Forces generated during actin-based propulsion: a direct measurement by micromanipulation

Actin dynamics: from nanoscale to microscale

The dynamic nature of actin in cells manifests itself constantly. Polymerization near the cell edge is balanced by depolymerization in the interior, externally induced actin polymerization is followed by depolymerization, and spontaneous oscillations of actin at the cell periphery are frequently seen. I discuss how mathematical modeling relates quantitative measures of actin dynamics to the rates of underlying molecular level processes. The dynamic properties addressed include the rate of actin assembly at the leading edge of a moving cell, the disassembly rates of intracellular actin networks, the polymerization time course in externally stimulated cells, and spontaneous spatiotemporal patterns formed by actin. Although several aspects of actin assembly have been clarified by increasingly sophisticated models, our understanding of rapid actin disassembly is limited, and the origins of nonmonotonic features in externally stimulated actin polymerization remain unclear. Theory has generated several concrete, testable hypotheses for the origins of spontaneous actin waves and cell-edge oscillations. The development and use of more biomimetic systems applicable to the geometry of a cell will be key to obtaining a quantitative understanding of actin dynamics in cells.

Force-velocity relation for actin-polymerization-driven motility from Brownian dynamics simulations

We report numerical simulation results for the force-velocity relation for actin-polymerization-driven motility. We use Brownian dynamics to solve a physically consistent formulation of the dendritic nucleation model with semiflexible filaments that self-assemble and push a disk. We find that at small loads, the disk speed is independent of load, whereas at high loads, the speed decreases and vanishes at a characteristic stall pressure. Our results demonstrate that at small loads, the velocity is controlled by the reaction rates, whereas at high loads the stall pressure is determined by the mechanical properties of the branched actin network. The behavior is consistent with experiments and with our recently proposed self-diffusiophoretic mechanism for actin-polymerization-driven motility. New in vitro experiments to measure the force-velocity relation are proposed.

The heel and toe of the cell's foot: a multifaceted approach for understanding the structure and dynamics of focal adhesions

Focal adhesions (FAs) are large clusters of transmembrane receptors of the integrin family and a multitude of associated cytoplasmic "plaque" proteins, which connect the extracellular matrix-bound receptors with the actin cytoskeleton. The formation of nearly stationary FAs defines a boundary between the dense and highly dynamic actin network in lamellipodium and the sparser and more diverse cytoskeletal organization in the lamella proper, creating a template for the organization of the entire actin network. The major "mechanical" and "sensory" functions of FAs; namely, the nucleation and regulation of the contractile, myosin-II-containing stress fibers and the mechanosensing of external surfaces depend, to a major extent, on the dynamics of molecular components within FAs. A central element in FA regulation concerns the positive feedback loop, based on the most intriguing feature of FAs; that is, their dependence on mechanical tension developing by the growing stress fibers. FAs grow in response to such tension, and rapidly disassemble upon its relaxation. In this article, we address the mechanistic relationships between the process of FA development, maturation and dissociation and the dynamic molecular events, which take place in different regions of the FA, primarily in the distal end of this structure (the "toe") and the proximal "heel," and discuss the central role of local mechanical forces in orchestrating the complex interplay between FAs and the actin system.

Role of E-cadherin in membrane-cortex interaction probed by nanotube extrusion

This study aims to define the role of E-cadherin (Ecad) engagement in cell-cell contact during membrane-cortex interaction. As a tool, we used a hydrodynamic membrane tube extrusion technique to characterize the mechanical interaction between the plasma membrane and the underlying cortical cytoskeleton. Cells were anchored on 4.5 microm beads coated with polylysine (PL) to obtain nonspecific cell adhesion or with an antibody against Ecad to mimic specific Ecad-mediated cell adhesion. We investigated tube length dynamics L(t) over time and through successive extrusions applied to the cell at regular time intervals. A constant slow velocity was observed for the first extrusion, for PL-attached cells. Subsequent extrusions had two phases: an initial high-velocity regime followed by a low-velocity regime. Successive extrusions gradually weakened the binding of the membrane around the tube neck to the underlying cortical cytoskeleton. Cells specifically attached via Ecad first exhibited a very low extrusion velocity regime followed by a faster extrusion regime similar to nonspecific extrusion. This indicates that Ecad strengthens the membrane-cortical cytoskeleton interaction, but only in a restricted area corresponding to the site of contact between the cell and the bead. Occasional giant "cortex" tubes were extruded with specifically anchored cells, demonstrating that the cortex remained tightly bound to the membrane through Ecad-mediated adhesion at the contact site.

How cofilin severs an actin filament

The actin regulatory protein, cofilin, promotes actin assembly dynamics by severing filaments and increasing the number of ends from which subunits add and dissociate. Recent studies provide biophysical descriptions of cooperative filament interactions in energetic, mechanical and structural terms. A one-dimensional Ising model with nearest-neighbor interactions permits thermodynamic analysis of cooperative binding and indicates that one or a few cofilin molecules can sever a filament. Binding and cooperative interactions are entropically driven. A significant fraction of the binding free energy results from the linked dissociation of filament-associated ions (polyelectrolyte effect), which modulate filament structure, stability and mechanics. The remaining binding free energy and essentially all of the cooperative free energy arise from the enhanced conformational dynamics of the cofilactin complex. Filament mechanics are modulated by cofilin such that cofilin-saturated filaments are approximately 10- to 20-fold more compliant in bending and twisting than bare filaments. Cofilin activity is well described by models in which discontinuities in topology, mechanics and conformational dynamics generate stress concentration and promote fracture at junctions of bare and decorated segments, analogous to the grain boundary fracture of crystalline materials and the thermally driven formation of shear transformation zones in colloidal glass.

