A minimal physical model captures the shapes of crawling cells

In silico analysis shows that dynamic changes in curvature guide cell migration over long distances

In vitro experiments have shown that cell scale curvatures influence cell migration; cells avoid convex hills and settle in concave valleys. However, it is not known whether dynamic changes in curvature can guide cell migration. This study extends a previous in-silico model to explore the effects over time of changing the substrate curvature on cell migration guidance. By simulating a dynamic surface curvature using traveling wave patterns, we investigate the influence of wave height and speed, and find that long-distance cell migration guidance can be achieved on specific wave patterns. We propose a mechanistic explanation of what we call dynamic curvotaxis and highlight those cellular features that may be involved. Our results open a new area of study for understanding cell mobility in dynamic environments, from single-cell in vitro experiments to multi-cellular in vivo mechanisms.

Spontaneous motion of a passive fluid droplet in an active microchannel

We numerically study the dynamics of a passive fluid droplet confined within a microchannel whose walls are covered with a thin layer of active gel. The latter represents a fluid of extensile material modelling, for example, a suspension of cytoskeletal filaments and molecular motors. Our results show that the layer is capable of producing a spontaneous flow triggering a rectilinear motion of the passive droplet. For a hybrid design (a single wall covered by the active layer), at the steady state the droplet attains an elliptical shape, resulting from an asymmetric saw-toothed structure of the velocity field. In contrast, if the active gel covers both walls, the velocity field exhibits a fully symmetric pattern considerably mitigating morphological deformations. We further show that the structure of the spontaneous flow in the microchannel can be controlled by the anchoring conditions of the active gel at the wall. These findings are also confirmed by selected 3D simulations. Our results may stimulate further research addressed to design novel microfludic devices whose functioning relies on the collective properties of active gels.

The analytical solution to the migration of an epithelial monolayer with a circular spreading front and its implications in the gap closure process

The coordinated behaviors of epithelial cells are widely observed in tissue development, such as re-epithelialization, tumor growth, and morphogenesis. In these processes, cells either migrate collectively or organize themselves into specific structures to serve certain purposes. In this work, we study a spreading epithelial monolayer whose migrating front encloses a circular gap in the monolayer center. Such tissue is usually used to mimic the wound healing process in vitro. We model the epithelial sheet as a layer of active viscous polar fluid. With an axisymmetric assumption, the model can be analytically solved under two special conditions, suggesting two possible spreading modes for the epithelial monolayer. Based on these two sets of analytical solutions, we assess the velocity of the spreading front affected by the gap size, the active intercellular contractility, and the purse-string contraction acting on the spreading edge. Several critical values exist in the model parameters for the initiation of the gap closure process, and the purse-string contraction plays a vital role in governing the gap closure kinetics. Finally, the instability of the morphology of the spreading front was studied. Numerical calculations show how the perturbated velocities and the growth rates vary with respect to different model parameters.

The crucial role of adhesion in the transmigration of active droplets through interstitial orifices

Active fluid droplets are a class of soft materials exhibiting autonomous motion sustained by an energy supply. Such systems have been shown to capture motility regimes typical of biological cells and are ideal candidates as building-block for the fabrication of soft biomimetic materials of interest in pharmacology, tissue engineering and lab on chip devices. While their behavior is well established in unconstrained environments, much less is known about their dynamics under strong confinement. Here, we numerically study the physics of a droplet of active polar fluid migrating within a microchannel hosting a constriction with adhesive properties, and report evidence of a striking variety of dynamic regimes and morphological features, whose properties crucially depend upon droplet speed and elasticity, degree of confinement within the constriction and adhesiveness to the pore. Our results suggest that non-uniform adhesion forces are instrumental in enabling the crossing through narrow orifices, in contrast to larger gaps where a careful balance between speed and elasticity is sufficient to guarantee the transition. These observations may be useful for improving the design of artificial micro-swimmers, of interest in material science and pharmaceutics, and potentially for cell sorting in microfluidic devices.

The nonlinear motion of cells subject to external forces

To develop a minimal model for a cell moving in a crowded environment such as in tissue, we investigate the response of a liquid drop of active matter moving on a flat rigid substrate to forces applied at its boundaries. We consider two different self-propulsion mechanisms, active stresses and treadmilling polymerisation, and we investigate how the active drop motion is altered by these surface forces. We find a highly non-linear response to forces that we characterise using drop velocity, drop shape, and the traction between the drop and the substrate. Each self-propulsion mechanism gives rise to two main modes of motion: a long thin drop with zero traction in the bulk, mostly occurring under strong stretching forces, and a parabolic drop with finite traction in the bulk, mostly occurring under strong squeezing forces. In each case there is a sharp transition between parabolic, and long thin drops as a function of the applied forces and indications of drop break-up where large forces stretch the drop.

Mechanochemical subcellular-element model of crawling cells

Constructing physical models of living cells and tissues is an extremely challenging task because of the high complexities of both intra- and intercellular processes. In addition, the force that a single cell generates vanishes in total due to the law of action and reaction. The typical mechanics of cell crawling involve periodic changes in the cell shape and in the adhesion characteristics of the cell to the substrate. However, the basic physical mechanisms by which a single cell coordinates these processes cooperatively to achieve autonomous migration are not yet well understood. To obtain a clearer grasp of how the intracellular force is converted to directional motion, we develop a basic mechanochemical model of a crawling cell based on subcellular elements with the focus on the dependence of the protrusion and contraction as well as the adhesion and de-adhesion processes on intracellular biochemical signals. By introducing reaction-diffusion equations that reproduce traveling waves of local chemical concentrations, we clarify that the chemical dependence of the cell-substrate adhesion dynamics determines the crawling direction and distance with one chemical wave. Finally, we also perform multipole analysis of the traction force to compare it with the experimental results. Our present work sheds light on how intracellular chemical reactions are converted to a directional cell migration under the force-free condition. Although the detailed mechanisms of actual cells are far more complicated than our simple model, we believe that this mechanochemical model is a good prototype for more realistic models.

