The variation of isometric energy rates with muscle length: a distribution-moment model analysis.

The Distribution-Moment Model of skeletal muscle, which has been enhanced recently to make possible the calculation of chemical energy release (E) and heat production (H) rates [1], is applied to isometric muscle. Under steady-state isometric conditions the model predicts a simple relation between the energy rates and the muscle length, namely (E/Emax) = (H/Hmax) = [1 + B alpha(symbol see text)]/[1 + B], where (symbol see text) is the ratio of muscle length to the "optimal" length at which maximal isometric tension is produced, and alpha (symbol see text) is a function numerically equal to the ratio of the tetanic isometric force to its maximum value. The single dimensionless constant in this relation, B, can be calculated from model parameters characterizing muscle dynamics at the optimum length, and has a value near unity for frog sartorius at 0 degrees C. The predicted behavior is shown to agree reasonably well with experimental measurements of heat production and phosphocreatine (PCr) hydrolysis. The model relates the isometric energy rates to PCr hydrolysis in (1) cross-bridge interactions, and (2) calcium pumping into the sarcoplasmic reticulum.     

A cross-bridge model that is able to explain mechanical and energetic properties of shortening muscle.

The responses of muscle to steady and stepwise shortening are simulated with a model in which actin-myosin cross-bridges cycle through two pathways distinct for the attachment-detachment kinetics and for the proportion of energy converted into work. Small step releases and steady shortening at low velocity (high load) favor the cycle implying approximately 5 nm sliding per cross-bridge interaction and approximately 100/s detachment-reattachment process; large step releases and steady shortening at high velocity (low load) favor the cycle implying approximately 10 nm sliding per cross-bridge interaction and approximately 20/s detachment-reattachment process. The model satisfactorily predicts specific mechanical properties of frog skeletal muscle, such as the rate of regeneration of the working stroke as measured by double-step release experiments and the transition to steady state during multiple step releases (staircase shortening). The rate of energy liberation under different mechanical conditions is correctly reproduced by the model. During steady shortening, the relation of energy liberation rate versus shortening speed attains a maximum (approximately 6 times the isometric rate) for shortening velocities lower than half the maximum velocity of shortening and declines for higher velocities. In addition, the model provides a clue for explaining how, in different muscle types, the higher the isometric maintenance heat, the higher the power output during steady shortening.     

Modelling concentric contraction of muscle using an improved cross-bridge model.

Despite its overwhelming acceptance in muscle research, the cross-bridge theory does not account for all phenomena observed during muscular contractions. A phenomenon which has received much attention in the biomechanics literature, but has evaded convincing explanation and is not accounted for in the formulation of the classic cross-bridge theory, is the persistent aftereffects of muscular length changes on force production. For example, following muscle shortening, the isometric force of a muscle is depressed for a long time period ( > 5 s) compared to the corresponding isometric force following no length change. In the present study, the classic cross-bridge model was modified in two ways in an attempt to account for the force depressions following muscle shortening. First, the steady-state force depressions following shortening were described by a single scalar variable: the work performed by the muscle during shortening; and second, the dynamic, history-dependent cross-bridge properties were described using a fading memory function. The proposed model was developed and tested for shortening of the cat soleus at constant speeds ranging from 4 to 32 mm/s, for shortening at changing speeds, and for shortening of different magnitudes ranging from 2 to 10 mm. The history-dependent forces during shortening and the steady-state force depressions following shortening were well captured with the modified cross-bridge model. The present model contains two mathematically simple adaptations to the classic cross-bridge model, and is the first such model to account for the long-lasting force depressions following muscle shortening using a single scalar variable.     

