Mysteries of muscle contraction

According to the cross-bridge theory, the steady-state isometric force of a muscle is given by the amount of actin-myosin filament overlap. However, it has been known for more than half a century that steady-state forces depend crucially on contractile history. Here, we examine history-dependent steady-state force production in view of the cross-bridge theory, available experimental evidence, and existing explanations for this phenomenon. This is done on various structural levels, ranging from the intact muscle to the myofibrillar and isolated contractile protein level, so that advantages and limitations of the various preparations can be fully exploited and overcome. Based on experimental evidence, we conclude that steady-state force following active muscle stretching is enhanced, and this enhancement has a passive and an active component. The active component is associated with the cross-bridge kinetics, and the passive component is associated with a calcium-dependent increase in titin stiffness.     

Geometrical factors influencing muscle force development. I. The effect of filament spacing upon axial forces.

The influence of geometry on the force and stiffness measured during muscle contraction at different sarcomere lengths is examined by using three specific models of muscle cross-bridge geometry which are based upon the double-hinge model of H. E. Huxley (Science [Wash. D.C.]. 1969, 164:1356-1366) extended to three dimensions. The force generated during muscle contraction depends upon the orientation of the individual cross-bridge force vectors and the distribution of the cross-bridges between various states. For the simplest models, in which filament separation has no effect upon cross-bridge distribution, it is shown that changes in force vectors accompanying changes in myofilament separation between sarcomere lengths 2.0 and 3.65 microgram in an intact frog skeletal muscle fiber have only a small effect upon axial force. The simplest models, therefore, produce a total axial force proportional to the overlap between the actin and myosin filaments and independent of filament separation. However, the analysis shows that it is possible to find assumptions that produce a cross-bridge model in which the axial force is not independent of filament spacing. It is also shown that for some modes of attachment of subfragment-1 (S1) to actin the azimuthal location of the actin site is important in determining the axial force. A mode of S1 attachment to actin similar to that deduced by Moore et al. (J. Mol. Biol., 1970, 50:279-294), however, exhibits rather constant cross-bridge behavior over a wide range of actin site location.     

On the theory of muscle contraction: filament extensibility and the development of isometric force and stiffness.

The newly discovered extensibility of actin and myosin filaments challenges the foundation of the theory of muscle mechanics. We have reformulated A. F. Huxley's sliding filament theory to explicitly take into account filament extensibility. During isometric force development, growing cross-bridge tractions transfer loads locally between filaments, causing them to extend and, therefore, to slide locally relative to one another. Even slight filament extensibility implies that 1) relative displacement between the two must be nonuniform along the region of filament overlap, 2) cross-bridge strain must vary systematically along the overlap region, and importantly, 3) the local shortening velocities, even at constant overall sarcomere length, reduce force below the level that would have developed if the filaments had been inextensible. The analysis shows that an extensible filament system with only two states (attached and detached) displays three important characteristics: 1) muscle stiffness leads force during force development; 2) cross-bridge stiffness is significantly higher than previously assessed by inextensible filament models; and 3) stiffness is prominently dissociated from the number of attached cross-bridges during force development. The analysis also implies that the local behavior of one myosin head must depend on the state of neighboring attachment sites. This coupling occurs exclusively through local sliding velocities, which can be significant, even during isometric force development. The resulting mechanical cooperativity is grounded in fiber mechanics and follows inevitably from filament extensibility.     

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.     

Length dependence of active force production in skeletal muscle.

The sliding filament and cross-bridge theories of muscle contraction provide discrete predictions of the tetanic force-length relationship of skeletal muscle that have been tested experimentally. The active force generated by a maximally activated single fiber (with sarcomere length control) is maximal when the filament overlap is optimized and is proportionally decreased when overlap is diminished. The force-length relationship is a static property of skeletal muscle and, therefore, it does not predict the consequences of dynamic contractions. Changes in sarcomere length during muscle contraction result in modulation of the active force that is not necessarily predicted by the cross-bridge theory. The results of in vivo studies of the force-length relationship suggest that muscles that operate on the ascending limb of the force-length relationship typically function in stretch-shortening cycle contractions, and muscles that operate on the descending limb typically function in shorten-stretch cycle contractions. The joint moments produced by a muscle depend on the moment arm and the sarcomere length of the muscle. Moment arm magnitude also affects the excursion (length change) of a muscle for a given change in joint angle, and the number of sarcomeres arranged in series within a muscle fiber determines the sarcomere length change associated with a given excursion.     