Incompressible Polar Active Fluids with Quenched Random Field Disorder in Dimensions d>2

We present a hydrodynamic theory of incompressible polar active fluids with quenched random field 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 nonzero 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.

Optimal transport and control of active drops

Understanding the complex patterns in space-time exhibited by active systems has been the subject of much interest in recent times. Complementing this forward problem is the inverse problem of controlling active matter. Here, we use optimal control theory to pose the problem of transporting a slender drop of an active fluid and determine the dynamical profile of the active stresses to move it with minimal viscous dissipation. By parametrizing the position and size of the drop using a low-order description based on lubrication theory, we uncover a natural "gather-move-spread" strategy that leads to an optimal bound on the maximum achievable displacement of the drop relative to its size. In the continuum setting, the competition between passive surface tension and active controls generates richer behavior with futile oscillations and complex drop morphologies that trade internal dissipation against the transport cost to select optimal strategies. Our work combines active hydrodynamics and optimal control in a tractable and interpretable framework and begins to pave the way for the spatiotemporal manipulation of active matter.

Dynamics of active liquid interfaces

Controlling interfaces of phase-separating fluid mixtures is key to the creation of diverse functional soft materials. Traditionally, this is accomplished with surface-modifying chemical agents. Using experiment and theory, we studied how mechanical activity shapes soft interfaces that separate an active and a passive fluid. Chaotic flows in the active fluid give rise to giant interfacial fluctuations and noninertial propagating active waves. At high activities, stresses disrupt interface continuity and drive droplet generation, producing an emulsion-like active state composed of finite-sized droplets. When in contact with a solid boundary, active interfaces exhibit nonequilibrium wetting transitions, in which the fluid climbs the wall against gravity. These results demonstrate the promise of mechanically driven interfaces for creating a new class of soft active matter.

Symmetry-breaking, motion and bistability of active drops through polarization-surface coupling

Cell crawling crucially depends on the collective dynamics of the acto-myosin cytoskeleton. However, it remains an open question to what extent cell polarization and persistent motion depend on continuous regulatory mechanisms and autonomous physical mechanisms. Experiments on cell fragments and theoretical considerations for active polar liquids have highlighted that physical mechanisms induce motility through splay and bend configurations in a nematic director field. Here, we employ a simple model, derived from basic thermodynamic principles, for active polar free-surface droplets to identify a different mechanism of motility. Namely, active stresses drive drop motion through spatial variations of polarization strength. This robustly induces parity-symmetry breaking and motility even for liquid ridges (2D drops) and adds to splay- and bend-driven pumping in 3D geometries. Intriguingly, then, stable polar moving and axisymmetric resting states may coexist, reminiscent of the interconversion of moving and resting keratocytes by external stimuli. The identified additional motility mode originates from a competition between the elastic bulk energy and the polarity control exerted by the drop surface. As it already breaks parity-symmetry for passive drops, the resulting back-forth asymmetry enables active stresses to effectively pump liquid and drop motion ensues.

Bacterial active matter

Bacteria are among the oldest and most abundant species on Earth. Bacteria successfully colonize diverse habitats and play a significant role in the oxygen, carbon, and nitrogen cycles. They also form human and animal microbiota and may become sources of pathogens and a cause of many infectious diseases. Suspensions of motile bacteria constitute one of the most studied examples of active matter: a broad class of non-equilibrium systems converting energy from the environment (e.g., chemical energy of the nutrient) into mechanical motion. Concentrated bacterial suspensions, often termed active fluids, exhibit complex collective behavior, such as large-scale turbulent-like motion (so-called bacterial turbulence) and swarming. The activity of bacteria also affects the effective viscosity and diffusivity of the suspension. This work reports on the progress in bacterial active matter from the physics viewpoint. It covers the key experimental results, provides a critical assessment of major theoretical approaches, and addresses the effects of visco-elasticity, liquid crystallinity, and external confinement on collective behavior in bacterial suspensions.

Combined Effect of Matrix Topography and Stiffness on Neutrophil Shape and Motility

The crawling behavior of leukocytes is driven by the cell morphology transition, which is a direct manifestation of molecular motor machinery. The topographical anisotropy and mechanical stiffness of the substrates are the main physical cues that affect leukocytes' shape generation and migratory responses. However, their combined effects on the cell morphology and motility have been poorly understood, particularly for neutrophils, which are the fastest reacting leukocytes against infections and wounds. Here, spatiotemporally correlated physical parameters are shown, which determine the neutrophil shape change during migratory processes, in response to surface topography and elasticity. Guided crawling and shape generation of individual neutrophils, activated by a uniform concentration of a chemoattractant, are analyzed by adopting elasticity-tunable micropatterning and live cell imaging techniques. Whole cell-level image analysis is performed based on a planar geometric quantification of cell shape and motility. The findings show that the pattern anisotropy and elastic modulus of the substrate induce synergic effects on the shape anisotropy, deformability, and polarization/alignment of crawling neutrophils. How the morphology-motility relationship is affected by different surface microstructures and stiffness is demonstrated. These results imply that the neutrophil shape-motility correlations can be utilized for controlling the immune cell functions with predefined physical microenvironments.

