RESEARCH ARTICLE SUMMARY◥

ACTIVE MATTER

Transition from turbulent to coherentflows in confined three-dimensionalactive fluidsKun-Ta Wu, Jean Bernard Hishamunda, Daniel T. N. Chen, Stephen J. DeCamp,Ya-Wen Chang, Alberto Fernández-Nieves, Seth Fraden,* Zvonimir Dogic*

INTRODUCTION: Conventional nonequili-brium systems are composed of inanimatecomponents whose dynamics is powered bythe external input of energy. For example, ina turbulent fluid, energy cascades down manylength scales before being dissipated. In com-parison, diverse nonequilibrium processes inliving organisms are powered at the micro-scopic scale by energy-transducing molecularprocesses. Energy injected at the smallest scalescascades up many levels of structural organiza-tion, collectively driving dynamics of subcellularorganelles, cells, tissues, and entire organisms.However, the fundamental principles by whichanimate components self-organize into activematerials andmachines capable of producingmacroscopicwork remainunknown. Elucidatingthese rules would not only provide insight intoorganization processes that take place in livingmatter but might lay the foundation for theengineering of self-organized machines com-posedof energy-consuminganimate componentsthat are capable of mimicking the propertiesof the living matter.

METHODS:We studied isotropic active fluidscomposed of filamentousmicrotubules, clustersof kinesin molecular motors, and depletingpolymers. The polymer bundles microtubules,

whereas the adenosine triphosphate (ATP)–fueled motion of kinesin clusters powers theirextension. The extensile bundles consist of op-positely aligned polar microtubules and thushave quadrupolar (nematic) symmetry. Theygenerate local active stresses that collectivelydrive mesoscale turbulent-like dynamics ofbulk active fluids. Upon ATP depletion, themotion of microscopic motors grinds to a halt;the turbulent-like dynamics of active fluidsceases, and one recovers the behavior of con-ventional gels.We confined such active isotropicfluids into three-dimensional (3D) toroids, disks,and other complex geometrieswhosedimensions’range frommicrometers tometers and studiedtheir self-organized dynamics. Using particletracking and image analysis, we simultaneouslyquantified the flow of the background fluid andthe structure of the activemicrotubule networkthat drives such fluid flows.

RESULTS: We demonstrate that 3D confine-ments and boundaries robustly transformturbulent-like dynamics of bulk active fluidsinto self-organized coherent macroscopic flowsthat persist on length scales ranging frommicrometers to meters and time scales ofhours. The transition from turbulent to a co-herently circulating state is not determined by

an inherent length scale of the active fluid butis rather controlled by a universal criterion thatis related to the aspect ratio of the confiningchannel. Coherent flows robustly form in chan-nels with square-like profiles and disappear asthe confining channels become too thin andwide or too tall and narrow. Consequently, thistransition to coherent flows is an intrinsically

3D phenomenon that isimpossible in systemswithreduced dimensionality.For toroidswhose channelwidth ismuchsmaller thanthe outer radius, the co-herent flows assume a

Poiseuille-like velocity profile. As the channelwidth becomes comparable with that of thetoroid outer diameter, the time-averaged flowvelocity profile becomes increasingly asym-metric. For disk-like confinements, the innertwo thirds of the fluid assumes rotationdynamicsthat is similar to that of a solid body. Analysisof the microtubule network structure revealsthat the transition to coherent flows is accom-panied by the increase in the thickness of thenematic layer that wets the confining surfaces.The spatial variation of the nematic layer canbe correlated to the velocity profiles of the self-organized flows. In mirror-symmetric geome-tries, the coherent flows can have either handed-ness. Ratchet-like chiral geometries establishgeometrical control over the flow direction.

DISCUSSION: Thousands of nanometer-sizedmolecular motors collectively generate a gra-dient in active stress, which powers fluid flowover meter scales. Our findings illustrate theessential role of boundaries in organizing thedynamics of active matter. In contrast to equ-ilibrium systems in which boundaries are alocal perturbation, in microtubule-based activefluid the influence of boundaries propagatesacross the entire system, regardless of its size.Our experiments also demonstrate that activeisotropic fluids with apolar symmetry can gen-erate large-scale motion and flows. From atechnology perspective, self-pumping activefluids set the stage for the engineering of softself-organizedmachines that directly transformchemical energy into mechanical work. From abiology perspective, our results provide insightinto collective many-body cellular phenomenasuch as cytoplasmic streaming, in which molec-ular motors generate local active stresses thatpower coherent flows of the entire cytoplasm,enhancing the nutrient transport that is es-sential for the development and survival ofmany organisms.▪

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1284 24 MARCH 2017 • VOL 355 ISSUE 6331 sciencemag.org SCIENCE

The list of author affiliations is available in the full article online.*Corresponding author. Email: fraden@brandeis.edu (S.F.);zdogic@brandeis.edu (Z.D.)Cite this article as K.-T. Wu et al., Science 355, eaal1979(2017). DOI: 10.1126/science.aal1979

Increasing the height of the annulus induces a transition from locally turbulent to globallycoherent flows of a confined active isotropic fluid.The left and right half-plane of each annulusillustrate the instantaneous and time-averaged flow and vorticity map of the self-organized flows.The transition to coherent flows is an intrinsically 3D phenomenon that is controlled by the aspectratio of the channel cross section and vanishes for channels that are either too shallow or too thin.

