Activity pulses induce spontaneous flow reversals in viscoelastic environments
Emmanuel L. C. VI M. Plan, Julia M. Yeomans, Amin Doostmohammadi
AActivity pulses induce spontaneous flow reversals in viscoelastic environments
Emmanuel L. C. VI M. Plan
Institute of Theoretical and Applied Research,Duy Tan University, Ha Noi 100 000, Viet Nam andFaculty of Natural Science, Duy Tan University, Da Nang 550 000, Viet Nam
Julia M. Yeomans
The Rudolf Peierls Centre for Theoretical Physics,Department of Physics, University of Oxford,Clarendon Laboratory, Oxford OX1 3PU, United Kingdom
Amin Doostmohammadi
The Niels Bohr Institute, University of Copenhagen,Blegdamsvej 17, 2100 Copenhagen, Denmark (Dated: February 8, 2021)Complex interactions between cellular systems and their surrounding extracellular matrices areemerging as important mechanical regulators of cell functions such as proliferation, motility, andcell death, and such cellular systems are often characterized by pulsating acto-myosin activities.Here, using an active gel model, we numerically explore the spontaneous flow generation by activitypulses in the presence of a viscoelastic medium. The results show that cross-talk between theactivity-induced deformations of the viscoelastic surroundings with the time-dependent response ofthe active medium to these deformations can lead to the reversal of spontaneously generated activeflows. We explain the mechanism behind this phenomenon based on the interaction between theactive flow and the viscoelastic medium. We show the importance of relaxation timescales of boththe polymers and the active particles and provide a phase-space over which such spontaneous flowreversals can be observed. Our results suggest new experiments investigating the role of controlledpulses of activity in living systems ensnared in complex mircoenvironments.
I. INTRODUCTION
The study of biological systems as active materials has made tremendous advances in the past decades [1–5]. The ‘activity’ describes the ability of living systems to extract chemical energy from their surroundingenvironment and convert it into mechanical work. This happens at the level of individual constituents ofthe matter in, for example, sperm cells thrusting forwards by the rotation of their flagella, bacterial self-propulsion, eukaryotic cells migrating within extracellular networks, and the cytoskeletal machinery insidecells that is powered by motor proteins. As such, the overarching theme in various kinds of living systems isthe local activity generation that drives the entire system far from thermodynamic equilibrium, resulting inthe collective patterns of motion observed in cellular tissues, bacterial colonies, and sub-cellular flows [3, 6, 7].The cross-talk between the mechanical micro-environment of living matter and this intrinsic ability ofliving systems to actively generate self-sustained motion governs pattern formation and self-organization inimportant biological processes including collective transport of sperm cells in confined tubes [8], shapingbacterial biofilms [9, 10], tissue regeneration [11], and sculpting organ development [12]. Not only doesthe mechanical micro-environment provide geometrical constraints for active materials [13], it is also oftenendowed with viscoelastic properties that allow for time-dependent responses to activity-induced stresses anddeformations [2, 14]. Significant examples are the extracellular matrices, surrounding cells and tissues thatplay a key role in cell death and proliferation, stem cell differentiation, cancer migration, and cell responseto drugs [15]. It is thus essential to understand the dynamic interconnection between the activity-inducedstresses and the mechanical response of the viscoelastic medium.Indeed, several recent studies have taken first attempts in this direction, showing that accounting forviscoelastic effects of the medium around living matter results in significant modification in the patterns ofmotion generated by continuous activity-induced stresses [16–22]. Importantly, however, in various biologicalcontexts the activity generation is not constant and continuous, but is rather characterized by changes in theactivity level and even activity pulses. Striking examples are the well-documented acto-myosin contractilitypulses that power the activity of epithelial cells and have been shown to be essential in tissue elongationduring development [23–25]. Therefore, here we examine the impact of activity pulses on the behavior of a r