A cross-bridge mechanism can explain the thixotropic short-range elastic component of relaxed frog skeletal muscle

Authors


  • Author's permanent address

    K. S. Campbell: Department of Physiology, 1300 University Avenue, University of Wisconsin-Madison, Madison, WI 53706, USA.

Corresponding author M. Lakie: Applied Physiology Research Group, School of Sport and Exercise Sciences, University of Birmingham, Birmingham B15 2TT, UK. Email: m.d.lakie@bham.ac.uk

Abstract

  • 1The passive tension and sarcomere length of relaxed frog skeletal muscle fibres were measured in response to imposed length stretches. The tension response to a constant-velocity stretch exhibited a clear discontinuity. Tension rose more rapidly during the initial ∼ 0.4 %L0 of the stretch than during the latter stages (where L0 is the resting length of the fibre). This initial tension response is attributed to the short-range elastic component (SREC).
  • 2The use of paired triangular stretches revealed that the maximum tension produced during the SREC response of the second stretch was significantly reduced by the first stretch. This history-dependent behaviour of the SREC reflects thixotropic stiffness changes that have been previously described in relaxed muscle.
  • 3The biphasic nature of the SREC tension response to movement was most apparent during the first imposed length change after a period at a fixed length, irrespective of the direction of movement.
  • 4If a relaxed muscle was subjected to an imposed triangular length change so that the muscle was initially stretched and subsequently shortened back to its original fibre length, the resting tension at the end of the stretch was reduced relative to its initial pre-stretch value. Following the end of the stretch, tension slowly increased towards its initial value but the tension recovery was not accompanied by a contemporaneous increase in sarcomere length. This finding suggests that the resting tension of a relaxed muscle fibre is not entirely due to passive elasticity. The results are compatible with the suggestion that a portion of the resting tension - the filamentary resting tension (FRT) - is produced by a low level of active force generation.
  • 5If a second identical stretch was imposed on the muscle at a time when the resting tension was reduced by the previous stretch, the maximal tension produced during the second stretch was the same as that produced during the first, despite the second stretch commencing from a lower initial resting tension.
  • 6In experiments using paired triangular length changes, an inter-stretch interval of zero did not produce a substantially greater thixotropic reduction in the second stretch elastic limit force than an inter-stretch interval in the range 0.5-1 s.
  • 7A theoretical model was developed in which the SREC and FRT arise as manifestations of a small number of slowly cycling cross-bridges linking the actin and myosin filaments of a relaxed skeletal muscle. The predictions of the model are compatible with many of the experimental observations. If the SREC and FRT are indeed due to cross-bridge activity, the model suggests that the cross-bridge attachment rate must increase during interfilamentary movement. A mechanism (based on misregistration between the actin binding sites and the myosin cross-bridges) by which this might arise is presented.

The precise control of posture and movement imposes severe constraints on the human neuromuscular system. In particular, to maintain stability, the motor control system must continually react to proprioceptive feedback as the body is perturbed by unpredictable external forces. In this type of control mechanism there is an inevitable time delay (due to e.g. nerve conduction and electromechanical coupling) between the sensing of the perturbation and the completion of the corrective movement. The delays in the feedback loop are destabilizing and may result in tremulous movement (McMahon, 1984). A small resistance to movement inherent in the muscle fibres themselves would provide mechanical damping. This would increase postural stability (Grillner, 1972) without markedly affecting the body's ability to produce large rapid movements under voluntary control.

Even when movements are generated it is often only a relatively small number of motor units in the agonist muscles which are activated. Such movements inevitably stretch the fibres of the antagonist muscle groups and passively shorten unrecruited fibres in the agonist muscles. The neuromuscular control system must allow for these passive mechanical properties if it is to control movements with precision. A better understanding of the mechanical properties of inactive muscle would thus be useful in a holistic description of the neuromuscular control system.

In 1968, D. K. Hill demonstrated that relaxed frog sartorii exhibited both a small actively generated resting tension - the filamentary resting tension (FRT) - and a disproportionately high resistance to the initial stages of an imposed length stretch - the short-range elastic component (SREC). Further work (e.g. Lännergren, 1971; Herbst, 1976; Lakie & Robson, 1988a, b, 1990; Campbell & Lakie, 1995a, 1996) has shown that both the initial resistance of the SREC and the tension generated by the FRT depend crucially on the mechanical history of the muscle. Prior movements applied to the muscle temporarily reduce its initial stiffness during a subsequent stretch by as much as 50 % (Lakie & Robson, 1988a, b). This behaviour, which characterizes a mechanical system in which the state depends on the history of movement, is thixotropic (Buchthal & Kaiser, 1951; Lakie, Walsh & Wright, 1984).

The precise mechanism underlying the SREC and FRT in relaxed frog skeletal muscle remains uncertain. Hill believed that the SREC and FRT were manifestations of a low level of cross-bridge activity in relaxed skeletal muscle. More recent work (although not specifically investigating the SREC and FRT) suggests that many of the mechanical properties of relaxed muscle may be attributed to either viscous and visco-elastic behaviour (Bagni, Cecchi, Colomo & Garzella, 1992, 1995; Mutungi & Ranatunga, 1996a, b) or to titin filaments (Linke, Bartoo, Ivemeyer & Pollack, 1996; Bartoo, Linke & Pollack, 1997). This paper describes some measurements of the SREC and FRT in relaxed frog skeletal muscle fibres. It also presents a simple theoretical model, the ‘Cross-bridge Population Displacement Mechanism’, which predicts the mechanical behaviour of a relaxed muscle in which the actin and myosin filaments are linked by a small number of slowly cycling cross-bridges. The predictions of the model are in good qualitative agreement with the experimental observations. It is undeniable that either visco-elastic mechanisms or titin filaments may dominate the mechanical properties of relaxed skeletal muscle under certain conditions. However, this work does show that the SREC and FRT are fully compatible with Hill's original cross-bridge hypothesis, and moreover, that the hypothesis provides a succinct explanation for many of the complex mechanical properties of relaxed skeletal muscle.

METHODS

Experimental measurements

Frogs (Rana temporaria and Rana pipiens) were stunned, killed by decapitation and pithed. Single twitch muscle fibres and small bundles of muscle fibres (2-6 twitch fibres) were isolated from the tibialis anterior, iliofibularis and semitendinosus muscles. The fibres were dissected in a polycarbonate trough containing standard Ringer solution (mM: NaCl, 115; KCl, 2.5; CaCl2, 1.8; NaH2PO4, 0.85; Na2HPO4, 2.15; pH 7.2). The entire dissecting trough was then transferred to the experimental apparatus and clamped to a stainless steel baseplate mounted on a Peltier-effect heat pump. The Peltier device was regulated by a feedback circuit which controlled the temperature of the bathing Ringer solution. The hooks of the force transducer and puller were lowered into the trough and the fibre connected between these using small clips cut from gold foil (thickness ∼25 μm). The clips crimped tightly around the tendinous sheets within 0.5 mm of each end of the fibre. The hooks (bent from stainless steel wire, AISI 316 (Goodfellow, UK), diameter 250 μm) were shaped to form a footplate which held the clips horizontal and helped to prevent the fibre from sagging under its own weight.

A number of different force transducers were used in these experiments. Each was constructed from an extended silicon beam sensor element (AE801, SensoNor, Norway) and had a resonant frequency in the range 400-1000 Hz. Although the resonant frequency of each modified sensor was not particularly high, the sensors were more than adequate for the stretch rise times (25 ms to 30 s) used in this work. The advantage of increased sensitivity obtained by extending the force transducer beam, outweighs the disadvantage of the accompanying reduction in resonant frequency for these experiments. The puller was a precision servo-controlled motor (Model 308B, Cambridge Technology Inc., Cambridge, MA, USA) with a modified, low inertia arm.

The fibres were illuminated near the force transducer with a 5 mW HeNe laser (LGK7267, Zeiss). A first-order diffracted beam was imaged through a cylindrical lens (focal length 0.3 m) onto a linear position-sensitive detector (SL76-1, United Detector Technology, Orlando, FL, USA) mounted 1.4 m distant from the fibre. This allowed measurement of the mean sarcomere length within the illuminated fibre segment (∼ 0.7 mm in length).

The puller system was interfaced with a PC (466/MX, Dell) through a combined ADC and DAC device (1401, Cambridge Electronic Design, UK). Pre-programmed stretches could be applied via the puller using the DAC module while the resulting force, fibre length and sarcomere length signals were recorded via the ADC module and saved to computer files for off-line analysis. A sampling frequency appropriate to the stretch duration was used, and ranged between 100 Hz and 10 kHz. The ADC and DAC operations were controlled by commercially available software (Spike2, Cambridge Electronic Design).

The experimental results revealed no obvious, consistent differences between preparations obtained from different muscles. Similar experimental trends were shown in many experiments on both single fibres and small bundles of fibres. There is, however, considerable variation in the passive stiffness of frog skeletal muscle e.g. seasonal changes of as much as 50 % (Robson, 1990) and, together with the difficulty of accurately normalizing for fibre cross-sectional areas, this significantly complicates comparisons between different fibre preparations. Consequently the data shown in this paper were derived from individual preparations.

