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The collective motion of epithelial cells is a fundamental biological process which plays a significant role in embryogenesis, wound healing and tumor metastasis. While it has been broadly investigated for over a decade both in vivo and in vitro, large scale coherent flocking phases remain underexplored and have so far been mostly described as fluid. In this work, we report a mode of large-scale collective motion for different epithelial cell types in vitro with distinctive new features. By tracking individual cells, we show that cells move over long time scales coherently not as a fluid, but as a polar elastic solid with negligible cell rearrangements. Our analysis reveals that this solid flocking phase exhibits signatures of long-range polar order, unprecedented in cellular systems, such as scale-free correlations, anomalously large density fluctuations, and shear waves. Based on a general theory of active polar solids, we argue that these features result from massless Goldstone modes, which, in contrast to polar fluids where they are generic, require the decoupling of global rotations of the polarity and in-plane elastic deformations in polar solids. We theoretically show and consistently observe in experiments that the fluctuations of elastic deformations diverge for large system size in such polar active solid phases, leading eventually to rupture and thus potentially loss of tissue integrity at large scales.

Growing experimental evidence shows that cell monolayers can induce long-lived perturbations to their environment, akin to footprints, which in turn influence the global dynamics of the system. Inspired by these observations, we propose a comprehensive theoretical framework to describe systems where an active field dynamically interacts with a non-advected footprint field, deposited by the active field. We derive the corresponding general hydrodynamics for both polar and nematic fields. Our findings reveal that the dynamic coupling to a footprint field induces remarkable effects absent in classical active hydrodynamics, such as symmetry-dependent modifications to the isotropic-ordered transition, which may manifest as either second-order or first-order, alterations in spontaneous flow transitions, potentially resulting in oscillating flows and rotating fields, and initial condition-dependent aging dynamics characterized by long-lived transient states. Our results suggest that footprint deposition could be a key mechanism determining the dynamical phases of cellular systems, or more generally active systems inducing long-lived perturbations to their environment.

We study the roughening of interfaces in phase-separated active suspensions on substrates. At large length and timescales, we show that the interfacial dynamics belongs to the |q|KPZ universality class discussed in Besse et al. Phys. Rev. Lett. 130, 187102 (2023). This holds despite the presence of long-ranged fluid flows. At early times, instead, the roughening exponents are the same as those in the presence of a momentum-conserving fluid. Surprisingly, when the effect of substrate friction can be ignored, the interface becomes random beyond a de Gennes-Taupin lengthscale which depends on the interfacial tension.

Chiral active materials display odd dynamical effects in both their elastic and viscous responses. We show that the most symmetric mesophase with two-dimensional odd elasticity in three dimensions is chiral, polar and columnar, with two-dimensional translational order in the plane perpendicular to the columns and no elastic restoring force for their relative sliding. We derive its hydrodynamic equations from those of a chiral active variant of model H. The most striking prediction of the odd dynamics is two distinct types of column oscillation whose frequencies do not vanish at zero wavenumber. In addition, activity leads to a buckling instability coming from the generic force-dipole active stress analogous to the mechanical Helfrich-Hurault instability in passive materials, while the chiral torque-dipole active stress fundamentally modifies the instability by the selection of helical column undulations., Comment: replaces arXiv:2306.03695; new results

Couplings between biochemical and mechanical processes have a profound impact on embryonic development. However, in-vitro studies capable of quantifying these interactions have remained elusive. Here, we investigate a synthetic system where a DNA reaction-diffusion (RD) front is advected by a turbulent flow generated by active matter (AM) flows in a quasi-one-dimensional geometry. Whereas the dynamics of simple RD fronts solely depend on the reaction and diffusion rates, we show that RD-AM front propagation is also influenced by the confinement geometry. We first experimentally dissected the different components of the reaction-diffusion-advection process by knocking out reaction or advection and observed how RD-AM allows for faster transport over large distances, avoiding dilution. We then show how confinement impacts active matter flow: while changes in instantaneous flow velocities are small; correlation times are dramatically increased with decreasing confinement. As a result, RD-AM front speed increased 3 to 9-fold compared to a RD one. This RD-AM experimental model provides a framework for the rational engineering of complex spatiotemporal processes observed in living systems. It will reinforce our understanding of how macro-scale patterns and structures emerge from microscopic components in non-equilibrium systems.

