Spontaneous symmetry breaking in active droplets provides a generic route to motility

Abstract

We explore a generic mechanism whereby a droplet of active matter acquires motility by the spontaneous breakdown of a discrete symmetry. The model we study offers a simple representation of a “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 mean sense of alignment of actin fibres. In the absence of polymerization and depolymerization processes (‘treadmilling’), the gel’s dynamics arises solely from the contractile motion of myosin motors; this should be unchanged when polarity is inverted. Our results suggest that motility can arise in the absence of treadmilling, by spontaneous symmetry breaking (SSB) of polarity inversion symmetry. Adapting our model to wall-bound cells in two dimensions, we find that as wall friction is reduced, treadmilling-induced motility falls but SSB-mediated motility rises. The latter might therefore be crucial in three dimensions where frictional forces are likely to be modest. At a supracellular level, the same generic mechanism can impart motility to aggregates of nonmotile but active bacteria; we show that SSB in this (extensile) case leads generically to rotational as well as translational motion.

Living cells can move themselves around in a variety of different conditions and environments, and they exploit a range of strategies and mechanisms to do so. Uncovering the generic pathways to cell motility remains central to many important processes ranging from wound healing and tissue development (1) to immunological response and diseases such as cancer (2). The best characterized case is that of a crawling cell on a planar 2D substrate or wall. Here motility is generally attributed to cytoskeletal actin filaments which polymerize at one end (+) and depolymerize at the other (-) in a process called treadmilling. So long as the system has nonzero polarity (where p is a unit tangent oriented from - to + and angle brackets denote an average over filaments), treadmilling leads to macroscopic motion. This process exploits a Brownian ratchet mechanism in which forward fluctuations of the cell perimeter are locked in by polmerization (3). However this mechanism requires a relatively solid anchor-point from which to propel the cell. For cells crawling on a 2D surface, this is provided by focal adhesions and other integrin-rich structures (4, 5).

In vivo, cells often move in a 3D environment such as an extracellular matrix and/or a tissue of cells (2, 6, 7). Especially in quasi-spherical motile cells, both integrin-rich structures and mechanical anchoring are less in evidence, and the mechanisms of motility in such 3D environments remain unclear (8). A recent study on tumor cells moving inside an elastic gel suggests that an important role in 3D locomotion may be played not by polymerization but by myosin contractility. This process can lead to collective internal flows of actin that may ultimately propel the cell forward (7). The contractile effect arises by a motor spanning two fibres causing an inward force pair (Fig. 1A) (9). As a result an active stress is created, usually modelled as with c the concentration of active material, while Greek suffices denote Cartesian directions, and is an activity parameter ( for contractile systems). There is therefore an important issue of principle: how exactly does a tensorial active stress result in a vectorial propulsion velocity?

Fig. 1.

(A) A minimal model for an active element or force dipole. The figure shows the quadrupolar nature of the hydrodynamic flow around an extensile element or pusher (left) and a contractile element or puller (right). (B) Contractile stress is created when a motor protein (myosin) pulls protein filaments (actin) together in the cytoskeleton. (C) Polymerization of the actin filaments gives rise to an effective “self-advection” velocity in the direction of the polarization vector (arrow).

Several studies use minimal models to address the fundamental physics of how activity imparts cell motility. Experimental progress has focused on “cell extracts” (10, 11): unregulated bags of cytoskeletal filaments (actin) and molecular motors (myosin), enclosed by a membrane. However, most of this work focuses on 2D crawling via the treadmill-ratchet mechanism described above. [3D systems are harder to study, and selective inhibition of the treadmill dynamics is biochemically difficult (12)]. On the modelling side, generic theories have been proposed to make contact with the cell-extract data, again focusing mainly on 2D crawling and the treadmill-ratchet mechanism (13–15). While the influence of myosin contractility on cell shape during locomotion has been addressed (14, 15), in 2D this has not so far been thought sufficient by itself to lead to motility.

