Titin-induced force enhancement and force depression: A ‘sticky-spring’ mechanism in muscle contractions?

The sliding filament and crossbridge theories do not suffice to explain a number of muscle experiments. For example, from the entire muscle to myofibrils, predictions of these theories were shown to underestimate the force output during and after active tissue stretch. The converse applies to active tissue shortening.In addition to the crossbridge cycle, we propose that another molecular mechanism is effective in sarcomere force generation. We suggest that, when due to activation, myosin binding sites are available on actin, the giant protein titin's PEVK region attaches itself to the actin filament at those sites. As a result, the molecular spring length is dramatically reduced. This leads to increased passive force when the sarcomere is stretched and to decreased or even negative passive force when the sarcomere shortens. Moreover, during shortening, the proposed mechanism interferes with active-force production by inhibiting crossbridges.Incorporation of a simple ‘sticky-spring’ mechanism model into a Hill-type model of sarcomere dynamics offers explanations for several force-enhancement and force-depression effects. For example, the increase of the sarcomere force compared to the force predicted solely by the sliding filament and crossbridge theories depends on the stretch amplitude and on the working range. The same applies to the decrease of sarcomere force during and after shortening. Using only literature data for its parameterization, the model predicts forces similar to experimental results.     

Lattice spacing changes accompanying isometric tension development in intact single muscle fibers.

The myosin lattice spacing of single intact muscle fibers of the frog, Rana temporaria, was studied in Ringer's solution (standard osmolarity 230 mOsm) and hyper- and hypotonic salines (1.4 and 0.8 times standard osmolarity respectively) in the relaxed state, during "fixed end" tetani, and during shortening, using synchrotron radiation. At standard tonicity, a tetanus was associated with an initial brief lattice expansion (and a small amount of sarcomere shortening), followed by a slow compression (unaccompanied by sarcomere length changes). In hypertonic saline (myosin lattice compressed by 8.1%), these spacing changes were suppressed, in hypotonic saline (lattice spacing increased by 7.5%), they were enhanced. During unloaded shortening of activated fibers, a rapid lattice expansion occurred at all tonicities, but became larger as tonicity was reduced. This expansion was caused in part by the change in length of the preparation, but also by a recoil of a stressed radial compliance associated with axial force. The lattice spacing during unloaded shortening was equal to or occasionally greater than predicted for a relaxed fiber at that sarcomere length, indicating that the lattice compression associated with activation is rapidly reversed upon loss of axial force. Lattice recompression occurred upon termination of shortening under standard and hypotonic conditions, but was almost absent under hypertonic conditions. These observations indicate that axial cross-bridge tension is associated with a compressive radial force in intact muscle fibers at full overlap; however, this radial force exhibits a much greater sensitivity to lattice spacing than does the axial force.     

Dynamic analysis of the structure and function of sarcomeres

We attempted to analyze the relationships between the steric structure of the sarcomere and its physiological functions by the use of a sarcomere model of muscle contraction, which includes the geometric arrangement of the thick and thin filaments of the sarcomere, as well as of the cross-bridges and actin sites. Motions of both cross-bridges and myofilaments were considered in terms of our three-state model of the elementary cycle under constraints caused by the steric structure of the sarcomere proposed by Huxley and Brown. Each cross-bridge moves in a molecular potential of our three-state model under the influence of the sliding motions of myofilaments. The sarcomere model described well the tension-velocity relation and isotonic transient processes quantitatively and consistently. In addition, it allowed independence of the no-load shortening velocity upon the overlap of the thick and thin filaments, although the motions of cross-bridges were not independent. Effects of the helical periodicities of the thick and thin filaments and of the number of cross-bridges upon muscle contraction were studied, and the conditions for smooth and efficient contraction of muscle were obtained.     

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.     