Homeostatic competition drives tumor growth and metastasis nucleation

We propose a mechanism for tumor growth emphasizing the role of homeostatic regulation and tissue stability. We show that competition between surface and bulk effects leads to the existence of a critical size that must be overcome by metastases to reach macroscopic sizes. This property can qualitatively explain the observed size distributions of metastases, while size-independent growth rates cannot account for clinical and experimental data. In addition, it potentially explains the observed preferential growth of metastases on tissue surfaces and membranes such as the pleural and peritoneal layers, suggests a mechanism underlying the seed and soil hypothesis introduced by Stephen Paget in 1889, and yields realistic values for metastatic inefficiency. We propose a number of key experiments to test these concepts. The homeostatic pressure as introduced in this work could constitute a quantitative, experimentally accessible measure for the metastatic potential of early malignant growths.

Mechanics model for actin-based motility

We present here a mechanics model for the force generation by actin polymerization. The possible adhesions between the actin filaments and the load surface, as well as the nucleation and capping of filament tips, are included in this model on top of the well-known elastic Brownian ratchet formulation. A closed form solution is provided from which the force-velocity relationship, summarizing the mechanics of polymerization, can be drawn. Model predictions on the velocity of moving beads driven by actin polymerization are consistent with experiment observations. This model also seems capable of explaining the enhanced actin-based motility of Listeria monocytogenes and beads by the presence of Vasodilator-stimulated phosphoprotein, as observed in recent experiments.

Force amplification response of actin filaments under confined compression

Actin protein is a major component of the cell cytoskeleton, and its ability to respond to external forces and generate propulsive forces through the polymerization of filaments is central to many cellular processes. The mechanisms governing actin's abilities are still not fully understood because of the difficulty in observing these processes at a molecular level. Here, we describe a technique for studying actin-surface interactions by using a surface forces apparatus that is able to directly visualize and quantify the collective forces generated when layers of noninterconnected, end-tethered actin filaments are confined between 2 (mica) surfaces. We also identify a force-response mechanism in which filaments not only stiffen under compression, which increases the bending modulus, but more importantly generates opposing forces that are larger than the compressive force. This elastic stiffening mechanism appears to require the presence of confining surfaces, enabling actin filaments to both sense and respond to compressive forces without additional mediating proteins, providing insight into the potential role compressive forces play in many actin and other motor protein-based phenomena.

Mathematics of cell motility: have we got its number?

Mathematical and computational modeling is rapidly becoming an essential research technique complementing traditional experimental biological methods. However, lack of standard modeling methods, difficulties of translating biological phenomena into mathematical language, and differences in biological and mathematical mentalities continue to hinder the scientific progress. Here we focus on one area-cell motility-characterized by an unusually high modeling activity, largely due to a vast amount of quantitative, biophysical data, 'modular' character of motility, and pioneering vision of the area's experimental leaders. In this review, after brief introduction to biology of cell movements, we discuss quantitative models of actin dynamics, protrusion, adhesion, contraction, and cell shape and movement that made an impact on the process of biological discovery. We also comment on modeling approaches and open questions.

Continuum modeling of forces in growing viscoelastic cytoskeletal networks

Mechanical properties of the living cell are important in cell movement, cell division, cancer development and cell signaling. There is considerable interest in measuring local mechanical properties of living materials and the living cytoskeleton using micromechanical techniques. However, living materials are constantly undergoing internal dynamics such as growth and remodeling. A modeling framework that combines mechanical deformations with cytoskeletal growth dynamics is necessary to describe cellular shape changes. The present paper develops a general finite deformation modeling approach that can treat the viscoelastic cytoskeleton. Given the growth dynamics in the cytoskeletal network and the relationship between deformation and stress, the shape of the network is computed in an incremental fashion. The growth dynamics of the cytoskeleton can be modeled as stress dependent. The result is a consistent treatment of overall cell deformation. The framework is applied to a growing 1-d bundle of actin filaments against an elastic cantilever, and a 2-d cell undergoing wave-like protrusion dynamics. In the latter example, mechanical forces on the cell adhesion are examined as a function of the protrusion dynamics.

Protrusion of a Virtual Model Lamellipodium by Actin Polymerization: A Coarse-grained Langevin Dynamics Model

We report the development of a coarse-grained Langevin dynamics model of a lamellipodium featuring growing F-actin filaments in order to study the effect of stiffness of the F-actin filament, the G-actin monomer concentration, and the number of polymerization sites on lamellipodium protrusion. The virtual lamellipodium is modeled as a low-aspect-ratio doubly capped cylinder formed by triangulated particles on its surface. It is assumed that F-actin filaments are firmly attached to a lamellipodium surface where polymerization sites are located, and actin polymerization takes place by connecting a G-actin particle to a polymerization site and to the first particle of a growing F-actin filament. It is found that there is an optimal number of polymerization sites for rapid lamellipodium protrusion. The maximum speed of lamellipodium protrusion is related to competition between the number of polymerization sites and the number of available G-actin particles, and the degree of pulling and holding of the lamellipodium surface by non-polymerizing actin filaments. The lamellipodium protrusion by actin polymerization displays saltatory motion exhibiting pseudo-thermal equilibrium: the lamellipodium speed distribution is Maxwellian in two dimensions but the lamellipodium motion is biased so that the lamellipodium speed in the direction of the lamellipodium motion is much larger than that normal to the lamellipodium motion.

