Cell motility driven by actin polymerization.

Certain kinds of cellular movements are apparently driven by actin polymerization. Examples include the lamellipodia of spreading and migrating embryonic cells, and the bacterium Listeria monocytogenes, that propels itself through its host's cytoplasm by constructing behind it a polymerized tail of cross-linked actin filaments. Peskin et al. (1993) formulated a model to explain how a polymerizing filament could rectify the Brownian motion of an object so as to produce unidirectional force (Peskin, C., G. Odell, and G. Oster. 1993. Cellular motions and thermal fluctuations: the Brownian ratchet. Biophys. J. 65:316-324). Their "Brownian ratchet" model assumed that the filament was stiff and that thermal fluctuations affected only the "load," i.e., the object being pushed. However, under many conditions of biological interest, the thermal fluctuations of the load are insufficient to produce the observed motions. Here we shall show that the thermal motions of the polymerizing filaments can produce a directed force. This "elastic Brownian ratchet" can explain quantitatively the propulsion of Listeria and the protrusive mechanics of lamellipodia. The model also explains how the polymerization process nucleates the orthogonal structure of the actin network in lamellipodia.     

Regulation of actin dynamics in rapidly moving cells: a quantitative analysis

We develop a mathematical model that describes key details of actin dynamics in protrusion associated with cell motility. The model is based on the dendritic-nucleation hypothesis for lamellipodial protrusion in nonmuscle cells such as keratocytes. We consider a set of partial differential equations for diffusion and reactions of sequestered actin complexes, nucleation, and growth by polymerization of barbed ends of actin filaments, as well as capping and depolymerization of the filaments. The mechanical aspect of protrusion is based on an elastic polymerization ratchet mechanism. An output of the model is a relationship between the protrusion velocity and the number of filament barbed ends pushing the membrane. Significantly, this relationship has a local maximum: too many barbed ends deplete the available monomer pool, too few are insufficient to generate protrusive force, so motility is stalled at either extreme. Our results suggest that to achieve rapid motility, some tuning of parameters affecting actin dynamics must be operating in the cell.     

Force-velocity measurements of a few growing actin filaments

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.     

Cooperative symmetry-breaking by actin polymerization in a model for cell motility

Polymerizing networks of actin filaments are capable of exerting significant mechanical forces, used by eukaryotic cells and their prokaryotic pathogens to change shape or to move. Here we show that small beads coated uniformly with a protein that catalyses actin polymerization are initially surrounded by symmetrical clouds of actin filaments. This symmetry is broken spontaneously, after which the beads undergo directional motion. We have developed a stochastic theory, in which each actin filament is modelled as an elastic brownian ratchet, that quantitatively accounts for the observed emergent symmetry-breaking behaviour. Symmetry-breaking can only occur for polymers that have a significant subunit off-rate, such as the biopolymers actin and tubulin.     

Models of the Collective Behavior of Proteins in Cells: Tubulin, Actin and Motor Proteins

One of the most important issues of molecular biophysics is the complex and multifunctional behavior of the cell's cytoskeleton. Interiors of living cells are structurally organized by the cytoskeleton networks of filamentous protein polymers: microtubules, actin and intermediate filaments with motor proteins providing force and directionality needed for transport processes. Microtubules (MT's) take active part in material transport within the cell, constitute the most rigid elements of the cell and hence found many uses in cell motility (e.g. flagella andcilia). At present there is, however, no quantitatively predictable explanation of how these important phenomena are orchestrated at a molecular level. Moreover, microtubules have been demonstrated to self-organize leading to pattern formation. We discuss here several models which attempt to shed light on the assembly of microtubules and their interactions with motor proteins. Subsequently, an overview of actin filaments and their properties isgiven with particular emphasis on actin assembly processes. The lengths of actin filaments have been reported that were formed by spontaneous polymerization of highly purified actin monomers after labeling with rhodamine-phalloidin. The length distributions are exponential with a mean of about 7 μm. This length is independent of the initial concentration of actin monomer, an observation inconsistent with a simple nucleation-elongation mechanism. However, with the addition of physically reasonable rates of filament annealing and fragmenting, a nucleation-elongation mechanism can reproduce the observed average length of filaments in two types of experiments: (1) filaments formed from a wide range of highly purified actin monomer concentrations, and (2) filaments formed from 24 mM actin over a range of CapZ concentrations. In the final part of the paper we briefly review the stochastic models used to describe the motion of motor proteins on protein filaments. The vast majority of these models are based on ratchet potentials with the presence of thermal noise and forcing due to ATP binding and a subsequent hydrolysis. Many outstanding questions remain to be quantitatively addressed on a molecular level in order to explain the structure-to-function relationship for the key elements of the cytoskeleton discussed in this review. This revised version was published online in July 2006 with corrections to the Cover Date.     