Modelling

A cross-bridge population displacement mechanism

A theoretical model has been developed in which the SREC and FRT arise as a manifestation of a small number of slowly cycling cross-bridges in relaxed muscle fibres. Briefly, the ‘Cross-bridge Population Displacement Mechanism’ consists of three components: a cross-bridge component, a parallel elastic component and a series elastic component. This three-component structure has previously been used to analyse experimental data (Haugen & Sten-Knudsen, 1981) but the dynamics of the cross-bridge component (and thus the model's predictive ability) have been developed during the course of this work. The two main assumptions underpinning the cross-bridge mechanism are that the detachment rate constant for an individual cross-bridge is dependent on the strain in its elastic linkage and that the attachment rate constant increases with the speed of interfilamentary movement (an effect which we attribute to a ‘movement enhancement’ mechanism). Full details of the Cross-bridge Population Displacement Mechanism are given in the Appendix to this paper.

The simulation algorithm was translated into source code for the C programming language and simulations were performed using a 66 MHz 486 PC. Simulating 1 s of the muscle's response took approximately 1 min of computation time.

RESULTS

Experimental measurements

Paired stretches

Figure 1 is a typical example of the tension and sarcomere length responses of a relaxed skeletal muscle fibre to a pair of identical constant-velocity triangular stretches. In agreement with previous results (e.g. Hill, 1968; Bagni et al. 1995) the tension response to the first stretch consisted of two distinct phases. Initially, the muscle presented considerable resistance to the imposed stretch and the tension rose sharply and approximately proportionately with the applied length change.

Figure 1.

Force and sarcomere length responses to two identical constant-velocity triangular stretches

Single relaxed semitendinosus muscle fibre. In all figures L0 is the fibre length which corresponds to a mean sarcomere length of 2.2 μm. Stretch length ∼ 0.01 L0. Stretch velocity ∼ 0.01 L0 s−1. Two stretches separated by 1.0 s. The fibre had been held at a constant length for 1 min before the first stretch was initiated. Tensions were measured from a baseline (arbitrarily assigned to be zero force) which corresponded to the mean resting tension in the 0.5 s period preceding the first stretch. Thus measured tension values could be less than, as well as greater than, the prevailing resting tension. Dashed lines show the tension baseline and the corresponding initial sarcomere length. The initial phase (SREC) proceeded until an elastic limit was reached. In the first stretch, this elastic limit occurred at a sarcomere length of 2.180 μm and a tension of 2.4 μN. The corresponding values in the second stretch were 2.180 μm and 1.2 μN. Temperature 5.0 °C.

As the length stretch continued there was a transient drop in tension. This drop was small compared with the overall tension response. The drop (while a common experimental finding e.g. Hill, 1968; Lännergren, 1971; Herbst, 1976) is not a feature which can be observed in all fibre preparations. Even if it is present at short sarcomere lengths it can be removed by stretching the muscle to a longer initial length (Haugen & Sten-Knudsen, 1981). It would thus appear to be critically dependent on the initial tension and/or length of the muscle.

Beyond this transient drop the tension continued to rise but at a slower rate and the tension reached a maximal value at the extreme stretch length. During the shortening phase of the triangular stretch, the tension fell with a concave profile and discontinuities in the tension response were much less pronounced than during the lengthening phase.

Although the fibre was restored to its original length at the end of the triangular stretch, the resting tension was reduced below its initial value. Much of this tension deficit was recovered during the first few hundred milliseconds after the end of the stretch but it is clear that neither the resting tension nor the sarcomere length recovered completely to their initial levels during the 1 s interval before a second identical stretch was imposed on the muscle. During the inter-stretch interval, fibre length was fixed, whereas sarcomere length progressively shortened with an associated rise in fibre tension. The rise in resting tension during the inter-stretch interval could only be attributed to an extension of passive elastic structures if sarcomere length increased simultaneously. The experimental results show that during the inter-stretch interval, sarcomere length shortened slightly, and thus the rise in tension cannot be attributed to an increased stretch of elastic non-cross-bridge components within the sarcomere. An increasing tension accompanied by sarcomeric shortening suggests to us that the contractile apparatus may be responsible.

In accordance with Hill's chosen nomenclature (1968) we refer to the initial phase of the tension response as the ‘short-range elastic component’ (SREC) and the length increment required to produce the tension transient as the elastic limit. An additional term, the elastic limit force, is defined in this work as the tension increment produced by the SREC phase of the response. The muscle's resting tension is considered to include both actively and passively generated components. Passive resting tension is elastic and reflects stretch of non-cross-bridge components. The actively generated portion of the resting tension is known as the filamentary resting tension (FRT). The tension in a relaxed muscle at any instant is thus the sum of the stress in the parallel elastic component and the force due to the attached cross-bridges. At physiological sarcomere lengths, the precise magnitudes of these components are difficult to determine but they are unlikely to be zero. As A. V. Hill (1952) pointed out, ‘the tension of a resting muscle diminishes continually with decreasing length; there is no sharp end point and it is impossible to define a length at which the tension becomes zero.‘

The SREC stiffness was calculated as a Young's Modulus from the increase in stress produced by the relative fibre length change during the initial linear phase of the response. The mean SREC stiffness calculated in these experiments was 1.98 × 105 N m−2 (s.e.m.± 0.44 × 105 N m−2, n= 16 preparations, stretch velocity ∼0.01 L0 s−1, where L0 is the fibre length which corresponds to a mean sarcomere length of 2.2 μm, minimum pre-stretch interval 60 s, temperature 5-7°C). This is in excellent agreement with other published values e.g. Hill (1968: frog sartorius, 1.3 × 105 N m−2, 20°C), Lännergren (1971: frog iliofibularis, 2.3 × 105 N m−2, 20- 24°C), Halpern & Moss (1976: frog semitendinosus, 2.0 × 105 N m−2, 3°C) and Lakie & Robson (1990: frog iliofibularis 1.8 × 105 N m−2, 3°C).

The second stretch produced a qualitatively similar biphasic tension response to the first (Fig. 1). However there were clear and consistent differences between the responses to the first and second stretches. In the second stretch, the SREC tension rose more slowly and from a slightly lower baseline than during the first stretch. Furthermore, the transient drop in tension at the elastic limit was no longer present and the elastic limit force was considerably reduced. In the two stretches, there was no consistent difference in either the maximal tension or the maximal sarcomere length attained.

The extent of the reduction of the second stretch elastic limit force depends on the time interval between the stretches. Figure 2 shows the recovery time course of the elastic limit force. The thixotropic reduction observed when the second stretch followed immediately after the first (i.e. an inter-stretch interval of 0 s) was not substantially greater than that produced by an inter-stretch interval in the range 0.5-1 s. As the inter-stretch interval was increased above 1 s, the elastic limit force recovered relatively slowly towards a saturating value. The finding that for inter-stretch intervals of greater than ∼1 s, the elastic limit force recovered at a progressively declining rate is in agreement with other published work (Buchthal & Kaiser, 1951; Lännergren, 1971; Lakie & Robson, 1988b). However, in the present experiments, the recovery in the elastic limit force for inter-stretch intervals of less than ∼1 s was slight. This consistent finding is a feature of the experimental results which does not appear to have been previously reported.

Figure 2.

SREC recovery following an initial stretch

Single semitendinosus muscle fibre. Individual trials were separated by a fixed period of 1 min and consisted of two identical triangular stretches of length ∼ 0.01 L0, velocity ∼ 0.01 L0 s−1 with a variable inter-stretch interval. Symbols show the mean elastic limit force ± standard deviation for the second stretch tension response at each inter-stretch interval. Dashed line shows the mean elastic limit force for the first stretch. This was independent of the inter-stretch interval. A minimum of three trials were performed at each inter-stretch interval. Temperature 5.5 °C. Four similar experiments were performed using single fibre preparations. With an inter-stretch interval of 0.5 s, the second stretch elastic limit force was 0.72 ± 0.09 (mean ± standard deviation, n= 4 preparations) of the corresponding first stretch value.

Any physical explanation proposed to underlie the SREC and FRT of relaxed skeletal muscle must account for these unusual mechanical properties. Figures 3 and 4 illustrate additional experimental evidence which imposes constraints on proposed theoretical mechanisms.

Figure 3 shows the tension and sarcomere length responses of an isolated single fibre to a pair of identical triangular stretches. The responses are plotted not against time but as an X-Y plot of tension against sarcomere length. This X-Y plot highlights the fact that the muscle is not purely elastic; in a purely elastic system tension is a single-valued function of length. Furthermore, the X-Y plot clearly shows that the FRT of the fibre was reduced by the first stretch and that it did not completely recover to its initial value in the 1 s interval between the imposed stretches. Any reduction in the maximal tension produced by each stretch was much less than the substantial reduction of around 1 μN in the FRT. This finding has been previously described (Campbell & Lakie, 1996).

Figure 3.

X-Y plot of force against sarcomere length for two constant-velocity triangular stretches

Single intact iliofibularis muscle fibre. Stretch length ∼ 0.011 L0. Stretch velocity ∼ 0.011 L0 s−1. Two stretches separated by 1.0 s. The fibre had been held at a constant length for 1 min before the first stretch was initiated. Temperature 6.0 °C.