We formulate the hydrodynamics of active columnar phases, with two-dimensional translational order in the plane perpendicular to the columns and no elastic restoring force for relative sliding of the columns, using the general formalism of an active model H$^*$. Our predictions include: two-dimensional odd elasticity coming from three-dimensional plasmon-like oscillations of the columns in chiral polar phases with a frequency that is independent of wavenumber and non-analytic; a buckling instability coming from the generic force-dipole active stress analogous to the mechanical Helfrich-Hurault instability in passive materials; the selection of helical column undulations by apolar chiral activity.

We analytically calculate the scaling exponents of a two-dimensional KPZ-like system: coherently moving incompressible polar active fluids. Using three different renormalization group approximation schemes, we obtain values for the ``roughness" exponent $\chi$ and anisotropy exponent $\zeta$ that are extremely near the known exact results. This implies our prediction for the previously completely unknown dynamic exponent $z$ is quantitatively accurate.

We study the moving phase of two-dimensional (2D) incompressible polar active fluids in the presence of both quenched and annealed disorder. We show that long-range polar order persists even in this defect-ridden two-dimensional system. We obtain the large-distance, long-time scaling laws of the velocity fluctuations using three distinct dynamic renormalization group schemes. These are an uncontrolled one-loop calculation in exactly two dimensions, and two $d=(d_c-\epsilon)$-expansions to $O(\epsilon)$, obtained by two different analytic continuations of our 2D model to higher spatial dimensions: a ``hard" continuation which has $d_c={7\over 3}$, and a ``soft" continuation with $d_c={5\over 2}$. Surprisingly, the quenched and annealed parts of the velocity correlation function have the same anisotropy exponent and the relaxational and propagating parts of the dispersion relation have the same dynamic exponent in the nonlinear theory even though they are distinct in the linearized theory. This is due to anomalous hydrodynamics. Furthermore, all three renormalization schemes yield very similar values for the universal exponents, and, therefore, we expect the numerical values we predict for them to be highly accurate., Comment: 39 pages, 11 figures

We present a hydrodynamic theory of incompressible polar active fluids with quenched disorder. This theory shows that such fluids can overcome the disruption caused by the quenched disorder and move coherently, in the sense of having a non-zero mean velocity in the hydrodynamic limit. However, the scaling behavior of this class of active systems cannot be described by linearized hydrodynamics in spatial dimensions between 2 and 5. Nonetheless, we obtain the exact dimension-dependent scaling exponents in these dimensions., Comment: 10 pages, 1 figure

We show that incompressible polar active fluids can exhibit an ordered, coherently moving phase even in the presence of quenched disorder in two dimensions. Unlike such active fluids with annealed (i.e., time-dependent) disorder only, which behave like equilibrium ferromagnets with long-range interactions, this robustness against quenched disorder is a fundamentally non-equilibrium phenomenon. The ordered state belongs to a new universality class, whose scaling laws we calculate using three different renormalization group schemes, which all give scaling exponents within 0.02 of each other, indicating that our results are quite accurate. Our predictions can be quantitatively tested in readily available artificial active systems, and imply that biological systems such as cell layers can move coherently in vivo, where disorder is inevitable., Comment: 6 pages

Active gels made of cytoskeletal proteins are valuable materials with attractive non-equilibrium properties such as spatial self-organization and self-propulsion. At least four typical routes to spatial patterning have been reported to date in different types of cytoskeletal active gels: bending and buckling instabilities in extensile systems, and global and local contraction instabilities in contractile gels. Here we report the observation of these four instabilities in a single type of active gel and we show that they are controlled by two parameters: the concentrations of ATP and depletion agent. We demonstrate that as the ATP concentration decreases, the concentration of passive motors increases until the gel undergoes a gelation transition. At this point, buckling is selected against bending, while global contraction is favored over local ones. Our observations are coherent with a hydrodynamic model of a viscoelastic active gel where the filaments are crosslinked with a characteristic time that diverges as the ATP concentration decreases. Our work thus provides a unified view of spatial instabilities in cytoskeletal active matter.