A Microscopic Formulation for the Actin-Driven Motion of Listeria in Curved Paths

Using a generalized Brownian ratchet model that accounts for the interactions of actin filaments with the surface of Listeria mediated by proteins like ActA and Arp2/3, we have developed a microscopic model for the movement of Listeria. Specifically, we show that a net torque can be generated within the comet tail, causing the bacteria to spin about its long axis, which in conjunction with spatially varying polymerization at the surface leads to motions of bacteria in curved paths that include circles, sinusoidal-like curves, translating figure eights, and serpentine shapes, as observed in recent experiments. A key ingredient in our formulation is the coupling between the motion of Listeria and the force-dependent rate of filament growth. For this reason, a numerical scheme was developed to determine the kinematic parameters of motion and stress distribution among filaments in a self-consistent manner. We find that a 5-15% variation in polymerization rates can lead to radii of curvatures of the order of 4-20 μm, measured in experiments. In a similar way, our results also show that most of the observed trajectories can be produced by a very low degree of correlation, <10%, among filament orientations. Since small fluctuations in polymerization rate, as well as filament orientation, can easily be induced by various factors, our findings here provide a reasonable explanation for why Listeria can travel along totally different paths under seemingly identical experimental conditions. Besides trajectories, stress distributions corresponding to different polymerization profiles are also presented. We have found that although some actin filaments generate propelling forces that push the bacteria forward, others can exert forces opposing the movement of Listeria, consistent with recent experimental observations.     

End versus side branching by Arp2/3 complex

We investigate the issue of end versus side branching of actin filaments by Arp2/3 complex, using a combination of analytic theory, polymerization assays, and quantitative modeling. The analytic theory shows that the effect of capping protein on the initial stages of actin polymerization in the presence of Arp2/3 complex depends strongly on whether new Arp2/3 complex-induced branches grow from the sides or ends of existing filaments. Motivated by these results, we measure and quantitatively model the kinetics of actin polymerization in the presence of activated Arp2/3 complex, for a range of concentrations of capping protein. Our model includes the most important types of events involving actin and actin-binding proteins, and can be adjusted to include end branching, side branching, or both. The side-branching model gives a better fit to the experimental data than the end-branching model. An end-plus-side model including both types of branching gives a moderate improvement in the quality of the fit. Another side-branching model, based on aging of subunits' capacity for branch formation, gives a significantly better fit than the end-plus-side model. We discuss implications for actin polymerization in 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.     

Model of reduction of actin polymerization forces by ATP hydrolysis

The effects of hydrolysis of ATP-actin to ADP-actin on actin polymerization-based force generation are calculated using a multifilament two-state Brownian ratchet model. The model treats an ensemble of rigid parallel filaments growing against a hard, inert, diffusing obstacle held in an optical trap. The filaments stochastically grow, depolymerize and undergo transitions between polymerizing and depolymerizing tip states. The parameters in the model are obtained from literature values and a fit to the measured dependence of the polymerization rate on the free-actin concentration. For more than two filaments, the stall force per filament near the critical concentration is much less than the equilibrium ATP-actin stall force. By reducing the availability of free monomers, the obstacle causes filament tips to convert to the depolymerizing state, so that only a small fraction of the filaments contact the obstacle at a given time.     

