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.     

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.     

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.     

Force generation by actin polymerization II: the elastic ratchet and tethered filaments

The motion of many intracellular pathogens is driven by the polymerization of actin filaments. The propulsive force developed by the polymerization process is thought to arise from the thermal motions of the polymerizing filament tips. Recent experiments suggest that the nucleation of actin filaments involves a phase when the filaments are attached to the pathogen surface by a protein complex. Here we extend the "elastic ratchet model" of Mogilner and Oster to incorporate these new findings. We apply this "tethered ratchet" model to derive the force-velocity relation for Listeria and discuss relations of our theoretical predictions to experimental measurements. We also discuss "symmetry breaking" dynamics observed in ActA-coated bead experiments, and the implications of the model for lamellipodial protrusion in migrating cells.     

A biomimetic motility assay provides insight into the mechanism of actin-based motility

Abiomimetic motility assay is used to analyze the mechanism of force production by site-directed polymerization of actin. Polystyrene microspheres, functionalized in a controlled fashion by the N-WASP protein, the ubiquitous activator of Arp2/3 complex, undergo actin-based propulsion in a medium that consists of five pure proteins. We have analyzed the dependence of velocity on N-WASP surface density, on the concentration of capping protein, and on external force. Movement was not slowed down by increasing the diameter of the beads (0.2 to 3 microm) nor by increasing the viscosity of the medium by 10(5)-fold. This important result shows that forces due to actin polymerization are balanced by internal forces due to transient attachment of filament ends at the surface. These forces are greater than the viscous drag. Using Alexa488-labeled Arp2/3, we show that Arp2/3 is incorporated in the actin tail like G-actin by barbed end branching of filaments at the bead surface, not by side branching, and that filaments are more densely branched upon increasing gelsolin concentration. These data support models in which the rates of filament branching and capping control velocity, and autocatalytic branching of filament ends, rather than filament nucleation, occurs at the particle surface.     

Loading history determines the velocity of actin-network growth

Directional polymerization of actin filaments in branched networks is one of the most powerful force-generating systems in eukaryotic cells. Growth of densely cross-linked actin networks drives cell crawling, intracellular transport of vesicles and organelles, and movement of intracellular pathogens such as Listeria monocytogenes. Using a modified atomic force microscope (AFM), we obtained force-velocity (Fv) measurements of growing actin networks in vitro until network elongation ceased at the stall force. We found that the growth velocity of a branched actin network against increasing forces is load-independent over a wide range of forces before a convex decline to stall. Surprisingly, when force was decreased on a growing network, the velocity increased to a value greater than the previous velocity, such that two or more stable growth velocities can exist at a single load. These results demonstrate that a single Fv relationship does not capture the complete behaviour of this system, unlike other molecular motors in cells, because the growth velocity depends on loading history rather than solely on the instantaneous load.     

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.     

Modulation of repulsive forces between neurofilaments by sidearm phosphorylation

Recent studies have advanced the notion that the axonal organization of neurofilaments (NFs) is based on mutual steric repulsion between the unstructured "sidearm" domains of adjacent NFs. Here, we present experimental evidence that these repulsive forces are modulated by the degree of sidearm phosphorylation. When NFs are sedimented into a gelatinous pellet, pellet volume falls with increasing ionic strength and enzymatic dephosphorylation; sedimentation of phosphorylated NFs in the presence of divalent cations also dramatically reduces pellet volume. Further, atomic force microscopy imaging of isolated mammalian NFs reveals robust exclusion of colloidal particles from the NF backbone that is reduced at high ionic strength and attenuated when the filaments are enzymatically dephosphorylated. Phosphate-phosphate repulsion on the NF sidearm appears to modulate NF excluded volume in a graded fashion, thereby controlling axonal NF organization through interfilament forces.     

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.     

