Spontaneous symmetry breaking in active dropletsprovides a generic route to motilityElsen Tjhung, Davide Marenduzzo1, and Michael E. Cates

SUPA, School of Physics and Astronomy, University of Edinburgh, Mayfield Road, Edinburgh EH9 3JZ, United Kingdom

Edited by T. C. Lubensky, University of Pennsylvania, Philadelphia, PA, and approved June 18, 2012 (received for review January 20, 2012)

We explore a generic mechanism whereby a droplet of active mat-ter acquires motility by the spontaneous breakdown of a discretesymmetry. The model we study offers a simple representation ofa “cell extract” comprising, e.g., a droplet of actomyosin solution.(Such extracts are used experimentally to model the cytoskeleton).Actomyosin is an active gel whose polarity describes the meansense of alignment of actin fibres. In the absence of polymerizationand depolymerization processes (‘treadmilling’), the gel’s dynamicsarises solely from the contractile motion of myosin motors; thisshould be unchanged when polarity is inverted. Our results sug-gest that motility can arise in the absence of treadmilling, by spon-taneous symmetry breaking (SSB) of polarity inversion symmetry.Adapting our model towall-bound cells in two dimensions, we findthat as wall friction is reduced, treadmilling-induced motility fallsbut SSB-mediated motility rises. The latter might therefore becrucial in three dimensions where frictional forces are likely to bemodest. At a supracellular level, the same generic mechanism canimpart motility to aggregates of nonmotile but active bacteria; weshow that SSB in this (extensile) case leads generically to rotationalas well as translational motion.

active gels ∣ cell motility ∣ phase transitions

Living cells can move themselves around in a variety of differentconditions and environments, and they exploit a range of stra-

tegies and mechanisms to do so. Uncovering the generic pathwaysto cell motility remains central to many important processesranging from wound healing and tissue development (1) to immu-nological response and diseases such as cancer (2). The best char-acterized case is that of a crawling cell on a planar 2D substrateor wall. Here motility is generally attributed to cytoskeletal actinfilaments which polymerize at one end ðþÞ and depolymerize atthe other ð−Þ in a process called treadmilling. So long as thesystem has nonzero polarity P ¼ hpi (where p is a unit tangentoriented from − to þ and angle brackets denote an average overfilaments), treadmilling leads to macroscopic motion. This processexploits a Brownian ratchet mechanism in which forward fluctua-tions of the cell perimeter are locked in by polmerization (3).However this mechanism requires a relatively solid anchor-pointfrom which to propel the cell. For cells crawling on a 2D surface,this is provided by focal adhesions and other integrin-rich struc-tures (4, 5).

In vivo, cells often move in a 3D environment such as anextracellular matrix and/or a tissue of cells (2, 6, 7). Especially inquasi-spherical motile cells, both integrin-rich structures andmechanical anchoring are less in evidence, and the mechanismsof motility in such 3D environments remain unclear (8). A recentstudy on tumor cells moving inside an elastic gel suggests that animportant role in 3D locomotion may be played not by polymer-ization but by myosin contractility. This process can lead to col-lective internal flows of actin that may ultimately propel the cellforward (7). The contractile effect arises by a motor spanning twofibres causing an inward force pair (Fig. 1A) (9). As a result anactive stress is created, usually modelled as σactive

αβ ¼ ζ̄cPαPβ withc the concentration of active material, while Greek suffices denoteCartesian directions, and ζ̄ is an activity parameter (ζ̄ > 0 for con-

tractile systems). There is therefore an important issue of princi-ple: how exactly does a tensorial active stress result in a vectorialpropulsion velocity?

Several studies use minimal models to address the fundamen-tal physics of how activity imparts cell motility. Experimentalprogress has focused on “cell extracts” (10, 11): unregulated bagsof cytoskeletal filaments (actin) and molecular motors (myosin),enclosed by a membrane. However, most of this work focuseson 2D crawling via the treadmill-ratchet mechanism describedabove. [3D systems are harder to study, and selective inhibitionof the treadmill dynamics is biochemically difficult (12)]. On themodelling side, generic theories have been proposed to makecontact with the cell-extract data, again focusing mainly on 2Dcrawling and the treadmill-ratchet mechanism (13–15). While theinfluence of myosin contractility on cell shape during locomotionhas been addressed (14, 15), in 2D this has not so far beenthought sufficient by itself to lead to motility.