Elongated active units cannot spontaneously break rotation symmetry in bulk fluids to form nematic or polar phases. This has led to the image of active suspensions as spontaneously evolving, spatiotemporally chaotic fluids. In contrast, I show that bulk active fluids have stable active nematic and polar states at fluid-fluid or fluid-air interfaces. The active flow-mediated long-range interactions that destroy the ordered phase in bulk, lead to long-range order at the interface. The active fluids have a surface ordering transition and form states with quiescent, ordered surfaces and a chaotic bulk. I further consider active units that are constrained to live at an interface to examine the minimal conditions for the existence of two-dimensional order in bulk three-dimensional fluids. In this case, immotile units do not order, but motile particles still form a long-range-ordered polar phase. This prediction of stable, uniaxial, active phases in bulk fluids may have functional consequences for active transport.

Spontaneous pattern formation in living systems is driven by reaction-diffusion chemistry and active mechanics. The feedback between chemical and mechanical forces is often essential to robust pattern formation, yet it remains poorly understood in general. In this analytical and numerical paper, we study an experimentally-motivated minimal model of coupling between reaction-diffusion and active matter: a propagating front of an autocatalytic and stress-generating species. In the absence of activity, the front is described by the the well-studied KPP equation. We find that front propagation is maintained even in active systems, with crucial differences: an extensile stress increases the front speed beyond a critical magnitude of the stress, while a contractile stress has no effect on the front speed but can generate a periodic instability in the high-concentration region behind the front. We expect our results to be useful in interpreting pattern formation in active systems with mechano-chemical coupling in vivo and in vitro., Comment: Supplemental movie available at: https://github.com/cdeljunco/rdam/blob/68237176bf8f5b7ff96abb79271dedaec5024c22/Plots/omega-7-movie.mp4. This manuscript contains a footnote about Fisher's racist and ableist motivations that was removed from the published version by the editors. The footnote contains references, so the reference numbers in this version will be different from the published version

In equilibrium liquid crystals, chirality leads to a variety of spectacular three-dimensional structures, but chiral and achiral phases with the same broken continuous symmetries have identical long-time, large-scale dynamics. In this paper, we demonstrate that chirality qualitatively modifies the dynamics of layered liquid crystals in active systems in both two and three dimensions due to an active "odder" elasticity. In three dimensions, we demonstrate that the hydrodynamics of active cholesterics differs fundamentally from smectic-A liquid crystals, unlike their equilibrium counterpart. This distinction can be used to engineer a columnar array of vortices, with anti-ferromagnetic vorticity alignment, that can be switched on and off by external strain. A two-dimensional chiral layered state -- an array of lines on an incompressible, free-standing film of chiral active fluid with a preferred normal direction -- is generically unstable. However, this instability can be tuned in easily realisable experimental settings, when the film is either on a substrate or in an ambient fluid.

