Stretch activation and nonlinear elasticity of muscle cross-bridges.

When active insect fibrillar flight muscle is stretched, its ATPase rate increases and it develops "negative viscosity," which allows it to perform oscillatory work. We use a six-state model for the cross-bridge cycle to show that such "stretch activation" may arise naturally as a nonlinear property of a cross-bridge interacting with a single attachment site on a thin filament. Attachment is treated as a thermally activated process in which elastic energy must be supplied to stretch or compress the cross-bridge spring. We find that stretch activation occurs at filament displacements where, before the power stroke, the spring is initially in compression rather than in tension. In that case, pulling the filaments relieves the initial compression and reduces the elastic energy required for attachment. The result is that the attachment rate is enhanced by stretching. The model also displays the "delayed tension" effect observed in length-step experiments. When the muscle is stretched suddenly, the power stroke responds very quickly, but there is a time lag before dissociation at the end of the cycle catches up with the increased attachment rate. This lag is responsible for the delayed tension and hence also for the negative viscosity.     

A comparison of the mechanical behavior of the cat soleus muscle with a distribution-moment model

A state-variable model for skeletal muscle, termed the "Distribution-Moment Model," is derived from A. F. Huxley's 1957 model of molecular contraction dynamics. The state variables are the muscle stretch and the three lowest-order moments of the bond-distribution function (which represent, respectively, the contractile tissue stiffness, the muscle force, and the elastic energy stored in the contractile tissue). The rate equations of the model are solved under various conditions, and compared to experimental results for the cat soleus muscle subjected to constant stimulation. The model predicts several observed effects, including yielding of the muscle force in constant velocity stretches, different "force-velocity relations" in isotonic and isovelocity experiments, and a decrease of peak force below the isometric level in small-amplitude sinusoidal stretches. Chemical energy and heat rates predicted by the model are also presented.     

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.     

Control of human eye movements: III. dynamic characteristics of the eye tracking mechanism

The properties of extraocular muscle are important in consideration of the control of human eye movements. A proposed model for human extraocular muscle is based on the anatomical and physiological evidence; it considers both the static and dynamic properties of active and passive muscle. The passive parallel elasticity was determined from the length-tension curves for passive muscle, while the active series elasticity was defined utilizing quick stretch results for active muscle. The characteristics of active muscle as the tension generator were computed from length-tension data; the force-velocity relationship was used to describe the viscosity of active muscle. Simulations using the muscle model accurately depicted the quick stretch experiments of both active and passive muscle as well as the isometric development of muscle force to a state of tentanus. The model will be incorporated into an overall representation of the extraocular plant mechanism in the immediately suceeding paper.     

History-dependent mechanical properties of permeabilized rat soleus muscle fibers.

Permeabilized rat soleus muscle fibers were subjected to repeated triangular length changes (paired ramp stretches/releases, 0.03 l(0), +/- 0.1 l(0) s(-1) imposed under sarcomere length control) to investigate whether the rate of stiffness recovery after movement increased with the level of Ca(2+) activation. Actively contracting fibers exhibited a characteristic tension response to stretch: tension rose sharply during the initial phase of the movement before dropping slightly to a plateau, which was maintained during the remainder of the stretch. When the fibers were stretched twice, the initial phase of the response was reduced by an amount that depended on both the level of Ca(2+) activation and the elapsed time since the first movement. Detailed analysis revealed three new and important findings. 1) The rates of stiffness and tension recovery and 2) the relative height of the tension plateau each increased with the level of Ca(2+) activation. 3) The tension plateau developed more quickly during the second stretch at high free Ca(2+) concentrations than at low. These findings are consistent with a cross-bridge mechanism but suggest that the rate of the force-generating power-stroke increases with the intracellular Ca(2+) concentration and cross-bridge strain.     

Control of human eye movements: I. modelling of extraocular muscle

The properties of extraocular muscle are important in consideration of the control of human eye movements. A proposed model for human extraocular muscle is based on the anatomical and physiological evidence; it considers both the static and dynamic properties of active and passive muscle. The passive parallel elasticity was determined from the length-tension curves for passive muscle, while the active series elasticity was defined utilizing quick stretch results for active muscle. The characteristics of active muscle as the tension generator were computed from length-tension data; the force-velocity relationship was used to describe the viscosity of active muscle. Simulations using the muscle model accurately depicted the quick stretch experiments of both active and passive muscle as well as the isometric development of muscle force to a state of tentanus. The model will be incorporated into an overall representation of the extraocular plant mechanism in the immediately suceeding paper.     

