Cross-bridge Model Research Articles

The cross-bridge cycle in muscle. Mechanical, biochemical, and structural studies on single skinned rabbit psoas fibers to characterize cross-bridge kinetics in muscle for correlation with the actomyosin-ATPase in solution.

A characteristic and important feature of myocardium is the modulation of tension when stimulated or possibly even when unstimulated. In addition, resistance to stretch and its variation in unstimulated heart muscle is an important factor in myocardial function. These features may occur in some new light when viewed from some recent advances in understanding of cross-bridge action and regulation of muscle. For this reason we give a short review of such advances. Firstly, we summarize some of our earlier results obtained in experiments designed to see whether and to what extent actomyosin ATPase data obtained in solution might apply in muscle. Secondly, we present a recently developed experimental approach to estimate the rate constants that determine the cycling of cross-bridges between weak-binding, 'non-force-generating' states and strong-binding, 'force-generating' states. The estimated rate constants confirm the prediction of cross-bridge models derived from in vitro studies that the step which is rate-limiting in solution also determines the rate of force-generation in the cross-bridge cycle in muscle. Experiments at various Ca++ concentrations imply that a major mechanism of regulation is the control of the transition from the weak-binding, 'non-force-generating' states to the strong-binding, 'force-generating' states while the number of activated interaction sites appears unchanged and always at its maximum. This implies that changes in the force-pCa relation cannot be interpreted without detailed analysis of cross-bridge kinetics, and that factors other than Ca++ may have the potential to modulate muscle activity, both in stimulated and unstimulated muscle, by affecting cross-bridge kinetics.

Effects of amrinone on the contractile behaviour of frog striated muscle fibres.

The effects of amrinone (0.005-2.1 mM) were studied on isolated single muscle fibres of the frog (0.6-4.3 degrees C). In the concentrations used amrinone potentiated the isometric twitch, increased the initial rate of force development and prolonged the time to peak twitch force and the time from peak force to half relaxation. There was a moderate but statistically significant (P less than 0.001) enhancement of the maximum tetanic force and a marked slowing of the relaxation phase of the tetanus. Amrinone depressed the velocity of unloaded shortening by 33 +/- 2% (mean +/- SE, n = 6) and reduced the curvature of the force-velocity relation leading to an increase of a/Po in Hill's (1938) hyperbolic equation. In accordance with the effects of amrinone on the force-velocity relation, there was a decrease in the rate of redevelopment of force after the fibre was released to zero tension during the tetanus plateau. The evidence suggests that the drug affects the contractile properties of frog muscle fibres by two different mechanisms: (I) amrinone affects the excitation-contraction coupling causing enhanced calcium release in response to membrane excitation and (2) amrinone exerts a direct effect on the kinetics of turnover of the myosin cross-bridges. The latter aspect of the drug effect was evaluated in terms of A.F. Huxley's (1957) cross-bridge model by means of a computer program.

Dissociation between mechanical performance and the cost of isometric tension maintenance in Lethocerus flight muscle.

A method for the continuous measurement of ATP hydrolysis (ATPase) by demembranated muscle fibres has been applied to isometrically held, glycerol-extracted flight muscle fibres from the water-bug Lethocerus, under conditions of high MgATP, neutral pH, and varying ionic strength, Ca2+ and extension. These variables caused parallel changes in isometric tension and ATPase. The slope of ATPase upon tension (delta ATPase/delta Tension; incremental tension cost) remained the same upon extension at either different Ca2+ concentrations or different ionic strength. Isometric activation by Ca2+ gave a higher incremental tension cost. The calculated mechanical rate constant of the work-producing process, measured by the small-amplitude behaviour, was increased by either Ca2+ or ionic strength, and little changed by extension; there was therefore a dissociation between its value and that of the incremental tension cost. The results appear to exclude a two-state crossbridge model for fibrillar insect flight muscle.

Intensification of the 5.9-nm actin layer line in contracting muscle.

