Confinement in fibrous environments positions and orients mitotic spindles

Accurate positioning of the mitotic spindle within the rounded cell body is critical to physiological maintenance. Adherent mitotic cells encounter confinement from neighboring cells or the extracellular matrix (ECM), which can cause rotation of mitotic spindles and, consequently, titling of the metaphase plate (MP). To understand the positioning and orientation of mitotic spindles under confinement by fibers (ECM-confinement), we use flexible ECM-mimicking nanofibers that allow natural rounding of the cell body while confining it to differing levels. Rounded mitotic bodies are anchored in place by actin retraction fibers (RFs) originating from adhesion clusters on the ECM-mimicking fibers. We discover the extent of ECM-confinement patterns RFs in 3D: triangular and band-like at low and high confinement, respectively. A stochastic Monte-Carlo simulation of the centrosome (CS), chromosome (CH), membrane interactions, and 3D arrangement of RFs on the mitotic body recovers MP tilting trends observed experimentally. Our mechanistic analysis reveals that the 3D shape of RFs is the primary driver of the MP rotation. Under high ECM-confinement, the fibers can mechanically pinch the cortex, causing the MP to have localized deformations at contact sites with fibers. Interestingly, high ECM-confinement leads to low and high MP tilts, which mechanistically depend upon the extent of cortical deformation, RF patterning, and MP position. We identify that cortical deformation and RFs work in tandem to limit MP tilt, while asymmetric positioning of MP leads to high tilts. Overall, we provide fundamental insights into how mitosis may proceed in fibrous ECM-confining microenvironments in vivo.

A minimal physical model for curvotaxis driven by curved protein complexes at the cell's leading edge

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 "curvotaxis." The origin and underlying mechanism that gives rise to this curvature sensitivity are not well understood. Here, we employ a "minimal cell" 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.

Polarization and motility of one-dimensional multi-cellular trains

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 ("trains") 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.

Experimental and theoretical model for the origin of coiling of cellular protrusions around fibers

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 cell's 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.

A minimal cell model for lamellipodia-based cellular dynamics and migration

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.

Modelling how curved active proteins and shear flow pattern cellular shape and motility

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 "minimal-cell" 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.

Theoretical model of membrane protrusions driven by curved active proteins

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.

A simple cognitive model explains movement decisions in zebrafish while following leaders

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 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], which was limited to a one-dimensional projection of the fish motion, we present here a model that describes the motion of the real fish 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 real fish 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.

Modelling animal contests based on spatio-temporal dynamics

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.

Mitotic outcomes and errors in fibrous environments

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 cortex-astral 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.

A theoretical model of efficient phagocytosis driven by curved membrane proteins and active cytoskeleton forces

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.

Ants resort to majority concession to reach democratic consensus in the presence of a persistent minority

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 "majority concession," 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.

Are cell jamming and unjamming essential in tissue development?

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.

Local actin dynamics couple speed and persistence in a cellular Potts model of cell migration

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 UCSP-even 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.

Active diffusion in oocytes nonspecifically centers large objects during prophase I and meiosis I

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.

Pair formation in insect swarms driven by adaptive long-range interactions

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.

Why a Large-Scale Mode Can Be Essential for Understanding Intracellular Actin Waves

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 activator-inhibitor 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.

Active Trap Model

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.

Excitable solitons: Annihilation, crossover, and nucleation of pulses in mass-conserving activator-inhibitor media

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.

Cellular Blebs and Membrane Invaginations Are Coupled through Membrane Tension Buffering

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.

Cell confinement reveals a branched-actin independent circuit for neutrophil polarity

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.

Cells display a high degree of plasticity in migration, but how polarity is organized in different microenvironments has remained unclear. This study uses mechanical perturbations to reveal that migration using actin-rich or bleb-based protrusions are both organized around Rac GTPase.

Theoretical study of vesicle shapes driven by coupling curved proteins and active cytoskeletal forces

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.

Guided by curvature: shaping cells by coupling curved membrane proteins and cytoskeletal forces

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.This article is part of the theme issue 'Self-organization in cell biology'.

Spatial Fluctuations at Vertices of Epithelial Layers: Quantification of Regulation by Rho Pathway

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.

Erratum to: Effective temperature of active fluids and sheared soft glassy materials

The authors were notified by their collaborators Golan Bel and Dan Wexler that the expression for the over-damped limit of the kinetic energy of the trapped particle, that they have used in the orignal paper, was in error. They provide the correct expressions in this erratum.

Signatures of motor susceptibility to forces in the dynamics of a tracer particle in an active gel

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.

Effective temperature of active fluids and sheared soft glassy materials

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.

Living Matter: Mesoscopic Active Materials

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.

Collective conflict resolution in groups on the move

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.

Bi-stability in cooperative transport by ants in the presence of obstacles

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.

Among animal groups, ants hold what may perhaps be the richest repertoire of collective behavior such as trail formation, nest excavation and food dissemination. Of the most intriguing of these behaviors is cooperative food transport, where many ants carry items that individuals cannot move. Using experiments and theory, we study cooperative transport when the motion is frustrated by an obstacle which contains a single narrow opening that leads to the nest. We find that the group exhibits two co-existing modes of motion that allow exploration of possible routes to overcome the obstacle: Either dwelling near the opening and attempting to pass the cargo through, or performing large excursions that can lead to obstacle circumvention. Previous studies have found that co-existing collective dynamic modes emerge when animal groups interact with constraints, however the origin of the phenomena remains unknown. Here, we provide a detailed theoretical explanation of the source of bi-stability and show how stochastic processes drive the transitions between the two dynamical modes.

Theory of Epithelial Cell Shape Transitions Induced by Mechanoactive Chemical Gradients

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.

Forces in inhomogeneous open active-particle systems

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.

Nonequilibrium mode-coupling theory for dense active systems of self-propelled particles

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 f₀ and a persistence time τ_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 τ_p. 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 τ_α, the α-relaxation time, behaves as τ_α ∼ f₀^-2γ, where γ = 1.74 is the MCT exponent for the passive system. τ_α may increase or decrease as a function of τ_p depending on the type of active force correlations, but the behavior is always governed by the same value of the exponent γ. Comparison with the numerical solution of the nonequilibrium MCT and simulation results give excellent agreement with scaling analysis.

Generalized Archimedes' principle in active fluids

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.

Stable swarming using adaptive long-range interactions

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.

Modeling collective cell migration in geometric confinement

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.

Geometrical Determinants of Neuronal Actin Waves

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.

Exclusion and Hierarchy of Time Scales Lead to Spatial Segregation of Molecular Motors in Cellular Protrusions

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.

Repulsive cues combined with physical barriers and cell-cell adhesion determine progenitor cell positioning during organogenesis

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.

The precise positioning of organ progenitor cells is essential for organ development and function. Here the authors use live imaging and mathematical modelling to show that the confinement of a motile progenitor cell population results from coupled physical barriers and cell-cell interactions.

Direct Cytoskeleton Forces Cause Membrane Softening in Red Blood Cells

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.

F-actin mechanics control spindle centring in the mouse zygote

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.

Pearling instability of membrane tubes driven by curved proteins and actin polymerization

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.

Self-organization of waves and pulse trains by molecular motors in cellular protrusions

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.

Physical model for the geometry of actin-based cellular protrusions

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.

Membrane-wrapping contributions to malaria parasite invasion of the human erythrocyte

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.

Modeling the dynamics of a tracer particle in an elastic active gel

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.

Modeling the finger instability in an expanding cell monolayer

Curvature-controlled cellular forces at the edge of an expanding monolayer are sufficient for the initiation and growth of finger-like instability.