Pulling-force generation by ensembles of polymerizing actin filaments.

The process by which actin polymerization generates pulling forces in cellular processes such as endocytosis is less well understood than pushing-force generation. To clarify the basic mechanisms of pulling-force generation, we perform stochastic polymerization simulations for a square array of polymerizing semiflexible actin filaments, having different interactions with the membrane. The filaments near the array center have a strong attractive component. Filament bending and actin-network elasticity are treated explicitly. We find that the outer filaments push on the membrane and the inner filaments pull, with a net balance of forces. The total calculated pulling force is maximized when the central filaments have a very deep potential well, and the outer filaments have no well. The steady-state force is unaffected by the gel rigidity, but equilibration takes longer for softer gels. The force distributions are flat over the pulling and pushing regions. Actin polymerization is enhanced by softening the gel or reducing the filament binding to the membrane. Filament-membrane detachment can occur for softer gels, even if the total binding energy of the filaments to the membrane is 100 [Formula: see text] or more. It propagates via a stress-concentration mechanism similar to that of a brittle crack in a solid, and the breaking stress is determined by a criterion similar to that of the 'Griffith' theory of crack propagation.

Load sharing in the growth of bundled biopolymers.

To elucidate the nature of load sharing in the growth of multiple biopolymers, we perform stochastic simulations of the growth of biopolymer bundles against obstacles under a broad range of conditions and varying assumptions. The obstacle motion due to thermal fluctuations is treated explicitly. We assume the "Perfect Brownian Ratchet" (PBR) model, in which the polymerization rate equals the free-filament rate as soon as the filament-obstacle distance exceeds the monomer size. Accurate closed-form formulas are obtained for the case of a rapidly moving obstacle. We find the following: (1) load sharing is usually sub-perfect in the sense that polymerization is slower than for a single filament carrying the same average force; (2) the sub-perfect behavior becomes significant at a total force proportional to the logarithm or the square root of the number of filaments, depending on the alignment of the filaments; (3) for the special case of slow barrier diffusion and low opposing force, an enhanced obstacle velocity for an increasing number of filaments is possible; (4) the obstacle velocity is very sensitive to the alignment of the filaments in the bundle, with a staggered alignment being an order of magnitude faster than an unstaggered one at forces of only 0.5 pN per filament for 20 filaments; (5) for large numbers of filaments, the power is maximized at a force well below 1 pN per filament; (6) for intermediate values of the obstacle diffusion coefficient, the shape of the force velocity relation is very similar to that for rapid obstacle diffusion.

Regimes of wave type patterning driven by refractory actin feedback: transition from static polarization to dynamic wave behaviour.

Patterns of waves, patches, and peaks of actin are observed experimentally in many living cells. Models of this phenomenon have been based on the interplay between filamentous actin (F-actin) and its nucleation promoting factors (NPFs) that activate the Arp2/3 complex. Here we present an alternative biologically-motivated model for F-actin-NPF interaction based on properties of GTPases acting as NPFs. GTPases (such as Cdc42, Rac) are known to promote actin nucleation, and to have active membrane-bound and inactive cytosolic forms. The model is a natural extension of a previous mathematical mini-model of small GTPases that generates static cell polarization. Like other modellers, we assume that F-actin negative feedback shapes the observed patterns by suppressing the trailing edge of NPF-generated wave-fronts, hence localizing the activity spatially. We find that our NPF-actin model generates a rich set of behaviours, spanning a transition from static polarization to single pulses, reflecting waves, wave trains, and oscillations localized at the cell edge. The model is developed with simplicity in mind to investigate the interaction between nucleation promoting factor kinetics and negative feedback. It explains distinct types of pattern initiation mechanisms, and identifies parameter regimes corresponding to distinct behaviours. We show that weak actin feedback yields static patterning, moderate feedback yields dynamical behaviour such as travelling waves, and strong feedback can lead to wave trains or total suppression of patterning. We use a recently introduced nonlinear bifurcation analysis to explore the parameter space of this model and predict its behaviour with simulations validating those results.

A model actin comet tail disassembling by severing.

