A quantitative analysis of contractility in active cytoskeletal protein networks

Cells actively produce contractile forces for a variety of processes including cytokinesis and motility. Contractility is known to rely on myosin II motors which convert chemical energy from ATP hydrolysis into forces on actin filaments. However, the basic physical principles of cell contractility remain poorly understood. We reconstitute contractility in a simplified model system of purified F-actin, muscle myosin II motors, and α-actinin cross-linkers. We show that contractility occurs above a threshold motor concentration and within a window of cross-linker concentrations. We also quantify the pore size of the bundled networks and find contractility to occur at a critical distance between the bundles. We propose a simple mechanism of contraction based on myosin filaments pulling neighboring bundles together into an aggregated structure. Observations of this reconstituted system in both bulk and low-dimensional geometries show that the contracting gels pull on and deform their surface with a contractile force of ∼1 μN, or ∼100 pN per F-actin bundle. Cytoplasmic extracts contracting in identical environments show a similar behavior and dependence on myosin as the reconstituted system. Our results suggest that cellular contractility can be sensitively regulated by tuning the (local) activity of molecular motors and the cross-linker density and binding affinity.     

The Effect of Myofilament Compliance on Kinetics of Force Generation by Myosin Motors in Muscle

We use the inhibitor of isometric force of skeletal muscle N-benzyl-p-toluene sulfonamide (BTS) to decrease, in a dose dependent way, the number of myosin motors attached to actin during the steady isometric contraction of single fibers from frog skeletal muscle (4°C, 2.1 μm sarcomere length). In this way we can reduce the strain in the myofilament compliance during the isometric tetanus (T₀) from 3.54 nm in the control solution (T_0,NR) to ∼0.5 nm in 1 μM BTS, where T₀ is reduced to ∼0.15 T_0,NR. The quick force recovery after a step release (1-3 nm per half-sarcomere) becomes faster with the increase of BTS concentration and the decrease of T₀. The simulation of quick force recovery with a multistate model of force generation, that adapts Huxley and Simmons model to account for both the high stiffness of the myosin motor (∼3 pN/nm) and the myofilament compliance, shows that the increase in the rate of quick force recovery by BTS is explained by the reduced strain in the myofilaments, consequent to the decrease in half-sarcomere force. The model estimates that i), for the same half-sarcomere release the state transition kinetics in the myosin motor are five times faster in the absence of filament compliance than in the control; and ii), the rate of force recovery from zero to T₀ is ∼6000/s in the absence of filament compliance.     

A kinetic model that explains the effect of inorganic phosphate on the mechanics and energetics of isometric contraction of fast skeletal muscle

A conventional five-step chemo-mechanical cycle of the myosin–actin ATPase reaction, which implies myosin detachment from actin upon release of hydrolysis products (ADP and phosphate, Pi) and binding of a new ATP molecule, is able to fit the [Pi] dependence of the force and number of myosin motors during isometric contraction of skeletal muscle. However, this scheme is not able to explain why the isometric ATPase rate of fast skeletal muscle is decreased by an increase in [Pi] much less than the number of motors. The question can be solved assuming the presence of a branch in the cycle: in isometric contraction, when the force generation process by the myosin motor is biased at the start of the working stroke, the motor can detach at an early stage of the ATPase cycle, with Pi still bound to its catalytic site, and then rapidly release the hydrolysis products and bind another ATP. In this way, the model predicts that in fast skeletal muscle the energetic cost of isometric contraction increases with [Pi]. The large dissociation constant of the product release in the branched pathway allows the isometric myosin–actin reaction to fit the equilibrium constant of the ATPase.     

Dynamics of myosin-driven skeletal muscle contraction: I. Steady-state force generation

Skeletal muscle contraction is a canonical example of motor-driven force generation. Despite the long history of research in this topic, a mechanistic explanation of the collective myosin force generation is lacking. We present a theoretical model of muscle contraction based on the conformational movements of individual myosins and experimentally measured chemical rate constants. Detailed mechanics of the myosin motor and the geometry of the sarcomere are taken into account. Two possible scenarios of force generation are examined. We find only one of the scenarios can give rise to a plausible contraction mechanism. We propose that the synchrony in muscle contraction is due to a force-dependent ADP release step. Computational results of a half sarcomere with 150 myosin heads can explain the experimentally measured force-velocity relationship and efficiency data. We predict that the number of working myosin motors increases as the load force is increased, thus showing synchrony among myosin motors during muscle contraction. We also find that titin molecules anchoring the thick filament are passive force generators in assisting muscle contraction.     

