Smooth muscle myosin cross-bridge interactions modulate actin filament sliding velocity in vitro

Although it is generally believed that phosphorylation of the regulatory light chain of myosin is required before smooth muscle can develop force, it is not known if the overall degree of phosphorylation can also modulate the rate at which cross-bridges cycle. To address this question, an in vitro motility assay was used to observe the motion of single actin filaments interacting with smooth muscle myosin copolymers composed of varying ratios of phosphorylated and unphosphorylated myosin. The results suggest that unphosphorylated myosin acts as a load to slow down the rate at which actin is moved by the faster cycling phosphorylated cross-bridges. Myosin that was chemically modified to generate a noncycling analogue of the "weakly" bound conformation was similarly able to slow down phosphorylated myosin. The observed modulation of actin velocity as a function of copolymer composition can be accounted for by a model based on mechanical interactions between cross-bridges.     

A physical model of ATP-induced actin-myosin movement in vitro.

The nature of the mechanism limiting the velocity of ATP-induced unidirectional movements of actin-myosin filaments in vitro is considered. In the sliding process two types of "cyclic" interactions between myosin heads and actin are involved, i.e., productive and nonproductive. In the productive interaction, myosin heads split ATP and generate a force which produces sliding between actin and myosin. In the nonproductive interaction "cycle," on the other hand, myosin heads rapidly attach to and detach from actin "reversibly," i.e., without splitting ATP or generating an active force. Such a nonproductive interaction "cycle" causes irreversible dissipation of sliding energy into heat, because the myosin cross-bridges during this interaction are passive elastic structures. This consideration has led us to postulate that such cross-bridges, in effect, exert viscous-like frictional drag on moving elements. Energetic considerations suggest that this frictional drag is much greater than the hydrodynamic viscous drag. We present a model in which the sliding velocity is limited by the balance between the force generated by myosin cross-bridges in the productive interaction and the frictional drag exerted by other myosin cross-bridges in the nonproductive interaction. The model is consistent with experimental findings of in vitro sliding, including the dependence of velocity on ATP concentration, as well as the sliding velocity of co-polymers of skeletal muscle myosin and phosphorylated and unphosphorylated smooth muscle myosins.     

Equilibrium muscle cross-bridge behavior. Theoretical considerations.

We have developed a model for the equilibrium attachment and detachment of myosin cross-bridges to actin that takes into account the possibility that a given cross-bridge can bind to one of a number of actin monomers, as seems likely, rather than to a site on only a single actin monomer, as is often assumed. The behavior of this multiple site model in response to constant velocity, as well as instantaneous stretches, was studied and the influence of system parameters on the force response explored. It was found that in the multiple site model the detachment rate constant has considerably greater influence on the mechanical response than the attachment rate constant. It is shown that one can obtain information about the detachment rate constants either by examining the relationship between the apparent stiffness and duration of stretch for constant velocity stretches or by examining the force-decay rate constants following an instantaneous stretch. The main effect of the attachment rate constant is to scale the mechanical response by influencing the number of attached cross-bridges. The significance of the modeling for the interpretation of experimental results is discussed.     

The mechanism of muscle contraction

Knowledge of the mechanism of contraction has been obtained from studies of the interaction of actin and myosin in solution, from an elucidation of the structure of muscle fibers, and from measurements of the mechanics and energetics of fiber contraction. Many of the states and the transition rates between them have been established for the hydrolysis of ATP by actin and myosin subfragments in solution. A major goal is to now understand how the kinetics of this interaction are altered when it occurs in the organized array of the myofibril. Early work on the structure of muscle suggested that changes in the orientation of myosin cross-bridges were responsible for the generation of force. More recently, fluorescent and paramagnetic probes attached to the cross-bridges have suggested that at least some domains of the cross-bridges do not change orientation during force generation. A number of properties of active cross-bridges have been defined by measurements of steady state contractions of fibers and by the transients which follow step changes in fiber length or tension. Taken together these studies have provided firm evidence that force is generated by a cyclic interaction in which a myosin cross-bridge attaches to actin, exerts force through a "powerstroke" of 12 nm, and is then released by the binding of ATP. The mechanism of this interaction at the molecular level remains unknown.     