Collective cell migration plays an essential role in various biological processes, such as development or cancer proliferation. While cell-cell interactions are clearly key determinants of collective cell migration -- in addition to individual cells self-propulsion -- the physical mechanisms that control the emergence of cell clustering and collective cell migration are still poorly understood. In particular, observations have shown that binary cell-cell collisions generally lead to anti-alignement of cell polarities and separation of pairs -- a process called contact inhibition of locomotion (CIL), which is expected to disfavor the formation of large scale cell clusters with coherent motion. Here, we adopt a joint experimental and theoretical approach to determine the large scale dynamics of cell assemblies from elementary pairwise cell-cell interaction rules. We quantify experimentally binary cell-cell interactions and show that they can be captured by a minimal equilibrium-like pairwise asymmetric aligning interaction potential that reproduces the CIL phenomenology. We identify its symmetry class, build the corresponding active hydrodynamic theory and show on general grounds that such asymmetric aligning interaction destroys large scale clustering and ordering, leading instead to a liquid-like microphase of cell clusters of finite size and short lived polarity, or to a fully dispersed isotropic phase. Finally, this shows that CIL-like asymmetric interactions in cellular systems -- or general active systems -- control cluster sizes and polarity, and can prevent large scale coarsening and long range polarity, except in the singular regime of dense confluent systems., Comment: 11 pages, 5 figures

I show that two-dimensional nematic order in an active fluid film can be stable and long-ranged if exchange of particles and momentum with an ambient three-dimensional fluid is allowed. Number-conserving films present an instability, with a fastest-growing mode independent of activity at high activities or low concentrations. Motility can stabilize long-range polar order on such immersed surfaces even when the swimmer number in the film is conserved. Active ordering in immersed surfaces is much richer than previously expected, with functional consequences for active transport, and suggests that a broader exploration of parameters in current and future experiments would display the new features predicted here.

Active materials, composed of internally driven particles, have properties that are qualitatively distinct from matter at thermal equilibrium. However, the most spectacular departures from equilibrium phase behaviour are thought to be confined to systems with polar or nematic asymmetry. In this paper, we show that such departures are also displayed in more symmetric phases such as hexatics if, in addition, the constituent particles have chiral asymmetry. We show that chiral active hexatics whose rotation rate does not depend on density have giant number fluctuations. If the rotation rate depends on density, the giant number fluctuations are suppressed due to a novel orientation-density sound mode with a linear dispersion which propagates even in the overdamped limit. However, we demonstrate that beyond a finite but large lengthscale, a chirality and activity-induced relevant nonlinearity invalidates the predictions of the linear theory and destroys the hexatic order. In addition, we show that activity modifies the interactions between defects in the active chiral hexatic phase, making them non-mutual. Finally, to demonstrate the generality of a chiral active hexatic phase we show that it results from the melting of chiral active crystals in finite systems.

As the constituent particles of a flock are polar and in a driven state, their interactions must, in general, be fore-aft asymmetric and non-reciprocal. Within a model that explicitly retains the classical spin angular momentum field of the particles we show that the resulting asymmetric contribution to interparticle torques, if large enough, leads to a buckling instability of the flock. Precisely this asymmetry also yields a natural mechanism for a difference between the speed of advection of information along the flock and the speed of the flock itself, concretely establishing that the absence of detailed balance, and not merely the breaking of Galilean invariance, is crucial for this distinction. To highlight this we construct a model of asymmetrically interacting spins fixed to lattice points and demonstrate that the speed of advection of polarisation remains non-zero. We delineate the conditions on parameters and wavenumber for the existence of the buckling instability. Our theory should be consequential for interpreting the behaviour of real animal groups as well as experimental studies of artificial flocks composed of polar motile rods on substrates., Comment: This paper supersedes our earlier version arXiv:1605.00981 and contains a substantially revised theoretical treatment

Multiple experiments on active systems consider oriented active suspensions on substrates or in chambers confined along one direction. The theories of polar and apolar phases in such geometries were considered in A. Maitra et al, arXiv 1901.01069 and A. Maitra et al., PNAS 115, 6934 (2018) respectively. However, the presence of quenched random disorder due to the substrate cannot be completely eliminated in many experimental contexts possibly masking the predictions from those theories. In this paper, I consider the effect of quenched orientational disorder on the phase behaviour of both polar and apolar suspensions on substrates. I show that polar suspensions have long-range order in two dimensions with anomalous number fluctuations, while their apolar counterparts have only short-range order, albeit with a correlation length that can increase with activity, and even more violent number fluctuations than active nematics without quenched disorder. These results should be of value in interpreting experiments on active suspensions on substrates with random disorder.