Effects of oxidation on the power of chemically skinned rat soleus fibres

Oxidation alters calcium sensitivity, and decreases maximum isometric force (Po) and shortening velocity (Vmax) of single muscle fibres. To examine the effect of oxidation on the curvature of the force-velocity relationship, which determines muscle power in addition to Po and Vmax, skinned rat type I fibres were maximally activated at 15°C in a solution with pCa 4.5 and subjected to isotonic contractions before and after 4-min incubation in 50 mM H?O? (n=10) or normal relaxing solution (n=3). In five oxidised and four control fibres the rate of force redevelopment (ktr), following a rapid release and re-stretch, was measured. This gives a measure of the sum of the rate constants for cross-bridge attachment (f) and detachment (g?): (f+g?). H?O? reduced Po, Vmax and ktr by 19%, 21% and 24% respectively (P<0.001), while the shape of the force-velocity relationship was unchanged. Fitting data to the Huxley cross-bridge model suggested that oxidation decreased both the rate constant for cross-bridge attachment (f), and detachment of negatively strained cross-bridges (g?), similar to the effect of reduced activation. This suggests that oxidative modification is a possible cause of the variation in contractile properties between muscle fibres of the same type.     

New insights into the behavior of muscle during active lengthening.

A muscle fiber was modeled as a series-connected string of sarcomeres, using an A. V. Hill type model for each sarcomere and allowing for some random variation in the properties of the sarcomeres. Applying stretches to this model led to the prediction that lengthening of active muscle on or beyond the plateau of the length tension curve will take place very nonuniformly, essentially by rapid, uncontrolled elongation of individual sarcomeres, one at a time, in order from the weakest toward the strongest. Such a "popped" sarcomere, at least in a single fiber, will be stretched to a length where there is no overlap between thick and thin filaments, and the tension is borne by passive components. This prediction allows modeling of many results that have previously been inexplicable, notably the permanent extra tension after stretch on the descending limb of the length tension curve, and the continued rise of tension during a continued stretch.     

Millisecond-scale biochemical response to change in strain

Muscle fiber contraction involves the cyclical interaction of myosin cross-bridges with actin filaments, linked to hydrolysis of ATP that provides the required energy. We show here the relationship between cross-bridge states, force generation, and Pi release during ramp stretches of active mammalian skeletal muscle fibers at 20°C. The results show that force and Pi release respond quickly to the application of stretch: force rises rapidly, whereas the rate of Pi release decreases abruptly and remains low for the duration of the stretch. These measurements show that biochemical change on the millisecond timescale accompanies the mechanical and structural responses in active muscle fibers. A cross-bridge model is used to simulate the effect of stretch on the distribution of actomyosin cross-bridges, force, and Pi release, with explicit inclusion of ATP, ADP, and Pi in the biochemical states and length-dependence of transitions. In the simulation, stretch causes rapid detachment and reattachment of cross-bridges without release of Pi or ATP hydrolysis.     

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.     

A non-cross-bridge stiffness in activated frog muscle fibers

Force responses to fast ramp stretches of various amplitude and velocity, applied during tetanic contractions, were measured in single intact fibers from frog tibialis anterior muscle. Experiments were performed at 14 degrees C at approximately 2.1 microm sarcomere length on fibers bathed in Ringer's solution containing various concentrations of 2,3-butanedione monoxime (BDM) to greatly reduce the isometric tension. The fast tension transient produced by the stretch was followed by a period, lasting until relaxation, during which the tension remained constant to a value that greatly exceeded the isometric tension. The excess of tension was termed "static tension," and the ratio between the force and the accompanying sarcomere length change was termed "static stiffness." The static stiffness was independent of the active tension developed by the fiber, and independent of stretch amplitude and stretching velocity in the whole range tested; it increased with sarcomere length in the range 2.1-2.8 microm, to decrease again at longer lengths. Static stiffness increased well ahead of tension during the tetanus rise, and fell ahead of tension during relaxation. These results suggest that activation increased the stiffness of some sarcomeric structure(s) outside the cross-bridges.     

