Simulation of cell motility that reproduces the force-velocity relationship

Liquid-like condensates mediate competition between actin branching and bundling

Cellular remodeling of actin networks underlies cell motility during key morphological events, from embryogenesis to metastasis. In these transformations, there is an inherent competition between actin branching and bundling, because steric clashes among branches create a mechanical barrier to bundling. Recently, liquid-like condensates consisting purely of proteins involved in either branching or bundling of the cytoskeleton have been found to catalyze their respective functions. Yet in the cell, proteins that drive branching and bundling are present simultaneously. In this complex environment, which factors determine whether a condensate drives filaments to branch or become bundled? To answer this question, we added the branched actin nucleator, Arp2/3, to condensates composed of VASP, an actin bundling protein. At low actin to VASP ratios, branching activity, mediated by Arp2/3, robustly inhibited VASP-mediated bundling of filaments, in agreement with agent-based simulations. In contrast, as the actin to VASP ratio increased, addition of Arp2/3 led to formation of aster-shaped structures, in which bundled filaments emerged from a branched actin core, analogous to filopodia emerging from a branched lamellipodial network. These results demonstrate that multi-component, liquid-like condensates can modulate the inherent competition between bundled and branched actin morphologies, leading to organized, higher-order structures, similar to those found in motile cells.

Liquid-like condensates mediate competition between actin branching and bundling

Cellular remodeling of actin networks underlies cell motility during key morphological events, from embryogenesis to metastasis. In these transformations there is an inherent competition between actin branching and bundling, because steric clashes among branches create a mechanical barrier to bundling. Recently, liquid-like condensates consisting purely of proteins involved in either branching or bundling of the cytoskeleton have been found to catalyze their respective functions. Yet in the cell, proteins that drive branching and bundling are present simultaneously. In this complex environment, which factors determine whether a condensate drives filaments to branch versus becoming bundled? To answer this question, we added the branched actin nucleator, Arp2/3, to condensates composed of VASP, an actin bundling protein. At low actin to VASP ratios, branching activity, mediated by Arp2/3, robustly inhibited VASP-mediated bundling of filaments, in agreement with agent-based simulations. In contrast, as the actin to VASP ratio increased, addition of Arp2/3 led to formation of aster-shaped structures, in which bundled filaments emerged from a branched actin core, analogous to filopodia emerging from a branched lamellipodial network. These results demonstrate that multi-component, liquid-like condensates can modulate the inherent competition between bundled and branched actin morphologies, leading to organized, higher-order structures, similar to those found in motile cells.

Positive, negative and controlled durotaxis

Cell migration is a physical process central to life. Among others, it regulates embryogenesis, tissue regeneration, and tumor growth. Therefore, understanding and controlling cell migration represent fundamental challenges in science. Specifically, the ability of cells to follow stiffness gradients, known as durotaxis, is ubiquitous across most cell types. Even so, certain cells follow positive stiffness gradients while others move along negative gradients. How the physical mechanisms involved in cell migration work to enable a wide range of durotactic responses is still poorly understood. Here, we provide a mechanistic rationale for durotaxis by integrating stochastic clutch models for cell adhesion with an active gel theory of cell migration. We show that positive and negative durotaxis found across cell types are explained by asymmetries in the cell adhesion dynamics. We rationalize durotaxis by asymmetric mechanotransduction in the cell adhesion behavior that further polarizes the intracellular retrograde flow and the protruding velocity at the cell membrane. Our theoretical framework confirms previous experimental observations and explains positive and negative durotaxis. Moreover, we show how durotaxis can be engineered to manipulate cell migration, which has important implications in biology, medicine, and bioengineering.

A mechanism with severing near barbed ends andannealing explains structure and dynamics of dendriticactin networks

Single molecule imaging has shown that part of actin disassembles within a few seconds after incorporation into the dendritic filament network in lamellipodia, suggestive of frequent destabilization near barbed ends. To investigate the mechanisms behind network remodeling, we created a stochastic model with polymerization, depolymerization, branching, capping, uncapping, severing, oligomer diffusion, annealing, and debranching. We find that filament severing, enhanced near barbed ends, can explain the single molecule actin lifetime distribution, if oligomer fragments reanneal to free ends with rate constants comparable to in vitro measurements. The same mechanism leads to actin networks consistent with measured filament, end, and branch concentrations. These networks undergo structural remodeling, leading to longer filaments away from the leading edge, at the +/- 35𝑜 orientation pattern. Imaging of actin speckle lifetimes at sub-second resolution verifies frequent disassembly of newly-assembled actin. We thus propose a unified mechanism that fits a diverse set of basic lamellipodia phenomenology.

Mechanistic insights into actin force generation during vesicle formation from cryo-electron tomography

Actin assembly provides force for a multitude of cellular processes. Compared to actin-assembly-based force production during cell migration, relatively little is understood about how actin assembly generates pulling forces for vesicle formation. Here, cryo-electron tomography identified actin filament number, organization, and orientation during clathrin-mediated endocytosis in human SK-MEL-2 cells, showing that force generation is robust despite variance in network organization. Actin dynamics simulations incorporating a measured branch angle indicate that sufficient force to drive membrane internalization is generated through polymerization and that assembly is triggered from ∼4 founding "mother" filaments, consistent with tomography data. Hip1R actin filament anchoring points are present along the entire endocytic invagination, where simulations show that it is key to pulling force generation, and along the neck, where it targets filament growth and makes internalization more robust. Actin organization described here allowed direct translation of structure to mechanism with broad implications for other actin-driven processes.