We use a numerical simulation to model an actin comet tail as it grows from the surface of a small object (a bead) and disassembles by severing. We explore the dependence of macroscopic properties such as the local tail radius and tail length on several controllable properties, namely the bead diameter, the bead velocity, the severing rate per unit length, and the actin gel mesh size. The model predicts an F-actin density with an initial exponential decay followed by an abrupt decay at the edge of the tail, and predicts that the comet tail diameter is constant along the length of the tail. The simulation results are used to fit a formula relating the comet tail length to the control parameters, and it is proposed that this formula offers a means to extract quantitative information on the actin gel mesh size and severing kinetics from simple macroscopic measurements.

Mechanisms of Cell Propulsion by Active Stresses.

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

The effects of filament aging and annealing on a model lamellipodium undergoing disassembly by severing.

We use numerical simulations to study the properties of a model lamellipodium as it disassembles by filament severing. The growing lamellipodium is modeled as a 2D or 3D periodic lattice of crosslinked actin filaments. At each time step a new layer of actin filaments is added at the membrane, existing filaments are severed stochastically and disconnected sections of the network are removed. Filament aging is modeled by including several different filament chemical states. Filament annealing is included by allowing existing filaments to grow new filaments. The properties of the model are studied as functions of the number of states and the severing and annealing rates. The network width is proportional to the sum of the average lifetimes of the states, and is well modeled by a simple kinetic theory. The length of the network scales linearly with actin concentration and has a finite width even at high severing protein concentrations. The edge of the growing network becomes sharper as either the number of states or the dimensionality is increased. Annealing increases the average length of the network, and the network length diverges at a critical annealing rate.

Actin polymerization overshoots induced by plus-end capping.

Transient polymerization beyond the steady state has been experimentally observed in in vitro actin polymerization time courses. These 'polymerization overshoots' have previously been described in terms of the time-dependent probabilities for binding distinct nucleotide hydrolysis states within subunits near the plus ends of actin filaments. We demonstrate a different type of overshoot dynamics where the plus-end contribution to polymerization steadily decreases relative to that of the minus end. This decrease occurs due to plus-end capping of an initial impulse of rapidly created actin filaments. We calculate the contribution of these dynamics to observed overshoot magnitudes using rate equations describing the concentration of polymerized actin. We find this contribution is highly sensitive to both initial filament concentration and plus-end capping rate. We develop an analytic formula that describes the magnitude of the overshoot as a function of these two key parameters. The overshoots we describe could be observed by current experimental methods for studying the effects of severing and branching mechanisms upon actin polymerization in the presence of plus-end capping and rapid nucleotide exchange. We also present a plausible cellular mechanism that could greatly amplify these overshoots in vivo.

Nonequilibrium actin polymerization treated by a truncated rate-equation method.

Actin polymerization time courses can exhibit rich nonequilibrium dynamics that have not yet been accurately described by simplified rate equations. Sophisticated stochastic simulations and elaborate recursion schemes have been used to model the nonequilibrium dynamics resulting from the hydrolysis and subsequent exchange of the nucleotide bound within the actin molecules. In this work, we use a truncation approach to derive a set of readily accessible deterministic rate equations which are significantly simpler than previous attempts at such modeling. These equations may be incorporated into whole-cell motility models which otherwise quickly become computationally inaccessible if polymerization of individual actin filaments is stochastically simulated within a virtual cell. Our equations accurately predict the relative concentrations of both monomeric and polymerized actin in differing nucleotide hydrolysis states throughout entire polymerization time courses nucleated via seed filaments. We extend our model to include the effects of capping protein. We also detail how our rate-equation method may be used to extract key parameters from experimental data.

Thermodynamically consistent treatment of the growth of a biopolymer in the presence of a smooth obstacle interaction potential.