Assembly and mechanosensory function of focal adhesions: experiments and models

Initial integrin-mediated cell-matrix adhesions (focal complexes) appear underneath the lamellipodia, in the regions of the "fast" centripetal flow driven by actin polymerization. Once formed, these adhesions convert the flow behind them into a "slow", myosin II-driven mode. Some focal complexes then turn into elongated focal adhesions (FAs) associated with contractile actomyosin bundles (stress fibers). Myosin II inhibition does not suppress formation of focal complexes but blocks their conversion into mature FAs and further FA growth. Application of external pulling force promotes FA growth even under conditions when myosin II activity is blocked. Thus, individual FAs behave as mechanosensors responding to the application of force by directional assembly. We proposed a thermodynamic model for the mechanosensitivity of FAs, taking into account that an elastic molecular aggregate subject to pulling forces tends to grow in the direction of force application by incorporating additional subunits. This simple model can explain a variety of processes typical of FA behavior. Assembly of FAs is triggered by the small G-protein Rho via activation of two major targets, Rho-associated kinase (ROCK) and the formin homology protein, Dia1. ROCK controls creation of myosin II-driven forces, while Dia1 is involved in the response of FAs to these forces. Expression of the active form of Dia1, allows the external force-induced assembly of mature FAs, even in conditions when Rho is inhibited. Conversely, downregulation of Dia1 by siRNA prevents FA maturation even if Rho is activated. Dia1 and other formins cap barbed (fast growing) ends of actin filaments, allowing insertion of the new actin monomers. We suggested a novel mechanism of such "leaky" capping based on an assumption of elasticity of the formin/barbed end complex. Our model predicts that formin-mediated actin polymerization should be greatly enhanced by application of external pulling force. Thus, the formin-actin complex might represent an elementary mechanosensing device responding to force by enhancement of actin assembly. In addition to its role in actin polymerization, Dia1 seems to be involved in formation of links between actin filaments and microtubules affecting microtubule dynamics. Alpha-tubulin deacetylase HDAC6 cooperates with Dia1 in formation of such links. Since microtubules are known to promote FA disassembly, the Dia1-mediated effect on microtubule dynamics may possibly play a role in the negative feedback loop controlling size and turnover of FAs.     

The polymerization ratchet model explains the force-velocity relation for growing microtubules

The polymerization of filamentous proteins generates mechanical forces which drive many cellular processes. Dogterom and Yurke measured the force-velocity relation generated by a single microtubule. They found that the force is generally in the range predicted by the “polymerization ratchet” mechanism, but the force-velocity relationship decreased faster than that theory predicted. Here we generalize the polymerization ratchet model to take into account the “subsidy effect” that arises because a microtubule consists of 13 protofilaments. With this generalization the model fits the experimental data well. The biological implications of the polymerization ratchet model are discussed. Received: 18 May 1998 / Revised version: 4 November 1998 / Accepted: 30 November 1998     

Quantitative Modeling and Optimization of Magnetic Tweezers

Magnetic tweezers are a powerful tool to manipulate single DNA or RNA molecules and to study nucleic acid-protein interactions in real time. Here, we have modeled the magnetic fields of permanent magnets in magnetic tweezers and computed the forces exerted on superparamagnetic beads from first principles. For simple, symmetric geometries the magnetic fields can be calculated semianalytically using the Biot-Savart law. For complicated geometries and in the presence of an iron yoke, we employ a finite-element three-dimensional PDE solver to numerically solve the magnetostatic problem. The theoretical predictions are in quantitative agreement with direct Hall-probe measurements of the magnetic field and with measurements of the force exerted on DNA-tethered beads. Using these predictive theories, we systematically explore the effects of magnet alignment, magnet spacing, magnet size, and of adding an iron yoke to the magnets on the forces that can be exerted on tethered particles. We find that the optimal configuration for maximal stretching forces is a vertically aligned pair of magnets, with a minimal gap between the magnets and minimal flow cell thickness. Following these principles, we present a configuration that allows one to apply ≥40 pN stretching forces on ≈1-μm tethered beads.     

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.     

