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.     

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.     

An activatable molecular spring reduces muscle tearing during extreme stretching

The purpose of this study was to determine failure stresses and failure lengths of actively and passively stretched myofibrils. As expected, myofibrils failed at average sarcomere lengths (about 6–7 μm) that vastly exceeded sarcomere lengths at which actin–myosin filament overlap ceases to exist (4 μm) and thus actin–myosin-based cross-bridge forces are zero at failure. Surprisingly, however, actively stretched myofibrils had much greater failure stresses and failure energies than passively stretched myofibrils, thereby providing compelling evidence for strong force production independent of actin–myosin-based cross-bridge forces. Follow-up experiments in which titin was deleted and cross-bridge formation was inhibited at high and low calcium concentrations point to titin as the regulator of this force, independent of calcium. The results of this study point to a mechanism of force production that reduces stretch-induced muscle damage at extreme length and limits injury and force loss within physiologically relevant ranges of sarcomere and muscle lengths.     

The stiffness of skeletal muscle in isometric contraction and rigor: the fraction of myosin heads bound to actin.

Step changes in length (between -3 and +5 nm per half-sarcomere) were imposed on isolated muscle fibers at the plateau of an isometric tetanus (tension T0) and on the same fibers in rigor after permeabilization of the sarcolemma, to determine stiffness of the half-sarcomere in the two conditions. To identify the contribution of actin filaments to the total half-sarcomere compliance (C), measurements were made at sarcomere lengths between 2.00 and 2.15 microm, where the number of myosin cross-bridges in the region of overlap between the myosin filament and the actin filament remains constant, and only the length of the nonoverlapped region of the actin filament changes with sarcomere length. At 2.1 microm sarcomere length, C was 3.9 nm T0(-1) in active isometric contraction and 2.6 nm T0(-1) in rigor. The actin filament compliance, estimated from the slope of the relation between C and sarcomere length, was 2.3 nm microm(-1) T0(-1). Recent x-ray diffraction experiments suggest that the myosin filament compliance is 1.3 nm microm(-1) T0(-1). With these values for filament compliance, the difference in half-sarcomere compliance between isometric contraction and rigor indicates that the fraction of myosin cross-bridges attached to actin in isometric contraction is not larger than 0.43, assuming that cross-bridge elasticity is the same in isometric contraction and rigor.     

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.     

Filament compliance effects can explain tension overshoots during force development

Spatially explicit stochastic simulations of myosin S1 heads attaching to a single actin filament were used to investigate the process of force development in contracting muscle. Filament compliance effects were incorporated by adjusting the spacing between adjacent actin binding sites and adjacent myosin heads in response to cross-bridge attachment/detachment events. Appropriate model parameters were determined by multi-dimensional optimization and used to simulate force development records corresponding to different levels of Ca(2+) activation. Simulations in which the spacing between both adjacent actin binding sites and adjacent myosin S1 heads changed by approximately 0.06 nm after cross-bridge attachment/detachment events 1), exhibited tension overshoots with a Ca(2+) dependence similar to that measured experimentally and 2), mimicked the observed k(tr)-relative tension relationship without invoking a Ca(2+)-dependent increase in the rate of cross-bridge state transitions. Tension did not overshoot its steady-state value in control simulations modeling rigid thick and thin filaments with otherwise identical parameters. These results underline the importance of filament geometry and actin binding site availability in quantitative theories of muscle contraction.     

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.     

Non-specific termination of simian virus 40 DNA replication.

Axial X-ray diffraction patterns have been studied from relaxed, contracted and rigor vertebrate striated muscles at different sarcomere lengths to determine which features of the patterns depend on the interaction of actin and myosin. The intensity of the myosin layer lines in a live, relaxed muscle is sometimes less in a stretched muscle than in the muscle at rest-length; the intensity depends not only on the sarcomere length but on the time that has elapsed since dissection of the muscle. The movement of cross-bridges giving rise to these intensity changes are not caused solely by the withdrawal of actin from the A-band.When a muscle contracts or passes into rigor many changes occur that are independent of the sarcomere length: the myosin layer lines decrease in intensity to about 30% of their initial value when the muscle contracts, and disappear completely when the muscle passes into rigor. Both in contracting and rigor muscles at all sarcomere lengths the spacings of the meridional reflections at 143 Å and 72 Å are 1% greater than from a live relaxed muscle at rest-length. It is deduced that the initial movement of cross-bridges from their positions in resting muscle does not depend on the interaction of each cross-bridge with actin, but on a conformational change in the backbone of the myosin filament: occurring as a result of activation. The possibility is discussed that the conformational change occurs because the myosin filament, like the actin filament, has an activation control mechanism. Finally, all the X-ray diffraction patterns are interpreted on a model in which the myosin filament can exist in one of two possible states: a relaxed state which gives a diffraction pattern with strong myosin layer lines and an axial spacing of 143.4 Å, and an activated state which gives no layer lines but a meridional spacing of 144.8 Å.     

