(r). (E) A characteristic length scale R emerges from g(r) [illustrated in (B)], which is plotted against average defect spacing l . Black line corresponds to R = l . Warmer color of a marker indicates a higher activity. d c d pﬃﬃﬃ 1=2 (F) Average defect spacing l is plotted against a , where the effective activity is defined as a ¼ a= k. eff eff semiquantitative agreement with our experimental observations (Fig. microscopic view that extensile systems are unstable to bend instability 5D). A shoulder is observed before the first peak of g(r); it can be ex- and low bend systems are prone to engender more defects. plained by inspecting the radial distribution functions corresponding We next consider defect orientation and study how it is coupled to to like-charge defects. defect separation. Figure 6 shows three types of defect pair, namely, In Fig. 5B, we differentiate + and − defects when calculating g(r)in +/− (Fig. 6A), +/+ (Fig. 6C), and −/− defects (Fig. 6D). By defining simulation. We see that a length scale R exists below which the radial the angle between defect orientations, q, one can prepare a probability distribution of ± defects deviates considerably from unity. While +/− heat map in terms of r and q. The definition of +1/2 defect orientation defect pairs exhibit a pronounced peak at distances below R , like- is illustrated in the insets of Fig. 6. Because a −1/2 defect has threefold charge defects exhibit short-range repulsions. The repulsive core symmetry, one has to choose one of its three branches to define its therefore shows up as the shoulder in the total g(r) seen in Fig. 5 (A orientation. For a +/− defect pair, we choose one branch as its orien- and D). By closely examining g(r) at distances between 10 and 50 mm, tation such that it is either parallel or antiparallel to the +1/2 defect’s we observe that g(r) for +/+ defect pairs reaches a plateau earlier than orientation [a minimizer of cos(|q|)]. For −/− defect pairs, we choose that for the −/− defect pair (Fig. 5C), implying that the average repul- one such that it makes the smallest angle with the defect position sive force between + defect pairs is weaker than that between − defect vector r (always pointing away from the defect of interest). We observe pairs. Higher-order peaks in g(r) at longer distances are clearly visible that for opposite charge defect pairs, defects tend to align with each and can be explained by the fact that chains of alternating ±1/2 defects other when they are close (see Fig. 6B for experimental images and are occasionally formed in these systems (Fig. 5D, inset). In Fig. 5E, we movie S10). There are two equally possible scenarios in a steady-state find that the emerging length scale R exhibits a linear relation with the system; in one, the +1/2 defect points toward the −1/2 defect (pre- average defect spacing l . Thus, spatial inhomogeneity in defect charge annihilation event), and in the other, the +1/2 defect points away from becomes important when the defect separation is below the average the −1/2 defect (post-proliferation event); similar scenarios are also spacing l . reported in passive liquid crystals (22). These findings also imply that l is a fundamental length scale that Unexpectedly, we find that there is a second stable regime for sets the system’s defect structure. To examine the effect of elastic anisot- which ±1/2 defects are antiparallel at slightly longer separations r.This ropy, we plotted l against an effective activity a ,defined as a ¼ indicates that when defect spacing is in some intermediate range, the d eff eff pﬃﬃﬃ a= k. Figure 5F shows that all data collapse onto a master curve. Be- far field dictated by the −1/2 defect aligns the +1/2 defect in an anti- cause at rest (0 activity), systems of different k are degenerate, bearing parallel fashion. We have found abundant experimental evidences, the same l = ∞ at equilibrium, we say that elastic anisotropy modifies some of which are shown in Fig. 6B and movie S10, in support of this the system’s activity rather than that activity modifies