Three-dimensional morphodynamic simulations of macropinocytic cups

Macropinocytosis refers to the non-specific uptake of extracellular fluid, which plays ubiquitous roles in cell growth, immune surveillance, and virus entry. Despite its widespread occurrence, it remains unclear how its initial cup-shaped plasma membrane extensions form without any external solid support, as opposed to the process of particle uptake during phagocytosis. Here, by developing a computational framework that describes the coupling between the bistable reaction-diffusion processes of active signaling patches and membrane deformation, we demonstrated that the protrusive force localized to the edge of the patches can give rise to a self-enclosing cup structure, without further assumptions of local bending or contraction. Efficient uptake requires a balance among the patch size, magnitude of protrusive force, and cortical tension. Furthermore, our model exhibits cyclic cup formation, coexistence of multiple cups, and cup-splitting, indicating that these complex morphologies self-organize via a common mutually-dependent process of reaction-diffusion and membrane deformation.

Design of nematic liquid crystals to control microscale dynamics

The dynamics of small particles, both living such as swimming bacteria and inanimate, such as colloidal spheres, has fascinated scientists for centuries. If one could learn how to control and streamline their chaotic motion, that would open technological opportunities in the transformation of stored or environmental energy into systematic motion, with applications in micro-robotics, transport of matter, guided morphogenesis. This review presents an approach to command microscale dynamics by replacing an isotropic medium with a liquid crystal. Orientational order and associated properties, such as elasticity, surface anchoring, and bulk anisotropy, enable new dynamic effects, ranging from the appearance and propagation of particle-like solitary waves to self-locomotion of an active droplet. By using photoalignment, the liquid crystal can be patterned into predesigned structures. In the presence of the electric field, these patterns enable the transport of solid and fluid particles through nonlinear electrokinetics rooted in anisotropy of conductivity and permittivity. Director patterns command the dynamics of swimming bacteria, guiding their trajectories, polarity of swimming, and distribution in space. This guidance is of a higher level of complexity than a simple following of the director by rod-like microorganisms. Namely, the director gradients mediate hydrodynamic interactions of bacteria to produce an active force and collective polar modes of swimming. The patterned director could also be engraved in a liquid crystal elastomer. When an elastomer coating is activated by heat or light, these patterns produce a deterministic surface topography. The director gradients define an activation force that shapes the elastomer in a manner similar to the active stresses triggering flows in active nematics. The patterned elastomer substrates could be used to define the orientation of cells in living tissues. The liquid-crystal guidance holds a major promise in achieving the goal of commanding microscale active flows.

Onsager's variational principle in active soft matter

Onsagers variational principle (OVP) was originally proposed by Lars Onsager in 1931 [L. Onsager, Phys. Rev., 1931, 37, 405]. This fundamental principle provides a very powerful tool for formulating thermodynamically consistent models. It can also be employed to find approximate solutions, especially in the study of soft matter dynamics. In this work, OVP is extended and applied to the dynamic modeling of active soft matter such as suspensions of bacteria and aggregates of animal cells. We first extend the general formulation of OVP to active matter dynamics where active forces are included as external non-conservative forces. We then use OVP to analyze the directional motion of individual active units: a molecular motor walking on a stiff biofilament and a toy two-sphere microswimmer. Next we use OVP to formulate a diffuse-interface model for an active polar droplet on a solid substrate. In addition to the generalized hydrodynamic equations for active polar fluids in the bulk region, we have also derived thermodynamically consistent boundary conditions. Finally, we consider the dynamics of a thin active polar droplet under the lubrication approximation. We use OVP to derive a generalized thin film equation and then employ OVP as an approximation tool to find the spreading laws for the thin active polar droplet. By incorporating the activity of biological systems into OVP, we develop a general approach to construct thermodynamically consistent models for better understanding the emergent behaviors of individual animal cells and cell aggregates or tissues.

The many behaviors of deformable active droplets

Active fluids consume fuel at the microscopic scale, converting this energy into forces that can drive macroscopic motions over scales far larger than their microscopic constituents. In some cases, the mechanisms that give rise to this phenomenon have been well characterized, and can explain experimentally observed behaviors in both bulk fluids and those confined in simple stationary geometries. More recently, active fluids have been encapsulated in viscous drops or elastic shells so as to interact with an outer environment or a deformable boundary. Such systems are not as well understood. In this work, we examine the behavior of droplets of an active nematic fluid. We study their linear stability about the isotropic equilibrium over a wide range of parameters, identifying regions in which different modes of instability dominate. Simulations of their full dynamics are used to identify their nonlinear behavior within each region. When a single mode dominates, the droplets behave simply: as rotors, swimmers, or extensors. When parameters are tuned so that multiple modes have nearly the same growth rate, a pantheon of modes appears, including zigzaggers, washing machines, wanderers, and pulsators.

Critical role of lipid membranes in polarization and migration of cells: a biophysical view

Cell migration plays vital roles in many biologically relevant processes such as tissue morphogenesis and cancer metastasis, and it has fascinated biophysicists over the past several decades. However, despite an increasing number of studies highlighting the orchestration of proteins involved in different signaling pathways, the functional roles of lipid membranes have been essentially overlooked. Lipid membranes are generally considered to be a functionless two-dimensional matrix of proteins, although many proteins regulating cell migration gain functions only after they are recruited to the membrane surface and self-organize their functional domains. In this review, we summarize how the logistical recruitment and release of proteins to and from lipid membranes coordinates complex spatiotemporal molecular processes. As predicted from the classical framework of the Smoluchowski equation of diffusion, lipid/protein membranes serve as a 2D reaction hub that contributes to the effective and robust regulation of polarization and migration of cells involving several competing pathways.

Dynamics of a cell motility model near the sharp interface limit

Phase-field models have recently had great success in describing the dynamic morphologies and motility of eukaryotic cells. In this work we investigate the minimal phase-field model introduced in Berlyand et al. (2017). Rigorous analysis of its sharp interface limit dynamics was completed in Mizuhara et al. (2016) and Mizuhara et al. (2019), where it was observed that persistent cell motion was not stable. In this work we numerically study the pre-limiting phase-field model near the sharp interface limit, to better understand this lack of persistent motion. We find that immobile, persistent, and rotating states are all exhibited in this minimal model, and investigate the loss of persistent motion in the sharp interface limit.