Here we provide a detailed computational study of the effects of active stresses in a minimal 2D model of an actomyosin cell extract. To confirm that our proposed motility mechanism remains pertinent in 3D, we additionally perform selective (computationally intensive) simulations in that case. Our simulation model comprises a droplet of an active fluid or gel (16, 17), confined by interfacial tension , and surrounded by a Newtonian host fluid. Our model equations are based on established continuum precepts and outlined in Materials and Methods. Myosin contractility is represented by an active stress as detailed above. This term is invariant under global polarity inversion as are, with treadmilling absent, the full equations of motion (for further discussions of their symmetries, refer to SI Appendix). We show then that when the activity parameter exceeds a given threshold, an initially circular or spherical droplet spontaneously breaks that inversion symmetry, leading to an elastic splay of the polarity field and to motion along ± P. This spontaneous symmetry breaking (SSB) manifests itself as a supercritical Hopf bifurcation, which can alternatively be viewed as a continuous nonequilibrium phase transition. (The threshold value depend on both , and an effective elastic constant κ penalizing distortions of the ordered polar state.) In 2D, the resulting motile droplets have crescent-like shapes similar to some crawling cells (18), whereas in 3D our model predicts both spherical and concave shapes. Representing wall friction by a depth-averaged drag term in 2D, we find this slows down the SSB-mediated motion, but does not stop it altogether unless a critical drag is exceeded.

We then introduce the treadmilling effect, which we represent by a self-advection parameter w. This parameter breaks the global P↔-P symmetry, but it does so directly, not spontaneously. (Note that in our model polarity is present even without treadmilling: see SI Appendix: (Section 1) for a discussion of the relevant physics.) This w is the speed at which, relative to the local suspending fluid (of velocity v(r,t)), each filament is self-propelled along its own tangent. Even though asymmetric polymerization does not lead directly to mass transport, for actomyosin the resulting mass flux wcP should capture, in a highly simplified manner, the preferential growth of filaments by addition of monomers at one end and loss at the other; see Fig. 1C. This simplified description is possible because we exclude the bath of monomers from the local mass density c(r,t) of active material. (We assume that on average, the monomers nonetheless keep up with the moving gel). In general, self-advection leads to density gradients, which in turn cause hydrodynamic flows v(r,t) in the direction opposite to P(r,t). This backflow severely limits the effectiveness of self-advection in creating motility, almost cancelling it for a droplet in free space as we discuss below. Our 2D study of the effect of wall friction shows however that high enough 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, our framework provides a broader generic approach to droplet motility. We can thus investigate what happens when we reverse the sign of the active stresses to consider the extensile case, (19, 20). The primary experimental relevance of this case is to suspensions of bacteria, which push fluid out along their major axes and draw it in around the equator; see Fig. 1A. In this context, w is the bacterial swim speed; although this is nonzero for motile 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 of such organisms can be created either by inducing an attraction between them (as our model effectively assumes) (22) or perhaps by confining them in an emulsion droplet. To attain nonzero P one further requires these particles to have net polar order [as opposed to a nematic state, for which there is orientational order, but equal numbers of particles with tangent ± p locally (see SI Appendix)]. Setting aside the possible difficulties in meeting all those requirements experimentally, we predict that such “shaker” bacterial droplets could again break the symmetry and start moving spontaneously. Intriguingly, the predicted trajectories in this extensile case are more complicated than those of the contractile model. This intricacy arises because, in extensile droplets, the SSB-mediated velocity forcing is in a direction perpendicular, rather than parallel, to the polarization vector P. Restoring nonzero w to describe the case of motile bacteria, the composition of these two motions leads to circular or spiralling trajectories of the droplet as a whole.

