Publications

Cell motility is fundamental to many biological processes, and cells exhibit a variety of migration patterns. Many motile cell types follow a universal law that connects their speed and persistency, a property that can originate from the intracellular transport of polarity cues due to the global actin retrograde flow. This mechanism was termed the "Universal Coupling between cell Speed and Persistency"(UCSP). Here we implement a simplified version of the UCSP mechanism in a coarse-grained "minimal-cell"model, which is composed of a three-dimensional vesicle that contains curved active proteins. This model spontaneously forms a lamellipodia-like motile cell shape, which is, however, sensitive and can depolarize into a nonmotile form due to random fluctuations or when interacting with external obstacles. The UCSP implementation introduces long-range inhibition, which stabilizes the motile phenotype. This allows our model to describe the robust polarity observed in cells and explain a large variety of cellular dynamics, such as the relation between cell speed and aspect ratio, cell-barrier scattering, and cellular oscillations in different types of geometric confinements.

Biological ensembles use collective intelligence to tackle challenges together, but suboptimal coordination can undermine the effectiveness of group cognition. Testing whether collective cognition exceeds that of the individual is often impractical since different organizational scales tend to face disjoint problems. One exception is the problem of navigating large loads through complex environments and toward a given target. People and ants stand out in their ability to efficiently perform this task not just individually but also as a group. This provides a rare opportunity to empirically compare problem-solving skills and cognitive traits across species and group sizes. Here, we challenge people and ants with the same \u201cpiano-movers\u201d load maneuvering puzzle and show that while ants perform more efficiently in larger groups, the opposite is true for humans. We find that although individual ants cannot grasp the global nature of the puzzle, their collective motion translates into emergent cognitive skills. They encode short-term memory in their internally ordered state and this allows for enhanced group performance. People comprehend the puzzle in a way that allows them to explore a reduced search space and, on average, outperform ants. However, when communication is restricted, groups of people resort to the most obvious maneuvers to facilitate consensus. This is reminiscent of ant behavior, and negatively impacts their performance. Our results exemplify how simple minds can easily enjoy scalability while complex brains require extensive communication to cooperate efficiently.

As wounds heal, embryos develop, cancer spreads, or asthma progresses, the cellular monolayer undergoes a glass transition between solid-like jammed and fluid-like flowing states. During some of these processes, the cells undergo an epithelial-to-mesenchymal transition (EMT): they acquire in-plane polarity and become motile. Thus, how motility drives the glassy dynamics in epithelial systems is critical for the EMT process. However, no analytical framework that is indispensable for deeper insights exists. Here, we develop such a theory inspired by a well-known glass theory. One crucial result of this work is that the confluency affects the effective persistence time-scale of active force, described by its rotational diffusivity, D^eff_r. D^eff_r differs from the bare rotational diffusivity, D_r, of the motile force due to cell shape dynamics, which acts to rectify the force dynamics: D^eff_r is equal to D_r when D_r is small and saturates when D_r is large. We test the theoretical prediction of D^eff_r and how it affects the relaxation dynamics in our simulations of the active Vertex model. This novel effect of D^eff_r is crucial to understanding the new and previously published simulation data of active glassy dynamics in epithelial monolayers.

In this study, we implement the deviatoric curvature model to examine dynamically triangulated surfaces with anisotropic membrane inclusions. The Monte-Carlo numerical scheme is devised to not only minimize the total bending energy of the membrane but also the in-plane nematic order of the inclusions by considering the mismatch between the curvature of the membrane and the intrinsic curvature of the inclusion. Neighboring inclusions can either attract with nearest-neighbor interaction or with a nematic interaction derived from liquid crystal theory. Orientational order determines whether vesicles fully covered with inclusions result in bulbs connected by necks or long tubes. Remarkably, when inclusions on vesicles with no vacancies interact non-nematically, a spontaneous local order can lead to a bulb transition which may have implications in cell or organelle division. Furthermore we find that average nematic order is inversely proportional to the number of thin necks formed in the vesicles. Our method shows good convergence and is suitable for further upgrades, for example to vesicles constrained by volume.

Cells often migrate on curved surfaces inside the body, such as curved tissues, blood vessels, or highly curved protrusions of other cells. Recent in vitro experiments provide clear evidence that motile cells are affected by the curvature of the substrate on which they migrate, preferring certain curvatures to others, termed \u201ccurvotaxis.\u201d The origin and underlying mechanism that gives rise to this curvature sensitivity are not well understood. Here, we employ a \u201cminimal cell\u201d model which is composed of a vesicle that contains curved membrane protein complexes, that exert protrusive forces on the membrane (representing the pressure due to actin polymerization). This minimal-cell model gives rise to spontaneous emergence of a motile phenotype, driven by a lamellipodia-like leading edge. By systematically screening the behavior of this model on different types of curved substrates (sinusoidal, cylinder, and tube), we show that minimal ingredients and energy terms capture the experimental data. The model recovers the observed migration on the sinusoidal substrate, where cells move along the grooves (minima), while avoiding motion along the ridges. In addition, the model predicts the tendency of cells to migrate circumferentially on convex substrates and axially on concave ones. Both of these predictions are verified experimentally, on several cell types. Altogether, our results identify the minimization of membrane-substrate adhesion energy and binding energy between the membrane protein complexes as key players of curvotaxis in cell migration.

Motile cells inside living tissues often encounter junctions, where their path branches into several alternative directions of migration. We present a theoretical model of cellular polarization for a cell migrating along a one-dimensional line, arriving at a symmetric Y junction and extending protrusions along the different paths that originate at the junction. The model predicts the spontaneous emergence of deterministic oscillations of growth and cellular polarization between competing protrusions during the directional decision-making process. The oscillations are modified by cellular noise but remain a dominant feature that affects the time it takes the cell to migrate across the junction. These predictions are confirmed experimentally for two different cell types (non-cancerous endothelial and cancerous glioma cells) migrating on a patterned network of thin adhesive lanes with junctions.

Animals must constantly make decisions on the move, such as when choosing among multiple options, or \u201ctargets,\u201d in space. Recent evidence suggests that this results from a recursive feedback between the (vectorial) neural representation of the targets and the resulting motion defined by this consensus, which then changes the egocentric neural representation of the options, and so on. Here we employ a simple model of this process both to explore how its dynamics accounts for the experimentally observed abruptly branching trajectories exhibited by animals during spatial decision-making, and to provide new insights into spatiotemporal computation. Essential neural dynamics, notably local excitation and long-range inhibition, are captured in our model via spin-system dynamics, with groups of Ising-spins representing neural \u201cactivity bumps\u201d corresponding to target directions (as in a neural ring-attractor network, for example). Analysis, employing a novel \u201cmean-field trajectory\u201d approach, reveals the nature of the spontaneous symmetry breaking bifurcations in the model that result in literal bifurcations in trajectory spaceand how it results in new geometric principles for spatiotemporal decision-making. We find that all bifurcation points, beyond the very first, fall on a small number of \u201cbifurcation curves.\u201d It is the spatial organization of these curves that is shown to be key to determining the shape of the trajectories, such as self-similar or space filling, exhibited during decision-making, irrespective of the trajectory's starting point. Furthermore, we find that a non-Euclidean (neural) representation of space (effectively an elliptic geometry) considerably reduces the number of bifurcation points in many geometrical configurations (including from an infinite number to only three), preventing endless indecision and promoting effective spatial decision-making. This suggests that a non-Euclidean neural representation of space may be expected to have evolved across species in order to facilitate spatial decision-making.

Active particle systems of interacting self-propelled particles offer a versatile framework for modeling complex systems. When employed to describe aspects of animal behavior, the complexity of animal movement and decision-making often requires the use of unique types of effective interactions between the particlesnotably nonreciprocal effective forces that do not obey the usual conservation laws of Newtonian mechanics. Here, we review two recent empirically motivated models of two very different types of animal behavior, where the behavior is described in terms of active particles which interact through nonreciprocal effective forces. The first model describes the dynamics of animal contests, wherein typically two rivals fight over a localized resource. The uniquely shaped effective potentials between the models \u201ccontestant particles\u201d manifest the adversarial nature of contest interactions and capture the dynamical essence of contest behavior in space and time. The second model describes the stabilization of cohesive swarms through long-range and adaptive gravity-like attraction. This \u201cadaptive gravity\u201d model explains the observed mass and velocity profiles of laboratory midge swarms. These examples demonstrate that theoretical models that use the framework of active particles to describe animal behavior can expand the scope of active-particle research as well as explain complex phenomena in animal behavior.

Collective cell migration, whereby cells adhere to form multi-cellular clusters that move as a single entity, play an important role in numerous biological processes, such as during development and cancer progression. Recent experimental work focused on migration of one-dimensional cellular clusters, confined to move along adhesive lanes, as a simple geometry in which to systematically study this complex system. One-dimensional migration also arises in the body when cells migrate along blood vessels, axonal projections, and narrow cavities between tissues. We explore here the modes of one-dimensional migration of cellular clusters (\u201ctrains\u201d) by implementing cell-cell interactions in a model of cell migration that contains a mechanism for spontaneous cell polarization. We go beyond simple phenomenological models of the cells as self-propelled particles by having the internal polarization of each cell depend on its interactions with the neighboring cells that directly affect the actin polymerization activity at the cell's leading edges. Both contact inhibition of locomotion and cryptic lamellipodia interactions between neighboring cells are introduced. We find that this model predicts multiple motility modes of the cell trains, which can have several different speeds for the same polarization pattern. Compared to experimental data, we find that Madin-Darby canine kidney cells are poised along the transition region where contact inhibition of locomotion and cryptic lamellipodia roughly balance each other, where collective migration speed is most sensitive to the values of the cell-cell interaction strength.

Protrusions at the leading-edge of a cell play an important role in sensing the extracellular cues during cellular spreading and motility. Recent studies provided indications that these protrusions wrap (coil) around the extracellular fibers. However, the physics of this coiling process, and the mechanisms that drive it, are not well understood. We present a combined theoretical and experimental study of the coiling of cellular protrusions on fibers of different geometry. Our theoretical model describes membrane protrusions that are produced by curved membrane proteins that recruit the protrusive forces of actin polymerization, and identifies the role of bending and adhesion energies in orienting the leading-edges of the protrusions along the azimuthal (coiling) direction. Our model predicts that the cells leading-edge coils on fibers with circular cross-section (above some critical radius), but the coiling ceases for flattened fibers of highly elliptical cross-section. These predictions are verified by 3D visualization and quantitation of coiling on suspended fibers using Dual-View light-sheet microscopy (diSPIM). Overall, we provide a theoretical framework, supported by experiments, which explains the physical origin of the coiling phenomenon.

One ubiquitous cellular structure for performing various tasks, such as spreading and migration over external surfaces, is the sheet-like protrusion called a lamellipodium, which propels the leading edge of the cell. Despite the detailed knowledge about the many components of this cellular structure, it is not yet fully understood how these components self-organize spatiotemporally to form lamellipodia. We review here recent theoretical works where we have demonstrated that membrane-bound protein complexes that have intrinsic curvature and recruit the protrusive forces of the cytoskeleton result in a simple, yet highly robust, organizing feedback mechanism that organizes the cytoskeleton and the membrane. This self-organization mechanism accounts for the formation of flat lamellipodia at the leading edge of cells spreading over adhesive substrates, allowing for the emergence of a polarized, motile 'minimal cell' model. The same mechanism describes how lamellipodia organize to drive robust engulfment of particles during phagocytosis and explains in simple physical terms the spreading and migration of cells over fibers and other curved surfaces. This Review highlights that despite the complexity of cellular composition, there might be simple general physical principles that are utilized by the cell to drive cellular shape dynamics.

Actin dynamics in cell motility, division, and phagocytosis is regulated by complex factors with multiple feedback loops, often leading to emergent dynamic patterns in the form of propagating waves of actin polymerization activity that are poorly understood. Many in the actin wave community have attempted to discern the underlying mechanisms using experiments and/or mathematical models and theory. Here, we survey methods and hypotheses for actin waves based on signaling networks, mechano-chemical effects, and transport characteristics, with examples drawn from Dictyostelium discoideum, human neutrophils, Caenorhabditis elegans, and Xenopus laevis oocytes. While experimentalists focus on the details of molecular components, theorists pose a central question of universality: Are there generic, model-independent, underlying principles, or just boundless cell-specific details? We argue that mathematical methods are equally important for understanding the emergence, evolution, and persistence of actin waves and conclude with a few challenges for future studies.

While moving, animals must frequently make decisions about their future travel direction, whether they are alone or in a group. Here we investigate this process for zebrafish (Danio rerio), which naturally move in cohesive groups. Employing state-of-the-art virtual reality, we study how real fish (RF) follow one or several moving, virtual conspecifics (leaders). These data are used to inform, and test, a model of social response that includes a process of explicit decision-making, whereby the fish can decide which of the virtual conspecifics to follow, or to follow in some average direction. This approach is in contrast with previous models where the direction of motion was based on a continuous computation, such as directional averaging. Building upon a simplified version of this model (Sridhar et al 2021 Proc. Natl Acad. Sci. 118 e2102157118), which was limited to a one-dimensional projection of the fish motion, we present here a model that describes the motion of the RF as it swims freely in two-dimensions. Motivated by experimental observations, the swim speed of the fish in this model uses a burst-and-coast swimming pattern, with the burst frequency being dependent on the distance of the fish from the followed conspecific(s). We demonstrate that this model is able to explain the observed spatial distribution of the RF behind the virtual conspecifics in the experiments, as a function of their average speed and number. In particular, the model naturally explains the observed critical bifurcations for a freely swimming fish, which appear in the spatial distributions whenever the fish makes a decision to follow only one of the virtual conspecifics, instead of following them as an averaged group. This model can provide the foundation for modeling a cohesive shoal of swimming fish, while explicitly describing their directional decision-making process at the individual level.

Eukaryotic cells intrinsically change their shape, by changing the composition of their membrane and by restructuring their underlying cytoskeleton. We present here further studies and extensions of a minimal physical model, describing a closed vesicle with mobile curved membrane protein complexes. The cytoskeletal forces describe the protrusive force due to actin polymerization which is recruited to the membrane by the curved protein complexes. We characterize the phase diagrams of this model, as function of the magnitude of the active forces, nearest-neighbor protein interactions and the proteins spontaneous curvature. It was previously shown that this model can explain the formation of lamellipodia-like flat protrusions, and here we explore the regimes where the model can also give rise to filopodia-like tubular protrusions. We extend the simulation with curved components of both convex and concave species, where we find the formation of complex ruffled clusters, as well as internalized invaginations that resemble the process of endocytosis and macropinocytosis. We alter the force model representing the cytoskeleton to simulate the effects of bundled instead of branched structure, resulting in shapes which resemble filopodia.

Cell spreading and motility on an adhesive substrate are driven by the active physical forces generated by the actin cytoskeleton. We have recently shown that coupling curved membrane complexes to protrusive forces, exerted by the actin polymerization that they recruit, provides a mechanism that can give rise to spontaneous membrane shapes and patterns. In the presence of an adhesive substrate, this model was shown to give rise to an emergent motile phenotype, resembling a motile cell. Here, we utilize this \u201cminimal-cell\u201d model to explore the impact of external shear flow on the cell shape and migration on a uniform adhesive flat substrate. We find that in the presence of shear the motile cell reorients such that its leading edge, where the curved active proteins aggregate, faces the shear flow. The flow-facing configuration is found to minimize the adhesion energy by allowing the cell to spread more efficiently over the substrate. For the non-motile vesicle shapes, we find that they mostly slide and roll with the shear flow. We compare these theoretical results with experimental observations, and suggest that the tendency of many cell types to move against the flow may arise from the very general, and non-cell-type-specific mechanism predicted by our model.

We present a general theoretical model for the spatio-temporal dynamics of animal contests. Inspired by interactions between physical particles, the model is formulated in terms of effective interaction potentials, which map typical elements of contest behaviour into empirically verifiable rules of contestant motion. This allows us to simulate the observable dynamics of contests in various realistic scenarios, notably in dyadic contests over a localized resource. Assessment strategies previously formulated in game-theoretic models, as well as the effects of fighting costs, can be described as variations in our model's parameters. Furthermore, the trends of contest duration associated with these assessment strategies can be derived and understood within the model. Detailed description of the contestants' motion enables the exploration of spatio-temporal properties of asymmetric contests, such as the emergence of chase dynamics. Overall, our framework aims to bridge the growing gap between empirical capabilities and theory in this widespread aspect of animal behaviour.

During mitosis, cells round up and utilize the interphase adhesion sites within the fibrous extracellular matrix (ECM) as guidance cues to orient the mitotic spindles. Here, using suspended ECM-mimicking nanofiber networks, we explore mitotic outcomes and error distribution for various interphase cell shapes. Elongated cells attached to single fibers through two focal adhesion clusters (FACs) at their extremities result in perfect spherical mitotic cell bodies that undergo significant 3-dimensional (3D) displacement while being held by retraction fibers (RFs). Increasing the number of parallel fibers increases FACs and retraction fiber-driven stability, leading to reduced 3D cell body movement, metaphase plate rotations, increased interkinetochore distances, and significantly faster division times. Interestingly, interphase kite shapes on a crosshatch pattern of four fibers undergo mitosis resembling single-fiber outcomes due to rounded bodies being primarily held in position by RFs from two perpendicular suspended fibers. We develop a cortexastral microtubule analytical model to capture the retraction fiber dependence of the metaphase plate rotations. We observe that reduced orientational stability, on single fibers, results in increased monopolar mitotic defects, while multipolar defects become dominant as the number of adhered fibers increases. We use a stochastic Monte Carlo simulation of centrosome, chromosome, and membrane interactions to explain the relationship between the observed propensity of monopolar and multipolar defects and the geometry of RFs. Overall, we establish that while bipolar mitosis is robust in fibrous environments, the nature of division errors in fibrous microenvironments is governed by interphase cell shapes and adhesion geometries.

The cell migration cycle, well-established in 2D, proceeds with forming new protrusive structures at the cell membrane and subsequent redistribution of contractile machinery. Three-dimensional (3D) environments are complex and composed of 1D fibers, and 1D fibers are shown to recapitulate essential features of 3D migration. However, the establishment of protrusive activity at the cell membrane and contractility in 1D fibrous environments remains partially understood. Here the role of membrane curvature regulator IRSp53 is examined as a coupler between actin filaments and plasma membrane during cell migration on single, suspended 1D fibers. IRSp53 depletion reduced cell-length spanning actin stress fibers that originate from the cell periphery, protrusive activity, and contractility, leading to uncoupling of the nucleus from cellular movements. A theoretical model capable of predicting the observed transition of IRSp53-depleted cells from rapid stick-slip migration to smooth and slower migration due to reduced actin polymerization at the cell edges is developed, which is verified by direct measurements of retrograde actin flow using speckle microscopy. Overall, it is found that IRSp53 mediates actin recruitment at the cellular tips leading to the establishment of cell-length spanning fibers, thus demonstrating a unique role of IRSp53 in controlling cell migration in 3D.