Radial forces within muscle fibers in rigor

Considering the widely accepted cross-bridge model of muscle contraction (Huxley. 1969. Science [Wash. D. C.]. 164:1356-1366), one would expect that attachment of angled cross-bridges would give rise to radial as well as longitudinal forces in the muscle fiber. These forces would tend, in most instances, to draw the myofilaments together and to cause the fiber to decrease in width. Using optical techniques, we have observed significant changes in the width of mechanically skinned frog muscle fibers when the fibers are put into rigor by deleting ATP from the bathing medium. Using a high molecular weight polymer polyvinylpyrrolidone (PVP-40; number average mol. wt. (Mn) = 40,000) in the bathing solution, we were able to estimate the magnitude of the radial forces by shrinking the relaxed fiber to the width observed with rigor induction. With rigor, fiber widths decreased up to approximately 10%, with shrinking being greater at shorter sarcomere spacing and at lower PVP concentrations. At higher PVP concentrations, some fibers actually swelled slightly. Radial pressures seen with rigor in 2 and 4% PVP ranged up to 8.9 x 10(3) N/m2. Upon rigor induction, fibers exerted a longitudinal force of approximately 1 x 10(5) N/m2 that was inhibited by high PVP concentrations (greater than or equal to 13%). In very high PVP concentrations (greater than or equal to 20%), fibers exerted an anomalous force, independent of ATP, which ranged up to 6 x 10(4) N/m2 at 60% PVP. Assuming that all the radial force is the result of cross- bridge attachment, we calculated that rigor cross-bridges exert a radial force of 0.2 x 1.2 x 10(-9) N per thick filament in sarcomeres near rest length. This force is of roughly the same order of magnitude as the longitudinal force per thick filament in rigor contraction or in maximal (calcium-activated) contraction of skinned fibers in ATP- containing solutions. Inasmuch as widths of fibers stretched well beyond overlap of thick and thin filaments decreased with rigor, other radially directed forces may be operating in parallel with cross-bridge forces.     

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.     

Thermodynamic features of myosin filament suspensions: implications for the modeling of muscle contraction.

The analysis of myosin filament suspensions shows that these solutions are characterized by highly nonideal behavior. From these data a model is constructed that allows us to predict that 1) when subjected to an increasing protein osmotic pressure, myosin filaments experience an elastic deformation, which is not linearly related to the acting force; and 2) at constant protein osmotic pressure, when the cross-bridges of the myosin filaments are subjected to an external, nonosmotic force parallel to the filament axis, they are deformed and the water activity coefficient is altered. As a consequence, in muscle, passive and active shortening of the sarcomere is expected to promote the change of the water-water and of the water-protein interactions. We thus propose to depict muscle contraction as a chemo-osmoelastic transduction, where the analysis of the energy partition during the power stroke requires consideration of the osmotic factor in addition to the chemoelastic ones.     

A mechanism for sarcomere breathing: volume change and advective flow within the myofilament lattice

During muscle contraction, myosin motors anchored to thick filaments bind to and slide actin thin filaments. These motors rely on energy derived from ATP, supplied, in part, by diffusion from the sarcoplasm to the interior of the lattice of actin and myosin filaments. The radial spacing of filaments in this lattice may change or remain constant during contraction. If the lattice is isovolumetric, it must expand when the muscle shortens. If, however, the spacing is constant or has a different pattern of axial and radial motion, then the lattice changes volume during contraction, driving fluid motion and assisting in the transport of molecules between the contractile lattice and the surrounding intracellular space. We first create an advective-diffusive-reaction flow model and show that the flow into and out of the sarcomere lattice would be significant in the absence of lattice expansion. Advective transport coupled to diffusion has the potential to substantially enhance metabolite exchange within the crowded sarcomere. Using time-resolved x-ray diffraction of contracting muscle, we next show that the contractile lattice is neither isovolumetric nor constant in spacing. Instead, lattice spacing is time varying, depends on activation, and can manifest as an effective time-varying Poisson ratio. The resulting fluid flow in the sarcomere lattice of synchronous insect flight muscles is even greater than expected for constant lattice spacing conditions. Lattice spacing depends on a variety of factors that produce radial force, including cross-bridges, titin-like molecules, and other structural proteins. Volume change and advective transport varies with the phase of muscle stimulation during periodic contraction but remains significant at all conditions. Although varying in magnitude, advective transport will occur in all cases in which the sarcomere is not isovolumetric. Akin to “breathing,” advective-diffusive transport in sarcomeres is sufficient to promote metabolite exchange and may play a role in the regulation of contraction itself.     

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.     

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.     