Here we provide a detailed computational study of the effectsof active stresses in a minimal 2D model of an actomyosin cellextract. To confirm that our proposed motility mechanism re-mains pertinent in 3D, we additionally perform selective (compu-tationally intensive) simulations in that case. Our simulationmodel comprises a droplet of an active fluid or gel (16, 17), con-fined by interfacial tension ~σ, and surrounded by a Newtonianhost fluid. Our model equations are based on established conti-nuum precepts and outlined in Materials and Methods. Myosincontractility is represented by an active stress as detailed above.This term is invariant under global polarity inversion as are, withtreadmilling absent, the full equations of motion (for further dis-cussions of their symmetries, refer to SI Appendix). We show thenthat when the activity parameter ζ̄ exceeds a given threshold, aninitially circular or spherical droplet spontaneously breaks thatinversion symmetry, leading to an elastic splay of the polarity fieldand to motion along �P. This spontaneous symmetry breaking(SSB) manifests itself as a supercritical Hopf bifurcation, whichcan alternatively be viewed as a continuous nonequilibrium phasetransition. (The threshold value depend on both ~σ, and an effec-tive elastic constant κ penalizing distortions of the ordered polarstate.) In 2D, the resulting motile droplets have crescent-likeshapes similar to some crawling cells (18), whereas in 3D ourmodel predicts both spherical and concave shapes. Representingwall friction by a depth-averaged drag term in 2D, we find thisslows down the SSB-mediated motion, but does not stop it alto-gether unless a critical drag is exceeded.

We then introduce the treadmilling effect, which we representby a self-advection parameter w. This parameter breaks the glo-bal P ↔ −P symmetry, but it does so directly, not spontaneously.(Note that in our model polarity is present even without tread-

Author contributions: E.T., D.M., and M.E.C. designed research, performed research,analyzed data, and wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.1To whom correspondence should be addressed. E-mail: dmarendu@ph.ed.ac.uk.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1200843109/-/DCSupplemental.

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milling: see SI Appendix: (Section 1) for a discussion of the rele-vant physics.) This w is the speed at which, relative to the localsuspending fluid (of velocity vðr; tÞ), each filament is self-pro-pelled along its own tangent. Even though asymmetric polymer-ization does not lead directly to mass transport, for actomyosinthe resulting mass flux wcP should capture, in a highly simplifiedmanner, the preferential growth of filaments by addition ofmonomers at one end and loss at the other; see Fig. 1C. Thissimplified description is possible because we exclude the bath ofmonomers from the local mass density cðr; tÞ of active material.(We assume that on average, the monomers nonetheless keep upwith the moving gel). In general, self-advection leads to densitygradients, which in turn cause hydrodynamic flows vðr; tÞ in thedirection opposite to Pðr; tÞ. This backflow severely limits theeffectiveness of self-advection in creating motility, almost cancel-ling it for a droplet in free space as we discuss below. Our 2Dstudy of the effect of wall friction shows however that high en-ough friction, by reducing v towards zero, restores treadmilling-induced motility with a speed that approaches wP.

Although our main focus is on actomyosin cell extracts, ourframework provides a broader generic approach to dropletmotility. We can thus investigate what happens when we reversethe sign of the active stresses to consider the extensile case, ζ ¼−ζ̄ > 0 (19, 20). The primary experimental relevance of this caseis to suspensions of bacteria, which push fluid out along theirmajor axes and draw it in around the equator; see Fig. 1A. In thiscontext, w is the bacterial swim speed; although this is nonzero formotile species, one may create virtually nonmotile mutants(called “shakers”) which still create extensile active stresses; e.g.,by excessively increasing the tumbling rate (21). A droplet ofsuch organisms can be created either by inducing an attractionbetween them (as our model effectively assumes) (22) or perhapsby confining them in an emulsion droplet. To attain nonzeroP one further requires these particles to have net polar order[as opposed to a nematic state, for which there is orientationalorder, but equal numbers of particles with tangent �p locally(see SI Appendix)]. Setting aside the possible difficulties in meet-ing all those requirements experimentally, we predict that such“shaker” bacterial droplets could again break the symmetry andstart moving spontaneously. Intriguingly, the predicted trajec-tories in this extensile case are more complicated than those ofthe contractile model. This intricacy arises because, in extensiledroplets, the SSB-mediated velocity forcing is in a directionperpendicular, rather than parallel, to the polarization vector P.