The length–tension curve in muscle depends on lattice spacing

Classic interpretations of the striated muscle length–tension curve focus on how force varies with overlap of thin (actin) and thick (myosin) filaments. New models of sarcomere geometry and experiments with skinned synchronous insect flight muscle suggest that changes in the radial distance between the actin and myosin filaments, the filament lattice spacing, are responsible for between 20% and 50% of the change in force seen between sarcomere lengths of 1.4 and 3.4 µm. Thus, lattice spacing is a significant force regulator, increasing the slope of muscle''s force–length dependence.     

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.     

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 evidence for the extensibility of actin and myosin filaments during muscle contraction.

To clarify the extensibility of thin actin and thick myosin filaments in muscle, we examined the spacings of actin and myosin filament-based reflections in x-ray diffraction patterns at high resolution during isometric contraction of frog skeletal muscles and steady lengthening of the active muscles using synchrotron radiation as an intense x-ray source and a storage phosphor plate as a high sensitivity, high resolution area detector. Spacing of the actin meridional reflection at approximately 1/2.7 nm-1, which corresponds to the axial rise per actin subunit in the thin filament, increased about 0.25% during isometric contraction of muscles at full overlap length of thick and thin filaments. The changes in muscles stretched to approximately half overlap of the filaments, when they were scaled linearly up to the full isometric tension, gave an increase of approximately 0.3%. Conversely, the spacing decreased by approximately 0.1% upon activation of muscles at nonoverlap length. Slow stretching of a contracting muscle increased tension and increased this spacing over the isometric contraction value. Scaled up to a 100% tension increase, this corresponds to a approximately 0.26% additional change, consistent with that of the initial isometric contraction. Taken together, the extensibility of the actin filament amounts to 3-4 nm of elongation when a muscle switches from relaxation to maximum isometric contraction. Axial spacings of the layer-line reflections at approximately 1/5.1 nm-1 and approximately 1/5.9 nm-1 corresponding to the pitches of the right- and left-handed genetic helices of the actin filament, showed similar changes to that of the meridional reflection during isometric contraction of muscles at full overlap. The spacing changes of these reflections, which also depend on the mechanical load on the muscle, indicate that elongation is accompanied by slight changes of the actin helical structure possibly because of the axial force exerted by the actomyosin cross-bridges. Additional small spacing changes of the myosin meridional reflections during length changes applied to contracting muscles represented an increase of approximately 0.26% (scaled up to a 100% tension increase) in the myosin periodicity, suggesting that such spacing changes correspond to a tension-related extension of the myosin filaments. Elongation of the myosin filament backbone amounts to approximately 2.1 nm per half sarcomere. The results indicate that a large part (approximately 70%) of the sarcomere compliance of an active muscle is caused by the extensibility of the actin and myosin filaments; 42% of the compliance resides in the actin filaments, and 27% of it is in the myosin filaments.     

Thick-to-Thin Filament Surface Distance Modulates Cross-Bridge Kinetics in Drosophila Flight Muscle

The demembranated (skinned) muscle fiber preparation is widely used to investigate muscle contraction because the intracellular ionic conditions can be precisely controlled. However, plasma membrane removal results in a loss of osmotic regulation, causing abnormal hydration of the myofilament lattice and its proteins. We investigated the structural and functional consequences of varied myofilament lattice spacing and protein hydration on cross-bridge rates of force development and detachment in Drosophila melanogaster indirect flight muscle, using x-ray diffraction to compare the lattice spacing of dissected, osmotically compressed skinned fibers to native muscle fibers in living flies. Osmolytes of different sizes and exclusion properties (Dextran T-500 and T-10) were used to differentially alter lattice spacing and protein hydration. At in vivo lattice spacing, cross-bridge attachment time (t_on) increased with higher osmotic pressures, consistent with a reduced cross-bridge detachment rate as myofilament protein hydration decreased. In contrast, in the swollen lattice, t_on decreased with higher osmotic pressures. These divergent responses were reconciled using a structural model that predicts t_on varies inversely with thick-to-thin filament surface distance, suggesting that cross-bridge rates of force development and detachment are modulated more by myofilament lattice geometry than protein hydration. Generalizing these findings, our results suggest that cross-bridge cycling rates slow as thick-to-thin filament surface distance decreases with sarcomere lengthening, and likewise, cross-bridge cycling rates increase during sarcomere shortening. Together, these structural changes may provide a mechanism for altering cross-bridge performance throughout a contraction-relaxation cycle.     