elastic anisotropy. prediction. For like-charge pairs (see Fig. 6, C and D), however, there Activity also breaks the symmetry of splay and bend. For the same ac- is only one stable regime in which defects are antiparallel (face to face tivity level, extensile systems of lower k engender more defects than or back to back) to each other. Note that the above calculations are those of higher k with the same K + K .Thisisconsistentwiththe similar to Fig. 3G in terms of understanding defect orientations, but 11 33 Kumar et al., Sci. Adv. 2018; 4 : eaat7779 12 October 2018 7of12 | SCIENCE ADVANCES RESEARCH ARTICLE Fig. 6. Analysis of defect orientational structures. Probability distribution as function of defect distance r and defect angle q (schematically defined in inset plots) for −2 +/− (A), +/+ (C), and −/− (D) defect pair. Inset images in (C) and (D) show experimental observations of antiparallel like-charge defects (activity level c = 0.005 mm and l =44 mm). (B) Typical structures of unlike-charge defect pairs observed in experiments. Top: Defect orientations tend to be parallel at short r. Bottom: Defect orienta- tions tend to be antiparallel at intermediate r. they are addressing different physics. In Fig. 3G, we examined the filament interactions give rise to uniaxial extensile stress. A previous dynamics of an isolated defect pair at low activity, when the active stress work has shown that contractile stress dominates in actomyosin systems is balanced by the elastic forces arising from existing defects. In contrast, as filament length increases (36) or with the addition of cross-linking in Fig. 6, we collected statistics for hundreds of interacting defect pairs in proteins (37). Further work will be needed to map out how the force the high-activity regime, where activity is dissipated by generating new generation by motor-filament interactions can be tuned by filament defects. Together, our findings indicate that defects in active systems can length, stiffness, and cross-linking (36). be described in terms of liquid state correlations, and that their interac- The similarities between actin and microtubule systems notwithstand- tions are anisotropic, with an interesting angular dependence that could ing, there are several quantitative differences that should be noted. First, potentially be used to engineer intricate transport mechanisms within activity-induced changes in defect shape have not been reported in these active materials. microtubule-based nematics. We expect that higher levels of activity may be needed to overcome the higher rigidity of microtubules, which are 1000-fold stiffer than actin filaments. Furthermore, another notable DISCUSSION difference between the two systems lies in the steady-state defect structure. Our work demonstrates the emergence of an active nematic in actin- In actin-myosin nematics, g(r) shows higher-order peaks, which indicate based LC driven by myosin II motors. This system closely resembles a strong interaction between defects; this more pronounced structure has the active nematics that are formed by microtubule filaments and kinesin not been reported in microtubule-kinesin experiments (38). A possible motors (13). One notable finding is that active nematics can be realized explanation for this difference might be the defect density, which is five- with punctate myosin filaments, which contain ~100 s of motor heads fold higher in actin than reported for microtubule-kinesin nematics. and are sparsely distributed. While the kinesin tetramers used to realize In summary, we have performed experiments and simulations on a active microtubule-based nematics have not been directly visualized, we quasi-2D active nematic LC composed of short actin filaments driven presume that they would be more homogeneously distributed across the by myosin motors. The clustering dynamics of myosin II motors have nematic. That stress inhomogeneities do not negatively affect the real- allowed us to investigate how the structure and dynamics of LCs vary ization of active nematics underscores that these