According to the cross-bridge model of muscle contraction, an interaction of myosin heads with interdigitating actin filaments produces tension. Although X-ray equatorial diffraction patterns of active (contracting) muscle show that the heads are in the vicinity of the actin filaments, structural proof of actual attachment of heads to actin during contraction has been elusive. We show here that during contraction of frog skeletal muscle, the 5.9-nm layer line arising from the genetic helix of actin is intensified by as much as 56% of the change which occurs when muscle enters rigor, using a two-dimensional X-ray detector. This provides strong structural evidence that myosin heads do in fact attach during contraction.

Contraction of rabbit skinned skeletal muscle fibers at low levels of magnesium adenosine triphosphate

The contractile properties of skinned single fibers from rabbit psoas muscle were investigated under conditions of low MgATP and no Ca2+ (i.e., less than 10(-8) M). At 1 microM MgATP, fibers shortened at a maximum velocity of 660 +/- 420 A/half sarcomere/s (n = 9), compared with 34,000 A/half sarcomere/s measured during maximum Ca2+-activation at 1 mM MgATP (Moss, R. L., 1982. J. Muscle Res. Cell. Motil ., 3:295-311). The observed dependence of Vmax on pMgATP between 7.0 and 5.3 was similar to that of actomyosin ATPase measured previously by Weber, A., R. Herz , and I. Reiss (1969, Biochemistry, 8:2266-2270). Isometric tension was found to vary with pMgATP in a manner much like that reported by Reuben , J. P., P. W. Brandt, M. Berman , and H. Grundfest (J. Gen. Physiol. 1971. 57:385-407). A simple cross-bridge model was developed to simulate contractile behaviour at both high and low levels of MgATP. It was found that the pMgATP dependence of Vmax and ATPase could be successfully modeled if the rate of detachment of the cross-bridge was made proportional to the concentration of MgATP. In the model, the similar dependence of Vmax and ATPase on pMgATP was derived from the fact that in this range of pMgATP every pass of a cross-bridge by an actin site resulted in an attachment-detachment cycle, and every such cycle caused hydrolysis of one molecule of ATP.

Force-velocity relation and stiffness in frog single muscle fibres during the rise of tension in an isometric tetanus.

The force-velocity (T-V) relation and the force-extension (T1) relation are determined at present times and at increasing isometric tensions during a tetanic contraction in frog single muscle fibres in which the passive compliance in series with the sarcomeres was made very small. The slope of the instantaneous T1 relation, the fibre stiffness, increases roughly in proportion to the level of the rising isometric tension at which the measurements were made. The value of V0 (the velocity of shortening under zero load) is time-independent, whereas the force T exerted during shortening at any velocity V lower than V0 increases gradually with time after the beginning of the tetanus volley and attains its steady state level before the isometric tension has attained the tetanus plateau and the fibre stiffness its final value. It is concluded that the delay of the development of the isometric tension and of the fibre stiffness with respect to the development of the T-V relation is determined by a specific factor of the contractile process. It is interesting to note that in a cross-bridge model of contraction, in which the value of the rate constant for cross-bridge formation is moderate, the recruitment of actin sites which is measured by the characteristics of the instantaneous T-V relation, is expected to lead significantly the actual cross-bridge formation, which is measured both by the instantaneous isometric tension and by the instantaneous stiffness.

The influence of velocity of length change on tension development in skeletal muscle: Model calculations and experimental results

Force responses obtained during constant velocity length changes on skeletal muscle tissue are simulated by means of two cross-bridge models proposed by Huxley and Simmons (1971, Nature 233, 533–538) and by Julian et al. (1974, Biophys. J. 14 546–562). An implicit method was used for the numerical approximation in the simulations. The simulated force transients due to constant velocity length changes are found to be in qualitative agreement with re-investigated experimental results obtained from the whole sartorius muscle of the frog. A non-linear tension transient is observed, dependent both on amplitude and on velocity of release revealing an inflexion which gives the transient a shoulder shape. When velocity is increased the inflexion occurs earlier and at a lower tension value. A non-linear transient is observed during stretches performed at moderate velocities. Force responses are found to deviate concavely downwards from a linear time course. Simulations, however, predict a rather linear tension transient for comparable velocities. Implications of the experimental findings are discussed for both models.