Structural changes in isometrically contracting insect flight muscle trapped following a mechanical perturbation

The application of rapidly applied length steps to actively contracting muscle is a classic method for synchronizing the response of myosin cross-bridges so that the average response of the ensemble can be measured. Alternatively, electron tomography (ET) is a technique that can report the structure of the individual members of the ensemble. We probed the structure of active myosin motors (cross-bridges) by applying 0.5% changes in length (either a stretch or a release) within 2 ms to isometrically contracting insect flight muscle (IFM) fibers followed after 5-6 ms by rapid freezing against a liquid helium cooled copper mirror. ET of freeze-substituted fibers, embedded and thin-sectioned, provides 3-D cross-bridge images, sorted by multivariate data analysis into ~40 classes, distinct in average structure, population size and lattice distribution. Individual actin subunits are resolved facilitating quasi-atomic modeling of each class average to determine its binding strength (weak or strong) to actin. ~98% of strong-binding acto-myosin attachments present after a length perturbation are confined to "target zones" of only two actin subunits located exactly midway between successive troponin complexes along each long-pitch helical repeat of actin. Significant changes in the types, distribution and structure of actin-myosin attachments occurred in a manner consistent with the mechanical transients. Most dramatic is near disappearance, after either length perturbation, of a class of weak-binding cross-bridges, attached within the target zone, that are highly likely to be precursors of strong-binding cross-bridges. These weak-binding cross-bridges were originally observed in isometrically contracting IFM. Their disappearance following a quick stretch or release can be explained by a recent kinetic model for muscle contraction, as behaviour consistent with their identification as precursors of strong-binding cross-bridges. The results provide a detailed model for contraction in IFM that may be applicable to contraction in other types of muscle.     

Influence of dispersion in myosin filament orientation and anisotropic filament contractions in smooth muscle

A new constitutive model for the biomechanical behaviour of smooth muscle tissue is proposed. The active muscle contraction is accomplished by the relative sliding between actin and myosin filaments, comprising contractile units in the smooth muscle cells. The orientation of the myosin filaments, and thereby the contractile units, are taken to exhibit a statistical dispersion around a preferred direction. The number of activated cross-bridges between the actin and myosin filaments governs the contractile force generated by the muscle and also the contraction speed. A strain-energy function is used to describe the mechanical behaviour of the smooth muscle tissue. Besides the active contractile apparatus, the mechanical model also incorporates a passive elastic part. The constitutive model was compared to histological and isometric tensile test results for smooth muscle tissue from swine carotid artery. In order to be able to predict the active stress at different muscle lengths, a filament dispersion significantly larger than the one observed experimentally was required. Furthermore, a comparison of the predicted active stress for a case of uniaxially oriented myosin filaments and a case of filaments with a dispersion based on the experimental histological data shows that the difference in generated stress is noticeable but limited. Thus, the results suggest that myosin filament dispersion alone cannot explain the increase in active muscle stress with increasing muscle stretch.     