Cross-bridge mechanism of residual force enhancement after stretching in a skeletal muscle

A muscle model that uses a modified Langevin equation with actomyosin potentials was used to describe the residual force enhancement after active stretching. Considering that the new model uses cross-bridge theory to describe the residual force enhancement, it is different from other models that use passive stretching elements. Residual force enhancement was simulated using a half sarcomere comprising 100 myosin molecules. In this paper, impulse is defined as the integral of an excess force from the steady isometric force over the time interval for which a stretch is applied. The impulse was calculated from the force response due to fast and slow muscle stretches to demonstrate the viscoelastic property of the cross-bridges. A cross-bridge mechanism was proposed as a way to describe the residual force enhancement on the basis of the impulse results with reference to the compliance of the actin filament. It was assumed that the period of the actin potential increased by 0.5% and the amplitude of the potential decreased by 0.5% when the half sarcomere was stretched by 10%. The residual force enhancement after 21.0% sarcomere stretching was 6.9% of the maximum isometric force of the muscle; this value was due to the increase in the number of cross-bridges.     

X-ray diffraction studies of the contractile mechanism in single muscle fibres

The molecular mechanism of muscle contraction was investigated in intact muscle fibres by X-ray diffraction. Changes in the intensities of the axial X-ray reflections produced by imposing rapid changes in fibre length establish the average conformation of the myosin heads during active isometric contraction, and show that the heads tilt during the elastic response to a change in fibre length and during the elementary force generating process: the working stroke. X-ray interference between the two arrays of myosin heads in each filament allows the axial motions of the heads following a sudden drop in force from the isometric level to be measured in situ with unprecedented precision. At low load, the average working stroke is 12 nm, which is consistent with crystallographic studies. The working stroke is smaller and slower at a higher load. The compliance of the actin and myosin filaments was also determined from the change in the axial spacings of the X-ray reflections following a force step, and shown to be responsible for most of the sarcomere compliance. The mechanical properties of the sarcomere depend on both the motor actions of the myosin heads and the compliance of the myosin and actin filaments.     

Is the Cross-Bridge Stiffness Proportional to Tension during Muscle Fiber Activation?

The cross-bridge stiffness can be used to estimate the number of S1 that are bound to actin during contraction, which is a critical parameter for elucidating the fundamental mechanism of the myosin motor. At present, the development of active tension and the increase in muscle stiffness due to S1 binding to actin are thought to be linearly related to the number of cross-bridges formed upon activation. The nonlinearity of total stiffness with respect to active force is thought to arise from the contribution of actin and myosin filament stiffness to total sarcomere elasticity. In this work, we reexamined the relation of total stiffness to tension during activation and during exposure to N-benzyl-p-toluene sulphonamide, an inhibitor of cross-bridge formation. In addition to filament and cross-bridge elasticity, our findings are best accounted for by the inclusion of an extra elasticity in parallel with the cross-bridges, which is formed upon activation but is insensitive to the subsequent level of cross-bridge formation. By analyzing the rupture tension of the muscle (an independent measure of cross-bridge formation) at different levels of activation, we found that this additional elasticity could be explained as the stiffness of a population of no-force-generating cross-bridges. These findings call into question the assumption that active force development can be taken as directly proportional to the cross-bridge number.     

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.     

Analysis of the effects of firing rate and synchronization on spike-triggered averaging of multidirectional motor unit torque

Spike-triggered averaging (STA) of muscle force transients has often been used to estimate motor unit contractile properties, using the discharge of a motor unit within the muscle as the triggering events. For motor units that exert torque about multiple degrees-of-freedom, STA has also been used to estimate motor unit pulling direction. It is well known that motor unit firing rate and weak synchronization of motor unit discharges with other motor units in the muscle can distort STA estimates of contractile properties, but the distortion of STA estimates of motor unit pulling direction has not been thoroughly evaluated. Here, we derive exact equations that predict that STA decouples firing rate and synchronization distortion when used to estimate motor unit pulling direction. We derive a framework for analyzing synchronization, consider whether the distortion due to synchronization can be removed from STA estimates of pulling direction, and show that there are distributions of motor unit pulling directions for which STA is insensitive to synchronization. We conclude that STA may give insight into how motoneuronal synchronization is organized with respect to motor unit pulling direction. Action Editor: David Terman     

X-ray evidence for radial cross-bridge movement and for the sliding filament model in actively contracting skeletal muscle