X i v : . [ c ond - m a t . s o f t ] F e b active matter surrounded by a viscoelastic medium.In order to investigate the fundamental impact of activity pulses we employ a continuum model of activematter based on the theory of active gels, which has proven very successful in describing several aspects of thephysics of active systems including acto-myosin dynamics at the cell cortex [5, 26], acto-myosin induced cellmotility [27, 28], actin retrograde flows [29, 30], and the topological characteristics of actin filaments [31, 32].One important prediction of active gel models is the emergence of spontaneous flow generation in a confinedactive gel [33], which has been further experimentally validated in different biological systems [34, 35] and isa generic feature of confined active materials. Here, we consider a simplified setup of an active gel confinedwithin viscoelastic surroundings and study the emergence of a spontaneous flow by introducing activity pulsesand varying the relaxation time of the viscoelastic medium. We show that introducing activity pulses canresult in the reversal of the spontaneous flow direction accompanied by the rearrangement of the orientationof active constituents. The mechanistic basis for this reversal is explained based on the feedback betweenthe active flows and the viscoelastic deformation, particularly in between activity pulses. We further providea simple model that reproduces the essential dynamics of the flow reversal and shows its dependence on therelevant time scales through a stability-diagram.The paper is organized as follows. In section II we describe the details of the simulation setup and introducethe governing equations of motion for the active gel, the surrounding viscoelastic medium, and the couplingbetween the two. The results of the simulation together with the physical mechanism of flow reversal andits associated phase-space are presented in section III. Finally concluding remarks and a discussion of thebroader impacts of the results are provided in section IV. II. METHODS
The active gel is simulated in two dimensions as a horizontal stripe within a passive viscoelastic region,and is differentiated via an indicator function φ whose value is φ = 1 in the active region and φ = 0 inthe passive viscoelastic region. The indicator function φ is only defined to distinguish between the activeand passive region and as such it is fixed, without any dynamical evolution. Activity and viscoelasticityare incorporated by introducing a generic two-phase model of active matter in viscoelastic domains [22, 36]which is summarised below. Active region
Following its success in describing the dynamics of the cell cytoskeleton, bacterial colonies and confluentcell layers, we use liquid crystal nematohydrodynamics to model the active region [4, 37, 38]. Within thisframework each active particle generates a dipolar flow field with axis along its direction of alignment. Thealignment direction is nematic, i.e. it has a head-tail symmetry. This can be captured on a coarse-grainedlevel by the order parameter tensor Q through its principal eigenvector, which describes the nematic directororientation, and the associated eigenvalue, which describes the degree of alignment.The free energy f Q for two-phase nematic models follows the Landau-De Gennes description f Q = A Q (cid:20) (cid:18) − η ( φ )3 (cid:19) Tr (cid:2) Q (cid:3) − η ( φ )3 Tr (cid:2) Q (cid:3) + η ( φ )4 Tr (cid:2) Q (cid:3) (cid:21) + K Q ∇ Q ) + L ( ∇ φ · Q · ∇ φ ) (1)where A Q describes the stability of the nematic or isotropic configurations, with the former being favouredwhen η > .
7. The elastic coefficient K Q penalizes gradients in Q , and a positive (negative) L enforcesnematic orientation parallel (perpendicular) to the active-viscoelastic interface.In the presence of a velocity field u , the nematic tensor is evolved according to the equation ∂ t Q + u ·∇ Q = S Q + Γ Q H Q where the left-hand side is the usual material advective derivative. The co-rotational term S Q = ( ξ D + Ω )( Q + I /
3) + ( Q + I / ξ D − Ω ) − ξ ( Q + I /