Soft channel formation and symmetry breaking in exotic active emulsions

We use computer simulations to study the morphology and rheological properties of a bidimensional emulsion resulting from a mixture of a passive isotropic fluid and an active contractile polar gel, in the presence of a surfactant that favours the emulsification of the two phases. By varying the intensity of the contractile activity and of an externally imposed shear flow, we find three possible morphologies. For low shear rates, a simple lamellar state is obtained. For intermediate activity and shear rate, an asymmetric state emerges, which is characterized by shear and concentration banding at the polar/isotropic interface. A further increment in the active forcing leads to the self-assembly of a soft channel where an isotropic fluid flows between two layers of active material. We characterize the stability of this state by performing a dynamical test varying the intensity of the active forcing and shear rate. Finally, we address the rheological properties of the system by measuring the effective shear viscosity, finding that this increases as active forcing is increased-so that the fluid thickens with activity.

Thin-film modeling of resting and moving active droplets

We propose a generic model for thin films and shallow drops of a polar active liquid that have a free surface and are in contact with a solid substrate. The model couples evolution equations for the film height and the local polarization in the form of a gradient dynamics supplemented with active stresses and fluxes. A wetting energy for a partially wetting liquid is incorporated allowing for motion of the liquid-solid-gas contact line. This gives a consistent basis for the description of drops of dense bacterial suspensions or compact aggregates of living cells on solid substrates. As example, we analyze the dynamics of two-dimensional active drops (i.e., ridges) and demonstrate how active forces compete with passive surface forces to shape droplets and drive their motion. In our simple two-dimensional scenario we find that defect structures within the polarization profile drastically influence the shape and motility of active droplets. Thus, we can observe a transition from resting to motile droplets via the elimination of defects in the polarization profile. Furthermore, droplet motility is modulated by strong active stresses. Contractile stresses even lead to topological changes, i.e., drop splitting, which is naturally encoded in the evolution equations.

How many ways a cell can move: the modes of self-propulsion of an active drop

Numerous physical models have been proposed to explain how cell motility emerges from internal activity, mostly focused on how crawling motion arises from internal processes. Here we offer a classification of self-propulsion mechanisms based on general physical principles, showing that crawling is not the only way for cells to move on a substrate. We consider a thin drop of active matter on a planar substrate and fully characterize its autonomous motion for all three possible sources of driving: (i) the stresses induced in the bulk by active components, which allow in particular tractionless motion, (ii) the self-propulsion of active components at the substrate, which gives rise to crawling motion, and (iii) a net capillary force, possibly self-generated, and coupled to internal activity. We determine travelling-wave solutions to the lubrication equations as a function of a dimensionless activity parameter for each mode of motion. Numerical simulations are used to characterize the drop motion over a wide range of activity magnitudes, and explicit analytical solutions in excellent agreement with the simulations are derived in the weak-activity regime.

Hydrodynamic effects on the motility of crawling eukaryotic cells

Eukaryotic cell motility is crucial during development, wound healing, the immune response, and cancer metastasis. Some eukaryotic cells can swim, but cells more commonly adhere to and crawl along the extracellular matrix. We study the relationship between hydrodynamics and adhesion that describe whether a cell is swimming, crawling, or combining these motions. Our simple model of a cell, based on the three-sphere swimmer, is capable of both swimming and crawling. As cell-matrix adhesion strength increases, the influence of hydrodynamics on migration diminishes. Cells with significant adhesion can crawl with speeds much larger than their nonadherent, swimming counterparts. We predict that, while most eukaryotic cells are in the strong-adhesion limit, increasing environment viscosity or decreasing cell-matrix adhesion could lead to significant hydrodynamic effects even in crawling cells. Signatures of hydrodynamic effects include a dependence of cell speed on the presence of a nearby substrate or interactions between noncontacting cells. These signatures will be suppressed at large adhesion strengths, but even strongly adherent cells will generate relevant fluid flows that will advect nearby passive particles and swimmers.

Motility and morphodynamics of confined cells

We introduce a minimal hydrodynamic model of polarization, migration, and deformation of a biological cell confined between two parallel surfaces. In our model, the cell is driven out of equilibrium by an active cytsokeleton force that acts on the membrane. The cell cytoplasm, described as a viscous droplet in the Darcy flow regime, contains a diffusive solute that actively transduces the applied cytoskeleton force. While fairly simple and analytically tractable, this quasi-two-dimensional model predicts a range of compelling dynamic behaviours. A linear stability analysis of the system reveals that solute activity first destabilizes a global polarization-translation mode, prompting cell motility through spontaneous symmetry breaking. At higher activity, the system crosses a series of Hopf bifurcations leading to coupled oscillations of droplet shape and solute concentration profiles. At the nonlinear level, we find traveling-wave solutions associated with unique polarized shapes that resemble experimental observations. Altogether, this model offers an analytical paradigm of active deformable systems in which viscous hydrodynamics are coupled to diffusive force transducers.

Active gel segment behaving as an active particle

We reduce a one-dimensional model of an active segment (AS), which is used, for instance, in the description of contraction-driven cell motility, to a zero-dimensional model of an active particle (AP) characterized by two internal degrees of freedom: position and polarity. Both models give rise to hysteretic force-velocity relations showing that an active agent can support two opposite polarities under the same external force and that it can maintain the same polarity while being dragged by external forces with opposite orientations. This double bistability results in a rich dynamic repertoire which we illustrate by studying static, stalled, motile, and periodically repolarizing regimes displayed by an active agent confined in a viscoelastic environment. We show that the AS and AP models can be calibrated to generate quantitatively similar dynamic responses.