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RESEARCH ARTICLE◥

ACTIVE MATTER

Transition from turbulent to coherentflows in confined three-dimensionalactive fluidsKun-Ta Wu,1 Jean Bernard Hishamunda,1 Daniel T. N. Chen,1 Stephen J. DeCamp,1

Ya-Wen Chang,2 Alberto Fernández-Nieves,2 Seth Fraden,1* Zvonimir Dogic1*

Transport of fluid through a pipe is essential for the operation of macroscale machines andmicrofluidic devices. Conventional fluids only flow in response to external pressure. Wedemonstrate that an active isotropic fluid, composed of microtubules and molecularmotors, autonomously flows through meter-long three-dimensional channels. We establishcontrol over the magnitude, velocity profile, and direction of the self-organized flowsand correlate these to the structure of the extensile microtubule bundles. The inherentlythree-dimensional transition from bulk-turbulent to confined-coherent flows occursconcomitantly with a transition in the bundle orientational order near the surface andis controlled by a scale-invariant criterion related to the channel profile.The nonequilibriumtransition of confined isotropic active fluids can be used to engineer self-organizedsoft machines.

Conventional nonequilibrium pattern-formingsystems, such as Rayleigh-Benard convec-tion, are powered by continuous injectionof energy through macroscopic externalboundaries (1). In comparison, hierarchi-

cally organized living matter is driven away fromequilibrium by the motion of microscopic energy-transducing animate constituents. Starting withmolecular motors and continuing up to subcel-lular organelles, cells, tissues, and entire orga-nisms, each biological element is constructedfrom energy-consuming components. At eachlevel of hierarchy, these animate elements pushand pull on each other, thus generating localactive stresses. Collectively, these stresses enablediverse functionalities that are essential for sur-vival of living organisms, such as cell divisionand motility (2–8). Under geometrical confine-ments, active building blocks such as motile cells,tissues, and organisms can also organize intocoherently moving nonequilibrium states (9–17).Creating active matter systems from the bot-tom up that mimic the remarkable propertiesof living matter remains a fundamental chal-lenge (18–20). Recent studies have revealed theemergence of diverse complex patterns in syn-thetic systems of active matter (21–26). The nextstep is to elucidate conditions that transformchaotic dynamics of these systems into coherentlong-ranged motion that can be used to harvest

energy and thus power various micromachines(27–32).Here, we study three-dimensional (3D) active

fluids and demonstrate an essential differencewith their conventional counterparts. The Navier-Stokes equations dictate that a conventional fluidcomposed of inanimate constituents will flowonly in response to an externally imposed bodyforce, or stress and pressure gradients (33). Thisis no longer true for active fluids. In living or-ganisms, the entire cellular interior can assumelarge-scale coherent flows in absence of any ex-ternally imposed stresses, a phenomenon knownas cytoplasmic streaming (34–36). Despite re-cent advances using living bacterial suspensions(12, 13, 17, 37), creating tunable synthetic activefluids that exhibit autonomous long-ranged flowson length scales large compared with those ofconstituent units remains a challenge. We usea 3D microtubule-based isotropic active fluidwhose bulk turbulent flows are driven by con-tinuous injection of energy through the linearmotion of the constituent kinesin motors (24, 38).We found that confinement robustly transformslocally turbulent dynamics of such active fluidsinto globally coherent flows that persist on me-ter scales. Our experiments demonstrate thatnonequilibrium transitions of synthetic activematerials can be used to engineer self-organizedmachines in which nanometer-sized molecularmotors collectively propel fluid on macroscopicscales.

Microtubule-based activeisotropic fluids

The active fluid we studied comprises micro-tubule filaments, kinesin motor clusters, and

depleting polymer (Fig. 1A) (24, 38). Kinesinmotors are bound into synthetic clusters withtetrameric streptavidin (39, 40). The depletingpolymer induces microtubule bundling (41),while the kinesin clusters simultaneously bindto and move along the neighboring filaments.For aligned microtubules with opposite polarity,kinesin clusters generate interfilament slidingthat drives the flow of the background fluid (Fig.1A) (42). At finite concentrations (≥0.5 mg/mL),active microtubule bundles form a self-rearranging3D isotropic network, whose dynamics compriserepeating cycles of bundles extending, buckling,fracturing, and annealing. Such dynamics ef-fectively drives the flow of the background fluidthat is mostly composed of an aqueous buffer(~99.9%). In bulk suspensions, microtubule-based active networks generate turbulent-likeflows that exhibit neither long-range order nornet material transport. The structure of theseflows and the associated vorticity fields are vi-sualized by doping active fluids with fluorescenttracer particles (Fig. 1D and Movie 1). Such analy-ses reveal a characteristic vortex size of ~100 mm,with an average lifetime of ~3 s.Energy-efficient kinesin motors allow for as-

sembly of bulk 3D active fluids that maintainsteady-state turbulent flows for up to 10 hours.This feature allows us to quantify the time-averageof various steady flows. Furthermore, filamen-tous microtubules effectively couple to the back-ground fluid. Therefore, even at volume fractionsas low as 0.05%, the microtubule network drivesthe bulk turbulent flows. At such low concen-trations, the network has a very large pore size(~10 mm); thus, its structure can be visualizedand quantitatively analyzed with optical micros-copy. This property allows us to correlate the

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1Department of Physics, Brandeis University, 415 SouthStreet, Waltham, MA 02453, USA. 2School of Physics,Georgia Institute of Technology, 837 State Street, Atlanta,GA 30339, USA.*Corresponding author. Email: fraden@brandeis.edu (S.F.); zdogic@brandeis.edu (Z.D.)

Movie 1. Motion of toroidally confined activemicrotubules (red) and tracer particles (green).Height and width of the toroid are 80 and 200 mm,respectively. The movie was taken with a confocalmicroscope, and microtubules and bead tracersare indicated in red and green, respectively. Tovisualize flow at a higher resolution, the concen-tration of traced beads is increased to ~0.01% v/v.Flow rates of microtubules and tracers reveal nomeasurable differences. Time stamp is hour: mi-nute: second.

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properties of the emergent flows to the struc-ture of the underlying active network.