Results

We first present results for the 2D contractile case, then briefly describe our findings in 3D, and finally give some further 2D results on extensile droplets. We initialized our simulation runs with a circular (or spherical) droplet within which the concentration of active material is taken to be a constant (c = c₀), with c = 0 outside. The polarization field P(r,t) within the droplet is initially uniform along the horizontal () axis and varies with concentration as magnitude with c_cr a threshold value for polarization onset. We choose c₀ > c_cr > 0 so that |P| is nonzero inside the droplet and zero (isotropic) outside, so that we assume the actomyosin network inside our active droplets is initially polarized. The system is then evolved via the equations of motion as specified in Materials and Methods (discussed further in SI Appendix: Section 1), with chosen values of the activity parameter and the self-advection (treadmilling) parameter w. On a relatively short time scale both the internal concentration and the polarization field relax towards the equilibrium values c_eq and P_eq (while still c = 0 and P = 0 externally) with some interfacial tension , set by minimization of our chosen free energy. (Our choice creates no anchoring of P at the surface so, in the absence of symmetry breaking, the polarization remains uniform; see Fig. 2A left. The effect of a soft anchoring is discussed in the SI Appendix: Section 1). Having made one such relaxed droplet, the dependence of its behavior on and/or w was systematically explored by incrementing those quantities and waiting for steady state, before incrementing again. Because we are primarily interested in trends and symmetry breaking phenomena, rather than quantitative predictions of where these will occur for specific materials, we report all results below in the natural units for lattice Boltzmann simulations (LBU); the connection between these and physical units is discussed in SI Appendix: Section 4.

Fig. 2.

(A) Steady state configurations of a contractile active droplet without self-advection. The (red) arrows show the polarization field P(r,t). Upon increasing the contractile activity , the droplet elongates perpendicular to P and then becomes unstable with respect to splay deformation at critical activity . When it splays, the droplet also spontaneously moves in the direction of the green arrow. The time evolution of the system is shown in Movie S1. (B) Left plot shows the velocity field of the droplet at which is quadrupolar, like that around a contractile element (Fig. 1A right). Right plot shows the velocity field of the splayed and moving active droplet which consists of two opposing vortices. The boundary of the droplet itself is given by the dashed line. (C) Bifurcation diagram showing spontaneous symmetry breaking from a uniform and stationary state to a splayed and moving state as the activity parameter is increased.

Contractile Stress Can Create Motility via SSB.

Although contractile motor stress and actin treadmilling are generally both present in motile cells, it is illuminating to study these two mechanisms separately. We first consider a droplet with no external drag term (no wall friction) and no treadmilling term (w = 0), and vary the activity parameter . For low activity , below some critical value , the droplet polarization field P remains aligned uniformly along its initial direction and the droplet remains stationary. However it becomes slightly elongated in the direction perpendicular to the polarization vector P as a result of the competition between the contractile stress and the interfacial tension (see Fig. 2A middle). In this regime, the contractile stress sets up a quadrupolar fluid flow around the droplet (Fig. 2B left), so that the whole droplet behaves as a large contractile element (compare Fig. 1A right). However it does not translate: there is no motility, and the droplet is a “shaker” rather than a “mover” (23).

As we increase beyond , the uniform polarization field P becomes unstable with respect to a splay deformation. This instability happens because the contractile stress is large enough to overcome the resistance to deformation mediated by the elastic constant κ. The splay creates a state in which neighboring vectors P either fan outwards (∇·P > 0) or inwards (∇·P < 0). The first is shown in Fig. 2A right; the second is found by first taking its mirror image and then reversing P. This choice is made at random, spontaneously breaking the global polarity inversion symmetry. As soon as this SSB happens, the droplet starts to move along the direction set by (∇·P)P =  ± P. This motion is attributable to the formation of a pair of flow vortices inside the droplet (Fig. 2B right). Such spontaneous propulsion is somewhat reminiscent of the self-electrophoretic motion of a vesicle with active membrane pumps in an ionic solution (24, 25).

We note that in bulk active contractile fluids, the state of uniform 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 here can be viewed as a manifestation of that bulk instability, albeit with two variations. First, spontaneous flows are present on both sides of our transition: as discussed above there is a quadrupolar flow field already in the nonmoving state. Second, in bulk the critical activity level is nonzero only in finite systems, for which the transition 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 more clearly, we plot the magnitude V_CM of the center of mass velocity of the droplet V_CM as a function of activity in Fig. 2C. This bifurcation diagram shows a continuous nonequilibrium transition from a stationary and uniform state to a moving and splayed state. Moreover, to within numerical accuracy the observations are consistent with a supercritical Hopf bifurcation, for which . This mean-field like exponent is perhaps unsurprising as there is no noise in our simulations. Accordingly it might change in the presence of activity-generated noise (28), depending on whether the bifurcation remains low-dimensional or acquires a many-body critical character.