systems are dominated as a function of internal activity. We characterize the motor-driven by long-range hydrodynamic and elastic effects. In both systems, motor- changes in structure and flows that arise in terms of the characteristic Kumar et al., Sci. Adv. 2018; 4 : eaat7779 12 October 2018 8of12 | SCIENCE ADVANCES RESEARCH ARTICLE correlation lengths and defect density as a function of motor density, coverslip using instant epoxy. Then, 3 ml of oil-surfactant solution was and find dependencies that are fully captured by nonequilibrium hy- added into the chamber and quickly pipetted out to leave a thin drodynamic simulations. Our combined theoretical and experimental coating. The sample was always imaged in the middle of the film over approach has allowed us to use the change in the +1/2 defect morphol- the camera field of view, which was about 200 mmby 250 mm, to make ogy induced by activity to reveal the change in effective bend elasticity sure that the sample remains in focus over this area, which is far away resulting from the microscopic stresses. We demonstrate that the activity from the edges. Imaging close to the edges was avoided. The polym- can fundamentally change the nature of defect pair interactions, from at- erization mixture was immediately added afterward. Thirty to 60 min tractive to effective repulsions, and we show that it is possible to control later, a thin layer of actin LC was formed. Myosin II motors were the relative defect speeds with motor concentration. The critical activity is added to the polymerization mixture at concentrations of 5 to 10 nM. shown to be a function of initial defect separation and relative orientation. The sample was imaged using an inverted microscope (Eclipse Ti-E; Our further calculations of correlations of defects show that their config- Nikon, Melville, NY) with a spinning disc confocal head (CSU-X, urations exhibit liquid-like structure and that the relative orientations of Yokagawa Electric, Musashino, Tokyo, Japan), equipped with a CMOS defect pairs become highly correlated when they are in close proximity. camera (Zyla-4.2 USB 3; Andor, Belfast, UK). A 40× 1.15 numerical aperture water-immersion objective (Apo LWD, Nikon) was used for imaging. Images were collected using 568- and 642-nm excitation for METHODS actin and myosin, respectively. Image acquisition was controlled by Experimental methods MetaMorph (Molecular Devices, Sunnyvale, CA). Proteins Image and data analysis Monomeric actin was purified from rabbit skeletal muscle acetone pow- The nematic director field was extracted the same way as in (22), der (Pel-Freez Biologicals, Rogers, AR) (39) and stored at −80°C in which used an algorithm that was described in detail in the methods G-buffer [2 mM tris-HCl (pH 8.0), 0.2 mM adenosine 5′-triphosphate section of Cetera et al.(24). The optical images were bandpass filtered (ATP), 0.2 mM CaCl , 0.2 mM dithiothreitol (DTT), and 0.005% and unsharp masked in ImageJ software (42) to remove noise and spa- NaN ]. Tetramethylrhodamine-6-maleimide (TMR) dye (Life Technol- tial irregularities in brightness. The image algorithm computes 2D fast ogies, Carlsbad, CA) was used to label actin. CP [mouse, with a HisTag, Fourier transform of a small local square sections (of side y)of the purified from bacteria (21); gift from the D. Kovar laboratory, The image and uses an orthogonal vector to calculate the local actin orien- University of Chicago, Chicago, IL] was used to regulate actin polym- tation. The sections were overlapped over a distance z to improve sta- erization and shorten the filament length. Skeletal muscle myosin II tistics. y and z are varied over 15 to 30 mmand 1to 3 mm, respectively, was purified from chicken breast (40) and labeled with Alexa-642 mal- for different images to minimize errors in the local director without eimide (Life Technologies, Carlsbad, CA) (41). changing the final director field. Experimental assay and microscopy Myosin puncta density was calculated using ImageJ software. The actin is polymerized in 1× F-buffer [10 mM imidazole (pH 7.5), Toward the end of the experiment, large clusters of myosin were 50 mM KCl, 0.2 mM EGTA, 1 mM MgCl , and 1 mM ATP]. To avoid not counted. Because the number of myosin polymers remains at least photobleaching, an oxygen-scavenging system [glucose (4.5 mg/ml), glu- 10-fold greater than that of myosin clusters, our results are insensitive cose oxidase (2.7 mg/ml; catalog no. 345486, Calbiochem, Billerica, to the choice of the cluster cutoff size. We calculated the mean l , x , d q MA), catalase (17,000 U/ml; catalog no. 02071, Sigma, St. Louis, MO), and x from overlapping 150-s intervals. We explored averaging over and 0.5 volume % b-mercaptoethanol] was added. Methylcellulose shorter time intervals and found that the trend in l was similar but, as [15 centipoise; 0.3 weight % (wt %)] was used as the crowding agent. expected, the SD increased (fig. S4). At the fastest rates of decrease, the Actin from frozen stocks stored in G-buffer was added to a final con- myosin density does not decrease over this interval but is within the centration of 2 mM with a ratio of 1:5 TMR-maleimide labeled/un- measurement error reported in Fig. 1C. The typical relaxation time of labeled actin monomer. Frozen CP stocks were thawed on ice and the actin nematic LC is given by t = gl /K,where g, l,and K are the added at the same time (6.7 and 3.3 nM for 1- and 2-mmlong actin rotational viscosity, the filament length, and the LC elastic modulus, filaments). We call this assay “polymerization mixture” from respectively. For g ~ 0.1 Pa∙s, l =1 mm, and K = 0.13 pN, we find that henceforth. Myosin II was mixed with phalloidin-stabilized F-actin t ~ 1 s. Thus, the LC structure achieves steady state on time scales at a 1:4 myosin/actin molar ratio in spin-down buffer (20 mM MOPS, much faster than the evolution of the myosin density. 500 mM KCl, 4 mM MgCl , 0.1 mM EGTA; pH 7.4) and centrifuged The active flows were quantified using particle image velocimetry for 30 min at 100,000g. The supernatant containing myosin with low (available at www.oceanwave.jp/softwares/mpiv/) to extract local affinity to F-actin was used in experiments, whereas the high-affinity velocity field, v. The orientational correlation length, x , was calculated g ðrÞ myosin was discarded. by computing∫dr , where g (r)= ⟨ cos[2(q − q )]⟩, indicating spatial 2 i j gð0Þ The experiment was performed in a glass cylinder (catalog no. ⟨v ð0Þ⋅v ðrÞ⟩ i j 09-552-22, Corning Inc.) glued to a coverslip (36). Coverslips were pairs i and j separated by a distance of r. Similarly, x ¼ ∫dr . v 2 ⟨v ⟩ cleaned by sonicating in water and ethanol. The surface was treated with triethoxy(octyl)silane in isopropanol to produce a hydrophobic Theory and modeling surface. To prepare a stable oil-water interface, PFPE-PEG-PFPE sur- Theoretical model factant (catalog no. 008, RAN Biotechnologies, Beverly, MA) was dis- The bulk free energy of the nematic LC, F, is defined as solved in Novec 7500 Engineered Fluid (3M, St. Paul, MN) to a concentration of 2 wt %. To prevent flows at the surface, a small Teflon F ¼ ∫ dVf þ ∫ dSf V bulk ∂V surf mask measuring 2 mm by 2 mm was placed on the treated coverslip before exposing it to ultraviolet-ozone for 10 min. The glass cylinder ¼ ∫ dVð f þ f Þþ ∫ dSf ð1Þ V ∂V was thoroughly cleaned with water and ethanol before gluing it to the LdG el surf Kumar et al., Sci. Adv. 2018; 4 : eaat7779 12 October 2018 9of12 | SCIENCE ADVANCES RESEARCH ARTICLE where f is the short-range free energy, f is the long-range elastic The constant x is related to the material’s aspect ratio, and G is LdG el energy, and f is the surface free energy due to anchoring. f is related to the rotational viscosity g of the system by G ¼ 2q =g surf LdG 1 1 given by a Landau–de Gennes expression of the form (30, 43) (45). The molecular field H, which drives the system toward thermo- dynamic equilibrium, is given by A U A U 0 0 2 3 st f ¼ 1 TrðQ Þ TrðQ Þ LdG dF 2 3 3 H ¼ ð8Þ dQ A U 0 2 þ ðTrðQ ÞÞ ð2Þ st where […] is a symmetric and traceless operator. When velocity is absent, that is, u(r) ≡ 