On the stall force for growing microtubules

The assembly of microtubules generates forces that play a role in cellular motility processes such as the motion of chromosomes during mitosis. Recently, Mogilner and Oster proposed a model for the growth of microtubules that agrees quantitatively with the force-velocity relation measured for individual microtubules. In addition, the authors predicted that the stall force for any polymer consisting of N independently growing protofilaments should increase as the square root of N. We simulated this model and found that the stall force increases linearly with N, and is in fact consistent with the maximum force predicted by thermodynamic arguments. We show that this discrepancy can be explained by a more careful treatment of the “off-term” in the Mogilner-Oster model. Received: 27 September 1999 / Revised version: 12 December 1999 / Accepted: 20 December 1999     

Slip sliding away: load-dependence of velocity generated by skeletal muscle myosin molecules in the laser trap

Skeletal muscle's ability to shorten and lengthen against a load is a fundamental property, presumably reflecting the inherent load-dependence of the myosin molecular motor. Here we report the velocity of a single actin filament translocated by a mini-ensemble of skeletal myosin approximately 8 heads under constant loads up to 15 pN in a laser trap assay. Actin filament velocity decreased with increasing load hyberbolically, with unloaded velocity and stall force differing by a factor of 2 with [ATP] (30 vs. 100 muM). Analysis of actin filament movement revealed that forward motion was punctuated with rapid backward 60-nm slips, with the slip frequency increasing with resistive load. At stall force, myosin-generated forward movement was balanced by backward slips, whereas at loads greater than stall, myosin could no longer sustain forward motion, resulting in negative velocities as in eccentric contractions of whole muscle. Thus, the force-velocity relationship of muscle reflects both the inherent load-dependence of the actomyosin interaction and the balance between forward and reverse motion observed at the molecular level.     

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.     

Sarcomere length dependence of the force-velocity relation in single frog muscle fibers.

The force-velocity relation of single frog fibers was measured at sarcomere lengths of 2.15, 2.65, and 3.15 microns. Sarcomere length was obtained on-line with a system that measures the distance between two markers attached to the surface of the fiber, approximately 800 microns apart. Maximal shortening velocity, determined by extrapolating the Hill equation, was similar at the three sarcomere lengths: 6.5, 6.0, and 5.7 microns/s at sarcomere lengths of 2.15, 2.65, and 3.15 microns, respectively. For loads not close to zero the shortening velocity decreased with increasing sarcomere length. This was the case when force was expressed as a percentage of the maximal force at optimal fiber length or as a percentage of the sarcomere-isometric force at the respective sarcomere lengths. The force-velocity relation was discontinuous around zero velocity: load clamps above the level that kept sarcomeres isometric resulted in stretch that was much slower than when the load was decreased below isometric by a similar amount. We fitted the force-velocity relation for slow shortening (less than 600 nm/s) and for slow stretch (less than 200 nm/s) with linear regression lines. At a sarcomere length of 2.15 microns the slopes of these lines was 8.6 times higher for shortening than for stretch. At 2.65 and 3.15 microns the values were 21.8 and 14.1, respectively. At a sarcomere length of 2.15 microm, the velocity of stretch abruptly increased at loads that were 160-170% of the sarcomere isometric load, i.e., the muscle yielded. However, at a sarcomere length of 2.65 and 3.15 microm yield was absent at such loads. Even the highest loads tested (260%) resulted in only slow stretch.(ABSTRACT TRUNCATED AT 250 WORDS)     

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.     

Pressure-Induced Changes in the Structure and Function of the Kinesin-Microtubule Complex

Kinesin-1 is an ATP-driven molecular motor that “walks” along a microtubule by working two heads in a “hand-over-hand” fashion. The stepping motion is well-coordinated by intermolecular interactions between the kinesin head and microtubule, and is sensitively changed by applied forces. We demonstrate that hydrostatic pressure works as an inhibitory action on kinesin motility. We developed a high-pressure microscope that enables the application of hydrostatic pressures of up to 200 MPa (2000 bar). Under high-pressure conditions, taxol-stabilized microtubules were shortened from both ends at the same speed. The sliding velocity of kinesin motors was reversibly changed by pressure, and reached half-maximal value at ∼100 MPa. The pressure-velocity relationship was very close to the force-velocity relationship of single kinesin molecules, suggesting a similar inhibitory mechanism on kinesin motility. Further analysis showed that the pressure mainly affects the stepping motion, but not the ATP binding reaction. The application of pressure is thought to enhance the structural fluctuation and/or association of water molecules with the exposed regions of the kinesin head and microtubule. These pressure-induced effects could prevent kinesin motors from completing the stepping motion.     