Sustained isometric contraction of skeletal muscle depleted of phosphocreatine

To evaluate the need for phosphocreatine as an energy reservoir to sustain isometric contraction of skeletal muscle, rats were depleted of phosphocreatine by feeding β-GPA (β-guanidinopropionate) as 1% of the diet. In the place of phosphocreatine, β-GPAP (phosphorylated β-GPA) accumulated to concentrations of 20–25 μmoles/g wet weight of muscle. Although the maximum isometric tension produced by the soleus was always less than that produced by the plantaris muscle, the maximum for either muscle was not significantly affected by feeding β-GPA. The endurance of experimental soleus muscles was prolonged, however. These muscles held 70% of their maximum isometric tension for 106 ± 40 seconds (mean ± SD, n = 4) whereas the value for five control muscles was 43 ± 18 seconds. With fatiguing, isometric contractions of control plantaris and soleus muscles, phosphocreatine concentrations decreased by 68–70%; in experimental muscles, the β-GPA concentration decreased less than 12%. This difference in phosphagen consumption demonstrates that skeletal muscle can sustain fatiguing, isometric contractions without using large amounts of phosphocreatine or a substitute phosphagen as an energy reservoir. Phosphocreatine hydrolysis during muscle contraction normally may serve some other purpose.     

Strain-dependent modulation of phosphate transients in rabbit skeletal muscle fibers.

When inorganic phosphate (Pi) is photogenerated from caged Pi during isometric contractions of glycerinated rabbit psoas muscle fibers, the released Pi binds to cross-bridges and reverses the working stroke of cross-bridges. The consequent force decline, the Pi-transient, is exponential and probes the kinetics of the power-stroke and Pi release. During muscle shortening, the fraction of attached cross-bridges and the average strain on them decreases (Ford, L. E., A.F. Huxley, and R.M. Simmons, 1977. Tension responses to sudden length change in stimulated frog muscle fibers near slack length. J. Physiol. (Lond.). 269:441-515; Ford, L. E., A. F. Huxley, and R.M. Simmons, 1985. Tension transients during steady state shortening of frog muscle fibers. J. Physiol. (Lond.). 361:131-150. To learn to what extent the Pi transient is strain dependent, muscle fibers were activated and shortened or lengthened at a fixed velocity during the photogeneration of Pi. The Pi transients observed during changes in muscle length showed three primary characteristics: 1) during shortening the Pi transient rate, Kpi, increased and its amplitude decreased with shortening velocity; Kpi increased linearly with velocity to > 110 s-1 at 0.3 muscle lengths per second (ML/s). 2) At a specific shortening velocity, increases in [Pi] produce increases in Kpi that are nonlinear with [Pi] and approach an asymptote. 3) During forced lengthening Kpi and the amplitude of the Pi transient are little different from the isometric contractions. These data can be approximated by a strain-dependent three-state cross-bridge model. The results show that the power stroke's rate is strain-dependent, and are consistent with biochemical studies indicating that the rate-limiting step at low strains is a transition from a weakly to a strongly bound cross-bridge state.     

Myocardial cross-bridge kinetics in transition to failure in Dahl salt-sensitive rats

The role of altered cross-bridge kinetics during the transition from cardiac hypertrophy to failure is poorly defined. We examined this in Dahl salt-sensitive (DS) rats, which develop hypertrophy and failure when fed a high-salt diet (HS). DS rats fed a low-salt diet were controls. Serial echocardiography disclosed compensated hypertrophy at 6 wk of HS, followed by progressive dilatation and impaired function. Mechanical properties of skinned left ventricular papillary muscle strips were analyzed at 6 wk of HS and then during failure (12 wk HS) by applying small amplitude (0.125%) length perturbations over a range of calcium concentrations. No differences in isometric tension-calcium relations or cross-bridge cycling kinetics or mechanical function were found at 6 wk. In contrast, 12 wk HS strips exhibited increased calcium sensitivity of isometric tension, decreased frequency of minimal dynamic stiffness, and a decreased range of frequencies over which cross bridges produce work and power. Thus the transition from hypertrophy to heart failure in DS rats is characterized by major changes in cross-bridge cycling kinetics and mechanical performance.     

Force enhancement without changes in cross-bridge turnover kinetics: the effect of EMD 57033.