Non-cross-bridge calcium-dependent stiffness in frog muscle fibers

At the end of the force transient elicited by a fast stretch applied to an activated frog muscle fiber, the force settles to a steady level exceeding the isometric level preceding the stretch. We showed previously that this excess of tension, referred to as "static tension," is due to the elongation of some elastic sarcomere structure, outside the cross bridges. The stiffness of this structure, "static stiffness," increased upon stimulation following a time course well distinct from tension and roughly similar to intracellular Ca²⁺ concentration. In the experiments reported here, we investigated the possible role of Ca²⁺ in static stiffness by comparing static stiffness measurements in the presence of Ca²⁺ release inhibitors (D600, Dantrolene, ²H₂O) and cross-bridge formation inhibitors [2,3-butanedione monoxime (BDM), hypertonicity]. Both series of agents inhibited tension; however, only D600, Dantrolene, and ²H₂O decreased at the same time static stiffness, whereas BDM and hypertonicity left static stiffness unaltered. These results indicate that Ca²⁺, in addition to promoting cross-bridge formation, increases the stiffness of an (unidentified) elastic structure of the sarcomere. This stiffness increase may help in maintaining the sarcomere length uniformity under conditions of instability. intact muscle fiber; static stiffness; tension inhibitors; titin     

Mechanical transients initiated by photolysis of caged ATP within fibers of insect fibrillar flight muscle

Kinetics of the cross-bridge cycle in insect fibrillar flight muscle have been measured using laser pulse photolysis of caged ATP and caged inorganic phosphate (Pi) to produce rapid step increases in the concentration of ATP and Pi within single glycerol-extracted fibers. Rapid photochemical liberation of 100 microM-1 mM ATP from caged ATP within a fiber caused relaxation in the absence of Ca2+ and initiated an active contraction in the presence of approximately 30 microM Ca2+. The apparent second order rate constant for detachment of rigor cross-bridges by ATP was between 5 x 10(4) and 2 x 10(5) M-1s-1. This rate is not appreciably sensitive to the Ca2+ or Pi concentrations or to rigor tension level. The value is within an order of magnitude of the analogous reaction rate constant measured with isolated actin and insect myosin subfragment-1 (1986. J. Muscle Res. Cell Motil. 7:179-192). In both the absence and presence of Ca2+ insect fibers showed evidence of transient cross-bridge reattachment after ATP-induced detachment from rigor, as found in corresponding experiments on rabbit psoas fibers. However, in contrast to results with rabbit fibers, tension traces of insect fibers starting at different rigor tensions did not converge to a common time course until late in the transients. This result suggests that the proportion of myosin cross-bridges that can reattach into force-generating states depends on stress or strain in the filament lattice. A steady 10-mM concentration of Pi markedly decreased the transient reattachment phase after caged ATP photolysis. Pi also decreased the amplitude of stretch activation after step stretches applied in the presence of Ca2+ and ATP. Photolysis of caged Pi during stretch activation abruptly terminated the development of tension. These results are consistent with a linkage between Pi release and the steps leading to force production in the cross-bridge cycle.     

Stiffness and tension during and after sudden length changes of glycerinated single insect fibrillar muscle fibres