A minimal computational model for three-dimensional cell migration

During migration, eukaryotic cells can continuously change their three-dimensional morphology, resulting in a highly dynamic and complex process. Further complicating this process is the observation that the same cell type can rapidly switch between different modes of migration. Modelling this complexity necessitates models that are able to track deforming membranes and that can capture the intracellular dynamics responsible for changes in migration modes. Here we develop an efficient three-dimensional computational model for cell migration, which couples cell mechanics to a simple intracellular activator-inhibitor signalling system. We compare the computational results to quantitative experiments using the social amoeba Dictyostelium discoideum. The model can reproduce the observed migration modes generated by varying either mechanical or biochemical model parameters and suggests a coupling between the substrate and the biomechanics of the cell.

Tractionless Self-Propulsion of Active Drops

We report on a new mode of self-propulsion exhibited by compact drops of active liquids on a substrate which, remarkably, is tractionless, i.e., which imparts no mechanical stress locally on the surface. We show, both analytically and by numerical simulation, that the equations of motion for an active nematic drop possess a simple self-propelling solution, with no traction on the solid surface and in which the direction of motion is controlled by the winding of the nematic director field across the drop height. The physics underlying this mode of motion has the same origins as that giving rise to the zero viscosity observed in bacterial suspensions. This topologically protected tractionless self-propusion provides a robust physical mechanism for efficient cell migration in crowded environments like tissues.

Rheology of active polar emulsions: from linear to unidirectional and inviscid flow, and intermittent viscosity

The rheological behaviour of an emulsion made of an active polar component and an isotropic passive fluid is studied by lattice Boltzmann methods. Different flow regimes are found by varying the values of the shear rate and extensile activity (occurring, e.g., in microtubule-motor suspensions). By increasing the activity, a first transition occurs from the linear flow regime to spontaneous persistent unidirectional macro-scale flow, followed by another transition either to a (low shear) intermittent flow regime with the coexistence of states with positive, negative, and vanishing apparent viscosity, or to a (high shear) symmetric shear thinning regime. The different behaviours can be explained in terms of the dynamics of the polarization field close to the walls. A maximum entropy production principle selects the most likely states in the intermittent regime.

Crawling in a Fluid

There is increasing evidence that mammalian cells not only crawl on substrates but can also swim in fluids. To elucidate the mechanisms of the onset of motility of cells in suspension, a model which couples actin and myosin kinetics to fluid flow is proposed and solved for a spherical shape. The swimming speed is extracted in terms of key parameters. We analytically find super- and subcritical bifurcations from a nonmotile to a motile state and also spontaneous polarity oscillations that arise from a Hopf bifurcation. Relaxing the spherical assumption, the obtained shapes show appealing trends.

Lattice Boltzmann methods and active fluids

We review the state of the art of active fluids with particular attention to hydrodynamic continuous models and to the use of Lattice Boltzmann Methods (LBM) in this field. We present the thermodynamics of active fluids, in terms of liquid crystals modelling adapted to describe large-scale organization of active systems, as well as other effective phenomenological models. We discuss how LBM can be implemented to solve the hydrodynamics of active matter, starting from the case of a simple fluid, for which we explicitly recover the continuous equations by means of Chapman-Enskog expansion. Going beyond this simple case, we summarize how LBM can be used to treat complex and active fluids. We then review recent developments concerning some relevant topics in active matter that have been studied by means of LBM: spontaneous flow, self-propelled droplets, active emulsions, rheology, active turbulence, and active colloids.

The matrix environmental and cell mechanical properties regulate cell migration and contribute to the invasive phenotype of cancer cells