Skeletal muscle performance determined by modulation of number of myosin motors rather than motor force or stroke size

Skeletal muscle can bear a high load at constant length, or shorten rapidly when the load is low. This force-velocity relationship is the primary determinant of muscle performance in vivo. Here we exploited the quasi-crystalline order of myosin II motors in muscle filaments to determine the molecular basis of this relationship by X-ray interference and mechanical measurements on intact single cells. We found that, during muscle shortening at a wide range of velocities, individual myosin motors maintain a force of about 6 pN while pulling an actin filament through a 6 nm stroke, then quickly detach when the motor reaches a critical conformation. Thus we show that the force-velocity relationship is primarily a result of a reduction in the number of motors attached to actin in each filament in proportion to the filament load. These results explain muscle performance and efficiency in terms of the molecular mechanism of the myosin motor.     

The Fraction of Myosin Motors That Participate in Isometric Contraction of Rabbit Muscle Fibers at Near-Physiological Temperature

The duty ratio, or the part of the working cycle in which a myosin molecule is strongly attached to actin, determines motor processivity and is required to evaluate the force generated by each molecule. In muscle, it is equal to the fraction of myosin heads that are strongly, or stereospecifically, bound to the thin filaments. Estimates of this fraction during isometric contraction based on stiffness measurements or the intensities of the equatorial or meridional x-ray reflections vary significantly. Here, we determined this value using the intensity of the first actin layer line, A1, in the low-angle x-ray diffraction patterns of permeable fibers from rabbit skeletal muscle. We calibrated the A1 intensity by considering that the intensity in the relaxed and rigor states corresponds to 0% and 100% of myosin heads bound to actin, respectively. The fibers maximally activated with Ca²⁺ at 4°C were heated to 31–34°C with a Joule temperature jump (T-jump). Rigor and relaxed-state measurements were obtained on the same fibers. The intensity of the inner part of A1 during isometric contraction compared with that in rigor corresponds to 41–43% stereospecifically bound myosin heads at near-physiological temperature, or an average force produced by a head of ∼6.3 pN.     

Isoforms Confer Characteristic Force Generation and Mechanosensation by Myosin II Filaments

Myosin II isoforms with varying mechanochemistry and filament size interact with filamentous actin (F-actin) arrays to generate contractile forces in muscle and nonmuscle cells. How myosin II force production is shaped by isoform-specific motor properties and environmental stiffness remains poorly understood. Here, we used computer simulations to analyze force production by an ensemble of myosin motors against an elastically tethered actin filament. We found that force output depends on two timescales: the duration of F-actin attachment, which varies sharply with the ensemble size, motor duty ratio, and external load; and the time to build force, which scales with the ensemble stall force, gliding speed, and environmental stiffness. Although force-dependent kinetics were not required to sense changes in stiffness, the myosin catch bond produced positive feedback between the attachment time and force to trigger switch-like transitions from transient attachments, generating small forces, to high-force-generating runs. Using parameters representative of skeletal muscle myosin, nonmuscle myosin IIB, and nonmuscle myosin IIA revealed three distinct regimes of behavior, respectively: 1) large assemblies of fast, low-duty ratio motors rapidly build stable forces over a large range of environmental stiffness; 2) ensembles of slow, high-duty ratio motors serve as high-affinity cross-links with force buildup times that exceed physiological timescales; and 3) small assemblies of low-duty ratio motors operating at intermediate speeds are poised to respond sharply to changes in mechanical context—at low force or stiffness, they serve as low-affinity cross-links, but they can transition to force production via the positive-feedback mechanism described above. Together, these results reveal how myosin isoform properties may be tuned to produce force and respond to mechanical cues in their environment.     

Forces measured with micro-fabricated cantilevers during actomyosin interactions produced by filaments containing different myosin isoforms and loop 1 structures

Background

There is evidence that the actin-activated ATP kinetics and the mechanical work produced by muscle myosin molecules are regulated by two surface loops, located near the ATP binding pocket (loop 1), and in a region that interfaces with actin (loop 2). These loops regulate force and velocity of contraction, and have been investigated mostly in single molecules. There is a lack of information of the work produced by myosin molecules ordered in filaments and working cooperatively, which is the actual muscle environment.

Methods

We use micro-fabricated cantilevers to measure forces produced by myosin filaments isolated from mollusk muscles, skeletal muscles, and smooth muscles containing variations in the structure of loop 1 (tonic and phasic myosins). We complemented the experiments with in-vitro assays to measure the velocity of actin motility.