0, Beris-Edwards equation (Eq. 7) reduces to Ginzburg-Landau equation Parameter U controls the magnitude of q , namely, the equilib- qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 3 8 rium scalar order parameter via q ¼ þ 1 .The elastic 4 4 3U ∂ Q ¼ GH energy f is written as (Q means ∂ Q ) el ij,k k ij To calculate the static structures of ±1/2 defects, we adopted the 1 1 above equation to solve for the Q-tensor at equilibrium. f ¼ L Q Q þ L Q Q 1 2 ij;k ij;k jk;k jl;l el Degenerate planar anchoring is implemented through a Fournier- 2 2 Galatola expression (46) that penalizes out-of-plane distortions of the Q tensor. The associated free energy expression is given by 1 1 þ L Q Q Q þ L Q Q ð3Þ 3 ij kl;i kl;j 4 ik;l jl;k 2 2 ⊥ ~ ~ f ¼ WðQ Q Þ ð9Þ surf If the system is uniaxial, the above equation is equivalent to the ~ ~ ~ Frank-Oseen expression where Q ¼ Q þðq =3ÞI and Q ¼ PQP. Here, P is the projection operator associated with the surface normal n as P = I − nn.The evo- 1 1 1 lution of the surface Q-field at one-constant approximation is governed 2 2 2 f ¼ K ð∇⋅nÞ þ K ðn⋅∇ nÞ þ K ðn ð∇ nÞÞ 11 22 33 by (47) 2 2 2 st ∂Q ∂f K ∇⋅½nð∇⋅nÞþ n ð∇ nÞ ð4Þ 24 surf ¼G L n⋅∇Q þ ð10Þ 2 s 1 ∂t ∂Q The L values in Eq. 3 can then be mapped to the K values in Eq. 4 via pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ where G = G/x withx ¼ L =A , namely, nematic coherence length. s N 1 0 1 1 Using an Einstein summation rule, the momentum equation for L ¼ K þ ðK K Þ 1 22 33 11 2q 3 the nematics can be written as (45, 48) rð∂ þ u ∂ Þu ¼ ∂ P þ h∂ ½∂ u þ ∂ u þð1 3∂ P Þ∂ u d t j j i j ij j i j j i r 0 g g i L ¼ ðK K Þ j 2 11 24 ð11Þ L ¼ ðK K Þ p a 3 33 11 The stress P = P + P consists of a passive and an active part. 2q The passive stress P is defined as L ¼ ðK KÞð5Þ 4 24 22 1 1 P ¼P d xH Q þ d x Q þ d H 0 ij ig gj gj gj gj ig ij By assuming a one elastic constant K = K = K = K ≡ K,one 11 22 33 24 3 3 has L ¼ L ≡ K=2q and L = L = L = 0. Point-wise, n is the eigen- 2 3 4 1 dF vector associated with the greatest eigenvalue of the Q-tensor at each þ2x Q þ d Q H ∂ Q þ Q H H Q ; ð12Þ ij ij ge ge j ge ig gj ig gj lattice point. 3 d∂ Q i ge To simulate the LC’s nonequilibrium dynamics, a hybrid lattice where h is the isotropic viscosity, and the hydrostatic pressure P is Boltzmann method was used to simultaneously solve a Beris-Edwards given by (49) equation and a momentum equation, which accounts for the hydro- dynamic effects. By introducing a velocity gradient W = ∂ u ,strain ij j i T T P ¼ rT f ð13Þ rate A =(W + W )/2, vorticity W =(W − W )/2, and a generalized 0 bulk advection term The temperature T is related to the speed of sound c by T ¼ c . SðW; QÞ¼ðxA þ WÞðQ þ I=3ÞþðQ þ I=3ÞðxA WÞ The active stress reads (50) 2xðQ þ I=3ÞTrðQWÞð6Þ P ¼aQ ð14Þ ij ij one can write the Beris-Edwards equation (44) according to in which a is the activity in the simulation. The stress becomes ð∂ þ u⋅∇ÞQ SðW; QÞ¼ GH ð7Þ extensile when a > 0 and contractile when a <0. Kumar et al., Sci. Adv. 2018; 4 : eaat7779 12 October 2018 10 of 12 | SCIENCE ADVANCES RESEARCH ARTICLE 8. S. Zhou, A. Sokolov, O. D. Lavrentovich, I. S. Aranson, Living liquid crystals. Proc. Natl. Numerical details Acad. Sci. U.S.A. 111, 1265–1270 (2014). We solve the evolution Eq. 7 using a finite difference method. The mo- 9. M. M. Genkin, A. Sokolov, O. D. Lavrentovich, I. S. Aranson, Topological defects in living mentum Eq. 11 is solved simultaneously via a lattice Boltzmann nematic ensnare swimming bacteria. Phys. Rev. X 7, 011029 (2017). method over a D3Q15 grid (51). The implementation of stress follows 10. C. Peng, T. Turiv, Y. Guo, Q.-H. Wei, O. D. Lavrentovich, Command of active matter by topological defects and patterns. Science 354, 882–885 (2016). the approach proposed by Guo et al.