The minimal structural unit of a solid tumor is a single cell or a cellular compartment such as the nucleus. A closer look inside the cells reveals that there are functional compartments or even structural domains determining the overall properties of a cell such as the mechanical phenotype. The mechanical interaction of these living cells leads to the complex organization such as compartments, tissues and organs of organisms including mammals. In contrast to passive non-living materials, living cells actively respond to the mechanical perturbations occurring in their microenvironment during diseases such as fibrosis and cancer. The transformation of single cancer cells in highly aggressive and hence malignant cancer cells during malignant cancer progression encompasses the basement membrane crossing, the invasion of connective tissue, the stroma microenvironments and transbarrier migration, which all require the immediate interaction of the aggressive and invasive cancer cells with the surrounding extracellular matrix environment including normal embedded neighboring cells. All these steps of the metastatic pathway seem to involve mechanical interactions between cancer cells and their microenvironment. The pathology of cancer due to a broad heterogeneity of cancer types is still not fully understood. Hence it is necessary to reveal the signaling pathways such as mechanotransduction pathways that seem to be commonly involved in the development and establishment of the metastatic and mechanical phenotype in several carcinoma cells. We still do not know whether there exist distinct metastatic genes regulating the progression of tumors. These metastatic genes may then be activated either during the progression of cancer by themselves on their migration path or in earlier stages of oncogenesis through activated oncogenes or inactivated tumor suppressor genes, both of which promote the metastatic phenotype. In more detail, the adhesion of cancer cells to their surrounding stroma induces the generation of intracellular contraction forces that deform their microenvironments by alignment of fibers. The amplitude of these forces can adapt to the mechanical properties of the microenvironment. Moreover, the adhesion strength of cancer cells seems to determine whether a cancer cell is able to migrate through connective tissue or across barriers such as the basement membrane or endothelial cell linings of blood or lymph vessels in order to metastasize. In turn, exposure of adherent cancer cells to physical forces, such as shear flow in vessels or compression forces around tumors, reinforces cell adhesion, regulates cell contractility and restructures the ordering of the local stroma matrix that leads subsequently to secretion of crosslinking proteins or matrix degrading enzymes. Hence invasive cancer cells alter the mechanical properties of their microenvironment. From a mechanobiological point-of-view, the recognized physical signals are transduced into biochemical signaling events that guide cellular responses such as cancer progression after the malignant transition of cancer cells from an epithelial and non-motile phenotype to a mesenchymal and motile (invasive) phenotype providing cellular motility. This transition can also be described as the physical attempt to relate this cancer cell transitional behavior to a T1 phase transition such as the jamming to unjamming transition. During the invasion of cancer cells, cell adaptation occurs to mechanical alterations of the local stroma, such as enhanced stroma upon fibrosis, and therefore we need to uncover underlying mechano-coupling and mechano-regulating functional processes that reinforce the invasion of cancer cells. Moreover, these mechanisms may also be responsible for the awakening of dormant residual cancer cells within the microenvironment. Physicists were initially tempted to consider the steps of the cancer metastasis cascade as single events caused by a single mechanical alteration of the overall properties of the cancer cell. However, this general and simple view has been challenged by the finding that several mechanical properties of cancer cells and their microenvironment influence each other and continuously contribute to tumor growth and cancer progression. In addition, basement membrane crossing, cell invasion and transbarrier migration during cancer progression is explained in physical terms by applying physical principles on living cells regardless of their complexity and individual differences of cancer types. As a novel approach, the impact of the individual microenvironment surrounding cancer cells is also included. Moreover, new theories and models are still needed to understand why certain cancers are malignant and aggressive, while others stay still benign. However, due to the broad variety of cancer types, there may be various pathways solely suitable for specific cancer types and distinct steps in the process of cancer progression. In this review, physical concepts and hypotheses of cancer initiation and progression including cancer cell basement membrane crossing, invasion and transbarrier migration are presented and discussed from a biophysical point-of-view. In addition, the crosstalk between cancer cells and a chronically altered microenvironment, such as fibrosis, is discussed including the basic physical concepts of fibrosis and the cellular responses to mechanical stress caused by the mechanically altered microenvironment. Here, is highlighted how biophysical approaches, both experimentally and theoretically, have an impact on classical hallmarks of cancer and fibrosis and how they contribute to the understanding of the regulation of cancer and its progression by sensing and responding to the physical environmental properties through mechanotransduction processes. Finally, this review discusses various physical models of cell migration such as blebbing, nuclear piston, protrusive force and unjamming transition migration modes and how they contribute to cancer progression. Moreover, these cellular migration modes are influenced by microenvironmental perturbances such as fibrosis that can induce mechanical alterations in cancer cells, which in turn may impact the environment. Hence, the classical hallmarks of cancer need to be refined by including biomechanical properties of cells, cell clusters and tissues and their microenvironment to understand mechano-regulatory processes within cancer cells and the entire organism.

Self-propulsion of an active polar drop

We investigate the self-propulsive motion of a drop containing an active polar field. The drop demonstrates spontaneous symmetry breaking from a uniform orientational order into a splay or bend instability depending on the types of active stress, namely, contractile or extensile, respectively. We develop an analytical theory of the mechanism of this instability, which has been observed only in numerical simulations. We show that both contractile and extensile active stresses result in the instability and self-propulsive motion. We also discuss asymmetry between contractile and extensile stresses and show that extensile active stress generates chaotic motion even under a simple model of the polarity field coupled with motion and deformation of the drop.

Shear-Induced First-Order Transition in Polar Liquid Crystals

The hydrodynamic theory of polar liquid crystals is widely used to describe biological active fluids as well as passive molecular materials. Depending on the "shear-alignment parameter", in passive or weakly active polar fluids under external shear, the polar order parameter p is either inclined to the flow at a fixed (Leslie) angle, or rotates continuously. Here, we study the role of an additional "shear-elongation parameter" that has been neglected in the recent literature and causes |p| to change under flow. We show that this effect can give rise to a shear-induced first-order phase transition from isotropic to polar, and significantly change the rheological properties of both active and passive polar fluids.

Cell motility dependence on adhesive wetting

Adhesive cell-substrate interactions are crucial for cell motility and are responsible for the necessary traction that propels cells. These interactions can also change the shape of the cell, analogous to liquid droplet wetting on adhesive substrates. To address how these shape changes affect cell migration and cell speed we model motility using deformable, 2D cross-sections of cells in which adhesion and frictional forces between cell and substrate can be varied separately. Our simulations show that increasing the adhesion results in increased spreading of cells and larger cell speeds. We propose an analytical model which shows that the cell speed is inversely proportional to an effective height of the cell and that increasing this height results in increased internal shear stress. The numerical and analytical results are confirmed in experiments on motile eukaryotic cells.

Lamellar ordering, droplet formation and phase inversion in exotic active emulsions

We study numerically the behaviour of a two-dimensional mixture of a passive isotropic fluid and an active polar gel, in the presence of a surfactant favouring emulsification. Focussing on parameters for which the underlying free energy favours the lamellar phase in the passive limit, we show that the interplay between nonequilibrium and thermodynamic forces creates a range of multifarious exotic emulsions. When the active component is contractile (e.g., an actomyosin solution), moderate activity enhances the efficiency of lamellar ordering, whereas strong activity favours the creation of passive droplets within an active matrix. For extensile activity (occurring, e.g., in microtubule-motor suspensions), instead, we observe an emulsion of spontaneously rotating droplets of different size. By tuning the overall composition, we can create high internal phase emulsions, which undergo sudden phase inversion when activity is switched off. Therefore, we find that activity provides a single control parameter to design composite materials with a strikingly rich range of morphologies.