Figure 4 shows the tension and sarcomere length responses of a single relaxed fibre to three identical consecutive triangular stretches. Although the second stretch SREC stiffness and elastic limit force values were reduced relative to the first stretch values, no systematic differences between the second and third stretch values were apparent. Indeed when more than three stretches were imposed on the muscle, no progressive differences were observed between the second and subsequent stretch responses. This finding suggests that the underlying physical mechanism is in an identical state at the end of the first, second and any subsequent stretch.

Figure 4.

Force and sarcomere length responses to three identical constant-velocity triangular stretches

Single intact iliofibularis muscle fibre. Stretch length ∼ 0.011 L0. Stretch velocity ∼ 0.011 L0 s−1. The fibre had been held at a constant length for 1 min before the first stretch was initiated. Temperature 5.0 °C.

Response to length changes where shortening preceded lengthening

The biphasic nature of the tension response is most sharply delineated if the muscle has been maintained at a constant length for a relatively long period of time. Using triangular stretches where lengthening preceded shortening, the SREC was always more obvious in the lengthening phase (Fig. 5, thick lines). However if the triangle was reversed and the muscle was initially shortened from its resting length, the discontinuities in the tension response were much more evident during the shortening phase (Fig. 5, thin lines).

Figure 5.

Force and sarcomere length responses to triangular stretches of opposite polarity

Single relaxed iliofibularis muscle fibre. Thick lines show the tension and sarcomere length responses to a triangular stretch where lengthening preceded shortening. Thin lines show responses to a triangular stretch where shortening preceded lengthening. Stretch magnitude ∼ 0.01 L0. Stretch velocity ∼± 0.01 L0 s−1. The fibre had been held at a constant length for 1 min before each stretch was initiated. Dashed lines show the tension baseline (defined as in Fig. 1) and the corresponding initial sarcomere length. Temperature 7.0 °C.

While the resting tension following a triangular stretch where lengthening preceded shortening was temporarily reduced below its initial value, in the converse situation where shortening preceded lengthening the resting tension was slightly increased at the end of the triangular stretch (Fig. 5).

Velocity dependence

Figure 6 shows the tension and sarcomere length responses of a relaxed single fibre to two triangular stretches of constant length but different velocity. Both the elastic limit force and the tension at the extreme stretch length were substantially greater for the faster stretch. It is clear therefore that the magnitude of the biphasic tension response is affected by the rate of stretching.

Figure 6.

Force and sarcomere length responses of the same fibre to two different velocity triangular stretches

Single iliofibularis muscle fibre. Stretch length ∼ 0.009 L0. Stretch velocities ∼ 0.009 and ∼ 0.09 L0 s−1. Note that the right-hand traces are drawn with a time base ten times faster than the left-hand traces. In both cases, the muscle fibre had been held at a constant length for 1 min before the stretch was initiated. Dashed lines show the tension baseline and the corresponding initial sarcomere length. Temperature 6.0 °C.

The precise nature of the velocity dependence of the tension responses of relaxed skeletal muscles to imposed movements may shed light on the responsible mechanisms. In the present experiments, tension responses were measured using stretches with velocities ranging from 5 × 10−4 to 2 L0 s−1. The maximum shortening velocity of frog skeletal muscles is of the order of 2 L0 s−1 near 5°C, and it seems likely that the range of stretch velocities used in the present experiments encompasses those which the intact muscle would be subjected to under normal physiological conditions.

Figure 7 shows the measured values of the SREC stiffness of a single iliofibularis muscle fibre for an approximately 1600-fold range of stretch velocities. The SREC stiffness approximately doubled over this velocity range increasing from ∼ 86 000 N m−2 at a stretch velocity of 7.5 × 10−4L0 s−1 to a value of ∼ 180 000 N m−2 at a stretch velocity of 1.2 L0 s−1.

Figure 7.

SREC stiffness for different stretch velocities

Intact iliofibularis muscle fibre. Triangular stretch length ∼ 0.009 L0. Trials were repeated at fixed intervals of 1 min. Stiffness expressed as Young's Modulus. Temperature 5.5 °C. The continuous line shows the best fit of a curvilinear relationship chosen to represent the trend of the experimental data. It is not the prediction of a theoretical model.

Measured values of the elastic limit force for stretch velocities ranging from 7.5 × 10−4 to 1.2 L0 s−1 are shown in Fig. 8. The elastic limit force increased approximately 5-fold from 1.3 to 6.9 μN as the velocity was raised by a factor of ∼1600.

Figure 8.

Elastic limit force for different stretch velocities

Single iliofibularis muscle fibre. Triangular stretch length ∼ 0.009 L0. Trials were repeated at fixed intervals of 1 min. Temperature 5.5 °C. As in Fig. 7 the continuous line represents the trend of the experimental data.

The measured elastic limit values of a bundle of five iliofibularis muscle fibres for stretch velocities ranging from 5.7 × 10−4 to 1.8 L0 s−1 are shown in Fig. 9. The elastic limit increased by a factor of approximately 3 over this velocity range.

Figure 9.

Elastic limit for different stretch velocities

Bundle of 5 intact iliofibularis muscle fibres. Triangular stretch length ∼ 0.015 L0. Trials were repeated at fixed intervals of 1 min. Temperature 5.0 °C. As in Fig. 7 the continuous line represents the trend of the experimental data.

If the SREC tension response was attributed solely to the acceleration of a viscous component within the sarcomere, the elastic limit force would be directly proportional to the stretch velocity i.e. it would increase linearly with stretch velocity from a value of zero at zero stretch velocity. Inspection of Figs 7 and 8 shows that neither the SREC stiffness nor the elastic limit force values extrapolate to zero at zero velocity. The SREC cannot be wholly attributed to a simple viscous mechanism. However, neither can the SREC be solely attributed to a simple elastic mechanism since the SREC stiffness (Fig. 7) and the elastic limit (Fig. 9) increase with stretch velocity. Hill (1968) and Lännergren (1971) also reported that the elastic limit increased with the velocity of stretching.

Modelling

Predictions of the Cross-bridge Population Displacement Mechanism

Paired stretches

This section presents some of the predictions of the Cross-bridge Population Displacement Mechanism. (The model itself is fully described in the Appendix to this paper.) Figure 10 shows the predicted tension and sarcomere length responses to two identical triangular stretches.

Figure 10.

Simulation of force, cross-bridge force, proportion of attached cross-bridges and sarcomere length responses to two identical constant-velocity triangular stretches

Stretch length 0.01 L0. Stretch velocity 0.01 L0 s−1. Two stretches separated by 1.0 s. Force simulations: thick line, overall muscle tension; thin line, internal cross-bridge force Fcb. The passive component of the tension response is represented by the difference between the overall muscle tension and the internal cross-bridge force. In the interests of clarity it has not been plotted in this diagram. As in the case of the experimental results, forces are measured as deviations from the prevailing resting tension at the beginning of the first stretch. Thus tension values can be less than as well as greater than zero. This diagram should be compared with the corresponding experimental results shown in Fig. 1. Dotted lines show the tension baseline, the initial sarcomere length and the proportion of cross-bridges attached at the commencement of the first stretch.

The model's prediction (Fig. 10) is in good qualitative agreement with the experimental results shown in Fig. 1; there is a clear SREC and a transient drop in tension at the elastic limit. Furthermore the tension at the end of the first stretch is reduced below the initial resting tension implying a temporary reduction in the FRT. The second stretch produces a similar biphasic tension response to the first but the SREC tension rises more slowly and from a lower base-line than during the first stretch. While the tension at the elastic limit is considerably reduced for the second stretch, the model predicts the same tension at the extreme stretch length for each of the two stretches.

The simulated values of the elastic limit stress (i.e. elastic limit force per unit area, 2750 N m−2) and the SREC stiffness (1.2 × 106 N m−2, Young's Modulus) are greater than the mean experimental values by factors of ∼ 5 and ∼ 6, respectively. Modifications to the present model which would improve the quantitative fit between these simulated parameters and the experimental results include (1) reducing the cross-bridge stiffness, (2) reducing the equilibrium number of attached cross-bridges and (3) altering the cross-bridge detachment rate so that it increases more quickly with cross-bridge strain than presently assumed (eqn (A9)). Although these modifications alter the simulated values of the SREC stiffness and elastic limit force towards the mean experimental values, they worsen other qualitative aspects of the fit between the experimental results and the simulated tension responses. It must be emphasized that the present simulations were developed to ascertain whether a slowly cycling cross-bridge hypothesis was broadly compatible with the thixotropic nature of the SREC tension response to imposed length changes of a physiological velocity. The simulated values of the SREC stiffness and elastic limit force are of the same order of magnitude as the mean experimental values. Consequently no serious attempts were made to improve this aspect of the model and the following simulations are intended to illustrate experimental trends rather to calculate specific numerical values.

Figure 11 shows the recovery time course of the SREC. It is also in reasonable qualitative agreement with the experimental data (Fig. 2). The second stretch elastic limit force is reduced relative to the first stretch value by an amount which depends on the inter-stretch interval. The maximum calculated reduction (∼42 %) is produced by an inter-stretch interval of 0.5 s. As the inter-stretch interval is increased above ∼2 s, the simulated elastic limit stress recovers at a declining rate towards a saturating value. However, for inter-stretch intervals of less than ∼1 s, the simulated second stretch elastic limit stress changes by only 4 %.