(52). The units are chosen as 11. R. Aditi Simha, S. Ramaswamy, Hydrodynamic fluctuations and instabilities in ordered follows: The unit length a is chosen to be a = x =1 mm, characteristic suspensions of self-propelled particles. Phys. Rev. Lett. 89, 058101 (2002). of the filament length, the characteristic viscosity is set to g =0.1 Pa∙s, 12. V. Narayan, S. Ramaswamy, N. Menon, Long-lived giant number fluctuations in a −11 and the force scale is made to be F =10 N. Other parameters are 0 swarming granular nematic. Science 317, 105–108 (2007). chosen to be A = 0.1, K =0.1, x = 0.8, G =0.13, h = 0.33, and U =3.5, 13. T. Sanchez, D. T. N. Chen, S. J. DeCamp, M. Heymann, Z. Dogic, Spontaneous motion in hierarchically assembled active matter. Nature 491, 431–434 (2012). leading to q ≈ 0.62. The simulation was performed in a rectangular 14. C. Dombrowski,L.Cisneros, S. Chatkaew,R.E.Goldstein,J.O.Kessler, Self- box. The boundary conditions in the xy plane are periodic with size concentration and large-scale coherence in bacterial dynamics. Phys. Rev. Lett. 93, [N , N ] = [250, 250]. Two confining walls were introduced in the x y 098103 (2004). z dimension, with strong degenerate planar anchoring, ensuring a 15. T. B. Saw, A. Doostmohammadi, V. Nier, L. Kocgozlu, S. Thampi, Y. Toyama, P. Marcq, C. T. Lim, J. M. Yeomans, B. Ladoux, Topological defects in epithelia govern cell death and quasi-2D system with z dimension 7 ≤ N ≤ 11. We refer the reader extrusion. Nature 544, 212–216 (2017). to (47) for additional details on the numerical methods used here. 16. K. Kawaguchi, R. Kageyama, M. Sano, Topological defects control collective dynamics in neural progenitor cell cultures. Nature 545, 327–331 (2017). 17. F. C. Keber, E. Loiseau, T. Sanchez, S. J. DeCamp, L. Giomi, M. J. Bowick, M. C. Marchetti, Z. Dogic, A. R. Bausch, Topology and dynamics of active nematic vesicles. Science 345, SUPPLEMENTARY MATERIALS 1135–1139 (2014). Supplementary material for this article is available at http://advances.sciencemag.org/cgi/ 18. P. Guillamat, J. Ignés-Mullol, F. Sagués, Control of active liquid crystals with a magnetic content/full/4/10/eaat7779/DC1 field. Proc. Natl. Acad. Sci. U.S.A. 113, 5498–5502 (2016). Fig. S1. Myosin motors cluster over time. 19. P. Guillamat, J. Ignés-Mullol, S. Shankar, M. C. Marchetti, F. Sagués, Probing the shear Fig. S2. Temporal behavior of root mean square velocity. viscosity of an active nematic film. Phys. Rev. E 94, 060602(R) (2016). Fig. S3. Myosin motors do not localize to defect cores. 20. P. W. Ellis, D. J. G. Pearce, Y.-W. Chang, G. Goldsztein, L. Giomi, A. Fernandez-Nieves, Fig. S4. Time averaging of defect spacing. Curvature-induced defect unbinding and dynamics in active nematic toroids. Nat. Phys. Fig. S5. Effect of flow alignment on the change of defect morphology. 14,85–90 (2018). Fig. S6. Director field associated with different defect orientations. 21. S. Palmgren, P. J. Ojala, M. A. Wear, J. A. Cooper, P. Lappalainen, Interactions with PIP , Supplementary Text ADP-actin monomers, and capping protein regulate the activity and localization of yeast Movie S1. Crowding of actin filaments (mean length, l =1 mm) on an oil-water interface twinfilin. J. Cell Biol. 155, 251–260 (2001). forming a 2D nematic LC. 22. R. Zhang, N. Kumar, J. L. Ross, M. L. Gardel, J. J. de Pablo, Interplay of structure, elasticity Movie S2. A dense film of actin filaments (red, l =1 mm) and myosin II (green) form an active and dynamics in actin-based nematic materials. Proc. Natl. Acad. Sci. U.S.A. 115, nematic. E124–E133 (2018). Movie S3. Generation of topological defect pairs in actomyosin-based active nematic 23. L. Giomi, M. J. Bowick, P. Mishra, R. Sknepnek, M. C. Marchetti, Defect dynamics in active (l =1 mm). nematics. Philos. Trans. R. Soc. Lond. A Math. Phys. Eng. Sci. 372, 20130365 (2014). Movie S4. Persistent motion of a +1/2 defect (solid symbol) along its orientation in our 24. M. Cetera, G. R. Ramirez-San Juan, P. W. Oakes, L. Lewellyn, M. J. Fairchild, G. Tanentzapf, actomyosin-based active nematics (l =1 mm); −1/2 (open symbol) defect remains immobile. M. L. Gardel, S. Horne-Badovinac, Epithelial rotation promotes the global alignment Movie S5. Fluorescence images of actin (left) and myosin (right) in actin nematic (l =2 mm). of contractile actin bundles during Drosophila egg chamber elongation. Nat. Commun. Movie S6. Time-lapse imaging of fluorescent actin in a passive nematic LC (l =1 mm, −2 5, 5511 (2014). c =0 mm ) showing annihilation of defect pairs. 