A quantitative model of intersarcomere dynamics during fixed-end contractions of single frog muscle fibers.

A numerical model of a muscle fiber as 400 sarcomeres, identical except for their initial lengths, was used to simulate fixed-end tetanic contractions of frog single fibers at sarcomere lengths above the optimum. The sarcomeres were represented by a lumped model, constructed from the passive and active sarcomere length-tension curves, the force-velocity curve, and the observed active elasticity of a single frog muscle fiber. An intersarcomere force was included to prevent large disparities in lengths of neighboring sarcomeres. The model duplicated the fast rise, slow creep rise, peak, and slow decline of tension seen in tetanic contractions of stretched living fibers. Decreasing the initial non-uniformity of sarcomere length reduced the rate of rise of tension during the creep phase, but did not decrease the peak tension reached. Limitations of the model, and other processes that might contribute to the shape of the fixed end tetanic tension record are discussed. Taking account of model and experimental results, it is concluded that the distinctive features of the tension records of fixed end tetanic contraction at lengths beyond optimum can be explained by internal motion within the fiber.     

Length-tension relationship of abdominal expiratory muscles: effect of emphysema

The present study examined the active and passive length-tension relationship of the abdominal expiratory muscles in vitro during electrically stimulated contractions. Studies were performed on isolated strips of transverse abdominis and external oblique muscle from nine adult hamsters with normal lung function. The effect of chronic hyperinflation on the two muscles was assessed in eight hamsters with elastase-induced emphysema. In normal animals the maximal active tension per cross-sectional area (Po) was equal in the two muscles. The absolute muscle fiber length at which Po occurred (Lo) was less for the external oblique than the transverse abdominis and the length-tension curve operated at shorter fiber lengths. However, the change in tension produced by an increase or decrease in muscle length expressed in relative terms (i.e., as %Lo) was greater for the transverse abdominis than the external oblique. Mean total lung capacity of emphysematous animals was 198% of control. Po of the transverse abdominis and external oblique were the same in emphysematous and control animals. However, Lo and the length-tension curve of the transverse abdominis occurred at shorter fiber lengths in emphysematous animals because of a reduction in the number of sarcomeres in series along the fiber. The length-tension curve and the number of sarcomeres in the external oblique was the same in emphysematous and control animals. These results in normal animals indicate that the magnitude of the change in active and passive tension produced by a change in muscle length differs in the transverse abdominis and external oblique. Moreover, chronic hyperinflation of the thorax produced by elastase injection alters the length-tension relationships of some but not all the expiratory muscles.     

Low-frequency depression of tension in the cat gastrocnemius muscle after eccentric exercise.

Subjecting a muscle to a series of eccentric contractions in which the contracting muscle is lengthened results in a number of changes in its mechanical properties. These include a fall in isometric tension that is particularly pronounced during low-frequency stimulation, a phenomenon known as low-frequency depression (LFD). Reports of LFD have not taken into account the shift in optimum length for active tension generation to longer muscle lengths that takes place after eccentric contractions. Given the length dependence of the stimulation frequency-tension curve, we tested the hypothesis that the change in this relationship after eccentric exercise is due to the shift in optimum length. We measured LFD by recording tension in response to a linearly increasing rate of stimulation of the nerve to medial gastrocnemius of anesthetized cats, over the range 0-100 pulses per second. Tension responses were measured before and after 50 eccentric contractions consisting of 6-mm stretches starting at 3 mm below optimum length and finishing at 3 mm above it. An index of LFD was derived from the tension responses to ramp stimulation. It was found that LFD after the eccentric contractions was partly, but not entirely, due to changes in the muscle's optimum length. An additional factor was the effect of fatigue. These observations led to the conclusion that the muscle length dependence of LFD was reduced by eccentric contractions. All of this means that after eccentric exercise the tension deficit at low rates of muscle activation is likely to be less severe than first thought.     