Processive capping by formin suggests a force-driven mechanism of actin polymerization

Regulation of actin polymerization is essential for cell functioning. Here, we predict a novel phenomenon-the force-driven polymerization of actin filaments mediated by proteins of the formin family. Formins localize to the barbed ends of actin filaments, but, in contrast to the standard capping proteins, allow for actin polymerization in the barbed direction. First, we show that the mechanism of such "leaky capping" can be understood in terms of the elasticity of the formin molecules. Second, we demonstrate that if a pulling force acts on the filament end via the leaky cap, the elastic stresses can drive actin polymerization. We estimate that a moderate pulling force of approximately 3.4 pN is sufficient to reduce the critical actin concentration required for barbed end polymerization by an order of magnitude. Furthermore, the pulling force increases the polymerization rate. The suggested mechanism of force-driven polymerization could be a key element in a variety of cellular mechanosensing devices.     

Organization of actin networks in intact filopodia

Filopodia are finger-like extensions of the cell surface that are involved in sensing the environment, in attachment of particles for phagocytosis, in anchorage of cells on a substratum, and in the response to chemoattractants or other guidance cues. Filopodia present an excellent model for actin-driven membrane protrusion. They grow at their tips by the assembly of actin and are stabilized along their length by a core of bundled actin filaments. To visualize actin networks in their native membrane-anchored state, filopodia of Dictyostelium cells were subjected to cryo-electron tomography. At the site of actin polymerization, a peculiar structure, the "terminal cone," is built of short filaments fixed with their distal end to the filopod's tip and with their proximal end to the flank of the filopod. The backbone of the filopodia consists of actin filaments that are shorter than the entire filopod and aligned in parallel or obliquely to the filopod's axis. We hypothesize that growth of the highly dynamic filopodia of Dictyostelium is accompanied by repetitive nucleation of actin polymerization at the filopod tip, followed by the rearrangement of filaments within the shaft.     

MARCKS-Related Protein Binds to Actin without Significantly Affecting Actin Polymerization or Network Structure

Actinis a 42-kDa protein which, due to its ability to polymerize into filaments (F-actin), is one of the major constituents of the cytoskeleton. It has been proposed that MARCKS (an acronym for myristoylated alanine-rich C kinase substrate) proteins play an important role in regulating the structure and mechanical properties of the actin cytoskeleton by cross-linking actin filaments. We have recently reported that peptides corresponding to the effector domain of MARCKS proteins promote actin polymerization and cause massive bundling of actin filaments. We now investigate the effect of MARCKS-related protein, a 20-kDa member of the MARCKS family, on both filament structure and the kinetics of actin polymerization in vitro. Our experiments document that MRP binds to F-actin with micromolar affinity and that the myristoyl chain at the N-terminus of MRP is not required for this interaction. In marked contrast to the effector peptide, binding of MRP is not accompanied by an acceleration of actin polymerization kinetics, and we also could not reliably observe an actin cross-linking activity of MRP.     

MARCKS-related protein binds to actin without significantly affecting actin polymerization or network structure. Myristoylated alanine-rich C kinase substrate

Actinis a 42-kDa protein which, due to its ability to polymerize into filaments (F-actin), is one of the major constituents of the cytoskeleton. It has been proposed that MARCKS (an acronym for myristoylated alanine-rich C kinase substrate) proteins play an important role in regulating the structure and mechanical properties of the actin cytoskeleton by cross-linking actin filaments. We have recently reported that peptides corresponding to the effector domain of MARCKS proteins promote actin polymerization and cause massive bundling of actin filaments. We now investigate the effect of MARCKS-related protein, a 20-kDa member of the MARCKS family, on both filament structure and the kinetics of actin polymerization in vitro. Our experiments document that MRP binds to F-actin with micromolar affinity and that the myristoyl chain at the N-terminus of MRP is not required for this interaction. In marked contrast to the effector peptide, binding of MRP is not accompanied by an acceleration of actin polymerization kinetics, and we also could not reliably observe an actin cross-linking activity of MRP.     