Can a bath of isotropic but active particles promote ordering of anisotropic but passive particles? In this paper, we uncover a fluctuation-driven mechanism by which this is possible. Somewhat counter-intuitively, we show that the passive particles tend to be more ordered upon increasing the noise-strength of the active isotropic bath. We first demonstrate this in a general dynamical model for a non-conserved order parameter (model A) coupled to an active isotropic field and then concentrate on two examples, i. a collection of polar rods on a substrate in an active isotropic bath and ii. a passive apolar suspension in a momentum conserved, actively forced but isotropic fluid which is relevant for current research in active systems. Our theory, which is relevant for understanding ordering transitions in out-of-equilibrium systems can be tested in experiments, for instance, by introducing a low concentration of passive rod-like objects in active isotropic fluids and, since it is applicable to any non-conserved dynamical field, may have applications far beyond active matter.

Active matter locally converts chemical energy into mechanical work and, for this reason, it provides new mechanisms of pattern formation. In particular, active gels made of protein motors and filaments are far-from-equilibrium systems that exhibit spontaneous flow,[Kruse2004, Voituriez2005] leading to active turbulence in two and three dimensions[Sanchez2012, Kumar2018] and coherent flow in three dimensions[Wu2017] (3D). Although these dynamic flows reveal a characteristic length scale resulting from the interplay between active forcing and passive restoring forces, the observation of static and long-range spatial patterns in active gels has remained elusive. In this work, we demonstrate that a 2D free-standing nematic active gel, formed spontaneously by depletion forces from a 3D solution of kinesin motors and microtubule filaments, actively buckles out-of-plane into a centimeter-sized periodic corrugated sheet that is stable for several days at low activity. Importantly, the corrugations are formed in the absence of flow and their wavelength and stability are controlled by the motor concentration, in agreement with a hydrodynamic theory. At higher activities these patterns are transient with the gel becoming turbulent at longer times. Our results underline the importance of both passive and active forces in shaping active gels and indicate that a static material can be sculpted through an active mechanism.

We present a comprehensive theory of the dynamics and fluctuations of a two-dimensional suspension of polar active particles in an incompressible fluid confined to a substrate. We show that, depending on the sign of a single parameter, a state with polar orientational order is anomalously stable (or anomalously unstable), with a nonzero relaxation (or growth) rate for angular fluctuations at zero wavenumber. This screening of the broken-symmetry mode in the stable state does lead to conventional rather than giant number fluctuations as argued by Bricard et al., Nature ${\bf 503}$, 95 (2013), but their bend instability in a splay-stable flock does not exist and the polar phase has long-range order in two dimensions. Our theory also describes confined three-dimensional thin-film suspensions of active polar particles as well as dense compressible active polar rods, and predicts a flocking transition without a banding instability

We present a complete analysis of the linearised dynamics of active solids with orientational order, taking into account a hitherto overlooked consequence of rotation invariance. Our predictions include the possibility of stable active solids with quasi-long-range order in two dimensions and long-range order in three dimensions, generic instability of the solid for one sign of active forcing, and the instability of the orientationally ordered phase in momentum-conserved systems for large active forcing irrespective of its sign.

We present in detail a Langevin formalism for constructing stochastic dynamical equations for active-matter systems coupled to a thermal bath. We apply the formalism to clarify issues of principle regarding the sources and signatures of nonequilibrium behaviour in a variety of polar and apolar single-particle systems and polar flocks. We show that distance from thermal equilibrium depends on how time-reversal is implemented and hence on the reference equilibrium state. We predict characteristic forms for the frequency-resolved entropy production for an active polar particle in a harmonic potential, which should be testable in experiments.