Cross-bridge duty cycle in isometric contraction of skeletal myofibrils

During interaction of actin with myosin, cross-bridges impart mechanical impulses to thin filaments resulting in rotations of actin monomers. Impulses are delivered on the average every tc seconds. A cross-bridge spends a fraction of this time (ts) strongly attached to actin, during which it generates force. The "duty cycle" (DC), defined as the fraction of the total cross-bridge cycle that myosin spends attached to actin in a force generating state (ts/ tc), is small for cross-bridges acting against zero load, like freely shortening muscle, and increases as the load rises. Here we report, for the first time, an attempt to measure DC of a single cross-bridge in muscle. A single actin molecule in a half-sarcomere was labeled with fluorescent phalloidin. Its orientation was measured by monitoring intensity of the polarized TIRF images. Actin changed orientation when a cross-bridge bound to it. During isometric contraction, but not during rigor, actin orientation oscillated between two values, corresponding to the actin-bound and actin-free state of the cross-bridge. The average ts and tc were 3.4 and 6 s, respectively. These results suggest that, in isometrically working muscle, cross-bridges spend about half of the cycle time attached to actin. The fact that 1/ tc was much smaller than the ATPase rate suggests that the bulk of the energy of ATP hydrolysis is used for purposes other than performance of mechanical work.     

Mathematical simulation of muscle cross-bridge cycle and force-velocity relationship

Muscle contraction underlies many essential functions such as breathing, heart beating, locomotion, regulation of blood pressure, and airway resistance. Active shortening of muscle is the result of cycling of myosin cross-bridges that leads to sliding of myosin filaments relative to actin filaments. In this study, we have developed a computer program that allows us to alter the rates of transitions between any cross-bridge-states in a stochastic cycle. The cross-bridge states within the cycle are divided into six attached (between myosin cross-bridges and actin filaments) states and one detached state. The population of cross-bridges in each of the states is determined by the transition rates throughout the cycle; differential equations describing the transitions are set up as a cyclic matrix. A method for rapidly obtaining steady-state exact solutions for the cyclic matrix has been developed to reduce computation time and avoid the divergence problem associated with numerical solutions. In the seven-state model, two power strokes are assumed for each cross-bridge cycle, one before the release of inorganic phosphate, and one after. The characteristic hyperbolic force-velocity relationship observed in muscle contraction can be reproduced by the model. Deviation from the single hyperbolic behavior at low velocities can be mimicked by allowing the rate of cross-bridge-attachment to vary with velocity. The effects of [ATP], [ADP], and [P(i)] are simulated by changing transition rates between specific states. The model has revealed new insights on how the force-velocity characteristics are related to the state transitions in the cross-bridge cycle.     

Rotation of the lever arm of Myosin in contracting skeletal muscle fiber measured by two-photon anisotropy

The rotation of the lever arm of myosin cross-bridges is believed to be responsible for muscle contraction. To resolve details of this rotation, it is necessary to observe a single cross-bridge. It is still impossible to do so in muscle fiber, but it is possible to investigate a small population of cross-bridges by simultaneously activating myosin in a femtoliter volume by rapid release of caged ATP. In earlier work, in which the number of observed cross-bridges was limited to approximately 600 by confocal microscopy, we were able to measure the rates of cross-bridge detachment and rebinding. However, we were unable to resolve the power stroke. We speculated that the reason for this was that the number of observed cross-bridges was too large. In an attempt to decrease this number, we used two-photon microscopy which permitted observation of approximately 1/2 as many cross-bridges as before with the same signal/noise ratio. With the two-photon excitation, the number of cross-bridges was small enough to resolve the beginning of the power stroke. The results indicated that the power stroke begins approximately 170 ms after the rigor cross-bridge first binds ATP.     

Altered kinetics of contraction in skeletal muscle fibers containing a mutant myosin regulatory light chain with reduced divalent cation binding.