3) Tr [ Q ∇ u ] describes nematic reorientation inresponse to both vorticity Ω and flow strain D , with the tumbling parameter ξ determining whether thedirectors align or tumble in the flow. Γ Q controls the speed of relaxation towards the free energy minimumdetermined by the molecular field H Q = − δf Q /δ Q . The typical nematic relaxation timescale t n whenconfined in a channel of width W is given by W / Γ Q K Q and the dynamical equation for Q can thus berewritten as ∂ t Q + u · ∇ Q = S Q + ( W /t n K Q ) H Q . (2)Since individual components of the active region generate dipolar forces with axis along their direction ofalignment, the corresponding active stress is proportional to the orientation tensor [1, 30, 37] σ active = − ζφ Q (3)such that gradients in the orientation field generate forces on the fluid and drive active flows. The activityparameter, ζ , measures the strength of the active driving. Viscoelastic region
The passive region is endowed with viscoelasticity that is described by the conformation tensor C , whichcharacterizes the polymer orientation by its principal eigenvector, and the (square of the) polymer length, byits trace. Here we use the Oldroyd-B model to reproduce simple viscoelastic effects, i.e. polymer relaxationlinear with respect to the elongation, and governed by a single relaxation time τ [39].The free energy associated with an Oldroyd fluid f C = A C (1 − φ )(Tr [ C − I ] − ln det C ) / , (4)governs the polymer relaxation to its unstretched equilibrium C = I (where I is the identity tensor). Here,the modulus of elasticity A C = ν/τ is the ratio of the polymer contribution to viscosity ν to the polymerrelaxation time τ . The corresponding molecular field H C = − δf C /δ C , appears in the dynamical equationgoverning the evolution of C : ∂ t C + u · ∇ C = S C + Γ C [ H C C + C (cid:62) H (cid:62) C ], where (cid:62) denotes the matrixtranspose. Similarly to Eq. (2) for the orientation tensor Q , the evolution of C accounts for the advectionof the polymer conformation C , its response to velocity gradients through S C = CΩ − ΩC + CD + D (cid:62) C (cid:62) ,and its relaxation to equilibrium at a rate Γ C = ν − . The viscoelastic timescale is clearly seen when thisequation of motion is simplified as ∂ t C + u · ∇ C = S C − τ (1 − φ )( C − I ) . (5)The polymer contribution to the stress, within the assumptions of the Oldroyd-B model, is σ polymer = A C (1 − φ )( C − I ) , (6)where the factor 1 − φ ensures that the polymer stress only acts within the passive region (where φ = 0). Coupling and simulation details
The active and viscoelastic regions interact with each other through the velocity field u which obeys theincompressible Navier-Stokes equations, ρ ( ∂ t u + u · ∇ u ) = − ∇ p + ∇ · σ , ( ∇ · u = ) , (7)where ρ is the fluid density, p is the pressure, and σ is the sum of viscous, capillary, elastic, active (Eq. (3)),and polymer (Eq. (6)) stresses. Both the active and the viscoelastic regions exert stresses on the fluid, andthe resulting velocity field couples the two regions through advective and co-rotational terms in Eqs. (2) and(5). The explicit forms of the other stresses are given in [22].Equations (2),(5), and (7) are evolved using a hybrid lattice Boltzmann method [40, 41]. The simulationdomain has dimensions L × H , is periodic in the x -direction, and has no-slip boundary conditions in the y -direction (see schematic in Fig. 1(a)). The dynamics is not affected by the length of the channel L becauseof periodicity; our results here use L = 10 , H = 100. The width of the active region is fixed at W = 20.In order to obtain a unidirectional flow the parameters for the active region are chosen to be A Q = 1 . K Q = 0 .
2, and we use ξ = 0 . η ( φ ) is chosen such that η = 2 .
95 within the active region φ = 1 [42], while the passive viscoelastic region, φ = 0, is in the isotropicphase with η < .
7. The polymer contribution to viscosity is fixed at ν = 1 and A C = 1 /τ . The value ofthe flow parameters are ρ = 1 , p = 0 .
25, and the Newtonian contribution to the viscosity is ν flow = 2 /
3. We a bcd
FIG. 1.
Schematic of problem set up and flow reversal mechanism. (a) The simulation domain indicating theactive region (gold) with black nematic directors and the viscoelastic region (silver) with polymers in grey. A steadyshear-like flow in both regions is shown in red ( ζ = 0 . , τ = 5000 , t n = 4000). (b)-(d) The mechanism behind theflow reversal: (b) The flow stretches the polymers and orients both polymers and nematic directors ( θ n ) to the Leslieangle; (c) When activity is turned off, the polymers relax and create a weaker but reversed flow, which reorients thedirectors; (d) When activity resumes, the directors either return to their original steady state or reverse directiondepending on the balance between their orientation and the residual polymer stress. also enforce weak nematic anchoring at the boundary, L = 0 .