Discrete mechanical model of lamellipodial actin network implements molecular clutch mechanism and generates arcs and microspikes

Mechanical forces, actin filament turnover, and adhesion to the extracellular environment regulate lamellipodial protrusions. Computational and mathematical models at the continuum level have been used to investigate the molecular clutch mechanism, calculating the stress profile through the lamellipodium and around focal adhesions. However, the forces and deformations of individual actin filaments have not been considered while interactions between actin networks and actin bundles is not easily accounted with such methods. We develop a filament-level model of a lamellipodial actin network undergoing retrograde flow using 3D Brownian dynamics. Retrograde flow is promoted in simulations by pushing forces from the leading edge (due to actin polymerization), pulling forces (due to molecular motors), and opposed by viscous drag in cytoplasm and focal adhesions. Simulated networks have densities similar to measurements in prior electron micrographs. Connectivity between individual actin segments is maintained by permanent and dynamic crosslinkers. Remodeling of the network occurs via the addition of single actin filaments near the leading edge and via filament bond severing. We investigated how several parameters affect the stress distribution, network deformation and retrograde flow speed. The model captures the decrease in retrograde flow upon increase of focal adhesion strength. The stress profile changes from compression to extension across the leading edge, with regions of filament bending around focal adhesions. The model reproduces the observed reduction in retrograde flow speed upon exposure to cytochalasin D, which halts actin polymerization. Changes in crosslinker concentration and dynamics, as well as in the orientation pattern of newly added filaments demonstrate the model's ability to generate bundles of filaments perpendicular (actin arcs) or parallel (microspikes) to the protruding direction.

The Actin Cytoskeleton as an Active Adaptive Material

Actin is the main protein used by biological cells to adapt their structure and mechanics to their needs. Cellular adaptation is made possible by molecular processes that strongly depend on mechanics. The actin cytoskeleton is also an active material that continuously consumes energy. This allows for dynamical processes that are possible only out of equilibrium and opens up the possibility for multiple layers of control that have evolved around this single protein.Here we discuss the actin cytoskeleton from the viewpoint of physics as an active adaptive material that can build structures superior to man-made soft matter systems. Not only can actin be used to build different network architectures on demand and in an adaptive manner, but it also exhibits the dynamical properties of feedback systems, like excitability, bistability, or oscillations. Therefore, it is a prime example of how biology couples physical structure and information flow and a role model for biology-inspired metamaterials.

A Tale of Two States: Normal and Transformed, With and Without Rigidity Sensing

For many years, major differences in morphology, motility, and mechanical characteristics have been observed between transformed cancer and normal cells. In this review, we consider these differences as linked to different states of normal and transformed cells that involve distinct mechanosensing and motility pathways. There is a strong correlation between repeated tissue healing and/or inflammation and the probability of cancer, both of which involve growth in adult tissues. Many factors are likely needed to enable growth, including the loss of rigidity sensing, but recent evidence indicates that microRNAs have important roles in causing the depletion of growth-suppressing proteins. One microRNA, miR-21, is overexpressed in many different tissues during both healing and cancer. Normal cells can become transformed by the depletion of cytoskeletal proteins that results in the loss of mechanosensing, particularly rigidity sensing. Conversely, the transformed state can be reversed by the expression of cytoskeletal proteins-without direct alteration of hormone receptor levels. In this review, we consider the different stereotypical forms of motility and mechanosensory systems. A major difference between normal and transformed cells involves a sensitivity of transformed cells to mechanical perturbations. Thus, understanding the different mechanical characteristics of transformed cells may enable new approaches to treating wound healing and cancer.

High Rac1 activity is functionally translated into cytosolic structures with unique nanoscale cytoskeletal architecture

Rac1 activation is at the core of signaling pathways regulating polarized cell migration. So far, it has not been possible to directly explore the structural changes triggered by Rac1 activation at the molecular level. Here, through a multiscale imaging workflow that combines biosensor imaging of Rac1 dynamics with electron cryotomography, we identified, within the crowded environment of eukaryotic cells, a unique nanoscale architecture of a flexible, signal-dependent actin structure. In cell regions with high Rac1 activity, we found a structural regime that spans from the ventral membrane up to a height of ∼60 nm above that membrane, composed of directionally unaligned, densely packed actin filaments, most shorter than 150 nm. This unique Rac1-induced morphology is markedly different from the dendritic network architecture in which relatively short filaments emanate from existing, longer actin filaments. These Rac1-mediated scaffold assemblies are devoid of large macromolecules such as ribosomes or other filament types, which are abundant at the periphery and within the remainder of the imaged volumes. Cessation of Rac1 activity induces a complete and rapid structural transition, leading to the absence of detectable remnants of such structures within 150 s, providing direct structural evidence for rapid actin filament network turnover induced by GTPase signaling events. It is tempting to speculate that this highly dynamical nanoscaffold system is sensitive to local spatial cues, thus serving to support the formation of more complex actin filament architectures-such as those mandated by epithelial-mesenchymal transition, for example-or resetting the region by completely dissipating.

Building a dendritic actin filament network branch by branch: models of filament orientation pattern and force generation in lamellipodia

We review mathematical and computational models of the structure, dynamics, and force generation properties of dendritic actin networks. These models have been motivated by the dendritic nucleation model, which provided a mechanistic picture of how the actin cytoskeleton system powers cell motility. We describe how they aimed to explain the self-organization of the branched network into a bimodal distribution of filament orientations peaked at 35° and - 35° with respect to the direction of membrane protrusion, as well as other patterns. Concave and convex force-velocity relationships were derived, depending on network organization, filament, and membrane elasticity and accounting for actin polymerization at the barbed end as a Brownian ratchet. This review also describes models that considered the kinetics and transport of actin and diffuse regulators and mechanical coupling to a substrate, together with explicit modeling of dendritic networks.

Structural organization and energy storage in crosslinked actin assemblies

During clathrin-mediated endocytosis in yeast cells, short actin filaments (< 200nm) and crosslinking protein fimbrin assemble to drive the internalization of the plasma membrane. However, the organization of the actin meshwork during endocytosis remains largely unknown. In addition, only a small fraction of the force necessary to elongate and pinch off vesicles can be accounted for by actin polymerization alone. In this paper, we used mathematical modeling to study the self-organization of rigid actin filaments in the presence of elastic crosslinkers in conditions relevant to endocytosis. We found that actin filaments condense into either a disordered meshwork or an ordered bundle depending on filament length and the mechanical and kinetic properties of the crosslinkers. Our simulations also demonstrated that these nanometer-scale actin structures can store a large amount of elastic energy within the crosslinkers (up to 10kBT per crosslinker). This conversion of binding energy into elastic energy is the consequence of geometric constraints created by the helical pitch of the actin filaments, which results in frustrated configurations of crosslinkers attached to filaments. We propose that this stored elastic energy can be used at a later time in the endocytic process. As a proof of principle, we presented a simple mechanism for sustained torque production by ordered detachment of crosslinkers from a pair of parallel filaments.