Equatorial X-ray diffraction patterns have been studied from muscles at rest, during contraction and in rigor. It is confirmed that the relative intensity ($\text{I}\text{1,0}\text{I}\text{1,1}$) of the two main equatorial reflections depends both on the sarcomere length and on the state of the muscle; in any one state the ratio $\text{I}\text{1,0}\text{I}\text{1,1}$ increases as the sarcomere length of the muscle increases, while at any fixed sarcomere length the ratio is smaller for contracting muscle than for resting muscle and smaller still for rigor muscles. The change of $\text{I}\text{1,0}\text{I}\text{1,1}$ with change of state at constant sarcomere length is interpreted as being due to radial movement of cross-bridges: the average movement during contraction being about 40% of that in rigor.Over the whole range of sarcomere length studied (between 1.8 and 2.7 μm) there was no evidence for any change in lattice spacing when a muscle contracts isometrically.Muscles were studied generating tension after they had shortened actively against a load. The lattice spacings and intensity ratio $\text{I}\text{1,0}\text{I}\text{1,1}$ both changed during active shortening in a way entirely consistent with the sliding filament theory of contraction.     

Straightening Out the Elasticity of Myosin Cross-Bridges

In a contracting muscle, myosin cross-bridges extending from thick filaments pull the interdigitating thin (actin-containing) filaments during cyclical ATP-driven interactions toward the center of the sarcomere, the structural unit of striated muscle. Cross-bridge attachments in the sarcomere have been reported to exhibit a similar stiffness under both positive and negative forces. However, in vitro measurements on filaments with a sparse complement of heads detected a decrease of the cross-bridge stiffness at negative forces attributed to the buckling of the subfragment 2 tail portion. Here, we review some old and new data that confirm that cross-bridge stiffness is nearly linear in the muscle filament lattice. The implications of high myosin stiffness at positive and negative strains are considered in muscle fibers and in nonmuscle intracellular cargo transport.     

Shortening-induced force depression is primarily caused by cross-bridges in strongly bound states

The steady-state isometric force following active muscle shortening is smaller than the corresponding force obtained for purely isometric contractions. This so-called residual force depression has been observed consistently for more than half a century, however its mechanism remains a matter of scientific debate. [Maréchal, G., Plaghki, L., 1979. The deficit of the isometric tetanic tension redeveloped after a release of frog muscle at a constant velocity. J. Gen. Physiol. 73, 453–467] suggested that force depression might be caused by alterations in the cross-bridge kinetics following muscle shortening, but there is no research studying force depression systematically for altered cross-bridge kinetic conditions. The purpose of this study was to investigate if force depression affects so-called weakly and strongly bound cross-bridges to the same degree. In order to achieve this aim, we modified the ratio of weakly to strongly bound cross-bridges with 2,3-butanedione monoxime (BDM) in single frog fibers. BDM inhibits the formation of strongly bound cross-bridges in a dose-dependent manner, thus the ratio of weakly to strongly bound cross-bridges could be altered in a systematic way. We found that the absolute amount of force depression was decreased by 50% while the relative amount was decreased by 12% in BDM exposed fibers compared to fibers in normal Ringer's solution. Furthermore, force depression was accompanied by a decrease in stiffness that was much greater in normal compared to BDM exposed fibers, leading to the conclusion that force depression was caused by an inhibition of cross-bridge attachment following fiber shortening and that this inhibition primarily affected cross-bridges in the strongly bound states.     

A new motor model representing the stretch-induced force enhancement and shortening-induced force depression in skeletal muscle

A motor model that consists of two Maxwell elements with a force generator and one Voigt element is proposed in this paper. The motor model can achieve a hyperbolic force-velocity relation when we alter weight functions applied to the Maxwell elements and the force generator. Rate coefficients are introduced to determine the weight function and to improve the motor performance and the time course of the motor force. The weight functions are used as a controller of the motor. We assume that the mechanical impulse applied to the motor affects the rate coefficients and found that the amount of the mechanical impulse is related to the amount of force depression following motor shortening and to the amount of force enhancement following motor stretching. The time courses of the motor force following shortening and stretching quantitatively resemble those in other muscle experiments. The maximum energy efficiency of the motor that we obtained was 50% with an ATP hydrolysis type and 25% with an AC-DC motor type.