05, for stability. This was verified to have noqualitative effect on the flow reversal dynamics.Each simulation begins with equilibriated polymers C = I and nematic directors with small randomperturbation about the x -axis. Active stresses are applied until the system establishes a steady-state, unidi-rectional flow, after which the activity is temporarily turned off at t off = 10 for a duration of d = t on − t off time steps. The equations are solved until the flow is re-established. III. RESULTS
When activity is turned off at a time t off , and turned back on at a time t on , the flow is re-established inthe same or, surprisingly, in the opposite direction. This is not a random choice but depends sensitively onthe parameters setting the relevant nematic and viscoelastic time scales. For example, the supplementaryMovie shows several successive reversals in the direction of the velocity (see Data Availability). We firstexplain how this dependence comes about, and then present a simple model which illustrates the underlyingphysics. The mechanism of the flow reversal is summarised in Fig. 1(b)–(d).Fig. 2 shows the variation of the mean velocity in the channel, (cid:104) u x (cid:105) , and the angle θ n that the meandirector field at the interface makes with the channel axis as a function of time for selected simulationparameters. Consider first Fig. 2(a) where there are no polymers in the passive region. Activity is switchedon at time t = 0. This drives the active nematic instability and active stresses resulting from gradients inthe director field set up a linear flow along the stripe. The spontaneous flow is stabilised by the channelinterfaces, and the resulting steady-state flow profile corresponds to directors aligning at the Leslie angle tothe local shear [43, 44]. When the activity is switched off the flow decays faster than the relaxation time ofthe nematic director. In the absence of polymers if activity is switched back on before the end of the decaythe residual director rotation ensures that the flow re-starts in the same direction as before.The presence of polymers changes the response of flow to activity pulses. In Fig. 2(b)–(d) the passivemedium is viscoelastic and the shear flow induced by the activity stretches the polymers thus storing energy.The initial build-up of the flow is similar to the no polymer case, but slower, as work is done to stretchthe polymers, and the steady-state value of the flow in the channel is lower because the polymers imposestresses that oppose the flow. This also means that after t off the decays of the velocity and director fields toequilibrium are faster and, in particular, stress imposed by the relaxation of the surrounding polymers maybe strong enough to reverse the flow in the channel.For example in Fig. 2(b) both the residual flow and the director have reversed at t on making a velocityreversal inevitable. By comparison, for the same polymer relaxation time but faster switching of activity inFig. 2(c), at t on the director distortion is still just positive but the velocity has been reversed by the polymerstresses. The activity tries to rebuild the flow in the original direction whereas the remaining elastic energystored in the surrounding polymers pushes the fluid in the opposite direction. The flow slowly reverses asthe instability is (just) overcome by the residual polymer stresses. The director distortion reverses and thevelocity slowly increases until it eventually attains its steady-state value, but in the opposite direction. Bycontrast Fig. 2(d) shows an example of faster polymer relaxation, compared to Fig. 2(c), where the polymerstress is not sufficiently strong to cause flow re-orientation and both nematic directors and mean velocityregain their original direction.Such a flow reversal mechanism that depends on stored polymer stresses and residual director orientationimplies that the relevant timescales here are the polymer relaxation time τ , the nematic relaxation time t n and the period of inactivity d = t on − t off . To further explain the phenomenon of flow reversal and highlightthe competition between these varying time scales, we construct simplified, space-independent, dynamicequations for the evolution of the active nematic and polymeric particles.To this end, we consider the dynamics of the alignment for the angles θ n , θ p formed by the active nematicdirectors and the polymer directors, respectively, with respect to the direction of a simple shear flow, u =( ˙ γy,
0) [45]: dθ n dt = ˙ γξ n cos 2 θ n − t n θ n , (8) dθ p dt = ˙ γξ p cos 2 θ p − τ θ p . (9)These equations are approximations of the orientation of rod-like particles with tumbling parameters ξ n , ξ p in response to a shear flow with rate ˙ γ . In the absence of shear, ˙ γ = 0, the angles will relax exponentially tozero.A time-dependent shear rate ˙ γ couples the two equations and can be determined from balancing theviscous stress σ viscous = 2 ν flow D , with the active (Eq. (3)), and polymer (Eq. (6)) stresses. Solving for theoff-diagonal term of the rate of strain tensor D and constructing the tensors Q ij = n i n j − δ ij / C ij = p i p j from the unit directors n = (cos θ n , sin θ n ) (and analogously for p ), we obtain the simple time-dependentshear ˙ γ = D xy = 12 ν flow (cid:16) ζQ xy − ντ C xy (cid:17) = 14 ν flow (cid:16) ζ sin 2 θ n − ντ sin 2 θ p (cid:17) , (10)in terms of the nematic and polymer directors. For simplicity, we fix ν flow = 2 / , ν = 1 and set ξ n = 1 . ξ p = 0 .
275 and take initial conditions θ n = θ p = 0 . ζ = 0 is turned off, the flow reverses in the opposite directiondue to the presence of the polymers. Moreover, a simulation of this simpler set of equations features the same -0.01-0.00500.0050.01 M ean v e l o c i t y u x a c t a: No Polymersb: d=10000, =10000c: d=5000, =10000d: d=5000, =3000 Time -4-2024 N e m a t i c ang l e n abc dabc dabc d Time -15-10-5051015 N e m a t i c ang l e n ( S i m p li f i ed m ode l ) e: d=400, =600f : d=200, =600g: d=200, =300 e f g FIG. 2.
