Energy transfer during stress relaxation of contracting frog muscle fibres

Abstract

1. A contracting muscle resists stretching with a force greater than the force it can exert at a constant length, T_o. If the muscle is kept active at the stretched length, the excess tension disappears, at first rapidly and then more slowly (stress relaxation). The present study is concerned with the first, fast tension decay. In particular, it is still debated if and to what extent the fast tension decay after a ramp stretch involves a conservation of the elastic energy stored during stretching into cross-bridge states of higher chemical energy.

2. Single muscle fibres of Rana temporaria and Rana esculenta were subjected to a short ramp stretch (∼15 nm per half-sarcomere at either 1.4 or 0.04 sarcomere lengths s⁻¹) on the plateau of the force-length relation at temperatures of 4 and 14°C. Immediately after the end of the stretch, or after discrete time intervals of fixed-end contraction and stress relaxation at the stretched length (Δt_isom= 0.5–300 ms), the fibre was released against a force ∼T_o. Fibre and sarcomere stiffness during the elastic recoil to T_o (phase 1) and the subsequent transient shortening against T_o (phase 2), which is expression of the work enhancement by stretch, were measured after different Δt_isom and compared with the corresponding fast tension decay during Δt_isom.

3. The amplitude of fast tension decay is large after the fast stretch, and small or nil after the slow stretch. Two exponential terms are necessary to fit the fast tension decay after the fast stretch at 4°C, whereas one is sufficient in the other cases. The rate constant of the dominant exponential term (0.1–0.2 ms⁻¹ at 4°C) increases with temperature with a temperature coefficient (Q₁₀) of ∼3.

4. After fast stretch, the fast tension decay during Δt_isom is accompanied in both species and at both temperatures by a corresponding increase in the amplitude of phase 2 shortening against T_o after Δt_isom: a maximum of ∼5 nm per half-sarcomere is attained when the fast tension decay is almost complete, i.e. 30 ms after the stretch at 4°C and 10 ms after the stretch at 14°C. After slow stretch, when fast tension decay is small or nil, the increase in phase 2 shortening is negligible.

5. The increase in phase 2 work during fast tension decay (ΔW_out) is a constant fraction of the elastic energy simultaneously set free by the recoil of the undamped elastic elements.

6. ΔW_out is accompanied by a decrease in stiffness, indicating that it is not due to a greater number of cross-bridges.

7. It is concluded that, during the fast tension decay following a fast ramp stretch, a transfer of energy occurs from the undamped elastic elements to damped elements within the sarcomeres by a temperature-dependent mechanism with a dominant rate constant consistent with the theory proposed by A. F. Huxley and R. M. Simmons in 1971.

Stretching an active muscle enhances its capability to do work during subsequent shortening (Cavagna et al. 1968, 1986, 1994; Edman et al. 1978; Sugi & Tsuchiya, 1981; Edman & Tsuchiya, 1996). Since the beginning of research into this area, it has been clear that the additional amount of work done after stretching was greater than the additional elastic energy stored during the stretch and released during shortening by the undamped elastic elements of muscle, and that some other mechanism had to be involved (Cavagna et al. 1968).

Shortening after a ramp stretch against a constant force equal to the maximum isometric force (T_o), developed on the plateau of the force-length relation, was taken as direct evidence of an absolute increase in the ability to do work as a consequence of the previous stretch. Since the force during shortening was constant, the work could not be due to the release of energy stored within the undamped elastic elements of tendons and sarcomeres (filaments and cross-bridges) and had to be due to the release of energy from a damped mechanism. The work done against T_o was found to be equal when measured at the fibre end or within a tendon-free segment of the fibre, indicating that it originated in the sarcomere context and not in the tendons (Cavagna et al. 1994). Damped elastic elements recruited during stretching because of sarcomere length inhomogeneity have been described by Morgan (1990) and Talbot & Morgan (1996). Proposed sources for the additional work done after stretching by the damped mechanism are: (1) a greater number of attached cross-bridges (Lombardi, 1998); (2) the recoil of passive visco-elastic elements within the sarcomere context due to sarcomere length inhomogeneity (Edman & Tsuchiya, 1996); and/or (3) a greater level of potential energy attained by the individual cross-bridges because of the stretch (Cavagna et al. 1986).