Morphogenetic degeneracies in the actomyosin cortex

One of the great challenges in biology is to understand the mechanisms by which morphogenetic processes arise from molecular activities. We investigated this problem in the context of actomyosin-based cortical flow in C. elegans zygotes, where large-scale flows emerge from the collective action of actomyosin filaments and actin binding proteins (ABPs). Large-scale flow dynamics can be captured by active gel theory by considering force balances and conservation laws in the actomyosin cortex. However, which molecular activities contribute to flow dynamics and large-scale physical properties such as viscosity and active torque is largely unknown. By performing a candidate RNAi screen of ABPs and actomyosin regulators we demonstrate that perturbing distinct molecular processes can lead to similar flow phenotypes. This is indicative for a ‘morphogenetic degeneracy’ where multiple molecular processes contribute to the same large-scale physical property. We speculate that morphogenetic degeneracies contribute to the robustness of bulk biological matter in development.

Simple Physical Model Unravels Influences of Chemokine on Shape Deformation and Migration of Human Hematopoietic Stem Cells

We studied the dynamic behavior of human hematopoietic stem cells (HSC) on the in vitro model of bone marrow surfaces in the absence and presence of chemokine (SDF1α). The deformation and migration of cells were investigated by varying the chemokine concentration and surface density of ligand molecules. Since HSC used in this study were primary cells extracted from the human umbilical cord blood, it is not possible to introduce molecular reporter systems before or during the live cell imaging. To account for the experimental observations, we propose a simple and general theoretical model for cell crawling. In contrast to other theoretical models reported previously, our model focuses on the nonlinear coupling between shape deformation and translational motion and is free from any molecular-level process. Therefore, it is ideally suited for the comparison with our experimental results. We have demonstrated that the results in the absence of SDF1α were well recapitulated by the linear model, while the nonlinear model is necessary to reproduce the elongated migration observed in the presence of SDF1α. The combination of the simple theoretical model and the label-free, live cell observations of human primary cells opens a large potential to numerically identify the differential effects of extrinsic factors such as chemokines, growth factors, and clinical drugs on dynamic phenotypes of primary cells.

Mechanical stress as a regulator of cell motility

The motility of a cell can be triggered or inhibited not only by an applied force but also by a mechanically neutral force couple. This type of loading, represented by an applied stress and commonly interpreted as either squeezing or stretching, can originate from extrinsic interaction of a cell with its neighbors. To quantify the effect of applied stresses on cell motility we use an analytically transparent one-dimensional model accounting for active myosin contraction and induced actin turnover. We show that stretching can polarize static cells and initiate cell motility while squeezing can symmetrize and arrest moving cells. We show further that sufficiently strong squeezing can lead to the loss of cell integrity. The overall behavior of the system depends on the two dimensionless parameters characterizing internal driving (chemical activity) and external loading (applied stress). We construct a phase diagram in this parameter space distinguishing between static, motile, and collapsed states. The obtained results are relevant for the mechanical understanding of contact inhibition and the epithelial-to-mesenchymal transition.

Cell migration in microengineered tumor environments

Recent advances in microengineered cell migration platforms are discussed critically with a focus on how cell migration is influenced by engineered tumor microenvironments, the medical relevance being to understand how tumor microenvironments may promote or suppress the progression of cancer. We first introduce key findings in cancer cell migration under the influence of the physical environment, which is systematically controlled by microengineering technology, followed by multi-cues of physico-chemical factors, which represent the complexity of the tumor environment. Recognizing that cancer cells constantly communicate not only with each other but also with tumor-associated cells such as vascular, fibroblast, and immune cells, and also with non-cellular components, it follows that cell motility in tumor microenvironments, especially metastasis via the invasion of cancer cells into the extracellular matrix and other tissues, is closely related to the malignancy of cancer-related mortality. Medical relevance of forefront research realized in microfabricated devices, such as single cell sorting based on the analysis of cell migration behavior, may assist personalized theragnostics based on the cell migration phenotype. Furthermore, we urge development of theory and numerical understanding of single or collective cell migration in microengineered platforms to gain new insights in cancer metastasis and in therapeutic strategies.

Flow of Deformable Droplets: Discontinuous Shear Thinning and Velocity Oscillations

We study the rheology of a suspension of soft deformable droplets subjected to a pressure-driven flow. Through computer simulations, we measure the apparent viscosity as a function of droplet concentration and pressure gradient, and provide evidence of a discontinuous shear thinning behavior, which occurs at a concentration-dependent value of the forcing. We further show that this response is associated with a nonequilibrium transition between a "hard" (or less deformable) phase, which is nearly jammed and flows very slowly, and a "soft" (or more deformable) phase, which flows much more easily. The soft phase is characterized by flow-induced time dependent shape deformations and internal currents, which are virtually absent in the hard phase. Close to the transition, we find sustained oscillations in both the droplet and fluid velocities. Polydisperse systems show similar phenomenology but with a smoother transition, and less regular oscillations.

A free-boundary model of a motile cell explains turning behavior

To understand shapes and movements of cells undergoing lamellipodial motility, we systematically explore minimal free-boundary models of actin-myosin contractility consisting of the force-balance and myosin transport equations. The models account for isotropic contraction proportional to myosin density, viscous stresses in the actin network, and constant-strength viscous-like adhesion. The contraction generates a spatially graded centripetal actin flow, which in turn reinforces the contraction via myosin redistribution and causes retraction of the lamellipodial boundary. Actin protrusion at the boundary counters the retraction, and the balance of the protrusion and retraction shapes the lamellipodium. The model analysis shows that initiation of motility critically depends on three dimensionless parameter combinations, which represent myosin-dependent contractility, a characteristic viscosity-adhesion length, and a rate of actin protrusion. When the contractility is sufficiently strong, cells break symmetry and move steadily along either straight or circular trajectories, and the motile behavior is sensitive to conditions at the cell boundary. Scanning of a model parameter space shows that the contractile mechanism of motility supports robust cell turning in conditions where short viscosity-adhesion lengths and fast protrusion cause an accumulation of myosin in a small region at the cell rear, destabilizing the axial symmetry of a moving cell.