Active hydrodynamic theories are a powerful tool to study the emergent ordered phases of internally driven particles such as bird flocks, bacterial suspension and their artificial analogues. While theories of orientationally ordered phases are by now well established, the effect of chirality on these phases is much less studied. In this paper, we present the first complete dynamical theory of orientationally ordered chiral particles in two-dimensional incompressible systems. We show that phase-coherent states of rotating chiral particles are remarkably stable in both momentum-conserved and non-conserved systems in contrast to their non-rotating counterparts. Furthermore, defect separation -- which drives chaotic flows in non-rotating active fluids -- is suppressed by intrinsic rotation of chiral active particles. We thus establish chirality as a source of dramatic stabilization in active systems, which could be key in interpreting the collective behaviours of some biological tissues, cytoskeletal systems and collections of bacteria.

Suspensions of actively driven anisotropic objects exhibit distinctively nonequilibrium behaviors, and current theories predict that they are incapable of sustaining orientational order at high activity. By contrast, here we show that nematic suspensions on a substrate can display order at arbitrarily high activity due to a previously unreported, potentially stabilizing active force. The resulting nonequilibrium ordered phase displays robust giant number fluctuations that cannot be suppressed even by an incompressible solvent. Our results apply to virtually all experimental assays used to investigate the active nematic ordering of self-propelled colloids, bacterial suspensions and the cytoskeleton, and have testable implications in interpreting their nonequilibrium behaviors

As the constituents of a flock are polar, one expects a fore-aft asymmetry in their interactions. We show here that the resulting antisymmetric part of the "exchange coupling" between a bird and its neighbours, if large enough, destabilizes the flock through spontaneous turning of the birds. The same asymmetry also yields a natural mechanism for a difference between the speed of advection of information along the flock and the speed of the flock itself. We show that the absence of detailed balance, and not merely the breaking of Galilean invariance, is responsible for this difference. We delineate the conditions on parameters and wavenumber for the existence of the turning instability. Lastly we present an alternative perspective based on flow-alignment effects in an active liquid crystal with turning inertia in contact with a momentum sink., Comment: 7 pages, 1 figure

We present a general dynamical theory of a membrane coupled to an actin cortex containing polymerizing filaments with active stresses and currents, and demonstrate that active membrane dynamics [Phys. Rev. Lett \textbf{84}, 3494 (2000)] and spontaneous shape oscillations emerge from this description. We also consider membrane instabilities and patterns induced by the presence of filaments with polar orientational correlations in the tangent plane of the membrane. The dynamical features we predict should be seen in a variety of cellular contexts involving the dynamics of the membrane-cytoskeleton composite and cytoskeletal extracts coupled to synthetic vesicles.

The nucleus of the eukaryotic cell functions amidst active cytoskeletal filaments, but its response to the stresses carried by these filaments is largely unexplored. We report here the results of studies of the translational and rotational dynamics of the nuclei of single fibroblast cells, with the effects of cell migration suppressed by plating onto fibronectin-coated micro-fabricated patterns. Patterns of the same area but different shapes and/or aspect ratio were used to study the effect of cell geometry on the dynamics. On circles, squares and equilateral triangles, the nucleus undergoes persistent rotational motion, while on high-aspect-ratio rectangles of the same area it moves only back and forth. The circle and the triangle showed respectively the largest and the smallest angular speed. We show that our observations can be understood through a hydrodynamic approach in which the nucleus is treated as a highly viscous inclusion residing in a less viscous fluid of orientable filaments endowed with active stresses. Lowering actin contractility selectively by introducing blebbistatin at low concentrations drastically reduced the speed and persistence time of the angular motion of the nucleus. Time-lapse imaging of actin revealed a correlated hydrodynamic flow around the nucleus, with profile and magnitude consistent with the results of our theoretical approach. Coherent intracellular flows and consequent nuclear rotation thus appear to be a generic property that cells must balance by specific mechanisms in order to maintain nuclear homeostasis.