The force-velocity relationship for the actin-based motility of Listeria monocytogenes

The intracellular movement of the bacterial pathogen Listeria monocytogenes has helped identify key molecular constituents of actin-based motility (recent reviews ). However, biophysical as well as biochemical data are required to understand how these molecules generate the forces that extrude eukaryotic membranes. For molecular motors and for muscle, force-velocity curves have provided key biophysical data to distinguish between mechanistic theories. Here we manipulate and measure the viscoelastic properties of tissue extracts to provide the first force-velocity curve for Listeria monocytogenes. We find that the force-velocity relationship is highly curved, almost biphasic, suggesting a high cooperativity between biochemical catalysis and force generation. Using high-resolution motion tracking in low-noise extracts, we find long trajectories composed exclusively of molecular-sized steps. Robust statistics from these trajectories show a correlation between the duration of steps and macroscopic Listeria speed, but not between average step size and speed. Collectively, our data indicate how the molecular properties of the Listeria polymerization engine regulate speed, and that regulation occurs during molecular-scale pauses.     

Clamped-filament elongation model for actin-based motors.

Although actin-based motility drives cell crawling and intracellular locomotion of organelles and certain pathogens, the underlying mechanism of force generation remains a mystery. Recent experiments demonstrated that Listeria exhibit episodes of 5.4-nm stepwise motion corresponding to the periodicity of the actin filament subunits, and extremely small positional fluctuations during the intermittent pauses [S. C. Kuo and J. L. McGrath. 2000. Nature. 407:1026-1029]. These findings suggest that motile bacteria remain firmly bound to actin filament ends as they elongate, a behavior that appears to rule out previous models for actin-based motility. We propose and analyze a new mechanochemical model (called the "Lock, Load & Fire" mechanism) for force generation by means of affinity-modulated, clamped-filament elongation. During the locking step, the filament's terminal ATP-containing subunit binds tightly to a clamp situated on the surface of a motile object; in the loading step, actin.ATP monomer(s) bind to the filament end, an event that triggers the firing step, wherein ATP hydrolysis on the clamped subunit attenuates the filament's affinity for the clamp. This last step initiates translocation of the new ATP-containing terminus to the clamp, whereupon another cycle begins anew. This model explains how surface-tethered filaments can grow while exerting flexural or tensile force on the motile surface. Moreover, stochastic simulations of the model reproduce the signature motions of Listeria. This elongation motor, which we term actoclampin, exploits actin's intrinsic ATPase activity to provide a simple, high-fidelity enzymatic reaction cycle for force production that does not require elongating filaments to dissociate from the motile surface. This mechanism may operate whenever actin polymerization is called upon to generate the forces that drive cell crawling or intracellular organelle motility.     

Characterization of protein capacity of nanocation exchanger particles as filling material for functional magnetic beads for bioseparation purposes

In this study we describe the synthesis and characterization of nanocation exchanger particles (NCEX) as the functional filling material for magnetic beads. Polystyrene NCEX particles were synthesized from styrene via a mini-emulsion polymerization. The coupling of cation exchanger groups was done with chlorosulfuric acid after the polymerization reaction. The NCEX particles have an average diameter of 160-260 nm. Their ion exchange capacity amounts up to 4.58 mval/g. In an adsorption experiment it was possible to adsorb 192 mg lysozyme/g NCEX. Depending on the equilibrium concentration of lysozyme in the bulk solution 70-85% of the attached protein was desorbed. NCEX particles were used to produce magnetic beads with cation exchanger properties. Therefore an innovative production process for the synthesis of magnetic beads from different single components was used. The produced magnetic beads contained 40 wt % NCEX material and showed an ion exchanger capacity of 2 mequiv/g. It was possible to adsorb 75 mg lysozyme/g magnetic beads with a maximum recovery rate of 95%.     

Reversible meso-scale smart polymer--protein particles of controlled sizes

Functionalized beads and particles in the size range of tens to hundreds of nanometers (nano- to meso-scale) are finding increased applications in the bioanalytical field. We show here that conjugates of streptavidin and the temperature-responsive polymer poly(N-isopropylacrylamide) (PNIPAAm), synthesized with low polydispersities by reversible addition--fragmentation chain transfer (RAFT) polymerization, rapidly formed mesoscale polymer--protein particles above the lower critical solution temperature (LCST). The average hydrodynamic diameters of these particles could be controlled between 250 nm to 900 nm by the choice of conjugate concentration and polymer molecular weight, and/or through control of the rate of temperature change. Once formed, the biohybrid particles were found to be stable for >16 h at the controlled size, unlike the free PNIPAAm which continued to aggregate and grow over time into very large and polydisperse aggregates. The reversibility between the smart polymer--protein particles and the free polymer--protein conjugates opens potential uses in traditional diagnostic formats and in microfluidic formats where the differential diffusive and physical properties might be exploited for separations, analyte concentration, and signal generation.     