Confinement-induced coherent flows

We found that 3D toroidal confinements can ef-fectively transform directionless turbulent flowsof bulk fluids into coherent circular currents ca-pable of transporting materials over macroscopicscales (Fig. 1, B and E). Similar phenomenawere alsoobserved for cylindrical confinements, demon-strating that toroidal geometry is not an essen-tial requirement for the formation of coherentflows (Fig. 1F). The autonomous circular cur-rents persisted for hours and ceased only afterthe available chemical energy [adenosine tri-phosphate (ATP)] is depleted (Fig. 1C). The flowcoherence was quantified by the order para-meterF ≡ hvq=jvji, where vq is the flow velocityalong the azimuthal direction (q^) and jvj is theflow speed. F = 1 indicates perfect circular flows,whereas F = 0 indicated no net transport alongthe tangential direction. We measured F ~ 0.6,and the magnitude of the order parameter re-mained fairly constant over the entire samplelifetime (Fig. 1C).

Not all geometries support coherent flows.For example, reducing the height of a toroid ora cylinder below a critical value completely sup-

pressed coherent flows and yielded turbulent-like dynamics (Movie 2 and fig. S1). To determinegeometries that support self-organized circular

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Fig. 1. Coherent macroscopic flows of confined active fluids. (A) The threemain constituents of an active fluid: microtubules; kinesin clusters, which powerinterfilament sliding; and pluronic micelles, which act as a depleting agent.(B) Schematic of a coherent flow that emerges when active fluids are con-fined in a toroid. (C) Timeevolution of the averagedazimuthal velocity (blue) andthe circulation order parameter (red) demonstrates that circular flows ceaseupon ATPdepletion. (D) A flowmap demonstrates that bulk active fluids exhibit

turbulent flows that do not transport material. The vector field indicates thelocal flow velocity normalized by themean flow speed.The colormap representsthe normalized vorticity distribution,with blue and red tones representing CCWand CW vorticities, respectively. Left and right halves are instant and time-averaged plots, respectively. (E) An active fluid in a toroidal confinement exhibitspersistent circular flows. (F) Coherent flows in a cylindrical confinement. Confine-ment heights are 1.3mm. (D) to (F) share the same scale, velocity, and color bars.

Movie 2. Coherent and incoherent flows in toroids and cylinders. Flows of active fluids in toroidal andcylindrical confinements that support and suppress coherent flows. Increasing confinement height in-duces a transition from incoherent to coherent flows. Heights of taller toroid (top left) and cylinder (topright) are 1.3 mm, and heights of shorter ones (bottom left and right) are 0.33 mm. Black vectors andcolor maps (left half in each panel) represent fluid velocity fields and vorticity distributions extracted fromthe motion of tracer beads (right half). Time stamp is hour: minute: second.

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flows, we varied the widths, heights, and radiiof toroids with rectangular cross sections as wellas radii and heights of the cylindrical geometries(Figs. 2A). From an extensive data set, we drawthe following observations: First, self-organizedflows are robust, occurring over a range of con-fining geometries whose volumes span morethan four orders of magnitude (from 0.006 to18 mL). This range could be larger because wehave determined neither the upper nor the lowervolume that supports circular flows. Second, theself-organized circular flows are equally likelyto be left- or right-handed. Third, for both to-roids and cylinders, coherent flows diminish asthe confining geometry becomes either too wideor too tall—that is, when the ratio of the long toshort side of the channel cross section becomesgreater than ~3. Sample-to-sample variation inthe circulation order parameter was too great todetermine whether the transition was continuousor discontinuous.To quantify this observation, we plotted the

circulation order parameter F as a function ofthe aspect ratio of toroids or cylinders: a ≡ (h –w)/Max(h, w), where h and w are the height andwidth, respectively, of either a cylinder or toroidand Max(h,w) represents their larger value (Fig.2B). Geometries with square-like profiles (a = 0)support robust flows. As the geometry becomeswider and/or shorter (a < 0), the circulating flowsonly formed for cross sections that were suf-ficiently symmetric (–0.6 < a < 0); otherwise, nonet flows were observed. In the complementaryregime of tall and narrow confinements (a > 0),we observed a second transition from coherentto turbulent flow as the confinement geometrybecame taller or narrower (a > 0.6). This obser-vation demonstrates the equivalency between theheight and width (or vorticity and gradient di-rection) of the confining geometry.We observed flows over a well-defined range

of the parameter a (–0.6 < a < 0.6). This cri-terion appears to be universal because it holdsfor both micrometer- and millimeter-scale con-finements as well as for toroidal and cylindri-cal geometries. The largest channels for whichthis criterion was established had a cross sectionof about a centimeter; further experimentationis required to establish whether there is an upperbound above which the criterion breaks down.This data suggest that the transition to coherentflows is not controlled by the intrinsic lengthscale of the bulk fluid, which is the vortex sizeof 100 mm (Fig. 1D), but rather by the dimen-sionless aspect ratio, a. For example, coherentcircular flows are observed in cylinders with w =3.3 mm, which is 30 times larger than the vortexsize of the bulk fluid. This is contrary to previousstudies that determined the need to confine flowssmaller than the intrinsic length scale of the bulksystem (12, 17). The criterion we uncovered dem-onstrates that the transition to coherent flowsis an intrinsic 3D phenomenon that cannot, evenin principle, exist in 2D systems. As the channelheight is reduced to the 2D limit (h → 0), theconfinement parameter reduces to a single value,a→ –1, which is in the regime in which coherent

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Fig. 2. Phase diagram and velocity profiles of coherent flows in toroids and cylinders. (A) A phasediagram indicating confinement geometries that support circular flows. The phase diagram is limitedto thin confining geometries, h ≤ 2w. Circles and dots represent toroidal and cylindrical confinements,respectively. (B) The circulation order parameter, F, as a function of the confinement aspect ratio, a ≡(h – w)/Max(h, w). a = 0 represents a square cross section. Blue and red dots represent toroids andcylinders, respectively. For a < 0.5, F was measured at midplane. For a > 0.5, F was measured at aquarter plane because of the finite working distance of the microscope objective. (C) Flow velocityprofiles in various toroidal confinements taken at a midplane. Outer diameters of toroids for blue, teal,and green curves are 4300 mm and for olive and red curves are 2000 mm. (Inset) Flow profiles of toroidswith width ≤300 mm. (D) Midplane flow profiles in various cylindrical confinements. (Inset) Flow profiles incylinders with radius ≤300 mm. The color bar on the right indicates the confinement heights.