The latch-bridge hypothesis of smooth muscle contraction

In contrast to striated muscle, both normalized force and shortening velocities are regulated functions of cross-bridge phosphorylation in smooth muscle. Physiologically this is manifested as relatively fast rates of contraction associated with transiently high levels of cross-bridge phosphorylation. In sustained contractions, Ca2+, cross-bridge phosphorylation, and ATP consumption rates fall, a phenomenon termed "latch". This review focuses on the Hai and Murphy (1988a) model that predicted the highly non-linear dependence of force on phosphorylation and a directly proportional dependence of shortening velocity on phosphorylation. This model hypothesized that (i) cross-bridge phosphorylation was obligatory for cross-bridge attachment, but also that (ii) dephosphorylation of an attached cross-bridge reduced its detachment rate. The resulting variety of cross-bridge cycles as predicted by the model could explain the observed dependencies of force and velocity on cross-bridge phosphorylation. New evidence supports modifications for more general applicability. First, myosin light chain phosphatase activity is regulated. Activation of myosin phosphatase is best demonstrated with inhibitory regulatory mechanisms acting via nitric oxide. The second modification of the model incorporates cooperativity in cross-bridge attachment to predict improved data on the dependence of force on phosphorylation. The molecular basis for cooperativity is unknown, but may involve thin filament proteins absent in striated muscle.     

Models in which many cross-bridges attach simultaneously can explain the filament movement per ATP split during muscle contraction

Contractile filaments in skeletal muscle are moved by less than 2 nm for each ATP used. If just one cross-bridge is attached to each thin filament at any instant then this distance represents the fundamental myosin cross-bridge step size (i.e. the distance one cross-bridge moves a thin filament in one ATP-splitting cycle). However, most contraction models assume many cross-bridges are attached at any instant along each thin filament. The purpose of this study was to establish whether the net filament sliding per ATP used could be explained quantitatively in terms of a cross-bridge model in which multiple cross-bridges are attached along each thin filament. It was found that the relationship between net filament sliding per ATP split and the load against which the muscle shortens is compatible with such a model and furthermore predicts that the cross-bridge step size is between 7.5 and 12.5 nm over most of the range of loads. These values were similar for different muscle fibre types.     

The Three Filament Model of Skeletal Muscle Stability and Force Production

Ever since the 1950s, muscle force regulation has been associated with the cross-bridge interactions between the two contractile filaments, actin and myosin. This gave rise to what is referred to as the "two-filament sarcomere model". This model does not predict eccentric muscle contractions well, produces instability of myosin alignment and force production on the descending limb of the force-length relationship, and cannot account for the vastly decreased ATP requirements of actively stretched muscles. Over the past decade, we and others, identified that a third myofilament, titin, plays an important role in stabilizing the sarcomere and the myosin filament. Here, we demonstrate additionally how titin is an active participant in muscle force regulation by changing its stiffness in an activation/force dependent manner and by binding to actin, thereby adjusting its free spring length. Therefore, we propose that skeletal muscle force regulation is based on a three filament model that includes titin, rather than a two filament model consisting only of actin and myosin filaments.     

Myosin filament polymerization and depolymerization in a model of partial length adaptation in airway smooth muscle

Length adaptation in airway smooth muscle (ASM) is attributed to reorganization of the cytoskeleton, and in particular the contractile elements. However, a constantly changing lung volume with tidal breathing (hence changing ASM length) is likely to restrict full adaptation of ASM for force generation. There is likely to be continuous length adaptation of ASM between states of incomplete or partial length adaption. We propose a new model that assimilates findings on myosin filament polymerization/depolymerization, partial length adaptation, isometric force, and shortening velocity to describe this continuous length adaptation process. In this model, the ASM adapts to an optimal force-generating capacity in a repeating cycle of events. Initially the myosin filament, shortened by prior length changes, associates with two longer actin filaments. The actin filaments are located adjacent to the myosin filaments, such that all myosin heads overlap with actin to permit maximal cross-bridge cycling. Since in this model the actin filaments are usually longer than myosin filaments, the excess length of the actin filament is located randomly with respect to the myosin filament. Once activated, the myosin filament elongates by polymerization along the actin filaments, with the growth limited by the overlap of the actin filaments. During relaxation, the myosin filaments dissociate from the actin filaments, and then the cycle repeats. This process causes a gradual adaptation of force and instantaneous adaptation of shortening velocity. Good agreement is found between model simulations and the experimental data depicting the relationship between force development, myosin filament density, or shortening velocity and length.     