Figure 11.

Simulation of the elastic limit force recovery following an initial stretch

Two identical stretches separated by a variable time delay. Stretch length 0.01 L0. Stretch velocity 0.01 L0 s−1. Symbols show the elastic limit force for the second stretch. Dashed line shows the elastic limit force for the first stretch. This was independent of the inter-stretch interval. This diagram should be compared with the corresponding experimental results shown in Fig. 2.

Velocity dependence

The velocity dependence of the simulations can also be examined. In Fig. 12, ▴ show the values of the SREC stiffness, elastic limit force and elastic limit predicted by the Cross-bridge Population Displacement Mechanism presented in the Appendix for stretch velocities ranging from 2 × 10−4 to 2 L0 s−1. The simulated SREC stiffness is independent of the imposed stretch velocity above 0.02 L0 s−1; the elastic limit stress and elastic limit increase approximately proportionally with the logarithm of the stretch velocity throughout the range of velocities investigated.

Figure 12.

Simulation of the velocity dependence of the tension responses

A, SREC stiffness (Young's Modulus). B, elastic limit stress. C, elastic limit. Stretch length 0.02 L0. ▴, values simulated by the Cross-bridge Population Displacement Mechanism presented in the Appendix. ^, simulated values when a viscous component is added in parallel with the cross-bridge and parallel elastic components shown in Fig. 13. The viscosity produces a force Fv where η represents the viscous coefficient (1.82 × 109 N m−3 s per half-sarcomere) and v is the interfilamentary sliding velocity in metres per second. The viscous component has a negligible effect on the simulated values for stretch velocities less than 0.02 L0 s−1.

The qualitative fit between the experimental results (Figs 7, 8 and 9) and the simulated values is improved by the addition of a viscous component in parallel with the parallel elastic and cross-bridge components shown in Fig. 13. The simulated values predicted by this adapted model are shown by ^ in Fig. 12. The addition of a viscous component causes the SREC stiffness and elastic limit force to increase more sharply with stretch velocity at the highest stretch velocities investigated. However, the viscosity has a negligible effect on the simulated tension responses at stretch velocities below 0.02 L0 s−1. The majority of the experiments investigating the thixotropic nature of the tension responses were performed using stretch velocities less than this value.

Figure 13.

Three-component model. A, parallel elastic and cross-bridge components in conjunction with a series elastic component. B, mathematical definitions

A, the series elastic component represents the tendon attachments. The parallel elastic component represents the effect of the sarcolemma, sarcoplasmic reticulum, titin filaments and other passive structures within the sarcomere. The parallel and series components are modelled as linear springs. The cross-bridge component is more complicated and simulates the cross-bridge interactions between actin and myosin filaments in a half-sarcomere. It acts both as a tension generator and a short-range elastic element. B, the model's state is defined by the parallel component length Xp, the parallel component stiffness kp, the series component length Xs, the series component stiffness ks and the cross-bridge force Fcb. The overall length X and tension F are defined in terms of these parameters. The parallel component length Xp corresponds to the mean half-sarcomere length of a real muscle.

DISCUSSION

The biphasic nature of the tension response

The experimental results presented here confirm that relaxed muscle fibres present a disproportionately high resistance to the initial stages of an imposed stretch. This biphasic behaviour was first fully described by D. K. Hill (1968). He named the initial phase of the tension response the short-range elastic component (SREC) and justified the term because the tension produced was almost linearly related to the size of the applied stretch up to an elastic limit of ∼0.2 %L0. The tension at the elastic limit was only affected to a minor degree by stretch velocity within the range of the relatively slow velocities that he used (1 × 10−6 to 0.2 L0 s−1).

Hill attributed the SREC to a low level of cross-bridge activation. He proposed a ‘working hypothesis’ in which a small proportion of long-lived cross-bridges are attached to the actin filaments. The elastic behaviour of the SREC is due to the spring-like properties of these linkages. The elasticity is short range because the cross-bridges can only be displaced by a small amount before their elastic limit is reached. These cross-bridges also generate an active tension - the filamentary resting tension (FRT).

Recent work (Bagni et al. 1992, 1995; Mutungi & Ranatunga, 1996a, b) has provided more information about the biphasic tension response. These experiments investigated the tension and sarcomere length responses of relaxed skeletal muscle fibres to imposed stretches and used generally faster stretches than Hill, e.g. Bagni et al. (1995): 2-250 L0 s−1; Mutungi & Ranatunga (1996b): 0.01-15 L0 s−1. At all stretch velocities, the measured tension response was biphasic and qualitatively similar to that described by Hill.

Bagni et al.‘s clear experimental records revealed that during the initial phase of the tension response, the sarcomere length lagged behind the imposed fibre length change. The sarcomere length accelerated throughout the initial phase of the tension response. The transition to the second phase of a less rapid rise in tension occurred when the sarcomere length had stopped accelerating and had attained a constant velocity. Bagni et al. concluded that the first phase of the tension response primarily resulted from the acceleration of a viscous element within the sarcomere. The complete tension response of a skinned muscle fibre to a single imposed ramp stretch could be accurately simulated with a three-element model consisting of a parallel combination of a viscosity, an elasticity and a visco-elastic element (a pure elasticity in series with a pure viscosity). This analysis suggests that the maximal tension produced during the initial phase of the response would be effectively proportional to the stretch velocity, i.e. it increases linearly with velocity but is zero at zero velocity.

Experimental results obtained from intact muscle fibres showed an additional tension contribution to the initial phase of the response which Bagni et al. attributed to the SREC. A graph of the maximal tension produced during the initial phase of the response against stretch velocity (Fig. 4 (squares) in Bagni et al. 1995) showed that the measured values increased linearly with stretch velocity but extrapolated to a positive intercept (rather than zero) at zero velocity. At the lower stretch velocities investigated by Bagni et al. this additional SREC component was a substantial fraction of the tension response that they measured.

The experiments of Hill and of Bagni et al. were performed using different types of preparations and different ranges of stretch velocities. However, both sets of experimental results suggest that the maximal tension produced during the initial phase of the response in intact muscle fibres increases with stretch velocity but is not zero at zero velocity. These results and the present experimental results suggest to us that the initial phase of the tension response may be dominated by elastic properties at low stretch velocities and by viscous properties at high stretch velocities.

A similar partly viscous, partly elastic tension response to stretch velocities of less than 2 L0 s−1 is suggested by Mutungi & Ranatunga's experimental results (1996b) for mammalian fibres.

Any complete theory underlying the biphasic tension response of relaxed muscle must explain not only the viscous and elastic components of the tension response but also their dependence on mechanical history (Fig. 2). The three-element model suggested by Bagni et al. to explain their experimental results does not provide a satisfactory explanation for all of the experimental results presented in this paper. Firstly Bagni et al.‘s three-element model does not explain the positive tension component which they attribute to the SREC. In the present experiments the equivalent positive intercept is a substantial part of the measured tension responses. Secondly, Bagni et al.‘s model cannot account for the thixotropic nature of the observed tension responses. Only the visco-elastic element could produce a tension response which is dependent on the prior history of movement. The visco-elastic relaxation time of ∼1 ms calculated from their results is at least two orders of magnitude too short to account for the thixotropic tension response shown in Fig. 2. This relaxation time is also too short to account for the long-lived reduction in FRT following the end of an imposed triangular stretch (Fig. 1). While viscous and visco-elastic systems must dominate the tension and sarcomere length responses at very high stretch velocities, they cannot be the dominant mechanisms underlying the experimental observations using the physiological stretch velocities described in this work.

A cross-bridge basis for the SREC and FRT?

At the relatively low stretch velocities described in this work, the SREC produces a tension which is approximately proportional to the applied length change below the elastic limit. This is characteristic of a response dominated by an elastic mechanism. Further evidence of the elastic nature of the SREC mechanism is the presence of a clear elastic limit in the tension response even when extremely slow stretches are used (e.g. 8 × 10−6L0 s−1, M. Lakie & K. S. Campbell, unpublished observations). A full explanation of the elasticity must account for its limited range, its mechanical history dependence, the FRT and its reduction following a triangular stretch. A small number of stable cross-bridges bound between the filaments in relaxed muscle can provide a satisfactory basis for this elasticity.

Is there any evidence that cross-bridges are bound between actin and myosin filaments in relaxed muscle? Experimental results suggest that a significant proportion of cross-bridges become attached to actin in skinned fibres at low ionic strength (Brenner, Schoenberg, Chalovich, Greene & Eisenberg, 1982; Schoenberg, 1988; Brenner, 1990; Tregear, Townes, Gabriel & Ellington, 1993). These cross-bridges are believed to correspond to weakly bound states and are in rapid equilibrium. Schoenberg (1985) showed that a distribution of these cycling cross-bridges, each acting as a linear spring with one attachment and one detachment rate constant, is mechanically analogous to a visco-elastic system with a relaxation time of ∼100 μs. These rapidly cycling cross-bridges cannot be responsible for the SREC. If the SREC is to be attributed to cross-bridges, it is essential that the responsible cross-bridges cycle only slowly.