We examined the kinetic properties of rabbit skinned skeletal muscle fibers in which the endogenous myosin regulatory light chain (RLC) was partially replaced with a mutant RLC (D47A) containing a point mutation within the Ca2+/Mg2+ binding site that severely reduced its affinity for divalent cations. We found that when approximately 50% of the endogenous RLC was replaced by the mutant, maximum tension declined to approximately 60% of control and the rate constant of active tension redevelopment (ktr) after mechanical disruption of cross-bridges was reduced to approximately 70% of control. This reduction in ktr was not an indirect effect on kinetics due to a reduced number of strongly bound myosin heads, because when the strongly binding cross-bridge analog N-ethylmaleimide-modified myosin subfragment1 (NEM-S1) was added to the fibers, there was no effect upon maximum ktr. Fiber stiffness declined after D47A exchange in a manner indicative of a decrease in the number of strongly bound cross-bridges, suggesting that the force per cross-bridge was not significantly affected by the presence of D47A RLC. In contrast to the effects on ktr, the rate of tension relaxation in steadily activated fibers after flash photolysis of the Ca2+ chelator diazo-2 increased by nearly twofold after D47A exchange. We conclude that the incorporation of the nondivalent cation-binding mutant of myosin RLC decreases the proportion of cycling cross-bridges in a force-generating state by decreasing the rate of formation of force-generating bridges and increasing the rate of detachment. These results suggest that divalent cation binding to myosin RLC plays an important role in modulating the kinetics of cross-bridge attachment and detachment.     

Modeling rigor cross-bridge patterns in muscle I. Initial studies of the rigor lattice of insect flight muscle.

We have undertaken some computer modeling studies of the cross-bridge observed by Reedy in insect flight muscle so that we investigate the geometric parameters that influence the attachment patterns of cross-bridges to actin filaments. We find that the appearance of double chevrons along an actin filament indicates that the cross-bridges are able to reach 10--14 nm axially, and about 90 degrees around the actin filament. Between three and five actin monomers are therefore available along each turn of one strand of actin helix for labeling by cross-bridges from an adjacent myosin filament. Reedy's flared X of four bridges, which appears rotated 60 degrees at successive levels on the thick filament, depends on the orientation of the actin filaments in the whole lattice as well as on the range of movement in each cross-bridge. Fairly accurate chevrons and flared X groupings can be modeled with a six-stranded myosin surface lattice. The 116-nm long repeat appears in our models as "beating" of the 14.5-nm myosin repeat and the 38.5-nm actin period. Fourier transforms of the labeled actin filaments indicate that the cross-bridges attach to each actin filament on average of 14.5 nm apart. The transform is sensitive to changes in the ease with which the cross-bridge can be distorted in different directions.     

Cross-bridge number, position, and angle in target zones of cryofixed isometrically active insect flight muscle

Electron micrographic tomograms of isometrically active insect flight muscle, freeze substituted after rapid freezing, show binding of single myosin heads at varying angles that is largely restricted to actin target zones every 38.7 nm. To quantify the parameters that govern this pattern, we measured the number and position of attached myosin heads by tracing cross-bridges through the three-dimensional tomogram from their origins on 14.5-nm-spaced shelves along the thick filament to their thin filament attachments in the target zones. The relationship between the probability of cross-bridge formation and axial offset between the shelf and target zone center was well fitted by a Gaussian distribution. One head of each myosin whose origin is close to an actin target zone forms a cross-bridge most of the time. The probability of cross-bridge formation remains high for myosin heads originating within 8 nm axially of the target zone center and is low outside 12 nm. We infer that most target zone cross-bridges are nearly perpendicular to the filaments (60% within 11 degrees ). The results suggest that in isometric contraction, most cross-bridges maintain tension near the beginning of their working stroke at angles near perpendicular to the filament axis. Moreover, in the absence of filament sliding, cross-bridges cannot change tilt angle while attached nor reach other target zones while detached, so may cycle repeatedly on and off the same actin target monomer.     

A dynamic model of smooth muscle contraction

A dynamic model of smooth muscle contraction is presented and is compared with the mechanical properties of vascular smooth muscle in the rat portal vein. The model is based on the sliding filament theory and the assumption that force is produced by cross-bridges extending from the myosin to the actin filaments. Thus, the fundamental aspects of the model are also potentially applicable to skeletal muscle. The main concept of the model is that the transfer of energy via the cross-bridges can be described as a 'friction clutch' mechanism. It is shown that a mathematical formulation of this concept gives rise to a model that agrees well with experimental observations on smooth muscle mechanics under isotonic as well as isometric conditions. It is noted that the model, without any ad hoc assumptions, displays a nonhyperbolic force-velocity relationship in its high-force portion and that it is able to maintain isometric force in conditions of reduced maximum contraction velocity. Both these findings are consistent with new experimental observations on smooth muscle mechanics cannot be accounted for by the classical Hill model.     