Phagocytosis is the process of engulfment and internalization of comparatively large particles by cells, and plays a central role in the functioning of our immune system. We study the process of phagocytosis by considering a simplified coarse grained model of a three-dimensional vesicle, having a uniform adhesion interaction with a rigid particle, and containing curved membrane-bound protein complexes or curved membrane nano-domains, which in turn recruit active cytoskeletal forces. Complete engulfment is achieved when the bending energy cost of the vesicle is balanced by the gain in the adhesion energy. The presence of curved (convex) proteins reduces the bending energy cost by self-organizing with a higher density at the highly curved leading edge of the engulfing membrane, which forms the circular rim of the phagocytic cup that wraps around the particle. This allows the engulfment to occur at much smaller adhesion strength. When the curved membrane-bound protein complexes locally recruit actin polymerization machinery, which leads to outward forces being exerted on the membrane, we found that engulfment is achieved more quickly and at a lower protein density. We consider spherical and non-spherical particles and found that non-spherical particles are more difficult to engulf in comparison to the spherical particles of the same surface area. For non-spherical particles, the engulfment time crucially depends on the initial orientation of the particles with respect to the vesicle. Our model offers a mechanism for the spontaneous self-organization of the actin cytoskeleton at the phagocytic cup, in good agreement with recent high-resolution experimental observations.

Cells engulf larger particles as a part of the immune response, and also during nutrient uptake, drug delivery and pathogen invasion, via the process known as \u201cphagocytosis\u201d. In this chapter, we discuss the mechanism of phagocytosis by considering a simplified coarse grained model of a three-dimensional vesicle, having uniform adhesion interaction with a rigid particle, and containing curved membrane-bound protein complexes (CMC), which in turn recruit active cytoskeletal forces. Complete engulfment is achieved when the bending energy cost of the vesicle is balanced by the gain in the adhesion energy. The presence of convex CMC reduces the bending energy cost by self-organizing with higher density at the highly curved leading edge of the engulfing membrane, which forms the circular rim of the phagocytic cup that wraps around the particle. This allows the engulfment to occur at reduced adhesion strength. When the CMC locally recruits actin polymerization that exerts outward forces on the membrane, we find that engulfment is achieved more quickly and at a lower density of CMC. We consider both spherical as well as non-spherical particles (spheroids, spherocylinders, dumb-bell etc.), and find that non-spherical particles are more difficult to engulf in comparison to the spherical particles of the same surface area. For non-spherical particles, the engulfment time crucially depends upon the initial orientation of the particles with respect to the vesicle. Our model offers a mechanism for the spontaneous self-organization of the actin cytoskeleton at the phagocytic cup, in good agreement with recent high-resolution experimental observations.

Eukaryotic cells have a flexible plasma membrane that allows them to change their shapes to fulfill various biological functions. To modify their shapes, cells can utilize specialized proteins that bind to the membrane and have an intrinsic shape that deforms it. In addition, the cells can exert forces on the membrane using their cytoskeleton, for example, by directing polymerization of actin biopolymers against the membrane at targeted locations. This chapter provides an introduction to the physical principles behind these two generic mechanisms of membrane deformations in living cells.

Cell shape can dynamically change to accommodate a variety of tasks. One mechanism for deforming the cellular membrane into the desired shape is through the use of curved membrane proteins (CMP). These proteins are often associated with the recruitment of the cytoskeleton, which then applies active forces that reshape the membrane, forming buds, necks, ruffles, or protrusions. In this review, we present the results of a simplified coarse-grained model of a vesicle with mobile CMP and active protrusive forces. We present the phase space of solutions for the passive and active force cases and find distinct classes of shapes: uniform, isolated buds, long finger-like protrusions, and pancake-like with proteins aggregated at the rim. The model is finally complemented with an inclusion of a uniform adhesion energy with a flat substrate, revealing that when curved proteins induce protrusive forces, a mechanism of efficient spreading is possible in the form of sheet-like lamellipodia, constituting a minimal system underlying cell motility.

Dense active matter, in the fluid or amorphous-solid form, has generated intense interest as a model for the dynamics inside living cells and multicellular systems. An extension of the random first-order transition theory (RFOT) to include activity was developed, whereby the activity of the individual particles was added to the free energy of the system in the form of the potential energy of an active particle, trapped by a harmonic potential that describes the effective confinement by the surrounding medium. This active-RFOT model was shown to successfully account for the dependence of the structural relaxation time in the active glass, extracted from simulations, as a function of the activity parameters: the magnitude of the active force (f ₀) and its persistence time (τ _p ). However, significant deviations were found in the limit of large activity (large f ₀ and/or τ _p ). Here we extend the active-RFOT model to high activity using an activity-dependent harmonic confining potential, which we solve self-consistently. The extended model predicts qualitative changes in the high activity regime, which agree with the results of simulations in both three-dimensional and two-dimensional models of active glass.

Cells remodel their cytoplasm with force-generating cytoskeletal motors. Their activity generates random forces that stir the cytoplasm, agitating and displacing membrane-bound organelles like the nucleus in somatic and germ cells. These forces are transmitted inside the nucleus, yet their consequences on liquid-like biomolecular condensates residing in the nucleus remain unexplored. Here, we probe experimentally and computationally diverse nuclear condensates, that include nuclear speckles, Cajal bodies, and nucleoli, during cytoplasmic remodeling of female germ cells named oocytes. We discover that growing mammalian oocytes deploy cytoplasmic forces to timely impose multiscale reorganization of nuclear condensates for the success of meiotic divisions. These cytoplasmic forces accelerate nuclear condensate collision-coalescence and molecular kinetics within condensates. Disrupting the forces decelerates nuclear condensate reorganization on both scales, which correlates with compromised condensate-associated mRNA processing and hindered oocyte divisions that drive female fertility. We establish that cytoplasmic forces can reorganize nuclear condensates in an evolutionary conserved fashion in insects. Our work implies that cells evolved a mechanism, based on cytoplasmic force tuning, to functionally regulate a broad range of nuclear condensates across scales. This finding opens new perspectives when studying condensate-associated pathologies like cancer, neurodegeneration and viral infections.

Social groups often need to overcome differences in individual interests and knowledge to reach consensus decisions. Here, we combine experiments and modeling to study conflict resolution in emigrating ant colonies during binary nest selection. We find that cohesive emigration, without fragmentation, is achieved only by intermediate-sized colonies. We then impose a conflict regarding the desired emigration target between colony subgroups. This is achieved using an automated selective gate system that manipulates the information accessible to each ant. Under this conflict, we find that individuals concede their potential benefit to promote social consensus. In particular, colonies resolve the conflict imposed by a persistent minority through \u201cmajority concession,\u201d wherein a majority of ants that hold first-hand knowledge regarding the superior quality nest choose to reside in the inferior one. This outcome is unlikely in social groups of selfish individuals and emphasizes the importance of group cohesion in eusocial societies.

Competition among animals for resources, notably food, territories, and mates, is ubiquitous at all scales of life. This competition is often resolved through contests among individuals, which are commonly understood according to their outcomes and in particular, how these outcomes depend on decision-making by the contestants. Because they are restricted to end-point predictions, these approaches cannot predict real-time or real-space dynamics of animal contest behavior. This limitation can be overcome by studying systems that feature typical contest behavior while being simple enough to track and model. Here, we propose to use such systems to construct a theoretical framework that describes real-time movements and behaviors of animal contestants. We study the spatiotemporal dynamics of contests in an orb-weaving spider, in which all the common elements of animal contests play out. The confined arena of the web, on which interactions are dominated by vibratory cues in a two-dimensional space, simplifies the analysis of interagent interactions. We ask whether these seemingly complex decision-makers can be modeled as interacting active particles responding only to effective forces of attraction and repulsion due to their interactions. By analyzing the emergent dynamics of \u201ccontestant particles,\u201d we provide mechanistic explanations for real-time dynamical aspects of animal contests, thereby explaining competitive advantages of larger competitors and demonstrating that complex decision-making need not be invoked in animal contests to achieve adaptive outcomes. Our results demonstrate that physics-based classification and modeling, in terms of effective rules of interaction, provide a powerful framework for understanding animal contest behaviors.

Choosing among spatially distributed options is a central challenge for animals, from deciding among alternative potential food sources or refuges to choosing with whom to associate. Using an integrated theoretical and experimental approach (employing immersive virtual reality), we consider the interplay between movement and vectorial integration during decision-making regarding two, or more, options in space. In computational models of this process, we reveal the occurrence of spontaneous and abrupt "critical" transitions (associated with specific geometrical relationships) whereby organisms spontaneously switch from averaging vectorial information among, to suddenly excluding one among, the remaining options. This bifurcation process repeats until only one option-the one ultimately selected-remains. Thus, we predict that the brain repeatedly breaks multichoice decisions into a series of binary decisions in space-time. Experiments with fruit flies, desert locusts, and larval zebrafish reveal that they exhibit these same bifurcations, demonstrating that across taxa and ecological contexts, there exist fundamental geometric principles that are essential to explain how, and why, animals move the way they do.

Mature red blood cells (RBCs) lack internal organelles and canonical defense mechanisms, making them both a fascinating host cell, in general, and an intriguing choice for the deadly malaria parasite Plasmodium falciparum (Pf), in particular. Pf, while growing inside its natural host, the human RBC, secretes multipurpose extracellular vesicles (EVs), yet their influence on this essential host cell remains unknown. Here we demonstrate that Pf parasites, cultured in fresh human donor blood, secrete within such EVs assembled and functional 20S proteasome complexes (EV-20S). The EV-20S proteasomes modulate the mechanical properties of naïve human RBCs by remodeling their cytoskeletal network. Furthermore, we identify four degradation targets of the secreted 20S proteasome, the phosphorylated cytoskeletal proteins β-adducin, ankyrin-1, dematin and Epb4.1. Overall, our findings reveal a previously unknown 20S proteasome secretion mechanism employed by the human malaria parasite, which primes RBCs for parasite invasion by altering membrane stiffness, to facilitate malaria parasite growth.

The last decade has seen a surge of evidence supporting the existence of the transition of the multicellular tissue from a collective material phase that is regarded as being jammed to a collective material phase that is regarded as being unjammed. The jammed phase is solid-like and effectively frozen, and therefore is associated with tissue homeostasis, rigidity, and mechanical stability. The unjammed phase, by contrast, is fluid-like and effectively melted, and therefore is associated with mechanical fluidity, plasticity and malleability that are required in dynamic multicellular processes that sculpt organ microstructure. Such multicellular sculpturing, for example, occurs during embryogenesis, growth and remodeling. Although unjamming and jamming events in the multicellular collective are reminiscent of those that occur in the inert granular collective, such as grain in a hopper that can flow or clog, the analogy is instructive but limited, and the implications for cell biology remain unclear. Here we ask, are the cellular jamming transition and its inverse the unjamming transition mere epiphenomena? That is, are they dispensable downstream events that accompany but neither cause nor quench these core multicellular processes? Drawing from selected examples in developmental biology, here we suggest the hypothesis that, to the contrary, the graded departure from a jammed phase enables controlled degrees of malleability as might be required in developmental dynamics. We further suggest that the coordinated approach to a jammed phase progressively slows those dynamics and ultimately enables long-term mechanical stability as might be required in the mature homeostatic multicellular tissue.

Cell migration is astoundingly diverse. Molecular signatures, cell-cell interactions, and environmental structures each play their part in shaping cell motion, yielding numerous morphologies and migration modes. Nevertheless, in recent years, a simple unifying law was found to describe cell migration across many different cell types and contexts: faster cells turn less frequently. This universal coupling between speed and persistence (UCSP) was explained by retrograde actin flow from front to back, but it remains unclear how this mechanism generalizes to cells with complex shapes and cells migrating in structured environments, which may not have a well-defined front-to-back orientation. Here, we present an in-depth characterization of an existing cellular Potts model, in which cells polarize dynamically from a combination of local actin dynamics (stimulating protrusions) and global membrane tension along the perimeter (inhibiting protrusions). We first show that the UCSP emerges spontaneously in this model through a cross talk of intracellular mechanisms, cell shape, and environmental constraints, resembling the dynamic nature of cell migration in vivo. Importantly, we find that local protrusion dynamics suffice to reproduce the UCSPeven in cases in which no clear global, front-to-back polarity exists. We then harness the spatial nature of the cellular Potts model to show how cell shape dynamics limit both the speed and persistence a cell can reach and how a rigid environment such as the skin can restrict cell motility even further. Our results broaden the range of potential mechanisms underlying the speed-persistence coupling that has emerged as a fundamental property of migrating cells.

Cooperative transport of large food loads by Paratrechina longicornis ants demands repeated decision-making. Inspired by the Evidence Accumulation (EA) model classically used to describe decision-making in the brain, we conducted a binary choice experiment where carrying ants rely on social information to choose between two paths. We found that the carried load performs a biased random walk that continuously alternates between the two options. We show that this motion constitutes a physical realization of the abstract EA model and exhibits an emergent version of the psychophysical Webers law. In contrast to the EA model, we found that the loads random step size is not fixed but, rather, varies with both evidence and circumstances. Using theoretical modeling we show that variable step size expands the scope of the EA model from isolated to sequential decisions. We hypothesize that this phenomenon may also be relevant in neuronal circuits that perform sequential decisions.

Eukaryotic cells adhere to extracellular matrix during the normal development of the organism, forming static adhesion as well as during cell motility. We study this process by considering a simplified coarse-grained model of a vesicle that has uniform adhesion energy with a flat substrate, mobile-curved membrane proteins and active forces. We find that a high concentration of curved proteins alone increases the spreading of the vesicle, by the self-organization of the curved proteins at the high-curvature vesiclesubstrate contact line, thereby reducing the bending energy penalty at the vesicle rim. This is most significant in the regime of low bare vesiclesubstrate adhesion. When these curved proteins induce protrusive forces, representing the actin cytoskeleton, we find efficient spreading, in the form of sheet-like lamellipodia. Finally, the same mechanism of spreading is found to include a minimal set of ingredients needed to give rise to motile phenotypes.Correction:In this article, the following addition has been made to the acknowledgments:\u201cN.S.G. acknowledges support by the Ben May Center for Theory and Computation.\u201dThe original article has been corrected.

To study the mechanisms controlling front-rear polarity in migrating cells, we used zebrafish primordial germ cells (PGCs) as an in vivo model. We find that polarity of bleb-driven migrating cells can be initiated at the cell front, as manifested by actin accumulation at the future leading edge and myosin-dependent retrograde actin flow toward the other side of the cell. In such cases, the definition of the cell front, from which bleb-inhibiting proteins such as Ezrin are depleted, precedes the establishment of the cell rear, where those proteins accumulate. Conversely, following cell division, the accumulation of Ezrin at the cleavage plane is the first sign for cell polarity and this aspect of the cell becomes the cell back. Together, the antagonistic interactions between the cell front and back lead to a robust polarization of the cell. Furthermore, we show that chemokine signaling can bias the establishment of the front-rear axis of the cell, thereby guiding the migrating cells toward sites of higher levels of the attractant. We compare these results to a theoretical model according to which a critical value of actin treadmilling flow can initiate a positive feedback loop that leads to the generation of the front-rear axis and to stable cell polarization. Together, our in vivo findings and the mathematical model, provide an explanation for the observed nonoriented migration of primordial germ cells in the absence of the guidance cue, as well as for the directed migration toward the region where the gonad develops.

Moving cells can sense and respond to physical features of the microenvironment; however, in vivo, the significance of tissue topography is mostly unknown. Here, we used Drosophila border cells, an established model for in vivo cell migration, to study how chemical and physical information influences path selection. Although chemical cues were thought to be sufficient, live imaging, genetics, modeling, and simulations show that microtopography is also important. Chemoattractants promote predominantly posterior movement, whereas tissue architecture presents orthogonal information, a path of least resistance concentrated near the center of the egg chamber. E-cadherin supplies a permissive haptotactic cue. Our results provide insight into how cells integrate and prioritize topographical, adhesive, and chemoattractant cues to choose one path among many.

In swarms of flying insects, the motions of individuals are largely uncoordinated with those of their neighbours, unlike the highly ordered motion of bird flocks. However, it has been observed that insects may transiently form pairs with synchronized relative motion while moving through the swarm. The origin of this phenomenon remains an open question. In particular, it is not known if pairing is a new behavioural process or whether it is a natural by-product of typical swarming behaviour. Here, using an 'adaptive-gravity' model that proposes that insects interact via long-range gravity-like acoustic attractions that are modulated by the total background sound (via 'adaptivity' or fold-change detection) and that reproduces measured features of real swarms, we show that pair formation can indeed occur without the introduction of additional behavioural rules. In the model, pairs form robustly whenever two insects happen to move together from the centre of the swarm (where the background sound is high) towards the swarm periphery (where the background sound is low). Due to adaptivity, the attraction between the pair increases as the background sound decreases, thereby forming a bound state since their relative kinetic energy is smaller than their pair-potential energy. When the pair moves into regions of high background sound, however, the process is reversed and the pair may break up. Our results suggest that pairing should appear generally in biological systems with long-range attraction and adaptive sensing, such as during chemotaxis-driven cellular swarming.

During migration, cells exhibit a rich variety of seemingly random migration patterns, which makes unraveling the underlying mechanisms that control cell migration a difficult challenge. For efficient migration, cells require a mechanism for polarization, so that traction forces are produced in the direction of motion, while adhesion is released to allow forward migration. To simplify the study of this process, cells have been studied when placed along one-dimensional tracks, where single cells exhibit both smooth and stick-slip migration modes. The stick-slip motility mode is characterized by protrusive motion at the cell front, coupled with a slow elongation of the cell, which is followed by a rapid retraction of the cell rear. In this study, we explore a minimal physical model that couples the force applied on the adhesion bonds to the length variations of the cell and to the traction forces applied by the polarized actin retrograde flow. We show that the rich spectrum of cell migration patterns emerges from this model as different deterministic dynamical phases. This result suggests a source for the large cell-to-cell variability (CCV) in cell migration patterns observed in single cells over time and within cell populations: fluctuations in the cellular components, such as adhesion strength or polymerization activity, can shift the cells from one migration mode to another, due to crossing the dynamical phase transition lines. Temporal noise is shown to drive random changes in the cellular polarization direction, which is enhanced during the stick-slip migration mode. The model contains an emergent critical length for cell polarization, whereby cells that retract below this length loose polarity, and are prone to making direction changes in migration. These results offer a new framework to explain experimental observations of migrating cells, resulting from noisy switching between underlying deterministic migration modes.