The regulation of actin polymerization and cross-linking in Dictyostelium

It is clear that the polymerization and organization of actin filament networks plays a critical role in numerous cellular processes. Inhibition of actin polymerization by pharmacological agents will completely prevent chemotactic motility, macropinocytosis, endocytosis, and phagocytosis. Recently there has been great progress in understanding the mechanisms that control the assembly and structure of the actin cytoskeleton. Members of the Rho family of GTPases have been identified as major players in the signal transduction pathway leading from a cell surface signal to actin polymerization. The Arp2/3 complex has been added to the list of means by which new actin filaments can be nucleated. However, it is clear that actin polymerization by Arp2/3 complex is not the whole story. In principle, the final structures formed by actin filaments will depend on factors such as: the length of actin filaments, the degree of branching, how they are cross-linked and the tensions imparted on them. In addition, the means by which actin polymerization generates protrusion of membranes is still controversial. A phagosome, filopodium and a lamellipodium all require polymerization of new actin filaments, but each has a characteristic morphology and cytoskeletal structure. In the following chapter, we will discuss actin polymerization and filament organization, especially as it relates to the machinery of phagocytosis in Dictyostelium.     

Occurrence of an actin-inserting domain in tensin.

We have previously described a protein called "insertin" that binds strongly to barbed ends of actin filaments and permits polymerization of actin filaments by insertion of actin monomers between the barbed ends and barbed end-bound insertin. We determined the amino acid sequence of insertin and found that the primary structure of insertin is almost identical to amino acid residues 862 to 1212 of the actin-binding protein tensin.     

Attachment conditions control actin filament buckling and the production of forces

Actin polymerization is the driving force for a large number of cellular processes. Formation of lamellipodia and filopodia at the leading edge of motile cells requires actin polymerization induced mechanical deformation of the plasma membrane. To generate different types of membrane protrusions, the mechanical properties of actin filaments can be constrained by interacting proteins. A striking example of such constraint is the buckling of actin filaments generated in vitro by the cooperative effect of a processive actin nucleating factor (formin) and a molecular motor (myosin II). We developed a physical model based on equations for an elastic rod that accounts for actin filament buckling. Both ends of the rod were maintained in a fixed position in space and we considered three sets of boundary conditions. The model qualitatively and quantitatively reproduces the shape distribution of actin filaments. We found that actin polymerization counterpoises a force in the range 0.4-1.6 pN for moderate end-to-end distance (approximately 1 microm) and could be as large as 10 pN for shorter distances. If the actin rod attachment includes a spring, we discovered that the stiffness must be in the range 0.1-1.2 pN/nm to account for the observed buckling.     

Mechanism of action of cytochalasin B on actin

Substoichiometric cytochalasin B (CB) inhibits both the rate of actin polymerization and the interaction of actin filaments in solution. The polymerization rate is reduced by inhibition of actin monomer addition to the "barbed" end of the filaments where monomers normally add more rapidly. 2 microM CB reduces the polymerization rate by up to 90%, but has little effect on the rate of monomer addition at the slow ("pointed") end of the filaments and no effect on the rate of filament annealing. Under most ionic conditions tested, 2 microM CB reduces the steady state high shear viscosity by 10-20% and increases the steady state monomer concentration by a factor of 2.5 or less. In addition to the effects on the polymerization process, 2 microM CB strongly reduces the low shear viscosity of actin filaments alone and actin filaments cross-linked by a variety of macromolecules. This may be due to inhibition of actin filament-filament interactions which normally contribute to network formation. Since the inhibition of monomer addition and of actin filament network formation have approximately the same CB concentration dependence, a common CB binding site, probably the barbed end of the filament, may be responsible for both effects.     