Results

Smooth muscle myosin filaments produced more force than skeletal and mollusk myosin filaments when normalized per filament overlap. Skeletal muscle myosin propelled actin filaments in a higher sliding velocity than smooth muscle myosin. The values for force and velocity were consistent with previous studies using myosin molecules, and suggest a close correlation with the myosin isoform and structure of surface loop 1.

General significance

The technique using micro-fabricated cantilevers to measure force of filaments allows for the investigation of the relation between myosin structure and contractility, allowing experiments to be conducted with an array of different myosin isoforms. Using the technique we observed that the work produced by myosin molecules is regulated by amino-acid sequences aligned in specific loops.     

Velocity of myosin-based actin sliding depends on attachment and detachment kinetics and reaches a maximum when myosin-binding sites on actin saturate

Molecular motors such as kinesin and myosin often work in groups to generate the directed movements and forces critical for many biological processes. Although much is known about how individual motors generate force and movement, surprisingly, little is known about the mechanisms underlying the macroscopic mechanics generated by multiple motors. For example, the observation that a saturating number, N, of myosin heads move an actin filament at a rate that is influenced by actin–myosin attachment and detachment kinetics is accounted for neither experimentally nor theoretically. To better understand the emergent mechanics of actin–myosin mechanochemistry, we use an in vitro motility assay to measure and correlate the N-dependence of actin sliding velocities, actin-activated ATPase activity, force generation against a mechanical load, and the calcium sensitivity of thin filament velocities. Our results show that both velocity and ATPase activity are strain dependent and that velocity becomes maximized with the saturation of myosin-binding sites on actin at a value that is 40% dependent on attachment kinetics and 60% dependent on detachment kinetics. These results support a chemical thermodynamic model for ensemble motor mechanochemistry and imply molecularly explicit mechanisms within this framework, challenging the assumption of independent force generation.     

Smooth muscle and skeletal muscle myosins produce similar unitary forces and displacements in the laser trap.

Purified smooth muscle myosin in the in vitro motility assay propels actin filaments at 1/10 the velocity, yet produces 3-4 times more force than skeletal muscle myosin. At the level of a single myosin molecule, these differences in force and actin filament velocity may be reflected in the size and duration of single motion and force-generating events, or in the kinetics of the cross-bridge cycle. Specifically, an increase in either unitary force or duty cycle may explain the enhanced force-generating capacity of smooth muscle myosin. Similarly, an increase in attached time or decrease in unitary displacement may explain the reduced actin filament velocity of smooth muscle myosin. To discriminate between these possibilities, we used a laser trap to measure unitary forces and displacements from single smooth and skeletal muscle myosin molecules. We analyzed our data using mean-variance analysis, which does not rely on scoring individual events by eye, and emphasizes periods in the data with constant properties. Both myosins demonstrated multiple but similar event populations with discrete peaks at approximately +11 and -11 nm in displacement, and 1.5 and 3.5 pN in force. Mean attached times for smooth muscle myosin were longer than for skeletal-muscle myosin. These results explain much of the difference in actin filament velocity between these myosins, and suggest that an increased duty cycle is responsible for the enhanced force-generating capacity of smooth over skeletal-muscle myosin.     

A model of muscle contraction based on the Langevin equation with actomyosin potentials

We propose a muscle contraction model that is essentially a model of the motion of myosin motors as described by a Langevin equation. This model involves one-dimensional numerical calculations wherein the total force is the sum of a viscous force proportional to the myosin head velocity, a white Gaussian noise produced by random forces and other potential forces originating from the actomyosin structure and intra-molecular charges. We calculate the velocity of a single myosin on an actin filament to be 4.9–49 μm/s, depending on the viscosity between the actomyosin molecules. A myosin filament with a hundred myosin heads is used to simulate the contractions of a half-sarcomere within the skeletal muscle. The force response due to a quick release in the isometric contraction is simulated using a process wherein crossbridges are changed forcibly from one state to another. In contrast, the force response to a quick stretch is simulated using purely mechanical characteristics. We simulate the force–velocity relation and energy efficiency in the isotonic contraction and adenosine triphosphate consumption. The simulation results are in good agreement with the experimental results. We show that the Langevin equation for the actomyosin potentials can be modified statistically to become an existing muscle model that uses Maxwell elements.     