Restoring nonzero w to describe the case of motile bacteria, thecomposition of these two motions leads to circular or spirallingtrajectories of the droplet as a whole.

ResultsWe first present results for the 2D contractile case, then brieflydescribe our findings in 3D, and finally give some further 2Dresults on extensile droplets. We initialized our simulation runswith a circular (or spherical) droplet within which the concentra-tion of active material is taken to be a constant (c ¼ c0), withc ¼ 0 outside. The polarization field Pðr; tÞ within the droplet isinitially uniform along the horizontal (x̂) axis and varies with con-

centration as magnitudeffiffiffiffiffiffiffiffiffiffiffiðc−ccrÞccr

qwith ccr a threshold value for

polarization onset. We choose c0 > ccr > 0 so that jPj is nonzeroinside the droplet and zero (isotropic) outside, so that we assumethe actomyosin network inside our active droplets is initiallypolarized. The system is then evolved via the equations of motionas specified in Materials and Methods (discussed further in SIAppendix: Section 1), with chosen values of the activity parameterζ̄ and the self-advection (treadmilling) parameter w. On a rela-tively short time scale both the internal concentration and thepolarization field relax towards the equilibrium values ceq andPeq (while still c ¼ 0 and P ¼ 0 externally) with some interfacialtension ~σ, set by minimization of our chosen free energy. (Ourchoice creates no anchoring of P at the surface so, in the absenceof symmetry breaking, the polarization remains uniform; seeFig. 2A left. The effect of a soft anchoring is discussed in theSI Appendix: Section 1). Having made one such relaxed droplet,the dependence of its behavior on ζ̄ and/or w was systematicallyexplored by incrementing those quantities and waiting for steadystate, before incrementing again. Because we are primarily inter-ested in trends and symmetry breaking phenomena, rather thanquantitative predictions of where these will occur for specific ma-terials, we report all results below in the natural units for latticeBoltzmann simulations (LBU); the connection between these andphysical units is discussed in SI Appendix: Section 4.

Contractile Stress Can Create Motility via SSB. Although contractilemotor stress and actin treadmilling are generally both present inmotile cells, it is illuminating to study these two mechanismsseparately. We first consider a droplet with no external drag term(no wall friction) and no treadmilling term (w ¼ 0), and vary theactivity parameter ζ̄ > 0. For low activity ζ̄, below some criticalvalue ζ̄c, the droplet polarization field P remains aligned uni-formly along its initial direction x̂ and the droplet remains station-ary. However it becomes slightly elongated in the directionperpendicular to the polarization vector P as a result of the com-petition between the contractile stress and the interfacial tension(see Fig. 2A middle). In this regime, the contractile stress sets upa quadrupolar fluid flow around the droplet (Fig. 2B left), so thatthe whole droplet behaves as a large contractile element (com-pare Fig. 1A right). However it does not translate: there is nomotility, and the droplet is a “shaker” rather than a “mover” (23).

As we increase ζ̄ beyond ζ̄c, the uniform polarization field Pbecomes unstable with respect to a splay deformation. This in-stability happens because the contractile stress is large enoughto overcome the resistance to deformation mediated by the elas-tic constant κ. The splay creates a state in which neighboring vec-tors P either fan outwards (∇ · P > 0) or inwards (∇ · P < 0). Thefirst is shown in Fig. 2A right; the second is found by first takingits mirror image and then reversing P. This choice is made atrandom, spontaneously breaking the global polarity inversionsymmetry. As soon as this SSB happens, the droplet starts tomove along the direction set by ð∇ · PÞP ¼ �P. This motion isattributable to the formation of a pair of flow vortices inside thedroplet (Fig. 2B right). Such spontaneous propulsion is somewhat

Fig. 1. (A) A minimal model for an active element or force dipole. The figureshows the quadrupolar nature of the hydrodynamic flow around an extensileelement or pusher (left) and a contractile element or puller (right). (B) Con-tractile stress is created when a motor protein (myosin) pulls protein fila-ments (actin) together in the cytoskeleton. (C) Polymerization of the actinfilaments gives rise to an effective “self-advection” velocity in the directionof the polarization vector (arrow).