Dynamic assessment of fibroblast mechanical activity during Rac-induced cell spreading in 3-D culture

The goal of this study was to determine the morphological and sub-cellular mechanical effects of Rac activation on fibroblasts within 3-D collagen matrices. Corneal fibroblasts were plated at low density inside 100 microm thick fibrillar collagen matrices and cultured for 1-2 days in serum-free media. Time-lapse imaging was then performed using Nomarski DIC. After an acclimation period, perfusion was switched to media containing PDGF. In some experiments, Y-27632 or blebbistatin were used to inhibit Rho-kinase (ROCK) or myosin II, respectively. PDGF activated Rac and induced cell spreading, which resulted in an increase in cell length, cell area, and the number of pseudopodial processes. Tractional forces were generated by extending pseudopodia, as indicated by centripetal displacement and realignment of collagen fibrils. Interestingly, the pattern of pseudopodial extension and local collagen fibril realignment was highly dependent upon the initial orientation of fibrils at the leading edge. Following ROCK or myosin II inhibition, significant ECM relaxation was observed, but small displacements of collagen fibrils continued to be detected at the tips of pseudopodia. Taken together, the data suggests that during Rac-induced cell spreading within 3-D matrices, there is a shift in the distribution of forces from the center to the periphery of corneal fibroblasts. ROCK mediates the generation of large myosin II-based tractional forces during cell spreading within 3-D collagen matrices, however residual forces can be generated at the tips of extending pseudopodia that are both ROCK and myosin II-independent.

Dynamic regimes and bifurcations in a model of actin-based motility

Propulsion by actin polymerization is widely used in cell motility. Here, we investigate a model of the brush range of an actin gel close to a propelled object, describing the force generation and the dynamics of the propagation velocity. We find transitions between stable steady states and relaxation oscillations when the attachment rate of actin filaments to the obstacle is varied. The oscillations set in at small values of the attachment rate via a homoclinic bifurcation. A second transition from a stable steady state to relaxation oscillations, found for higher values of the attachment rate, occurs via a supercritical Hopf bifurcation. The behavior of the model near the second transition is similar that of a system undergoing a canard explosion. Consequently, we observe excitable dynamics also. The model further exhibits bistability between stationary states or stationary states and limit cycles. Therefore, the brush of actin filament ends appears to have a much richer dynamics than was assumed until now.

New proposed mechanism of actin-polymerization-driven motility

We present the first numerical simulation of actin-driven propulsion by elastic filaments. Specifically, we use a Brownian dynamics formulation of the dendritic nucleation model of actin-driven propulsion. We show that the model leads to a self-assembled network that exerts forces on a disk and pushes it with an average speed. This simulation approach is the first to observe a speed that varies nonmonotonically with the concentration of branching proteins (Arp2/3), capping protein, and depolymerization rate, in accord with experimental observations. Our results suggest a new interpretation of the origin of motility. When we estimate the speed that this mechanism would produce in a system with realistic rate constants and concentrations as well as fluid flow, we obtain a value that is within an order-of-magnitude of the polymerization speed deduced from experiments.

Models for actin polymerization motors

Actin polymerization drives cell membrane protrusions and the propulsion of intracellular pathogens. The molecular mechanisms driving actin polymerization are not yet fully understood. Various mathematical models have been proposed to explain how cells convert chemical energy released upon actin polymerization into a pushing force on a surface. These models have attempted to explain puzzling properties of actin-based motility, including persistent attachment of the network to the membrane during propulsion and the interesting trajectories of propelled particles. These models fall generally into two classes: those requiring filament (+)-ends to fluctuate freely from the membrane to add subunits, and those where filaments elongate with their (+)-ends persistently associated with surface through filament end-tracking proteins ("actoclampin" models). This review compares and contrasts the key predictions of these two classes of models with regard to force-velocity profiles, and evaluates them with respect to experiments with biomimetic particles, and the experimental evidence on the role of end-tracking proteins such as formins and nucleation-promoting factors in actin-based motility.

Viscous-fingering-like instability of cell fragments

We present a novel flow instability that can arise in thin films of cytoskeletal fluids if the friction with the substrate on which the film lies is sufficiently strong. We consider a two-dimensional, membrane-bound fragment containing actin filaments that polymerize at the edge and depolymerize in the fragment. Performing a linear stability analysis of the initial state due to perturbations of the fragment boundary, we find, in the limit of large friction, that the perturbed actin velocity and pressure fields obey the same laws governing the viscous fingering instability of an interface between immiscible fluids in a Hele-Shaw cell. A remarkable feature of this instability is that it is independent of the strength of the interaction between actin filaments and myosin motors.

Curvature and torsion in growing actin networks

Intracellular pathogens such as Listeria monocytogenes and Rickettsia rickettsii move within a host cell by polymerizing a comet-tail of actin fibers that ultimately pushes the cell forward. This dense network of cross-linked actin polymers typically exhibits a striking curvature that causes bacteria to move in gently looping paths. Theoretically, tail curvature has been linked to details of motility by considering force and torque balances from a finite number of polymerizing filaments. Here we track beads coated with a prokaryotic activator of actin polymerization in three dimensions to directly quantify the curvature and torsion of bead motility paths. We find that bead paths are more likely to have low rather than high curvature at any given time. Furthermore, path curvature changes very slowly in time, with an autocorrelation decay time of 200 s. Paths with a small radius of curvature, therefore, remain so for an extended period resulting in loops when confined to two dimensions. When allowed to explore a three-dimensional (3D) space, path loops are less evident. Finally, we quantify the torsion in the bead paths and show that beads do not exhibit a significant left- or right-handed bias to their motion in 3D. These results suggest that paths of actin-propelled objects may be attributed to slow changes in curvature, possibly associated with filament debranching, rather than a fixed torque.