Mechanical stress as a regulator of cell motility

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

Vinculin forms a directionally asymmetric catch bond with F-actin

Vinculin is an actin-binding protein thought to reinforce cell-cell and cell-matrix adhesions. However, how mechanical load affects the vinculin-F-actin bond is unclear. Using a single-molecule optical trap assay, we found that vinculin forms a force-dependent catch bond with F-actin through its tail domain, but with lifetimes that depend strongly on the direction of the applied force. Force toward the pointed (-) end of the actin filament resulted in a bond that was maximally stable at 8 piconewtons, with a mean lifetime (12 seconds) 10 times as long as the mean lifetime when force was applied toward the barbed (+) end. A computational model of lamellipodial actin dynamics suggests that the directionality of the vinculin-F-actin bond could establish long-range order in the actin cytoskeleton. The directional and force-stabilized binding of vinculin to F-actin may be a mechanism by which adhesion complexes maintain front-rear asymmetry in migrating cells.

Mathematical model suitable for efficient simulation of thin semi-flexible polymers in complex environments

We present an alternative approach to simulations of semi-flexible polymers. In contrast with the usual bead-rod compromise between bead-spring and rigid rod models, we use deformable cylindrical segments as basic units of the polymer. The length of each segment is not preserved with end points diffusing under constraints keeping the polymer chain nature intact. The model allows the simulation of tension transport and elasticity properties. In particular we describe a new cooperative regime in the relaxation of the polymer from its fully elongated configuration.

Mechanical properties of branched actin filaments

Cells moving on a two dimensional substrate generate motion by polymerizing actin filament networks inside a flat membrane protrusion. New filaments are generated by branching off existing ones, giving rise to branched network structures. We investigate the force-extension relation of branched filaments, grafted on an elastic structure at one end and pushing with the free ends against the leading edge cell membrane. Single filaments are modeled as worm-like chains, whose thermal bending fluctuations are restricted by the leading edge cell membrane, resulting in an effective force. Branching can increase the stiffness considerably; however the effect depends on branch point position and filament orientation, being most pronounced for intermediate tilt angles and intermediate branch point positions. We describe filament networks without cross-linkers to focus on the effect of branching. We use randomly positioned branch points, as generated in the process of treadmilling, and orientation distributions as measured in lamellipodia. These networks reproduce both the weak and strong force response of lamellipodia as measured in force-velocity experiments. We compare properties of branched and unbranched networks. The ratio of the network average of the force per branched filament to the average force per unbranched filament depends on the orientation distribution of the filaments. The ratio exhibits compression dependence and may go up to about 4.5 in networks with a narrow orientation distribution. With orientation distributions measured in lamellipodia, it is about two and essentially independent from network compression, graft elasticity and filament persistence length.

The flatness of Lamellipodia explained by the interaction between actin dynamics and membrane deformation

The crawling motility of many cell types relies on lamellipodia, flat protrusions spreading on flat substrates but (on cells in suspension) also growing into three-dimensional space. Lamellipodia consist of a plasma membrane wrapped around an oriented actin filament meshwork. It is well known that the actin density is controlled by coordinated polymerization, branching, and capping processes, but the mechanisms producing the small aspect ratios of lamellipodia (hundreds of nm thickness vs. several μm lateral and inward extension) remain unclear. The main hypothesis of this work is a strong influence of the local geometry of the plasma membrane on the actin dynamics. This is motivated by observations of co-localization of proteins with I-BAR domains (like IRSp53) with polymerization and branching agents along the membrane. The I-BAR domains are known to bind to the membrane and to prefer and promote membrane curvature. This hypothesis is translated into a stochastic mathematical model where branching and capping rates, and polymerization speeds depend on the local membrane geometry and branching directions are influenced by the principal curvature directions. This requires the knowledge of the deformation of the membrane, being described in a quasi-stationary approximation by minimization of a modified Helfrich energy, subject to the actin filaments acting as obstacles. Simulations with this model predict pieces of flat lamellipodia without any prescribed geometric restrictions.

Power transduction of actin filaments ratcheting in vitro against a load

The actin cytoskeleton has the unique capability of producing pushing forces at the leading edge of motile cells without the implication of molecular motors. This phenomenon has been extensively studied theoretically, and molecular models, including the widely known Brownian ratchet, have been proposed. However, supporting experimental work is lacking, due in part to hardly accessible molecular length scales. We designed an experiment to directly probe the mechanism of force generation in a setup where a population of actin filaments grows against a load applied by magnetic microparticles. The filaments, arranged in stiff bundles by fascin, are constrained to point toward the applied load. In this protrusion-like geometry, we are able to directly measure the velocity of filament elongation and its dependence on force. Using numerical simulations, we provide evidence that our experimental data are consistent with a Brownian ratchet-based model. We further demonstrate the existence of a force regime far below stalling where the mechanical power transduced by the ratcheting filaments to the load is maximal. The actin machinery in migrating cells may tune the number of filaments at the leading edge to work in this force regime.

Branching influences force-velocity curves and length fluctuations in actin networks

We investigate collective dynamics of branched actin networks growing against a rigid movable wall constrained by a resistive force. Computing the force velocity relations, we show that the stall force of such networks depends not only on the average number of filaments touching the wall, but also on the amount of fluctuation of the leading edge of the network. These differences arise due to differences in the network architecture, namely, distance between two adjacent branching points and the initial distance of the starting filament from the wall, with their relative magnitudes influencing the nature of the force velocity curves (convex versus concave). We also show that the introduction of branching results in nonmonotonic diffusion constant, a quantity that measures the growth in length fluctuation of the leading edge of the network, as a function of externally applied force. Together our results demonstrate how the collective dynamics of a branched network differs from that of a parallel filament network.