Emergence of spontaneous flow reversals. (Top) The mean velocity (cid:104) u x (cid:105) act in the active stripe and(middle) the nematic angle at the interface θ n as a function of time for the full simulations ( t n = 4000), and (bottom)the nematic angle for the simplified model ( t n = 150). (a) black: no polymers, activity switching time d = 10 ; (b)red: activity switching time d = 10 , polymer relaxation time τ = 10 ; (c) dashed: activity switching time d = 5000,polymer relaxation time τ = 10 ; (d) blue: activity switching time d = 5000, polymer relaxation time τ = 3000. Thecyan curves show the behaviour in the absence of reactivation. (e) red: activity switching time d = 400, polymerrelaxation time τ = 600; (f) dashed: activity switching time d = 200, polymer relaxation time τ = 600; (g) blue:activity switching time d = 200, polymer relaxation time τ = 300. The insets show a zoom in on the period ofinactivity and immediately after reactivation. essential reason for the reversal as in the full simulations: there exists sufficient polymer stress to reorientthe nematics during the period between activity pulses. Figures 2(e)–(g) show that the nematic angle θ n exhibits trajectories similar to the result from the full equations of motion, Figs. 2(b)–(d). The reversedflow can drive the nematic angle negative during the period of inactivity, which is a sufficient condition for aflow reversal. Moreover, even if θ n remains positive when the activity is switched on, the polymer stress canstill overcome the activity (i.e. ζ sin 2 θ n < ν sin 2 θ p /τ ) and continue to drive the nematic director to reverse /t n d /t n bbbb FIG. 3.
Phase diagram of the (a) simplified equations and (b) full simulations.
Blue points indicate flowreversal while red points indicate non-reversal. Different markers represent different nematic relaxation timescales:(a) t n = 140 ( (cid:53) ); t n = 150 (+); t n = 160 ( (cid:3) ); t n = 180 ( (cid:5) ); (b) t n ≈ (cid:52) ); t n = 5000 (+); t n = 4000 ( × ); t n ≈ ◦ ). The dashed lines are (a) τ /t n > . , d/t n > . , dτ /t n > τ /t n > . , d/t n > . , dτ /t n > . direction.Using the simple model we examined the phase space of flow reversal for varying time scales of nematicand polymer relaxations, t n and τ respectively, as well as the activity switching time d = t on − t off (Fig. 3(a)).In particular, we observe that the reversal can be ensured if three dimensionless ratios are sufficiently large:(i) d/t n : nematic directors have enough time to relax(ii) τ /t n : polymers retain enough energy to reverse nematic orientation when θ n ≈ dτ /t n : residual polymer stress upon moment of reactivation can help reverse nematic orientation ifcondition (i) alone is insufficientThese three constraints are illustrated as dashed black lines on the flow reversal phase diagram in the d/t n - τ /t n phase space and clearly distinguish the parameter space with ( blue markers ) and without ( red markers )flow reversal.We also plot, in Fig. 3(b), the similar phase diagram obtained from full simulations of the active gel sur-rounded by a viscoelastic medium showing close qualitative agreement with the simple model, and indicatingthat the parameter region for flow reversal is suitably captured by these three constraints. The quantitativedifference between the simplified model and the full simulations is expected because in the simplified modelthere is no space dependence and we only account for polymer orientation in the polymer stress, whereas thefull simulations are space-dependent and have a polymer stress (Eq. (6)) that depends on polymer elongation,which is higher when τ is larger. Notwithstanding these limitations, the close qualitative agreement of theangle profiles and the phase diagram obtained from the simple model with those from the full hydrodynamicsimulations of spatio-temporal evolution of the active nematics and polymeric fluids confirms the importanceof the time scale constraints for the flow reversal and the underlying physics of stress balance during theperiod of inactivity. IV. CONCLUSION
In this report we present how, in the absence of any external forcing, activity pulses in living matterinteracting with a viscoelastic environment can spontaneously generate flow reversals. Based on a well-documented active gel theory, a spontaneous steady flow of active matter is achieved even while in contactwith polymer-laden surrounding. This flow stretches the polymers near the interface which, in betweenperiods of activity, relax and produce a weak backflow that may determine the flow direction upon resumptionof the active driving.The well-established spontaneous flow generation of confined active matter relies on the level of activity.Here the spontaneous flow reversals hinge on several timescales: the polymer relaxation time, the intervalbetween activity pulses, and the relaxation dynamics of nematic active matter, as shown in the phasediagram. Indeed the need for sufficient polymer stretching and feedback as well as quick nematic reorderinghighlight the time-dependent viscoelastic response in between activity pulses. Our work emphasizes not onlythe importance of accounting for a viscoelastic environment, but also the involvement of several timescalesarising from both active matter and its surroundings.Our theoretical work invites several experiments to be performed in order to confirm our predictionsand deepen our understanding of the role of a viscoelastic environment on the dynamics of living matter.For instance, cells confined in a monolayer have been shown to display nematic behaviour and exhibitunidirectional flows in channels [46]. They could be repeatedly subjected to cell inhibitors or uncouplers [47–49] to simulate gaps between activity pulses. Another option would be to regulate the movement of a colonyof elongated bacteria by utilizing phototactic methods [50]. Our result thus holds potential in understandingmechanotaxis and motivating the use of viscoelastic media to control living matter at microscopic scales.