Sarcomere lattice geometry influences cooperative myosin binding in muscle

In muscle, force emerges from myosin binding with actin (forming a cross-bridge). This actomyosin binding depends upon myofilament geometry, kinetics of thin-filament Ca²⁺ activation, and kinetics of cross-bridge cycling. Binding occurs within a compliant network of protein filaments where there is mechanical coupling between myosins along the thick-filament backbone and between actin monomers along the thin filament. Such mechanical coupling precludes using ordinary differential equation models when examining the effects of lattice geometry, kinetics, or compliance on force production. This study uses two stochastically driven, spatially explicit models to predict levels of cross-bridge binding, force, thin-filament Ca²⁺ activation, and ATP utilization. One model incorporates the 2-to-1 ratio of thin to thick filaments of vertebrate striated muscle (multi-filament model), while the other comprises only one thick and one thin filament (two-filament model). Simulations comparing these models show that the multi-filament predictions of force, fractional cross-bridge binding, and cross-bridge turnover are more consistent with published experimental values. Furthermore, the values predicted by the multi-filament model are greater than those values predicted by the two-filament model. These increases are larger than the relative increase of potential inter-filament interactions in the multi-filament model versus the two-filament model. This amplification of coordinated cross-bridge binding and cycling indicates a mechanism of cooperativity that depends on sarcomere lattice geometry, specifically the ratio and arrangement of myofilaments.     

Geometrical factors influencing muscle force development. II. Radial forces.

If the subfragment-2 (S2) portion of the myosin cross-bridge to actin does not lie parallel to the myofilament axes then when a muscle fiber contracts, there will be a radial component to the cross-bridge force. When the subfragment-1 (S1) portion of the cross-bridge attaches to actin with its long axis projecting through the filament axis, the magnitude of the radial force depends upon the azimuthal location of the actin site, but when the attachment of the S1 to actin is slewed, as in the reconstruction of Moore et al. (J. Mol. Biol., 1970, 50:279-294), then for a single cross-bridge the radial component of the cross-bridge force is not quite so sensitive to actin site location and is approximately 0.1 the axial component. In both cases, the ratio of the radial to axial force decreases with decreasing filament separation. If the radial-axial force ratio for each cross-bridge is approximately 0.1, then at full overlap in a frog skeletal muscle fiber the radial component of the cross-bridge force accompanying full activation will exert a compressive pressure of approximately 5 X 10(-3) atm. This would have little effect upon an intact muscle fiber where the volume constraints are likely osmotic, but it might produce a 1-2% change in filament spacing in a "skinned" muscle fiber from which the sarcolemma had been removed. These computations assume that the S2 link between the S1 head and the myosin filament does not support a bending moment of shear. If it does, then the radial component of the cross-bridge will be either greater or less, depending on the specific cross-bridge geometry.     

Calcium alone does not fully activate the thin filament for S1 binding to rigor myofibrils.

Skeletal muscle contraction is regulated by calcium via troponin and tropomyosin and appears to involve cooperative activation of cross-bridge binding to actin. We studied the regulation of fluorescent myosin subfragment 1 (fS1) binding to rigor myofibrils over a wide range of fS1 and calcium levels using highly sensitive imaging techniques. At low calcium and low fS1, the fluorescence was restricted to the actin-myosin overlap region. At high calcium and very low fS1, the fluorescence was still predominantly in the overlap region. The ratio of nonoverlap to overlap fluorescence intensity showed that increases in the fS1 level resulted in a shift in maximum fluorescence from the overlap to the nonoverlap region at both low and high calcium; this transition occurred at lower fS1 levels in myofibrils with high calcium. At a fixed fS1 level, increases in calcium also resulted in a shift in maximum fluorescence from the overlap region to the nonoverlap region. These results suggest that calcium alone does not fully activate the thin filament for rigor S1 binding and that, even at high calcium, the thin filament is not activated along its entire length.     

Axial and radial forces of cross-bridges depend on lattice spacing

Nearly all mechanochemical models of the cross-bridge treat myosin as a simple linear spring arranged parallel to the contractile filaments. These single-spring models cannot account for the radial force that muscle generates (orthogonal to the long axis of the myofilaments) or the effects of changes in filament lattice spacing. We describe a more complex myosin cross-bridge model that uses multiple springs to replicate myosin's force-generating power stroke and account for the effects of lattice spacing and radial force. The four springs which comprise this model (the 4sXB) correspond to the mechanically relevant portions of myosin's structure. As occurs in vivo, the 4sXB's state-transition kinetics and force-production dynamics vary with lattice spacing. Additionally, we describe a simpler two-spring cross-bridge (2sXB) model which produces results similar to those of the 4sXB model. Unlike the 4sXB model, the 2sXB model requires no iterative techniques, making it more computationally efficient. The rate at which both multi-spring cross-bridges bind and generate force decreases as lattice spacing grows. The axial force generated by each cross-bridge as it undergoes a power stroke increases as lattice spacing grows. The radial force that a cross-bridge produces as it undergoes a power stroke varies from expansive to compressive as lattice spacing increases. Importantly, these results mirror those for intact, contracting muscle force production.     