During the last decade, intracellular actin waves have attracted much attention due to their essential role in various cellular functions, ranging from motility to cytokinesis. Experimental methods have advanced significantly and can capture the dynamics of actin waves over a large range of spatio-temporal scales. However, the corresponding coarse-grained theory mostly avoids the full complexity of this multi-scale phenomenon. In this perspective, we focus on a minimal continuum model of activatorinhibitor type and highlight the qualitative role of mass conservation, which is typically overlooked. Specifically, our interest is to connect between the mathematical mechanisms of pattern formation in the presence of a large-scale mode, due to mass conservation, and distinct behaviors of actin waves. View Full-Text

Cells adhered to an external solid substrate are observed to exhibit rich dynamics of actin structures on the basal membrane, which are distinct from those observed on the dorsal (free) membrane. Here we explore the dynamics of curved membrane proteins, or protein complexes, that recruit actin polymerization when the membrane is confined by the solid substrate. Such curved proteins can induce the spontaneous formation of membrane protrusions on the dorsal side of cells. However, on the basal side of the cells, such protrusions can only extend as far as the solid substrate and this constraint can convert such protrusions into propagating wave-like structures. We also demonstrate that adhesion molecules can stabilize localized protrusions that resemble some features of podosomes. This coupling of curvature and actin forces may underlie the differences in the observed actin-membrane dynamics between the basal and dorsal sides of adhered cells.

Motivated by the dynamics of particles embedded in active gels, both in vitro and inside the cytoskeleton of living cells, we study an active generalization of the classical trap model. We demonstrate that activity leads to dramatic modifications in the diffusion compared to the thermal case: the mean square displacement becomes subdiffusive, spreading as a power law in time, when the trap depth distribution is a Gaussian and is slower than any power law when it is drawn from an exponential distribution. The results are derived for a simple, exactly solvable, case of harmonic traps. We then argue that the results are robust for more realistic trap shapes when the activity is strong.

Nucleus centering in mouse oocytes results from a gradient of actin-positive vesicle activity and is essential for developmental success. Here, we analyze 3D model simulations to demonstrate how a gradient in the persistence of actin-positive vesicles can center objects of different sizes. We test model predictions by tracking the transport of exogenous passive tracers. The gradient of activity induces a centering force, akin to an effective pressure gradient, leading to the centering of oil droplets with velocities comparable to nuclear ones. Simulations and experimental measurements show that passive particles subjected to the gradient exhibit biased diffusion toward the center. Strikingly, we observe that the centering mechanism is maintained in meiosis I despite chromosome movement in the opposite direction; thus, it can counteract a process that specifically off-centers the spindle. In conclusion, our findings reconcile how common molecular players can participate in the two opposing functions of chromosome centering versus off-centering.

Previous work has suggested that disordered swarms of flying insects can be well modeled as self-gravitating systems, as long as the "gravitational"interaction is adaptive. Motivated by this work, we compare the predictions of the classic, mean-field King model for isothermal globular clusters to observations of insect swarms. Detailed numerical simulations of regular and adaptive gravity allow us to expose the features of the swarms' density and velocity profiles that are due to long-range interactions and are captured by the King model phenomenology, and those that are due to adaptivity and short-range repulsion. Our results provide further support for adaptive gravity as a model for swarms.

Excitable pulses are among the most widespread dynamical patterns that occur in many different systems, ranging from biological cells to chemical reactions and ecological populations. Traditionally, the mutual annihilation of two colliding pulses is regarded as their prototypical signature. Here we show that colliding excitable pulses may exhibit solitonlike crossover and pulse nucleation if the system obeys a mass conservation constraint. In contrast to previous observations in systems without mass conservation, these alternative collision scenarios are robustly observed over a wide range of parameters. We demonstrate our findings using a model of intracellular actin waves since, on time scales of wave propagations over the cell scale, cells obey conservation of actin monomers. The results provide a key concept to understand the ubiquitous occurrence of actin waves in cells, suggesting why they are so common, and why their dynamics is robust and long-lived.

The dynamics of active particles is of interest at many levels and is the focus of theoretical and experimental research. There have been many attempts to describe the dynamics of particles affected by random active forces in terms of an effective temperature. This kind of description is tempting due to the similarities (or lack thereof) to systems in or near thermal equilibrium. However, the generality and validity of the effective temperature is not yet fully understood. Here we study the dynamics of trapped particles subjected to both thermal and active forces. The particles are not overdamped. Expressions for the effective temperature due to the potential and kinetic energies are derived, and they differ from each other. A third possible effective temperature can be derived from the escape time of the particle from the trap, using a Kramers-like expression for the mean escape time. We find that over a large fraction of the parameter space, the potential energy effective temperature is in agreement with the escape temperature, while the kinetic effective temperature only agrees with the former two in the overdamped limit. Moreover, we show that the specific implementation of the random active force, and not only its first two moments and the two point autocorrelation function, affects the escape-time distribution.

Bleb-type cellular protrusions play key roles in a range of biological processes. It was recently found that bleb growth is facilitated by a local supply of membrane from tubular invaginations, but the interplay between the expanding bleb and the membrane tubes remains poorly understood. On the one hand, the membrane area stored in tubes may serve as a reservoir for bleb expansion. On the other hand, the sequestering of excess membrane in stabilized invaginations may effectively increase the cell membrane tension, which suppresses spontaneous protrusions. Here, we investigate this duality through physical modeling and in vivo experiments. In agreement with observations, our model describes the transition into a tube-flattening mode of bleb expansion while also predicting that the blebbing rate is impaired by elevating the concentration of the curved membrane proteins that form the tubes. We show both theoretically and experimentally that the stabilizing effect of tubes could be counterbalanced by the cortical myosin contractility. Our results largely suggest that proteins able to induce membrane tubulation, such as those containing N-BAR domains, can buffer the effective membrane tension-a master regulator of all cell deformations.

Migratory cells use distinct motility modes to navigate different microenvironments, but it is unclear whether these modes rely on the same core set of polarity components. To investigate this, we disrupted actin-related protein 2/3 (Arp2/3) and the WASP-family verprolin homologous protein (WAVE) complex, which assemble branched actin networks that are essential for neutrophil polarity and motility in standard adherent conditions. Surprisingly, confinement rescues polarity and movement of neutrophils lacking these components, revealing a processive bleb-based protrusion program that is mechanistically distinct from the branched actin-based protrusion program but shares some of the same core components and underlying molecular logic. We further find that the restriction of protrusion growth to one site does not always respond to membrane tension directly, as previously thought, but may rely on closely linked properties such as local membrane curvature. Our work reveals a hidden circuit for neutrophil polarity and indicates that cells have distinct molecular mechanisms for polarization that dominate in different microenvironments.

In multi-cellular organisms, the migration of cohesive clusters of cells containing many individual cells is a common occurrence. Examples include the migration of cells during processes such as the development of the embryo, wound healing, immune response, and the spread of cancer. The migration process depends not only on the traction forces applied by the cluster on its surroundings, in order to move, but also on the viscoelastic properties of both the surrounding matrix and the migrating cellular cluster. Characterizing the viscoelastic properties of the cluster, its environment and the forces within the cluster, in great detail, is difficult both invitro and certainly in-vivo. We review here several examples where theoretical studies using simplified models can be used to gain insights into the basic underlying mechanisms that control the cellular migration patterns.

Eukaryote cells have a flexible shape, which dynamically changes according to the function performed by the cell. One mechanism for deforming the cell membrane into the desired shape is through the expression of curved membrane proteins. Furthermore, these curved membrane proteins are often associated with the recruitment of the cytoskeleton, which then applies active forces that deform the membrane. This coupling between curvature and activity was previously explored theoretically in the linear limit of small deformations, and low dimensionality. Here we explore the unrestricted shapes of vesicles that contain active curved membrane proteins, in three-dimensions, using Monte-Carlo numerical simulations. The activity of the proteins is in the form of protrusive forces that push the membrane outwards, as may arise from the cytoskeleton of the cell due to actin or microtubule polymerization occurring near the membrane. For proteins that have an isotropic convex shape, the additional protrusive force enhances their tendency to aggregate and form membrane protrusions (buds). In addition, we find another transition from deformed spheres with necklace type aggregates, to flat pancake-shaped vesicles, where the curved proteins line the outer rim. This second transition is driven by the active forces, coupled to the spontaneous curvature, and the resulting configurations may shed light on the formation of sheet-like protrusions and lamellipodia of adhered and motile cells.

We study a model for the motion of a tracer particle inside an active gel, exposing the properties of the van Hove distribution of the particle displacements. Active events of a typical force magnitude can give rise to non-Gaussian distributions having exponential tails or side peaks. The side peaks are predicted to appear when the local bulk elasticity of the gel is large enough and few active sources are dominant. We explain the regimes of the different distributions and study the structure of the side peaks for active sources that are susceptible to the elastic stress that they cause inside the gel. We show how the van Hove distribution is altered by both the duty cycle of the active sources and their susceptibility, and suggest it as a sensitive probe to analyze microrheology data in active systems with restoring elastic forces.

An introduction to the physical properties of living active matter at the mesoscopic scale (tens of nanometers to micrometers) and their unique features compared with dead, nonactive matter is presented. This field of research is increasingly denoted as biological physics where physics includes chemical physics, soft matter physics, hydrodynamics, mechanics, and the related engineering sciences. The focus is on the emergent properties of these systems and their collective behavior, which results in active self-organization and how they relate to cellular-level biological function. These include locomotion (cell motility and migration) forces that give rise to cell division, the growth and form of cellular assemblies in development, the beating of heart cells, and the effects of mechanical perturbations such as shear flow (in the bloodstream) or adhesion to other cells or tissues. An introduction to the fundamental concepts and theory with selected experimental examples related to the authors' own research is presented, including red-blood-cell membrane fluctuations, motion of the nucleus within an egg cell, self-contracting acto-myosin gels, and structure and beating of heart cells (cardiomyocytes), including how they can be driven by an oscillating, mechanical probe.

The dynamics within active fluids, driven by internal activity of the self-propelled particles, is a subject of intense study in non-equilibrium physics. These systems have been explored using simulations, where the motion of a passive tracer particle is followed. Similar studies have been carried out for a soft glassy material that is driven by shearing its boundaries. In both types of systems the non-equilibrium motion have been quantified by defining a set of "effective temperatures", using both the tracer particle kinetic energy and the fluctuation-dissipation relation. We demonstrate that these effective temperatures extracted from the many-body simulations fit analytical expressions that are obtained for a single active particle inside a visco-elastic fluid. This result provides testable predictions and suggests a unified description for the dynamics inside active systems.

Certain malignant cancer cells form clusters in a chemoattractant gradient, which can spontaneously show three different phases of motion: translational, rotational, and random. Guided by our experiments on the motion of two-dimensional clusters in vitro, we developed an agent-based model in which the cells form a cohesive cluster due to attractive and alignment interactions. We find that when cells at the cluster rim are more motile, all three phases of motion coexist, in agreement with our observations. Using the model, we show that the transitions between different phases are driven by competition between an ordered rim and a disordered core accompanied by the creation and annihilation of topological defects in the velocity field. The model makes specific predictions, which we verify with our experimental data. Our results suggest that heterogeneous behavior of individuals, based on local environment, can lead to novel, experimentally observed phases of collective motion.

Background/Aims: Transient nanometric cholesterol-and sphingolipid-enriched domains, called rafts, are characterized by higher lipid order as compared to surrounding lipids. Here, we asked whether the seminal concept of highly ordered rafts could be refined with the presence of lipid domains exhibiting different enrichment in cholesterol and sphingomyelin and association with erythrocyte curvature areas. We also investigated how differences in lipid order between domains and surrounding membrane (bulk) are regulated and whether changes in order differences could participate to erythrocyte deformation and vesiculation. Methods: We used the fluorescent hydration-and membrane packing-sensitive probe Laurdan to determine by imaging mode the Generalized Polarization (GP) values of lipid domains vs the surrounding membrane. Results: Laurdan revealed the majority of sphingomyelin-enriched domains associated to low erythrocyte curvature areas and part of the cholesterol-enriched domains associated with high curvature. Both lipid domains were less ordered than the surrounding lipids in erythrocytes at resting state. Upon erythrocyte deformation (elliptocytes and stimulation of calcium exchanges) or membrane vesiculation (storage at 4°C), lipid domains became more ordered than the bulk. Upon aging and in membrane fragility diseases (spherocytosis), an increase in the difference of lipid order between domains and the surrounding lipids contributed to the initiation of domain vesiculation. Conclusion: The critical role of domain-bulk differential lipid order modulation for erythrocyte reshaping is discussed in relation with the pressure exerted by the cytoskeleton on the membrane.

How does nonequilibrium activity modify the approach to a glass? This is an important question, since many experiments reveal the near-glassy nature of the cell interior, remodeled by activity. However, different simulations of dense assemblies of active particles, parametrized by a self-propulsion force, f₀, and persistence time, τp, appear to make contradictory predictions about the influence of activity on characteristic features of glass, such as fragility. This calls for a broad conceptual framework to understand active glasses; here, we extend the random first-order transition (RFOT) theory to a dense assembly of self-propelled particles. We compute the active contribution to the configurational entropy through an effective model of a single particle in a caging potential. This simple active extension of RFOT provides excellent quantitative fits to existing simulation results. We find that whereas f₀ always inhibits glassiness, the effect of τp is more subtle and depends on the microscopic details of activity. In doing so, the theory automatically resolves the apparent contradiction between the simulation models. The theory also makes several testable predictions, which we verify by both existing and new simulation data, and should be viewed as a step toward a more rigorous analytical treatment of active glass.

Anyone who has moved furniture together with friends will appreciate that cooperative transport requires some non-trivial communication. Yet ants are adept at collectively moving objects several times their size. How they do so has long been a subject of research, but recent advances have suggested that this communication occurs through the forces the ants exert on the load. This implies that the collective transport problem can be mapped to an Ising model, in which decisions by individual ants are described by spin flips. Within this framework, the group is poised in the vicinity of the transition between uncoordinated and coordinated motion. It thus profits from both internal coordination and maximal responsiveness to external information, mediated by temporarily informed leader ants. Here, we review the implications of these findings for cooperative transport, and discuss the way in which a more complete multiscale understanding of such systems would require the development of a new formalism that combines statistical physics of interacting particles with the cognitive capabilities of individuals.

Eukaryote cells have flexible membranes that allow them to have a variety of dynamical shapes. The shapes of the cells serve important biological functions, both for cells within an intact tissue, and during embryogenesis and cellular motility. How cells control their shapes and the structures that they form on their surface has been a subject of intensive biological research, exposing the building blocks that cells use to deform their membranes. These processes have also drawn the interest of theoretical physicists, aiming to develop models based on physics, chemistry and nonlinear dynamics. Such models explore quantitatively different possible mechanisms that the cells can employ to initiate the spontaneous formation of shapes and patterns on their membranes. We review here theoretical work where one such class of mechanisms was investigated: the coupling between curved membrane proteins, and the cytoskeletal forces that they recruit. Theory indicates that this coupling gives rise to a rich variety of membrane shapes and dynamics, while experiments indicate that this mechanism appears to drive many cellular shape changes.

To cooperatively carry large food items to the nest, individual ants conform their efforts and coordinate their motion. Throughout this expedition, collective motion is driven both by internal interactions between the carrying ants and a response to newly arrived informed ants that orient the cargo towards the nest. During the transport process, the carrying group must overcome obstacles that block their path to the nest. Here, we investigate the dynamics of cooperative transport, when the motion of the ants is frustrated by a linear obstacle that obstructs the motion of the cargo. The obstacle contains a narrow opening that serves as the only available passage to the nest, and through which single ants can pass but not with the cargo. We provide an analytical model for the ant-cargo system in the constrained environment that predicts a bi-stable dynamic behavior between an oscillatory mode of motion along the obstacle and a convergent mode of motion near the opening. Using both experiments and simulations, we show how for small cargo sizes, the system exhibits spontaneous transitions between these two modes of motion due to fluctuations in the applied force on the cargo. The bi-stability provides two possible problem solving strategies for overcoming the obstacle, either by attempting to pass through the opening, or take large excursions to circumvent the obstacle.

Active diffusion of intracellular components is emerging as an important process in cell biology. This process is mediated by complex assemblies of molecular motors and cytoskeletal filaments that drive force generation in the cytoplasm and facilitate enhanced motion. The kinetics of molecular motors have been precisely characterized in vitro by single molecule approaches, but their in vivo behavior remains elusive. Here, we study the active diffusion of vesicles in mouse oocytes, where this process plays a key role in nuclear positioning during development, and combine an experimental and theoretical framework to extract molecular-scale force kinetics (force, power stroke, and velocity) of the in vivo active process. Assuming a single dominant process, we find that the nonequilibrium activity induces rapid kicks of duration τ ∼ 300 μs resulting in an average force of F ∼ 0.4 pN on vesicles in in vivo oocytes, remarkably similar to the kinetics of in vitro myosin-V. Our results reveal that measuring in vivo active fluctuations allows extraction of the molecular-scale activity in agreement with single-molecule studies and demonstrates a mesoscopic framework to access force kinetics.

Collective decision-making regarding direction of travel is observed during natural motion of animal and cellular groups. This phenomenon is exemplified, in the simplest case, by a group that contains two informed subgroups that hold conflicting preferred directions of motion. Under such circumstances, simulations, subsequently supported by experimental data with birds and primates, have demonstrated that the resulting motion is either towards a compromise direction or towards one of the preferred targets (even when the two subgroups are equal in size). However, the nature of this transition is not well understood. We present a theoretical study that combines simulations and a spin model for mobile animal groups, the latter providing an equilibrium representation, and exact solution in the thermodynamic limit. This allows us to identify the nature of this transition at a critical angular difference between the two preferred directions: in both flocking and spin models the transition coincides with the change in the group dynamics from Brownian to persistent collective motion. The groups undergo this transition as the number of uninformed individuals (those in the group that do not exhibit a directional preference) increases, which acts as an inverse of the temperature (noise) of the spin model. When the two informed subgroups are not equal in size, there is a tendency for the group to reach the target preferred by the larger subgroup. We find that the spin model captures effectively the essence of the collective decision-making transition and allows us to reveal a noise-dependent trade-off between the decision-making speed and the ability to achieve majority (democratic) consensus.