A force based model of individual cell migration with discrete attachment sites and random switching terms

A force based model of cell migration is presented which gives new insight into the importance of the dynamics of cell binding to the substrate. The main features of the model are the focus on discrete attachment dynamics, the treatment of the cellular forces as springs, and an incorporation of the stochastic nature of the attachment sites. One goal of the model is to capture the effect of the random binding and unbinding of cell attachments on global cell motion. Simulations reveal one of the most important factor influencing cell speed is the duration of the attachment to the substrate. The model captures the correct velocity and force relationships for several cell types.

Plasma membrane--cortical cytoskeleton interactions: a cell biology approach with biophysical considerations

From a biophysical standpoint, the interface between the cell membrane and the cytoskeleton is an intriguing site where a "two-dimensional fluid" interacts with an exceedingly complex three-dimensional protein meshwork. The membrane is a key regulator of the cytoskeleton, which not only provides docking sites for cytoskeletal elements through transmembrane proteins, lipid binding-based, and electrostatic interactions, but also serves as the source of the signaling events and molecules that control cytoskeletal organization and remolding. Conversely, the cytoskeleton is a key determinant of the biophysical and biochemical properties of the membrane, including its shape, tension, movement, composition, as well as the mobility, partitioning, and recycling of its constituents. From a cell biological standpoint, the membrane-cytoskeleton interplay underlies--as a central executor and/or regulator--a multitude of complex processes including chemical and mechanical signal transduction, motility/migration, endo-/exo-/phagocytosis, and other forms of membrane traffic, cell-cell, and cell-matrix adhesion. The aim of this article is to provide an overview of the tight structural and functional coupling between the membrane and the cytoskeleton. As biophysical approaches, both theoretical and experimental, proved to be instrumental for our understanding of the membrane/cytoskeleton interplay, this review will "oscillate" between the cell biological phenomena and the corresponding biophysical principles and considerations. After describing the types of connections between the membrane and the cytoskeleton, we will focus on a few key physical parameters and processes (force generation, curvature, tension, and surface charge) and will discuss how these contribute to a variety of fundamental cell biological functions.

Control of actin-based motility through localized actin binding

A wide variety of cell biological and biomimetic systems use actin polymerization to drive motility. It has been suggested that an object such as a bacterium can propel itself by self-assembling a high concentration of actin behind it, if it is repelled by actin. However, it is also known that it is essential for the moving object to bind actin. Therefore, a key question is how the actin tail can propel an object when it both binds and repels the object. We present a physically consistent Brownian dynamics model for actin-based motility that includes the minimal components of the dendritic nucleation model and allows for both attractive and repulsive interactions between actin and a moveable disc. We find that the concentration gradient of filamentous actin generated by polymerization is sufficient to propel the object, even with moderately strong binding interactions. Additionally, actin binding can act as a biophysical cap, and may directly control motility through modulation of network growth. Overall, this mechanism is robust in that it can drive motility against a load up to a stall pressure that depends on the Young's modulus of the actin network and can explain several aspects of actin-based motility.

Molecular transport modulates the adaptive response of branched actin networks to an external force

Actin networks are an integral part of the cytoskeleton of eukaryotic cells and play an essential role in determining cellular shape and movement. Understanding the underlying mechanism of actin network assembly is of fundamental importance. We developed in this work a minimal motility model and performed stochastic simulations to study mechanical regulation of the growth dynamics of lamellipodia-like branched actin networks, characterized by various force-velocity relations. In such networks, the treadmilling process leads to a concentration gradient of G-actin, and thus G-actin transport is essential to effective actin network assembly. We first explore how capping protein modulates force-velocity relations and then discuss how actin transport due to diffusion and facilitated transport such as advective flow tunes the growth dynamics of the branched actin network. Our work demonstrates the important role of molecular transport in determining the adaptive response of the actin network to an external force.

Mathematical modeling of eukaryotic cell migration: insights beyond experiments

A migrating cell is a molecular machine made of tens of thousands of short-lived and interacting parts. Understanding migration means understanding the self-organization of these parts into a system of functional units. This task is one of tackling complexity: First, the system integrates numerous chemical and mechanical component processes. Second, these processes are connected in feedback interactions and over a large range of spatial and temporal scales. Third, many processes are stochastic, which leads to heterogeneous migration behaviors. Early on in the research of cell migration it became evident that this complexity exceeds human intuition. Thus, the cell migration community has led the charge to build mathematical models that could integrate the diverse experimental observations and measurements in consistent frameworks, first in conceptual and more recently in molecularly explicit models. The main goal of this review is to sift through a series of important conceptual and explicit mathematical models of cell migration and to evaluate their contribution to the field in their ability to integrate critical experimental data.

Crawling of a driven adherent membrane

We discuss motions of an elastic N × M membrane model whose constituents can bind reversibly with strength ε to adhesive sites of a flat substrate. One of the edges of the membrane ("front") is driven in one direction at rate constant p by N stochastically treadmilling short parallel lines ("cortex"). The main conclusions derived from Monte Carlo studies of this model are the following: (a) Since the polymerizing cortex pushes only the leading edge of the membrane, the major part of the membranes is dragged behind. Therefore, the locomotion of the membrane can be described by frictional sliding processes which are asymmetrically distributed between front and rear of the membrane. A signature of this asymmetry is the difference between the life times of adhesion bonds at front and rear, τ(1) and τ(M), respectively, where τ(1) ≫ τ(M). (b) There are four characteristic times for the membrane motion: The first time, T(0) ~ τ(M) ~ e(aε), is the resting time where the displacement of the membrane is practically zero. The second time, T(p) ~ τ(1) ~ M, is the friction time which characterizes the time between two consecutive ruptures of adhesion bonds at the front, and which signalizes the onset of drift ("protrusion") at the leading edge. The third time, T(r) ~ M(γ(ε)) (γ > 1), characterizes the "retraction" of the trailing edge, which is the retarded response to the pulling leading edge. The fourth time, T(L) ~ M(2), is the growth time for fluctuation of the end-to-end distance. (c) The separation of time scales, T(r)/T(p) ~ M(γ(ε) - 1), leads to stretched fluctuations of the end-to-end distance, which are considered as stochastic cycles of protrusion and retraction on the time scale of T(L). (d) The drift velocity v obeys anomalous scaling, v/p~f(p(1/γ(ε))M), where f(z) ~ const. for small drag pM ≪ 1, and f(z) ~ z(-γ(ε)) for pM ≫ 1, which implies v~M(-γ(ε)). These results may also turn out to be useful for the (more difficult) problem of understanding the protrusion-retraction cycle of crawling biological cells. We compare our model and our results to previous two-particle theories for membrane protrusion and to known stochastic friction models.