Performance of a population of independent filaments in lamellipodial protrusion

Actin polymerization is responsible for moving a wide variety of loads, from the protrusion of membrane-bound filopodia and lamellipodia of immune, cancer, and other motile cells, to the propulsion of some intracellular pathogens. A universal explanation of the forces and velocities generated by these systems has been hampered by a lack of understanding in how a population of independent filaments pushes these loads. Protrusion of a lamellipodium by the very filaments supporting the membrane load is thought to operate by the Brownian ratchet mechanism, with overall organization governed by the dendritic-nucleation/array-treadmilling model. We have incorporated these two models into a two-dimensional, stochastic computer model of lamellipodial protrusion, and studied how force and velocity generation varied under different assumptions. Performance is very sensitive to the extent to which the work of protrusion is shared among individual polymerization events within the filament population. Three identified mechanisms promote this "work-sharing": 1), Most systems, including lamellipodia, utilize a self-organizing distribution of filament-load distances which serves to decrease the effective size of a monomer and dramatically improve performance. 2), A flexible membrane allows for consistent performance over wide leading edges. 3), Finally, very flexible filaments are capable of sharing work very uniformly, and therefore, of near-perfect theoretical performance. Transient tethering to the lamellipodial membrane limits their efficacy, however, and mandates a minimum filament stiffness. Overall, we estimate lamellipodia to operate with 40-nm bending-length filaments and low characteristic tether forces. Modeled lamellipodia exhibit sigmoidal force-velocity relationships and share the work of protrusion only moderately well among filaments, performing at approximately one-half of theoretical force and velocity maximums. At this level of work-sharing, the natural monomer size is optimal for generating velocity.

The asymmetric self-assembly mechanism of adherens junctions: a cellular push-pull unit

To form adherens junctions (AJ), cells first establish contact by sending out lamellipodia onto neighboring cells. We investigated the role of contacting cells in AJ assembly by studying an asymmetric AJ motif: finger-like AJ extending across the cell-cell interface. Using a cytoskeleton replica and immunofluorescence, we observed that actin bundles embedded in the lamellipodia are co-localized with stress fibers in the neighboring cell at the AJ. This suggests that donor lamellipodia present actin fingers, which are stabilized by acceptor lamellae via acto-myosin contractility. Indeed, we show that changes in actin network geometry promoted by Rac overexpression lead to corresponding changes in AJ morphology. Moreover, contractility inhibition and enhancement (via drugs or local traction) lead respectively to the disappearance and further growth of AJ fingers. Thus, we propose that receiving lamellae exert a local pull on AJ, promoting further polymerization of the donor actin bundles. In spite of different compositions, AJ and focal contacts both act as cellular mechanosensors.

Arp2/3 controls the motile behavior of N-WASP-functionalized GUVs and modulates N-WASP surface distribution by mediating transient links with actin filaments

Spatially controlled assembly of actin in branched filaments generates cell protrusions or the propulsion of intracellular vesicles and pathogens. The propulsive movement of giant unilamellar vesicles (GUVs) functionalized by N-WASP (full-length or truncated) is reconstituted in a biochemically controlled medium, and analyzed using phase contrast and fluorescence microscopy to elucidate the links between membrane components and the actin cytoskeleton that determine motile behavior. Actin-based propulsion displays a continuous regime or a periodic saltatory regime. The transition between the two regimes is controlled by the concentration of Arp2/3 complex, which branches filaments by interacting with N-WASP at the liposome surface. Saltatory motion is linked to cycles in the distribution of N-WASP at the membrane between a homogeneous and a segregated state. Comparison of the changes in distribution of N-WASP, Arp2/3, and actin during propulsion demonstrates that actin filaments bind to N-WASP, and that these bonds are transitory. This interaction, mediated by Arp2/3, drives N-WASP segregation. VC-fragments of N-WASP, that interact more weakly than N-WASP with the Arp2/3 complex, segregate less than N-WASP at the rear of the GUVs. GUV propulsion is inhibited by the presence of VCA-actin covalent complex, showing that the release of actin from the nucleator is required for movement. The balance between segregation and free diffusion determines whether continuous movement can be sustained. Computed surface distributions of N-WASP, derived from a theoretical description of this segregation-diffusion mechanism, account satisfactorily for the measured density profiles of N-WASP, Arp2/3 complex, and actin.

Depolymerization-driven flow in nematode spermatozoa relates crawling speed to size and shape

Cell crawling is an inherently physical process that includes protrusion of the leading edge, adhesion to the substrate, and advance of the trailing cell body. Research into advance of the cell body has focused on actomyosin contraction, with cytoskeletal disassembly regarded as incidental, rather than causative; however, extracts from nematode spermatozoa, which use Major Sperm Protein rather than actin, provide at least one example where cytoskeletal disassembly apparently generates force in the absence of molecular motors. To test whether depolymerization can explain force production during nematode sperm crawling, we constructed a mathematical model that simultaneously describes the dynamics of both the cytoskeleton and the cytosol. We also performed corresponding experiments using motile Caenorhabditis elegans spermatozoa. Our experiments reveal that crawling speed is an increasing function of both cell size and anterior-posterior elongation. The quantitative, depolymerization-driven model robustly predicts that cell speed should increase with cell size and yields a cytoskeletal disassembly rate that is consistent with previous measurements. Notably, the model requires anisotropic elasticity, with the cell being stiffer along the direction of motion, to accurately reproduce the dependence of speed on elongation. Our simulations also predict that speed should increase with cytoskeletal anisotropy and disassembly rate.