Changes in the mechanical properties of human and amphibian muscle after eccentric exercise

Following a series of eccentric contractions, that is stretching of the muscle while generating active tension, the length-tension relationship of isolated amphibian muscle has been shown to shift towards longer muscle lengths (Katz 1939; Wood et al. 1993). Here we report observations of electrically stimulated ankle extensor muscles of nine human subjects, demonstrating a similar shift in optimum angle for torque generation [3.9 (1.5)°] following exercise on an inclined treadmill that involved eccentric contractions in one leg. (All values are means with the SEMs in parentheses.) The shift in the unexercised, control leg was significantly less [mean 0.4 (0.7)°P < 0.05]. Correlated with this shift was a drop in torque [25.1 (5.6)% for the experimental leg; 1.6 (0.7)% for the control leg, P < 0.002]. Optimum angles returned to pre-exercise values by 2 days post-exercise, while torque took a week to recover. A similar shift in optimum length [12 (1.3)% of rest length] was obtained for five toad (Bufo marinus) sartorius muscles subjected to 25 eccentric contractions. Isometrically contracted control muscles showed a smaller shift [3.5 (1.6)%, n = 5]. Accompanying the shift was a drop in tension of 46 (3)% after the eccentric contractions [control isometric, 23 (6)%, P < 0.0001]. By 5 h after the eccentric contractions the shift had returned to control values, while tension had not recovered. When viewed with an electron microscope, sartorius muscles fixed immediately after the eccentric contractions exhibited many small, and a few larger, regions of myofilament disruption. In muscles fixed 5 h after the contractions, no small regions of disruption were visible, and the number of large regions was no greater than in those muscles fixed immediately after the eccentric contractions. These disruptions are interpreted as the cause of the shift in length-tension relationship. Accepted: 9 January 1997     

Damage to different motor units from active lengthening of the medial gastrocnemius muscle of the cat.

Slow-twitch motor units in the medial gastrocnemius muscle of the anesthetized cat were found to have an average optimum length for active tension that was 0.8 +/- 0.5 (SE) mm longer than the whole muscle optimum. For fast-twitch units (time to peak < 50 ms), the average optimum was 1.3 +/- 0.3 mm shorter than the whole muscle optimum. After the muscle had been subjected to 10 stretches while maximally activated, beginning at the whole muscle optimum length, the optimum lengths of the 27 fast-twitch motor units shifted significantly further in the direction of longer muscle lengths (mean 4.3 +/- 0.3 mm) than for the eight slow-twitch units (2.1 +/- 0.4 mm). A shift in the muscle's length-tension relation was interpreted as being due to sarcomere disruption. Statistical analysis showed that a motor unit's optimum length for a contraction, relative to the whole muscle optimum, was a better indicator of the unit's susceptibility to damage from active lengthenings than was motor unit type.     

Differential serial sarcomere number adaptations in knee extensor muscles of rats is contraction type dependent.

Sarcomerogenesis, or the addition of sarcomeres in series within a fiber, has a profound impact on the performance of a muscle by increasing its contractile velocity and power. Sarcomerogenesis may provide a beneficial adaptation to prevent injury when a muscle consistently works at long lengths, accounting for the repeated-bout effect. The association between eccentric exercise, sarcomerogenesis and the repeated-bout effect has been proposed to depend on damage, where regeneration allows sarcomeres to work at shorter lengths for a given muscle-tendon unit length. To gain additional insight into this phenomenon, we measured fiber dynamics directly in the vastus lateralis (VL) muscle of rats during uphill and downhill walking, and we measured serial sarcomere number in the VL and vastus intermedius (VI) after chronic training on either a decline or incline grade. We found that the knee extensor muscles of uphill walking rats undergo repeated active concentric contractions, and therefore they suffer no contraction-induced injury. Conversely, the knee extensor muscles during downhill walking undergo repeated active eccentric contractions. Serial sarcomere numbers change differently for the uphill and downhill exercise groups, and for the VL and VI muscles. Short muscle lengths for uphill concentric-biased contractions result in a loss of serial sarcomeres, and long muscle lengths for downhill eccentric-biased contractions result in a gain of serial sarcomeres.     