Asymmetry between pushing and pulling for crawling cells

Eukaryotic cells possess motility mechanisms allowing them not only to self-propel but also to exert forces on obstacles (to push) and to carry cargoes (to pull). To study the inherent asymmetry between active pushing and pulling we model a crawling acto-myosin cell extract as a one-dimensional layer of active gel subjected to external forces. We show that pushing is controlled by protrusion and that the macroscopic signature of the protrusion dominated motility mechanism is concavity of the force-velocity relation. In contrast, pulling is driven by protrusion only at small values of the pulling force and it is replaced by contraction when the pulling force is sufficiently large. This leads to more complex convex-concave structure of the force-velocity relation; in particular, competition between protrusion and contraction can produce negative mobility in a biologically relevant range. The model illustrates active readjustment of the force generating machinery in response to changes in the dipole structure of external forces. The possibility of switching between complementary active mechanisms implies that if necessary "pushers" can replace "pullers" and vice versa.

Branching and capping determine the force-velocity relationships of branching actin networks

A branching actin network is the major engine that drives cell motility. A measure of the effectiveness of an engine is the velocity the engine is able to produce at a given resistance-the force-velocity relationship. Concave force-velocity relationships consist of a force-insensitive region, indicative of an adaptive response. In contrast, convex force-velocity relationships would reflect a passive response. Even in in vitro experiments, branching actin networks can exhibit both concave and convex force-velocity curves. However, the exact mechanism that can explain both force-velocity curves is not yet known. We carried out an agent-based stochastic simulation to explore such a mechanism. We discovered an emergent behavior of a branching actin network: Upon resistance, it remodels itself by increasing the number of filaments growing in contact with the load. The remodeling is favored by branching events and limited by capping. The force-velocity relationship hinges on the relative time-scale between the intrinsic kinetics of the branching actin network and the loading. Shortly after encountering resistance (∼seconds), the force-velocity relationship of the actin network is always convex, as it does not have enough time to remodel itself. A concave force-velocity relationship requires network remodeling at longer time-scales (∼tens of seconds to minutes) and the faster branching event relative to capping. Furthermore, our model explains the observed hysteresis in the force-velocity relationship of actin networks. Our model thus establishes a unified mechanism that can account for both convex and concave force-velocity relationships observed in branching actin networks.

Distributed actin turnover in the lamellipodium and FRAP kinetics

Studies of actin dynamics at the leading edge of motile cells with single-molecule speckle (SiMS) microscopy have shown a broad distribution of EGFP-actin speckle lifetimes and indicated actin polymerization and depolymerization over an extended region. Other experiments using FRAP with the same EGFP-actin as a probe have suggested, by contrast, that polymerization occurs exclusively at the leading edge. We performed FRAP experiments on XTC cells to compare SiMS to FRAP on the same cell type. We used speckle statistics obtained by SiMS to model the steady-state distribution and kinetics of actin in the lamellipodium. We demonstrate that a model with a single diffuse actin species is in good agreement with FRAP experiments. A model including two species of diffuse actin provides an even better agreement. The second species consists of slowly diffusing oligomers that associate to the F-actin network throughout the lamellipodium or break up into monomers after a characteristic time. Our work motivates studies to test the presence and composition of slowly diffusing actin species that may contribute to local remodeling of the actin network and increase the amount of soluble actin.

Emergent complexity of the cytoskeleton: from single filaments to tissue

Despite their overwhelming complexity, living cells display a high degree of internal mechanical and functional organization which can largely be attributed to the intracellular biopolymer scaffold, the cytoskeleton. Being a very complex system far from thermodynamic equilibrium, the cytoskeleton's ability to organize is at the same time challenging and fascinating. The extensive amounts of frequently interacting cellular building blocks and their inherent multifunctionality permits highly adaptive behavior and obstructs a purely reductionist approach. Nevertheless (and despite the field's relative novelty), the physics approach has already proved to be extremely successful in revealing very fundamental concepts of cytoskeleton organization and behavior. This review aims at introducing the physics of the cytoskeleton ranging from single biopolymer filaments to multicellular organisms. Throughout this wide range of phenomena, the focus is set on the intertwined nature of the different physical scales (levels of complexity) that give rise to numerous emergent properties by means of self-organization or self-assembly.

Migration of cells in a social context

In multicellular organisms and complex ecosystems, cells migrate in a social context. Whereas this is essential for the basic processes of life, the influence of neighboring cells on the individual remains poorly understood. Previous work on isolated cells has observed a stereotypical migratory behavior characterized by short-time directional persistence with long-time random movement. We discovered a much richer dynamic in the social context, with significant variations in directionality, displacement, and speed, which are all modulated by local cell density. We developed a mathematical model based on the experimentally identified "cellular traffic rules" and basic physics that revealed that these emergent behaviors are caused by the interplay of single-cell properties and intercellular interactions, the latter being dominated by a pseudopod formation bias mediated by secreted chemicals and pseudopod collapse following collisions. The model demonstrates how aspects of complex biology can be explained by simple rules of physics and constitutes a rapid test bed for future studies of collective migration of individual cells.