Time course of isotonic shortening and the underlying contraction mechanism in airway smooth muscle

Although the structure of the contractile unit in smooth muscle is poorly understood, some of the mechanical properties of the muscle suggest that a sliding-filament mechanism, similar to that in striated muscle, is also operative in smooth muscle. To test the applicability of this mechanism to smooth muscle function, we have constructed a mathematical model based on a hypothetical structure of the smooth muscle contractile unit: a side-polar myosin filament sandwiched by actin filaments, each attached to the equivalent of a Z disk. Model prediction of isotonic shortening as a function of time was compared with data from experiments using ovine tracheal smooth muscle. After equilibration and establishment of in situ length, the muscle was stimulated with ACh (100 μM) until force reached a plateau. The muscle was then allowed to shorten isotonically against various loads. From the experimental records, length-force and force-velocity relationships were obtained. Integration of the hyperbolic force-velocity relationship and the linear length-force relationship yielded an exponential function that approximated the time course of isotonic shortening generated by the modeled sliding-filament mechanism. However, to obtain an accurate fit, it was necessary to incorporate a viscoelastic element in series with the sliding-filament mechanism. The results suggest that a large portion of the shortening is due to filament sliding associated with muscle activation and that a small portion is due to continued deformation associated with an element that shows viscoelastic or power-law creep after a step change in force.     

Parenchymal tethering, airway wall stiffness, and the dynamics of bronchoconstriction.

We do not yet have a good quantitative understanding of how the force-velocity properties of airway smooth muscle interact with the opposing loads of parenchymal tethering and airway wall stiffness to produce the dynamics of bronchoconstriction. We therefore developed a two-dimensional computational model of a dynamically narrowing airway embedded in uniformly elastic lung parenchyma and compared the predictions of the model to published measurements of airway resistance made in rats and rabbits during the development of bronchoconstriction following a bolus injection of methacholine. The model accurately reproduced the experimental time-courses of airway resistance as a function of both lung inflation pressure and tidal volume. The model also showed that the stiffness of the airway wall is similar in rats and rabbits, and significantly greater than that of the lung parenchyma. Our results indicate that the main features of the dynamical nature of bronchoconstriction in vivo can be understood in terms of the classic Hill force-velocity relationship operating against elastic loads provided by the surrounding lung parenchyma and an airway wall that is stiffer than the parenchyma.     

Prediction of the cardiac muscle force-velocity relation from its force-time and force-length relations

The force-velocity relation for cardiac muscle fibers can be calculated from a proposed constitutive law based on force-time and force-length data. The calculated force-velocity relation agrees quite well with the measured force-velocity relation obtained from a quick release of sarcomere controlled rat cardiac trabeculae. The theory confirms the measured linear relationship between maximal velocity of sarcomere shortening and sarcomere length. The implication is that the force-velocity relation is not an independent property, and therefore need not be explicitly included as a rheological element in the constitutive law.     

Force-induced bidirectional stepping of cytoplasmic dynein

Cytoplasmic dynein is a minus-end-directed microtubule motor whose mechanism of movement remains poorly understood. Here, we use optical tweezers to examine the force-dependent stepping behavior of yeast cytoplasmic dynein. We find that dynein primarily advances in 8 nm increments but takes other sized steps (4-24 nm) as well. An opposing force induces more frequent backward stepping by dynein, and the motor walks backward toward the microtubule plus end at loads above its stall force of 7 pN. Remarkably, in the absence of ATP, dynein steps processively along microtubules under an external load, with less force required for minus-end- than for plus-end-directed movement. This nucleotide-independent walking reveals that force alone can drive repetitive microtubule detachment-attachment cycles of dynein's motor domains. These results suggest a model for how dynein's two motor domains coordinate their activities during normal processive motility and provide new clues for understanding dynein-based motility in living cells.     

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.     