Fig. 3. Controlling flow of coherent handedness. (A) The handedness of the circular flows is controlledby decorating the toroid’s outer surface with counterclockwise or clockwise ratchets. The vector andcolor maps represent the local flow velocity and vorticity. (B) Velocity field and vorticity maps demon-strate that each notch induces a stationary vortex. (C) Midplane flow profiles in CCW and CW ratchetswith varying notch numbers. Backflow from ratchets occurs near the outer edge, radius (r) ≈ 2300 mm.All channel heights are 1.3 mm. (D) The circulation order parameter as a function of the number of notches.

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flows do not occur (Fig. 2B). In other words, thescale-invariant criterion does not exist in twodimensions because there is only one channeldimension (w) perpendicular to the flow. In con-trast, in 3D systems, the cross section has twodimensions (h, w), making it possible to con-struct a dimensionless aspect ratio that controlsthe onset of the coherent flows.

Structure of the coherent flows

To quantify the spatial structure of the self-organized circular flows, we measured time-averaged azimuthal velocity profiles hvqðrÞi inboth toroidal and cylindrical geometries. Forsmall toroidal widths (w < 1000 mm), coherentflows exhibited Poiseuille-like symmetric pro-files, with a peak velocity at the channel midpointand decreasing to zero at the boundary becauseof the no-slip condition (Fig. 2C). The peak veloc-ity increased with channel height. For a 330-mm-high and 650-mm-wide toroid, the maximum

velocity was ~4 mm/s. Increasing the height to1300 mm, for the same width, increased the peakvelocity to ~10 mm/s. For the largest toroidsstudied, peak velocities reached ~17 mm/s, whereasfor smallest micrometer-sized ones, peak ve-locities were as small as ~1 mm/s (Fig. 2C, inset).The asymmetry of the time-averaged velocityprofile increased with increasing toroid width.The asymmetry became especially pronouncedfor cylindrical geometries in which the velocityincreased from zero at the disk center, growinglinearly to a maximum value before decreasingrapidly to zero at the outer edge (Fig. 2D). Sucha distinguishing feature is universal for bothmicrometer- and millimeter-sized cylinders, indi-cating again that the flow structure is indepen-dent of the confinement size.

Controlling handedness of coherent flows

Inmirror-symmetric toroidsandcylinders, coherentflows are equally likely to be either counterclockwise

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Fig. 4. Circulating flows in closed tracks. (A) Velocity profiles of coherent flows in racetrack geometriesare independent of the length of the straight segment (solid curves). (B) Coherent flow in a 200-mm-longracetrack. Because of limited observation windows, we only observed the blue and red segments whoseflow profiles are dashed same-color curves in (A). The fluids in these segments flow as fast as in anequivalent toroidwhosew= 1mmand h= 1.3mm [dashed black curve in (A)]. (C) Self-organized flows cansolve a 2Dmaze consisting of a closed loop (brown) and 10 dead-end branches (black).We observed onlythe blue and red segments in the close loop and the green segment in one branch.Their flow profiles arethe dotted same-color curves in (A). Dead-end branches inhibit net transport, whereas the closed loopsupports a persistent circulating flow. All channel heights are 1.3 mm.

Movie 3. Coherent flows in 12- and 1-tooth CCWand CW ratchets. Decorating the toroid outer sur-face with notches induced coherent flows in thedirection of same handedness. Even one notch issufficient to control flow direction.The notch sizesare ~300 mm. Top two ratchets (left, CCW; right,CW) have 12 teeth, and bottom ones (left, CCW;right, CW) have only 1 tooth. Heights are 1.3 mm.Time stamp is hour: minute: second.

Movie 4. Coherent flows in concentric micrometer-sized toroids.Toroids’ outer radii range from 175 to1525 mm.Varying the radii does not alter flow rates,implying that coherent flow is independent of channelcurvature.Tracks are shaded in blue and red for CCWand CWcoherent flows. Channel heights are 60 mm.Time stamp is hour: minute: second.

Movie 5. Coherent flows in racetracks withvarying circumferences and an equivalent to-roid.Varyingcircumferences does not alter flow rate,implying that coherent flow is independent of channellength.The lengths of straight paths from left to rightare 17,200, 12,900, 4300, and 0 mm, respectively.Heights andwidths in the racetracks and toroid areboth 1.3 mm.Time stamp is hour: minute: second.

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(CCW) or clockwise (CW). To control the flowhandedness, we decorated the toroid outer edgewith sawtooth notches, finding that a CCWgeom-etry induces flows of same handedness and viceversa (Fig. 3A andMovie 3) (43, 44). In contrast, dec-orating inner edges with the same pattern leadsto flows with opposite handedness (fig. S2, A andB). Coherent flows could be completely suppressedby decorating both inner and outer surfaces withnotches of the same chirality (fig. S2, C and D).Decreasing the number of notches increased thecirculation order parameter,F, and even a singlenotch controlled the flow direction (Fig. 3D). Flowmaps reveal that each notch creates a stationary vor-tex (Fig. 3B), and the back-flow associated with thisvortexdecreases the overall flowcoherency (Fig. 3C).We explored how the self-organized flow rates

scale with the channel curvature and length, byeither changing the toroid radii or stretchingtoroids into shapes resembling racetracks whilemaintaining the same cross section. The toroidalradius (curvature) does not affect the propertiesof the self-organized flows (Movie 4 and fig. S3).For racetrack geometries, we varied the total cir-cumference length from 14 to 48mm (Fig. 4A andMovie 5). Flow profiles in these geometries werenearly identical, demonstrating not only that theflow rate is independent of the pipe length butalso that the straight and curved segments of aracetrack provide equal driving power. To furtherconfirm this finding, we designed a 20-cm-longtrack (Fig. 4B). Coherent flow persisted in such alengthy track; its rate remained nearly the sameas that in a toroid of equivalent cross-section butshorter circumference (Fig. 4A).