An electron microscopic and optical diffraction analysis of the structure of Limulus telson muscle thick filaments

Long, thick filaments (greater than 4.0 micrometer) rapidly and gently isolated from fresh, unstimulated Limulus muscle by an improved procedure have been examined by electron microscopy and optical diffraction. Images of negatively stained filaments appear highly periodic with a well-preserved myosin cross-bridge array. Optical diffraction patterns of the electron micrographs show a wealth of detail and are consistent with a myosin helical repeat of 43.8 nm, similar to that observed by x-ray diffraction. Analysis of the optical diffraction patterns, in conjunction with the appearance in electron micrographs of the filaments, supports a model for the filament in which the myosin cross-bridges are arranged on a four-stranded helix, with 12 cross-bridges per turn or each helix, thus giving an axial repeat every third level of cross-bridges (43.8 nm).     

Millisecond-scale biochemical response to change in strain

Muscle fiber contraction involves the cyclical interaction of myosin cross-bridges with actin filaments, linked to hydrolysis of ATP that provides the required energy. We show here the relationship between cross-bridge states, force generation, and Pi release during ramp stretches of active mammalian skeletal muscle fibers at 20°C. The results show that force and Pi release respond quickly to the application of stretch: force rises rapidly, whereas the rate of Pi release decreases abruptly and remains low for the duration of the stretch. These measurements show that biochemical change on the millisecond timescale accompanies the mechanical and structural responses in active muscle fibers. A cross-bridge model is used to simulate the effect of stretch on the distribution of actomyosin cross-bridges, force, and Pi release, with explicit inclusion of ATP, ADP, and Pi in the biochemical states and length-dependence of transitions. In the simulation, stretch causes rapid detachment and reattachment of cross-bridges without release of Pi or ATP hydrolysis.     

Recent X-ray Diffraction and Electron Microscope Studies of Striated Muscle

The sliding filament model for muscular contraction supposes that an appropriately directed force is developed between the actin and myosin filaments by some process in which the cross-bridges are involved. The cross-bridges between the filaments are believed to represent the parts of the myosin molecules which possess the active sites for ATPase activity and actin-binding ability, and project out sidewise from the backbone of the thick filaments. The arrangement of the cross-bridges is now being studied by improved low-angle X-ray diffraction techniques, which show that in a resting muscle, they are arranged approximately but not exactly in a helical pattern, and that there are other structural features of the thick filaments which give rise to additional long periodicities shown up by the X-ray diagram. The actin filaments also contain helically arranged subunits, and both the subunit repeat and the helical repeat are different from those in the myosin filaments. Diffraction diagrams can be obtained from muscles in rigor (when permanent attachment of the cross-bridges to the actin subunits takes place) and now, taking advantage of the great increase in the speed of recording, from actively contracting muscles. These show that changes in the arrangement of the cross-bridges are produced under both these conditions and are no doubt associated in contraction with the development of force. Thus configurational changes of the myosin component in muscle have been demonstrated: these take place without any significant over-all change in the length of the filaments.     