While there is no direct evidence for the presence of long-lasting cross-bridges in relaxed muscle, there is a strong relationship between the SREC and the intracellular Ca2+ concentration. Moss, Sollins & Julian (1976) showed that the size of the SREC in skinned skeletal and cardiac muscle fibres was dependent on the concentration of Ca2+ in the fluid bathing the myofibrils. This Ca2+ dependence supports the hypothesis that the SREC is a manifestation of cross-bridge activity because Ca2+ is known to control cross-bridge cycling in contracting muscle. Further supporting evidence comes from the work of Isaacson (1969) who demonstrated that Ringer solution made hypertonic with sucrose elevates the intracellular Ca2+ concentration in frog sartorius muscles. This increase in Ca2+ concentration may underlie the increase in the magnitude of both the SREC and the FRT in such hypertonic Ringer solutions (Hill, 1968; Lännergren & Noth, 1973).

Although Hill suggested that hypertonicity caused a mechanically induced increase in cross-bridge interaction between the actin and myosin filaments, Lännergren & Noth (1973) have shown that hypertonic Ringer solutions partially activate the contractile apparatus. They also showed that tetracaine substantially reduced (but did not abolish) the increase in FRT produced by hypertonicity without affecting the increase in the SREC. This finding was interpreted as implying that the SREC and FRT could not share a common cross-bridge origin. However, it is now known that tetracaine is a ryanodine receptor blocker and inhibits the calcium-induced calcium release mechanism (Endo, 1992). Tetracaine will not eliminate the increase in intracellular Ca2+ concentration due to the hypertonic medium but does prevent the internal amplification of Ca2+ by the calcium-induced calcium release mechanism. While the hypertonic Ringer solution elevates the intracellular Ca2+ concentration sufficiently to promote increased cross-bridge attachment (and thus a larger SREC), tetracaine may prevent intracellular Ca2+ escalating to a level which causes substantial cross-bridge force production.

It is difficult to measure with certainty the intracellular concentration of free Ca2+ in a relaxed muscle fibre. Recent estimates using calcium-sensitive electrodes suggest a value of ∼0.16 μM (Rüegg, 1992). A skinned fibre preparation at this Ca2+ concentration would develop around 25 % of its maximal tension (Julian, 1971). While the calcium kinetics in an intact fibre may be different it seems entirely possible that there may be sufficient free Ca2+ under relaxed conditions to permit a low level of cross-bridge cycling. This would have metabolic implications. Raising the intracellular Ca2+ concentration would produce a large increase in cross-bridge turnover and a corresponding increase in the muscle's metabolic rate. Conversely active removal of Ca2+ from the cytoplasm would reduce cross-bridge turnover but would use energy to pump Ca2+. The minimal resting metabolic rate of the muscle may represent a compromise between the energy required to remove Ca2+ from the cytoplasm and the energy consumed by cross-bridge cycling.

We hypothesize that the SREC and FRT are actively generated by the calcium-dependent binding of cross-bridges between the actin and myosin filaments of relaxed muscle. It is possible to estimate how many cross-bridges are actually attached between the filaments if it is assumed (as in the case of tetanically stimulated muscle) that the stiffness of a relaxed muscle is proportional to the number of attached cross-bridges. The initial stiffness of the SREC is approximately 1 % of that of tetanically stimulated muscle. If it is assumed that between 50 and 75 % of cross-bridges are attached to thin filaments during a maximal isometric contraction (Bershitsky et al. 1997), this simple analysis suggests that the SREC and the FRT are manifestations of around 0.5-0.75 % of the total cross-bridge number.

Our experimental results impose two constraints on the behaviour of these cross-bridges if they are responsible for the SREC. The first constraint relates to the rate of attachment of cross-bridges. When the muscle is stretched beyond the elastic limit forcibly detaching cross-bridges, tension is maintained at or near the elastic limit level rather than collapsing to near zero (Fig. 1). As Hill pointed out, this behaviour requires that any forcibly detached cross-bridges are rapidly replaced to produce what he referred to as a ‘frictional resistance’. Thus cross-bridges are required to attach swiftly during interfilamentary movement. This implies that the cross-bridge attachment rate during movement must be high.

Since the stiffness of the SREC at any time is assumed to be controlled by the number of cross-bridges bound between the filaments, the thixotropic reduction in the size of the SREC following prior movement (Figs 1 and 2) may be explained if the number of bound cross-bridges is temporarily reduced by the perturbation. As the SREC recovers only slowly after prior movement (Fig. 2) the cross-bridge attachment rate following a perturbation must be relatively slow. There are thus conflicting requirements for a fast cross-bridge attachment rate during a stretch and a slow cross-bridge attachment rate between stretches. This can be resolved if the probability of cross-bridge attachment is increased during interfilamentary movement as a result of a ‘movement enhancement’ mechanism (Campbell & Lakie, 1995b). A number of mechanisms by which this might arise can be envisaged. Hill (1968) suggested that forcibly detached cross-bridges might immediately reattach further along the actin filament. A similar mechanism has been proposed for contracting muscle by Lombardi & Piazzesi (1990). Alternatively, it is possible to speculate that interfilamentary movement might ‘dislodge’ some tropomyosin molecules, temporarily exposing an increased number of binding sites on the actin filament, or that movement of the sarcoplasmic reticulum releases calcium ions into the sarcoplasm, thereby increasing cross-bridge activity. The present simulations are based on an alternative geometrical argument relating to misregistration between the cross-bridges on the myosin filaments and the binding sites on the actin filaments - see Appendix for details.

The second constraint is imposed by the form of the velocity dependence of the SREC. If the cross-bridges possessed both a single attachment and a single detachment rate constant, they would produce a visco-elastic response to stretch (Schoenberg, 1985). During imposed ramp stretches, the cross-bridges would produce a force which rose to a saturating plateau value directly proportional to the stretch velocity. Inspection of Fig. 8 shows that while the elastic limit force increases with stretch velocity it is very far from being directly proportional to it, i.e. doubling the stretch velocity produces less than double the initial elastic limit force. A simple way of explaining the relatively weak velocity dependence of the SREC is if cross-bridges detach more rapidly when pulled to more strained configurations. This hypothesis (that the cross-bridge detachment rate is dependent on strain) is consistent with most theories of active muscle contraction, e.g. Huxley (1957).

The maximal tensions produced by paired stretches are almost identical, despite the reduction in the prevailing resting tension at the commencement of the second stretch (Figs 1 and 3). As the two stretches commence at identical fibre lengths, the reduction in prevailing tension must reflect changes in actively generated tension (FRT), rather than in the tension produced by the passive stretch of non-cross-bridge components. If the imposed stretch produces the same tension increment during the first and second stretches, the maximal tension should be depressed to the same extent as the resting tension. Figures 1 and 3 clearly show that any reduction in the maximal tensions produced by the two stretches is much less than the substantial reduction of around 1 μN in the FRT. The hypothesis that the SREC and FRT are manifestations of a low level of cross-bridge activity in relaxed muscle provides a possible explanation for this result. After a long period with no interfilamentary movement, the cross-bridges linking the actin and myosin filaments evolve to a stable equilibrium distribution (Fig. 15). An applied stretch displaces these cross-bridges and distorts the equilibrium distribution. If the stretch is sufficiently long, the mechanical memory of the initial equilibrium distribution is lost and the continuing stretch maintains a skewed but steady-state dynamic distribution (Fig. 15). After the first stretch, the cross-bridge distribution takes many seconds to re-attain its initial equilibrium distribution. This altering distribution underlies the slow recovery of the FRT. The second applied stretch skews this partially recovered population which eventually re-attains the same steady-state dynamic distribution as during the first stretch. Once this steady-state distribution is achieved the tension responses for the two stretches are identical. The cross-bridge distributions are different at the beginning and end of the first stretch. However, the distributions are pulled into the same state at the end of the first, second and any subsequent identical stretch. This mechanism thus explains the difference between the tension responses to the first and second stretches, and the similarity of the tension responses to the second and third stretches shown in Fig. 4.

Figure 15.

Redistribution of the cross-bridge population during an imposed stretch

The six graphs show an illustrative example of the redistribution of the attached cross-bridges during an imposed stretch. Y-axes show the number of attached cross-bridges per nanometre as a fraction of the total cross-bridge number; X-axes show the corresponding cross-bridge displacements. The graphs illustrate the cross-bridge distribution at (from left to right, top to bottom) 0.0 s, 0.1 s, 0.2 s, 0.4 s, 0.6 s and 0.8 s, respectively, after the start of an imposed stretch. Stretch velocity 0.01 L0 s−1 (i.e. the filaments would be displaced at a relative velocity of 11 nm s−1 if the series elastic component was infinitely stiff). The cross-bridge population is redistributed from a symmetrical equilibrium profile around x0 at time t= 0, to a steady-state distribution maintained by the sustained stretch.