Mathematical modeling of cell migration

Mathematical modeling has become increasingly important in many areas of biology during the past two decades, and the area of cell migration and motility has seen significant contributions from a wide range of modeling approaches. In this chapter, we cover examples from the broad range of work in this area, emphasizing the models' biological significance and the relationships between them. We focus on three specific areas: cell protrusion, cell adhesion, and retraction/whole-cell models. At the end of this chapter, we provide our perspective on issues that future models and experiments should consider in order to advance the boundaries of this field.

Stress field in actin gel growing on spherical substrate

Polymerization of actin to form an elastic gel is one of the main mechanisms responsible for cellular motility. The particular problem addressed here stems from the need to model theoretically the growth of actin gel under controlled conditions, as observed in experiments. A biomimetic in vitro system which consists of a spherical latex bead, coated by the enzymatic protein ActA, and a reconstituted cytoplasm within which such beads are placed, induces polymerization of actin on the surface of the bead in the form of successive elastic thin spherical layers. Each newly formed layer pushes outward, and is pushed inward by, the already formed spherical layers which altogether constitute an elastic spherical shell of thickness h varying with time. Thus, a stress field is created in the shell which in turn affects the rate of polymerization as well as that of dissociation of actin gel. Given this bio-chemo-mechanical coupling, the accurate determination of the stress field becomes a subject of great importance for the understanding of the process, and it is the main objective of this work. The problem is addressed by first assuming appropriate constitutive laws for the actin gel elastic material, and then solving the only non-trivial stress equilibrium differential equation along the radial direction assuming spherical symmetry. A linear and a non-linear constitutive model for isotropic elasticity is used, appropriate for small and finite strains, respectively, and the solution is found in closed analytical forms in both cases. Two important conclusions are reached. First, the stress field depends strongly on the compressibility of the actin gel medium via the value of the Poisson ratio, for both linear and non-linear analysis. Second, the linear and non-linear solutions are very close for small strains, but they diverge progressively as the strains increase from small to large. Guided by available experimental data on the observed strain levels, the analytical results are illustrated by selected graphs of stress variation along the radial direction. At the end some comments and suggestions on the bio-chemo-mechanical coupling of actin gel growth and resorption are presented, where the role of properly defined joint isotropic invariants of stress and a unit vector along the predominant direction of free ends of actin filaments at the polymerization site is introduced.

Properties of the force exerted by filopodia and lamellipodia and the involvement of cytoskeletal components

During neuronal differentiation, lamellipodia and filopodia explore the environment in search for the correct path to the axon's final destination. Although the motion of lamellipodia and filopodia has been characterized to an extent, little is known about the force they exert. In this study, we used optical tweezers to measure the force exerted by filopodia and lamellipodia with a millisecond temporal resolution. We found that a single filopodium exerts a force not exceeding 3 pN, whereas lamellipodia can exert a force up to 20 pN. Using metabolic inhibitors, we showed that no force is produced in the absence of actin polymerization and that development of forces larger than 3 pN requires microtubule polymerization. These results show that actin polymerization is necessary for force production and demonstrate that not only do neurons process information, but they also act on their environment exerting forces varying from tenths pN to tens of pN.

ARF1-mediated actin polymerization produces movement of artificial vesicles

Vesicular trafficking and actin dynamics on Golgi membranes are both regulated by ADP-ribosylation factor 1 (ARF1) through the recruitment of various effectors, including vesicular coats. Actin assembly on Golgi membranes contributes to the architecture of the Golgi complex, vesicle formation, and trafficking and is mediated by ARF1 through a cascade that leads to Arp2/3 complex activation. Here we addressed the role of Golgi actin downstream of ARF1 by using a biomimetic assay consisting of liposomes of defined lipid composition, carrying an activated form of ARF1 incubated in cytosolic cell extracts. We observed actin polymerization around the liposomes resulting in thick actin shells and actin comet tails that pushed the ARF1 liposomes forward. The assay was used to characterize the ARF1-dependent pathway, leading to actin polymerization, and confirmed a dependency on CDC42 and its downstream effector N-WASP. Overall, this study demonstrates that actin polymerization driven by the complex multicomponent signaling cascade of the Golgi apparatus can be reproduced with a biomimetic system. Moreover, our results are consistent with the view that actin-based force generation at the site of vesicle formation contributes to the mechanism of fission. In addition to its well established function in coat recruitment, the ARF1 machinery also might produce movement- and fission-promoting forces through actin polymerization.

Load fluctuations drive actin network growth

The growth of actin filament networks is a fundamental biological process that drives a variety of cellular and intracellular motions. During motility, eukaryotic cells and intracellular pathogens are propelled by actin networks organized by nucleation-promoting factors that trigger the formation of nascent filaments off the side of existing filaments in the network. A Brownian ratchet (BR) mechanism has been proposed to couple actin polymerization to cellular movements, whereby thermal motions are rectified by the addition of actin monomers at the end of growing filaments. Here, by following actin-propelled microspheres using three-dimensional laser tracking, we find that beads adhered to the growing network move via an object-fluctuating BR. Velocity varies with the amplitude of thermal fluctuation and inversely with viscosity as predicted for a BR. In addition, motion is saltatory with a broad distribution of step sizes that is correlated in time. These data point to a model in which thermal fluctuations of the microsphere or entire actin network, and not individual filaments, govern motility. This conclusion is supported by Monte Carlo simulations of an adhesion-based BR and suggests an important role for membrane tension in the control of actin-based cellular protrusions.