Energy storage by the damped mechanism may occur both during stretching and after stretching. This last possibility was suggested by the finding that the capability of the sarcomeres to shorten against the maximum isometric tension T_o was retained, in spite of the large reduction in force, during 5-20 ms of isometric (fixed-end) contraction after a large ramp stretch (Cavagna et al. 1986, 1994).

The force greater than T_o attained during stretching leads to a storage of elastic potential energy within the undamped elastic elements in excess of that stored during an isometric contraction. When the muscle is kept active at the stretched length, the force falls towards T_o during stress relaxation (often called ‘tension recovery’). In a fixed-end contraction, this fall in force necessarily involves a shortening of the undamped elements, and as a consequence, a release of elastic energy which may be either totally dissipated as heat or stored in part in damped elements within the sarcomere context. The relative amount of energy stored in these damped elements during or after the stretch depends on the characteristics of the stretch. For example, in a small, 100 μs step stretch, most of the energy storage in the damped elements must occur after the stretch. At the other extreme, a large ramp stretch would favour energy storage in the damped elements during the stretch, when a steady state condition is approached. However, as mentioned above, it is possible that even in this condition some energy storage within the damped elements may occur after the stretch.

In the present study, we used ramp stretches of small amplitude in order to analyse energy storage both in a non-steady-state condition (after fast stretches, which would favour energy storage by the damped mechanism after the stretch), and in a steady-state condition (after very slow stretches, which would favour energy storage by the damped mechanism during the stretch). Using small ramp stretches instead of large ramp stretches reduces the possibility of inducing sarcomere length inhomogeneity during the stretch (Morgan, 1990), with recruitment of strained visco-elastic elements which are an obvious explanation for the enhanced work capability induced by stretching (Edman & Tsuchiya, 1996).

METHODS

Apparatus and procedures

The apparatus and procedures used in the present study have been described in detail by Cavagna (1993) and Cavagna et al. (1994), and will be only summarized here. The present study includes experiments on single frog fibres of Rana esculenta from Rome (cross-sectional area, A = 7590 ± 2490 μm²; slack length at the end of the ramp stretch, l_f,o= 5.43 ± 0.41 mm; n = 13 fibres) and of Rana temporaria from Ireland (A = 15 750 ± 6330 μm²; l_f,o= 5.65 ± 0.68 mm; n = 12 fibres). Mean values ±s.d. are reported throughout; comparisons between groups were made using a single-factor analysis of variance (ANOVA) (Excel v8.0).

Single fibres were dissected from the caput laterale of the tibialis anterior muscle after killing the frog by decapitation and pithing (following the official regulation of the European Community Council, direction 86/609/EEC, assimilated into Italian Government Legislation). Fibres were mounted horizontally in Ringer solution at 4.1 ± 0.3 °C (n = 264 tetani) and 14.1 ± 0.1 °C (n = 57 tetani), with one end attached to a force transducer (Huxley & Lombardi, 1980) and the other attached to a linear electromagnetic coil motor. The motor position was adjusted so that the relaxed sarcomere length, measured at three or more positions along the fibre, was 2.19 μm (this corresponded to isometrically contracting sarcomere lengths, l_s,o, of 2.15 ± 0.03 μm (n = 13) for R. esculenta and 2.13 ± 0.03 μm (n = 12) for R. temporaria).