25. R. Zhang, Y. Zhou, M. Rahimi, J. J. de Pablo, Dynamic structure of active nematic shells. Movie S7. Time-lapse imaging of fluorescent actin in an active nematic LC (l =1 mm, −2 Nat. Commun. 7, 13483 (2016). c = 0.01 mm ) showing defect repulsion. 26. E. J. Hemingway, P. Mishra, M. C. Marchetti, S. M. Fielding, Correlation lengths in Movie S8. Time-lapse imaging of fluorescent actin in an active nematic LC (l =1 mm, −2 hydrodynamic models of active nematics. Soft Matter 12, 7943–7952 (2016). c = 0.0015 mm ) showing defect stalling, where the defect pair separation (indicated by open 27. S. D. Hudson, E. L. Thomas, Frank elastic-constant anisotropy measured from symbols) does not change significantly over the course of 3 min. transmission-electron-microscope images of disclinations. Phys. Rev. Lett. 62, 1993–1996 Movie S9. Simulation movie of defect dynamics in a quasi-2D active nematic LC. (1989). Movie S10. Time-lapse imaging of ± defect pair dynamics in active nematic LC (l =1 mm, −2 28. S. Zhou, S. V. Shiyanovskii, H.-S. Park, O. D. Lavrentovich, Fine structure of the topological c = 0.0015 mm ) showing that defect orientations change from roughly antiparallel at large defect cores studied for disclinations in lyotropic chromonic liquid crystals. Nat. Commun. separation to parallel when close, consistent with the structural analysis of defects in 8, 14974 (2017). simulations. 29. Y. Hatwalne, S. Ramaswamy, M. Rao, R. A. Simha, Rheology of active-particle suspensions. Phys. Rev. Lett. 92, 118101 (2004). 30. P.-G. de Gennes, J. Prost, The Physics of Liquid Crystals (Oxford Univ. Press Inc., 1995). 31. M. Kleman, O. D. Lavrentovich, Soft Matter Physics (Springer, 2001). REFERENCES AND NOTES 32. G. Tóth, C. Denniston, J. M. Yeomans, Hydrodynamics of topological defects in nematic 1. S. Gueron, S. A. Levin, D. I. Rubenstein, The dynamics of herds: From individuals to liquid crystals. Phys. Rev. Lett. 88, 105504 (2002). aggregations. J. Theor. Biol. 182,85–98 (1996). 33. L. Giomi, M. J. Bowick, X. Ma, M. C. Marchetti, Defect annihilation and proliferation in 2. A. Cavagna, I. Giardina, Bird flocks as condensed matter. Annu. Rev. Condens. Matter Phys. active nematics. Phys. Rev. Lett. 110, 228101 (2013). 5, 183–207 (2014). 34. A. J. Vromans, L. Giomi, Orientational properties of nematic disclinations. Soft Matter 12, 3. D. Needleman, Z. Dogic, Active matter at the interface between materials science and cell 6490–6495 (2016). biology. Nat. Rev. Mater. 2, 17048 (2017). 35. X. Tang, J. V. Selinger, Orientation of topological defects in 2D nematic liquid crystals. 4. N. Kumar, H. Soni, S. Ramaswamy, A. K. Sood, Flocking at a distance in active granular Soft Matter 13, 5481–5490 (2017). matter. Nat. Commun. 5, 4688 (2014). 36. S. Stam, S. L. Freedman, S. Banerjee, K. L. Weirich, A. R. Dinner, M. L. Gardel, Filament 5. W. F. Paxton, K. C. Kistler, C. C. Olmeda, A. Sen, S. K. St. Angelo, Y. Cao, T. E. Mallouk, rigidity and connectivity tune the deformation modes of active biopolymer networks. P. E. Lammert, V. H. Crespi, Catalytic nanomotors: Autonomous movement of striped Proc. Natl. Acad. Sci. U.S.A. 114, E10037–E10045 (2017). nanorods. J. Am. Chem. Soc. 126, 13424–13431 (2004). 37. M. P. Murrell, M. L. Gardel, F-actin buckling coordinates contractility and severing 6. J. Alicea, L. Balents, M. P. A. Fisher, A. Paramekanti, L. Radzihovsky, Transition to zero in a biomimetic actomyosin cortex. Proc. Natl. Acad. Sci. U.S.A. 109,20820–20825 resistance in a two-dimensional electron gas driven with microwaves. Phys. Rev. B 71, (2012). 235322 (2005). 38. S. J. DeCamp, G. S. Redner, A. Baskaran, M. F. Hagan, Z. Dogic, Orientational order of 7. M. C. Marchetti, J. F. Joanny, S. Ramaswamy, T. B. Liverpool, J. Prost, M. Rao, R. Aditi Simha, motile defects in active nematics. Nat. Mater. 