Cell shape is determined by a balance of intrinsic properties of the cell as well as its mechanochemical environment. Inhomogeneous shape changes underlie many morphogenetic events and involve spatial gradients in active cellular forces induced by complex chemical signaling. Here, we introduce a mechanochemical model based on the notion that cell shape changes may be induced by external diffusible biomolecules that influence cellular contractility (or equivalently, adhesions) in a concentration-dependent manner?and whose spatial profile in turn is affected by cell shape. We map out theoretically the possible interplay between chemical concentration and cellular structure. Besides providing a direct route to spatial gradients in cell shape profiles in tissues, we show that the dependence on cell shape helps create robust mechanochemical gradients.

In living matter, shape fluctuations induced by acto-myosin are usually studied in vitro via reconstituted gels, whose properties are controlled by changing the concentrations of actin, myosin, and cross-linkers. Such an approach deliberately avoids consideration of the complexity of biochemical signaling inherent to living systems. Acto-myosin activity inside living cells is mainly regulated by the Rho signaling pathway, which is composed of multiple layers of coupled activators and inhibitors. Here, we investigate how such a pathway controls the dynamics of confluent epithelial tissues by tracking the displacements of the junction points between cells. Using a phenomenological model to analyze the vertex fluctuations, we rationalize the effects of different Rho signaling targets on the emergent tissue activity by quantifying the effective diffusion coefficient, and the persistence time and length of the fluctuations. Our results reveal an unanticipated correlation between layers of activation/inhibition and spatial fluctuations within tissues. Overall, this work connects regulation via biochemical signaling with mesoscopic spatial fluctuations, with potential application to the study of structural rearrangements in epithelial tissues.

The physics of active systems of self-propelled particles, in the regime of a dense liquid state, is an open puzzle of great current interest, both for statistical physics and because such systems appear in many biological contexts. We develop a nonequilibrium mode-coupling theory (MCT) for such systems, where activity is included as a colored noise with the particles having a self-propulsion force f(0) and a persistence time tau(p). Using the extended MCT and a generalized fluctuation-dissipation theorem, we calculate the effective temperature T-eff of the active fluid. The nonequilibrium nature of the systems is manifested through a time-dependent T-eff that approaches a constant in the long-time limit, which depends on the activity parameters f0 and tp. We find, phenomenologically, that this long-time limit is captured by the potential energy of a single, trapped active particle (STAP). Through a scaling analysis close to the MCT glass transition point, we show that tau(a), the alpha-relaxation time, behaves as tau(a) similar to f(0) (-2 gamma), where gamma = 1.74 is the MCT exponent for the passive system. ta may increase or decrease as a function of tp depending on the type of active force correlations, but the behavior is always governed by the same value of the exponent g. Comparison with the numerical solution of the nonequilibrium MCT and simulation results give excellent agreement with scaling analysis.

We study the force that noninteracting pointlike active particles apply to a symmetric inert object in the presence of a gradient of activity and particle sources and sinks. We consider two simple patterns of sources and sinks that are common in biological systems. We analytically solve a one-dimensional model designed to emulate higher-dimensional systems, and study a two-dimensional model by numerical simulations. We specify when the particle flux due to the creation and annihilation of particles can act to smooth the density profile that is induced by a gradient in the velocity of the active particles, and find the net resultant force due to both the gradient in activity and the particle flux. These results are compared qualitatively to observations of nuclear motion inside the oocyte, that is driven by a gradient in activity of actin-coated vesicles.

We show how a gradient in the motility properties of noninteracting pointlike active particles can cause a pressure gradient that pushes a large inert object. We calculate the force on an object inside a system of active particles with position-dependent motion parameters, in one and two dimensions, and show that a modified Archimedes' principle is satisfied. We characterize the system, both in terms of the model parameters and in terms of experimentally measurable quantities: the spatial profiles of the density, velocity and pressure. This theoretical analysis is motivated by recent experiments, which showed that the nucleus of a mouse oocyte (immature egg cell) moves from the cortex to the center due to a gradient of activity of vesicles propelled by molecular motors; it more generally applies to artificial systems of controlled localized activity.

During macropinocytosis, cells remodel their morphologies for the uptake of extracellular matter. This endocytotic mechanism relies on the collapse and closure of precursory structures, which are propagating actin-based, ring-shaped vertical undulations at the dorsal (top) cell membrane, a.k.a. circular dorsal ruffles (CDRs). As such, CDRs are essential to a range of vital and pathogenic processes alike. Here we show, based on both experimental data and theoretical analysis, that CDRs are propagating fronts of actin polymerization in a bistable system. The theory relies on a novel mass-conserving reaction-diffusion model, which associates the expansion and contraction of waves to distinct counter-propagating front solutions. Moreover, the model predicts that under a change in parameters (for example, biochemical conditions) CDRs may be pinned and fluctuate near the cell boundary or exhibit complex spiral wave dynamics due to a wave instability. We observe both phenomena also in our experiments indicating the conditions for which macropinocytosis is suppressed.

Monolayer expansion has generated great interest as a model system to study collective cell migration. During such an expansion the culture front often develops 'fingers', which we have recently modeled using a proposed feedback between the curvature of the monolayer's leading edge and the outward motility of the edge cells. We show that this model is able to explain the puzzling observed increase of collective cellular migration speed of a monolayer expanding into thin stripes, as well as describe the behavior within different confining geometries that were recently observed in experiments. These comparisons give support to the model and emphasize the role played by the edge cells and the edge shape during collective cell motion.

Sensory mechanisms in biology, from cells to humans, have the property of adaptivity, whereby the response produced by the sensor is adapted to the overall amplitude of the signal, reducing the sensitivity in the presence of strong stimulus, while increasing it when it is weak. This property is inherently energy consuming and a manifestation of the nonequilibrium nature of living organisms. We explore here how adaptivity affects the effective forces that organisms feel due to others in the context of a uniform swarm, in both two and three dimensions. The interactions between the individuals are taken to be attractive and long-range and of power-law form. We find that the effects of adaptivity inside the swarm are dramatic, where the effective forces decrease (or remain constant) with increasing swarm density. Linear stability analysis demonstrates how this property prevents collapse (Jeans instability), when the forces are adaptive. Adaptivity therefore endows swarms with a natural mechanism for self-stabilization.

Hippocampal neurons produce in their early stages of growth propagative, actin-rich dynamical structures called actin waves. The directional motion of actin waves from the soma to the tip of neuronal extensions has been associated with net forward growth, and ultimately with the specification of neurites into axon and dendrites. Here, geometrical cues are used to control actin wave dynamics by constraining neurons on adhesive stripes of various widths. A key observable, the average time between the production of consecutive actin waves, or mean inter-wave interval (IWI), was identified. It scales with the neurite width, and more precisely with the width of the proximal segment close to the soma. In addition, the IWI is independent of the total number of neurites. These two results suggest a mechanistic model of actin wave production, by which the material conveyed by actin waves is assembled in the soma until it reaches the threshold leading to the initiation and propagation of a new actin wave. Based on these observations, we formulate a predictive theoretical description of actin wave-driven neuronal growth and polarization, which consistently accounts for different sets of experiments.

Molecular motors that carry cargo along biopolymer filaments within cells play a crucial role in the functioning of the cell. In particular, these motors are essential for the formation and maintenance of the cellular protrusions that play key roles in motility and specific functionalities, such as the stereocilia in hair cells. Typically, there are several species of motors, carrying different cargos, that share the same track. Furthermore, it was observed that in the mature stereocilia, the different motors occupy well-segregated bands as a function of distance from the tip. We use a totally asymmetric exclusion process model with two-and three-motor species, to study the conditions that give rise to such spatial patterns. We find that the well-segregated bands appear for motors with a strong hierarchy of attachment or detachment rates. This is a striking example of pattern formation in nonequilibrium, low-dimensional systems.

Collective motion by animal groups is affected by internal interactions, external constraints, and the influx of information. A quantitative understanding of how these different factors give rise to different modes of collective motion is, at present, lacking. Here, we study how ants that cooperatively transport a large food item react to an obstacle blocking their path. Combining experiments with a statistical physics model of mechanically coupled active agents, we show that the constraint induces a deterministic collective oscillatory mode that facilitates obstacle circumvention. We provide direct experimental evidence, backed by theory, that this motion is an emergent group effect that does not require any behavioral changes at the individual level. We trace these relaxation oscillations to the interplay between two forces; informed ants pull the load toward the nest whereas uninformed ants contribute to the motions persistence along the tangential direction. The models predictions that oscillations appear above a critical system size, that the group can spontaneously transition into its ordered phase, and that the system can exhibit complete rotations are all verified experimentally. We expect that similar oscillatory modes emerge in collective motion scenarios where the structure of the environment imposes conflicts between individually held information and the groups tendency for cohesiveness.

Cell migration paths are generally described as random walks, associated with both intrinsic and extrinsic noise. However, complex cell locomotion is not merely related to such fluctuations, but is often determined by the underlying machinery. Cell motility is driven mechanically by actin and myosin, two molecular components that generate contractile forces. Other cell functions make use of the same components and, therefore, will compete with the migratory apparatus. Here, we propose a physical model of such a competitive system, namely dendritic cells whose antigen capture function and migratory ability are coupled by myosin II. The model predicts that this coupling gives rise to a dynamic instability, whereby cells switch from persistent migration to unidirectional self-oscillation, through a Hopf bifurcation. Cells can then switch to periodic polarity reversals through a homoclinic bifurcation. These predicted dynamic regimes are characterized by robust features that we identify through in vitro trajectories of dendritic cells over long timescales and distances. We expect that competition for limited resources in other migrating cell types can lead to similar deterministic migration modes.

Living organisms are inherently out-of-equilibrium systems. We employ recent developments in stochastic energetics and rely on a minimal microscopic model to predict the amount of mechanical energy dissipated by such dynamics. Our model includes complex rheological effects and nonequilibrium stochastic forces. By performing active microrheology and tracking micron-sized vesicles in the cytoplasm of living oocytes, we provide unprecedented measurements of the spectrum of dissipated energy. We show that our model is fully consistent with the experimental data, and we use it to offer predictions for the injection and dissipation energy scales involved in active fluctuations.

The collective motion of groups of animals emerges from the net effect of the interactions between individual members of the group. In many cases, such as birds, fish, or ungulates, these interactions are mediated by sensory stimuli that predominantly arise from nearby neighbors. But not all stimuli in animal groups are short range. Here, we consider mating swarms of midges, which are thought to interact primarily via long-range acoustic stimuli. We exploit the similarity in form between the decay of acoustic and gravitational sources to build a model for swarm behavior. By accounting for the adaptive nature of the midges' acoustic sensing, we show that our 'adaptive gravity' model makes mean-field predictions that agree well with experimental observations of laboratory swarms. Our results highlight the role of sensory mechanisms and interaction range in collective animal behavior. Additionally, the adaptive interactions that we present here open a new class of equations of motion, which may appear in other biological contexts.

Red blood cells, or erythrocytes, are seen to flicker under optical microscopy, a phenomenon initially described as thermal fluctuations of the cell membrane. But recent studies have suggested the involvement of non-equilibrium processes, without definitively ruling out equilibrium interpretations. Using active and passive microrheology to directly compare the membrane response and fluctuations on single erythrocytes, we report here a violation of the fluctuation-dissipation relation, which is a direct demonstration of the non-equilibrium nature of flickering. With an analytical model of the composite erythrocyte membrane and realistic stochastic simulations, we show that several molecular mechanisms may explain the active fluctuations, and we predict their kinetics. We demonstrate that tangential metabolic activity in the network formed by spectrin, a cytoskeletal protein, can generate curvature-mediated active membrane motions. We also show that other active membrane processes represented by direct normal force dipoles may explain the observed membrane activity. Our findings provide solid experimental and theoretical frameworks for future investigations of the origin and function of active motion in cells.

The precise positioning of organ progenitor cells constitutes an essential, yet poorly understood step during organogenesis. Using primordial germ cells that participate in gonad formation, we present the developmental mechanisms maintaining a motile progenitor cell population at the site where the organ develops. Employing high-resolution live-cell microscopy, we find that repulsive cues coupled with physical barriers confine the cells to the correct bilateral positions. This analysis revealed that cell polarity changes on interaction with the physical barrier and that the establishment of compact clusters involves increased cell-cell interaction time. Using particle-based simulations, we demonstrate the role of reflecting barriers, from which cells turn away on contact, and the importance of proper cell-cell adhesion level for maintaining the tight cell clusters and their correct positioning at the target region. The combination of these developmental and cellular mechanisms prevents organ fusion, controls organ positioning and is thus critical for its proper function.

A long-standing question in collective cell migration has been what might be the relative advantage of forming a cluster over migrating individually. Does an increase in the size of a collectively migrating group of cells enable them to sample the chemical gradient over a greater distance because the difference between front and rear of a cluster would be greater than for single cells? We combined theoretical modeling with experiments to study collective migration of the border cells in-between nurse cells in the Drosophila egg chamber. We discovered that cluster size is positively correlated with migration speed, up to a particular point above which speed plummets. This may be due to the effect of viscous drag from surrounding nurse cells together with confinement of all of the cells within a stiff extracellular matrix. The model predicts no relationship between cluster size and velocity for cells moving on a flat surface, in contrast to movement within a 3D environment. Our analyses also suggest that the overall chemoattractant profile in the egg chamber is likely to be exponential, with the highest concentration in the oocyte. These findings provide insights into collective chemotaxis by combining theoretical modeling with experimentation.

Unconventional myosins belong to a class of molecular motors that walk processively inside cellular protrusions towards the tips, on top of actin filament. Surprisingly, in addition, they also form retrograde moving self-organized aggregates. The qualitative properties of these aggregates are recapitulated by a mass conserving reaction-diffusion-advection model and admit two distinct families of modes: traveling waves and pulse trains. Unlike the traveling waves that are generated by a linear instability, pulses are nonlinear structures that propagate on top of linearly stable uniform backgrounds. Asymptotic analysis of isolated pulses via a simplified reaction-diffusion-advection variant on large periodic domains, allows to draw qualitative trends for pulse properties, such as the amplitude, width, and propagation speed. The results agree well with numerical integrations and are related to available empirical observations.

Mitotic spindle position relies on interactions between astral microtubules nucleated by centrosomes and a rigid cortex. Some cells, such as mouse oocytes, do not possess centrosomes and astral microtubules. These cells rely only on actin and on a soft cortex to position their spindle off-centre and undergo asymmetric divisions. While the first mouse embryonic division also occurs in the absence of centrosomes, it is symmetric and not much is known on how the spindle is positioned at the exact cell centre. Using interdisciplinary approaches, we demonstrate that zygotic spindle positioning follows a three-step process: (1) coarse centring of pronuclei relying on the dynamics of an F-actin/Myosin-Vb meshwork; (2) fine centring of the metaphase plate depending on a high cortical tension; (3) passive maintenance at the cell centre. Altogether, we show that F-actin-dependent mechanics operate the switch between asymmetric to symmetric division required at the oocyte to embryo transition.

Membrane deformation inside living cells is crucial for the proper shaping of various intracellular organelles and is necessary during the fission/fusion processes that allow membrane recycling and transport (e.g. endocytosis). Proteins that induce membrane curvature play a key role in such processes, mostly by adsorbing to the membrane and forming a scaffold that deforms the membrane according to the curvature of the proteins. In this paper we explore the possibility of membrane tube destabilization through a pearling mechanism enabled by the combined effects of the adsorbed curved proteins and the actin polymerization that they recruit. The pearling instability can serve as the initiation for fission of the tube into vesicles. We find that adsorbed curved proteins are more likely to stabilize the tubes, while the actin polymerization can provide the additional constrictive force needed for the robust instability. We discuss the relevance of the theoretical results to in vivo and in vitro experiments.

Although collective cell motion plays an important role, for example during wound healing, embryogenesis, or cancer progression, the fundamental rules governing this motion are still not well understood, in particular at high cell density. We study here the motion of human bronchial epithelial cells within a monolayer, over long times. We observe that, as the monolayer ages, the cells slow down monotonously, while the velocity correlation length first increases as the cells slow down but eventually decreases at the slowest motions. By comparing experiments, analytic model, and detailed particle-based simulations, we shed light on this biological amorphous solidification process, demonstrating that the observed dynamics can be explained as a consequence of the combined maturation and strengthening of cell-cell and cell-substrate adhesions. Surprisingly, the increase of cell surface density due to proliferation is only secondary in this process. This analysis is confirmed with two other cell types. The very general relations between the mean cell velocity and velocity correlation lengths, which apply for aggregates of self-propelled particles, as well as motile cells, can possibly be used to discriminate between various parameter changes in vivo, from noninvasive microscopy data.

Collective migration of cells is of fundamental importance for a number of biological functions such as tissue development and regeneration, wound healing and cancer metastasis. The movement of cell groups consisting of multiple cells connected by cell-cell junctions depends on both extracellular and intercellular contacts. Epithelial cell assemblies are thus regulated by a cross-talk between cell-substrate and cell-cell interactions. Here, we investigated the onset of collective migration in groups of cells as they expand from few cells into large colonies as a function of extra-cellular matrix (ECM) protein coating. By varying the amount of extracellular matrix proteins (ECM) presented to the cells, we observe that the mode of colony expansion as well as their overall geometry is strongly dependent on substrate adhesiveness. On high ECM protein coated surfaces, cells at the edges of the colonies are well spread exhibiting large outward-pointing protrusive activity whereas cellular colonies display more circular and convex shapes on less adhesive surfaces. Actin structures at the edge of the colonies also show different organizations with the formation of lamellipodial structures on highly adhesive surfaces and a pluricellular actin cable on less adhesive ones. The analysis of traction forces and cell velocities within the cellular assemblies confirm these results. By increasing ECM protein density, cells exert higher traction forces together with a higher outward motility at the edges. Furthermore, tuning cell-cell adhesion of epithelial cell lines modified the mode of expansion of the colonies. Finally, we used a recently developed computational model to recapitulate the emergent experimental behaviors of expanding cell colonies and extract that the main observed differences are dependent on the different cell-substrate interactions. Overall, our data suggest that switching behaviors of epithelial cell assemblies results of a tug-of-war between friction forces at cell-substrate interface and cell-cell interactions.

Collective motion occurs in many biological processes, such as wound healing, tumor invasion and embryogenesis. Experiments of cell monolayer migration have revealed the spontaneous formation of finger-like instabilities, with leader cells at their tips. We present a particle-based model for collective cell migration, based on several elements that have been found experimentally to influence cellular movement. Inside the bulk we include velocity alignment interactions between neighboring cells. At the border contour of the layer we introduce the following additional forces: surface-elasticity restoring force, curvature-dependent positive feedback, and contractile acto-myosin cables. We find that the curvature-driven instability at the layer edge is necessary and sufficient for the formation of cellular fingers, which are in good agreement with experimental observations.