ACKNOWLEDGEMENT AND FUNDING
We thank Sally Horne-Badovinac for helpful discussions. A.D. acknowledges support from the NovoNordisk Foundation (grant no. NNF18SA0035142), Villum Fonden (grant no. 29476), Danish Councilfor Independent Research, Natural Sciences (DFF-117155-1001), and funding from the European Union’sHorizon 2020 research and innovation program under the Marie Sk(cid:32)lodowska-Curie grant agreement no.847523 (INTERACTIONS).
DATA AVAILABILITY
The codes used for this work are available in https://github.com/elcplan/Activity_pulses_flow_reversal-Plan_Doostmohammadi_Yeomans.git . The supplementary movie (in .gif and .mp4 format) is alsoavailable in this repository.
COMPETING INTERESTS
The authors declare no competing interests.
AUTHORS CONTRIBUTION
All authors contributed in the conceptualisation, data analysis, and the preparation and submission of themanuscript. [1] Marchetti MC, Joanny JF, Ramaswamy S, Liverpool TB, Prost J, Rao M, Aditi Simha R. 2013 Hydrodynamicsof soft active matter.
Rev. Mod. Phys. , 1143-1189. (doi:10.1103/RevModPhys.85.1143)[2] Bechinger C, Di Leonardo R, L¨owen H, Reichhardt C, Volpe G, Volpe G. 2016 Active particles in complex andcrowded environments. Rev. Mod. Phys. , 045006. (doi:10.1103/RevModPhys.88.045006)[3] Needleman D, Dogic Z. 2017 Active matter at the interface between materials science and cell biology. Nat. Rev.Mater. , 17048. (doi:10.1038/natrevmats.2017.48)[4] Doostmohammadi A, Ign´es-Mullol J, Yeomans JM, Sagu´es F. 2018 Active nematics. Nat. Comm. , 3246.(doi:10.1038/s41467-018-05666-8)[5] J¨ulicher F, Grill SW, Salbreux G. 2018 Hydrodynamic theory of active matter. Rep. Prog. Phys. , 076601.(doi:10.1088/1361-6633/aab6bb)[6] Elgeti J, Winkler RG, Gompper G. 2015 Physics of microswimmers-single particle motion and collective behavior:a review. Rep. Prog. Phys. , 056601. (doi:10.1088/0034-4885/78/5/056601) [7] Ladoux B, M`ege RM. 2017 Mechanobiology of collective cell behaviours. Nat. Rev. Mol. Cell Bio. , 743-757.(doi:10.1038/nrm.2017.98)[8] Hook KA, Weber WD, Fisher HS. 2020 Post-copulatory sexual selection is associated with sperm aggregatequality in Peromyscus mice. bioRxiv. (doi:10.1101/2020.02.08.939975)[9] Vidakovic L, Singh PK, Hartmann R, Nadell CD, Drescher K. 2018 Dynamic biofilm architecture confersindividual and collective mechanisms of viral protection.
Nat. Microbiol. , 26-31. (doi:10.1038/s41564-017-0050-1)[10] Meacock OJ, Doostmohammadi A, Foster KR, Yeomans JM, Durham WM. 2020 Bacteria solve the problem ofcrowding by moving slowly. Nat. Phys. (doi:10.1038/s41567-020-01070-6)[11] Friedl P, Gilmour D. 2009 Collective cell migration in morphogenesis, regeneration and cancer.
Nat. Rev. Mol.Cell Bio. , 445-457. (doi:10.1038/nrm2720)[12] Maroudas-Sacks Y, Garion L, Shani-Zerbib L, Livshits A, Braun E, Keren K. 2020 Topological defects in thenematic order of actin fibers as organization centers of Hydra morphogenesis.