A model of flagellar movement based on cooperative dynamics of dynein-tubulin cross-bridges

A theoretical model based on molecular mechanisms of both dynein cross-bridges and radial spokes is used to study bend propagation by eukaryotic flagella. Though nine outer doublets are arranged within an axoneme, a simplified model with four doublets is constructed on the assumption that cross-bridges between two of the four doublets are opposed to those between the other two, corresponding to the geometric array of cross-bridges on the 6-9 and the 1-4 doublets in the axoneme. We also assume that external viscosity is zero, whereas internal viscosity is non-zero in order to reduce numerical complexity. For demonstrating flagellar movement, computer simulations are available by dividing a long flagellum into many straight segments. Considering the fact that dynein cross-bridge spacing is almost equal to attachment site spacing, we may use a localized cross-bridge distribution along attachment sites in each straight segment. Dynamics of cross-bridges are determined by a three-state model, and effects of radial spokes are represented by a periodic mechanical potential whose periodicity is considered to be a stroke distance of the radial spoke. First of all, we examine the model of a short segment to know basic properties of the system. Changing parameters relating to "activation" of cross-bridges, our model demonstrates various phenomena; for example "excitable properties with threshold phenomena" and "limit cycle oscillation". Here, "activation" and "inactivation" (i.e. switching mechanisms) between a pair of oppositely-directed cross-bridges are essential for generation of excitable or oscillatory properties. Next, the model for a flagellar segment is incorporated into a flagellum with a whole length to show bending movement. When excitable properties of cross-bridges, not oscillatory properties, are provided along the length of the flagellum and elastic links between filaments are presented at the base, then our model can demonstrate self-organization of bending waves as well as wave propagation without special feedback control by the curvature of the flagellum. Here, "cooperative interaction" between adjacent short segments, based on "cooperative dynamics" of cross-bridges, is important for wave propagation.     

Cooperative mechanisms of thin filament activation and their contribution to the myocardial contractile function: Assessment in a mathematical model

A mathematical model was used for comparative analysis of the contribution to the myocardial mechanical activity of two potentially possible variants of the cooperative influence of myosin cross-bridges on calcium activation of sarcomere actin filaments. One of these variants implies that the cooperative action of the cross-bridge on the affinity of troponin C for calcium is localized within the functional group A₇TmTn (seven adjacent globular actin monomers, tropomyosin, and one troponin complex TnC + TnI + TnT) where this bridge is attached. The second variant is based on the assumption that cross-bridges may influence the troponin C affinity for calcium also in neighboring A₇TmTn groups (and the closer the group is positioned relative to the bridge, the stronger is the influence on the CaTnC complex affinity in this group). The contribution of each of these two variants to the active mechanical behavior of the cardiac muscle in the contraction-relaxation cycle was assessed. It turned out that adequate simulation of the muscle mechanical activity is provided only by the second variant. Thus, the results of modeling argue in favor of the existence of just this variant of cooperativity.     

Compliant realignment of binding sites in muscle: transient behavior and mechanical tuning.

The presence of compliance in the lattice of filaments in muscle raises a number of concerns about how one accounts for force generation in the context of the cross-bridge cycle--binding site motions and coupling between cross-bridges confound more traditional analyses. To explore these issues, we developed a spatially explicit, mechanochemical model of skeletal muscle contraction. With a simple three-state model of the cross-bridge cycle, we used a Monte Carlo simulation to compute the instantaneous balance of forces throughout the filament lattice, accounting for both thin and thick filament distortions in response to cross-bridge forces. This approach is compared to more traditional mass action kinetic models (in the form of coupled partial differential equations) that assume filament inextensibility. We also monitored instantaneous force generation, ATP utilization, and the dynamics of the cross-bridge cycle in simulations of step changes in length and variations in shortening velocity. Three critical results emerge from our analyses: 1) there is a significant realignment of actin-binding sites in response to cross-bridge forces, 2) this realignment recruits additional cross-bridge binding, and 3) we predict mechanical behaviors that are consistent with experimental results for velocity and length transients. Binding site realignment depends on the relative compliance of the filament lattice and cross-bridges, and within the measured range of these parameters, gives rise to a sharply tuned peak for force generation. Such mechanical tuning at the molecular level is the result of mechanical coupling between individual cross-bridges, mediated by thick filament deformations, and the resultant realignment of binding sites on the thin filament.