Can Quick Release Experiments Reveal the Muscle Structure? A Bionic Approach

The goal of this study was to understand the macroscopic mechanical structure and function of biological muscle with respect to its dynamic role in the contraction.A recently published muscle model,deriving the hyperbolic force-velocity relation from first-order mechanical principles,predicts different force-velocity operating points for different load situations.With anew approach,this model could be simplified and thus,transferred into a numerical simulation and a hardware experiment.Two types of quick release experiments were performed in simulation and with the hardware setup,which represent two extreme cases of the contraction dynamics:against a constant force (isotonic) and against an inertial mass.Both experiments revealed hyperbolic or hyperbolic-like force-velocity relations.Interestingly,the analytical model not only predicts these extreme cases,but also additionally all contraction states in between.It was possible to validate these predictions with the numerical model and the hardware experiment.These results prove that the origin of the hyperbolic force-velocity relation can be mechanically explained on a macroscopic level by the dynamical interaction of three mechanical elements.The implications for the interpretation of biological muscle experiments and the realization of muscle-like bionic actuators are discussed.     

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.     

Cross-bridge cycling theories cannot explain high-speed lengthening behavior in frog muscle.

The Huxley 1957 model of cross-bridge cycling accounts for the shortening force-velocity curve of striated muscle with great precision. For forced lengthening, however, the model diverges from experimental results. This paper examines whether it is possible to bring the model into better agreement with experiments, and if so what must be assumed about the mechanical capabilities of cross-bridges. Of particular interest is how introduction of a maximum allowable cross-bridge strain, as has been suggested by some experiments, affects the predictions of the model. Because some differences in the models are apparent only at high stretch velocities, we acquired new force-velocity data to permit a comparison with experiment. Using whole, isolated frog sartorius muscles at 2 degrees C, we stretched active muscle at speeds up to and exceeding 2 Vmax. Force during stretch was always greater than the peak isometric level, even during the fastest stretches, and was approximately independent of velocity for stretches faster than 0.5 Vmax. Although certain modifications to the model brought it into closer correspondence with the experiments, the accompanying requirements on cross-bridge extensibility were unreasonable. We suggest (both in this paper and the one that follows) that sarcomere inhomogeneities, which have been implicated in such phenomena as "tension creep" and "permanent extra tension," may also play an important role in determining the basic force-velocity characteristics of muscle.     

Torque generated by the flagellar motor of Escherichia coli.

Cells of the bacterium Escherichia coli were tethered and spun in a high-frequency rotating electric field at a series of discrete field strengths. This was done first at low field strengths, then at field strengths generating speeds high enough to disrupt motor function, and finally at low field strengths. Comparison of the initial and final speed versus applied-torque plots yielded relative motor torque. For backward rotation, motor torque rose steeply at speeds close to zero, peaking, on average, at about 2.2 times the stall torque. For forward rotation, motor torque remained approximately constant up to speeds of about 60% of the zero-torque speed. Then the torque dropped linearly with speed, crossed zero, and reached a minimum, on average, at about -1.7 times the stall torque. The zero-torque speed increased with temperature (about 90 Hz at 11 degrees C, 140 Hz at 16 degrees C, and 290 Hz at 23 degrees C), while other parameters remained approximately constant. Sometimes the motor slipped at either extreme (delivered constant torque over a range of speeds), but eventually it broke. Similar results were obtained whether motors broke catastrophically (suddenly and completely) or progressively or were de-energized by brief treatment with an uncoupler. These results are consistent with a tightly coupled ratchet mechanism, provided that elastic deformation of force-generating elements is limited by a stop and that mechanical components yield at high applied torques.     

Suppressor analysis of the MotB(D33E) mutation to probe bacterial flagellar motor dynamics coupled with proton translocation

MotA and MotB form the stator of the proton-driven bacterial flagellar motor, which conducts protons and couples proton flow with motor rotation. Asp-33 of Salmonella enterica serovar Typhimurium MotB, which is a putative proton-binding site, is critical for torque generation. However, the mechanism of energy coupling remains unknown. Here, we carried out genetic and motility analysis of a slowly motile motB(D33E) mutant and its pseudorevertants. We first confirmed that the poor motility of the motB(D33E) mutant is due to neither protein instability, mislocalization, nor impaired interaction with MotA. We isolated 17 pseudorevertants and identified the suppressor mutations in the transmembrane helices TM2 and TM3 of MotA and in TM and the periplasmic domain of MotB. The stall torque produced by the motB(D33E) mutant motor was about half of the wild-type level, while those for the pseudorevertants were recovered nearly to the wild-type levels. However, the high-speed rotations of the motors under low-load conditions were still significantly impaired, suggesting that the rate of proton translocation is still severely limited at high speed. These results suggest that the second-site mutations recover a torque generation step involving stator-rotor interactions coupled with protonation/deprotonation of Glu-33 but not maximum proton conductivity.     