Mesoscopic model of actin-based propulsion

Two theoretical models dominate current understanding of actin-based propulsion: microscopic polymerization ratchet model predicts that growing and writhing actin filaments generate forces and movements, while macroscopic elastic propulsion model suggests that deformation and stress of growing actin gel are responsible for the propulsion. We examine both experimentally and computationally the 2D movement of ellipsoidal beads propelled by actin tails and show that neither of the two models can explain the observed bistability of the orientation of the beads. To explain the data, we develop a 2D hybrid mesoscopic model by reconciling these two models such that individual actin filaments undergoing nucleation, elongation, attachment, detachment and capping are embedded into the boundary of a node-spring viscoelastic network representing the macroscopic actin gel. Stochastic simulations of this ‘in silico’ actin network show that the combined effects of the macroscopic elastic deformation and microscopic ratchets can explain the observed bistable orientation of the actin-propelled ellipsoidal beads. To test the theory further, we analyze observed distribution of the curvatures of the trajectories and show that the hybrid model's predictions fit the data. Finally, we demonstrate that the model can explain both concave-up and concave-down force-velocity relations for growing actin networks depending on the characteristic time scale and network recoil. To summarize, we propose that both microscopic polymerization ratchets and macroscopic stresses of the deformable actin network are responsible for the force and movement generation.

There are two major ideas about how actin networks generate force against an obstacle: one is that the force comes directly from the elongation and bending of individual actin filaments against the surface of the obstacle; the other is that a growing actin gel can build up stress around the obstacle to squeeze it forward. Neither of the two models can explain why actin-propelled ellipsoidal beads move with equal bias toward long- and short-axes. We propose a hybrid model by combining those two ideas so that individual actin filaments are embedded into the boundary of a deformable actin gel. Simulations of this model show that the combined effects of pushing from individual filaments and squeezing from the actin network explain the observed bi-orientation of ellipsoidal beads as well as the curvature of trajectories of spherical beads and the force-velocity relation of actin networks.

Toward the virtual cell: automated approaches to building models of subcellular organization "learned" from microscopy images

We review state-of-the-art computational methods for constructing, from image data, generative statistical models of cellular and nuclear shapes and the arrangement of subcellular structures and proteins within them. These automated approaches allow consistent analysis of images of cells for the purposes of learning the range of possible phenotypes, discriminating between them, and informing further investigation. Such models can also provide realistic geometry and initial protein locations to simulations in order to better understand cellular and subcellular processes. To determine the structures of cellular components and how proteins and other molecules are distributed among them, the generative modeling approach described here can be coupled with high throughput imaging technology to infer and represent subcellular organization from data with few a priori assumptions. We also discuss potential improvements to these methods and future directions for research.

Actin filament elasticity and retrograde flow shape the force-velocity relation of motile cells

Cells migrate through a crowded environment during processes such as metastasis or wound healing, and must generate and withstand substantial forces. The cellular motility responses to environmental forces are represented by their force-velocity relation, which has been measured for fish keratocytes but remains unexplained. Even pN opposing forces slow down lamellipodium motion by three orders of magnitude. At larger opposing forces, the retrograde flow of the actin network accelerates until it compensates for polymerization, and cell motion stalls. Subsequently, the lamellipodium adapts to the stalled state. We present a mechanism quantitatively explaining the cell's force-velocity relation and its changes upon application of drugs that hinder actin polymerization or actomyosin-based contractility. Elastic properties of filaments, close to the lamellipodium leading edge, and retrograde flow shape the force-velocity relation. To our knowledge, our results shed new light on how these migratory responses are regulated, and on the mechanics and structure of the lamellipodium.

Excitable actin dynamics in lamellipodial protrusion and retraction

Many animal cells initiate crawling by protruding lamellipodia, consisting of a dense network of actin filaments, at their leading edge. We imaged XTC cells that exhibit flat lamellipodia on poly-L-lysine-coated coverslips. Using active contours, we tracked the leading edge and measured the total amount of F-actin by summing the pixel intensities within a 5-μm band. We observed protrusion and retraction with period 130-200 s and local wavelike features. Positive (negative) velocities correlated with minimum (maximum) integrated actin concentration. Approximately constant retrograde flow indicated that protrusions and retractions were driven by fluctuations of the actin polymerization rate. We present a model of these actin dynamics as an excitable system in which a diffusive, autocatalytic activator causes actin polymerization; F-actin accumulation in turn inhibits further activator accumulation. Simulations of the model reproduced the pattern of actin polymerization seen in experiments. To explore the model's assumption of an autocatalytic activation mechanism, we imaged cells expressing markers for both F-actin and the p21 subunit of the Arp2/3 complex. We found that integrated Arp2/3-complex concentrations spike several seconds before spikes of F-actin concentration. This suggests that the Arp2/3 complex participates in an activation mechanism that includes additional diffuse components. Response of cells to stimulation by fetal calf serum could be reproduced by the model, further supporting the proposed dynamical picture.

A review of models of fluctuating protrusion and retraction patterns at the leading edge of motile cells

A characteristic feature of motile cells as they undergo a change in motile behavior is the development of fluctuating exploratory motions of the leading edge, driven by actin polymerization. We review quantitative models of these protrusion and retraction phenomena. Theoretical studies have been motivated by advances in experimental and computational methods that allow controlled perturbations, single molecule imaging, and analysis of spatiotemporal correlations in microscopic images. To explain oscillations and waves of the leading edge, most theoretical models propose nonlinear interactions and feedback mechanisms among different components of the actin cytoskeleton system. These mechanisms include curvature-sensing membrane proteins, myosin contraction, and autocatalytic biochemical reaction kinetics. We discuss how the combination of experimental studies with modeling promises to quantify the relative importance of these biochemical and biophysical processes at the leading edge and to evaluate their generality across cell types and extracellular environments.

Actin branching in the initiation and maintenance of lamellipodia

Using correlated live cell imaging and electron tomography we found that actin branch junctions in protruding and treadmilling lamellipodia are not concentrated at the front as previously supposed, but link actin filament subsets in which there is a continuum of distances from a junction to the filament plus ends, up to at least 1 µm. When branch sites were observed closely spaced on the same filament their separation was commonly a multiple of the actin helical repeat of 36 nm. Image averaging of branch junctions in the tomograms yielded a model for the in vivo branch at 2.9 nm resolution, which compared closely to that derived for the in vitro actin - Arp2/3 complex. Lamellipodia initiation was monitored in an intracellular wound-healing model and involved branching from the sides of actin filaments oriented parallel to the plasmalemma. Many filament plus ends, presumably capped, terminated behind the lamellipodium tip and localized on the dorsal and ventral surfaces of the actin network. These findings reveal how branching events initiate and maintain a network of actin filaments of variable length and provide the first structural model of the branch junction in vivo. A possible role of filament capping in generating the lamellipodium leaflet is discussed and a mathematical model of protrusion is also presented.