The thiadiazinon derivative EMD 57033 has been found previously in cardiac muscle to increase isometric force generation without a proportional increase in fiber ATPase, thus causing a reduction in tension cost. To analyze the mechanism by which EMD 57033 affects the contractile system, we studied its effects on isometric force, isometric fiber ATPase, the rate constant of force redevelopment (k(redev)), active fiber stiffness, and its effect on Fo, which is the force contribution of a cross-bridge in the force-generating states. We used chemically skinned fibers of the rabbit psoas muscle. It was found that with 50 microM EMD 57033, isometric force increases by more than 50%, whereas Kredev, active stiffness, and isometric fiber ATPase increase by at most 10%. The results show that EMD 57033 causes no changes in cross-bridge turnover kinetics and no changes in active fiber stiffness that would result in a large enough increase in occupancy of the force-generating states to account for the increase in active force. However, plots of force versus length change recorded during stretches and releases (T plots) indicate that in the presence of EMD 57033 the y(o) value (x axis intercept) for the cross-bridges becomes more negative while its absolute value increases. This might suggest a larger cross-bridge strain as the basis for increased active force. Analysis of T plots with and without EMD 57033 shows that the increase in cross-bridge strain is not due to a redistribution of cross-bridges among different force-generating states favoring states of larger strain. Instead, it reflects an increased cross-bridge strain in the main force-generating state. The direct effect of EMD 57033 on the force contribution of cross-bridges in the force-generating states represents an alternative mechanism for a positive inotropic intervention.     

A Novel Three-Filament Model of Force Generation in Eccentric Contraction of Skeletal Muscles

We propose and examine a three filament model of skeletal muscle force generation, thereby extending classical cross-bridge models by involving titin-actin interaction upon active force production. In regions with optimal actin-myosin overlap, the model does not alter energy and force predictions of cross-bridge models for isometric contractions. However, in contrast to cross-bridge models, the three filament model accurately predicts history-dependent force generation in half sarcomeres for eccentric and concentric contractions, and predicts the activation-dependent forces for stretches beyond actin-myosin filament overlap.     

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.     

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.     

Extra calcium on shortening in barnacle muscle. Is the decrease in calcium binding related to decreased cross-bridge attachment,force, or length?

Barnacle single muscle fibers were microinjected with the calcium-specific photoprotein aequorin. We have previously shown (Ridgway, E. B., and A. M. Gordon, 1984, Journal of General Physiology, 83:75-104) that when barnacle fibers are stimulated under voltage clamp and length control and allowed to shorten during the declining phase of the calcium transient, extra myoplasmic calcium is observed. The time course of the extra calcium for shortening steps at different times during the calcium transient is intermediate between those of free calcium and muscle force. Furthermore, the amplitude increases with an increased stimulus, calcium transient, and force. Therefore, the extra calcium probably comes from the activating sites on the myofilaments, possibly as a result of changes in calcium binding by the activating sites. The change in calcium binding may be due, in turn, to the change in muscle length and/or muscle force and/or cross-bridge attachment per se. In the present article, we show that the amount of the extra calcium depends on the initial muscle length, declining at shorter lengths. This suggests length-dependent calcium binding. The relation between initial length and extra calcium, however, parallels that between initial length and peak active force. The ratio of extra calcium to active force is therefore virtually independent of initial length. These data do not distinguish between a direct effect of length on calcium binding and an indirect effect owing to changes in cross-bridge attachment and force through some geometrical factor. The amount of extra calcium increases with the size of the shortening step, tending toward saturation for steps of greater than or equal to 10%. This experiment suggests that calcium binding depends on muscle force or cross-bridge attachment, not just length (if at all). There is much less extra calcium seen with shortening steps at high force when the high force results from stretch of the active muscle than when it results from increased stimulation of muscle.(ABSTRACT TRUNCATED AT 400 WORDS)     

Stretch of active muscle during the declining phase of the calcium transient produces biphasic changes in calcium binding to the activating sites

