Multiscale Polar Theory of Microtubule and Motor-Protein Assemblies

Tong Gao,1 Robert Blackwell,2 Matthew A. Glaser,2 M. D. Betterton,2,* and Michael J. Shelley1,†1Courant Institute of Mathematical Sciences, New York University, New York, New York 10012, USA

2Department of Physics and Liquid Crystal Materials Research Center and Biofrontiers Institute, University of Colorado,Boulder, Colorado 80309, USA

(Received 30 June 2014; published 27 January 2015)

Microtubules and motor proteins are building blocks of self-organized subcellular biological structuressuch as the mitotic spindle and the centrosomal microtubule array. These same ingredients can form new“bioactive” liquid-crystalline fluids that are intrinsically out of equilibrium and which display complexflows and defect dynamics. It is not yet well understood how microscopic activity, which involves polarity-dependent interactions between motor proteins and microtubules, yields such larger-scale dynamicalstructures. In our multiscale theory, Brownian dynamics simulations of polar microtubule ensembles drivenby cross-linking motors allow us to study microscopic organization and stresses. Polarity sorting and cross-link relaxation emerge as two polar-specific sources of active destabilizing stress. On larger length scales,our continuum Doi-Onsager theory captures the hydrodynamic flows generated by polarity-dependentactive stresses. The results connect local polar structure to flow structures and defect dynamics.

DOI: 10.1103/PhysRevLett.114.048101 PACS numbers: 87.10.-e, 47.57.E-

Nonequilibrium materials composed of self-drivenconstituents—active matter—present novel physics tounderstand and may one day provide new technologiessuch as autonomously moving and self-healing materials[1–5]. One central example is mixtures of cytoskeletalfilaments and molecular motors, which are important fortheir ability to form self-assembled cellular structures, suchas the mitotic spindle and cell cortex. Reduced in vitrosystems show that biofilament and motor-protein mixturescan form self-organized patterns, such as vortices andasters, reminiscent of cellular structures [6–8]. Recently,Sanchez et al. [9] synthesized mixtures of microtubules(MTs), multimeric kinesin-1 motor complexes, AdenosineTriphosphate (ATP), and a depletant. In bulk, extended MTbundles spontaneously form, which continuously stretch,bend, and fracture, leading to large-scale flows. Whencondensed onto an oil-water interface, the MTs form anematically ordered active surface characterized by turbu-lentlike motions and motile disclination defects.Understanding reduced filament-motor systems is an

important step towards comprehending more complexactive systems. Therefore, theoretical studies have inves-tigated aspects of MT–motor-protein assemblies at differ-ent scales [10–14]. Inspired by the experiments of Sanchezet al. [9], Giomi et al. [15,16] and Thampi et al. [17–19]have studied liquid crystal hydrodynamic models driven byan apolar active stress [20]. While apolar models reproducequalitative features of these experiments, MTs have polarityand cross-linking motors move directionally; hence,aligned MTs must have different interactions than anti-aligned MTs, and activity-driven material stresses andfluxes should reflect the polarity of these interactions.We investigate this through multiscale modeling, first

discovering two separate microscopic sources of activeand extensile stresses, one induced by motor-driven polar-ity sorting of antialigned MTs, and another from relaxationof cross-link tethers between polar-aligned MTs. Weformulate a Doi-Onsager model [21–24] with fluxes andstresses reflecting these effects, and use this to study theinterfacial experiments of Sanchez et al. [9]. Simulationsshow persistent folding flows and defect birth and annihi-lation, arising from active stresses occupying geometricallydistinct regions. Having properly accounted for drag of thebounding fluids, we find a well-defined characteristiclength scale from linear theory, which agrees well withfeature sizes in our simulations.We outline the basic model in Fig. 1. Every MT has

a plus-end-oriented director p, the same length l, anddiameter b [Fig. 1(a)]. Nearby MTs are coupled by activeplus-end-directed cross-links consisting of two motors

pLv

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FIG. 1 (color online). (a) Schematic of a cluster of polar-alignedand antialigned MTs, with their plus ends marked by red rings.Cross-linking motors walk on neighboring MTs at speed v and(b) exert springlike forces, with a linear force-velocity relation.(c) An antialigned MT pair being polarity sorted by active cross-links. The left- (right-)pointing MTmoves right (left) with velocityvL (vR). (d) A polar-aligned MT pair upon which cross-link forcesare relaxing due to the force-velocity relation. In both, the grayarrows characterize the magnitude of an induced extensile stress.