To understand shapes and movements of simple motile cells, we systematically explore minimal models describing a cell as a two-dimensional actin-myosin gel with a free boundary. The models account for actin-myosin contraction balanced by viscous stresses in the actin gel and uniform adhesion. The myosin contraction causes the lamellipodial boundary to retract. Actin protrusion at the boundary counters the retraction, and the balance of protrusion and retraction shapes the cell. The models reproduce a variety of motile shapes observed experimentally. The analysis shows that the mechanical state of a cell depends on a small number of parameters. We find that when the contractility is sufficiently strong, cells break symmetry and move steadily along either straight or circular trajectory. Scanning model parameters shows that the contractile mechanism of motility supports robust cell turning behavior in conditions where deformable actin gel and fast protrusion destabilize the axial symmetry of a moving cell.

Minimal model of directed cell motility on patterned substrates

Crawling cell motility is vital to many biological processes such as wound healing and the immune response. Using a minimal model we investigate the effects of patterned substrate adhesiveness and biophysical cell parameters on the direction of cell motion. We show that cells with low adhesion site formation rates may move perpendicular to adhesive stripes while those with high adhesion site formation rates results in motility only parallel to the substrate stripes. We explore the effects of varying the substrate pattern geometry and the strength of actin polymerization on the directionality of the crawling cell. These results reveal that high strength of actin polymerization results in motion perpendicular to substrate stripes only when the substrate is relatively nonadhesive; in particular, this suggests potential applications in motile cell sorting and guiding on engineered substrates.

Motility of active nematic films driven by "active anchoring"

We provide a minimal model for an active nematic film in contact with both a solid substrate and a passive isotropic fluid, and explore its dynamics in one and two dimensions using a combination of hybrid Lattice Boltzmann simulations and analytical calculations. By imposing nematic anchoring at the substrate while active flows induce a preferred alignment at the interface ("active anchoring"), we demonstrate that directed fluid flow spontaneously emerges in cases where the two anchoring types are opposing. In one dimension, our model reduces to an analogue of a loaded elastic column. Here, the transition from a stationary to a motile state is akin to the buckling bifurcation, but offers the possibility to reverse the flow direction for a given set of parameters and boundary conditions solely by changing initial conditions. The two-dimensional variant of our model allows for additional tangential instabilities, and it is found that undulations form in the interface above a threshold activity. Our results might be relevant for the design of active microfluidic geometries or curvature-guided self-assembly.

Anchoring-driven spontaneous rotations in active gel droplets

We study the dynamics of an active gel droplet with imposed orientational anchoring (normal or planar) at its surface. We find that if the activity is large enough droplets subject to strong anchoring spontaneously start to rotate, with the sense of rotation randomly selected by fluctuations. Contractile droplets rotate only for planar anchoring and extensile ones only for normal anchoring. This is because such a combination leads to a pair of stable elastic deformations which creates an active torque to power the rotation. Interestingly, under these conditions there is a conflict between the anchoring promoted thermodynamically and that favoured by activity. By tuning activity and anchoring strength, we find a wealth of qualitatively different droplet morphologies and spatiotemporal patterns, encompassing steady rotations, oscillations, and more irregular trajectories. The spontaneous rotations we observe are fundamentally different from previously reported instances of rotating defects in active fluids as they require the presence of strong enough anchoring and entail significant droplet shape deformations.

Chase-and-run dynamics in cell motility and the molecular rupture of interacting active elastic dimers

Cell migration in morphogenesis and cancer metastasis typically involves interplay between different cell types. We construct and study a minimal, one-dimensional model composed of two different motile cells with each cell represented as an active elastic dimer. The interaction between the two cells via cadherins is modeled as a spring that can rupture beyond a threshold force as it undergoes dynamic loading from the interacting motile cells. We obtain a phase diagram consisting of chase-and-run dynamics and clumping dynamics as a function of the stiffness of the interaction spring and the threshold force and, therefore, posit that active rupture, or rupture via active forces, is a mechanosensitive means to regulate dynamics between cells. Since the parameters in the model differentiate between N- and E-cadherins, we make predictions for the interactions between a placodelike cell and a neural crestlike cell in a microchannel as well as discuss how our results inform chase-and-run dynamics found in a group of placode cells interacting with a group of neural crest cells. In particular, an argument was made in the latter case that the feedback between cadherins and cell-substrate interaction via integrins was necessary to obtain the chase-and-run behavior. Based on our two-cell results, we argue that this feedback accentuates, but is not necessary for, the chase-and-run behavior.

Contractile and chiral activities codetermine the helicity of swimming droplet trajectories

Active fluids are a class of nonequilibrium systems where energy is injected into the system continuously by the constituent particles themselves. Many examples, such as bacterial suspensions and actomyosin networks, are intrinsically chiral at a local scale, so that their activity involves torque dipoles alongside the force dipoles usually considered. Although many aspects of active fluids have been studied, the effects of chirality on them are much less known. Here, we study by computer simulation the dynamics of an unstructured droplet of chiral active fluid in three dimensions. Our model considers only the simplest possible combination of chiral and achiral active stresses, yet this leads to an unprecedented range of complex motilities, including oscillatory swimming, helical swimming, and run-and-tumble motion. Strikingly, whereas the chirality of helical swimming is the same as the microscopic chirality of torque dipoles in one regime, the two are opposite in another. Some of the features of these motility modes resemble those of some single-celled protozoa, suggesting that underlying mechanisms may be shared by some biological systems and synthetic active droplets.