Coherent flows in complex confinements

The self-organized coherent flows are remark-ably robust self-organized patterns. To determinelimits of their stability, we explored emergentdynamics in geometries of increasing complex-

ities. First, coherent flows developed not onlyin simple toroids but also in a complex maze-likegeometry comprising a closed loop and manydead-end branches (Fig. 4C). Upon filling suchgeometry with active fluid, coherent flows de-veloped in the closed loop while remaining tur-bulent in the dead-end branches. However, unlikeracetrack geometries, the flow rate was ~50%slower than in an equivalent toroid, indicatingthat the dead-end branches disturb the self-organized flows (Fig. 4A). Second, we exploredflows that emerge in two or three adjoiningtoroids and found that such geometries supportmultiple self-organized states (Fig. 5 and Movie6) (45). For example, in one state, flows in the twoadjoining toroids had opposite handedness andwere thus reinforced in the mutually adjoiningregion (Fig. 5A). In another state, the flow orga-nized along the outer edge of the toroid doublet,therefore suppressing coherence in the adjoiningregion (Fig. 5B). Similarly, in triplet geometry weobserved either flow circulating along the outeredge, leaving the center quiescent (Fig. 5C), orCW flow in one toroid and CCW flow in the othertwo (Fig. 5D). Third, coherent flows also developedin confinements with sharp corners, such as asquare-like geometry (Fig. 5E). However, sharpcorners induced formation of stationary vortices,which generated back-flow, thus decreasing thecirculation order parameter, when comparedwith a toroid with an equivalent cross section.Last, we found that the self-organized flows arenot limited to microfluidic channels. For example,active fluids exhibited macroscopic flows bothwithin a torus inscribed within a finite yield-stress elastomer (fig. S4) (46), as well as a 1.1-m-long plastic tube (Fig. 6).

Structure of the microtubule network

To elucidate the microscopic mechanism thatdrives coherent flows, we imaged the micro-

tubule network structure of active fluids in bothturbulent and coherent states and extracted theorientational distribution function (ODF) of theconstituent bundle segments (Fig. 7A andMovies7 and 8) (47). In the channel center, the measuredODFs were isotropic in both the coherent andturbulent states (indistinguishable from ODFsof bulk active fluids) (Fig. 7B). Emergence of thecoherent flows was correlatedwith the change inthe thickness of the nematic layer that wets thesurfaces. We measured the spatial variation ofthe nematic order parameter in the directionperpendicular to the confining wall (Fig. 7C).In turbulent states, the nematic order decays awayfrom the boundaries with a characteristic length

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Fig. 5. Coherent flows in miscellaneous confinements. (A and B) Two distinct flow configurationsin a pair of partially overlapping toroids. Confinement heights are ~60 mm. (C and D) Flow paths inthree partially overlapping toroids. Confinement heights are ~60 mm. (E) Coherent flows in a square-like confinement. Channel height is ~75 mm.

Fig. 6. Active fluids flow on macroscopic scales.An active fluid confined within a 1.1-m-long flexibletube with a circular cross section of 0.86 mm ex-hibits persistent flows.The red segment highlightsthe tube portion where fluid flows were observedand quantified.

Movie 6. Circulations in two (doublet) and three(triplet) connected toroids. Fluids in the doubletcan develop a CCW and CW circulations (top left)or two CCW circulations (top right). In three ad-joining toroids, fluids can flow in the outer edgewhile leaving center quiescent (bottom left) or de-velop two CCW and one CW circulations (bottomright). Channel height is ~60 mm.Time stamp ishour: minute: second.

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scale (l ~ 30 mm). Transition to coherent flows isaccompanied with an increase in the thicknessof the surface-induced nematic layer (l ~ 100 mm).Similar to fluid flow profiles (Fig. 2C), lateralprofiles of nematic order were asymmetric for cir-culating toroids with wide cross sections, whereasthey were symmetric for narrow ones (Fig. 7C,inset), implying that the shear rates generated bycoherent flows drive the filament alignment. Theangle between the confining surface and the self-organized nematic layer had a finite nonzerovalue of ~20° (Fig. 7B), suggesting that activestresses generated by the spatially distorted wall-bounded nematic layer push against the no-slipboundary and thus propel the fluid forward (48).

Discussion

Wedemonstrated that confining boundaries trans-form turbulent dynamics of bulk isotropic activefluids into long-ranged coherent flows. Our re-sults suggest a mechanism by which this transi-tion takes place. In conventional isotropic liquidcrystals, the bounding surface breaks the local

symmetry and locally aligns the constituent par-ticles (49). In active systems, turbulent flowsenhance this effect, magnifying and extendingthe wall-induced nematic order. We suggest thatthe nematic-wetting layer aligns to the surfaceat an oblique angle, thus generating an active sur-face stress that induces further flows. In turn,these flows further enhance and stabilize theorientation of the wetting layer. This positive feed-back loop yields a cooperative nonequilibriumtransition to globally coherent flows, reminiscentof a phenomenon known as “backflow” in con-ventional liquid crystals, in which mutually re-inforcing couplingbetween the reorientingnematicdirector and the fluid velocity leads to emergentperiodic structures (50, 51).Emergence of coherent flows is determined by