Modulation of passive force in single skeletal muscle fibres

In this study, we investigated the effects of activation and stretch on the passive force-sarcomere length relationship in skeletal muscle. Single fibres from the lumbrical muscle of frogs were placed at varying sarcomere lengths on the descending limb of the force-sarcomere length relationship, and tetanic contractions, active stretches and passive stretches (amplitudes of ca 10% of fibre length at a speed of 40% fibre length/s) were performed. The passive forces following stretch of an activated fibre were higher than the forces measured after isometric contractions or after stretches of a passive fibre at the corresponding sarcomere length. This effect was more pronounced at increased sarcomere lengths, and the passive force-sarcomere length relationship following active stretch was shifted upwards on the force axis compared with the corresponding relationship obtained following isometric contractions or passive stretches. These results provide strong evidence for an increase in passive force that is mediated by a length-dependent combination of stretch and activation, while activation or stretch alone does not produce this effect. Based on these results and recently published findings of the effects of Ca2+ on titin stiffness, we propose that the observed increase in passive force is caused by the molecular spring titin.     

Correlation between active and passive isometric force and intramuscular pressure in the isolated rabbit tibialis anterior muscle

The purpose of this study was to quantify the relationship between intramuscular pressure (IMP) and muscle force during isometric muscle contraction of the rabbit tibialis anterior (TA) absent the effect of either bone or fascia. To quantify this relationship, length-tension experiments were performed on the isolated TA of the New Zealand White rabbit (mass=2.5+/-0.5kg, n=12). The knee was fixed in a custom jig, the distal tendon of the TA was attached to a servomotor, and a 360 microm fiber optic pressure transducer was inserted into the TA. The peroneal nerve was stimulated to define optimal length (L(0)). The length-tension curve was created using 40Hz isometric contractions with 2-min rest intervals between each contraction. Measurements began at L(0)-50%L(f) and progressed to L(0)+50%L(f), changing the length-tension in 5% L(f) increments after each contraction. Qualitatively, the length-tension curve for isometric contractions was mimicked by the length-pressure curve for both active and passive conditions. Linear regression was performed individually for each animal for the ascending and descending limb of the length-tension curve and for active and passive conditions. Pressure-force coefficients of determination ranged from 0.138-0.963 for the active ascending limb and 0.343-0.947 for the active descending limb. Passive pressure coefficients of determination ranged from 0.045-0.842 for the ascending limb and 0.672-0.982 for the descending limb. These data indicate that IMP measurement provide a fairly accurate index of relative muscle force, especially at muscle lengths longer than optimal.     

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.     

Length-tension relation in Limulus striated muscle

Laser diffraction techniques coupled with simultaneous tension measurements were used to determine the length-tension relation in intact, small (0.5-mm thick, 10-mm wide, 20-25-mm long) bundles of a Limulus (horseshoe crab) striated muscle, the telson levator muscle. This muscle differs from the model vertebrate systems in that the thick filaments are not of a constant length, but shorten from 4.9 to approximately 2.0 micrometers as the sarcomeres shorten from 7 to 3 micrometers. In the Limulus muscle, the length-tension relation plateaued to an average maximum tension of 0.34 N/mm2 at a sarcomere length of 6.5 micrometers (Lo) to 8.0 micrometers. In the sarcomere length range from 3.8 to 12.5 micrometers, the muscle developed 50% or more of the maximum tension. When the sarcomere lengths are normalized (expressed as L/Lo) and the Limulus data are compared to those from frog muscle, it is apparent that Limulus muscle develops tension over a relatively greater range of sarcomere lengths.     

Passive and active tension in single cardiac myofibrils.