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.     

Cross-bridge attachment during high-speed active shortening of skinned fibers of the rabbit psoas muscle: implications for cross-bridge action during maximum velocity of filament sliding

To characterize the kinetics of cross-bridge attachment to actin during unloaded contraction (maximum velocity of filament sliding), ramp-shaped stretches with different stretch-velocities (2-40,000 nm per half-sarcomere per s) were applied to actively contracting skinned fibers of the rabbit psoas muscle. Apparent fiber stiffness observed during such stretches was plotted versus the speed of the imposed stretch (stiffness-speed relation) to derive the rate constants for cross-bridge dissociation from actin. The stiffness-speed relation obtained for unloaded shortening conditions was shifted by about two orders of magnitude to faster stretch velocities compared to isometric conditions and was almost identical to the stiffness-speed relation observed in the presence of MgATPgammaS at high Ca(2+) concentrations, i.e., under conditions where cross-bridges are weakly attached to the fully Ca(2+) activated thin filaments. These data together with several control experiments suggest that, in contrast to previous assumptions, most of the fiber stiffness observed during high-speed shortening results from weak cross-bridge attachment to actin. The fraction of strongly attached cross-bridges during unloaded shortening appears to be as low as some 1-5% of the fraction present during isometric contraction. This is about an order of magnitude less than previous estimates in which contribution of weak cross-bridge attachment to observed fiber stiffness was not considered. Our findings imply that 1) the interaction distance of strongly attached cross-bridges during high-speed shortening is well within the range consistent with conventional cross-bridge models, i.e., that no repetitive power strokes need to be assumed, and 2) that a significant part of the negative forces that limit the maximum speed of filament sliding might originate from weak cross-bridge interactions with actin.     

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.     

Thin filament length regulation in striated muscle sarcomeres: pointed-end dynamics go beyond a nebulin ruler

The actin (thin) filaments in striated muscle are highly regulated and precisely specified in length to optimally overlap with the myosin (thick) filaments for efficient myofibril contraction. Here, we review and critically discuss recent evidence for how thin filament lengths are controlled in vertebrate skeletal, vertebrate cardiac, and invertebrate (arthropod) sarcomeres. Regulation of actin polymerization dynamics at the slow-growing (pointed) ends by the capping protein tropomodulin provides a unified explanation for how thin filament lengths are physiologically optimized in all three muscle types. Nebulin, a large protein thought to specify thin filament lengths in vertebrate skeletal muscle through a ruler mechanism, may not control pointed-end actin dynamics directly, but instead may stabilize a large core region of the thin filament. We suggest that this stabilizing function for nebulin modifies the lengths primarily specified by pointed-end actin dynamics to generate uniform filament lengths in vertebrate skeletal muscle. We suggest that nebulette, a small homolog of nebulin, may stabilize a correspondingly shorter core region and allow individual thin filament lengths to vary according to working sarcomere lengths in vertebrate cardiac muscle. We present a unified model for thin filament length regulation where these two mechanisms cooperate to tailor thin filament lengths for specific contractile environments in diverse muscles.     

Myosin filament sliding through the Z-disc relates striated muscle fibre structure to function

Striated muscle contraction requires intricate interactions of microstructures. The classic textbook assumption that myosin filaments are compressed at the meshed Z-disc during striated muscle fibre contraction conflicts with experimental evidence. For example, myosin filaments are too stiff to be compressed sufficiently by the muscular force, and, unlike compressed springs, the muscle fibres do not restore their resting length after contractions to short lengths. Further, the dependence of a fibre''s maximum contraction velocity on sarcomere length is unexplained to date. In this paper, we present a structurally consistent model of sarcomere contraction that reconciles these findings with the well-accepted sliding filament and crossbridge theories. The few required model parameters are taken from the literature or obtained from reasoning based on structural arguments. In our model, the transition from hexagonal to tetragonal actin filament arrangement near the Z-disc together with a thoughtful titin arrangement enables myosin filament sliding through the Z-disc. This sliding leads to swivelled crossbridges in the adjacent half-sarcomere that dampen contraction. With no fitting of parameters required, the model predicts straightforwardly the fibre''s entire force–length behaviour and the dependence of the maximum contraction velocity on sarcomere length. Our model enables a structurally and functionally consistent view of the contractile machinery of the striated fibre with possible implications for muscle diseases and evolution.     

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.