a universal scale-invariant criterion, which is re-lated to the aspect ratio of the confining channelcross section rather than the overall size of theconfining geometry. In particular, coherent flowstake place when the channel cross section is re-latively symmetric—that is, thewidth and height of

the channel are within a factor of 3 of each other.Understanding the onset of large-scale flows is acentral focus of active matter research. One classof theoreticalmodels predicts an active Fréedericksztransition from quiescent to coherent flows formonodomain active liquid crystals (27, 28, 52, 53).There are several notable differences betweenour observations and the active Fréedericksz transi-tion. First, our phenomenon is inherently 3D. Thatthe aspect ratio alone, and not the absolute channelsize, controls the onset of coherent flows forbidsthis transition from taking place in systems withlower dimensionality. This restriction does nothold for the active Fréedericksz transition, whichcan occur in two dimensions. Second, in our ex-periments the coherent flows form in an activeisotropic fluid, instead of a liquid crystalline one.Third, the flows we observed also emerged fromthe turbulent state,whereas the active Fréedericksztransition predicts the formation of such flowsfrom a quiescent, single-domain nematic.

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Fig. 7. Transition to coherent flows correlates with the thickness of the nematic-wetting layer.(A) (Top) Fluorescence images of amicrotubule network that drives the coherent and turbulent fluid flowstaken at the channel midplane. (Bottom) Reconstruction of the microtubule network that drives flow ofactive fluids in both the coherent and turbulent phase. (B) ODFs of microtubule bundle segments in thecoherent (blue) and turbulent (red) state shown in (A).The solid curves indicate ODFs near the channel’souter boundary, whereas the dashed ones are measured at the channel center [solid and dashed boxes,respectively, in (A)].The gray curve represents an isotropic suspension confined in a cuboid whosewidth =18,000 mm, length = 14,000 mm, and height =90 mm. (C) NOPprofile across the channel width. (Inset) NOPprofiles across 800-mm-wide toroids in coherent (light blue) and turbulent (pink) states.

Movie 7. Coherent (left) and incoherent (right)motion of active microtubule networks con-fined in toroids. In a thin toroid (right, h = 100 mm),microtubules move irregularly. Increasing confine-ment height to h = 400 mm triggers a self-organizedtransition from turbulent to coherent flows (left).Time stamp is hour: minute: second.

Movie 8. Demonstration of using SOAX to an-alyze 3D bundle structure. Magenta curves rep-resent snakes that profile microtubule bundles;green dots represent their intersections. By usingSOAX, bundle structures can be extracted fromtheir confocal z-scanned images. Height, width,and outer diameter of the toroid are 100, 200, and2000 mm, respectively.

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The boundary-generated active stresses powercoherent flows. This bears similarity to electro-osmosis, in which an external electric field createsboundary-layer stresses that lead to plug flow(54). The time-averaged coherent flows of activematter confined in cylinders, in which the in-terior fluid away from the interface moves as asolid body, resemble such plug flows. However,there are also notable differences. The velocityof electroosmotic plug flow is independent ofchannel size when the boundary layer is small.In contrast, we observed that the coherent activeflow velocity varieswith channel size. Additionally,active flows in tori resemble Poisuille flow, in-dicating pressure-driven flow.The possible relevance of our mechanism to

cell biology still has to be ascertained. In particular,coherent cytoplasmic flows in living cells arepowered by cytoskeletal filaments that are per-manently affixed at the cellular cortex (34–36),whereas in our system, the active elements aredispersed throughout the sample interior anddynamically self-organize at the solid boundariesthat confine the fluid.We demonstrate that nonequilibrium transi-

tions in active matter can be used for the robustassembly of self-organized machines that pro-duce useful work by powering large-scale fluidflow. These machines are fueled by the collectivemotion of the constituent nanometer-sizedmotorsand are thus evocative of living organisms. How-ever, the goals of activematter research are not onlyto recreate materials found in nature but addition-ally are to discover and elucidate the full range ofdynamical organization that is possible in livingand nonliving systems alike. Toward this end, ex-plaining the scale-invariant criterion for the onsetof coherent flows and their velocity profiles re-presents a challenge for active matter theory.

Materials and methodsAssembling kinesin clusters

Weusedkinesinmotors dimers comprising a 401-amino acid N-terminal domain derived fromDrosophila melanogaster kinesin labeledwith a6-his tag and a biotin tag. Kinesin motors werepurified according to a previously publishedpro-tocol (55). An isolated kinesinmoves along a singlemicrotubule protofilament toward its plus end.Linkingkinesins intomultimotorclustersenablesthem to bindmultiplemicrotubulesanddrive in-terfilament sliding. The biotin-labeled motorswere assembled into multimotor clusters usingstreptavidin tetramers. To assemble kinesin clusters,we mixed 1.5 mM kinesin motors with 1.8 mMstreptavidin (Invitrogen, S-888) at a 1:1.2 ratio inmicrotubule M2B buffer (M2B: 80 mM PIPES,pH 6.8, 1 mM EGTA, 2 mM MgCl2) containing120 mM dithiothreitol (DTT). The mixture wasthen incubated for 30 min at 4°C, before beingaliquoted and stored long-term at -80°C.

Microtubule polymerization

Tubulin monomers were purified from bovinebrains with two cycles of polymerizations anddepolymerizations in a high salt buffer (1 MPIPES buffer), followed with one cycle in M2B

buffer (56). To induce polymerization, the tubulinsuspension (8 mg/mL in M2B) was mixed with600 mMGMCPP (guanosine-5′-[(a,b)-methyleno]triphosphate, Jena Biosciences, NU-4056) and1 mM DTT: For fluorescence imaging, 3% of tu-bulin monomers were labeled with Alexa-Fluor647 (Invitrogen, A-20006). The tubulin mixturewas first incubated for 30 min at 37°C, followedby an annealing step at room temperature for sixhours. The resulting microtubules have an aver-age length of ~1 mm (24). The polymerizedmicro-tubules were stored at -80°C.