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reminiscent of the self-electrophoretic motion of a vesicle withactive membrane pumps in an ionic solution (24, 25).

We note that in bulk active contractile fluids, the state of uni-form P is also generically unstable to splay fluctuations in 1D(19, 26, 27), which then leads to the onset of spontaneous flow.At one level, the SSB-induced motility transition described herecan be viewed as a manifestation of that bulk instability, albeitwith two variations. First, spontaneous flows are present on bothsides of our transition: as discussed above there is a quadrupolarflow field already in the nonmoving state. Second, in bulk the cri-tical activity level is nonzero only in finite systems, for which thetransition is discontinuous, unlike ours (see below and Fig. 2C),and the resulting velocity field much more complicated (17, 20).

To illustrate our symmetry breaking motility mechanism moreclearly, we plot the magnitude VCM of the center of mass velocityof the droplet VCM as a function of activity ζ̄ in Fig. 2C. This

bifurcation diagram shows a continuous nonequilibrium transi-tion from a stationary and uniform state to a moving and splayedstate. Moreover, to within numerical accuracy the observationsare consistent with a supercritical Hopf bifurcation, for whichVCM ∼ ðζ̄ − ζ̄cÞ0.5. This mean-field like exponent is perhaps un-surprising as there is no noise in our simulations. Accordingly itmight change in the presence of activity-generated noise (28),depending on whether the bifurcation remains low-dimensionalor acquires a many-body critical character.

SSB-Induced Motility Is Diminished by Friction.The results above arefor a droplet in 2D surrounded by Newtonian fluid. To betterdescribe experiments involving cell-crawling on a substrate, wenow consider an additional frictional force between the solid walland the cell. To add friction within our 2D continuum model, weintroduce an additional force density f friction ¼ −γv to the mo-

Fig. 2. (A) Steady state configurations of a contractileactive droplet without self-advection. The (red) arrows showthe polarization field Pðr; tÞ. Upon increasing the contractileactivity ζ̄, the droplet elongates perpendicular to P and thenbecomes unstable with respect to splay deformation at cri-tical activity ζ̄c. When it splays, the droplet also sponta-neously moves in the direction of the green arrow. Thetime evolution of the system is shown in Movie S1. (B) Leftplot shows the velocity field of the droplet at ζ̄ < ζ̄c which isquadrupolar, like that around a contractile element (Fig. 1Aright). Right plot shows the velocity field of the splayedand moving active droplet which consists of two opposingvortices. The boundary of the droplet itself is given by thedashed line. (C) Bifurcation diagram showing spontaneoussymmetry breaking from a uniform and stationary stateto a splayed and moving state as the activity parameter ζ̄is increased.

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mentum balance equation (see Materials and Methods and SIAppendix). Here γ is an effective coefficient of friction whichdepends on whether we have nonslip or partial-slip boundaryconditions on the substrate and also on the thickness of the cell.It may represent conventional friction and/or a coarse-grainedmodel of focal adhesions and other localized mechanical con-tacts. The presence of this friction will significantly quench thehydrodynamic flow v. Because our SSB-induced motility requiresa hydrodynamic vortex flow inside the droplet, the frictional forcecan greatly reduce the droplet’s velocity, bringing it to rest beyonda critical level γc, at which point the symmetry is restored; thevalue of γc depends on activity and other model parameters. Thisbehavior is apparent from a plot of VCM against γ in Fig. 3.

Self-Advective Motility Is Enhanced by Friction.We now consider thecase when there is only treadmilling (modeled as self-advectionw > 0) but no contractile stress (ζ̄ ¼ 0). As discussed previously,the droplet will move along the polarization vector P with speedVCM < w. Because there is no spontaneous symmetry breakinginvolved, translational motion occurs for any nonzero w, in con-trast to the threshold behavior seen for contractile SSB-motility.The most interesting aspect is the role of the friction parameter γ.Again we plot the velocity of the droplet VCM as a function of γ inFig. 3. In contrast to the previous case, motility is significantlyenhanced by the presence of friction. Indeed, in the limit γ → ∞,we have v ¼ 0 and VCM → wP (see Fig. 3 inset).

The intersection of the two plots of droplet velocity vs. friction(found respectively by switching off activity or self-advection) de-fines a characteristic friction scale eγ. For γ < eγ contractile stressesdominate cell motility, while for γ > eγ, self-advection is dominant.