We investigate the effect of filament-obstacle interactions on the force-velocity relation of growing biopolymers, via calculations explicitly treating obstacle diffusion and stochastic addition and subtraction of subunits. We first show that the instantaneous subunit on- and off-rates satisfy a rigorous thermodynamic relationship determined by the filament-obstacle interaction potential, which has been violated by several calculations in the literature. The instantaneous rates depend not only on the average force on the obstacle but also on the shape of the potential on the nanometer length scale. Basing obstacle-induced reduction of the on-rate entirely on the force, as previous work has often done, is thermodynamically inconsistent and can overestimate the stall force, sometimes by more than a factor of two. We perform simulations and analytic calculations of the force-velocity relation satisfying the thermodynamic relationship. The force-velocity relation can deviate strongly from the Brownian-Ratchet predictions. For shallow potential wells of depth ∼5k_{B}T, which might correspond to transient filament-membrane attachments, the velocity drops more rapidly than predicted by the Brownian-Ratchet model, in some cases by as much as a factor of 50 at an opposing force of only 1 pN. On the other hand, the zero-force velocity is much less affected than would be expected from naive use of the Boltzmann factor. Furthermore, the growth velocity has a surprisingly strong dependence on the obstacle diffusion coefficient even when the dimensionless diffusion coefficient is large. For deep potential wells, as might result from strong filament-membrane links, both the on- and off-rates are reduced significantly, slowing polymerization. Such potentials can sustain pulling forces while polymerizing but only if the attractive well is relatively flat over a region comparable to or greater than the monomer size. For double-well potentials, which have such a flat region, the slowing of polymerization by external pushing force is almost linear up to the stall force in some parameter ranges.

Actin polymerization overshoots and ATP hydrolysis as assayed by pyrene fluorescence.

We investigate via stochastic simulation the overshoots observed in the fluorescence intensity of pyrene-labeled actin during rapid polymerization. We show that previous assumptions about pyrene intensity that ignore the intensity differences between subunits in different ATP hydrolysis states are not consistent with experimental data. This strong sensitivity of intensity to hydrolysis state implies that a measured pyrene intensity curve does not immediately reveal the true polymerization kinetics. We show that there is an optimal range of hydrolysis and phosphate release rate combinations simultaneously consistent with measured polymerization data from previously published severing and Arp2/3 complex-induced branching experiments. Within this range, we find that the pyrene intensity curves are described very accurately by the following average relative intensity coefficients: 0.37 for F-ATP actin; 0.55 for F-ADP + P(i) actin; and 0.75 for F-ADP actin. Finally, we present an analytic formula, which properly accounts for the sensitivity of the pyrene assay to hydrolysis state, for estimation of the concentration of free barbed ends from pyrene intensity curves.

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.

First-contact time to a patch in a multidimensional potential well.

The escape of a diffusing particle from a potential well is an important aspect of many dynamic processes in chemistry, physics, and biology, and such an escape process often involves finding a restricted region or patch in a multidimensional potential well. We study an idealized model of this process via simulation and analytic theory. By combining results from special cases having either high symmetry or zero potential, we obtain a simple formula for the first-contact time for a particle moving to a boundary patch in an arbitrary number of dimensions. We apply this formula in two, three, and six dimensions. The predicted dependences of the first-contact time on the well depth and patch size are compared to results from simulations, and close agreement is found. We extend the theory to calculate the first-contact time between two particles in separate harmonic potential wells. As an application of this extended theory, we calculate the first-contact time for two parallel semiflexible biopolymer filaments and compare these results to previous simulations.

Actin growth profile in clathrin-mediated endocytosis.

Clathrin-mediated endocytosis in yeast is driven by a protein patch containing close to 100 different types of proteins. Among the proteins are 5000-10000 copies of polymerized actin, and successful endocytosis requires growth of the actin network. Since it is not known exactly how actin network growth drives endocytosis, we calculate the spatial distribution of actin growth required to generate the force that drives the process. First, we establish the force distribution that must be supplied by actin growth, by combining membrane-bending profiles obtained via electron microscopy with established theories of membrane mechanics. Next, we determine the profile of actin growth, using a continuum mechanics approach and an iterative procedure starting with an actin growth profile obtained from a linear analysis. The profile has fairly constant growth outside a central hole of radius 45-50 nm, but very little growth in this hole. This growth profile can reproduce the required forces if the actin shear modulus exceeds 80 kPa, and the growing filaments can exert very large polymerization forces. The growth profile prediction could be tested via electron-microscopy or super-resolution experiments in which the turgor pressure is suddenly turned off.

Contractile stress generation by actomyosin gels.