In voltage-clamped barnacle single muscle fibers, muscle shortening during the declining phase of the calcium transient increases myoplasmic calcium. This extra calcium is probably released from the activating sites by a change in affinity when cross-bridges break (Gordon, A. M., and E. B. Ridgway, 1987. J. Gen. Physiol. 90:321-340). Stretching the muscle at similar times causes a more complex response, a rapid increase in intracellular calcium followed by a transient decrease. The amplitudes of both phases increase with the rate and amplitude of stretch. The rapid increase, however, appears only when the muscle is stretched more than approximately 0.4%. This is above the length change that produces the breakpoint in the force record during a ramp stretch. This positive phase in response to large stretches is similar to that seen on equivalent shortening at the same point in the contraction. For stretches at different times during the calcium transient, the peak amplitude of the positive phase has a time course that is delayed relative to the calcium transient, while the peak decrease during the negative phase has an earlier time course that is more similar to the calcium transient. The amplitudes of both phases increase with increasing strength of stimulation and consequent force. When the initial muscle the active force. A large decrease in length (which drops the active force to zero) decreases the extra calcium seen on a subsequent restretch. After such a shortening step, the extra calcium on stretch recovers (50 ms half time) toward the control level with the same time course as the redeveloped force. Conversely, stretching an active fiber decreases the extra calcium on a subsequent shortening step that is imposed shortly afterward. Enhanced calcium binding due to increased length alone cannot explain our data. We hypothesize that the calcium affinity of the activating sites increases with cross-bridge attachment and further with cross-bridge strain. This accounts for the biphasic response to stretch as follows: cross-bridges detached by stretch first decrease calcium affinity, then upon reattachment increase calcium affinity due to the strained configuration brought on by the stretch. The experiments suggest that cross-bridge attachment and strain can modify calcium binding to the activating sites in intact muscle.     

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.     

Cross-bridge induced force enhancement?

When a muscle is stretched while activated, its steady-state isometric force following stretch is greater than the corresponding purely isometric force. This so-called residual force enhancement (RFE) has been observed for half a century, yet its mechanism remains unknown. Recent experiments suggest that RFE is not caused by non-uniformities in sarcomere lengths, as had been assumed for a long time, and cannot be explained primarily with increases in passive force, but is directly related to the kinetics of the cross-bridge cycle. Specifically, it has been suggested that stretching an attached cross-bridge increases its dwell time and duty ratio; therefore, the proportion of attached cross-bridges in a muscle would be increased by stretch, thereby causing RFE. A three bead laser trap setup was used for testing single cross-bridge (myosin II) interactions with actin. Upon attachment of a cross-bridge, a stretch or shortening of the cross-bridge was applied with a force of about 1.0 pN. The hypothesis that stretching a single cross-bridge increases its dwell time and duty ratio was rejected. However, stretching caused an increase in the average steady-state force per cross-bridge (3.4±0.4 pN; n=433) compared to shortening (1.9±0.3 pN; n=689). Therefore, based on the results of this study, RFE cannot be explained by an increased duty ratio and the associated increase in proportion of attached cross-bridges, but might be associated with an increased force per cross-bridge.     

Heat, mechanics, and myosin ATPase in normal and hypertrophied heart muscle

In this paper we review our previous work on the myothermic economy of isometric force production in compensated cardiac hypertrophy secondary to pulmonary artery constriction (pressure overload) and/or thyrotoxicosis (volume overload). Hypertrophy-induced changes in isotonic and isometric twitch mechanics are correlated with accompanying changes in actin-activated myosin ATPase and heat liberation. Heat measurements were made with rapid, high-sensitivity thermopiles on right ventricular papillary muscles from normal and hypertrophied rabbit hearts. Total activity-related heat was separated into initial and recovery heat. Initial heat was separated into a tension-dependent component (TDH) relating to cross-bridge activity, and a tension-independent component (TIH) relating to excitation-contraction coupling. There were oppositely directed changes in most parameters studied in pressure overload hypertrophy (P) as compared with thyrotoxic hypertrophy (T). Thus, in P there was depression (30-50% in the rate of isometric force production, mechanical Vmax, TDH and TDH rate, myosin ATPase, TIH, and prolongation in time-to-peak twitch tension, whereas in T all parameters were oppositely changed except for no change in TIH. Thyrotoxicosis following pressure overload reversed the P-induced changes in all parameters. There was a direct, linear relation between in vitro actin-activated myosin ATPase and in vivo TDH. However, TDH per unit twitch tension or tension-time integral varied inversely with ATPase, making force production more economical than normal in P muscles and less economical than normal in T muscles. These cellular changes beneficially equip P hearts for slow, high-pressure, economical pumping the T hearts for fast, high-volume, uneconomical pumping. The differences are similar to those between slow and fast skeletal muscle and between neonatal and adult skeletal muscle. The mechanism of these changes is discussed in terms of an enzyme kinetic scheme of chemomechanical coupling in actomyosin interaction.     