Torque-speed relationships of Na+-driven chimeric flagellar motors in Escherichia coli

The bacterial flagellar motor is a rotary motor in the cell envelope of bacteria that couples ion flow across the cytoplasmic membrane to torque generation by independent stators anchored to the cell wall. The recent observation of stepwise rotation of a Na⁺-driven chimeric motor in Escherichia coli promises to reveal the mechanism of the motor in unprecedented detail. We measured torque-speed relationships of this chimeric motor using back focal plane interferometry of polystyrene beads attached to flagellar filaments in the presence of high sodium-motive force (85 mM Na⁺). With full expression of stator proteins the torque-speed curve had the same shape as those of wild-type E. coli and Vibrio alginolyticus motors: the torque is approximately constant (at ∼ 2200 pN nm) from stall up to a “knee” speed of ∼ 420 Hz, and then falls linearly with speed, extrapolating to zero torque at ∼ 910 Hz. Motors containing one to five stators generated ∼ 200 pN nm per stator at speeds up to ∼ 100 Hz/stator; the knee speed in 4- and 5-stator motors is not significantly slower than in the fully induced motor. This is consistent with the hypothesis that the absolute torque depends on stator number, but the speed dependence does not. In motors with point mutations in either of two critical conserved charged residues in the cytoplasmic domain of PomA, R88A and R232E, the zero-torque speed was reduced to ∼ 400 Hz. The torque at low speed was unchanged by mutation R88A but was reduced to ∼ 1500 pN nm by R232E. These results, interpreted using a simple kinetic model, indicate that the basic mechanism of torque generation is the same regardless of stator type and coupling ion and that the electrostatic interaction between stator and rotor proteins is related to the torque-speed relationship.     

An electromechanical model of myosin molecular motors

There is a long-running debate on the working mechanism of myosin molecular motors, which, by interacting with actin filaments, convert the chemical energy of ATP into a variety of mechanical work. After the development of technologies for observing and manipulating individual working molecules, experimental results negating the widely accepted 'lever-arm hypothesis' have been reported. In this paper, based on the experimental results so far accumulated, an alternative hypothesis is proposed, in which motor molecules are modelled as electromechanical components that interact with each other through electrostatic force. Electrostatic attractive force between myosin and actin is assumed to cause a conformational change in the myosin head during the attachment process. An elastic energy resulting from the conformational change then produces the power stroke. The energy released at the ATP hydrolysis is mainly used to detach the myosin head from actin filaments. The mechanism presented in this paper is compatible with the experimental results contradictory to the previous theories. It also explains the behavior of myosins V and VI, which are engaged in cellular transport and move processively along actin filaments.     

Single-molecule adhesion forces and attachment lifetimes of myosin-I phosphoinositide interactions

Phosphoinositides regulate the activities and localization of many cytoskeletal proteins involved in crucial biological processes, including membrane-cytoskeleton adhesion. Yet little is known about the mechanics of protein-phosphoinositide interactions, or about the membrane-attachment mechanics of any peripheral membrane proteins. Myosin-Ic (myo1c) is a molecular motor that links membranes to the cytoskeleton via phosphoinositide binding, so it is particularly important to understand the mechanics of its membrane attachment. We used optical tweezers to measure the strength and attachment lifetime of single myo1c molecules as they bind beads coated with a bilayer of 2% phosphatidylinositol 4,5-bisphosphate and 98% phosphatidylcholine. Adhesion forces measured under ramp-load ranged between 5.5 and 16 pN at loading rates between 250 and 1800 pN/s. Dissociation rates increased linearly with constant force (0.3-2.5 pN), with rates exceeding 360 s⁻¹ at 2.5 pN. Attachment lifetimes calculated from adhesion force measurements were loading-rate-dependent, suggesting nonadiabatic behavior during pulling. The adhesion forces of myo1c with phosphoinositides are greater than the motors stall forces and are within twofold of the force required to extract a lipid molecule from the membrane. However, attachment durations are short-lived, suggesting that phosphoinositides alone do not provide the mechanical stability required to anchor myo1c to membranes during multiple ATPase cycles.     

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.     

Robust mechanosensing and tension generation by myosin VI

Myosin VI is a molecular motor that is thought to function both as a transporter and as a cytoskeletal anchor in vivo. Here we use optical tweezers to examine force generation by single molecules of myosin VI under physiological nucleotide concentrations. We find that myosin VI is an efficient transporter at loads of up to ∼ 2 pN but acts as a cytoskeletal anchor at higher loads. Our data and the resulting model are consistent with an indirect coupling of global structural motions to nucleotide binding and release. The model provides a mechanism by which load may regulate the dual functions of myosin VI in vivo. Our results suggest that myosin VI kinetics are tuned such that the motor maintains a consistent level of mechanical tension within the cell, a property potentially shared by other mechanosensitive proteins.     