Nat. Phys. (doi:10.1038/s41567-020-01083-1)[13] Xi W, Saw TB, Delacour D, Lim CT, Ladoux B. 2019 Material approaches to active tissue mechanics.
Nat. Rev.Mater. , 23-44. (doi:10.1038/s41578-018-0066-z)[14] Liu S, Shankar S, Marchetti MC, Wu Y. 2020 Viscoelastic control of spatiotemporal order in bacterial activematter. arXiv . (2007.16206)[15] Chaudhuri O, Cooper-White J, Janmey PA, Mooney DJ, Shenoy VB. 2020 Effects of extracellular matrixviscoelasticity on cellular behaviour. Nat. , 535-546. (doi:10.1038/s41586-020-2612-2)[16] Bozorgi Y, Underhill PT. 2014 Effects of elasticity on the nonlinear collective dynamics of self-propelled particles.
J. Non-Newton. Fluid Mech. , 69-77. (doi:10.1016/j.jnnfm.2014.09.016)[17] Patteson AE, Gopinath A, Goulian M, Arratia PE. 2015 Running and tumbling with
E. coli in polymericsolutions.
Sci. Rep. , 15761. (doi:10.1038/srep15761)[18] Li G, Ardekani AM. 2016 Collective motion of microorganisms in a viscoelastic fluid. Phys. Rev. Lett. ,118001. (doi:10.1103/PhysRevLett.117.118001)[19] Z¨ottl A, Yeomans JM. 2019 Enhanced bacterial swimming speeds in macromelecular polymer solutions.
Nat.Phys. , 554-558. (doi:https://doi.org/10.1038/s41567-019-0454-3)[20] Nam S, Chaudhuri O. 2018 Mitotic cells generate protrusive extracellular forces to divide in three-dimensionalmicroenvironments. Nat. Phys. , 621–628. (doi:10.1038/s41567-018-0092-1)[21] Narinder N, Bechinger C, Gomez-Solano JR. 2018 Memory-induced transition from a persistent random walk tocircular motion for achiral microswimmers. Phys. Rev. Lett. , 078003. (doi:10.1103/PhysRevLett.121.078003)[22] Plan ELCVM, Yeomans JM, Doostmohammadi A. 2020 Active matter in a viscoelastic environment.
Phys. Rev.Fluids , 023102. (doi:10.1103/PhysRevFluids.5.023102)[23] Martin AC, Kaschube M, Wieschaus EF. 2009 Pulsed contractions of an actin-myosin network drive apicalconstriction. Nat. , 495-499. (doi:10.1038/nature07522)[24] He L, Wang X, Tang HL, Montell DJ. 2010 Tissue elongation requires oscillating contractions of a basal acto-myosin network.
Nat. Cell Bio. , 1133-1142. (doi:10.1038/ncb2124)[25] Gorfinkiel N. 2016 From actomyosin oscillations to tissue-level deformations. Dev. Dynam. , 268-275.(doi:10.1002/dvdy.24363)[26] Naganathan SR, F¨urthauer S, Nishikawa M, J¨ulicher F, Grill SW. 2014 Active torque generation by the acto-myosin cell cortex drives left-right symmetry breaking.
Elife. , e04165. (doi:10.7554/eLife.04165)[27] Tjhung E, Cates ME, Marenduzzo D. 2017 Contractile and chiral activities codetermine the helicity of swimmingdroplet trajectories. P. Natl. Acad. Sci. USA , 4631-4636. (doi:10.1073/pnas.1619960114)[28] Banerjee S, Gardel ML, Schwarz US. 2020 The actin cytoskeleton as an active adaptive material.
Annu. Rev.Cond. Matter Phys. , 421-439. (doi:10.1146/annurev-conmatphys-031218-013231)[29] J¨ulicher F, Kruse K, Prost J, Joanny JF. 2007 Active behavior of the cytoskeleton. Phys. Rep. , 3-28.(doi:10.1016/j.physrep.2007.02.018)[30] Prost J, J¨ulicher F, Joanny JF. 2015 Active gel physics.