Kinetic evidence for insertion of actin monomers between the barbed ends of actin filaments and barbed end-bound insertin, a protein purified from smooth muscle

An actin polymerization-retarding protein was isolated from chicken gizzard smooth muscle. This protein copurified with vinculin on DEAE-cellulose and gel filtration columns. The polymerization-retarding protein could be separated from vinculin by hydroxylapatite chromatography. The isolated polymerization-retarding protein lost its activity within a few days, but was stable for weeks when it was not separated from vinculin. We termed the polymerization-retarding protein "insertin". Because of the instability of the isolated insertin, we investigated the effect of insertin-vinculin on actin polymerization. Insertin-vinculin retarded nucleated actin polymerization maximally fivefold. Polymerization at the pointed ends of gelsolin-capped actin filaments was not affected by insertin-vinculin, suggesting that insertin-vinculin binds to the barbed ends, but not to the pointed ends, of actin filaments. Retarded polymerization was observed even if the actin monomer concentration was between the critical concentrations of the ends of treadmilling actin filaments. As at this low monomer concentration the pointed ends depolymerize, monomers appeared to be inserted at the barbed ends between the terminal subunit and barbed end-bound insertin molecules. Insertin-vinculin was found not to increase the actin monomer concentration to the value of the pointed ends. These observations support the conclusion that insertin is not a barbed end-capping protein but an actin monomer-inserting protein. According to a quantitative analysis of the kinetic data, all observations could be explained by a model in which two insertin molecules were assumed to bind co-operatively to the barbed ends of actin filaments. Actin monomers were found to be inserted between the barbed ends and barbed end-bound insertin molecules at a rate of about 1 x 10(6) M-1 s-1. Insertin may be an essential part of the machinery of molecules that permit treadmilling of actin filaments in living cells by insertion of actin molecules between membranes and actin filaments.     

Rate of binding of tropomyosin to actin filaments

The decrease of the rate of actin polymerization by tropomyosin molecules which bind near the ends of actin filaments was analyzed in terms of the rate of binding of tropomyosin to actin filaments. Monomeric actin was polymerized onto actin filaments in the presence of various concentrations of tropomyosin. At high concentrations of monomeric actin (c1) and low tropomyosin concentrations (ct) (c1/ct greater than 10), actin polymerization was not retarded by tropomyosin because actin polymerization was faster than binding of tropomyosin to actin filaments. At low actin concentrations and high tropomyosin concentrations (c1/ct less than 5), the rate of elongation of actin filaments was decreased because actin polymerization was slower than binding of tropomyosin at the ends of actin filaments. The results were quantitatively analyzed by a model in which it was assumed that actin-bound tropomyosin molecules which extend beyond the ends of actin filaments retard association of actin monomers with filament ends. Under the experimental conditions (100 mM KCl, 1 mM MgCl2, pH 7.5, 25 degrees C), the rate constant for binding of tropomyosin to actin filaments turned out to be about 2.5 X 10(6) to 4 X 10(6) M-1 S-1.     

Annealing accounts for the length of actin filaments formed by spontaneous polymerization

We measured the lengths of actin filaments formed by spontaneous polymerization of highly purified actin monomers by fluorescence microscopy after labeling with rhodamine-phalloidin. The length distributions are exponential with a mean of approximately 7 microm (2600 subunits). This length is independent of the initial concentration of actin monomer, an observation inconsistent with a simple nucleation-elongation mechanism. However, with the addition of physically reasonable rates of filament annealing and fragmenting, a nucleation-elongation mechanism can reproduce the observed average length of filaments in two types of experiments: 1) filaments formed from a wide range of highly purified actin monomer concentrations, and 2) filaments formed from 24 microM actin over a range of CapZ concentrations.