Actin-based cellular protrusions are an ubiquitous feature of cells, performing a variety of critical functions ranging from cell-cell communication to cell motility. The formation and maintenance of these protrusions relies on the transport of proteins via myosin motors, to the protrusion tip. While tip-directed motion leads to accumulation of motors (and their molecular cargo) at the protrusion tip, it is observed that motors also form rearward moving, periodic and isolated aggregates. The origins and mechanisms of these aggregates, and whether they are important for the recycling of motors, remain open puzzles. Motivated by novel myosin-XV experiments, a mass conserving reaction-diffusion-advection model is proposed. The model incorporates a non-linear cooperative interaction between motors, which converts them between an active and an inactive state. Specifically, the type of aggregate formed (traveling waves or pulse-trains) is linked to the kinetics of motors at the protrusion tip which is introduced by a boundary condition. These pattern selection mechanisms are found not only to qualitatively agree with empirical observations but open new vistas to the transport phenomena by molecular motors in general.

Closure of wounds and gaps in tissues is fundamental for the correct development and physiology of multicellular organisms and, when misregulated, may lead to inflammation and tumorigenesis. To re-establish tissue integrity, epithelial cells exhibit coordinated motion into the void by active crawling on the substrate and by constricting a supracellular actomyosin cable. Coexistence of these two mechanisms strongly depends on the environment. However, the nature of their coupling remains elusive because of the complexity of the overall process. Here we demonstrate that epithelial gap geometry in both in vitro and in vivo regulates these collective mechanisms. In addition, the mechanical coupling between actomyosin cable contraction and cell crawling acts as a large-scale regulator to control the dynamics of gap closure. Finally, our computational modelling clarifies the respective roles of the two mechanisms during this process, providing a robust and universal mechanism to explain how epithelial tissues restore their integrity.

To cooperatively transport a large load, it is important that carriers conform in their efforts and align their forces. A downside of behavioural conformism is that it may decrease the group's responsiveness to external information. Combining experiment and theory, we show how ants optimize collective transport. On the single-ant scale, optimization stems from decision rules that balance individuality and compliance. Macroscopically, these rules poise the system at the transition between random walk and ballistic motion where the collective response to the steering of a single informed ant is maximized. We relate this peak in response to the divergence of susceptibility at a phase transition. Our theoretical models predict that the ant-load system can be transitioned through the critical point of this mesoscopic system by varying its size; we present experiments supporting these predictions. Our findings show that efficient group-level processes can arise from transient amplification of individual-based knowledge.

Cochlear hair cell bundles, made up of 10s to 100s of individual stereocilia, are essential for hearing, and even relatively minor structural changes, due to mutations or injuries, can result in total deafness. Consistent with its specialized role, the staircase geometry (SCG) of hair cell bundles presents one of the most striking, intricate, and precise organizations of actin-based cellular shapes. Composed of rows of actin-filled stereocilia with increasing lengths, the hair cell's staircase-shaped bundle is formed from a progenitor field of smaller, thinner, and uniformly spaced microvilli with relatively invariant lengths. While recent genetic studies have provided a significant increase in information on the multitude of stereocilia protein components, there is currently no model that integrates the basic physical forces and biochemical processes necessary to explain the emergence of the SCG. We propose such a model derived from the biophysical and biochemical characteristics of actin-based protrusions. We demonstrate that polarization of the cell's apical surface, due to the lateral polarization of the entire epithelial layer, plays a key role in promoting SCG formation. Furthermore, our model explains many distinct features of the manifestations of SCG in different species and in the presence of various deafness-associated mutations.

Formation of patterns during development has been a long-standing puzzle. Alan Turing proposed chemical gradients as a solution to the problem (1), and many chemical signals that pattern cells have since been found. However, only recently have roles for mechanical forces in patterning become apparent (26). In PNAS, Hannezo et al. (7) present and test a biophysical model involving three key elementsactin, myosin II, and anisotropic "effective friction" arising from interactions with the extracellular matrix (ECM)that recapitulates the formation of the periodic subcellular actin bundles that coherently span several cells to form rings in the developing Drosophila tracheal (airway) tubes (Fig. 1A). Strikingly, although the mechanism of ring formation was previously unknown, Hannezo et al.s model predicts that formation of the bundles, as well as their periodicity and orientation, will arise within each cell through self-organization. Experimental tests of predictions of the model show that it correctly describes multiple unexpected behaviors of the system in vivo, including imperfections in the actin rings (Fig. 1 B and C) and the formation of only a single unanchored actin ring per cell when the ECM is eliminated. Hannezo et al.s work provides a mechanistic basis for understanding formation of patterned actin ring structures in Drosophila and other species, and highlights the potential of the ECM to influence actin organization through mechanical rather than biochemical signaling interactions.

The internal dynamics of active gels both in artificial (in vitro) model systems and inside the cytoskeleton of living cells has been extensively studied with experiments of recent years. These dynamics are probed using tracer particles embedded in the network of biopolymers together with molecular motors, and distinct nonthermal behavior is observed. We present a theoretical model of the dynamics of a trapped active particle, which allows us to quantify the deviations from equilibrium behavior, using both analytic and numerical calculations. We map the different regimes of dynamics in this system and highlight the different manifestations of activity: breakdown of the virial theorem and equipartition, different elasticity-dependent "effective temperatures," and distinct non-Gaussian distributions. Our results shed light on puzzling observations in active gel experiments and provide physical interpretation of existing observations, as well as predictions for future studies.

Erythrocytes are flexible cells specialized in the systemic transport of oxygen in vertebrates. This physiological function is connected to their outstanding ability to deform in passing through narrow capillaries. In recent years, there has been an influx of experimental evidence of enhanced cell-shape fluctuations related to metabolically driven activity of the erythroid membrane skeleton. However, no direct observation of the active cytoskeleton forces has yet been reported to our knowledge. Here, we show experimental evidence of the presence of temporally correlated forces superposed over the thermal fluctuations of the erythrocyte membrane. These forces are ATP-dependent and drive enhanced flickering motions in human erythrocytes. Theoretical analyses provide support for a direct force exerted on the membrane by the cytoskeleton nodes as pulses of well-defined average duration. In addition, such metabolically regulated active forces cause global membrane softening, a mechanical attribute related to the functional erythroid deformability.

We propose a model for the dynamics of a probe embedded in a living cell, where both thermal fluctuations and nonequilibrium activity coexist. The model is based on a confining harmonic potential describing the elastic cytoskeletal matrix, which undergoes random active hops as a result of the nonequilibrium rearrangements within the cell. We describe the probe's statistics and we bring forth quantities affected by the nonequilibrium activity. We find an excellent agreement between the predictions of our model and experimental results for tracers inside living cells. Finally, we exploit our model to arrive at quantitative predictions for the parameters characterizing nonequilibrium activity, such as the typical time scale of the activity and the amplitude of the active fluctuations.

Cell movement has essential functions in development, immunity, and cancer. Various cell migration patterns have been reported, but no general rule has emerged so far. Here, we show on the basis of experimental data in vitro and in vivo that cell persistence, which quantifies the straightness of trajectories, is robustly coupled to cell migration speed. We suggest that this universal coupling constitutes a generic law of cell migration, which originates in the advection of polarity cues by an actin cytoskeleton undergoing flows at the cellular scale. Our analysis relies on a theoretical model that we validate by measuring the persistence of cells upon modulation of actin flow speeds and upon optogenetic manipulation of the binding of an actin regulator to actin filaments. Beyond the quantitative prediction of the coupling, the model yields a generic phase diagram of cellular trajectories, which recapitulates the full range of observed migration patterns.

In somatic cells, the position of the cell centroid is dictated by the centrosome. The centrosome is instrumental in nucleus positioning, the two structures being physically connected. Mouse oocytes have no centrosomes, yet harbour centrally located nuclei. We demonstrate how oocytes define their geometric centre in the absence of centrosomes. Using live imaging of oocytes, knockout for the formin 2 actin nucleator, with off-centred nuclei, together with optical trapping and modelling, we discover an unprecedented mode of nucleus positioning. We document how active diffusion of actin-coated vesicles, driven by myosin Vb, generates a pressure gradient and a propulsion force sufficient to move the oocyte nucleus. It promotes fluidization of the cytoplasm, contributing to nucleus directional movement towards the centre. Our results highlight the potential of active diffusion, a prominent source of intracellular transport, able to move large organelles such as nuclei, providing in vivo evidence of its biological function.

Circular Dorsal Ruffles (CDRs) have been known for decades, but the mechanism that organizes these actin waves remains unclear. In this article we systematically analyze the dynamics of CDRs on fibroblasts with respect to characteristics of current models of actin waves. We studied CDRs on heterogeneously shaped cells and on cells that we forced into disk-like morphology. We show that CDRs exhibit phenomena such as periodic cycles of formation, spiral patterns, and mutual wave annihilations that are in accord with an active medium description of CDRs. On cells of controlled morphologies, CDRs exhibit extremely regular patterns of repeated wave formation and propagation, whereas on random-shaped cells the dynamics seem to be dominated by the limited availability of a reactive species. We show that theoretical models of reaction-diffusion type incorporating conserved species capture partially the behavior we observe in our data.

Collective cell migration is a widespread biological phenomenon, whereby groups of highly coordinated, adherent cells move in a polarized fashion [1, 2]. This migration mode is a hallmark of tissue morphogenesis during development and repair and of solid tumor dissemination [1]. In addition to circulating as solitary cells, lymphoid malignancies can assemble into tissues as multicellular aggregates [3]. Whether malignant lymphocytes are capable of coordinating their motility in the context of chemokine gradients is, however, unknown. Here, we show that, upon exposure to CCL19 or CXCL12 gradients, malignant B and T lymphocytes assemble into clusters that migrate directionally and display a wider chemotactic sensitivity than individual cells. Physical modeling recapitulates cluster motility statistics and shows that intracluster cell cohesion results in noise reduction and enhanced directionality. Quantitative image analysis reveals that cluster migration runs are periodically interrupted by transitory rotation and random phases that favor leader cell turnover. Additionally, internalization of CCR7 in leader cells is accompanied by protrusion retraction, loss of polarity, and the ensuing replacement by new leader cells. These mechanisms ensure sustained forward migration and resistance to chemorepulsion, a behavior of individual cells exposed to steep CCL19 gradients that depends on CCR7 endocytosis. Thus, coordinated cluster dynamics confer distinct chemotactic properties, highlighting unexpected features of lymphoid cell migration.

Cell mechanics control the outcome of cell division. In mitosis, external forces applied on a stiff cortex direct spindle orientation and morphogenesis. During oocyte meiosis on the contrary, spindle positioning depends on cortex softening. How changes in cortical organization induce cortex softening has not yet been addressed. Furthermore, the range of tension that allows spindle migration remains unknown. Here, using artificial manipulation of mouse oocyte cortex as well as theoretical modelling, we show that cortical tension has to be tightly regulated to allow off-center spindle positioning: a too low or too high cortical tension both lead to unsuccessful spindle migration. We demonstrate that the decrease in cortical tension required for spindle positioning is fine-tuned by a branched F-actin network that triggers the delocalization of myosin-II from the cortex, which sheds new light on the interplay between actin network architecture and cortex tension.

What do we mean by \u201ccollective cell migration\u201d? The scope of this phenomenon includes different forms of motion of groups of cells moving together. In this review, we focus only on eukaryotic cells that belong to a multicellular organism, thereby not treating the collective motion of bacteria. Most cells of a multicellular adult organism are rather stationary, but situations of collective migration of cells arise during the normal embryonic development process [1-3] and the physiological responses during wound healing or immune response. It also plays an important role during pathologies such as cancer metastasis [4]. During these processes, cells have to be both motile and have a certain degree of adhesion to one another [5], so that the collectivity of the migration is maintained.

Active fluctuations, driven by processes that consume ATP, are prevalent in living cells and are mostly driven by different forms of molecular motors. Such motors often move and transmit forces along biopolymers, which in general can be treated as semiflexible chains. We present a theoretical analysis of the active (out of thermal equilibrium) fluctuation of semiflexible polymers, using both analytical and simulation methods. We find that enhanced diffusion, even superdiffusive, occurs in a well-defined temporal regime, defined by the thermal modes of the chain and the typical timescale of the activity. In addition, we find a dynamic resonance-like condition between the elastic modes of the chain and the duration of the active force, which leads to enhanced spatial correlation of local displacements. These results are in qualitative agreement with observations of cytoskeletal biopolymers, and were recently observed for the dynamics of chromatin in interphase cells. We therefore propose that the interplay between elasticity and activity is driving long-range correlations in our model system, and may also be manifest inside living cells.

Actin-based cellular protrusions are a ubiquitous feature of cell morphology, e.g., filopodia and microvilli, serving a huge variety of functions. Despite this, there is still no comprehensive model for the mechanisms that determine the geometry of these protrusions. We present here a detailed computational model that addresses a combination of multiple biochemical and physical processes involved in the dynamic regulation of the shape of these protrusions. We specifically explore the role of actin polymerization in determining both the height and width of the protrusions. Furthermore, we show that our generalized model can explain multiple morphological features of these systems, and account for the effects of specific proteins and mutations.

The blood stage malaria parasite, the merozoite, has a small window of opportunity during which it must successfully target and invade a human erythrocyte. The process of invasion is nonetheless remarkably rapid. To date, mechanistic models of invasion have focused predominantly on the parasite actomyosin motor contribution to the energetics of entry. Here, we have conducted a numerical analysis using dimensions for an archetypal merozoite to predict the respective contributions of the host-parasite interactions to invasion, in particular the role of membrane wrapping. Our theoretical modeling demonstrates that erythrocyte membrane wrapping alone, as a function of merozoite adhesive and shape properties, is sufficient to entirely account for the first key step of the invasion process, that of merozoite reorientation to its apex and tight adhesive linkage between the two cells. Next, parasite-induced reorganization of the erythrocyte cytoskeleton and release of parasite-derived membrane can also account for a considerable energetic portion of actual invasion itself, through membrane wrapping. Thus, contrary to the prevailing dogma, wrapping by the erythrocyte combined with parasite-derived membrane release can markedly reduce the expected contributions of the merozoite actomyosin motor to invasion. We therefore propose that invasion is a balance between parasite and host cell contributions, evolved toward maximal efficient use of biophysical forces between the two cells.

The ability of cells to coordinately migrate in groups is crucial to enable them to travel long distances during embryonic development, wound healing and tumorigenesis, but the fundamental mechanisms underlying intercellular coordination during collective cell migration remain elusive despite considerable research efforts. A novel analytical framework is introduced here to explicitly detect and quantify cell clusters that move coordinately in a monolayer. The analysis combines and associates vast amount of spatiotemporal data across multiple experiments into transparent quantitative measures to report the emergence of new modes of organized behavior during collective migration of tumor and epithelial cells in wound healing assays. First, we discovered the emergence of a wave of coordinated migration propagating backward from the wound front, which reflects formation of clusters of coordinately migrating cells that are generated further away from the wound edge and disintegrate close to the advancing front. This wave emerges in both normal and tumor cells, and is amplified by Met activation with hepatocyte growth factor/scatter factor. Second, Met activation was found to induce coinciding waves of cellular acceleration and stretching, which in turn trigger the emergence of a backward propagating wave of directional migration with about an hour phase lag. Assessments of the relations between the waves revealed that amplified coordinated migration is associated with the emergence of directional migration. Taken together, our data and simplified modeling-based assessments suggest that increased velocity leads to enhanced coordination: higher motility arises due to acceleration and stretching that seems to increase directionality by temporarily diminishing the velocity components orthogonal to the direction defined by the monolayer geometry. Spatial and temporal accumulation of directionality thus defines coordination. The findings offer new insight and suggest a basic cellular mechanism for long-term cell guidance and intercellular communication during collective cell migration.

Molecular motors are involved in key transport processes inside actin-based cellular protrusions. The motors carry cargo proteins to the protrusion tip which participate in regulating the actin polymerization and play a key role in facilitating the growth and formation of such protrusions. It is observed that the motors accumulate at the tips of cellular protrusions and form aggregates that are found to drift towards the protrusion base at the rate of actin treadmilling. We present a one-dimensional driven lattice model, where motors become inactive after delivering their cargo at the tip, or by loosing their cargo to a cargoless neighbor. The results suggest that the experimental observations may be explained by the formation of traffic jams that form at the tip. The model is solved using a novel application of mean-field and shock analysis. We find a new class of shocks that undergo intermittent collapses. Extensions with attachment and detachment events and relevance to experiments are briefly described.

Neurons acquire their functional and morphological axo-dendritic polarity by extending, from competing minor processes (neurites), one long axon among numerous dendrites. We employed complementary sets of micropatterns built from 2 and 6 μm wide stripes of various lengths to constrain hippocampal neuron shapes. Using these geometries, we have (i) limited the number of neuronal extensions to obtain a minimal in vitro system of bipolar neurons and (ii) controlled the neurite width during growth by the generation of a progressive cell shape asymmetry on either side of the cellular body. From this geometrical approach, we gained a high level of control of each neurite length and of the localization of axonal specification. To analyze these results, we developed a model based on a width and polarization dependent neurite elongation rate and on the existence of a critical neurite length that sets the axonal fate. Our data on the four series of micro-patterns developed for this study are described by a single set of growth parameters, well supported by experiments. The control of neuronal shapes by adhesive micro-patterns thereby offers a novel paradigm to follow the dynamical process of neurite lengthening and competition through the process of axonal polarization. This journal is

Daniel Cohen and colleagues report that galvanotaxis, where living cells, both eukaryote and prokaryote, undergo motion in response to weak electric fields, can be used to control the collective migration of cells on a flat substrate. Using lithography to design the shape of the electrodes and the geometric constraints of the cellular layer, the approach of Cohen and colleagues allows for the exploration of cellular dynamics with very complex spatiotemporal regulation. From the electric field oscillations, the authors were able to measure the time that it takes for the collective motion of the layer to set in, that is, for all the patches to merge into a single coherent migration flow. This is found to be of the order of 10-30 min, which is the time it takes for cellular orientational ordering to propagate across the length of the rotating patches. Cohen and collaborators find that the leader cells are rather oblivious to the applied electric field and that they do not respond to it as do cells within the bulk of the layer.