The biphasic nature of the tension response is most sharply delineated if the muscle has been maintained at a constant length for a relatively long period of time before the length change is imposed. Hill (1968), measuring the tension responses to imposed ramp stretches, observed equally clear discontinuities with shortening or lengthening stretches. In the present experiments, using triangular stretches where lengthening preceded shortening, the SREC was always more obvious in the lengthening phase (Fig. 5). However, if the triangle was reversed and the muscle was initially shortened from its resting length, the discontinuities were much more evident during the shortening phase (Fig. 5). Thus, the SREC is most evident during the first imposed length change, regardless of the direction of movement. This symmetrical behaviour, which to our knowledge has not previously been satisfactorily explained, is predicted by the Cross-bridge Population Displacement Mechanism.

When a muscle is subjected to an imposed length change, the magnitude of the cross-bridge force increases and, if the stretch continues for a sufficient period, is maintained at a constant level. This change in cross-bridge force represents a redistribution of cross-bridges from the equilibrium distribution to the steady-state dynamic distribution (Fig. 15) produced by the imposed length change. It is greatest when the cross-bridges are displaced from the narrow stable equilibrium distribution (Fig. 15). This explains the sharp SREC response to the first movement after a long time interval at a fixed length. The biphasic nature of the tension response is less obvious if the cross-bridges are redistributed from the broad steady-state distribution produced by movement. The Cross-bridge Population Displacement Mechanism confers on resting muscle a time-dependent elasticity which is self-resetting at all physiological lengths.

The Cross-bridge Population Displacement Mechanism

The model presented in this work predicts that the initial phase of the tension response to imposed stretches increases with stretch velocity but is decreased by prior movement. The form of the predictions of the model for the mechanical history dependence (Fig. 11) and the velocity dependence (Fig. 12) of the tension responses are in general agreement with the experimental results described in the present work (Figs 2, 7, 8 and 9) and elsewhere in the literature, e.g. Hill (1968) and Lännergren (1971).

The model also provides possible explanations for (1) a period of sarcomere length acceleration which was sometimes observed during the initial phase of the tension responses (Figs 1 and 10) and (2) the relationship between the SREC and the FRT.

Thixotropy

If a muscle is subjected to two identical stretches, the elastic limit force produced during the second stretch depends on the inter-stretch interval. If the two stretches are separated by less than ∼10 s the elastic limit force produced during the second stretch is reduced relative to that of the first (Fig. 2). The model attributes this thixotropic effect to a temporary reduction in the number of cross-bridges bound between the filaments. During an imposed stretch interfilamentary displacement skews the cross-bridge distribution (Fig. 15), and although highly strained cross-bridges are continually detaching, movement enhancement ensures that they are rapidly replaced by other less strained cross-bridges so that the number of attached cross-bridges remains constant (Fig. 10). At the end of the stretch a large number of cross-bridges are left in highly strained states. These cross-bridges detach. However, since there is relatively little interfilamentary movement, the movement enhancement mechanism can no longer provide rapid replacements for these detaching cross-bridges and the number of cross-bridges bound between the filaments falls slightly (Fig. 10). If the muscle is held at a constant length for a long period of time, the cross-bridge population gradually redevelops and evolves towards its stable equilibrium distribution (Fig. 15). However, this takes many seconds and if a second stretch is imposed before the population has completely redeveloped, the initial effect of the imposed stretch is to displace a reduced number of cross-bridges. This produces a reduced SREC.

The Cross-bridge Population Displacement Mechanism predicts that the reduction in the elastic limit force for inter-stretch intervals of zero is not substantially greater than that for short inter-stretch intervals in the range 0.5-1 s (Fig. 11). This prediction is supported by the experimental results (Fig. 2). An inter-stretch interval of 0.5 s produces the greatest simulated thixotropic reduction in the elastic limit force (Fig. 11) because the strained cross-bridges do not detach instantaneously once the imposed movement stops. It requires a short time interval for the strained cross-bridge population to fall. The maximum SREC reduction is produced by an inter-stretch interval which is long enough for highly strained cross-bridges to detach and yet short enough that the cross-bridge population does not recover significantly between the stretches.

Velocity dependence

The Cross-bridge Population Displacement Mechanism was developed to investigate whether Hill's cross-bridge hypothesis could be extended to account for the thixotropic nature of the tension responses to paired stretches. The model produced a satisfactory simulation of the thixotropic nature of the tension responses (Fig. 10). However, it is also desirable that the model satisfactorily simulates the form of the velocity dependence of the SREC. Inspection of Figs 7 and 8 shows that at the higher stretch velocities examined in this work, the relative increase in the measured values of the SREC stiffness and elastic limit force with stretch velocity is greater than that predicted by the cross-bridge mechanism (Fig. 12, ▴). The qualitative fit of the Cross-bridge Population Displacement Mechanism predictions (Fig. 12) to the velocity dependence data at high stretch velocities is improved if a viscous component is added in parallel with the cross-bridge and parallel elastic components shown in Fig. 13. This addition does not substantially change the simulated values of the SREC stiffness, elastic limit force and elastic limit at stretch velocities less than 0.02 L0 s−1. Neither does the viscous component affect the thixotropic nature of the simulated tension responses.

It must be emphasized that in the present model, the biphasic nature of the SREC tension response is attributed solely to the skewing of the cross-bridge distribution to the steady-state dynamic distribution produced by the imposed movement. At velocities much higher than those investigated here, the viscous resistance would dominate the simulated tension responses. However, at stretch velocities less than 0.02 L0 s−1, the viscous resistance is negligible compared with that produced by the cross-bridge component. Thus for stretch velocities less than 0.02 L0 s−1, the velocity dependence of the tension responses can be attributed solely to the effect of stretch velocity on the skewing of the cross-bridge distribution. In Fig. 12, the velocity dependence of the SREC parameters simulated with (^) and without (▴) the viscous component have been plotted separately to clarify this point.

The effect of stretch velocity on the displacement of the cross-bridge distribution is relatively complicated. Fast stretches displace cross-bridges further before they detach than slow stretches. This explains why the simulated elastic limit increases with stretch velocity (Fig. 12C). Examination of the parameters simulated without a viscous component shows that the effect of stretch velocity on the SREC stiffness is relatively weak (Fig. 12A). This is because the stiffness of the cross-bridge component is dominated by the number of cross-bridges bound at the beginning of the imposed movement - a number which cannot be influenced by the stretch itself. Thus to some extent, the velocity dependence of the elastic limit force (Fig. 12B) at stretch velocities less than 0.02 L0 s−1 is characteristic of that of a predominantly elastic system with an elastic limit which increases with the velocity of stretching.

Sarcomere length acceleration

Bagni et al. (1992, 1995) showed that sarcomere length accelerates throughout the initial stages of a fast stretch. Sarcomere length acceleration could sometimes be observed in the present experiments with stretch velocities as low as 0.01 L0 s−1. It appeared to be influenced by the prior mechanical history and was normally more apparent in the first of repeated stretches. It is interesting to note that the present model suggests that (at least for the slower stretches used in the present work) sarcomere length could accelerate as a result of changes in the cross-bridge force.

If the muscle is held in a fixed position for a long time, the cross-bridge population of the three-component model evolves to its stable equilibrium distribution (Fig. 15). If a length stretch is imposed on the muscle, the cross-bridge component initially presents considerable resistance to movement. The apparent stiffness of the combination of the parallel elastic element and the cross-bridges is relatively high and the applied stretch produces a relatively small increase in the length of the parallel elastic element. A significant proportion of the applied length change extends the series elastic element.

As the stretch continues, cross-bridges start to detach more rapidly due to their increased strain (Fig. 10). The combined stiffness of the parallel elastic element and the reduced cross-bridge population is now less than at the beginning of the stretch and the continuing fibre length change produces a proportionately greater elongation in the parallel elastic element than at the beginning of the stretch. This is sarcomere length acceleration.

If a second stretch is imposed on the muscle before the cross-bridge population has fully redeveloped (Fig. 10), the cross-bridge component will produce less resistance to the initial stages of the imposed stretch. The parallel elastic element will extend more rapidly at the beginning of the stretch and the subsequent sarcomere length acceleration will be reduced.

The model thus provides a possible explanation both for sarcomere length acceleration and its dependence on the mechanical history of the muscle during stretches of a physiological velocity.

The relationship between the SREC and FRT

The time course of the redevelopment of the SREC and FRT after an imposed stretch are clearly different (Fig. 1 and Hufschmidt & Schwaller, 1987). This disparity has been used as evidence to suggest that the SREC, the FRT, or both, must be generated by structures other than cross-bridges, e.g. Helber (1980) and Lännergren (1971).

A further strength of the Cross-bridge Population Displacement Mechanism is that it provides an explanation for this difference. The reduction in resting tension following a triangular stretch is largely because the mean cross-bridge displacement is negative; most of the cross-bridges are compressed. Tension recovery occurs as a result of two distinct mechanisms. Immediately after the stretch, negatively strained cross-bridges detach rapidly and the cross-bridge force rises. Thus the early phase of rising tension corresponds to a redistribution of the cross-bridge population. A second slower phase of the tension rise is due to the reattachment of cross-bridges as the population reattains its stable distribution.

The time course of the redevelopment of the FRT is controlled largely by the rate at which the cross-bridge distribution re-evolves to a symmetrical distribution around x0. The stiffness of the SREC is proportional to the number of cross-bridges attached between the filaments at any time. Thus, the SREC and FRT, while sharing a common origin, redevelop at different rates.