Microtubule-based active fluids

To assemble active fluids, 1.3 mg/mLmicrotubulesuspension was mixed with kinesin-streptavidinmotor clusters in a high-saltM2B (M2B + 3.9 mMMgCl2) (24). Kinesinmotors hydrolyze adenosinetriphosphate (ATP) into anadenosinediphosphate(ADP) while stepping toward the microtubuleplus end. To fuel suchmotion we added 1.4 mMATP. To maintain constant ATP concentrationand steady dynamics in active fluids, we have in-corporated an ATP regeneration system compris-ing 26mMphosphoenolpyruvate (PEP)and2.8%v/vstockpyruvatekinase/lactatedehydrogenaseenzymes(PK/LDH) (Sigma, P-0294) (57). PK consumes PEPand converts ADP back to ATP, thusmaintainingsteadyATP concentration. To reduce photo-bleachingeffectsduring fluorescence imaging,weadded2mMtrolox (Sigma, 238813) and an oxygen scavengingenzymemixture comprised of 0.22mg/mLglucoseoxidase (Sigma, G2133), 0.038 mg/mL catalase(Sigma, C40), and 3.3 mg/mL glucose. To sta-bilize proteinswe added 5.5mMDTTas a reducingagent. As the depletion agent we used 2% w/wPluronic F127, which assembles into ~20 nmmi-celles (58). Finally, to track the flow induced byactive fluid, we added ~0.0004% v/v Alexa 488-labeled 3-mm polystyrene particles, stabilizedby Pluronic F127. Upon mixing all of the aboveingredients, nonequilibrium dynamics of activefluids can last for hours, until both ATP and PEPare depleted.

Engineering microfluidic chips

Millimeter-scale (≥0.3mm)devicesweremanufac-tured by directlymilling a 2-mm-thick COCplaque(TOPAS Advanced Polymers) with a CNC end-milling machine, which reads G-code translatedfrom CAD with Cut2D (Vectric). Micron-scaledevices were manufactured by embossing COCplaques (thermoplastic polymers) with patternedpolydimethylsiloxane (PDMS, thermosetting poly-mers) slabs at ~180°C (59). The slabs weremoldedby casting uncured PDMS resin (1:10 ratio ofcuring agent to base, Dow Corning) on a standardSU8 master and baked at 70°C overnight. Themaster was photo-lithographedwith photomasksprinted from CAD. Millimeter- and micron-scaledevices were subsequently cleaned with ethanolby a ~5-min sonication, followed by a ~20-minsonication in water. To protect proteins fromdirect surface contact, the COC chips were pre-coated with polymer brushes. Since COC is hydro-phobic we coated its surface with Pluronic by anovernight incubation in 2% w/w F127 in water.

ClampingTo confine active microtubules in a predesignedconfinement, the patterned COC substrate wasloadedwith activemicrotubules, and subsequentlycovered with a 101-mm thick COC film (TOPASAdvanced Polymers). To prevent leakage or sam-ple drying, the COC layers were sandwiched be-tween a pair of aluminum frames held togetherby screws and bolts (Fig. S5). The middle of thealuminum frames had a one-inch window toserve as an observationwindow. To prevent chipsfrom falling through the windows, glass slideswere placedacross thewindows. To even clampingpressure distribution, thin PDMS slabs wereinserted between the COC chips and glass slides.After clamping, the samples remained sealedandhydrated for days.Microfluidic chamberswerereused after cleaning with ethanol and 20-minsonication in water.

Analysis of fluid flows

Motion of active fluid flows was quantified bytracking the motion of tracer beads. To measurevelocity field, v(r, q, t), we analyzed sequentialimages of tracer beads with particle image ve-locimetry (PIV, PIVlab version 1.4) (60). Takingcurl of velocity field yields vorticity distribution,c(r, q, t) ≡ [∇ × v]z. Flows are faster in a largerchannel (Fig. 2C, 2D). To compare flow fields andvorticity maps among samples, we normalize ve-locity fields by their instant spatially averagedspeed:Vðr; q; tÞ ≡ vðr; q; tÞ=hjvðr; q; tÞjir;q, andvorticity distributions by triple their instant stan-dard deviations: C(r, q, t) ≡ c(r, q, t)/[3s(t)], wheres(t) is the standard deviation of a vorticity dis-tribution at time, t. Tomeasure flow profile acrossa channel, tracers were trackedwith Lagrangianparticle tracking (61). Flow profiles were deter-mined by averaging the q-component of theirvelocities, vq over time (t), and orientations (q

^):hvqðr; q; tÞiq;t . To characterize flow coherency,we defined circulation order parameter,F asF ≡hvq=jvji, where v is tracer velocity. F = 1 impliesa perfect counter-clockwise flow (q^); F = –1,clockwise (−q^). F = 0 implies either randomflows or perfect radial flows ðr̂Þ(not seen).Elastomer confinements

To ensure that the development of coherentflows is not exclusive to COC-based microfluidicchannels, we have confined active fluids in 3Dtori inscribed in an elastomer (62). The elastomerconsisted of 26% w/w 10-cst PDMS oil (Sigma-Aldrich, 378321-250ML) and 74% w/w elastomer(Dow Corning 9041 silicone elastomer blend). Toinscribe a torus, we inserted a 27-gauge bluntneedle (SAI Infusion Technologies, B27-150) intothe elastomer, followed by rotating the elastomerwhile pumping active fluids through the needle.The pumped fluid formed a torus after one fullrevolution. The torus radius could be tuned bychanging the distance between the needle tip andthe elastomer rotation axis; the torus thickness bycontrolling the pumped fluid volume. Subsequently,the needle was extracted from the elastomer,yielding an isolated torus sealed within the elas-tomer (Fig. S4). Active fluid contained Pluronic

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that coated the inner torus surface, preventingmicrotubules adsorption. Unlike COC, the elas-tomer contains PDMS oil so the surface of thetorus confinement is fluid.