SSB-Induced Motility Is Diminished by Friction.

The results above are for a droplet in 2D surrounded by Newtonian fluid. To better describe experiments involving cell-crawling on a substrate, we now consider an additional frictional force between the solid wall and the cell. To add friction within our 2D continuum model, we introduce an additional force density f_friction = -γv to the momentum balance equation (see Materials and Methods and SI Appendix). Here γ is an effective coefficient of friction which depends on whether we have nonslip or partial-slip boundary conditions on the substrate and also on the thickness of the cell. It may represent conventional friction and/or a coarse-grained model of focal adhesions and other localized mechanical contacts. The presence of this friction will significantly quench the hydrodynamic flow v. Because our SSB-induced motility requires a hydrodynamic vortex flow inside the droplet, the frictional force can greatly reduce the droplet’s velocity, bringing it to rest beyond a critical level γ_c, at which point the symmetry is restored; the value of γ_c depends on activity and other model parameters. This behavior is apparent from a plot of V_CM against γ in Fig. 3.

Fig. 3.

Representative plots of droplet velocity V_CM against frictional coefficient γ for motile droplets driven by: contractile stress only (solid red line) and polymerization/self-advection only (dot-dashed blue line). The inset shows the polymerization-only driven motility in the limit of large friction in which the droplet velocity approaches “polymerization” speed wP.

Self-Advective Motility Is Enhanced by Friction.

We now consider the case when there is only treadmilling (modeled as self-advection w > 0) but no contractile stress (). As discussed previously, the droplet will move along the polarization vector P with speed V_CM < w. Because there is no spontaneous symmetry breaking involved, translational motion occurs for any nonzero w, in contrast 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 V_CM as a function of γ in Fig. 3. In contrast to the previous case, motility is significantly enhanced by the presence of friction. Indeed, in the limit γ → ∞, we have v = 0 and V_CM → 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) defines a characteristic friction scale . For contractile stresses dominate cell motility, while for , 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 other hand, some recent experiments (29) directly identify spontaneous symmetry breaking of the actomyosin network as the initiator of polarized cell motility in keratocytes. Our work emphasizes that spontaneous breaking of global polarity inversion symmetry arises from contractile motor activity, not from treadmilling. It is therefore arguable that the role of motor activity in 2D motility has so far been underestimated. We note however that an equivalent discrete symmetry breaking would create motility if a pure treadmilling state of zero , but finite nematic order (), spontaneously acquires polarity locally (see SI Appendix).

3D Droplets Show a Window of SSB-Induced Motility.

In the context of experiments on 3D tumor cells, it has been argued that motility is driven primarily by contractile stress (2, 7), suggesting that the low-friction limit of our model prevails here. This possibility accords with the much diminished part played by focal adhesions in 3D (8). For the 3D case we therefore neglect the friction term, and run selected simulations to confirm that the SSB route to motility remains operative.

Fig. 4 shows steady state polarization fields inside a 3D contractile droplet with increasing values of the activity magnitude from A to D. As expected, the first steady state encountered is a symmetric but deformed immotile droplet (Fig. 4A). The active stress contracts the droplet along ± P resulting in a lenticular shape. As we increase beyond a critical , splay instability spontaneously breaks symmetry and causes the droplet to move along (∇.P)P just as in the 2D case. The droplet shape is concave and (as in the lenticular case) both it and the flow field resemble that created by rotating the 2D droplet about the P axis (Fig. 4B). The resulting hydrodynamic flow (Fig. 4E) therefore corresponds to a toroidal vortex ring. It would be interesting to see how these predictions compare with intracellular actin and fluid flow maps that might in future be measured for cells moving in 3D environments, for instance those studied in ref. 2.