Reorganisation of the dendritic actin network during cancer cell migration and invasion

Invasion of cancer cells into surrounding tissues has a causal role in tumour progression and is an initial step in tumour metastasis. It requires cell migration, which is driven by the polymerisation of actin within two distinct structures, lamellipodia and filopodia, and attachment to the extracellular matrix through actin-rich adhesive structures. Podosomes and invadopodia are modified adhesive structures that not only establish contact with the substratum, but are also involved in matrix degradation leading to invasion. Actin dynamics and organisation are tightly regulated processes responsible for the range of different and specific cellular functions in response to various stimuli. This review explores the mechanistic basis of tumour cell invasion by focusing on the reorganisation of the dendritic actin network. Actin filaments are flexible structures that are poorly able to resist bending forces, causing them to bend rather than push when encountering obstacles. During migration, cells overcome this problem either by creating a dense array of short-branched filaments as found in lamellipodia, or by bundling filaments as found in filopodia. Here we discuss the possible switch mechanism for the two modes of actin organisation and the advantages of each in the perspective of cell migration and invasion during tumour metastasis.

Mathematical Models of Cell Motility

Cell motility is an essential biological action in the creation, operation and maintenance of our bodies. Developing mathematical models elucidating cell motility will greatly advance our understanding of this fundamental biological process. With accurate models it is possible to explore many permutations of the same event and concisely investigate their outcome. While great advancements have been made in experimental studies of cell motility, it now has somewhat fallen on mathematical models to taking a leading role in future developments. The obvious reason for this is the complexity of cell motility. Employing the processing power of today's computers will give researches the ability to run complex biophysical and biochemical scenarios, without the inherent difficulty and time associated with in vitro investigations. Before any great advancement can be made, the basics of cell motility will have to be well-defined. Without this, complicated mathematical models will be hindered by their inherent conjecture. This review will look at current mathematical investigations of cell motility, explore the reasoning behind such work and conclude with how best to advance this interesting and challenging research area.

Stress fluctuations and motion of cytoskeletal-bound markers

Cytoskeletal (CSK) dynamics such as remodeling and reorganization can be studied by tracking the spontaneous motion of CSK-bound particles. Particle motion is thought to be driven by local, ATP-dependent intracellular force fluctuations due to polymerization processes and motor proteins, and to be impeded by a viscoelastic, metastable cytoskeletal network. The mechanisms that link particle motion to force fluctuations and the CSK dynamics remain unclear. We report simultaneous measurements of the spontaneous motion of CSK-bound particles and of cellular force fluctuations. Cellular force fluctuations were measured by tracking fluorescent markers embedded in an elastic polyacrylamide hydrogel substrate that served as an extracellular matrix (ECM). The motion of CSK-bound particles and markers embedded in the ECM showed both persistence and superdiffusive behavior. Moreover, the movements of CSK-bound beads were temporally and spatially correlated with force fluctuations in the ECM. The findings suggest that the spontaneous motion of CSK-bound beads is driven not by random, local stress fluctuations within a viscoelastic continuum or cage, but rather by stress fluctuations within a tensed and constantly remodeling CSK network that transmits stresses over considerable distances to the ECM.

The forces behind cell movement

Cell movement is a complex phenomenon primarily driven by the actin network beneath the cell membrane, and can be divided into three general components: protrusion of the leading edge of the cell, adhesion of the leading edge and deadhesion at the cell body and rear, and cytoskeletal contraction to pull the cell forward. Each of these steps is driven by physical forces generated by unique segments of the cytoskeleton. This review examines the specific physics underlying these phases of cell movement and the origins of the forces that drive locomotion.

A kinematic description of the trajectories of Listeria monocytogenes propelled by actin comet tails

The bacterial pathogen Listeria monocytogenes propels itself in the cytoplasm of the infected cells by forming a filamentous comet tail assembled by the polymerization of the cytoskeletal protein actin. Although a great deal is known about the molecular processes that lead to actin-based movement, most macroscale aspects of motion, including the nature of the trajectories traced out by the motile bacteria, are not well understood. Here, we present 2D trajectories of Listeria moving between a glass-slide and coverslip in a Xenopus frog egg extract motility assay. We observe that the bacteria move in a number of fascinating geometrical trajectories, including winding S curves, translating figure eights, small- and large-amplitude sine curves, serpentine shapes, circles, and a variety of spirals. We then develop a dynamic model that provides a unified description of these seemingly unrelated trajectories. A key ingredient of the model is a torque (not included in any microscopic models of which we are aware) that arises from the rotation of the propulsive force about the body axis of the bacterium. We show that a large variety of trajectories with a rich mathematical structure are obtained by varying the rate at which the propulsive force moves about the long axis. The trajectories of bacteria executing both steady and saltatory motion are found to be in excellent agreement with the predictions of our dynamic model. When the constraints that lead to planar motion are removed, our model predicts motion along regular helical trajectories, observed in recent experiments.

Actin based processes that could determine the cytoplasmic architecture of plant cells

Actin polymerisation can generate forces that are necessary for cell movement, such as the propulsion of a class of bacteria, including Listeria, and the protrusion of migrating animal cells. Force generation by the actin cytoskeleton in plant cells has not been studied. One process in plant cells that is likely to depend on actin-based force generation is the organisation of the cytoplasm. We compare the function of actin binding proteins of three well-studied mammalian models that depend on actin-based force generation with the function of their homologues in plants. We predict the possible role of these proteins, and thus the role of actin-based force generation, in the production of cytoplasmic organisation in plant cells.