In most experiments on 2D crawling of cells/cell extracts (5),the involvement of focal adhesions suggests that the high friction(treadmilling dominated) limit generically prevails. On the otherhand, some recent experiments (29) directly identify spontaneoussymmetry breaking of the actomyosin network as the initiator ofpolarized cell motility in keratocytes. Our work emphasizes thatspontaneous breaking of global polarity inversion symmetry arisesfrom contractile motor activity, not from treadmilling. It is there-fore arguable that the role of motor activity in 2D motility has sofar been underestimated. We note however that an equivalent dis-crete symmetry breaking would create motility if a pure treadmill-ing state of zeroPα ¼ hpαi, but finite nematic order (hpαpβi − δαβ∕3 ≠ 0), spontaneously acquires polarity locally (see SI Appendix).

3D Droplets Show a Window of SSB-Induced Motility. In the contextof experiments on 3D tumor cells, it has been argued that motilityis driven primarily by contractile stress (2, 7), suggesting that thelow-friction limit of our model prevails here. This possibilityaccords with the much diminished part played by focal adhesionsin 3D (8). For the 3D case we therefore neglect the friction term,and run selected simulations to confirm that the SSB route tomotility remains operative.

Fig. 4 shows steady state polarization fields inside a 3D con-tractile droplet with increasing values of the activity magnitudefrom A to D. As expected, the first steady state encountered is asymmetric but deformed immotile droplet (Fig. 4A). The activestress contracts the droplet along �P resulting in a lenticularshape. As we increase ζ̄ beyond a critical ζ̄c, splay instability spon-taneously breaks symmetry and causes the droplet to move alongð∇:PÞP just as in the 2D case. The droplet shape is concave and(as in the lenticular case) both it and the flow field resemble thatcreated by rotating the 2D droplet about the P axis (Fig. 4B). Theresulting hydrodynamic flow (Fig. 4E) therefore corresponds toa toroidal vortex ring. It would be interesting to see how thesepredictions compare with intracellular actin and fluid flow mapsthat might in future be measured for cells moving in 3D environ-ments, for instance those studied in ref. 2.

Interestingly, as ζ̄ is increased further, the droplet becomesincreasingly spherical (Fig. 4C), and finally symmetry is restored,creating an immotile spherical droplet with a ‘hedgehog’ defect[of topological charge 1 as dictated by the polar ordering (30)] atthe center (Fig. 4D).

Extensile SSB Creates Transverse or Circular Motility.Our final resultsare for 2D extensile droplets (Fig. 1A). Continuum descriptions

Fig. 3. Representative plots of droplet velocity VCM against frictional coef-ficient γ for motile droplets driven by: contractile stress only (solid red line)and polymerization/self-advection only (dot-dashed blue line). The insetshows the polymerization-only driven motility in the limit of large frictionin which the droplet velocity approaches “polymerization” speed wp.

Fig. 4. Steady state conformations in 3D con-tractile droplets without self-advection on in-creasing activity ζ̄ from (A) to (D). (B) and (C) aremotile as indicated while (A) and (D) are sta-tionary. (E) shows the toroidal fluid flow insidethe motile droplet of steady state B. The timeevolution of the droplets in (B) and (D) areshown in Movies S2 and S3 respectively.

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of uniform extensile active fluids are widely used to describedense bacterial suspensions (19, 23, 31). These results may there-fore be relevant to bacterial droplets formed by aggregation in thepresence of attractive forces (22), or possibly by confinement ofbacteria within a droplet emulsion. Again, for fixed thermody-namic parameters controlling the elasticity and interfacial tensionof the confined material, one can vary the activity parameterζ ¼ −ζ̄ (now positive) and the self-advection parameter w whichcorresponds to the swimming speed of individual bacteria and isnonzero for “movers” but zero for “shakers” (19, 26).