Possible objections to a cross-bridge basis for the SREC

It has long been known that the size of the SREC is not proportional to filament overlap (Hill, 1968). This observation has been used to suggest that the SREC cannot arise from a cross-bridge mechanism (Sandow, 1970). Experimental evidence shows that the SREC stiffness and tension actually peak in frog muscle fibres at a sarcomere length of around 3 μm (Haugen & Sten-Knudsen, 1981) rather than around 2.2 μm where maximal active tension is generated. This is despite the fact that filament overlap is reduced to around 50 % of its maximum value at 3 μm. If cross-bridge activity in relaxed muscle was only proportional to the number of actin binding sites within interacting range of cross-bridge heads, then the size of a cross-bridge-generated SREC would scale directly with filament overlap. However, at low Ca2+ concentrations the sarcomere length for optimal tension generation is increased to around 3 μm (Endo, 1973). This length would lie on the descending limb of the length- tension relationship obtained during maximal activation and suggests that the number of active cross-bridges at low Ca2+ concentrations is not simply proportional to filament overlap. The length sensitivity of the contractile apparatus to Ca2+ has been reviewed by Stephenson & Wendt (1984). The increase in calcium sensitivity at long sarcomere lengths may be a consequence of a number of mechanisms, e.g. length-dependent changes in myofilament lattice spacing or the differential distribution of troponin-C affinity for Ca2+ along the thin filament (Martyn, Coby, Huntsman & Gordon, 1993). Thus the overlap dependence of a cross-bridge-generated SREC may be quite different from that of active tension.

Similar uncertainties complicate the analysis of experiments utilizing 2,3-butanedione 2-monoxime (BDM). BDM is believed to depress, reversibly, actively generated tension both by reducing calcium release and by suppressing cross-bridge cycling. The mode of action appears to be different in different muscles and in those of different species (Lyster & Stephenson, 1995). BDM does not appear to influence the tension response of relaxed rat skeletal muscle to stretch (Mutungi & Ranatunga, 1996b). Before clear conclusions can be drawn from this result it is vital that the precise mode of action of BDM, whether through its effect on calcium release or on cross-bridge cycling, is established in rat skeletal muscle. If BDM mainly affects calcium release in activated muscle this may not significantly change resting calcium levels in relaxed muscle. Alternatively, if BDM only stops a proportion of bound cross-bridges undergoing a power-stroke and generating tension it may not have a significant effect on the SREC.

Hill (1968) concluded that since the SREC was considerably less stiff than either the tendons or the sliding filaments ‘the material responsible for the elastic response must be located between the filaments’. Strictly speaking, Hill's argument does not lead inevitably to this conclusion and the SREC response could, in principle, arise from any connection between the Z-lines of the sarcomere. The significance of this clarification is increased following recent suggestions (Mutungi & Ranatunga, 1996a, b; Linke et al. 1996; Bartoo et al. 1997) that the visco-elastic properties of relaxed skeletal muscle are dominated by titin filaments (the exceptionally large proteins which link the thick myosin filaments to their adjacent Z-lines).

We believe that titin filaments may bear a substantial portion of the resting tension at long sarcomere lengths (Horowits, Kempner, Bisher & Podolsky, 1986; Keller, 1997) but cannot wholly account for the properties of the SREC and FRT described in this paper. The evidence supporting this viewpoint is compelling. First, if titin controlled the mechanical history-dependent FRT, it would have to bear tension at the short (∼2.2 μm) sarcomere lengths at which the present experiments were performed. The experimental evidence suggests that this is not the case. Radiation-induced damage to titin causes a dose-dependent reduction in the resting tension of rabbit psoas muscles stretched beyond a sarcomere length of 2.6 μm but does not affect the mean sarcomere length of a slack fibre (Horowits et al. 1986). Furthermore, the segmental extension model (Wang, McCarter, Wright, Beverley & Ramirez-Mitchell, 1991) suggests that titin behaves as a dual-stage molecular spring so that moderate stretches extend the I-band portion of titin with an accompanying exponential rise in tension (Linke et al. 1996). However, Wang et al. (1991) reported that skeletal muscles fibres showed no significant increase in passive resting tension below sarcomere lengths of 2.5 μm. These findings imply that titin does not normally contribute to passive resting tension near slack length. Secondly, if titin produced the biphasic SREC tension response it would have to be disproportionately stiff for small movements. Titin would have to have the characteristics of a softening spring. Again, the present experimental evidence is not consistent with this behaviour. Recent experiments (Tskhovrebova, Trinick, Sleep & Simmons, 1997) in which the force-extension relationship of a single titin molecule was measured using an optical-tweezers technique show that titin behaves as a non-linear stiffening elastic element for dynamic stretches. Tskhovrebova et al. (1997) interpreted their results as suggesting that at short lengths, the I-band region of titin adopted a random-coil configuration. Imposed stretches first straightened the molecule and then extended the polypeptide chain of the PEVK region (Tskhovrebova & Trinick, 1997). At extreme stretch lengths, there was the possibility of the stiffer immunoglobulin domains unfolding (Labeit & Kolmerer, 1995; Keller, 1997). In summary, experimental results suggest that during the initial stages of an imposed stretch titin produces less resistance to movement than it does during the latter stages. Titin molecules thus appear to possess the opposite mechanical properties to those required to generate the SREC.

An alternative judgement has been reached by Bartoo et al. (1997) who concluded that the passive mechanical properties (though not specifically the SREC and FRT) of relaxed skeletal and cardiac muscle fibres were dominated by titin filaments at all sarcomere lengths. If this is indeed the case, titin must possess complicated short-range elastic properties. Bartoo et al. measured the tension response to sinusoidal length perturbations of varying amplitude. The apparent stiffness of the myofibrils increased as the size of the length perturbation was reduced. This amplitude dependence persisted even when the mean sarcomere length was increased to a length at which there was ‘negligible thick and thin filament overlap’ (Bartoo et al. 1997) and the authors thus concluded that the effect could not be due to cross-bridge activity. However careful inspection of their Fig. 1 shows that the proportional increase in stiffness was greatest at the short sarcomere lengths corresponding to full filament overlap. (For oscillations ranging from 80 to 5 nm per half-sarcomere, stiffness increased by 260 % at a sarcomere length of 1.90 μm. The corresponding increase at 3.61 μm was just 40 %.) This striking dependence on filament overlap strongly suggests to us that the phenomenon may be partially attributed to the forcible detachment of stable cross-bridges by the imposed length perturbations. The residual 40 % increase at long sarcomere lengths could either arise from titin (as suggested by Bartoo et al.) or, if there were sarcomere length inhomogeneities, from a degree of filament overlap in some sarcomeres. Bartoo et al.‘s observations of an increased stiffness for small movements in myofibrils are very similar to those described by Buchthal & Kaiser (1951) in intact fibres. If, as we believe is likely, the increased stiffness for small movements is synonymous with the SREC, it may arise either from titin (but see comments above) or from cross-bridges. However, we entirely agree with Bartoo et al. that the behaviour cannot arise as a result of weakly binding rapidly cycling cross-bridges - it is an essential requirement of the Cross-bridge Population Displacement Mechanism that the responsible cross-bridges cycle only slowly - see ‘A cross-bridge basis for the SREC and FRT?’ for further discussion.

Conclusion

We conclude that the SREC and FRT are manifestations of a low level of cross-bridge activity in relaxed muscle. Although the proposed dynamics of the cross-bridge distribution are quite complicated, a useful simplification is that the initial stage of the tension response to stretch is dominated by the time at rest, while the latter stage is dominated by the stretch velocity. The existence of bonds linking the actin and myosin filaments, and the presence of a FRT may imply a small but significant energy consumption in relaxed muscles. The molecular motors of muscle may be idling rather than switched off when the muscle is relaxed.

Appendix

A cross-bridge population displacement mechanism

The theoretical model presented in this work consists of three mechanical components: a cross-bridge component, a parallel elastic component and a series elastic component (Fig. 13).

It is axiomatic that tension is continuous throughout the three-component model. Following the nomenclature defined in Fig. 13, the tension in the muscle is given as

display math((A1))

subject to the constraint that

display math((A2))

The thixotropic SREC response arises as a result of time-dependent changes in the cross-bridge force Fcb. The cross-bridge force is calculated by simulating the behaviour of slowly cycling cross-bridges, each of which acts as a linear spring of stiffness kcb, as the actin and myosin filaments are displaced relative to each other by interfilamentary movement. Although in reality cross-bridge interactions would occur at many sites along the actin and myosin filaments, the simulation regards all cross-bridge linkages as equivalent and the state of an individual cross-bridge is defined solely by its displacement x from the position corresponding to zero strain.

Under equilibrium conditions the cross-bridges produce a small force (the FRT) which acts in such a direction as to increase filament overlap. In this model it is assumed that the FRT arises as a result of a slight binding asymmetry. It is hypothesized that under equilibrium conditions cross-bridges are bound symmetrically around a mean displacement x0 so that an undisturbed population of n attached cross-bridges produces a cross-bridge force nkcbx0. An alternative explanation for the FRT might be that a small proportion of the attached cross-bridges undergoes a power stroke and generates tension as a result of a conformational change.