Fig. 4.

Steady state conformations in 3D contractile droplets without self-advection on increasing activity from (A) to (D). (B) and (C) are motile as indicated while (A) and (D) are stationary. (E) shows the toroidal fluid flow inside the motile droplet of steady state B. The time evolution of the droplets in (B) and (D) are shown in Movies S2 and S3 respectively.

Interestingly, as is increased further, the droplet becomes increasingly 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)] at the center (Fig. 4D).

Extensile SSB Creates Transverse or Circular Motility.

Our final results are for 2D extensile droplets (Fig. 1A). Continuum descriptions of uniform extensile active fluids are widely used to describe dense bacterial suspensions (19, 23, 31). These results may therefore be relevant to bacterial droplets formed by aggregation in the presence of attractive forces (22), or possibly by confinement of bacteria within a droplet emulsion. Again, for fixed thermodynamic parameters controlling the elasticity and interfacial tension of the confined material, one can vary the activity parameter (now positive) and the self-advection parameter w which corresponds to the swimming speed of individual bacteria and is nonzero for “movers” but zero for “shakers” (19, 26).

Fig. 5A shows steady state configurations of a 2D purely extensile (w = 0) droplet at different ranges of activity. For ζ < ζ_c1 the droplet remains stationary but again elongates symmetrically, this time along the direction of the polarization field ± P. The extensile stress creates the quadrupolar flow field expected of a large, extensile shaker. For ζ beyond the critical value ζ_c1, the droplet again becomes unstable, but now with respect to bend deformation as opposed to splay. This instability gives rise to a horizontal vortex pair inside the droplet as opposed to the vertical one in the contractile case (Fig. 2B right). This flow field causes the droplet to move in a direction set by the sense of bending as P × (∇ × P) (which can be upwards or downwards according to Fig. 5A). As can be seen from Fig. 5A this stationary-to-motile transition can still be characterized as a continuous SSB transition, however, the droplet speed V_CM attains a maximum at intermediate ζ before falling to zero again. Beyond a second critical activity threshold (ζ_c2 in Fig. 5A), the polarization pattern oscillates continuously while remaining symmetric at all times and consequently, the droplet again becomes nonmotile.

Fig. 5.

(A) Plot of center of mass velocity against activity () for extensile droplet without self-advection. It shows continuous transitions from stationary to motile and then from motile to oscillatory at critical activity ζ_c1 and ζ_c2 respectively. Also shown are the steady state polarization field P for the stationary and motile case. The movie of the time evolution of the system is shown in Movie S4. (B) The presence of both extensile stress and self-advection leads to an outward spiral trajectory (solid green lines). Also shown are the snapshots of the polarization field at different timesteps (red arrows).

Turning finally to the case of nonzero self-advection (w > 0) we find that this, combined with the extensile motility, can give rise to an intriguing spiralling motion (see Fig. 5B). This motion arises because the SSB-induced motility is at right angles to the polarization (Fig. 5B right). On the other hand, self-advection everywhere transports material along P(r,t) locally; because the polarization field is spontaneously curved, this by itself would give a circular droplet orbit. Compounding these two motions typically leads to outward spiral trajectories as shown in Fig. 5B. This outcome contrasts with the contractile case where both the SSB-induced and the self-advective motion (the latter averaged over the droplet configuration) point either together or oppositely along the ± P direction and only straight line motion can result.

Discussion and Conclusions

Our simulations of contractile droplets can be viewed as a simple in silico analogue of the in vitro cell-extract studies that have been used to dissect the biophysical ingredients of motility. Our simulation work powerfully complements these studies, by allowing us to isolate the role of contractile (motor) activity in cell locomotion: this is very difficult in the laboratory, where current strategies for inhibiting polymerization dynamics (treadmilling) can severely impair other key subcellular processes (12).