Mobility of Molecular Motors Regulates Contractile Behaviors of Actin Networks

Cells generate mechanical forces primarily from interactions between F-actin, cross-linking proteins, myosin motors, and other actin-binding proteins in the cytoskeleton. To understand how molecular interactions between the cytoskeletal elements generate forces, a number of in vitro experiments have been performed but are limited in their ability to accurately reproduce the diversity of motor mobility. In myosin motility assays, myosin heads are fixed on a surface and glide F-actin. By contrast, in reconstituted gels, the motion of both myosin and F-actin is unrestricted. Because only these two extreme conditions have been used, the importance of mobility of motors for network behaviors has remained unclear. In this study, to illuminate the impacts of motor mobility on the contractile behaviors of the actin cytoskeleton, we employed an agent-based computational model based on Brownian dynamics. We find that if motors can bind to only one F-actin like myosin I, networks are most contractile at intermediate mobility. In this case, less motor mobility helps motors stably pull F-actins to generate tensile forces, whereas higher motor mobility allows F-actins to aggregate into larger clustering structures. The optimal intermediate motor mobility depends on the stall force and affinity of motors that are regulated by mechanochemical rates. In addition, we find that the role of motor mobility can vary drastically if motors can bind to a pair of F-actins. A network can exhibit large contraction with high motor mobility because motors bound to antiparallel pairs of F-actins can exert similar forces regardless of their mobility. Results from this study imply that the mobility of molecular motors may critically regulate contractile behaviors of actin networks in cells.     

Local heating of molecular motors using single carbon nanotubes

Temperature globally affects all chemical processes and biomolecules in living cells. Elevating the temperature of an entire cell accelerates so many biomolecular reactions simultaneously that it is difficult to distinguish the various mechanisms involved. The ability to localize temperature changes to the nanometer range within a cell could provide a powerful new tool for regulating biomolecular activity at the level of individual molecules. The search for a nanoheater for biological research has prompted experiments with carbon nanotubes (CNTs), which have the highest conductivity of any known material. The adsorption of skeletal muscle myosin molecules along the length of single multi-walled CNTs (~10 μm) has allowed researchers to observe the ATP-driven sliding of fluorescently labeled actin filaments. In one study, red-laser irradiation focused on one end of a myosin-coated CNT was used to heat myosin motors locally without directly heating the surrounding water; this laser irradiation instantly accelerated the actin-filament sliding speeds from ~6 to ~12 μm/s in a reversible manner, indicating a local, real-time heating of myosin motors by approximately Δ12 K. Calculation of heat transfer using the finite element method, based on the estimated temperature along a single CNT with a diameter of 170 nm, indicated a high thermal conductivity of ~1540 Wm^?1K^?1 in solution, consistent with values measured in vacuum in earlier studies. Temperature distribution indicated by half-decrease distances was ~3660 nm along the length of the CNT and ~250 nm perpendicular to the length. These results suggest that single-CNT-based heating at the nanometer- or micrometer-range could be used to regulate various biomolecules in many areas of biological, physical, and chemical research.     

Evidence for pre- and post-power stroke of cross-bridges of contracting skeletal myofibrils

We examined the orientational fluctuations of a small number of myosin molecules (approximately three) in working skeletal muscle myofibrils. Myosin light chain 1 (LC1) was labeled with a fluorescent dye and exchanged with the native LC1 of skeletal muscle myofibrils cross-linked with 1-ethyl-3-[3(dimethylamino) propyl] carbodiimide to prevent shortening. We observed a small volume within the A-band (∼10⁻¹⁵ L) by confocal microscopy, and measured cyclic fluctuations in the orientation of the myosin neck (containing LC1) by recording the parallel and perpendicular components of fluorescent light emitted by the fluorescently labeled myosin LC1. Histograms of orientational fluctuations from fluorescent molecules in rigor were represented by a single Gaussian distribution. In contrast, histograms from contracting muscles were best fit by at least two Gaussians. These results provide direct evidence that cross-bridges in working skeletal muscle assume two distinct conformations, presumably corresponding to the pre- and post-power-stroke states.     

Calcium regulation of myosin-I tension sensing

Myo1b is a myosin that is exquisitely sensitive to tension. Its actin-attachment lifetime increases > 50-fold when its working stroke is opposed by 1 pN of force. The long attachment lifetime of myo1b under load raises the question: how are actin attachments that last >50 s in the presence of force regulated? Like most myosins, forces are transmitted to the myo1b motor through a light-chain binding domain that is structurally stabilized by calmodulin, a calcium-binding protein. Thus, we examined the effect of calcium on myo1b motility using ensemble and single-molecule techniques. Calcium accelerates key biochemical transitions on the ATPase pathway, decreases the working-stroke displacement, and greatly reduces the ability of myo1b to sense tension. Thus, calcium provides an effective mechanism for inhibiting motility and terminating long-duration attachments.