Rapid length changes were applied (within 0.2 ms or 0.4 ms) to single isometrically contracted glycerol extracted muscle fibres of the dorsal longitudinal muscle ofLethocerus maximus suspended in an Ca²⁺ and ATP containing solution at 20–23‡ C. Force transients and the fibre stiffness were measured during and after rapid length changes. At length changesbelow 0.5% of the initial fibre length (∼ 2.4 Μm sarcomere length) the mechanical transients were characterized as follows: (1) After stretch and after release the force regains at least partly the value of tension before the length change within a quick phase of tension recovery. The quick phase induced by stretch was nearly completed within 1–2 ms. (2) A pulse in length of 1.5 ms duration, i.e., a stretch followed by a release to the initial length or a release followed by a stretch to the initial length, was applied to the fibre. The force transient induced by this procedure regains after the second length change the value of the isometric tension before the procedure. (3) The stiffness was constant during each length change of the “pulse” and was equal during the first and the second length changes. These findings are predicted by the muscle contraction model of Huxley and Simmons (1971): The identical force before and after a length pulse may indicate that the rotation of cross bridges after the first length change is followed by a rotation into the original position after the second length change. The constancy of the stiffness during the length changes may indicate a Hookean elastic element of the cross bridge. The similarity of the stiffness during the first and the second length changes, i.e., before and after the quick phase, gives evidence that the quick phases after stretch and after release are not accompanied by a change in the net number of attached cross bridges. If stretches ofmore than 0.5% of the initial length were applied, the mechanical transient of the muscle fibre changed as follows: (1) An ultra fast tension decay phase (duration < 0.4 ms) was observed in addition to the slower decay phase induced by the smaller stretches. (2) If the initial stretch was followed by a release to the initial length, no fast recovery phase was observed, which returns the force to the value before the stretch. The reduced tension value persists for a longer period in time than 10 ms. (3) If the muscle was stretched and released repetitively an ultra fast quick phase was induced only by the first stretch. (4) The stiffness increased during stretch, but was found to be the same in the isometrically contracting muscle and after the quick tension decay phase following a large stretch. These findings indicate that the contraction model of Huxley and Simmons has to be extended by a further process additional to cross bridge rotation in case of large stretches (> 0.5%L ini). The findings are taken to indicate a rapid detachment and reattachment of overstrained cross bridges, i.e., a cross bridge slippage induced by large stretches.     

Force enhancement and relaxation rates after stretch of activated muscle fibres

The residual force enhancement following muscle stretch might be associated with an increase in the proportion of attached cross-bridges, as supported by stiffness measurements. In this case, it could be caused by an increase in the attachment or a decrease in the detachment rate of cross-bridges, or a combination of the two. The purpose of this study was to investigate if the stretch-induced force enhancement is related to cross-bridge attachment/detachment kinetics. Single muscle fibres dissected from the lumbrical muscle of frog were place at a length approximately 20% longer than the plateau of the force-length relationship; they were maximally activated, and after full isometric force was reached, ramp stretches were imposed with amplitudes of 5 and 10% fibre length, at a speed of 40% fibre length s(-1). Experiments were performed in Ringer's solution, and with the addition of 2, 5 and 10 nM of 2,3-butanedione monoxime (BDM), a drug that places cross-bridges in a pre-power-stroke, state, inhibiting force production. The total force following stretch was higher than the corresponding force measured after isometric contraction at the corresponding length. This residual force enhancement was accompanied by an increase relaxation time. BDM, which decreases force production during isometric contractions, considerably increased the relative levels of force enhancement. BDM also increased relaxation times after stretch, beyond the levels observed during reference contractions in Ringer's solution, and beyond isometric control tests at the corresponding BDM concentrations. Together, these results support the idea that force enhancement is caused, at least in part, by a decrease in cross-bridge detachment rates, as manifested by the increased relaxation times following fibre stretch.     

Changes in force-velocity properties of trachealis due to oscillatory strains.

The physically dynamic environment of the lung constantly modulates the mechanical properties of airway smooth muscle. In vitro experiments have shown that contractility of the muscle is compromised by oscillatory strains, perhaps through disruption of cross-bridge interaction and organization of the contractile filaments. To understand the mechanism by which oscillation affects contractility, functional changes of the muscle in terms of force-velocity relationship were assessed before and after imposition of length oscillation in both relaxed and activated states. The oscillation protocol was designed to reduce isometric force by 15-20%, followed by measurement of force-velocity properties. Maximal velocity and power changed by +8 and -14%, respectively, after oscillation applied in the relaxed state and changed by -15 and -25%, respectively, after oscillation applied during contraction. A simple model of reduced activation could not account for the results; neither could the results be explained satisfactorily by the current cross-bridge theory of contraction. The results, however, could be explained if the possibility of reorganization of the contractile filaments due to oscillatory strains was considered.     

Active force inhibition and stretch-induced force enhancement in frog muscle treated with BDM.