Modeling morphodynamic phenotypes and dynamic regimes of cell motion

Many cellular processes and signaling pathways converge onto cell morphology and cell motion, which share important components. The mechanisms used for propulsion could also be responsible for shape changes, if they are capable of generating the rich observed variety of dynamic regimes. Additionally, the analysis of cell shape changes in space and time promises insight into the state of the cytoskeleton and signaling pathways controlling it. While this has been obvious for some time by now, little effort has been made to systematically and quantitatively explore this source of information. First pioneering experimental work revealed morphodynamic phenotypes which can be associated with dynamic regimes like oscillations and excitability. Here, we review the current state of modeling of morphodynamic phenotypes, the experimental results and discuss the ideas on the mechanisms driving shape changes which are suggested by modeling.

Mechanochemical model of cell migration on substrates of varying stiffness

Cells propel themselves along a substrate by organizing structures at the leading edge called lamellipodia that contain the actin network, myosin, integrin, and other proteins. In this article, we describe a quantitative model that couples the response of stretch-sensitive proteins in the lamellipodia to the dynamics of the actin cytoskeleton, therefore allowing the cell to respond to different substrate stiffnesses. Using this model, we predict the various phases of dynamics possible, including continuous protrusion, unstable retractions leading to ruffling, and periodic protrusion-retraction cycles. We explain the necessary conditions for each type of migratory behavior to occur. In particular, we show that, for periodic protrusion-retraction cycles to occur, the stiffness of the substrate must be high, the myosin-dependent maturation rate of nascent to focal adhesions must be high, and the myosin-independent integrin activation rate must be low. In addition, we also predict the dynamics expected at a given substrate stiffness, leading to a quantitative explanation of experimental data that showed that periodic protrusion-retraction cycles disappear when cells are placed on soft substrates. We also suggest experiments with downregulating α actinin and/or talin and upregulating p130Cas and make predictions on what types of migratory dynamics will be observed.

THE BROWNIAN RATCHET REVISITED: DIFFUSION FORMALISM, POLYMER-BARRIER ATTRACTIONS, AND MULTIPLE FILAMENTOUS BUNDLE GROWTH

Actin polymerization driven stochastic movement of the bacteria Listeria monocytogenes is often measured using single-particle tracking (SPT) methodology and analyzed in terms of statistics. Experimental results suggested a dynamic association between the growing actin filaments and the propelled bacteria. Based on an alternative mathematical formalism for a Brownian ratchet (BR), we introduce such an attractive interaction into the one-dimensional BR model and show that its effect is equivalent to an external resistant force on the bacterium. Such a force significantly reduces the Brownian motion of a driven bacterium, and accentuates the stepping due to polymerization. We then consider the growth, with and without a barrier, of a filamentous bundle consisting of N identical filaments. It is shown that the bundle grows with a similar rate as a single filament in the absence of a load, but can oppose N times the external force under the stalling condition. A set of relationships describing the velocity of the bacterium movement (Vz) and its apparent diffusivity (Dz) as functions of the resistant force (F) and the number of filaments in a bundle (N) are obtained. The theoretical study suggests methods for data analysis in future experiments with applied external resistant force.

Model for self-polarization and motility of keratocyte fragments

Computational modelling of cell motility on substrates is a formidable challenge; regulatory pathways are intertwined and forces that influence cell motion are not fully quantified. Additional challenges arise from the need to describe a moving deformable cell boundary. Here, we present a simple mathematical model coupling cell shape dynamics, treated by the phase-field approach, to a vector field describing the mean orientation (polarization) of the actin filament network. The model successfully reproduces the primary phenomenology of cell motility: discontinuous onset of motion, diversity of cell shapes and shape oscillations. The results are in qualitative agreement with recent experiments on motility of keratocyte cells and cell fragments. The asymmetry of the shapes is captured to a large extent in this simple model, which may prove useful for the interpretation of experiments.

Mechanisms of Cell Propulsion by Active Stresses

The mechanisms by which cytoskeletal flows and cell-substrate interactions interact to generate cell motion are explored using a simplified model of the cytoskeleton as a viscous gel containing active stresses. This model yields explicit general results relating cell speed and traction forces to the distributions of active stress and cell-substrate friction. It is found that 1) the cell velocity is given by a function that quantifies the asymmetry of the active-stress distribution, 2) gradients in cell-substrate friction can induce motion even when the active stresses are symmetrically distributed, 3) the traction-force dipole is enhanced by protrusive stresses near the cell edges or contractile stresses near the center of the cell, and 4) the cell velocity depends biphasically on the cell-substrate adhesion strength if active stress is enhanced by adhesion. Specific experimental tests of the calculated dependences are proposed.

Keratocyte lamellipodial protrusion is characterized by a concave force-velocity relation

We report on the characterization of actin driven lamellipodial protrusion forces and velocities in keratocytes. A vertically mounted glass fiber acted as a flexible barrier positioned in front of migrating keratocytes with parallel phase contrast microscopy. A laser beam was coupled into the fiber and allowed detecting the position of the fiber by a segmented photodiode. Calibration of the fiber was carried out with the thermal oscillation method. Deflection and force signals were measured during lamellipodial protrusion. Velocity was constant during initial contact whereas loading force increased until finally the cell was stalled at higher forces. Stall forces were on the order of 2.9 ± 0.6 nN, which corresponds to a stall pressure of 2.7 ± 1.6 nN/μm(2). Assuming a density of actin filaments of 240 filaments per μm, we can estimate a stall force per actin filament of 1.7 ± 0.8 pN. To check for adaption of the cell against an external force, we let the cell push toward the glass fiber several times. On the timescale of the experiment (∼1 min), however, the cell did not adapt to previous loading events.