An expanded latch-bridge model of protein kinase C-mediated smooth muscle contraction.

A thin-filament-regulated latch-bridge model of smooth muscle contraction is proposed to integrate thin-filament-based inhibition of actomyosin ATPase activity with myosin phosphorylation in the regulation of smooth muscle mechanics. The model included two latch-bridge cycles, one of which was identical to the four-state model as proposed by Hai and Murphy (Am J Physiol Cell Physiol 255: C86-C94, 1988), whereas the ultraslow cross-bridge cycle has lower cross-bridge cycling rates. The model-fitted phorbol ester induced slow contractions at constant myosin phosphorylation and predicted steeper dependence of force on myosin phosphorylation in phorbol ester-stimulated smooth muscle. By shifting cross bridges between the two latch-bridge cycles, the model predicts that a smooth muscle cell can either maintain force at extremely low-energy cost or change its contractile state rapidly, if necessary. Depending on the fraction of cross bridges engaged in the ultraslow latch-bridge cycle, the model predicted biphasic kinetics of smooth muscle mechanics and variable steady-state dependencies of force and shortening velocity on myosin phosphorylation. These results suggest that thin-filament-based regulatory proteins may function as tuners of actomyosin ATPase activity, thus allowing a smooth muscle cell to have two discrete cross-bridge cycles with different cross-bridge cycling rates.     

Geometrical restrictions on cross-bridge motion and sliding filament dynamics

To ensure asynchronous motion and continuous force development in a sliding filament mechanism when the cross-bridge configurational cycle is geometrically constant and uniform, the repeat distances of the cross-bridge set and the active site set must be different. When the two sets overlap they will then form an interference pattern—the elements are in phase in some regions, out of phase in others. The geometrical restrictions that this puts on the cross-bridge cycle are discussed. One consequence may be that only a discrete set of modes of motion are possible in active filament sliding, the number of modes being determined by the energy supply rate and the internal friction in the system.     

The Mechanism of Spontaneous Oscillatory Contractions in Skeletal Muscle

Most striated muscles generate steady contractile tension when activated, but some preparations, notably cardiac myocytes and slow-twitch fibers, may show spontaneous oscillatory contractions (SPOC) at low levels of activation. We have provided what we believe is new evidence that SPOC is a property of the contractile system at low actin-myosin affinity, whether caused by a thin-filament regulatory system or by other means. We present a quantitative single-sarcomere model for isotonic SPOC in skeletal muscle with three basic ingredients: i), actin and myosin filaments initially in partial overlap, ii), stretch activation by length-dependent changes in the lattice spacing, and iii), viscoelastic passive tension. Modeling examples are given for slow-twitch and fast-twitch fibers, with periods of 10 s and 4 s respectively. Isotonic SPOC occurs in a narrow domain of parameter values, with small minimum and maximum values for actin-myosin affinity, a minimum amount of passive tension, and a maximum transient response rate that explains why SPOC is favored in slow-twitch fibers. The model also predicts the contractile, relaxed and SPOC phases as a function of phosphate and ADP levels. The single-sarcomere model can also be applied to a whole fiber under auxotonic and fixed-end conditions if the remaining sarcomeres are treated as a viscoelastic load. Here the model predicts an upper limit for the load stiffness that leads to SPOC; this limit lies above the equivalent loads expected from the rest of the fiber.     

Nonlinear Actomyosin Elasticity in Muscle?