Nat Phys. , 111-117. (doi:10.1038/nphys3224)[31] Kumar N, Zhang R, de Pablo JJ, Gardel ML. 2018 Tunable structure and dynamics of active liquid crystals. Sci. Adv. , eaat7779. (doi:10.1126/sciadv.aat7779)[32] Zhang R, Kumar N, Ross JL, Gardel ML, De Pablo JJ. 2018 Interplay of structure, elasticity, and dynamics inactin-based nematic materials. P. Natl. Acad. Sci. USA , E124-E133. (doi:10.1073/pnas.1713832115)[33] Voituriez R, Joanny JF, Prost J. 2005 Spontaneous flow transition in active polar gels.
Europhys. Lett. , 404.(doi:10.1209/epl/i2004-10501-2)[34] Duclos G, Blanch-Mercader C, Yashunsky V, Salbreux G, Joanny JF, Prost J, et al. 2018 Spontaneous shearflow in confined cellular nematics. Nat. Phys. , 728-732. (doi:10.1038/s41567-018-0099-7)[35] Hardo¨uin J, Hughes R, Doostmohammadi A, Laurent J, Lopez-Leon T, Yeomans JM, Ign´es-Mullol J, Sagu´esF. 2019 Reconfigurable flows and defect landscape of confined active nematics. Comm. Phys. , 121.(doi:10.1038/s42005-019-0221-x) [36] Hemingway EJ, Cates ME, Fielding SM. 2016 Viscoelastic and elastomeric active matter: Linear instability andnonlinear dynamics. Phys. Rev. E
93, 032702. (doi:10.1103/PhysRevE.93.032702)[37] Ramaswamy S. 2010 The mechanics and statistics of active matter.
Annu. Rev. Cond. Matter Phys. , 323-345.(doi:10.1146/annurev-conmatphys-070909-104101)[38] Doostmohammadi A, Thampi SP, Yeomans JM. 2016 Defect-mediated morphologies in growing cell colonies. Phys. Rev. Lett. , 048102. (doi:10.1103/PhysRevLett.117.048102)[39] Benzi R, Ching ESC. 2018 Polymers in Fluid Flows.
Annu. Rev. Cond. Matter Phys. , 163-181.(doi:10.1146/annurev-conmatphys-033117-053913)[40] Marenduzzo D, Orlandini E, Yeomans JM. 2007 Hydrodynamics and rheology of active liquid crystals: anumerical investigation. Phys. Rev. Lett. , 118102. (doi:10.1103/PhysRevLett.98.118102)[41] Thampi SP, Golestanian R, Yeomans JM. 2014 Instabilities and topological defects in active nematics. Europhys.Lett. , 18001. (doi:10.1209/0295-5075/105/18001)[42] Chandragiri S, Doostmohammadi A, Yeomans JM, Thampi SP. 2019 Active transport in a channel: stabilisationby flow or thermodynamics.
Soft matter , 1597-1604. (doi:10.1039/C8SM02103A)[43] De Gennes PG, Prost J. 1993 The physics of liquid crystals , 2nd edn. Oxford, NY: Oxford University Press.[44] Thijssen K, Doostmohammadi A. 2020 Binding self-propelled topological defects in active turbulence.
Phys.Rev. Res. , 042008(R). (doi:10.1103/PhysRevResearch.2.042008)[45] Aigouy B, Farhadifar R, Staple DB, Sagner A, R¨oper JC, J¨ulicher F, Sagner A. 2010 Cell flow reorients the axisof planar polarity in the wing epithelium of Drosophila . Cell , 773-786. (doi:10.1016/j.cell.2010.07.042)[46] Stedden CG, Menegas W, Zajac AL, Williams AM, Cheng S, ¨Ozkan E, Horne-Badovinac S. 2019 Planar-polarizedsemaphorin-5c and Plexin A promote the collective migration of epithelial cells in
Drosophila . Curr. Bio. ,908-920. (doi:10.1016/j.cub.2019.01.049)[47] Jacobelli J, Chmura SA, Buxton DB, Davis MM, Krummel MF. 2004 A single class II myosin modulates T cellmotility and stopping, but not synapse formation. Nat. Immunol. , 531-538. (doi:10.1038/ni1065)[48] Kolega J. 2006 The role of myosin II motor activity in distributing myosin asymmetrically and coupling protrusiveactivity to cell translocation. Mol. Bio. Cell. , 4435-4445. (doi:10.1091/mbc.e06-05-0431)[49] Dewangan NK, Conrad JC. 2020 Bacterial motility enhances adhesion to oil droplets. Soft Matter , 8237-8244.(doi:10.1039/D0SM00944J)[50] Wilde A, Mullineaux CW. 2017 Light-controlled motility in prokaryotes and the problem of directional lightperception. FEMS Microbiol. Rev.41