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connected by a springlike tether. Motor velocities arecontrolled by a piecewise linear force-velocity relation[Fig. 1(b)]. For antialigned MTs [Fig. 1(c)], the two motorsmove in opposite directions, stretching the tether to slidethe MTs towards their minus ends, which is termed polaritysorting [10]. Conversely, for polar-aligned MTs the twomotors move in the same direction, with little or no netsliding, and the retarding force on the leading motor causesstretched tethers to relax [Fig. 1(d)].Microscopic model.—We first perform 2D Brownian

dynamics (BD) Monte Carlo (MC) simulations of MTsdriven by explicit motors with binding and unbindingkinetics [25]. The main purpose is to quantify local MTpair interactions, with long-ranged hydrodynamicsneglected due to its high computational cost. We representMTs as perfectly rigid rods, and assume a reservoir of idealmotors at fixed chemical potential. The motors bind to(unbind from) two filaments simultaneously, and unbindimmediately upon reaching the plus end of either MT. Atequilibrium, the average number of motors cross-linkingMTs i and j is hNiji ∼ ρ2

Rdsi

Rdsj exp f−½ucðsi; sjÞ=

kBT�g, where ρ is the linear binding-site density on a singleMT, uc is the quadratic potential for cross-link extension,and si;j parametrizes the MTarclength [25]. The number ofmotors that bind or unbind is sampled from a Poissondistribution with the correct average number of events ineach time interval so that the equilibrium distribution isrecovered for static cross-links. Bound motors are insertedby first selecting pairs of MTs then sampling from theappropriate bivariate normal distribution to choose motorend points. The motor on each cross-link end pointmoves with a linear force-velocity relation [40]:v ¼ vmmax (0;minð1; 1þ f=fsÞ), where f is the magni-tude of the cross-linking force, vm is the maximum trans-location velocity, and fs is the stall force. After the MCcycle, we compute all the forces and torques from motors,short-range repulsion, anisotropic local fluid drag by thesolvent, and random thermal forces, to evolve MT positionsand orientations forward in time [25,41]. The BD MCsimulations are nondimensionalized using the length b,energy kBT, and time τ ¼ D=b2, where D is the diffusioncoefficient of a sphere of diameter b. Our model is similarto that of Head et al. [14], but new in our work arealgorithmic improvements for handling cross-links andneglect of filament elasticity that allows us to simulatelarger systems and measure the stress tensor.Extensile stress and its origins.—Figure 2 illustrates the

long-time behavior of MTs in the BD MC simulations (seevideo S1 in the Supplemental Material [25]). For twoMTs iand j with orientations pi and pj and center-of-massdisplacement rij, the longitudinal displacement is sij ¼ð1=2Þrij · ½pi þ sgnðpi · pjÞpj�. For antialigned MT pairs(pi · pj < 0), sij is negativewhen theMTpair is contracting,and becomes positive when the MT pair is extending (seeFig. 1). When cross-links are static or on polar-aligned MTs(pi · pj ≥ 0), the distribution of cross-links is symmetric

about sij ¼ 0 [Fig. 2(a)]. However, for motors on anti-aligned MTs, configurations that generate pair extension(those with positive values of sij) are more likely.Motor motion alters the distribution of cross-link exten-

sion rc [Fig. 2(b)]. The minimum rc ≈ 1 is due to MT stericinteractions. For antialigned (polar-aligned) pairs, cross-link relaxation shifts the distribution toward larger (smaller)extension. The bulk material stress tensor is Σb ¼ðNkBT=VÞIþ ð1=VÞhPN

i Wii for N interacting MTs ina volume V, with Wi ¼ ð1=2ÞPN

j≠i rijFij the single-MTvirial tensor [25,42]. Over a wide range of motor param-eters, the time-averaged bulk stress tensor Σb is anisotropic,with larger components in the average MT alignmentdirection. Denoting the alignment direction by y, the stressdifference Σyy