Fig. 5A shows steady state configurations of a 2D purely exten-sile (w ¼ 0) droplet at different ranges of activity. For ζ < ζc1 thedroplet remains stationary but again elongates symmetrically, thistime along the direction of the polarization field �P. The exten-sile stress creates the quadrupolar flow field expected of a large,extensile shaker. For ζ beyond the critical value ζc1, the dropletagain becomes unstable, but now with respect to bend deforma-tion as opposed to splay. This instability gives rise to a horizontalvortex pair inside the droplet as opposed to the vertical one in thecontractile case (Fig. 2B right). This flow field causes the dropletto move in a direction set by the sense of bending as P × ð∇ × PÞ(which can be upwards or downwards according to Fig. 5A). Ascan be seen from Fig. 5A this stationary-to-motile transitioncan still be characterized as a continuous SSB transition, how-ever, the droplet speed VCM attains a maximum at intermediateζ before falling to zero again. Beyond a second critical activitythreshold (ζc2 in Fig. 5A), the polarization pattern oscillatescontinuously while remaining symmetric at all times and conse-quently, the droplet again becomes nonmotile.

Turning finally to the case of nonzero self-advection (w > 0)we find that this, combined with the extensile motility, can giverise to an intriguing spiralling motion (see Fig. 5B). This motionarises because the SSB-induced motility is at right angles to thepolarization (Fig. 5B right). On the other hand, self-advectioneverywhere transports material along Pðr; tÞ locally; because thepolarization field is spontaneously curved, this by itself would givea circular droplet orbit. Compounding these two motions typi-cally leads to outward spiral trajectories as shown in Fig. 5B. Thisoutcome contrasts with the contractile case where both the SSB-induced and the self-advective motion (the latter averaged overthe droplet configuration) point either together or oppositelyalong the �P direction and only straight line motion can result.

Discussion and ConclusionsOur simulations of contractile droplets can be viewed as a simplein silico analogue of the in vitro cell-extract studies that have beenused to dissect the biophysical ingredients of motility. Our simu-lation work powerfully complements these studies, by allowingus to isolate the role of contractile (motor) activity in cell loco-motion: this is very difficult in the laboratory, where current stra-tegies for inhibiting polymerization dynamics (treadmilling) canseverely impair other key subcellular processes (12).

Our approach likewise complements those of previous theories(14, 15, 32) which have concentrated on treadmilling as the maindriver of motility. We have shown in 2D that contractile stressesalone can not only shape the rear of a crawling cell (Fig. 2), butalso create motility itself, provided that the motor forces are largeenough to create an asymmetric circulatory flow as in Fig. 2B.Interestingly, keratocyte cells crawling on glass seemingly do ex-ploit myosin activity to set up an intracellular actin flow in therear of a cell which ultimately polarizes it and makes it motile(29). Nonetheless, our study of the effects of a frictional term,which promotes the motility created by treadmilling but inhibitsthat caused by contractile stress, lends support to the view thatcell crawling on a wall is usually dominated by the treadmilling.

In contrast, in 3D cell motility, recent work suggests that tread-milling plays at most a minor role (2, 7). In the 3D case, therefore,our work describes a simple and compelling mechanism for how

spontaneous translational motion can in principle arise solely bythe action of a contractile stress. The onset of motility requiresspontaneous symmetry breaking, mediated in our case by splay de-formation in response to that stress. Our 2D and 3D simulationsgo beyond the 2D theory of ref. 7 by addressing the dynamics ofthe polarization field. We do however make some important sim-plifications: our droplets are confined only by interfacial tensionnot by an elastic membrane; we treat treadmilling as a simple self-advection; and we do not address any direct transition betweennematic and polar order, despite assuming polarity inversionsymmetry at thermodynamic level. Improving the model in theserespects will require a more detailed microscopic derivation whichwe shall leave to future work. To test whether our model indeedcaptures the biophysics of 3D cell motility, it would be exciting tovisualize experimentally the detailed cytoskeletal organization andflow fields; e.g., for cells moving through matrigel (2, 7).

Our generic framework is not limited to contractile actomyosinnetworks. Indeed we have discussed the case of extensile droplets,possibly relevant to aggregates or emulsions of active but immo-tile bacteria; here translational motility arises by spontaneous

Fig. 5. (A) Plot of center of mass velocity against activity (ζ ¼ −ζ̄ > 0) forextensile droplet without self-advection. It shows continuous transitions fromstationary to motile and then from motile to oscillatory at critical activity ζc1and ζc2 respectively. Also shown are the steady state polarization field P forthe stationary and motile case. The movie of the time evolution of the systemis shown in Movie S4. (B) The presence of both extensile stress and self-advec-tion leads to an outward spiral trajectory (solid green lines). Also shown arethe snapshots of the polarization field at different timesteps (red arrows).