Tube confinements

To investigate the broader applicability of ourfindings we loaded active fluids into a plastic tube[Scientific Commodities Inc., BB31695-PE/5, (Fig.6)]. The tube length was 1,060mm; its outer andinner diameters were 1.32 and 0.86 mm, respec-tively. Tube ends were joined with a 50-mm long20 gauge metallic needle (SAI infusion technolo-gies, B20-150). The needle outer and inner diame-ters were 0.91 and 0.60 mm, respectively. Toreduce evaporation from gaps between the nee-dle and tube, we applied vacuum grease (DowCorning). For imaging purpose the tube was fixedon a glass slide with tape. The tube cross-sectionalprofile was circular, distorting tracer images dueto index of refractionmismatch. To reduce such dis-tortions, the tubewas immersed inuncuredUVglue.

Analyzing structure of active networks

To correlate fluid flow with underlying active gelstructure we z-scanmicrotubule networks withconfocal microscopy when in coherent and tur-bulent states. From themidplane slice (thickness =10 mm) we use a filament tracking algorithm toextract bundle structures, represented as windingsnakes (magenta curves in Fig. S6A and Movie 8)(44). Snakes are composed of unit-length bundlesegments whose rq__-plane orientations are givenby angle Y, where -p/2 ≤ Y ≤ p/2. To measuretime-averaged distributions ofY__, the 3D confocalimages of thenetworks structurewere continuouslyacquired, and fromhere anglesY__ were extracted.The snake reconstruction algorithmhas a bias thatyields a slight peak aroundY__~0 and p/2, even forsuspensions that are known to be isotropic (bluecurve in Fig. S6B).When the imageswere rotatedby –45° the peaks also shifted by ~45°, demon-strating the biased detection associated with thereconstruction algorithm. To remove this artificialbias we rotated the images from 0° to 360°, by 30°(Fig. S6C). From these images, we extracted andstacked bundle orientations. The stacked orien-tations yielded a probability orientation distribu-tion towithin a fewpercent (gray curve). Repeatingthe same experiments with different samplesyielded similar distributions (black curve). Thismethodwas applied to eliminate the bias of thereconstruction algorithm, for both coherent andincoherent samples (Fig. 7, A and B).To characterize bundle orientational order we

measured the scalar nematic order parameter(NOP), the largest eigenvalue of the nematic orderparameter tensor,Qij, definedasQij ≡ hn^in^j−dij=2i,where n̂ ≡ cosðYÞ(q^)+sin(Y) r̂ is the director ofa bundle segment, and dij is Kronecker deltafunction (Fig. S6A). To characterize NOP spatialdistribution we measure its radial profile, NOP(d0); the corresponding tensor profile Qij (d0) ismeasured by taking the average over t and q__:Qijðd0Þ ¼ h⋯it;q. In either coherent or incoherentstates the NOP profile decays from boundaries.The decay length l is defined by where NOP is

reduced by a factor of three: NOP (distance l froma boundary) ≡NOP (at the boundary)/3 (Fig. 7C).

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ACKNOWLEDGMENTS

This work was primarily supported by the U.S. Department of Energy,Office of Basic Energy Sciences, through award DE-SC0010432TDD(K.T.W., S.J.D., and Z.D.). We also acknowledge support by NSF-MRSEC-1420382 (J.B.H. and S.F.). We acknowledge use of theBrandeis Materials Research Science and Engineering Center(MRSEC) optical, microfluidic, and biosynthesis facilities supported byNSF-MRSEC-1420382. Analysis of microtubule networks wassupported by Templeton Foundation grant 57392 (D.T.N.C). Weacknowledge B. Rogers for use of his confocal microscope. We aregrateful to D. Needleman for providing historical perspective onself-organized machines. K.-T.W. performed the research; J.B.H.acquired data in micrometer confinements; K.-T.W., J.B.H., S.F., andZ.D. analyzed the data; D.T.N.C. analyzed the network structure;Z.D. and S.F. designed the research; S.J.D, K.-T.W, Y.W.C., and A.F.-N.observed the initial effects; and K.-T.W., S.F., and Z.D. wrote themanuscript. All authors have reviewed the manuscript.

SUPPLEMENTARY MATERIALS

www.sciencemag.org/content/355/6331/eaal1979/suppl/DC1Materials and MethodsFigs. S1 to S6

12 October 2016; accepted 10 February 201710.1126/science.aal1979

Wu et al., Science 355, eaal1979 (2017) 24 March 2017 9 of 9

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Transition from turbulent to coherent flows in confined three-dimensional active fluids

Seth Fraden and Zvonimir DogicKun-Ta Wu, Jean Bernard Hishamunda, Daniel T. N. Chen, Stephen J. DeCamp, Ya-Wen Chang, Alberto Fernández-Nieves,

DOI: 10.1126/science.aal1979 (6331), eaal1979.355Science 

, this issue p. eaal1979; see also p. 1262Scienceof the geometry was chosen appropriately, flows were observed for a wide range of system dimensions.motion, either clockwise or counterclockwise. Oddly, this behavior was independent of scale; as long as the aspect ratioconfined in periodic toroidal channels and cylindrical domains, the flow was organized and persisted in a unidirectional molecular motor kinesin, confined in a variety of microfluidic geometries (see the Perspective by Morozov). Whenemergence of spontaneous directional flows in active fluids containing a suspension of microtubules and clusters of the

analyzed theet al.transport may be isotropic or governed by the presence of specific chemical gradients. Wu The transport of ordinary fluids tends to be driven by pressure differentials, whereas for active or biological matter,

Go with the changing flow

ARTICLE TOOLS http://science.sciencemag.org/content/355/6331/eaal1979

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REFERENCES

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