The tension generated by randomly distributed myosin minifilaments in an actin gel is evaluated using a rigorous theorem relating the surface forces acting on the gel to the forces exerted by the myosins. The maximum tension generated per myosin depends strongly on the lengths of the myosin minifilaments and the actin filaments. The result is used to place an upper bound on the tension that can be generated during cytokinesis. It is found that actomyosin contraction by itself generates too little force for ring contraction during cytokinesis unless the actin filaments are tightly crosslinked into inextensible units much longer than a single actin filament.

Model study of protein unfolding by interfaces.

We study interface-induced protein unfolding on hydrophobic and polar interfaces by means of a two-dimensional lattice model and an exhaustive enumeration ground-state structure search, for a set of model proteins of length 20 residues. We compare the effects of the two types of interfaces, and search for criteria that influence the retention of a protein's native-state structure upon adsorption. We find that the unfolding proceeds by a large, sudden loss of native contacts. The unfolding at polar interfaces exhibits similar behavior to that at hydrophobic interfaces but with a much weaker interface coupling strength. Further, we find that the resistance of proteins to unfolding in our model is positively correlated with the magnitude of the folding energy in the native-state structure, the thermal stability (or energy gap) for that structure, and the interface energy for native-state adsorption. We find these factors to be of roughly equal importance.

Effects of molecular-scale processes on observable growth properties of actin networks.

The properties of actin network growth against a flat obstacle are studied using several different sets of molecular-level assumptions regarding filament growth and nucleation. These assumptions are incorporated into a multifilament methodology which treats both the distribution of filament orientations and bending of filaments. Three single-filament force-generation mechanisms in the literature are compared within this framework. Each mechanism is treated using two different filament nucleation modes, namely, spontaneous nucleation and branching off pre-existing filaments. We find that the shape of the force-velocity relation depends mainly on the ratio of the thermodynamic and mechanical stall forces of the filaments. If the thermodynamic stall force greatly exceeds the mechanical stall force, the velocity drops abruptly to zero when the mechanical stall force is reached; otherwise, it goes more gradually to zero. In addition, branching nucleation gives a steeper increase in the filament number with opposing force than spontaneous nucleation does. Finally, the zero-force velocity of the obstacle as a function of the detachment and capping rates differs significantly between the different single-filament growth mechanisms. Experiments are proposed to use these differences to discriminate between the network growth models.

Stimulation of actin polymerization by filament severing.

The extent and dynamics of actin polymerization in solution are calculated as functions of the filament severing rate, using a simple model of in vitro polymerization. The model is solved by both analytic theory and stochastic-growth simulation. The results show that severing essentially always enhances actin polymerization by freeing up barbed ends, if barbed-end cappers are present. Severing has much weaker effects if only pointed-end cappers are present. In the early stages of polymerization, the polymerized-actin concentration grows exponentially as a function of time. The exponential growth rate is given in terms of the severing rate, and the latter is given in terms of the maximum slope in a polymerization time course. Severing and branching are found to act synergistically.

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.

Energetics and dynamics of constrained actin filament bundling.

The formation of filopodia-like bundles from a dendritic actin network has been observed to occur in vitro as a result of branching induced by Arp2/3 complex. We study both the energetics and dynamics of actin filament bundling in such a network to evaluate their relative importance in bundle formation processes. Our model considers two semiflexible actin filaments fixed at one end and free at the other, described using a normal-mode approximation. This model is studied by both Brownian dynamics and free-energy minimization methods. Remarkably, even short filaments can bundle at separations comparable to their lengths. In the dynamic simulations, we evaluate the time required for the filaments to interact and bind, and examine the dependence of this bundling time on the filament length, the distance between the filament bases, and the cross-linking energy. In most cases, bundling occurs in a second or less. Beyond a certain critical distance, we find that the bundling time increases very rapidly with increasing interfilament separation and/or decreasing filament length. For most of the cases we have studied, the energetics results for this critical distance are similar to those obtained from dynamics simulations run for 10 s, suggesting that beyond this timescale, energetics, rather than kinetic constraints, determine whether or not bundling occurs. Over a broad range of conditions, we find that the times required for bundling from a network are compatible with experimental observations.