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symmetry breaking only at intermediate activity, and is mediatedby bend rather than splay deformation. The addition of self-advection along the bent polarization field then leads in additionto rotational motion. We note that rotational and translationalmotility of small bacterial aggregates was recently observed, butattributed to a somewhat different mechanism where symmetry isbroken by frozen-in statistical fluctuations rather than SSB (22).

Finally, our hydrodynamic equations of motion, or close var-iants of these, might in some cases be applicable to concentratedeukaryotic cell masses such as tissue (33). Within a tissue eachcell exerts forces on its neighbors which at the lowest order con-tinuum level creates a certain density of force dipoles (33); thevelocity field v then describes the slow migration of cells insidethe tissue. It is intriguing to note that the large-scale tissue flow inanimal cells during gastrulation may break the symmetry to formvortices, similarly to our active droplets. An example is the case ofthe so-called “polonaise movements” which are observed in thedeveloping chick embryo (1), and which are important to formthe correct supercellular structure. In this context a “polariza-tion” field is sometimes used to describe the orientation of indi-vidual cells (34). The relation, if any, between the onset of thisvortex flow and that seen in our droplets remains to be explored.

Materials and MethodsWe briefly outline here the hydrodynamic model used in this work (moredetails are in SI Appendix). We consider a fluid, comprising amixture of activematerial and solvent, with constant total mass density ρ. The hydrodynamicvariables whose dynamics we monitor are: (i) the concentration of activematerial cðr; tÞ, (ii) the fluid velocity vðr; tÞ (with incompressibility requiring∇:v ¼ 0), and (iii) the polarization field Pðr; tÞ ¼ hpi as defined previously.

Although an active droplet is a nonequilibrium system, we introduce thefollowing free energy functional to describe its equilibrium physics in thepassive limit of zero activity:

F½c; P� ¼Z

d3rfV ðcÞ þ k2j∇cj2 − α

2

ðc − ccrÞccr

jPj2 þ α4jPj4

þ κ2ð∇PÞ2g [1]

Here α > 0 is a phenomenological free energy amplitude, k determines thedroplet interfacial tension, and κ is an effective elastic constant. This choice ofF½c; P� leads to a continuous isotropic-to-polar transition at c ¼ ccr. To confinethe active material into a droplet, we choose: VðcÞ ¼ a

4c 4crc2ðc − c0Þ2 and set

c0 > ccr. This potential creates two free energy minima corresponding to aphase of pure passive solvent (external to the droplet, c ¼ 0 and P ¼ 0) anda polar active phase (inside the droplet, c ¼ ceq > ccr and P ¼ Peq).

Treating the active material as locally conserved, the time evolution ofthe concentration field cðr; tÞ can then be written as a convective-diffusionequation:

∂c∂t

þ ∇ ·�cðvþ wPÞ −M∇

δFδc

�¼ 0; [2]

where M is a thermodynamic mobility parameter and w is the self-advectionparameter.

The dynamics of the polarization field Pðr; tÞ follows an “active nematic”evolution (20), given by

∂P∂t

þ ððvþ wPÞ · ∇ÞP ¼ −Ω · Pþ ξv · P −1

ΓδFδP

; [3]

where v and Ω are the symmetric and antisymmetric parts of the velocity gra-dient tensor∇v. Γ is the rotational viscosity and ξ is related to the geometry ofthe active particles (16).

Force balance is ensured through the Navier-Stokes equation,

ρð∂∕∂t þ v · ∇Þv ¼ −∇P þ ∇ · σtotal − γv; where P is the isotropic pressure, −γvis the friction force per unit volume and σtotal is the total stress in the fluidwhich includes viscous, elastic/Ericksen, interfacial and “active” stresses (seeSI Appendix for details and for a discussion of the effects of additional terms,allowed by symmetry, in the equations of motion). The active stress is σactive

αβ ¼ζ̄cPαPβ (23) where ζ̄ > 0 for contractile activity and ζ ¼ −ζ̄ > 0 for extensile.

To solve these equations in 2D and 3D, we performed hybrid lattice Boltz-mann simulations, as done previously for other active flows (17, 20).

ACKNOWLEDGMENTS. We thank R. Voituriez for very useful discussions. E.T.thanks SUPA for a Prize Studentship and M.E.C. holds a Royal SocietyResearch Professorship.

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