Reaction-diffusion models have been used to describe pattern formation on the cellular scale, and traditionally do not include feedback between cellular shape changes and biochemical reactions. We introduce here a distinct reaction-diffusion-elasticity approach: The reaction-diffusion part describes bistability between two actin orientations, coupled to the elastic energy of the cell membrane deformations. This coupling supports spatially localized patterns, even when such solutions do not exist in the uncoupled self-inhibited reaction-diffusion system. We apply this concept to describe the nonlinear (threshold driven) initiation mechanism of actin-based cellular protrusions and provide support by several experimental observations.

Molecular motors are involved in key transport processes in the cell. Many of these motors can switch from an active to a nonactive state, either spontaneously or depending on their interaction with other molecules. When active, the motors move processively along the filaments, while when inactive they are stationary. We treat here the simple case of spontaneously switching motors, between the active and inactive states, along an open linear track. We use our recent analogy with vehicular traffic, where we go beyond the mean-field description. We map the phase diagram of this system, and find that it clearly breaks the symmetry between the different phases, as compared to the standard total asymmetric exclusion process. We make several predictions that may be testable using molecular motors in vitro and in living cells.

At mitosis onset, cortical tension increases and cells round up, ensuring correct spindle morphogenesis and orientation. Thus, cortical tension sets up the geometric requirements of cell division. On the contrary, cortical tension decreases during meiotic divisions in mouse oocytes, a puzzling observation because oocytes are round cells, stable in shape, that actively position their spindles. We investigated the pathway leading to reduction in cortical tension and its significance for spindle positioning. We document a previously uncharacterized Arp2/3-dependent thickening of the cortical F-actin essential for first meiotic spindle migration to the cortex. Using micropipette aspiration, we show that cortical tension decreases during meiosis I, resulting from myosin-II exclusion from the cortex, and that cortical F-actin thickening promotes cortical plasticity. These events soften and relax the cortex. They are triggered by the Mos-MAPK pathway and coordinated temporally. Artificial cortex stiffening and theoretical modelling demonstrate that a soft cortex is essential for meiotic spindle positioning.

Collective behavior refers to the emergence of complex migration patterns over scales larger than those of the individual elements constituting a system. It plays a pivotal role in biological systems in regulating various processes such as gastrulation, morphogenesis and tissue organization. Here, by combining experimental approaches and numerical modeling, we explore the role of cell density (\u201ccrowding\u201d), strength of intercellular adhesion (\u201ccohesion\u201d) and boundary conditions imposed by extracellular matrix (ECM) proteins (\u201cconstraints\u201d) in regulating the emergence of collective behavior within epithelial cell sheets. Our results show that the geometrical confinement of cells into well-defined circles induces a persistent, coordinated and synchronized rotation of cells that depends on cell density. The speed of such rotating large-scale movements slows down as the density increases. Furthermore, such collective rotation behavior depends on the size of the micropatterned circles: we observe a rotating motion of the overall cell population in the same direction for sizes of up to 200 μm. The rotating cells move as a solid body, with a uniform angular velocity. Interestingly, this upper limit leads to length scales that are similar to the natural correlation length observed for unconfined epithelial cell sheets. This behavior is strongly altered in cells that present a downregulation of adherens junctions and in cancerous cell types. We anticipate that our system provides a simple and easy approach to investigate collective cell behavior in a well-controlled and systematic manner.

The forces that arise from the actin cytoskeleton play a crucial role in determining the cell shape. These include protrusive forces due to actin polymerization and adhesion to the external matrix. A theoretical model for the cellular shapes resulting from the feedback between the membrane shape and the forces acting on the membrane, mediated by curvature-sensitive membrane complexes of a convex shape is presented. In previous theoretical chapters the regimes of linear instability were investigated where spontaneous formation of cellular protrusions is initiated. Here we calculate the evolution of a two dimensional cell contour beyond the linear regime and determine the final steady-state shapes arising within the model. We find that shapes driven by adhesion or by actin polymerization (lamellipodia) have very different morphologies, as observed in cells. Furthermore, we find that as the strength of the protrusive forces diminish, the system approaches a stabilization of a periodic pattern of protrusions. This result can provide an explanation for a number of puzzling experimental observations regarding cellular shape dependence on the properties of the extra-cellular matrix.

Cells exhibit propagating membrane waves which involve the actin cytoskeleton. One type of such membranal waves are Circular Dorsal Ruffles (CDR) which are related to endocytosis and receptor internalization. Experimentally, CDRs have been associated with membrane bound activators of actin polymerization of concave shape. Experimental evidence is presented for the localization of convex membrane proteins in these structures, and their insensitivity to inhibition of myosin II contractility in immortalized mouse embryo fibroblasts cell cultures. These observations lead us to propose a theoretical model which explains the formation of these waves due to the interplay between complexes that contain activators of actin polymerization and membrane-bound curved proteins of both types of curvature (concave and convex). Our model predicts that the activity of both types of curved proteins is essential for sustaining propagating waves, which are abolished when one type of curved activator is removed. Within this model waves are initiated when the level of actin polymerization induced by the curved activators is higher than some threshold value, which allows the cell to control CDR formation. It is demonstrated that the model can explain many features of CDRs, and give several testable predictions. This work demonstrates the importance of curved membrane proteins in organizing the actin cytoskeleton and cell shape.

We study the dynamics and patterning of polar contractile filaments on the surface of a cylindrical cell using active hydrodynamic equations that incorporate couplings between curvature and filament orientation. Cables and rings spontaneously emerge as steady state configurations on the cylinder, and can be stationary or moving, helical or tilted segments moving along helical trajectories. We observe phase transitions in the steady state patterns upon changing cell diameter or motor-driven activity and make several testable predictions. Our results are relevant to the dynamics and patterning of a variety of active biopolymers in cylindrical cells. DOI: 10.1103/PhysRevLett.110.168104

The mechanisms underlying the collective motion of molecular motors in living cells are not yet fully understood. One such open puzzle is the observed pulses of backward-moving myosin-X in the filopodia structure. Motivated by this phenomenon we introduce two generalizations of the 'total asymmetric exclusion process' (TASEP) that might be relevant to the formation of such pulses. The first is adding a nearest-neighbours attractive interaction between motors, while the second is adding an internal degree of freedom corresponding to a processive and immobile form of the motors. Switching between the two states occurs stochastically, without a conservation law. Both models show strong deviations from the mean field behaviour and lack particle-hole symmetry. We use approximations borrowed from the research on vehicular traffic models to calculate the current and jam size distribution in a system with periodic boundary conditions and introduce a novel modification to one of these approximation schemes.

Successful cytokinesis requires proper assembly of the contractile actomyosin ring, its stable positioning on the cell surface and proper constriction. Over the years, many of the key molecular components and regulators of the assembly and positioning of the actomyosin ring have been elucidated. Here we show that cell geometry and mechanics play a crucial role in the stable positioning and uniform constriction of the contractile ring. Contractile rings that assemble in locally spherical regions of cells are unstable and slip towards the poles. By contrast, actomyosin rings that assemble on locally cylindrical portions of the cell under the same conditions do not slip, but uniformly constrict the cell surface. The stability of the rings and the dynamics of ring slippage can be described by a simple mechanical model. Using fluorescence imaging, we verify some of the quantitative predictions of the model. Our study reveals an intimate interplay between geometry and actomyosin dynamics, which are likely to apply in a variety of cellular contexts.

Contractile function of striated muscle cells depends crucially on the almost crystalline order of actin and myosin filaments in myofibrils, but the physical mechanisms that lead to myofibril assembly remains ill-defined. Passive diffusive sorting of actin filaments into sarcomeric order is kinetically impossible, suggesting a pivotal role of active processes in sarcomeric pattern formation. Using a one-dimensional computational model of an initially unstriated actin bundle, we show that actin filament treadmilling in the presence of processive plus-end crosslinking provides a simple and robust mechanism for the polarity sorting of actin filaments as well as for the correct localization of myosin filaments. We propose that the coalescence of crosslinked actin clusters could be key for sarcomeric pattern formation. In our simulations, sarcomere spacing is set by filament length prompting tight length control already at early stages of pattern formation. The proposed mechanism could be generic and apply both to premyofibrils and nascent myofibrils in developing muscle cells as well as possibly to striated stress-fibers in non-muscle cells.

Lateral heterogeneity of cell membranes has been demonstrated in numerous studies showing anomalous diffusion of membrane proteins; it has been explained by models and experiments suggesting dynamic barriers to free diffusion, that temporarily confine membrane proteins into microscopic patches. This picture, however, comes short of explaining a steady-state patchy distribution of proteins, in face of the transient opening of the barriers. In our previous work we directly imaged persistent clusters of MHC-I, a type I transmembrane protein, and proposed a model of a dynamic equilibrium between proteins newly delivered to the cell surface by vesicle traffic, temporary confinement by dynamic barriers to lateral diffusion, and dispersion of the clusters by diffusion over the dynamic barriers. Our model predicted that the clusters are dynamic, appearing when an exocytic vesicle fuses with the plasma membrane and dispersing with a typical lifetime that depends on lateral diffusion and the dynamics of barriers. In a subsequent work, we showed this to be the case. Here we test another prediction of the model, and show that changing the stability of actin barriers to lateral diffusion changes cluster lifetimes. We also develop a model for the distribution of cluster lifetimes, consistent with the function of barriers to lateral diffusion in maintaining MHC-I clusters.

In many bacterial species, the protein FtsZ forms a cytoskeletal ring that marks the future division site and scaffolds the division machinery. In rod-shaped bacteria, most frequently membrane-attached FtsZ rings or ring fragments are reported and occasionally helices. By contrast, axial FtsZ clusters have never been reported. In this paper, we investigate theoretically how dynamic FtsZ aggregates align in rod-shaped bacteria. We study systematically different physical mechanisms that affect the alignment of FtsZ polymers using a computational model that relies on autocatalytic aggregation of FtsZ filaments at the membrane. Our study identifies a general tool kit of physical and geometrical mechanisms by which rod-shaped cells align biopolymer aggregates. Our analysis compares the relative impact of each mechanism on the circumferential alignment of FtsZ as observed in rod-shaped bacteria. We determine spontaneous curvature of FtsZ polymers and axial confinement of FtsZ on the membrane as the strongest factors. Including Min oscillations in our model, we find that these stabilize axial and helical clusters on short time scales, but promote the formation of an FtsZ ring at the cell middle at longer times. This effect could provide an explanation to the long standing puzzle of transiently observed oscillating FtsZ helices in Escherichia coli cells prior to cell division.

Recent studies have demonstrated that actin filaments are not crucial for the short-term stability of tubular membrane protrusions originating from the cell surface. It has also been demonstrated that prominin nanodomains and curvature inducing I-BAR proteins could account for the stability of the membrane protrusion. Here we constructed an axisymmetric model of a membrane protrusion that excludes actin filaments in order to investigate the contributions of prominin nanodomains (rafts) and I-BAR proteins to the membrane protrusion stability. It was demonstrated that prominin nanodomains and I-BAR proteins can stabilize the membrane protrusion only over a specific range of spontaneous curvature. On the other hand, high spontaneous curvature and/or high density of I-BAR proteins could lead to system instability and to non-uniform contraction in the radial direction of the membrane protrusion. In agreement with previous studies, it was also shown that the isotropic bending energy of lipids is not sufficient to explain the stability of the observed tubular membrane protrusion without actin filaments.

Cortactin is involved in invadopodia and podosome formation [1], pathogens and endosome motility [2], and persistent lamellipodia protrusion [3, 4]; its overexpression enhances cellular motility and metastatic activity [5-8]. Several mechanisms have been proposed to explain cortactin's role in Arp2/3-driven actin polymerization [9, 10], yet its direct role in cell movement remains unclear. We use a biomimetic system to study the mechanism of cortactin-mediated regulation of actin-driven motility [11]. We tested the role of different cortactin variants that interact with Arp2/3 complex and actin filaments distinctively. We show that wild-type cortactin significantly enhances the bead velocity at low concentrations. Single filament experiments show that cortactin has no significant effect on actin polymerization and branch stability, whereas it strongly affects the branching rate driven by Wiskott-Aldrich syndrome protein (WASP)-VCA fragment and Arp2/3 complex. These results lead us to propose that cortactin plays a critical role in translating actin polymerization at a bead surface into motion, by releasing WASP-VCA from the new branching site. This enhanced release has two major effects: it increases the turnover rate of branching per WASP molecule, and it decreases the friction-like force caused by the binding of the moving surface with respect to the growing actin network.

Bacterial cell division takes place in three phases: Z-ring formation at midcell, followed by divisome assembly and building of the septum per se. Using time-lapse microscopy of live bacteria and a high-precision cell edge detection method, we have previously found the true time for the onset of septation, τ_c, and the time between consecutive divisions, τ_g. Here, we combine the above method with measuring the dynamics of the FtsZ-GFP distribution in individual Escherichia coli cells to determine the Z-ring positioning time, τ_z. To analyze the FtsZ-GFP distribution along the cell, we used the integral fluorescence profile (IFP), which was obtained by integrating the fluorescence intensity across the cell width. We showed that the IFP may be approximated by an exponential peak and followed the peak evolution throughout the cell cycle, to find a quantitative criterion for the positioning of the Z-ring and hence the value of τ_z. We defined τ_z as the transition from oscillatory to stable behavior of the mean IFP position. This criterion was corroborated by comparison of the experimental results to a theoretical model for the FtsZ dynamics, driven by Min oscillations. We found that τ_z < τ_c for all the cells that were analyzed. Moreover, our data suggested that τ_z is independent of τ_c, τ_g and the cell length at birth, L ₀. These results are consistent with the current understanding of the Z-ring positioning and cell septation processes.

There is a body of literature that describes the geometry and the physics of filopodia using either stochastic models or partial differential equations and elasticity and coarse-grained theory. Comparatively, there is a paucity of models focusing on the regulation of the network of proteins that control the formation of different actin structures. Using a combination of in-vivo and in-vitro experiments together with a system of ordinary differential equations, we focused on a small number of well-characterized, interacting molecules involved in actin-dependent filopodia formation: the actin remodeler Eps8, whose capping and bundling activities are a function of its ligands, Abi-1 and IRSp53, respectively; VASP and Capping Protein (CP), which exert antagonistic functions in controlling filament elongation. The model emphasizes the essential role of complexes that contain the membrane deforming protein IRSp53, in the process of filopodia initiation. This model accurately accounted for all observations, including a seemingly paradoxical result whereby genetic removal of Eps8 reduced filopodia in HeLa, but increased them in hippocampal neurons, and generated quantitative predictions, which were experimentally verified. The model further permitted us to explain how filopodia are generated in different cellular contexts, depending on the dynamic interaction established by Eps8, IRSp53 and VASP with actin filaments, thus revealing an unexpected plasticity of the signaling network that governs the multifunctional activities of its components in the formation of filopodia.

Biologically driven nonequilibrium fluctuations are often characterized by their non-Gaussianity or by an "effective temperature", which is frequency dependent and higher than the ambient temperature. We address these two measures theoretically by examining a randomly kicked particle, with a variable number of kicking motors, and show how these two indicators of nonequilibrium behavior can contradict. Our results are compared with new experiments on shape fluctuations of red-blood cell membranes, and demonstrate how the physical nature of the motors in this system can be revealed using these global measures of nonequilibrium.

Research studies conducted by Tambe et al. report an analysis of the coupling between cellular motion and mechanical forces in a continuous two-dimensional cell culture in vitro. They generate high-resolution maps for the measurement of the local stress field that the moving cells exert on the underlying elastic substrate. Under normal conditions the result is a cellular layer where the cytoskeleton of each cell is self-organized to maintain strong adhesions with its neighboring cells. Tambe et al. also find that the long-range ordering in the cellular traction forces and motion is communicated by strong cell-cell adhesions, cancer cells that move independently do not exhibit plithotaxis. It is suggested that controlling plithotaxis could potentially allow us to induce faster wound healing or impede metastasis. The observed behavior of a dense layer of cells, with its highly anisotropic field of intercellular forces and cell motion, resembles the dynamics observed in amorphous systems such as colloids, glasses, and granular materials.

Dynamin is a protein that plays a key role in the transport and recycling of membrane tubes and vesicles within a living cell. This protein adsorbs from solution to PIP₂-containing membranes, and on these tubes it forms curved oligomers that condense into tight helical domains of uniform radius. The dynamics of this process is treated here in terms of the linear stability of a continuum model, whereby membrane-mediated interactions are shown to drive the spontaneous nucleation of condensed dynamin domains. We furthermore show that the deformation of the membrane outside the dynamin domains induces an energy barrier that can hinder the full coalescence of neighboring growing domains. We compare these calculations to experimental observations on dynamin dynamics in vitro.

The forces that arise from the actin cytoskeleton play a crucial role in determining the cell shape. These include protrusive forces due to actin polymerization and adhesion to the external matrix. We present here a theoretical model for the cellular shapes resulting from the feedback between the membrane shape and the forces acting on the membrane, mediated by curvature-sensitive membrane complexes of a convex shape. In previous theoretical studies we have investigated the regimes of linear instability where spontaneous formation of cellular protrusions is initiated. Here we calculate the evolution of a two dimensional cell contour beyond the linear regime and determine the final steady-state shapes arising within the model. We find that shapes driven by adhesion or by actin polymerization (lamellipodia) have very different morphologies, as observed in cells. Furthermore, we find that as the strength of the protrusive forces diminish, the system approaches a stabilization of a periodic pattern of protrusions. This result can provide an explanation for a number of puzzling experimental observations regarding cellular shape dependence on the properties of the extra-cellular matrix.

The cytoskeletal protein FtsZ polymerizes to a ring structure (Z ring) at the inner cytoplasmic membrane that marks the future division site and scaffolds the division machinery in many bacterial species. FtsZ is known to polymerize in the presence of GTP into single-stranded protofilaments. In vivo, FtsZ polymers become associated with the cytoplasmic membrane via interaction with the membrane-binding proteins FtsA and ZipA. The FtsZ ring structure is highly dynamic and undergoes constantly polymerization and depolymerization processes and exchange with the cytoplasmic pool. In this theoretical study, we consider a scenario of Z ring self-organization via self-enhanced attachment of FtsZ polymers due to end-to-end interactions and lateral interactions of FtsZ polymers on the membrane. With the assumption of exclusively circumferential polymer orientations, we derive coarse-grained equations for the dynamics of the pool of cytoplasmic and membrane-bound FtsZ. To capture stochastic effects expected in the system due to low particle numbers, we simulate our computational model using a Gillespie-type algorithm. We obtain ring- and arc-shaped aggregations of FtsZ polymers on the membrane as a function of monomer numbers in the cell. In particular, our model predicts the number of FtsZ rings forming in the cell as a function of cell geometry and FtsZ concentration. We also calculate the time of FtsZ ring localization to the midplane in the presence of Min oscillations. Finally, we demonstrate that the assumptions and results of our model are confirmed by 3D reconstructions of fluorescently-labeled FtsZ structures in E. coli that we obtained.