Analysis of actin filament network organization in lamellipodia by comparing experimental and simulated images

Protrusion of lamellipodia during cell migration depends on the assembly of actin network. The assembly mechanism, based on dendritic filament branching, has been investigated in reconstituted in vitro systems, but little is known about the dynamical and structural properties of the actin network in the lamellipodia of migrating cells. The length and orientation of filaments are difficult to measure directly in either optical or electron microscopy images because of the high filament density and overlapping of individual filaments. Here, we use the non-uniformity of optical images of the lamellipodia to extract information about the structural and dynamical properties of the underlying actin network. To determine the relationship between the image features and the properties of the network, we performed simulations of actin network assembly, based on the hypothesis of stochastic branching and capping of filaments, and produced computed `fluorescence' and `electron microscopy' images of the simulated network. By varying simulation parameters, in particular the actin filament density, length and orientation, we closely reproduced the contrast and the characteristic diagonal criss-cross pattern observed in the experimental optical images. Thus, matching the images of the simulated network to the experimental images allowed us to estimate parameters of actin filament network in lamellipodia.

Differential force microscope for long time-scale biophysical measurements

Force microscopy techniques including optical trapping, magnetic tweezers, and atomic force microscopy (AFM) have facilitated quantification of forces and distances on the molecular scale. However, sensitivity and stability limitations have prevented the application of these techniques to biophysical systems that generate large forces over long times, such as actin filament networks. Growth of actin networks drives cellular shape change and generates nano-Newtons of force over time scales of minutes to hours, and consequently network growth properties have been difficult to study. Here, we present an AFM-based differential force microscope with integrated epifluorescence imaging in which two adjacent cantilevers on the same rigid support are used to provide increased measurement stability. We demonstrate 14 nm displacement control over measurement times of 3 hours and apply the instrument to quantify actin network growth in vitro under controlled loads. By measuring both network length and total network fluorescence simultaneously, we show that the average cross-sectional density of the growing network remains constant under static loads. The differential force microscope presented here provides a sensitive method for quantifying force and displacement with long time-scale stability that is useful for measurements of slow biophysical processes in whole cells or in reconstituted molecular systems in vitro.

Lamellipodial actin mechanically links myosin activity with adhesion-site formation

Cell motility proceeds by cycles of edge protrusion, adhesion, and retraction. Whether these functions are coordinated by biochemical or biomechanical processes is unknown. We find that myosin II pulls the rear of the lamellipodial actin network, causing upward bending, edge retraction, and initiation of new adhesion sites. The network then separates from the edge and condenses over the myosin. Protrusion resumes as lamellipodial actin regenerates from the front and extends rearward until it reaches newly assembled myosin, initiating the next cycle. Upward bending, observed by evanescence and electron microscopy, results in ruffle formation when adhesion strength is low. Correlative fluorescence and electron microscopy shows that the regenerating lamellipodium forms a cohesive, separable layer of actin above the lamellum. Thus, actin polymerization periodically builds a mechanical link, the lamellipodium, connecting myosin motors with the initiation of adhesion sites, suggesting that the major functions driving motility are coordinated by a biomechanical process.

Mechanism of actin network attachment to moving membranes: barbed end capture by N-WASP WH2 domains

Actin filament networks exert protrusive and attachment forces on membranes and thereby drive membrane deformation and movement. Here, we show that N-WASP WH2 domains play a previously unanticipated role in vesicle movement by transiently attaching actin filament barbed ends to the membrane. To dissect the attachment mechanism, we reconstituted the propulsive motility of lipid-coated glass beads, using purified soluble proteins. N-WASP WH2 mutants assembled actin comet tails and initiated movement, but the comet tails catastrophically detached from the membrane. When presented on the surface of a lipid-coated bead, WH2 domains were sufficient to maintain comet tail attachment. In v-Src-transformed fibroblasts, N-WASP WH2 mutants were severely defective in the formation of circular podosome arrays. In addition to creating an attachment force, interactions between WH2 domains and barbed ends may locally amplify signals for dendritic actin nucleation.

Force of an actin spring

Cellular movements are produced by forces. Typically, cytoskeletal proteins such as microtubules and actin filaments generate forces via polymerization or in conjunction with molecular motors. However, the fertilization of a Limulus polyphemus egg involves a third type of actin-based cellular engine--a biological spring. During the acrosome reaction, a 60-microm long coiled and twisted bundle of actin filaments straightens and extends from a sperm cell, penetrating the vitelline layer surrounding the egg. A subtle overtwist of 0.2 degrees /subunit underlies the mechanochemical basis for the extension of this actin spring. Upon calcium activation, this conformational strain energy is converted to mechanical work, generating the force required to extend the bundle through the vitelline layer. In this article, we stall the extension of the acrosome bundle in agarose gels of different concentrations. From the stall forces, we estimate a maximum force of 2 nN and a puncturing pressure of 1.6 MPa. We show the maximum force of extension is three times larger than the force required to puncture the vitelline layer. Thus, the elastic strain energy stored in the acrosome bundle is more than sufficient to power the acrosome reaction through the egg envelope.

Direct measurement of force generation by actin filament polymerization using an optical trap

Actin filament polymerization generates force for protrusion of the leading edge in motile cells. In protrusive structures, multiple actin filaments are arranged in cross-linked webs (as in lamellipodia or pseudopodia) or parallel bundles (as in filopodia). We have used an optical trap to directly measure the forces generated by elongation of a few parallel-growing actin filaments brought into apposition with a rigid barrier, mimicking the geometry of filopodial protrusion. We find that the growth of approximately eight actin parallel-growing filaments can be stalled by relatively small applied load forces on the order of 1 pN, consistent with the theoretical load required to stall the elongation of a single filament under our conditions. Indeed, large length fluctuations during the stall phase indicate that only the longest actin filament in the bundle is in contact with the barrier at any given time. These results suggest that force generation by small actin bundles is limited by a dynamic instability of single actin filaments, and therefore living cells must use actin-associated factors to suppress this instability to generate substantial forces by elongation of parallel bundles of actin filaments.