The role of magnesium adenosine triphosphate in the contractile kinetics of insect fibrillar flight muscle.

The changes in tension produced by small step or sinusoidal changes of length have been measured for chemically skinned flight muscle fibres of the giant tropical water bug Lethocerus at MgATP concentrations in the range 0.1-10 mM. In the presence of calcium ion concentrations of about 20 microM, the rates of the rapid mechanical processes observed were found to increase with increasing MgATP, exhibiting saturation with an apparent half-saturation constant lying between 0.1 and 1 mM MgATP, depending upon the conditions used. Under the same conditions, an increase in MgATP concentration was found to lead to a slight decrease in the isometric tension generated by the preparations. The results are discussed with reference to some current crossbridge models of muscle contraction.

Organization of contractile fibers in smooth muscle.

The smooth muscle of vertebrates is chiefly a mural tissue of the large and small hollow organ systems. The “smoothness” of this musculature presumably describes the impression of a uniformity in the optical properties of the cytoplasm that contrasts with the regular optical anisotropy that is characteristic of the striated muscles. The quality of optical uniformity is obtained from a labyrinth of three distinct systems of cytoplasmic filaments that comprise the bulk of the contractile apparatus, and a major objective in contemporary morphological analysis of smooth muscle is to elucidate the molecular organization of the constituent filaments and define the symmetry of the functional elements. Structural studies have been undertaken primarily with the view of establishing correlations between the composition and organization of the filaments in smooth muscle fibers and the structural composition of the sarcomere in striated muscle (Shoenberg and Needham, 1976; Chamley-Camp- bell et al., 1979; Small and Sobieszek, 1980; Somlyo, 1980). The basis for this approach is provided by some quantitative and especially qualitative similarities in the traditionally defined functional characteristics shared by smooth and striated muscle, and by the remarkable correlations between structural changes and functional variation that are formulated or otherwise implicit in the sliding filament concept and the cross-bridge model of contraction for the striated muscles.

A sliding-filament cross-bridge ensemble model of muscle contraction for mechanical transients

A high-efficiency mechanochemical model of contraction for striated muscle is developed via the formalism of statistical mechanics. The myosin cross-bridges of the half-sarcomere ensemble cycle through five biochemical states. The structural components of the sliding-filament system include extensible myosin S2-units and indefinite arrays of equivalent actin sites. ATP ligand exchanges provide the far-from-equilibrium thermodynamic driving potential for cross-bridge cycling. The strain-dependent rate constants obey the self-consistency requirements of detailed balance. The mathematical solution for the linear time-varying system of equations of cross-bridge biokinetics employs discrete-time state transition matrices. Computer simulations describe the time evolution of ultrastructural (macromolecular) phenomena and their ensemble (macroscopic) averages. Modest fidelity for mechanical transients of muscle under many varied protocols, including conditions of fatique, is obtained between the simulations and their experimental counterpart.

Calcium model for mammalian skeletal muscle.

A model is presented describing quantitatively the events between excitation and force development in skeletal muscle. It consists of a calcium mediated activation model (c.m.a.m.) in series with a force generator model (f.g.m.). The c.m.a.m. was based on intracellular processes such as cisternal Ca-release, Ca-troponin interaction and Ca-uptake by SR. Ca-troponin complex concentration is an output of the c.m.a.m. and input of the f.g.m., the latter being a simplified actinmyosin cross-bridge model. Simulated and measured forces were compared for the rat slow (soleus) muscle. With the present structure and parameter values there is reasonable agreement between simulated and measured forces for single- and double-pulse responses. The parameters of the f.g.m. mainly determine the rising phase of the twitch since the Ca-release is brief and the binding of Ca to troponin is fast. In relaxation the Ca-troponin interaction and the Ca transporting ATPase parameters are also important. The behaviour of the muscle after a second action potential depends strongly on the level of Ca occupancy for both the troponin and Ca transporting ATPase.