Before the ramp stretch, the relaxed fibre was shortened by the length of the stretch, a conditioning twitch was given, and the fibre was then stimulated to tetanus isometrically (fixed-end contraction) attaining the maximum isometric tension T_o. The isometric stress S_o developed during the first tetanus in the ∼4 °C experiments was 155.4 ± 37.4 kN m⁻² (n = 13) for R. esculenta and 167.6 ± 45.6 kN m⁻² (n = 12) for R. temporaria; the force attained during the last tetanus was on average ∼4 % less. In the experiments where the temperature was increased to ∼14 °C, the isometric stress increased on average 1.82-fold in R. esculenta and 1.48-fold in R. temporaria.

When tetanic tension reached a plateau, the fibre was subjected to the ramp stretch. This always had the same amplitude and was imposed at one of two different speeds (0.09 mm at either 9.4 mm s⁻¹ or 0.3 mm s⁻¹). The corresponding strain, 1.69 ± 0.17 % for the fibre (n = 293), and 1.37 ± 0.28 % for the sarcomeres (n = 163), was usually just sufficient to reach the maximum tension without appreciable ‘give’ (Fig. 1). The high lengthening velocity corresponded to 1.73 ± 0.17 fibre lengths s⁻¹ (n = 221), and 1.35 ± 0.28 sarcomere lengths s⁻¹ (n = 120). The low lengthening velocity corresponded to 0.05 ± 0.01 fibre lengths s⁻¹ (n = 72), and 0.04 ± 0.01 sarcomere lengths s⁻¹ (n = 43). After stretching, the fibre was released to a force, T_iso, about equal to the isometric tension T_o developed before the stretch (T_iso/T_o= 1.03 ± 0.03, n = 293). Release took place either immediately after the end of the ramp or after a pre-set time interval, Δt_isom, of 0.5, 1, 2, 3, 5, 7, 10, 20, 30, 50, 100 or 300 ms. In some experiments, the fibre was released, without previous stretch, from a state of isometric contraction down to ∼0.9 T_o. The average sarcomere length changes within a fixed-length segment of the fibre delimited by two laser spots were measured with a striation follower from the difference in the number of sarcomeres moving in and out of the segment (Huxley et al. 1981). The segment, 0.6-1.5 mm in length, was located between the centre and the force transducer end of the fibre. The length changes of the whole fibre were measured simultaneously from the motor position.

Figure 1. Example of experimental tracings from which stretch amplitude, phase 1 elastic recoil and phase 2 shortening against T_o were determined.

Oscilloscope traces of length and tension changes recorded from one fibre at 4.0 °C (top panels) and 14.1 °C (bottom panels), using fast ramp stretches (1.81 fibre lengths s⁻¹, left panels) or slow ramp stretches (0.05 fibre lengths s⁻¹, right panels), are shown as a function of time on a slow time scale (three top traces in each panel) or a fast time scale (three bottom traces in each panel). When available, the sarcomere length changes measured by a striation follower (thin records) are shown directly beneath the fibre length changes measured by the position of a linear coil motor. The left-hand traces in each panel show tetani in which the fibre is released to T_o immediately after the end of the ramp stretch, whereas the right-hand traces show tetani in which the fibre is released to T_o after an isometric delay, Δt_isom, of either 30 ms at the low temperature or 10 ms at the high temperature. Note that: (i) the amplitude of isotonic shortening against T_o (phase 2) increases with Δt_isom after fast stretches when T/T_o is high, but not after slow stretches when T/T_o is low; (ii) T/T_o is higher at lower temperature than at higher temperature; and (iii) phase 2 duration decreases with temperature (see also Fig. 2). The characteristics of the Rana temporaria fibre are: contracting sarcomere length, l_s,o= 2.18 μm; relaxed fibre length and cross-section at the end of the ramp, l_f,o= 5.10 mm; and A = 7770 μm². h.s., half-sarcomere.