Cyclic interactions between myosin II motor domains and actin filaments that are powered by turnover of ATP underlie muscle contraction and have key roles in motility of nonmuscle cells. The elastic characteristics of actin-myosin cross-bridges are central in the force-generating process, and disturbances in these properties may lead to disease. Although the prevailing paradigm is that the cross-bridge elasticity is linear (Hookean), recent single-molecule studies suggest otherwise. Despite convincing evidence for substantial nonlinearity of the cross-bridge elasticity in the single-molecule work, this finding has had limited influence on muscle physiology and physiology of other ordered cellular actin-myosin ensembles. Here, we use a biophysical modeling approach to close the gap between single molecules and physiology. The model is used for analysis of available experimental results in the light of possible nonlinearity of the cross-bridge elasticity. We consider results obtained both under rigor conditions (in the absence of ATP) and during active muscle contraction. Our results suggest that a wide range of experimental findings from mechanical experiments on muscle cells are consistent with nonlinear actin-myosin elasticity similar to that previously found in single molecules. Indeed, the introduction of nonlinear cross-bridge elasticity into the model improves the reproduction of key experimental results and eliminates the need for force dependence of the ATP-induced detachment rate, consistent with observations in other single-molecule studies. The findings have significant implications for the understanding of key features of actin-myosin-based production of force and motion in living cells, particularly in muscle, and for the interpretation of experimental results that rely on stiffness measurements on cells or myofibrils.     

Direct tests of muscle cross-bridge theories: predictions of a Brownian dumbbell model for position-dependent cross-bridge lifetimes and step sizes with an optically trapped actin filament.

Force and displacement events from a single myosin molecule interacting with an actin filament suspended between optically trapped beads (Finer, J. T., R. M. Simmons, and J. A. Spudich. 1994. Nature. 368:113-119) can be interpreted in terms of a generalized cross-bridge model that includes the effects of Brownian forces on the beads. Steady-state distributions of force and displacement can be obtained directly from a generalized Smoluchowski equation for Brownian motion of the actin-bead "dumbbell," and time series from Monte Carlo simulations of the corresponding Langevin equation. When the frequency spectrum of Brownian motion extends beyond cross-bridge transition rates, the inverse mean lifetimes of force/displacement pulses are given by cross-bridge rate constants averaged over a Boltzmann distribution of Brownian noise. These averaged rate constants reflect the strain-dependence of the rate constants for the stationary filament, most faithfully at high trap stiffness. Hence, measurements of the lifetimes and displacements of single events as a function of the resting position of the dumbbell can provide a direct test of different cross-bridge theories of muscle contraction. Quantitative demonstrations are given for Huxley models with 1) faster binding or 2) slower dissociation at positive cross-bridge strain. Predictions for other models can be inferred from the averaging procedure.     

Proposed role of the M-band in sarcomere mechanics and mechano-sensing: a model study

In sarcomeres of striated muscles the middle parts of adjacent thick filaments are connected to each other by the M-band proteins. To understand the role of the M-band in sarcomere mechanics a model of forces which pull a thick filament to opposite Z-disks of a sarcomere is considered. Forces of actin-myosin cross-bridges, I-band titin segments and the M-band are accounted for. A continual expression for the M-band force is obtained assuming that the M-band proteins which connect neighbor thick filaments have nonlinear elastic properties. On the ascending and descending limbs of the force-length diagram cross-bridge forces tend to destabilize sarcomere while titin tries to restore its symmetric configuration. When destabilizing cross-bridge force exceeds a critical limit, symmetric configuration of a sarcomere becomes unstable and the M-band buckles. Stiffness of the M-band increases stability only if the M-band is anchored to the extra-sarcomere cytoskeleton. Realistic magnitudes of the M-band buckling require that the M-band proteins have essentially nonlinear elasticity. The buckling may explain the M-band bending and axial misalignment of the thick filaments observed in contracting muscle. We hypothesize that the buckling stretches the titin protein kinase domain localized in the M-band being the signal for mechanical control of gene expression and protein turnover in striated muscle.     

Modulation of contractile activation in skeletal muscle by a calcium-insensitive troponin C mutant

Calcium controls the level of muscle activation via interactions with the troponin complex. Replacement of the native, skeletal calcium-binding subunit of troponin, troponin C, with mixtures of functional cardiac and mutant cardiac troponin C insensitive to calcium and permanently inactive provides a novel method to alter the number of myosin cross-bridges capable of binding to the actin filament. Extraction of skeletal troponin C and replacement with functional and mutant cardiac troponin C were used to evaluate the relationship between the extent of thin filament activation (fractional calcium binding), isometric force, and the rate of force generation in muscle fibers independent of the calcium concentration. The experiments showed a direct, linear relationship between force and the number of cross-bridges attaching to the thin filament. Further, above 35% maximal isometric activation, following partial replacement with mixtures of cardiac and mutant troponin C, the rate of force generation was independent of the number of actin sites available for cross-bridge interaction at saturating calcium concentrations. This contrasts with the marked decrease in the rate of force generation when force was reduced by decreasing the calcium concentration. The results are consistent with hypotheses proposing that calcium controls the transition between weakly and strongly bound cross-bridge states.