Stepwise sarcomere shortening: analysis by high-speed cinemicrography.

Sarcomere shortening in striated muscle appears to follow a regionally synchronized staircase-like time course not anticipated in some cross-bridge models. The visualization method used has been criticized as subject to Bragg diffraction effects. Two independent optical methods were used to visualize a muscle during contraction; agreement between the stepwise behavior observed with the two methods suggests that the phenomenon is genuine.

Force-velocity relation in deuterium oxide-treated frog single muscle fibres during the rise of tension in an isometric tetanus.

1. The force-velocity (P-V) relation from a single fibres isolated from the semitendinosus muscle of the frog was determined at pre-set times during the rise of tension and the plateau of isometric tetani. The controlled-velocity release method was used. Experiments were performed at about 2.25 micrometers sarcomere length and at 3-4 degrees C or at 19-21 degrees C. 2. Replacing H2O with D2O resulted in a rapid large reduction of the peak twitch tension and of the speed of development of twitch and tetanic tensions. The tetanic tension (P0) was usually reduced, in certain fibres to as low as 5% of the value in H2O-Ringer solution. 3. The depression of twitch and tetanus characteristics was followed by a recovery, the duration of which varied greatly in different fibres. During the recovery period previous conditioning activity potentiated the tetanus characteristics. 4. After the end of the recovery period in D2O-Ringer solution both the peak twitch tension and the speed of development of tetanic tension was still greatly depressed, whereas the value of P0 was slightly greater than in H2O-Ringer. The speed of rise of isometric tension after a quick release imposed at the tetanus plateau was reduced in D2O-Ringer, usually to about 50% of the value in H2O-Ringer. 5. D2O increased the development time of the P-V relation and produced a conspicuous increase in the degree of its curvature. The value of V0 (the velocity of shortening at zero load) was not significantly depressed by D2O and it was the same independent both of the time after the beginning of stimulation and of the isometric tension at which the measurement was made. The P-V relation attained its final characteristics before the isometric tension reached the plateau. During the recovery period in D2O-Ringer, at the plateau of isometric tetani of different size, the relative force exerted at a given velocity of shortening was constant. 6. In D2O-treated fibres, NO3- and caffeine (i) potentiated the peak twitch tension and the speed of development of tetanic tension without affecting significantly the speed of the redevelopment of tension after a quick release imposed at the tetanus plateau and (ii) reduced the development time of the P-V relation, but did not affect either the degree of its curvature or the value of V0 and P0. 7. The results are discussed by assuming that the release of Ca2+ from the sarcoplasmic reticulum is a rate-limiting process for the development of activation and in turn for the development of isometric tension. In terms of the cross-bridge model of Huxley (1957), the time or Ca2+-dependent factor of activation appears to be the recruitment of actin sites for cross-bridge formation, whereas the value of the rate constants regulating the cross-bridge kinetics appears to be time and Ca2+-independent.

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.

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.

Cross-bridge model of muscle contraction. Quantitative analysis

We recently presented, in a qualitative manner, a cross-bridge model of muscle contraction which was based on a biochemical kinetic cycle for the actomyosin ATPase activity. This cross-bridge model consisted of two cross-bridge states detached from actin and two cross-bridge states attached to actin. In the present paper, we attempt to fit this model quantitatively to both biochemical and physiological data. We find that the resulting complete cross-bridge model is able to account reasonably well for both the isometric transient data observed when a muscle is subjected to a sudden change in length and for the relationship between the velocity of muscle contraction in vivo and the actomyosin ATPase activity in vitro. This model also illustrates the interrelationship between biochemical and physiological data necessary for the development of a complete cross-bridge model of muscle contraction.

Cross-bridge behavior in rigor muscle.