RecA force generation by hydrolysis waves

We present a simple theory of the dynamics of force generation by RecA during homologous strand exchange and a continuous, deterministic mathematical model of the proposed process. Calculations show that force generation is possible in this model for certain reasonable values of the parameters. We predict the shape of the force-velocity curve for the Holliday junction, which exhibits a distinctive kink at large retarding force, and suggest experiments which should distinguish between the proposed model and other models in the literature.     

The energetics, chemistry, and mechanics of a processive motor protein

When kinesin moves along microtubule, it can occasionally malfunction and make a backward step. Recent single molecule experiments on moving kinesin have revealed that the forward to backward step ratio depends exponentially on the load force. We introduce a model of a Brownian step that accounts for recorded data with great accuracy. We find that the forward to backward step ratio does not depend on any structural features of the kinesin. The stepping statistics appear fully determined by the 8 nanometer stepsize, the energy that drives the step, and k(B)T, which is the natural "quantum" of thermal energy. With this model we next analyze the energetics of the Brownian stepper. We derive force-velocity relations for the vicinity of the "static head" case, which is when the applied force is close to the stopping force. We also derive force-velocity relations for the close-to-equilibrium case, i.e. a small load and a small ATP-ADP chemical potential.     

High motility reduces grazing mortality of planktonic bacteria

We tested the impact of bacterial swimming speed on the survival of planktonic bacteria in the presence of protozoan grazers. Grazing experiments with three common bacterivorous nanoflagellates revealed low clearance rates for highly motile bacteria. High-resolution video microscopy demonstrated that the number of predator-prey contacts increased with bacterial swimming speed, but ingestion rates dropped at speeds of >25 microm s(-1) as a result of handling problems with highly motile cells. Comparative studies of a moderately motile strain (<25 microm s(-1)) and a highly motile strain (>45 microm s(-1)) further revealed changes in the bacterial swimming speed distribution due to speed-selective flagellate grazing. Better long-term survival of the highly motile strain was indicated by fourfold-higher bacterial numbers in the presence of grazing compared to the moderately motile strain. Putative constraints of maintaining high swimming speeds were tested at high growth rates and under starvation with the following results: (i) for two out of three strains increased growth rate resulted in larger and slower bacterial cells, and (ii) starved cells became smaller but maintained their swimming speeds. Combined data sets for bacterial swimming speed and cell size revealed highest grazing losses for moderately motile bacteria with a cell size between 0.2 and 0.4 microm(3). Grazing mortality was lowest for cells of >0.5 microm(3) and small, highly motile bacteria. Survival efficiencies of >95% for the ultramicrobacterial isolate CP-1 (< or =0.1 microm(3), >50 microm s(-1)) illustrated the combined protective action of small cell size and high motility. Our findings suggest that motility has an important adaptive function in the survival of planktonic bacteria during protozoan grazing.     

Influence of partial activation on force-velocity properties of frog skinned muscle fibers in millimolar magnesium ion

Segments of briefly glycerinated muscle fibers from Rana pipiens were activated rapidly by a brief exposure to 2.5 mM free calcium followed by a solution containing calcium buffered with EGTA to produce the desired level of force. Steps to isotonic loads were made using a servomotor, usually 3-5 s after the onset of activation. The relative isotonic forces (P/P0) and velocities from contractions obtained under similar circumstances were grouped together and fitted with hyperbolic functions. Under the condition of 6 mM MgCl2 and 5 mM ATP, there was no significant difference in the relative force-velocity relations obtained at full activation compared with those obtained at partial activation when developed force was approximately 40% of its full value. Control experiments showed that a variety of factors did not alter either the relative force-velocity relations or the finding that partial activation did not change these properties. The factors investigated included the decline in force that occurs with each successive contraction of skinned fibers, the segment length (over a range of 1-3 mm), the sarcomere length (over a range of 1.9-2.2 microns), the magnesium ion concentration (26 microM and 1.4 mM were tested), the ATP concentration, the presence of free calcium, and the age of the preparation (up to 30 h). Attempts to repeat earlier experiments by others showing a dependence of shortening velocity on activation were unsuccessful because the low ionic strength used in those experiments caused the fibers to break after a few contractions. The main conclusion, that the shortening velocity is independent of the level of activation, is consistent with the hypothesis that the cross-bridges act independently and that activating calcium acts only as an all-or-none switch for individual cross-bridge attachment sites, and does not otherwise influence the kinetics of cross-bridge movement.