Cross-bridges continually attach to and detach from binding sites on the actin filament. If n (x,t) is defined as the cross-bridge number density (so that n(x,t)δx represents the number of cross-bridges attached with displacements between x and xx) it follows that (see for example Schoenberg, 1985):

display math((A3))

where A(x) is the attachment rate constant for cross-bridges binding with displacements between x and xx, D(x) is the corresponding detachment rate constant, N is the total number of cross-bridges available to interact and v is the interfilamentary velocity (defined as positive for stretch).

The number density of attached cross-bridges n(x,t) can be calculated by integrating eqn (A3) once the initial conditions have been specified. In turn, the cross-bridge force Fcb(t) can be calculated from the expression

display math((A4))

Since eqn (A1) defines the parallel and series elastic component lengths Xp and Xs in terms of the overall length X, the cross-bridge force Fcb and the (known) parallel and series elastic stiffnesses kp and ks, integration of eqn (A3) allows the state of the three-component structure (and thus the muscle's tension and sarcomere length) to be calculated at any instant during an imposed length change.

An assumption inherent in this model is that the cross-bridge attachment rate increases during interfilamentary movement - an effect which we describe as movement enhancement. The present model assumes that cross-bridge binding in relaxed muscle is partly limited by misregistration between cross-bridges on the myosin filament and binding sites on the actin filament. Interfilamentary movement enhances cross-bridge attachment by increasing the probability of a cross-bridge passing within interaction range of a binding site to which it can attach in a given time interval. Some alternative mechanisms which might underlie movement enhancement are considered in the Discussion. Figure 14 shows a section of the actin and myosin filaments.

Figure 14.

Movement enhancement mechanism

A, actin binding sites which are potentially available to interact with the myosin filament are separated by a mean distance y. Cross-bridges which can bind to an available actin site are separated by z. Each cross-bridge has an interaction range w. B, if the filaments are moved a relative distance ΔXp (in either direction) in unit time, the attachment rate constant is increased proportionally with the distance moved by each cross-bridge. Full details of the proposed ‘movement enhancement’ are found in the text.

Actin binding sites are separated by a mean distance y; cross-bridges are separated by a mean distance z. Each cross-bridge has a finite interaction range w and can only bind to an actin binding site within ½w of the cross-bridge's mean position. Individual actin binding sites are not continually available for cross-bridge attachment. Rather, individual sites cycle at a slow rate through active (i.e. available for cross-bridge binding) and inactive (i.e. unavailable) states. This might occur as a result of changes in the local Ca2+ concentration.

If a binding site is within interaction range of a cross-bridge, the rate constant for attachment occurring with a cross-bridge displacement between x and xx is given by a(x). Consequently the attachment rate constant A(x) for cross-bridges with displacements between x and xx binding between stationary filaments is given by the function A0(x) which is the product of the rate constant a(x) and the proportion of binding sites within interaction range of cross-bridges

display math((A5))

It is assumed that wz so that (when the filaments are stationary) there is a low probability of a cross-bridge being within interaction range if a binding site becomes temporarily available. However, the probability of an active binding site passing within range of a cross-bridge to which it can attach is increased by interfilamentary movement. If the filaments move a relative distance ΔXp in a time interval of 1 s, an additional proportion ΔXp/z of binding sites are exposed to cross-bridges. The attachment rate is increased accordingly. Thus

display math((A6))

where v’ is the interfilamentary velocity expressed as the number of interaction ranges traversed per second. It is assumed that although the chances of an individual cross-bridge binding to the actin filament are low, if attachment can take place, it does so effectively instantaneously, i.e. on a very short time scale relative to the velocity of interfilamentary movement. Consequently, the rate of cross-bridge attachment is a reflection of the number of cross-bridges that have passed within interaction range of active binding sites and is unaffected by the time for which each cross-bridge remained within range. (This assumption applies only to the present model of relaxed muscle in which it is assumed that only some cross-bridges and some actin sites (perhaps those oriented towards each other) can interact. The situation will be different in contracting muscle and the assumption would not apply.)

The attachment rate constant A0(x) is assumed to be a Gaussian function centred on x0 with the standard deviation σ set by the cross-bridge stiffness. Thus

display math((A7))

ψ is the maximum attachment rate (at x=x0), k is Boltzmann's constant (1.38 × 10−23 J K−1), and T is the temperature (in K). When the filaments are stationary, the attachment rate constant for cross-bridges binding with any displacement x is therefore given by the parameter β where

display math((A8))

The detachment rate constant of an individual cross-bridge is set by an activation energy which is decreased by the strain energy stored in the elastic cross-bridge linkage. Thus if the cross-bridges are assumed to detach symmetrically around a mean equilibrium displacement x0, the detachment rate constant D(x) is of the form

display math((A9))

where D0 represents the finite rate constant for a cross-bridge attached at x=x0. If D0 is low, cross-bridges attached near x=x0 are stable and long-lived, while detaching more rapidly if they are pulled to higher strains by interfilamentary movement.

For the purposes of the computer simulations, the cross-bridge distribution was approximated by an array of points. Each element of the array ni represented the number of cross-bridges attached with displacements in the range xi - ½Δx to xi+½Δx. Equation (A3) was solved using a numerical approach in which the attachment and detachment of cross-bridges was calculated separately from the movement of existing attached cross-bridges. Cross-bridge attachment and detachment was calculated by integrating the first two terms on the right-hand side of eqn (A3) using Euler's method (Press, Teukolsky, Vetterling & Flannery, 1992) with an iterative time step Δt. The movement of existing attached cross-bridges was controlled by calculating the interfilamentary movement which occurred during each time step and shifting the array elements ni by an appropriate number of places in the requisite direction. This procedure helped to minimize the effect of truncation errors inherent in the computational procedures. The attachment and detachment rate constants were considered constant within the interval Δx. In the majority of simulations presented in this work Δx= 0.0075 nm and Δt= 1 ms. Simulations were always performed with Δx and Δt sufficiently small that further reductions did not affect the numerical results of the computer simulations. The resting length of the fibre was assumed to correspond to a mean sarcomere length of 2.20 μm.

Figure 15 shows an illustrative example of the redistribution of the cross-bridge population during an imposed stretch from the stable equilibrium profile to the skewed but steady-state distribution maintained by the sustained interfilamentary movement.

The model is summarized in Table 1.

Table 1. Propositions of the Cross-bridge Population Displacement Mechanism
(1)In a relaxed muscle, the actin and myosin filaments are linked by a relatively small number of cross-bridges, each of which behaves as a linear spring for both extension and compression.
(2)Only cross-bridges with displacements corresponding to a defined interaction range can bind between the filaments.
(3)The attachment probability is increased by interfilamentary movement.
(4)Unstrained cross-bridges are stable and relatively long-lived, whereas cross-bridges with high strains detach more rapidly.
(5)Interfilamentary displacement skews the cross-bridge distribution.
(6)The mean cross-bridge displacement of an undisturbed population is slightly positive. This bias produces a force which acts to increase filament overlap and contributes to the muscle's resting tension.

Preliminary simulations revealed that the thixotropic nature of the SREC tension response was not critically dependent on the precise manner in which the attachment and detachment rate constants varied as functions of the cross-bridge displacement x. The parameters used in the present simulations (defined in Table 2) seem plausible but no claim is made that they produce the optimal fit with the experimental data.

Table 2. Model parameters
ParameterDefinitionValueUnits
  1. a k pand kswere assumed to be proportional to the cross-sectional area of the preparation. The value of kpwas chosen to produce an appropriate rise in tension beyond the elastic limit relative to the cross-bridge response.bx0 was chosen to produce a FRT which was a comparable proportion of the SREC tension response to that evident from the experimental results.cWith a Gaussian attachment rate constant, the full interaction width extends to x=±∞, although the probability of attachment occurring at extreme values of x is vanishingly small. w (which controls the magnitude of movement enhancement) was arbitrarily set to 3 nm. This range encompasses 81% of the attachment probability and 94% of the attached cross-bridges under equilibrium conditions. The parameters y and z (Fig. 14) do not appear explicitly in eqn (6). However, it is assumed that w << z (only one cross-bridge can interact with a given binding site) and that wy(there is a low probability of a cross-bridge being within interaction range when the filaments are at rest). With w= 3 nm, the above conditions are satisfied if it is assumed that z= 43 nm (i.e. the distance between cross-bridges projecting from the thick filament in the same direction) and that y= 37 nm (i.e. the distance separating cross-overs between the two right-handed helical strands of actin; Edman, Månsson & Caputo, 1997). d Flitney & Hirst (1978).

k cb Cross-bridge stiffness3.0 × 10−3N m−1
k p Parallel elastic component stiffness3.0 × 1011 aN m−1per m2fibre area
k s Series elastic component stiffness1.5 × 1013 aN m−1per m2fibre area
x0 Binding asymmetry1.0 × 10−10bm
w Interaction width3.0 × 10−9cm
βAttachment rate constant0.001s−1
D0 Detachment rate constant0.12s−1
N Cross-bridge number density5 × 1016dm−2per half-sarcomere
k Boltzmann's constant1.38 × 10−23J K−1
TTemperature278K

Acknowledgements

K. S. C. was supported by a Wellcome Prize Studentship.

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