14, 1110–1115 (2015). Hydrodynamics of soft active matter. Rev. Mod. Phys. 85, 1143–1189 (2013). Kumar et al., Sci. Adv. 2018; 4 : eaat7779 12 October 2018 11 of 12 | SCIENCE ADVANCES RESEARCH ARTICLE 39. J. A. Spudich, S. Watt, The regulation of rabbit skeletal muscle contraction: I. Biochemical 51. Z. Guo, C. Shu, Lattice Boltzmann Method and Its Applications in Engineering (World studies of the interaction of the tropomyosin-troponin complex with actin and the Scientific Publishing Company, ed. 1, 2013). proteolytic fragments of myosin. J. Biol. Chem. 246, 4866–4871 (1971). 52. Z. Guo, C. Zheng, B. Shi, Discrete lattice effects on the forcing term in the lattice 40. S. S. Margossian, S. Lowey, Preparation of myosin and its subfragments from rabbit Boltzmann method. Phys. Rev. E 65, 046308 (2002). skeletal muscle, in Structural and Contractile Proteins Part B: The Contractile Apparatus and the Cytoskeleton. Methods in Enzymology, D. W. Frederiksen, L. W. Cunningham, Eds. Acknowledgments: We thank K. Weirich for useful discussions and purified proteins, (Academic Press, 1982), vol. 85, pp. 55–71. S. Stam for assisting with experiments, and P. Oakes for helping in director field analysis. 41. A. B. Verkhovsky, G. G. Borisy, Non-sarcomeric mode of myosin II organization in the Funding: This work was supported primarily by The University of Chicago Materials fibroblast lamellum. J. Cell Biol. 123, 637–652 (1993). Research Science and Engineering Center, which is funded by the NSF under award number 42. W. S. Rasband, ImageJ (U.S. National Institutes of Health, 1997–2009). DMR-1420709. M.L.G. acknowledges support from ARO MURI grant W911NF1410403. 43. L. Landau, E. Lifshitz, Statistical Physics (Pergamon Press, ed. 3, 1980). J.J.d.P. acknowledges support from NSF grant DMR-1710318. N.K. acknowledges the Yen 44. A. N. Beris, B. J. Edwards, Thermodynamics of Flowing Systems with Internal Microstructure Fellowship of the Institute for Biophysical Dynamics, The University of Chicago. R.Z. is (Oxford Univ. Press, 1994). grateful for the support of The University of Chicago Research Computing Center for 45. C. Denniston, E. Orlandini, J. M. Yeomans, Lattice Boltzmann simulations of liquid crystal assistance with the calculations carried out in this work. Author contributions: N.K., R.Z., hydrodynamics. Phys. Rev. E 63, 056702 (2001). J.J.d.P., and M.L.G. designed research; N.K. performed experiments, and R.Z. carried out 46. J.-B. Fournier, P. Galatola, Modeling planar degenerate wetting and anchoring in nematic simulations; N.K. and R.Z. analyzed data; and N.K., R.Z., J.J.d.P., and M.L.G. wrote the paper. liquid crystals. Europhys. Lett. 72, 403–409 (2005). Competing interests: The authors declare that they have no competing interests. Data 47. R. Zhang, T. Roberts, I. S. Aranson, J. J. de Pablo, Lattice Boltzmann simulation of asymmetric and materials availability: All data needed to evaluate the conclusions in the paper flow in nematic liquid crystals with finite anchoring. J. Chem. Phys. 144, 084905 (2016). are present in the paper and/or the Supplementary Materials. Additional data related to this 48. C. Denniston, D. Marenduzzo, E. Orlandini, J. M. Yeomans, Lattice Boltzmann algorithm paper may be requested from the authors. for three-dimensional liquid-crystal hydrodynamics. Philos. Trans. R. Soc. Lond. A 362, 1745–1754 (2004). Submitted 4 April 2018 49. J.-i. Fukuda, H. Yokoyama, M. Yoneya, H. Stark, Interaction between particles in a nematic Accepted 31 August 2018 liquid crystal: Numerical study using the Landau-de Gennes continuum theory. Mol. Cryst. Published 12 October 2018 Liq. Cryst. 435,63–74 (2005). 10.1126/sciadv.aat7779 50. D. Marenduzzo, E. Orlandini, M. E. Cates, J. M. Yeomans, Steady-state hydrodynamic instabilities of active liquid crystals: Hybrid lattice Boltzmann simulations. Phys. Rev. E 76, Citation: N. Kumar, R. Zhang, J. J. de Pablo, M. L. Gardel, Tunable structure and dynamics of 031921 (2007). active liquid crystals. Sci. Adv. 4, eaat7779 (2018). Kumar et al., Sci. Adv. 2018; 4 : eaat7779 12 October 2018 12 of 12
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