Our approach likewise complements those of previous theories (14, 15, 32) which have concentrated on treadmilling as the main driver of motility. We have shown in 2D that contractile stresses alone can not only shape the rear of a crawling cell (Fig. 2), but also create motility itself, provided that the motor forces are large enough to create an asymmetric circulatory flow as in Fig. 2B. Interestingly, keratocyte cells crawling on glass seemingly do exploit myosin activity to set up an intracellular actin flow in the rear 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 inhibits that caused by contractile stress, lends support to the view that cell crawling on a wall is usually dominated by the treadmilling.

In contrast, in 3D cell motility, recent work suggests that treadmilling 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 by the action of a contractile stress. The onset of motility requires spontaneous symmetry breaking, mediated in our case by splay deformation in response to that stress. Our 2D and 3D simulations go beyond the 2D theory of ref. 7 by addressing the dynamics of the polarization field. We do however make some important simplifications: our droplets are confined only by interfacial tension not by an elastic membrane; we treat treadmilling as a simple self-advection; and we do not address any direct transition between nematic and polar order, despite assuming polarity inversion symmetry at thermodynamic level. Improving the model in these respects will require a more detailed microscopic derivation which we shall leave to future work. To test whether our model indeed captures the biophysics of 3D cell motility, it would be exciting to visualize experimentally the detailed cytoskeletal organization and flow fields; e.g., for cells moving through matrigel (2, 7).

Our generic framework is not limited to contractile actomyosin networks. Indeed we have discussed the case of extensile droplets, possibly relevant to aggregates or emulsions of active but immotile bacteria; here translational motility arises by spontaneous symmetry breaking only at intermediate activity, and is mediated by bend rather than splay deformation. The addition of self-advection along the bent polarization field then leads in addition to rotational motion. We note that rotational and translational motility of small bacterial aggregates was recently observed, but attributed to a somewhat different mechanism where symmetry is broken by frozen-in statistical fluctuations rather than SSB (22).

Finally, our hydrodynamic equations of motion, or close variants of these, might in some cases be applicable to concentrated eukaryotic cell masses such as tissue (33). Within a tissue each cell exerts forces on its neighbors which at the lowest order continuum level creates a certain density of force dipoles (33); the velocity field v then describes the slow migration of cells inside the tissue. It is intriguing to note that the large-scale tissue flow in animal cells during gastrulation may break the symmetry to form vortices, similarly to our active droplets. An example is the case of the so-called “polonaise movements” which are observed in the developing chick embryo (1), and which are important to form the correct supercellular structure. In this context a “polarization” field is sometimes used to describe the orientation of individual cells (34). The relation, if any, between the onset of this vortex flow and that seen in our droplets remains to be explored.

Materials and Methods

We briefly outline here the hydrodynamic model used in this work (more details are in SI Appendix). We consider a fluid, comprising a mixture of active material and solvent, with constant total mass density ρ. The hydrodynamic variables whose dynamics we monitor are: (i) the concentration of active material c(r,t), (ii) the fluid velocity v(r,t) (with incompressibility requiring ∇.v = 0), and (iii) the polarization field as defined previously.

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

[1]

Here α > 0 is a phenomenological free energy amplitude, k determines the droplet interfacial tension, and κ is an effective elastic constant. This choice of F[c,P] leads to a continuous isotropic-to-polar transition at c = c_cr. To confine the active material into a droplet, we choose: and set c₀ > c_cr. This potential creates two free energy minima corresponding to a phase of pure passive solvent (external to the droplet, c = 0 and P = 0) and a polar active phase (inside the droplet, c = c_eq > c_cr and P = P_eq).

Treating the active material as locally conserved, the time evolution of the concentration field c(r,t) can then be written as a convective-diffusion equation:

[2]

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

The dynamics of the polarization field P(r,t) follows an “active nematic” evolution (20), given by

[3]

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

Force balance is ensured through the Navier-Stokes equation, where P is the isotropic pressure, -γv is the friction force per unit volume and is the total stress in the fluid which includes viscous, elastic/Ericksen, interfacial and “active” stresses (see SI 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 (23) where for contractile activity and for extensile.

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