Single myofibrils were isolated from chemically skinned rabbit heart and mounted in an apparatus described previously (Fearn et al., 1993; Linke et al., 1993). We measured the passive length-tension relation and active isometric force, both normalized to cross sectional area. Myofibrillar cross sectional area was calculated based on measurements of myofibril diameter from both phase-contrast images and electron micrographs. Passive tension values up to sarcomere lengths of approximately 2.2 microns were similar to those reported in larger cardiac muscle specimens. Thus, the element responsible for most, if not all, passive force of cardiac muscle at physiological sarcomere lengths appears to reside within the myofibrils. Above 2.2 microns, passive tension continued to rise, but not as steeply as reported in multicellular preparations. Apparently, structures other than the myofibrils become increasingly important in determining the magnitude of passive tension at these stretched lengths. Knowing the myofibrillar component of passive tension allowed us to infer the stress-strain relation of titin, the polypeptide thought to support passive force in the sarcomere. The elastic modulus of titin is 3.5 x 10(6) dyn cm-2, a value similar to that reported for elastin. Maximum active isometric tension in the single myofibril at sarcomere lengths of 2.1-2.3 microns was 145 +/- 35 mN/mm2 (mean +/- SD; n = 15). This value is comparable with that measured in fixed-end contractions of larger cardiac specimens, when the amount of nonmyofibrillar space in those preparations is considered. However, it is about 4 times lower than the maximum active tension previously measured in single skeletal myofibrils under similar conditions (Bartoo et al., 1993).     

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.     

Length-force characteristics of aponeurosis in passive muscle and during isometric and slow dynamic contractions of rat gastrocnemius muscle

Length-force characteristics of aponeurosis of rat gastrocnemius medialis muscle and achilles tendon were studied for passive and active muscle. Active muscle performed isometric as well as slow concentric and eccentric contractions at low velocity. For isometric conditions, different aponeurosis and tendon length-force characteristics were found between passive and active muscle: At comparable low levels of force longer aponeuroses were encountered in passive than in active muscle. Similar results were found for achilles tendon, but the magnitude of the length change involved was smaller than for aponeurosis. For active muscle, no differences of aponeurosis length- force characteristics could be distinguished between the isometric contractions and a slow concentric contraction. Indications that such differences of aponeurosis length-force characteristics may exist between slow concentric and eccentric contractions were found. It is concluded that, for gastrocnemius medialis muscle, aponeurosis and tendon length - force characteristics may be quite variable depending on recent history of muscle length and activity.     

Whole muscle length-tension relationships are accurately modeled as scaled sarcomeres in rabbit hindlimb muscles

An a priori model of the whole active muscle length-tension relationship was constructed utilizing only myofilament length and serial sarcomere number for rabbit tibialis anterior (TA), extensor digitorum longus (EDL), and extensor digitorum II (EDII) muscles. Passive tension was modeled with a two-element Hill-type model. Experimental length-tension relations were then measured for each of these muscles and compared to predictions. The model was able to accurately capture the active-tension characteristics of experimentally-measured data for all muscles (ICC=0.88 ± 0.03). Despite their varied architecture, no differences in predicted versus experimental correlations were observed among muscles. In addition, the model demonstrated that excursion, quantified by full-width-at-half-maximum (FWHM) of the active length-tension relationship, scaled linearly (slope=0.68) with normalized muscle fiber length. Experimental and theoretical FWHM values agreed well with an intraclass correlation coefficient of 0.99 (p<0.001). In contrast to active tension, the passive tension model deviated from experimentally-measured values and thus, was not an accurate predictor of passive tension (ICC=0.70 ± 0.07). These data demonstrate that modeling muscle as a scaled sarcomere provides accurate active functional but not passive functional predictions for rabbit TA, EDL, and EDII muscles and call into question the need for more complex modeling assumptions often proposed.     