Reversible stress softening of actin networks

The mechanical properties of cells play an essential role in numerous physiological processes. Organized networks of semiflexible actin filaments determine cell stiffness and transmit force during mechanotransduction, cytokinesis, cell motility and other cellular shape changes. Although numerous actin-binding proteins have been identified that organize networks, the mechanical properties of actin networks with physiological architectures and concentrations have been difficult to measure quantitatively. Studies of mechanical properties in vitro have found that crosslinked networks of actin filaments formed in solution exhibit stress stiffening arising from the entropic elasticity of individual filaments or crosslinkers resisting extension. Here we report reversible stress-softening behaviour in actin networks reconstituted in vitro that suggests a critical role for filaments resisting compression. Using a modified atomic force microscope to probe dendritic actin networks (like those formed in the lamellipodia of motile cells), we observe stress stiffening followed by a regime of reversible stress softening at higher loads. This softening behaviour can be explained by elastic buckling of individual filaments under compression that avoids catastrophic fracture of the network. The observation of both stress stiffening and softening suggests a complex interplay between entropic and enthalpic elasticity in determining the mechanical properties of actin networks.

Growth of attached actin filaments

In several studies of actin-based cellular motility, the barbed ends of actin filaments have been observed to be attached to moving obstacles. Filament growth in the presence of such filament-obstacle interactions is studied via Brownian dynamics simulations of a three-dimensional energy-based model. We find that with a binding energy greater than 24k B T and a highly directional force field, a single actin filament is able to push a small obstacle for over a second at a speed of half of the free filament elongation rate. These results are consistent with experimental observations of plastic beads in cell extracts. Calculations of an external force acting on a single-filament-pushed obstacle show that for typical in vitro free-actin concentrations, a 3pN pulling force maximizes the obstacle speed, while a 4pN pushing force almost stops the obstacle. Extension of the model to treat beads propelled by many filaments suggests that most of the propulsive force could be generated by attached filaments.

Direct measurement of the lamellipodial protrusive force in a migrating cell

There has been a great deal of interest in the mechanism of lamellipodial protrusion (Pollard, T., and G. Borisy. 2003. Cell. 112:453-465). However, one of this mechanism's endpoints, the force of protrusion, has never been directly measured. We place an atomic force microscopy cantilever in the path of a migrating keratocyte. The deflection of the cantilever, which occurs over a period of approximately 10 s, provides a direct measure of the force exerted by the lamellipodial leading edge. Stall forces are consistent with approximately 100 polymerizing actin filaments per micrometer of the leading edge, each working as an elastic Brownian ratchet and generating a force of several piconewtons. However, the force-velocity curves obtained from this measurement, in which velocity drops sharply under very small loads, is not sensitive to low loading forces, and finally stalls rapidly at large loads, are not consistent with current theoretical models for the actin polymerization force. Rather, the curves indicate that the protrusive force generation is a complex multiphase process involving actin and adhesion dynamics.

Deformations in actin comets from rocketing beads

The mechanical and dynamical properties of the actin network are essential for many cellular processes like motility or division, and there is a growing body of evidence that they are also important for adhesion and trafficking. The leading edge of migrating cells is pushed out by the polymerization of actin networks, a process orchestrated by cross-linkers and other actin-binding proteins. In vitro physical characterizations show that these same proteins control the elastic properties of actin gels. Here we use a biomimetic system of Listeria monocytogenes, beads coated with an activator of actin polymerization, to assess the role of various actin-binding proteins in propulsion. We find that the properties of actin-based movement are clearly affected by the presence of cross-linkers. By monitoring the evolution of marked parts of the comet, we provide direct experimental evidence that the actin gel continuously undergoes deformations during the growth of the comet. Depending on the protein composition in the motility medium, deformations arise from either gel elasticity or monomer diffusion through the actin comet. Our findings demonstrate that actin-based movement is governed by the mechanical properties of the actin network, which are fine-tuned by proteins involved in actin dynamics and assembly.

Entropic forces generated by grafted semiflexible polymers

The entropic force exerted by the Brownian fluctuations of a grafted semiflexible polymer upon a rigid smooth wall are calculated both analytically and by Monte Carlo simulations. Such forces are thought to play an important role for several cellular phenomena, in particular, the physics of actin-polymerization-driven cell motility and movement of bacteria like Listeria. In the stiff limit, where the persistence length of the polymer is larger than its contour length, we find that the entropic force shows scaling behavior. We identify the characteristic length scales and the explicit form of the scaling functions. In certain asymptotic regimes, we give simple analytical expressions which describe the full results to a very high numerical accuracy. Depending on the constraints imposed on the transverse fluctuations of the filament, there are characteristic differences in the functional form of the entropic forces. In a two-dimensional geometry, the entropic force exhibits a marked peak.

Actin-based propulsion of a microswimmer

A simple hydrodynamic model of actin-based propulsion of microparticles in dilute cell-free cytoplasmic extracts is presented. Under the basic assumption that actin polymerization at the particle surface acts as a force dipole, pushing apart the load and the free (nonanchored) actin tail, the propulsive velocity of the microparticle is determined as a function of the tail length, porosity, and particle shape. The anticipated velocities of the cargo displacement and the rearward motion of the tail are in good agreement with recently reported results of biomimetic experiments. A more detailed analysis of the particle-tail hydrodynamic interaction is presented and compared to the prediction of the simplified model.