b − Σxxb is positive, which corresponds to an

extensile stress. The stress difference can be expressed as asum of pair interactions, with each ij pair contributing astresslet Sij, prior to division by the bulk volume. Theaverage pair stresslet S [green symbols in Fig. 2(c)]increases with the motor run length l up to a maximumwhen the typical motor run length is the MT length. Herel ¼ vm=k0l, the typical distance a motor travels during onebinding event, with k0 a base binding rate of motors.Increasing l further leads to decreasing S because themotors rapidly move to the ends of the MTs and unbind.The extensile stress from antialigned pair interactions

arises from asymmetries during polarity sorting. If a MTpair begins sliding when the two minus ends touch andslide with a force proportional to pair overlap until the twoplus ends meet, then the total extensile stresslet would bezero. Two effects break this symmetry. First, MTs areunlikely to begin interacting exactly when their minus endsmeet, decreasing the range of negative sij over whichsliding occurs. Second, more motors are bound on average

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FIG. 2 (color online). Results of the BD MC particle simu-lations. (a) Histogram of cross-link occupancy as a function of theparticle pair longitudinal displacement sij. (b) Histogram ofcross-link extension rc. (c) Variation of extensile pair stressletS (unit of force × length) with motor run length l. (d) Typicalvariation of extensile pair stresslet with local polarity mi.

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during extension [Fig. 2(a)]. Figure 2(d) shows a typicalcurve of S as a function of mi, where the local polarorientational order parameter mi varies between 1 (allneighboring MTs are polar aligned) and −1 (all neighborsare antialigned). A maximum for mi occurs near −1because polarity sorting is the dominant source of pairwiseextensile stress. As mi increases, S drops with approximatelinearity, at least away from the two isolated peaks thatclose examination shows are related to strong stericinteractions of nearly parallel MTs: nearly, but not exactly,parallel MTs experience aligning torques due to cross-link-mediated attraction; the resulting steric collisions tend topromote pair extension that increases the extensile stress.To understand the surprising and counterintuitive result

that S remains positive even for polar-aligned pairs, weconsider cross-link relaxation on perfectly parallel fila-ments. When cross-links are active, the force of a longi-tudinally stretched cross-link opposes the leading motor,slowing it, and pulls forward on the trailing motor. Thiscauses a slight but significant shift in the distribution ofcross-link extension toward smaller values relative to thestatic cross-link case [Fig. 2(b)]. With cross-linking motors,the cross-link-induced contractile stress along the MTalignment direction is decreased, while there is no changein the transverse stress induced by cross-links. This leads toa net anisotropic extensile stress in the alignment direction.When varying system parameters, we find that the extensilestresslet of polar-aligned MT pairs is typically 2–5 timessmaller than that of antialigned pairs.From microscopic to macroscopic models.—To coarse

grain the BD MC simulation results, we introduce a distri-bution function Ψðx;p; tÞ of MT center-of-mass positionsx and polar orientation vectors p (jpj ¼ 1) and describethe particle dynamics in terms of the concentrationΦ ¼ R

p Ψ,the polarity vector q ¼ R

p Ψp=Φ, the second-moment tensorD ¼ R

p Ψpp, and the tensor order parameter tensorQ ¼ D=Φ − I=d, with d ¼ 2 or 3 the spatial dimension.We first consider a nematically ordered local cluster of

MTs undergoing polarity sorting (Fig. 1), with n MTspointing rightwards (labeled R) and m MTs pointingleftwards (labeled L). When the motor ends move at acharacteristic speed vw, for an antipolar MT pair, a minus-end-directed sliding is induced. Using Stokesian slender-body theory [43], we find the velocities of the left- andrightward pointing MTs [25]: vL ¼ 2nvw=ðnþmÞ; vR ¼−2mvw=ðnþmÞ. This expression shows that the speed ofeach population depends on how many opposingMTs thereare to pull against, with their drag as the anchor, and theirrelative velocity fixed at vL − vR ¼ 2vw. When consideringa more general orientation distribution, a similar calculation[25] yields _x ¼ q − p as the translational flux for MTs.Slender-body theory also yields the forces each rod exerts

on the fluid, and hence, the induced “extra stress” tensor bypolarity sorting can be expressed in dimensional form asσaa ¼ ðηvwl2=VcÞαaa(mn=ðmþ nÞ)pp [25,44]. Here Vc isthe cluster volume and αaa ¼ s=l with s is the signed