Force generation upon T cell receptor engagement

T cells are major players of adaptive immune response in mammals. Recognition of an antigenic peptide in association with the major histocompatibility complex at the surface of an antigen presenting cell (APC) is a specific and sensitive process whose mechanism is not fully understood. The potential contribution of mechanical forces in the T cell activation process is increasingly debated, although these forces are scarcely defined and hold only limited experimental evidence. In this work, we have implemented a biomembrane force probe (BFP) setup and a model APC to explore the nature and the characteristics of the mechanical forces potentially generated upon engagement of the T cell receptor (TCR) and/or lymphocyte function-associated antigen-1 (LFA-1). We show that upon contact with a model APC coated with antibodies towards TCR-CD3, after a short latency, the T cell developed a timed sequence of pushing and pulling forces against its target. These processes were defined by their initial constant growth velocity and loading rate (force increase per unit of time). LFA-1 engagement together with TCR-CD3 reduced the growing speed during the pushing phase without triggering the same mechanical behavior when engaged alone. Intracellular Ca(2+) concentration ([Ca(2+)](i)) was monitored simultaneously to verify the cell commitment in the activation process. [Ca(2+)](i) increased a few tens of seconds after the beginning of the pushing phase although no strong correlation appeared between the two events. The pushing phase was driven by actin polymerization. Tuning the BFP mechanical properties, we could show that the loading rate during the pulling phase increased with the target stiffness. This indicated that a mechanosensing mechanism is implemented in the early steps of the activation process. We provide here the first quantified description of force generation sequence upon local bidimensional engagement of TCR-CD3 and discuss its potential role in a T cell mechanically-regulated activation process.

Force-velocity measurements of a few growing actin filaments

The authors propose a new mechanism for actin-based force generation based on results using chains of actin-grafted magnetic colloids.

The polymerization of actin in filaments generates forces that play a pivotal role in many cellular processes. We introduce a novel technique to determine the force-velocity relation when a few independent anchored filaments grow between magnetic colloidal particles. When a magnetic field is applied, the colloidal particles assemble into chains under controlled loading or spacing. As the filaments elongate, the beads separate, allowing the force-velocity curve to be precisely measured. In the widely accepted Brownian ratchet model, the transduced force is associated with the slowing down of the on-rate polymerization. Unexpectedly, in our experiments, filaments are shown to grow at the same rate as when they are free in solution. However, as they elongate, filaments are more confined in the interspace between beads. Higher repulsive forces result from this higher confinement, which is associated with a lower entropy. In this mechanism, the production of force is not controlled by the polymerization rate, but is a consequence of the restriction of filaments' orientational fluctuations at their attachment point.

Actin self-assembles into filaments, and this produces forces that deform cell membranes in a large number of motile processes. While physical measurements have been performed of the force produced by growth of either a single filament or a large intricate array of filaments organized in an active macroscopic gel, these measurements don't provide a clear picture of how force is produced by the assembly of each filament within a complex structure. The present study explores a situation between these two extremes by measuring the force produced by the assembly of a small number of filaments. We developed a method in which actin filaments grow from the surface of magnetic beads that are aligned by a controlled magnetic field. The distance between beads in a chain-like arrangement increases with time when the force is kept constant. We observe that the growth of actin filaments is not affected by the load, in contrast to the widely accepted “Brownian ratchet model.” Instead, our results suggest that the surface opposite growing filaments imposes restrictions on the rotational fluctuations of a filament at its free hinge anchoring point, inducing a repulsive force. The confinement of filaments increases as they grow, and this in turn increases the repulsive force developed by their growth. This entropy-based mechanism may operate during motile processes when actin networks are loosely organized.

Robust organizational principles of protrusive biopolymer networks in migrating living cells

Cell migration is associated with the dynamic protrusion of a thin actin-based cytoskeletal extension at the cell front, which has been shown to consist of two different substructures, the leading lamellipodium and the subsequent lamellum. While the formation of the lamellipodium is increasingly well understood, organizational principles underlying the emergence of the lamellum are just beginning to be unraveled. We report here on a 1D mathematical model which describes the reaction-diffusion processes of a polarized actin network in steady state, and reproduces essential characteristics of the lamellipodium-lamellum system. We observe a steep gradient in filament lengths at the protruding edge, a local depolymerization maximum a few microns behind the edge, as well as a differential dominance of the network destabilizer ADF/cofilin and the stabilizer tropomyosin. We identify simple and robust organizational principles giving rise to the derived network characteristics, uncoupled from the specifics of any molecular implementation, and thus plausibly valid across cell types. An analysis of network length dependence on physico-chemical system parameters implies that to limit array treadmilling to cellular dimensions, network growth has to be truncated by mechanisms other than aging-induced depolymerization, e.g., by myosin-associated network dissociation at the transition to the cell body. Our work contributes to the analytical understanding of the cytoskeletal extension's bisection into lamellipodium and lamellum and sheds light on how cells organize their molecular machinery to achieve motility.

Leading-edge-gel coupling in lamellipodium motion

We present a model for actin-based motility that combines the dynamics of the semiflexible region at the leading edge of the lamellipodium with actomyosin gel properties in the bulk described by the theory of active polar gels. We calculate the velocity of the lamellipodium determined by the interaction of the gel and adhesion with forces in the semiflexible region. The stationary concave force-velocity relation of the model reproduces experimental results. We suggest that it is determined by retrograde flow at small forces and gel formation and retrograde flow at large ones. The variety of dynamic regimes of the semiflexible region reproducing experimentally observed morphodynamics is conserved when we couple the leading edge to the gel.

Critical review: cellular mechanobiology and amoeboid migration

Cell motility is important for tissue homeostasis and plays a central role in various pathologies, notably inflammation and cancer. Research into the critical processes involved in cell migration has so far mostly focused on cell adhesion and proteolytic degradation of the extracellular matrix. However, pharmacological interference with these processes only partially blocks cell motility in vivo. In this review we summarize the arising evidence that the mechanical properties of the cell body have a major role to play in cell motility--especially in a low-adhesion, amoeboid-like migration mode in three-dimensional tissue structures. We summarize the processes determining cell mechanics and discuss relevant measurement technologies including their applications in medical cell biology.