There is evidence that the stretch-induced residual force enhancement observed in skeletal muscles is associated with 1) cross-bridge dynamics and 2) an increase in passive force. The purpose of this study was to characterize the total and passive force enhancement and to evaluate whether these phenomena may be associated with a slow detachment of cross bridges. Single fibers from frog lumbrical muscles were placed at a length 20% longer than the plateau of the force-length relationship, and active and passive stretches (amplitudes of 5 and 10% of fiber length and at a speed of 40% fiber length/s) were performed. Experiments were conducted in Ringer solution and with the addition of 2, 5, and 10 mM of 2,3-butanedione monoxime (BDM), a cross-bridge inhibitor. The steady-state active and passive isometric forces after stretch of an activated fiber were higher than the corresponding forces measured after isometric contractions or passive stretches. BDM decreased the absolute isometric force and increased the total force enhancement in all conditions investigated. These results suggest that total force enhancement is directly associated with cross-bridge kinetics. Addition of 2 mM BDM did not change the passive force enhancement after 5 and 10% stretches. Addition of 5 and 10 mM did not change (5% stretches) or increased (10% stretches) the passive force enhancement. Increasing stretch amplitudes and increasing concentrations of BDM caused relaxation after stretch to be slower, and because passive force enhancement is increased at the greatest stretch amplitudes and the highest BDM concentrations, it appears that passive force enhancement may be related to slow-detaching cross bridges.     

The effects of aponeurosis geometry on strain injury susceptibility explored with a 3D muscle model

In the musculoskeletal system, some muscles are injured more frequently than others. For example, the biceps femoris longhead (BFLH) is the most commonly injured hamstring muscle. It is thought that acute injuries result from large strains within the muscle tissue, but the mechanism behind this type of strain injury is still poorly understood. The purpose of this study was to build computational models to analyze the stretch distributions within the BFLH muscle and to explore the effects of aponeurosis geometry on the magnitude and location of peak stretches within the model. We created a three-dimensional finite element (FE) model of the BFLH based on magnetic resonance (MR) images. We also created a series of simplified models with a similar geometry to the MR-based model. We analyzed the stretches predicted by the MR-based model during lengthening contractions to determine the region of peak local fiber stretch. The peak along-fiber stretch was 1.64 and was located adjacent to the proximal myotendinous junction (MTJ). In contrast, the average along-fiber stretch across all the muscle tissue was 0.95. By analyzing the simple models, we found that varying the dimensions of the aponeuroses (width, length, and thickness) had a substantial impact on the location and magnitude of peak stretches within the muscle. Specifically, the difference in widths between the proximal and distal aponeurosis in the BFLH contributed most to the location and magnitude of peak stretch, as decreasing the proximal aponeurosis width by 80% increased peak average stretches along the proximal MTJ by greater than 60% while slightly decreasing stretches along the distal MTJ. These results suggest that the aponeurosis morphology of the BFLH plays a significant role in determining stretch distributions throughout the muscle. Furthermore, this study introduces the new hypothesis that aponeurosis widths may be important in determining muscle injury susceptibility.     

Modeling residual force enhancement with generic cross-bridge models

The interaction of actin and myosin through cross-bridges explains much of muscle behavior. However, some properties of muscle, such as residual force enhancement, cannot be explained by current cross-bridge models. There is ongoing debate whether conceptual cross-bridge models, as pioneered by Huxley (A.F. Huxley, Muscle structure and theories of contraction, Prog. Biophys. Biophys. Chem. 7 (1957) 255) could, if suitably modified, fit experimental data showing residual force enhancement. Here we prove that there are only two ways to explain residual force enhancement with these ‘traditional’ cross-bridge models: the first requires cross-bridges to become stuck on actin (the stuck cross-bridge model) while the second requires that cross-bridges that are pulled off beyond a critical strain enter a ‘new’ unbound state that leads to a new force-producing cycle (the multi-cycle model). Stuck cross-bridge models cannot fit the velocity and stretch amplitude dependence of residual force enhancement, while the multi-cycle models can. The results of this theoretical analysis demonstrate that current kinetic models of cross-bridge action cannot explain the experimentally observed residual force enhancement. Either cross-bridges in the force-enhanced state follow a different kinetic cycle than cross-bridges in a ‘normal’ force state, or the assumptions underlying traditional cross-bridge models must be violated during experiments that show residual force enhancement.