distance between the centers of mass of the p and −poriented subclusters. For the extra stress due to cross-linkrelaxation, we lack a simple first-principles model of polar-aligned MTs, though the number of polar-pair interactionsscales as m2 þ n2. Given that the antialigned andpolar-aligned stresses are of the same order [Fig. 2(d)],we assume the form σpa ¼ ðηvwl2=VcÞðαpa=2Þ(ðm2 þ n2Þ=ðmþ nÞ)pp. Thus, we are able to extract the (negative)values of αaa;pa by comparing the antialigned and polar-aligned pair stresslet strengths [Saa;pa ¼ ηvwl2αaa;pa=ðmþ nÞ, and vw is taken as vm] with the BD MCsimulations. Again, we construct the dimensionless3D extra stress fromD andΦqq (i.e., the simplest symmetrictensors quadratic in p) as Σa¼ΣaaþΣpa¼ðαaa=2ÞðD−ΦqqÞþðαpa=2ÞðDþΦqqÞ. The first (second) termcaptures active stress production via polarity sorting(cross-link relaxation) and exactly reproduces the form ofσaa (σpa).We further account for particle rotation as well as steric

interactions and couple MTmotion with a background flowto study the effect of long-range hydrodynamic interactionsabsent in the BD MC simulations through a continuumpolar fluid model [21,22,24,25,45,46]. Since in theSanchez et al. experiments [9] the active material isconfined to an interface between oil and water, we assumea thin layer of suspension immersed in the bulk viscousliquid and close the system by solving the hydrodynamiccoupling between the surface flow U and external fluidmotions through a velocity-stress relation in Fourier space:U ¼ ði=2ÞðI − k kÞðΣekÞ, where k ¼ k=k is the normal-ized 2D wave vector [25].Defects and polarity.—Assuming 2D periodic boundary

conditions and using a Fourier pseudospectral numericalmethod [22,25], we simulated our model over long times. Inregions of flow instability, we find persistently unsteadyturbulentlike flows that are correlated with continualgenesis, propagation, and annihilation of�1=2 order defectpairs (see videos S2–S5 in the Supplemental Material [25]).In Fig. 3(a), the defects exist in regions of small nematicorder (dark blue), and are born as opposing pairs in elongated“incipient crack” regions, qualitatively similar to the struc-tures found in both experiments and apolar models [15–19].Moreover, as shown in Fig. 3(b), the polarity field developsconsiderable spatial variation with regions of high and lowpolar order jqj (video S5 in the SupplementalMaterial [25]).The two active stresses vary in strength depending on thelocal polarity—the polar-aligned (antialigned) stress is largein regions of high (low) polar order [Figs. 3(c) and 3(d)]—and hence are largest, respectively, in their complementaryregions. The circles in Fig. 3(b) encircleþ1=2-order defects,showing the sharp variation of the polarity field aroundthem, with the gradients of active stresses there yieldinglarge active force as shown in Fig. 3(e).We further simulated the results of a photobleaching

experiment in which a circular region is exposed tohigh-intensity laser light to inactivate the fluorescent

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molecules on the corresponding MTs [47] [Figs. 3(f)–3(h)].In a small high-polarity region (marked A in Fig. 3), littleor no polarity sorting occurs, and the photobleachedspot remains approximately circular [Fig. 3(f) top, andFig. 3(g)] over longer times. In a low-polarity region ofhigh nematic order (marked B in Fig. 3), strong polaritysorting of antialigned MTs causes a photobleached spot toseparate into two lobes, showing decreased bleaching. Thistype of experiment probes the local polarity field, and hencethe origins of active stress.Hydrodynamic instabilities and characteristic length.—

A key observation in our simulations as well as other activefluid systems [18,48–50] is that defect pairs are generatedalong elongated cracks that themselves develop fromregions of high polar order. Here we performed a linearstability analysis for a nematically ordered homogeneous