Properties of the rigor state in muscle can be explained by a simple cross-bridge model, of the type which has been suggested for active muscle, in which detachment of cross-bridges by ATP is excluded. Two attached cross-bridge states, with distinct force vs. distortion relationships, are required, in addition to a detached state, but the attached cross-bridge states in rigor muscle appear to differ significantly from the attached cross-bridge states in active muscle. The stability of the rigor force maintained in muscle under isometric conditions does not require exceptional stability of the attached cross-bridges, if the positions in which attachment of cross-bridges is allowed are limited so that the attachment of cross-bridges in positions which have minimum free energy is excluded. This explanation of the stability of the rigor state may also be applicable to the maintenance of stable rigor waves on flagella.

Coupling between the enzymatic site of myosin and the mechanical output of muscle

When Mg-imido-ATP binds to the active site of myosin in glycerol-extracted insect muscle fibres it causes a rapid, fully reversible and stress-independent increase in rest length of 2 nm per half sarcomere, whilst the isotonic stiffness remains within 2% of the rigor value. Mg-ADP and H-imido-ATP bind but have less mechanical effect and also less effect on the equatorial X-ray diffraction pattern. Coupling of binding to length change is quantitatively reversible, since stress of 200 nm per fibre doubles the amount of either Mg-imido-ATP or Mg-ATP bound at low concentrations but has no effect on Mg-ADP or H-imido-ATP binding. Linkage between the nucleotide and actin binding sites is shown by studies with subfragment-1 in solution. By fluorometric titration the nucleotide affinity for subfragment-1 was shown to increase in the order of H-imido-ATP, Mg-ADP and Mg-imido-ATP, while by a sedimentation technique the affinity of actin for subfragment-1-nucleotide was shown to decrease in the same order. The data are interpreted in terms of a simple cross-bridge model and their relevance to the mechanism of muscle contraction is discussed.

The sarcomere length-tension relation in skeletal muscle.

Tension development during isometric tetani in single fibers of frog semitendinosus muscle occurs in three phases: (a) in initial fast-rise phase; (b) a slow-rise phase; and (c) a plateau, which lasts greater than 10 s. The slow-rise phase has previously been assumed to rise out of a progressive increase of sarcomere length dispersion along the fiber (Gordon et al. 1966. J. Physiol. [Lond.]. 184:143--169;184:170--192). Consequently, the "true" tetanic tension has been considered to be the one existing before the onset of the slow-rise phase; this is obtained by extrapolating the slowly rising tension back to the start of the tetanus. In the study by Gordon et al. (1966. J. Physiol. [Lond.] 184:170--192), as well as in the present study, the relation between this extrapolated tension and sarcomere length gave the familiar linear descending limb of the length-tension relation. We tested the assumption that the slow rise of tension was due to a progressive increase in sarcomere length dispersion. During the fast rise, the slow rise, and the plateau of tension, the sarcomere length dispersion at any area along the muscle was less than 4% of the average sarcomere length. Therefore, a progressive increase of sarcomere length dispersion during contraction appears unable to account for the slow rise of tetanic tension. A sarcomere length-tension relation was constructed from the levels of tension and sarcomere length measured during the plateau. Tension was independent of sarcomere length between 1.9 and 2.6 microgram, and declined to 50% maximal at 3.4 microgram. This result is difficult to reconcile with the cross-bridge model of force generation.

A cross-bridge model for inotropism as revealed by stiffness measurements in cardiac muscle

Stiffness measurements obtained by means of rapid length changes performed according to Huxley and Simmons (23) showed that the series elasticity of the living frog myocardium obeys Hooke's law and alters in proportion to isometric tension. The same results had previously been reported for glycerol-extracted heart muscle (15, 16). Under conditions of postive inotropism caused by application of noradrenaline, adrenaline or increased extracellular Ca++ concentration, the proportionality between tension and stiffness is maintained (13). As there is strong evidence that the series elasticity of heart muscle resides in the cross-bridges (17, 24) this means that systolic force development and positive inotropism are due to the same process, namely a recruitment of "activated" cross-bridges (an increase in the number of cross-bridges attached to actin at any moment). This rules out the two-component model proposed by Sonnenblick in which a non-linear series elastic element was postulated.