We have quantitatively and systemically measured the morphologies and dynamics of fluctuations in human RBC membranes using a full-field laser interferometry technique that accurately measures dynamic membrane fluctuations. We present conclusive evidence that the presence of adenosine 5'-triphosphate (ATP) facilitates nonequilibrium dynamic fluctuations in the RBC membrane and that these fluctuations are highly correlated with specific regions in the biconcave shape of RBCs. Spatial analysis reveals that these nonequilibrium membrane fluctuations are enhanced at the scale of the spectrin mesh size. Our results indicate the presence of dynamic remodeling in the RBC membrane cortex powered by ATP, which results in nonequilibrium membrane fluctuations. (This conference proceeding paper is primary based on our recent publication - please refer to YK Park et al., Proc. Nat. Acad. Sci., 107, 1289 (2010) for details.)

Human red blood cells (RBCs) lack the actin-myosin-microtubule cytoskeleton that is responsible for shape changes in other cells. Nevertheless, they can display highly dynamic local deformations in response to external perturbations, such as those that occur during the process of apical alignment preceding merozoite invasion in malaria. Moreover, after lysis in divalent cation-free media, the isolated membranes of ruptured ghosts show spontaneous inside-out curling motions at the free edges of the lytic hole, leading to inside-out vesiculation. The molecular mechanisms that drive these rapid shape changes are unknown. Here, we propose a molecular model in which the spectrin filaments of the RBC cortical cytoskeleton control the sign and dynamics of membrane curvature depending on two types of spectrin filaments. Type I spectrin filaments that are grafted at one end, or at both ends but not connected to the rest of the cytoskeleton, induce a concave spontaneous curvature. Type II spectrin filaments that are grafted at both ends to the cytoskeleton induce a local convex spontaneous curvature. Computer simulations of the model reveal that curling, as experimentally observed, can be obtained either by an overall excess of type I filaments throughout the cell, or by the flux of such filaments toward the curling edges. Divalent cations have been shown to arrest the curling process and Ca2+ ions have also been implicated in local membrane deformations during merozoite invasion. These effects can be replicated in our model by attributing the divalent cation effects to increased filament membrane binding. This process converts the curl-inducing loose filaments into fully bound filaments that arrest curling. The same basic mechanism can be shown to account for Ca2+- induced local and dynamic membrane deformations in intact RBCs. The implications of these results in terms of RBC membrane dynamics under physiological, pathological, and experimental conditions are discussed.

Cells exhibit propagating membrane waves which involve the actin cytoskeleton. One type of such membranal waves are Circular Dorsal Ruffles (CDR) which are related to endocytosis and receptor internalization. Experimentally, CDRs have been associated with membrane bound activators of actin polymerization of concave shape. We present experimental evidence for the localization of convex membrane proteins in these structures, and their insensitivity to inhibition of myosin II contractility in immortalized mouse embryo fibroblasts cell cultures. These observations lead us to propose a theoretical model which explains the formation of these waves due to the interplay between complexes that contain activators of actin polymerization and membrane-bound curved proteins of both types of curvature (concave and convex). Our model predicts that the activity of both types of curved proteins is essential for sustaining propagating waves, which are abolished when one type of curved activator is removed. Within this model waves are initiated when the level of actin polymerization induced by the curved activators is higher than some threshold value, which allows the cell to control CDR formation. We demonstrate that the model can explain many features of CDRs, and give several testable predictions. This work demonstrates the importance of curved membrane proteins in organizing the actin cytoskeleton and cell shape.

Human red blood cells (RBCs) lack the actin-myosin-microtubule cytoskeleton that is responsible for shape changes in other cells. Nevertheless, they can display highly dynamic local deformations in response to external perturbations, such as those that occur during the process of apical alignment preceding merozoite invasion in malaria. Moreover, after lysis in divalent cation-free media, the isolated membranes of ruptured ghosts show spontaneous inside-out curling motions at the free edges of the lytic hole, leading to inside-out vesiculation. The molecular mechanisms that drive these rapid shape changes are unknown. Here, we propose a molecular model in which the spectrin filaments of the RBC cortical cytoskeleton control the sign and dynamics of membrane curvature depending on whether the ends of the filaments are free or anchored to the bilayer. Computer simulations of the model reveal that curling, as experimentally observed, can be obtained either by an overall excess of weakly-bound filaments throughout the cell, or by the flux of such filaments toward the curling edges. Divalent cations have been shown to arrest the curling process, and Ca²⁺ ions have also been implicated in local membrane deformations during merozoite invasion. These effects can be replicated in our model by attributing the divalent cation effects to increased filament-membrane binding. This process converts the curl-inducing loose filaments into fully bound filaments that arrest curling. The same basic mechanism can be shown to account for Ca²⁺-induced local and dynamic membrane deformations in intact RBCs. The implications of these results in terms of RBC membrane dynamics under physiological, pathological, and experimental conditions is discussed.

A hydrophobic mismatch between protein length and membrane thickness can lead to a modification of protein conformation, function, and oligomerization. To study the role of hydrophobic mismatch, we have measured the change in mobility of transmembrane peptides possessing a hydrophobic helix of various length d_π in lipid membranes of giant vesicles. We also used a model system where the hydrophobic thickness of the bilayers, h, can be tuned at will. We precisely measured the diffusion coefficient of the embedded peptides and gained access to the apparent, size of diffusing objects. For bilayers thinner than d_π the diffusion coefficient decreases, and the derived characteristic sizes are larger than the peptide radii. Previous studies suggest that peptides accommodate by tilting. This scenario was confirmed by ATR-FTTR spectroscopy. As the membrane thickness increases, the value of the diffusion coefficient increases to reach a maximum at h ≈ d_π We show that this variation in diffusion coefficient is consistent with a decrease in peptide tilt. To do so, we have derived a relation between the diffusion coefficient and the tilt angle, and we used this relation to derive the peptide tilt from our diffusion measurements. As the membrane thickness increases, the peptides raise (i.e., their tilt is reduced) and reach an upright, position and a maximal mobility for h ≈ d_π Using accessibility measurements, we show that when the membrane becomes too thick, the peptide polar heads sink into the interfacial region. Surprisingly, this "pinching" behavior does not hinder the lateral diffusion of the transmembrane peptides. Ultimately, a break in the peptide transmembrane anchorage is observed and is revealed by a "jump" in the D values.

Collective cell migration is of great significance in many biological processes. The goal of this work is to give a physical model for the dynamics of cell migration during the wound healing response. Experiments demonstrate that an initially uniform cell-culture monolayer expands In a nonuniform manner, developing fingerlike shapes. These fingerlike shapes of the cell culture front are composed of columns of cells that move collectively. We propose a physical model to explain this phenomenon, based on the notion of dynamic instability. In this model, we treat the first layers of cells at the front of the moving cell culture as a continuous one-dimensional membrane (contour), with the usual elasticity of a membrane: curvature and surface-tension. This membrane is active, due to the forces of cellular motility of the cells, and we propose that this motility is related to the local curvature of the culture interface; larger convex curvature correlates with a stronger cellular motility force. This shape-force relation gives rise to a dynamic instability, which we then compare to the patterns observed in the wound healing experiments.

The remarkable deformability of the human red blood cell (RBC) results from the coupled dynamic response of the phospholipid bilayer and the spectrin molecular network. Here we present quantitative connections between spectrin morphology and membrane fluctuations of human RBCs by using dynamic full-field laser interferometry techniques. We present conclusive evidence that the presence of adenosine 5-triphosphate (ATP) facilitates non-equilibrium dynamic fluctuations in the RBC membrane that are highly correlated with the biconcave shape of RBCs. Spatial analysis of the fluctuations reveals that these non-equilibrium membrane vibrations are enhanced at the scale of spectrin mesh size. Our results indicate that the dynamic remodeling of the coupled membranes powered by ATP results in non-equilibrium membrane fluctuations manifesting from both metabolic and thermal energies and also maintains the biconcave shape of RBCs.

The assembly and budding of a new virus is a fundamental step in retroviral replication. Yet, despite substantial progress in the structural and biochemical characterization of retroviral budding, the underlying physical mechanism remains poorly understood, particularly with respect to the mechanism by which the virus overcomes the energy barrier associated with the formation of high membrane curvature during viral budding. Using atomic force, fluorescence, and transmission electron microscopy, we find that both human immunodeficiency virus and Moloney murine leukemia virus remodel the actin cytoskeleton of their host. These actin-filamentous structures assemble simultaneously with or immediately after the beginning of budding, and disappear as soon as the nascent virus is released from the cell membrane. Analysis of sections of cryopreserved virus-infected cells by transmission electron microscopy reveals similar actin filament structures emerging from every nascent virus. Substitution of the nucleocapsid domain implicated in actin binding by a leucine-zipper domain results in the budding of virus-like particles without remodeling of the cell's cytoskeleton. Notably, viruses carrying the modified nucleocapsid domains bud more slowly by an order of magnitude compared to the wild-type. The results of this study show that retroviruses utilize the cell cytoskeleton to expedite their assembly and budding.

We propose a mechanism for the formation of membrane oscillations and traveling waves, which arise due to the coupling between the actin cytoskeleton and the calcium flux through the membrane. In our model, the fluid cell membrane has a mobile but constant population of proteins with a convex spontaneous curvature, which act as nucleators of actin polymerization and adhesion. Such a continuum model couples the forces of cell-substrate adhesion, actin polymerization, membrane curvature, and the flux of calcium through the membrane. Linear stability analysis shows that sufficiently strong coupling among the calcium, membrane, and protein dynamics may induce robust traveling waves on the membrane. This result was checked for a reduced feedback scheme and is compared to the results without the effects of calcium, where permanent phase separation without waves or oscillations is obtained. The model results are compared to the published observations of calcium waves in cell membranes, and a number of testable predictions are proposed.

Collective motion of cell cultures is a process of great interest, as it occurs during morphogenesis, wound healing, and tumor metastasis. During these processes cell cultures move due to the traction forces induced by the individual cells on the surrounding matrix. A recent study [Trepat, et al. (2009). Nat. Phys. 5, 426-430] measured for the first time the traction forces driving collective cell migration and found that they arise throughout the cell culture. The leading 5-10 rows of cell do play a major role in directing the motion of the rest of the culture by having a distinct outwards traction. Fluctuations in the traction forces are an order of magnitude larger than the resultant directional traction at the culture edge and, furthermore, have an exponential distribution. Such exponential distributions are observed for the sizes of adhesion domains within cells, the traction forces produced by single cells, and even in nonbiological nonequilibrium systems, such as sheared granular materials. We discuss these observations and their implications for our understanding of cellular flows within a continuous culture.

The distribution of widths of axons was recently investigated, and was found to have a distinct peak at an optimized value. The optimized axon width at the peak may arise from the conflicting demands of minimizing energy consumption and assuring signal transmission reliability. The distribution around this optimized value is found to have a distinct non-Gaussian shape, with an exponential "tail". We propose here a mechanical model whereby this distribution arises from the interplay between the elastic energy of the membrane surrounding the axon core, the osmotic pressure induced by the neurofilaments inside the axon bulk, and active processes that remodel the microtubules and neurofilaments inside the axon. The axon's radius of curvature can be determined by the cell's control of the osmotic pressure difference across the membrane, the membrane tension or by changing the composition of the different components of the membrane. We find that the osmotic pressure, determined by the neurofilaments, seems to be the dominant control parameter.

We calculate the influence of a flexible network of long-chain proteins, which is anchored to a fluid membrane, on protein diffusion in this membrane. This is a model for the cortical cytoskeleton and the lipid bilayer of the red blood cell, which we apply to predict the influence of the cytoskeleton on the diffusion coefficient of a mobile band 3 protein. Using the pressure field that the cytoskeleton exerts on the membrane, from the steric repulsion between the diffusing protein and the cytoskeletal filaments, we define a potential landscape for the diffusion within the bilayer. We study the changes to the diffusion coefficient on removal of one type of anchor proteins, e.g., in several hemolytic anemias, as well as for isotropic and anisotropic stretching of the cytoskeleton. We predict an overall increase of the diffusion for a smaller number of anchor proteins and increased diffusion for anisotropic stretching in the direction of the stretch, because of the decrease in the spatial frequency as well as in the height of the potential barriers.

Networks of cylindrical membrane tubules appear in many intra-cellular organelles, such as Golgi, ER and mitochondria, and have also been recreated artificially in vitro. These tubules are pulled by molecular motors along filaments of the cytoskeleton, including both actin and microtubules. We propose here a model that is an extension of the thermodynamic equilibrium treatment of such membrane tubules. We treat the active, motors-induced motion of the tubule free ends through the use of an "effective temperature" describing this motion. This mean-field treatment allows us to calculate the effects of such active motion on the phase diagram of the tubules, demonstrating the control that the cell can exert on the morphology of intracellular membrane networks. We compare these results to recent observations of the ER network in cells.

The erythrocyte cytoskeleton is composed of a two-dimensional network of flexible proteins that is anchored to the cell membrane, and determines cell shape. The cytoskeleton also affects the diffusion and distribution of membrane proteins and lipids through direct interactions and steric effects. Here, we present a unified model which describes how the coupling of the local interactions of the cytoskeleton with the bilayer exerts control over the process of membrane vesiculation. In this model, a disturbance of the band 3-ankyrin anchoring complexes leads to increased compression and rigidity of the spectrin cytoskeleton, leading to buckling of the phospholipid bilayer, resulting in vesicle formation. The predictions of this model on size and protein composition of vesicles are confirmed by the available data, especially data of vesicles that are generated during aging in vivo and in blood bank storage conditions. Finally, we suggest some future theoretical elaborations of this model, as well as the experimental approaches for testing it.

The role of the coupling between the shape of membrane-bound filaments and the membrane is demonstrated for the dynamics of FtsZ rings on cylindrical membranes. Filaments with an arc-like spontaneous curvature, and a possible added active contractile force, are shown to spontaneously condense into tight rings, associated with a local inward deformation of the membrane. The long-range membrane-mediated interactions are attractive at short ring-ring separations, inducing further coarsening dynamics, whereby smaller rings merge to form larger and fewer rings that deform the membrane more strongly. At the same time, these interactions induce a potential barrier that can suppress further ring coalescence at a separation of about seven times the cylinder radius. These results of the model are in very good agreement with recent in-vitro experiments on the dynamics of FtsZ filaments in cylindrical liposomes. These results emphasize the important role of long-range membrane-mediated interactions in the organization of cytoskeletal elements at the membrane.

We present a physical model that describes the active localization of actin-regulating proteins inside stereocilia during steady-state conditions. The mechanism of localization is through the interplay of free diffusion and directed motion, which is driven by coupling to the treadmilling actin filaments and to myosin motors that move along the actin filaments. The resulting localization of both the molecular motors and their cargo is calculated, and is found to have an exponential (or steeper) profile. This localization can be at the base (driven by actin retrograde flow and minus-end myosin motors), or at the stereocilia tip (driven by plus-end myosin motors). The localization of proteins that influence the actin depolymerization and polymerization rates allow us to describe the narrow shape of the stereocilia base, and the observed increase of the actin polymerization rate with the stereocilia height.

In this letter we describe how membrane inclusions that have a spontaneous curvature, will be convected on the membrane due to the propagation of membrane waves. We calculate the Stokes drift of such particles and the effect on their overall density field. We solve analytically for a uniform sinusoidal wave in the absence of diffusion, and using simulations for the more realistic case of decaying waves with diffusion. In the latter case we provide some good analytic approximations. A variety of such membrane waves that propagate over a significant proportion of the cell surface exists in living cells, and we therefore show that they can play a role in transporting membrane proteins. Copyright (C) EPLA, 2008

Propagating waves on the surface of cells, over many micrometers, involve active forces. We investigate here the mechanical excitation of such waves when the membrane is perturbed by an external oscillatory force. The external perturbation may trigger the propagation of such waves away from the force application. This scheme is then suggested as a method to probe the properties of the excitable medium of the cell, and learn about the mechanisms that drive the wave propagation. We then apply these ideas to a specific model of active cellular membrane waves, demonstrating how the response of the system to the external perturbation depends on the properties of the model. The most outstanding feature that we find is that the excited waves exhibit a resonance phenomenon at the frequency corresponding to the tendency of the system to develop a linear instability. Mechanical excitation of membrane waves in cells at different frequencies can therefore be used to characterize the properties of the mechanism underlying the existence of these waves.

The length distribution of cytoskeletal filaments is an important physical parameter, which can modulate physiological cell functions. In both eukaryotic and prokaryotic cells various biological cytoskeletal polymers form supramolecular structures due to short-range forces induced mainly by molecular crowding or cross linking proteins, bur their in vivo length distribution remains difficult to measure. In general, based on experimental evidence and mathematical modeling of actin filaments in aqueous solutions, the steady state length distribution of fibrous proteins is believed to be exponential. We performed in vitro TIRF- and electron-microscopy to demonstrate that in the presence of short-range forces, which are an integral part of any living cell, the steady state length distributions of the eukaryotic cytoskeletal biopolymer actin, its prokaryotic homolog ParM and microtubule homolog FtsZ deviate from the classical exponential and are either double-exponential or Gaussian, as recent theoretical modeling predicts. Double exponential or Gaussian distributions opposed to exponential can change for example the visco-elastic properties of actin networks within the cell, influence cell motility by decreasing the amount of free ends at the leading edge of the cell or effect the assembly of FtsZ into the bacterial Z-ring thus modulating membrane constriction.

The formation of bundles composed of actin filaments and cross-linking proteins is an essential process in the maintenance of the cells' cytoskeleton. It has also been recreated by in-vitro experiments, where actin networks are routinely produced to mimic and study the cellular structures. It has been observed that these bundles seem to have a well-defined width distribution, which has not been adequately described theoretically. We propose here that packing defects of the filaments, quenched and random, contribute an effective repulsion that counters the cross-linking adhesion energy and leads to a well-defined bundle width. This is a two-dimensional strain-field version of the classic Rayleigh instability of charged droplets.