base state [25]. As shown in Fig. 4(a), the analysis reveals awave number of maximal growth kcr along this direc-tion, with kcr growing with approximate linearity inα ¼ αaa þ αpa. Also the plane wave vector of maximalgrowth is aligned with the nematic director [θ ¼ 0 inFig. 4(a) inset]. By contrast, when solving a Stokesequation forced by a bulk stress, the velocity-stress relationin Fourier space becomes u ¼ ði=kÞðI − k kÞðΣekÞ.Comparing to our surface model, the factor of k in thedenominator profoundly changes the nature of systemstability, giving that maximal growth occurs at k ¼ 0 forthe bulk model, and so not producing a characteristic lengthscale [22,51]. (This was also noted in Ref. [52] in theirstudy of swimmers confined to immersed thin films, whileRef. [53] shows that adding substrate friction changeslength scale selection in 2D active nematic models.)Figure 4(b) shows the nonlinear results of this linear

instability. A series of cracks form along y, associated withmoving fluid jets and bending of nematic field lines. InFig. 4(b) inset, the spatial variations of the velocity field arein excellent agreement with the velocity eigenmode asso-ciated with kcr for the linearized system. The distancebetween these cracks matches the half-wavelength, i.e.,λcr ¼ π=kcr, which is in fact representative of the character-istic of the full dynamics of motile defects [Fig. 3(a)]. Atlate times, Fig. 4(c) shows that these cracks lose stabilityand eventually “break” to form defect pairs.Discussion.—We have explored other aspects of our

model system. For example, when turning off hydrody-namics in our kinetic model, we find polar lanes emergingas in our BD MC model. This arises from a slow instability(consistent with the BD MC model) when compared withhydrodynamic instabilities. We find that either active stress(aa or pa) taken individually will produce qualitativelysimilar flows and defect dynamics. Hence, the qualitativenature of the large-scale dynamics does not by itself isolatethe precise origins of a destabilizing stress. An interestingaspect of our BD MC study is that active stresses are

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FIG. 3 (color online). Snapshots of streamingMT nematics on animmersed interface. (a) The nematic director field n superimposedon the color map of the scalar order parameter (twice the positiveeigenvalue of Q); λcr is a calculated characteristic length betweenthe cracks. (b) The polarity vector field q superimposed upon itsmagnitude. (c),(d) Polarity-dependent active stress magnitudes,showing principal eigenvalues of the active stresses [Σaa in (c)and Σpa in (d)]. (e) The vector field of the active force fa ¼ ∇ · Σa

superimposed upon its magnitude. In (a)–(e), circular areas labeledA and B mark regions of high and low polarity, respectively.Positions of þ1=2-order defects are marked by circles. (f)–(h)Predicted results of a photobleaching experiment of fluorescentMTs for a bleached spot in a region of high polar order [(g), area A],and in a region of low polar order [(h), area B]. Arrows representMTs with arrowheads denoting plus ends. A dimensionless time isused with scale b=νvw, where ν is the effective volume fraction.

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FIG. 4 (color online). Linear stability analysis (a) and nonlinearsimulation (b),(c) for strongly antialigned MTs. (a) The real partof the growth rate as a function of wave number k for severalvalues of α (α ¼ αaa þ αpa). Here kcr corresponds to a maximumgrowth rate. Inset: Real part of the growth rate as a function ofwave angle θwhen fixing k ¼ kcr. (b) Crack formation. Inset: Thefluid velocity vector field (blue) and the eigenmode (red line)associated with kcr. (c) Genesis of defects at late times.

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extensile, which is very different from the contractilityobserved in actin-myosin gels [54]. This is likely relatedto the rigid MTs being in a nematically ordered state. Whilewe applied our multiscale polar model to study experimentsof synthesized active fluids, similar but more elaboratedmodels might serve as a principled basis fromwhich to studybiological systems such as the eukaryotic mitotic spindle.

We thank D. Chen and D. Needleman for usefuldiscussions. This work was funded by National ScienceFoundation Grants No. DMR-0820341 (NYU MRSEC)(T. G., M. J. S.), No. DMS-0920930 (M. J. S.), No. EF-ATB-1137822 (M. D. B.), No. DMR-0847685 (M. D. B.),and No. DMR-0820579 (CU MRSEC) (M. A. G.); U.S.Department of Energy Grant No. DE-FG02-88ER25053(T. G., M. J. S.); National Institutes of Health GrantNo. R01 GM104976-03 (M. D. B., M. J. S.). The use ofthe Janus supercomputer was supported by NationalScience Foundation Grant No. CNS-0821794.

*Corresponding author.Meredith.Betterton@colorado.edu

†Corresponding author.shelley@cims.nyu.edu

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