Temperature,contractility and high energy phosphates in anoxic fish heart muscle

Summary Isolated, electrically paced ventricular tissue of rainbow trout, Oncorhynchus mykiss, was examined at 20 and 10°C for the effects of different metabolic inhibitions on isometric force development and cellular content of phosphocreatine, creatine, ATP, ADP and AMP. At 20 relative to 10°C, twitch force was the same, but both twitch development and relaxation occurred over a shorter time and at a considerably higher maximal rate. Inhibition of cellular respiration caused twitch force and phosphocreatine to decrease, both about twice as fast at 20 as at 10°C. This doubling of energy degradation, i.e. in decrease of phosphocreatine, ATP, and loss of twitch force also occurred in preparations in which the energy liberation was totally blocked by iodoacetate in combination with N₂ and cyanide; both anaerobic energy degradation and anaerobic energy liberation expressed as lactate production were doubled. The similar effect of temperature on degradation and liberation of energy might explain why loss of twitch force during a 1-h period of anoxia was the same at both temperatures. The latter result was also found in the myocardium of eel Anguilla anguilla. In spite of its large influence on the time-course of twitch force development, the difference in temperature had no evident effects on the relationship between twitch force and phosphocreatine.Abbreviation Cr_t total creatine (creatine and phosphocreatine) - EDTA ethylenediminetetra-acetate - IAA iodoacetate - PCr phosphocreatine - TPT time-to-peak force - TR ₇₅ time for relaxation - V _F maximal rate of force development - V _R maximal rate of relaxation     

Distributed Representations for Actin-Myosin Interaction in the Oscillatory Contraction of Muscle

In this paper we suggest and test a specific hypothesis relating the attachment-detachment cycle of cross bridges between actin (I) and myosin (A) filaments to the measured length-tension dynamics of active insect fibrillar flight muscle. It is first shown that if local A-filament strain perturbs the rate constants in the cross-bridge cycle appropriately, then exponentially delayed tension changes can follow imposed changes of length; the latter phenomenon is sufficient for the work-producing property of fibrillar muscle, as measured with small-signal forcing of length and at low Ca²⁺ concentration, and possibly for related effects described recently in frog striated muscle. It is not clear a priori that the above explanation of work production by fibrillar muscle will remain tenable when the viscoelastic complexity of the heterogeneous sarcomere is taken into account. However, White's (1967) recent mechanical and electron microscope study of the passive dynamics of glycerinated fibrillar muscle has produced a model of the distributed viscoeleastic structure sufficiently explicit that alternative schemes for cross-bridge force generation in this muscle can now be tested more critically than previously. Therefore, we derive and solve third-order partial-differential equations which relate local interfilament shear forces associated with the perturbed cross-bridge cycles to the over-all length-tension dynamics of an idealized sarcomere. We then show (a) that the starting hypothesis can account approximately for the small-signal dynamics of glycerinated muscle in the work-producing state over two decades of frequency and (b) that the rate constants for cross-bridge formation and breakage, restricted solely by fitting of the model to the mechanical data, determine a cycling rate of cross bridges in the model compatible with recent measurements of ATP hydrolysis rate vs. stretch in this muscle. Finally, the formulation is extended tentatively to the large-signal nonlinear case, and shown to compare favorably with previous suggestions for the origin of the work-producing dynamics of fibrillar flight muscle.     

Energetics of activation in frog skeletal muscle at sarcomere lengths beyond myofilament overlap.

Experiments were designed to gain information about the effects of extremely long sarcomere lengths (greater than 3.8 microns) on muscle activation. The amount of energy liberated in an isometric twitch by muscles stretched to sarcomere lengths where myofilament overlap is vanishingly small (greater than 3.6 microns) is thought to be an indirect measure of the Ca2+ cycled during contraction. The effects of altering sarcomere length from 3.8 to 4.3 microns on the amount of Ca2+ cycled was measured using twitch energy liberation as an indicator of the Ca2+ cycled. Twitch energy liberation decreased by approximately 20% over this sarcomere length region, suggesting that the amount of Ca2+ released by a single action potential is not altered dramatically when a muscle is stretched to extreme lengths.     

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.