Force-dependent detachment of kinesin-2 biases track switching at cytoskeletal filament intersections

Intracellular trafficking of organelles often involves cytoskeletal track switching. Organelles such as melanosomes are transported by multiple motors including kinesin-2, dynein, and myosin-V, which drive switching between microtubules and actin filaments during dispersion and aggregation. Here, we used optical trapping to determine the unitary and ensemble forces of kinesin-2, and to reconstitute cargo switching at cytoskeletal intersections in a minimal system with kinesin-2 and myosin-V motors bound to beads. Single kinesin-2 motors exerted forces up to ～5 pN, similar to kinesin-1. However, kinesin-2 motors were more likely to detach at submaximal forces, and the duration of force maintenance was short as compared to kinesin-1. In multimotor assays, force increased with kinesin-2 density but was not affected by the presence of myosin-V. In crossed filament assays, switching frequencies of motor-bound beads were dependent on the starting track. At equal average forces, beads tended to switch from microtubules onto overlying actin filaments consistent with the relatively faster detachment of kinesin-2 at near-maximal forces. Thus, in addition to relative force, switching probability at filament intersections is determined by the dynamics of motor-filament interaction, such as the quick detachment of kinesin-2 under load. This may enable fine-tuning of filament switching in the cell.     

Dictyostelium cells' cytoplasm as an active viscoplastic body

We applied a recently developed microrheology technique based on colloidal magnetic tweezers to measure local viscoelastic moduli and active forces in cells of Dictyostelium discoideum. The active transport of nonmagnetic beads taken up by phagocytosis was analyzed by single particle tracking, which allowed us to measure the length of straight steps and the corresponding velocities of the movements. The motion consists of a superposition of nearly straight long-range steps (step length in the micrometer range) and local random walks (step widths about 0.1 microm). The velocities for the former type of motion range from 1 to 3 microm/s. They decrease with increasing bead size and are attributed to rapid active transport along microtubuli. The short-range local motions exhibit velocities of less than 0.5 microm/s and reflect the internal dynamics of the cytoplasm. Viscoelastic response curves were measured by application of force pulses with amplitudes varying between 50 pN and 400 pN. Analysis of the response curves in terms of mechanical equivalent circuits yielded cytoplasmic viscosities varying between 10 and 350 Pa s. Simultaneous analysis of the response curves and of the bead trajectories showed that the motion of the beads is determined by the local yield stress within the cytoplasmic scaffold and cisternae, which varies between sigma = 30 Pa and 250 Pa. The motion of intracellular particles is interpreted in terms of viscoplastic behavior and the apparent viscosity is a measure of the reciprocal rate of bond breakage within the cytoplasmatic network. The viscoelastic moduli are interpreted as dynamic quantities which depend sensitively on the amplitude of the forces, and the rate of bond breakage is determined by the Arrhenius-Kramers law with the activation energy being reduced by the work performed by the applied force. In agreement with previous work, we provide evidence that the myosin II-deficient cells exhibit higher yield stresses, suggesting that the function of myosin II as a cross-linker is taken over by the other (non-active) cross-linkers.     

Sensitive bondforce measurements of ligand-receptor pairs with magnetic beads

The bondforces between biotinylated surfaces and streptavidin or avidin coated beads are investigated by a magnetic field based manipulation system for magnetic microbeads. The magnetic field is generated by currents through a set of conducting lines, and its gradient exerts a force onto the magnetic beads. The force can be increased until the bond between the bead and the surface breaks. Consistent with other groups we found two conformations for both investigated bonds. The measured bondforces for the two conformations are for Streptavidin-Biotin: 55.9 and 244.7 fN and for Avidin-Biotin: 15.9 and 58.4 fN. These very low bondforces (10-100 times smaller than earlier measurements) match to the extremely low loading rate of about 1 fN/s. This new technique thus allows to investigate biomolecular bonds by extremely low forces.