Hair cell stereocilia are apical membrane protrusions filled with uniformly polarized actin filament bundles. Protein tyrosine phosphatase receptor Q (PTPRQ), a membrane protein with extracellular fibronectin repeats has been shown to localize at the stereocilia base and the apical hair cell surface, and to be essential for stereocilia integrity. We analyzed the distribution of PTPRQ and a possible mechanism for its compartmentalization. Using immunofluorescence we demonstrate that PTPRQ is compartmentalized at the stereocilia base with a decaying gradient from base to apex. This distribution can be explained by a model of transport directed toward the stereocilia base, which counteracts diffusion of the molecules. By mathematical analysis, we show that this counter transport is consistent with the minus end-directed movement of myosin VI along the stereocilia actin filaments. Myosin VI is localized at the stereocilia base, and exogenously expressed myosin VI and PTPRQ colocalize in the perinuclear endosomes in COS-7 cells. In myosin VI-deficient mice, PTPRQ is distributed along the entire stereocilia. PTPRQ-deficient mice show a pattern of stereocilia disruption that is similar to that reported in myosin VI-deficient mice, where the predominant features are loss of tapered base, and fusion of adjacent stereocilia. Thin section and freeze-etching electron microscopy showed that localization of PTPRQ coincides with the presence of a dense cell surface coat. Our results suggest that PTPRQ and myosin VI form a complex that dynamically maintains the organization of the cell surface coat at the stereocilia base and helps maintain the structure of the overall stereocilia bundle.

Bundles of filamentous actin form the primary building blocks of a broad range of cytoskeletal structures, including filopodia, stereocilia and microvilli. In each case, the cell uses specific associated proteins to tailor the dynamics, dimensions and mechanical properties of the bundles to suit a specific cellular function. While the length distribution of actin bundles was extensively studied, almost nothing is known about the thickness distribution. Here, we use high-resolution cryo-TEM to measure the thickness distribution of actin/fascin bundles, in vitro. We find that the thickness distribution has a prominent peak, with an exponential tail, supporting a scenario of an initial fast formation of a disc-like nucleus of short actin filaments, which only later elongates. The bundle thicknesses at steady state are found to follow the distribution of the initial nuclei indicating that no lateral coalescence occurs. Our results show that the distribution of bundles thicknesses can be controlled by monitoring the initial nucleation process. In vivo, this is done by using specific regulatory proteins complexes.

We present a physical mechanism to describe initiation of the contractile ring during cell division. The model couples the membrane curvature with the contractile forces produced by protein clusters attached to the membrane. These protein clusters are mobile on the membrane and possess either an isotropic or an anisotropic spontaneous curvature. Our results show that under these conditions the contraction force gives rise to an instability that corresponds in a closed cellular system to the initiation of the contractile ring. We find a quantization of this process at distinct length-scales, which we compare to available data for different types of eukaryote cells.

Formation of protrusions and protein segregation on the membrane is of a great importance for the functioning of the living cell. This is most evident in recent experiments that show the effects of the mechanical properties of the surrounding substrate on cell morphology. We propose a mechanism for the formation of membrane protrusions and protein phase separation, which may lay behind this effect. In our model, the fluid cell membrane has a mobile but constant population of proteins with a convex spontaneous curvature. Our basic assumption is that these membrane proteins represent small adhesion complexes, and also include proteins that activate actin polymerization. Such a continuum model couples the membrane and protein dynamics, including cell-substrate adhesion and protrusive actin force. Linear stability analysis shows that sufficiently strong adhesion energy and actin polymerization force can bring about phase separation of the membrane protein and the appearance of protrusions. Specifically, this occurs when the spontaneous curvature and aggregation potential alone (passive system) do not cause phase separation. Finite-size patterns may appear in the regime where the spontaneous curvature energy is a strong factor. Different instability characteristics are calculated for the various regimes, and are compared to various types of observed protrusions and phase separations, both in living cells and in artificial model systems. A number of testable predictions are proposed.

The polymerized network of the cytoskeleton of the red-blood cell (RBC) contains different protein components that maintain its overall integrity and attachment to the lipid bilayer. One of these key components is the band 3-ankyrin complex that attaches the spectrin filaments to the fluid bilayer. Defects in this particular component result in the shape transformation called spherocytosis, through the shedding of membrane nano-vesicles. We show here that this transition and membrane shedding can be explained through the increased stiffness of the network when the band 3-ankyrin complexes are removed. ATP-induced transient dissociations lead to network softening, which offsets the stiffening to some extent, and causes increased fragility of these mutant cells, as is observed.

The fluctuation spectra and the intermembrane interaction of two membranes at a fixed average distance are investigated. Each membrane can either be a fluid or a solid membrane, and in isolation, its fluctuations are described by a bare or a wave-vector-dependent bending modulus, respectively. The membranes interact via their excluded-volume interaction; the average distance is maintained by an external, homogeneous pressure. For strong coupling, the fluctuations can be described by a single, effective membrane that combines the elastic properties. For weak coupling, the fluctuations of the individual, noninteracting membranes are recovered. The case of a composite membrane consisting of one fluid and one solid membrane can serve as a microscopic model for the plasma membrane and cytoskeleton of the red blood cell. We find that, despite the complex microstructure of bilayers and cytoskeletons in a real cell, the fluctuations with wavelengths λ 400 nm are well described by the fluctuations of a single, polymerized membrane (provided that there are no inhomogeneities of the microstructure). The model is applied to the fluctuation data of discocytes ("normal" red blood cells), a stomatocyte, and an echinocyte. The elastic parameters of the membrane and an effective temperature that quantifies active, metabolically driven fluctuations are extracted from the experiments.

Several cell types, among them red blood cells, have a cortical, two-dimensional (2D) network of filaments sparsely attached to their lipid bilayer. In many mammalian cells, this 2D polymer network is connected to an underlying 3D, more rigid cytoskeleton. In this paper, we consider the pressure exerted by the thermally fluctuating, cortical network of filaments on the bilayer and predict the bilayer deformations that are induced by this pressure. We treat the filaments as flexible polymers and calculate the pressure that a network of such linear chains exerts on the bilayer; we then minimize the bilayer shape in order to predict the resulting local deformations. We compare our predictions with membrane deformations observed in electron micrographs of red blood cells. The polymer pressure along with the resulting membrane deformation can lead to compartmentalization, regulate in-plane diffusion and may influence protein sorting as well as transmit signals to the polymerization of the underlying 3D cytoskeleton.

We present a model which couples the membrane with the protrusive forces of actin polymerization and contractile forces of molecular motors, such as myosin. The actin polymerization at the membrane is activated by freely diffusing membrane proteins that have a spontaneous curvature. Molecular motors are recruited to the polymerizing actin filaments, from a constant reservoir, and produce a contractile force. All the forces and variables are treated in the linear limit. Our results show that for convex membrane proteins the myosin activity gives rise to robust transverse membrane waves, similar to those observed on different cells.

The length distribution of tread-milling actin filaments has been measured in vitro, in the presence of severing and cross-linking proteins. The cross-linking proteins induce the formation of bundles, and we present here a kinetic model for this system which takes into account the inherent asymmetry and non-equilibrium nature of the tread-milling actin filaments. The effects of actin cross-linkers is to make the length distribution of filaments inside bundles narrower compared to free filaments. We get good agreement with the available experimental data.

In red blood cells there is a cortical cytoskeleton; a two-dimensional elastic network of membrane-attached proteins. We describe, using a simple model, how the metabolic activity of the cell, through the consumption of ATP, controls the stiffness of this elastic network. The unusual mechanical property of active strain softening is described and compared to experimental data. As a by-product of this activity there is also an active contribution to the amplitude of membrane fluctuations. We model this membrane as a field of independent â\u20acœcurvature motors,â\u20ac and calculate the spectrum of active fluctuations. We find that the active cytoskeleton contributes to the amplitude of the membrane height fluctuations at intermediate wavelengths, as observed experimentally.

During the aging of the red-blood cell, or under conditions of extreme echinocytosis, membrane is shed from the cell plasma membrane in the form of nanovesicles. We propose that this process is the result of the self-adaptation of the membrane surface area to the elastic stress imposed by the spectrin cytoskeleton, via the local buckling of membrane under increasing cytoskeleton stiffness. This model introduces the concept of force balance as a regulatory process at the cell membrane and quantitatively reproduces the rate of area loss in aging red-blood cells.

We calculate the size distribution of two-dimensional aggregates, for different simple dynamical growth models. The resulting size distributions of these domains, at steady state, are shown to depend strongly on the mode of domain growth. We then compare to the measured size-distribution of focal-adhesion domains. Using our calculation and the measured exponential distribution of focal-adhesion domain lengths can be used to test the validity of recent models proposed to describe the dynamics of these complexes in adhering cells.

The biological function of transmembrane proteins is closely related to their insertion, which has most often been studied through their lateral mobility. For >30 years, it has been thought that hardly any information on the size of the diffusing object can be extracted from such experiments. Indeed, the hydrodynamic model developed by Saffman and Delbrück predicts a weak, logarithmic dependence of the diffusion coefficient D with the radius K of the protein. Despite widespread use, its validity has never been thoroughly investigated. To check this model, we measured the diffusion coefficients of various peptides and transmembrane proteins, incorporated into giant unilamellar vesicles of 1-stearoyl-2-oleoyl-sn-glycero-3-phosphocholine (SOPC) or in model bilayers of tunable thickness. We show in this work that, for several integral proteins spanning a large range of sizes, the diffusion coefficient is strongly linked to the protein dimensions. A heuristic model results in a Stokes-like expression for D, (D ∝ 1/R), which fits literature data as well as ours. Diffusion measurement is then a fast and fruitful method; it allows determining the oligomerization degree of proteins or studying lipid-protein and protein-protein interactions within bilayers.

We put forward a theoretical model for the morphological transitions of templated mesoporous materials. These materials consist of a mixture of surfactant molecules and inorganic compounds which evolve dynamically upon mixing to form different morphologies depending on the composition and conditions at which mixing occurs. Our theoretical analysis is based on the assumption that adsorption of the inorganic compounds onto mesoscopic assemblies of surfactant molecules changes the effective interactions between the surfactant molecules, consequently lowering the spontaneous curvature of the surfactant layer and inducing morphological changes in the system. On the basis of a mean field phase diagram, we are able to follow the trajectories of the system starting with different initial conditions, and predict the final morphology of the product. In a typical scenario, the reduction in the spontaneous curvature leads first to a smooth transition from compact spherical micelles to elongated wormlike micelles. In the second stage, the layer of inorganic material coating the micelles gives rise to attractive inter-micellar interactions that eventually induce a collapse of the system into a closely packed hexagonal array of coated cylinders. Other pathways may lead to different structures including disordered bicontinuous and ordered cubic phases. The model is in good qualitative agreement with experimental observations.

A motile cell, when stimulated, shows a dramatic increase in the activity of its membrane, manifested by the appearance of dynamic membrane structures such as lamellipodia, filopodia, and membrane ruffles. The external stimulus turns on membrane bound activators, like Cdc42 and PIP2, which cause increased branching and polymerization of the actin cytoskeleton in their vicinity leading to a local protrusive force on the membrane. The emergence of the complex membrane structures is a result of the coupling between the dynamics of the membrane, the activators, and the protrusive forces. We present a simple model that treats the dynamics of a membrane under the action of actin polymerization forces that depend on the local density of freely diffusing activators on the membrane. We show that, depending on the spontaneous membrane curvature associated with the activators, the resulting membrane motion can be wavelike, corresponding to membrane ruffling and actin waves, or unstable, indicating the tendency of filopodia to form. Our model also quantitatively explains a variety of related experimental observations and makes several testable predictions.

Many different cell types have dynamic protrusions, called microvilli, on their surface. We model these structures as arising from the balance between the force of actin polymerization and the restoring force of the membrane. From this simple model we calculate the distribution function of microvilli heights for several cells. We further describe the phase diagram and the resulting morphology of the microvilli aggregates on the cell surface.

We analyze theoretically the effects of curvature on the diffusion in a fluid membrane, within the Saffman-Delbrück hydrodynamic model. We calculate the effect of curvature on the intrinsic fluidity of a membrane through changes in its thickness, for both static or fluctuating curvature. We treat both thermal curvature fluctuations, and fluctuations due to active processes. Such curvature fluctuations increase the average membrane thickness and diminish the projected area, thereby decreasing the diffusion coefficient. This calculation allows us to predict the effect of shear flow on the membrane diffusion, and to compare to observations on living cells.

We extend a model for nonthermal membrane undulations driven by active (adenosine triphosphate-dependent or light-harvesting) membrane proteins [N. Gov, Phys. Rev. Lett. 93, 268104 (2004)]. The present model accounts for the fact that proteins can diffuse laterally across the membrane surface and that individual proteins are expected to exert forces preferentially in one normal direction over the other (due to their orientation within the bilayer). The addition of these effects alters the scaling of fluctuation amplitudes with system size. Additionally, theoretical arguments and dynamic simulations both suggest that, in certain regimes, the probability distribution of fluctuation amplitudes is expected to be non-Gaussian (in contrast to thermal systems).

Experimental measurements of the shape, fluctuations and adhesion properties of biological cells were analyzed. Particular emphasis was placed on the role of the cytoskeletal and cell elasticity and the shape and adhesion of elastic cells was contrasted with fluid-filled vesicles. The shapes of cells that adhere to a substrate were strongly determined by the cytoskeletal elasticity that can be varied experimentally by drugs that depolymerize the cytoskeleton. It was shown that cell adhesion strongly depends on the forces exerted on the adhesion sites by the tension of the cytoskeleton.

We show theoretically how adenosine 5-triphosphate (ATP)-induced dynamic dissociations of spectrin filaments (from each other and from the membrane) in the cytoskeleton network of red blood cells (RBC) can explain in a unified manner both the measured fluctuation amplitude as well as the observed shape transformations as a function of intracellular ATP concentration. Static defects can be induced by external stresses such as those present when RBCs pass through small capillaries. We suggest that the partially freed actin at these defect sites may explain the activation of the CFTR membrane-bound protein and the subsequent release of ATP by RBCs subjected to deformations. Our theoretical predictions can be tested by experiments that measure the correlation between variations in the binding of actin to spectrin, the activity of CFTR, and the amount of ATP released.

The height undulations of a membrane was analyzed due to fluctuations in the force generated by membrane-bound proteins that induce normal motion or bending. The main features of these active proteins are that they 'kick' the membrane locally and independently of each other. The study also compare the results of experiments on red blood cells and vesicles with incorporated active proton pumps. The results show that the active fluctuation are inversely proportional to the viscosity of the surrounding fluids. In addition, the height fluctuations of the active membranes exhibit general properties, irrespective of the energy source.

We calculate the hydrodynamic interaction Lambda(k) (Oseen interaction kernel) and relaxation frequency omega(k) for the fluctuations of a membrane that is harmonically bounded to a permeable or impermeable wall. We show that due to the confining wall there is an increase in the effective viscosity of the fluid surrounding the membrane. This has been observed in experiments on confined membranes, such as lamellar phases and the red-blood cell membrane. Our results allow a quantitative analysis of these experiments, in terms of the strength of the membrane confining potential and dislocations.

We analyze the thermal fluctuations of fluid membranes in the presence of periodic confining harmonic potentials. This is a simple model of the biologically important, inhomogeneous attachment of the cytoskeleton to the external, fluid membrane of the cell. We study a two-dimensional checkerboard potential as well as one-dimensional, sinusoidal and periodic highly localized, [Formula presented] function potentials. The membranes are described by an energy functional that includes the curvature bending modulus of the membrane and the harmonic external potential. We predict the magnitude of the membrane shape fluctuations. The sinusoidal potentials give a spontaneous surface tension, and an emergent intermediate-range order in the membrane undulations. The [Formula presented] function potentials induce a renormalization of the curvature modulus, with perfect pinning at the [Formula presented] potential sites. After spatial averaging, the [Formula presented]-function potentials also give rise to an effective surface tension. Finally, we compare these results with measurements of the fluctuations of the red-blood cell membrane, which shows the effects of cytoskeleton attachment to the cellular membrane.

A quantum solid is intrinsically 'restless', in the sense that atoms continuously vibrate about their position and exchange places even at the absolute zero of temperature. The archetypal quantum solid is low density solid helium. This paper describes some recent experiments done on solid He which illuminate the distinction of a quantum solid from a classical one, and relate some of these properties to new theoretical predictions.

Recent neutron scattering experiments [T. Markovich, E. Polturak, J. Bossy, and E. Farhi, Phys. Rev. Lett. 88, 195301 (2002)] on solid (formula presented) discovered opticlike mode in the bcc phase. This excitation was predicted by a recently proposed model that describes the correlated atomic zero-point motion in bcc Helium in terms of dynamic electric dipole moments. Modulations of the relative phase of these dipoles between different atoms describes the anomalously soft (formula presented) phonon and two new opticlike modes, one of which was recently found in the neutron scattering experiments. In this work we show that the correlated dipolar interactions can be written as velocity-dependent interactions. This formulation then results in a modified sum rule for the (formula presented) phonon, in good agreement with recent (unpublished) data. This is a striking example of velocity-dependent interactions appearing on a microscopic level in a simple condensed-matter system.

We analyze theoretically both the static and dynamic fluctuation spectra of the red blood cell in a unified manner, using a simple model of the composite membrane. In this model, the two-dimensional spectrin network that forms the cytoskeleton is treated as a rigid shell, located at a fixed, average distance from the lipid bilayer. The cytoskeleton thereby confines both the static and dynamic fluctuations of the lipid bilayer. The sparse connections of the cytoskeleton and bilayer induce a surface tension, for wavelengths larger than the bilayer persistence length. The predictions of the model give a consistent account for both the wave vector and frequency dependence of the experimental data.

We propose a new model for the nature of the nucleation of solid from the superfluid phases of ⁴He and ³He. Unique to the superfluid phases the solid nucleation involves an extremely fast solidification front. We depart from the usual "quasi-static" treatment of solid nucleation by proposing that the nucleation of a solid seed is helped by the simultaneous nucleation of vortex-loops in the superfluid around it. It is the composite entity which is nucleated out of the over-pressurized liquid. This occurs when the local release of pressure creates a velocity field in the superfluid which in turn facilitates the nucleation of vortex-loops. The kinetic energy gain of this process balances the surface tension, as the solid surface is quickly covered by many vortex-loops ("hairy snow-ball"). We show that this scenario gives good agreement with many experiments on heterogeneous nucleation, where the energy barrier is found to differ with the classical theory of homogeneous nucleation by 8 orders of magnitude. We propose several experiments that could show the involvement of vortices with solid nucleation.

We propose a model for the nature of the low-temperature phase of a geometrically frustrated antiferromagnet with a Kagomé lattice, SrCr_8-xGa_4+xO₁₉. We propose that the long-range dipolar interaction between the magnetic Cr³⁺ ions introduces correlations in their zero-point dynamics. The dipolar ground state has the spins performing correlated oscillations in a coherent state with a well defined global phase. We calculate the magnon excitations of such a dipolar array and find good agreement with the spin-wave velocities inferred from measurements of the specific heat. We can further explain the unusual muon spin-relaxation signal of this phase.