The dynamical stability of isometrically contracting muscle and its relation to transient force response

The length-tension relation for tetanically contracting muscle indicates that for lengths longer than the resting length, the muscle should be dynamically unstable; i.e. some sarcomeres should lengthen and others shorten. This behavior is not observed experimentally. The theoretical behavior of muscle in this respect is determined both by the length-tension curve, and by the response of muscle to rapid changes in its mechanical state. In this paper it is shown that the force generated after a small step change in length is related to the dynamical stability of muscle. By means of a simple model, the behavior of isometrically contracting muscle is predicted based on in vitro mechanical studies and classical control theory. It is found that inhomogeneities in sarcomere length can develop only after many seconds, and that this relative stability is due entirely to the presence of the muscle transients.     

A mechanism accounting for independence on starting length of tension increase in ramp stretches of active skeletal muscle at short half-sarcomere lengths

Based on previous experimental results of independence on starting length of the tension gradient in constant-velocity stretches of active skeletal muscle at muscle lengths including the ascending limb and the plateau of the tension-length relation, a possible physiological mechanism determining the tension increase in lengthening active muscle is discussed. Considering the sliding filament theory, it is suggested that the tension-length relation of a half-sarcomere in lengthening contractions is different from that in isometric contractions. The assumed mechanism predicts, among others, that the thick filament retains its shortened length in lengthening contractions starting from a half-sarcomere length where this filament is compressed. An example model is implemented and checked with simulations.     

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.     

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.     

Passive tension in cardiac muscle: contribution of collagen, titin, microtubules, and intermediate filaments.

The passive tension-sarcomere length relation of rat cardiac muscle was investigated by studying passive (or not activated) single myocytes and trabeculae. The contribution of collagen, titin, microtubules, and intermediate filaments to tension and stiffness was investigated by measuring (1) the effects of KCl/KI extraction on both trabeculae and single myocytes, (2) the effect of trypsin digestion on single myocytes, and (3) the effect of colchicine on single myocytes. It was found that over the working range of sarcomeres in the heart (lengths approximately 1.9-2.2 microns), collagen and titin are the most important contributors to passive tension with titin dominating at the shorter end of the working range and collagen at longer lengths. Microtubules made a modest contribution to passive tension in some cells, but on average their contribution was not significant. Finally, intermediate filaments contributed about 10% to passive tension of trabeculae at sarcomere lengths from approximately 1.9 to 2.1 microns, and their contribution dropped to only a few percent at longer lengths. At physiological sarcomere lengths of the heart, cardiac titin developed much higher tensions (> 20-fold) than did skeletal muscle titin at comparable lengths. This might be related to the finding that cardiac titin has a molecular mass of 2.5 MDa, 0.3-0.5 MDa smaller than titin of mammalian skeletal muscle, which is predicted to result in a much shorter extensible titin segment in the I-band of cardiac muscle. Passive stress plotted versus the strain of the extensible titin segment showed that the stress-strain relationships are similar in cardiac and skeletal muscle. The difference in passive stress between cardiac and skeletal muscle at the sarcomere level predominantly resulted from much higher strains of the I-segment of cardiac titin at a given sarcomere length. By expressing a smaller titin isoform, without changing the properties of the molecule itself, cardiac muscle is able to develop significant levels of passive tension at physiological sarcomere lengths.     

Resting Sarcomere Length-Tension Relation in Living Frog Heart

The sarcomere pattern and tension of isolated resting frog atrial trabeculae were continuously monitored. In the absence of any resting tension the sarcomere lengths varied with the diameter of the trabeculae. In over 75 % of the trabeculae the value exceeded 2.05 µm, the estimated in vivo length of the thin filaments, and it was never less than 1.89 µm. When the trabeculae were stretched the increase in length of the central undamaged portion could be completely accounted for by an increase in sarcomere length. The width of the A band was constant only at sarcomere lengths between 2.3 and 2.6 µm it decreased at smaller and increased at larger sarcomere lengths. A group of spontaneously active cells stretched the sarcomeres in cells in series to longer lengths than could be produced by passive tension applied to the ends of the trabeculae, but they did not influence the sarcomeres of adjacent cells. It is proposed that the connective tissue is a major factor in determining sarcomere length and that there are interactions between thick and thin filaments in resting muscles.