The feedback system which determined motor position was under fibre length control during the isometric contraction and the ramp stretch, and was switched to force control during the isotonic release so that the isotonic force applied to the fibre was as near as possible to T_o. The succession of events during calibration of the striation follower, the pre-experimental sarcomere and fibre length adjustments, as well as fibre stimulation, oscilloscope triggering, length and force feedback to the motor, and data collection and analysis during the experiment, were all under microcomputer control. In ∼40 % of records, the striation follower measurements of sarcomere length changes could not be used because of oscillations during the elastic recoil and/or discontinuities (jumps) in the signal.

Measurement of stiffness during elastic recoil and of the subsequent transient shortening against T_o

The force exerted by the fibre, the fibre length changes (measured from motor position) and the sarcomere length changes within the selected segment (measured by the striation follower) were recorded simultaneously on two four-channel digital oscilloscopes (Nicolet 4094, Madison, WI, USA). A fast time scale (5 μs per point), was used to record the elastic recoil simultaneous with the fall in force to T_o (phase 1, lasting 138 ± 42 μs, n = 293) and the beginning of the subsequent isotonic shortening against T_o. A slow time scale (500 μs per point for fibre length measurements and 50-200 μs per point for sarcomere length measurements) was used to record the whole of the isotonic shortening against T_o (phase 2). The experimental records were stored on disk (Nicolet XF-44).

Measurements of phase 1 elastic recoil and phase 2 shortening against T_o were made directly by the microcomputer from the oscilloscope tracings. Points used for the calculations were selected by eye using cursor controls. Sarcomere and fibre ‘step’ stiffness (Mantovani et al. 1999) were measured during the elastic recoil to T_o (or to ∼0.9 T_o in the case of the control experiments performed without pre-stretch). Stiffness was normalized by dividing the isometric stress before the ramp stretch into the Young's modulus; this results in a dimensionless value which takes into account variations in the number of attached cross-bridges in different tetani. Phase 2 shortening was determined by subtracting the final length measured on the slow oscilloscope trace from the initial length measured on the fast trace at the end of the elastic recoil (Fig. 1). Any offset between the two oscilloscope traces (fast and slow) was taken into account.

Kinetics of fast tension decay during stress relaxation

One aim of the present study was to compare the kinetics of the fast phase of stress relaxation with the ability of the fibre to shorten against T_o at different time intervals since the end of the stretch, while stress relaxation was taking place. To this end, tension decay was recorded continuously in one tetanus for a 300 ms time period of isometric contraction after the end of the ramp stretch, whereas in other tetani on the same fibre, phase 2 shortening was measured after release to T_o at discrete time intervals during the same period. The 300 ms interval was much larger than the time required to complete the fast phase of stress relaxation (see insets in Fig. 3) and encompassed the relevant changes of the phase 2 velocity transient against T_o taking place after the ramp stretch (Fig. 2).

Figure 3. Kinetics of the fast phase of stress relaxation and simultaneous increase of phase 2 shortening against T_o after fast ramp stretches.

The left panels refer to one fibre of Rana temporaria and the right panels refer to one fibre of Rana esculenta at 4 °C (top) and 14 °C (bottom). The upper right inset in each panel shows the oscilloscope record of stress relaxation during Δt_isom= 300 ms after a fast stretch. The first 50 ms of this record are expanded below the inset: the circles (superposed on the experimental points) are calculated from eqn (1). The first two exponential terms of eqn (1) at 4 °C (top), and the second term of eqn (1) at 14 °C (bottom) are plotted separately to indicate the component(s) of the fast phase of stress relaxation. The slow phase of stress relaxation (third term of eqn (1)) is not plotted separately because it is not pertinent to the present study. The bottom graph in each panel indicates the phase 2 shortening against T_o measured when release to T_o took place after the different Δt_isom intervals indicated on the abscissa. The lines through the symbols are traced by eye. Note that: (i) in all cases, phase 2 shortening increases simultaneously with tension decay during the fast phase of stress relaxation, attaining a maximum near the end of it; and (ii) both tension decay and phase 2 increment are accelerated approximately threefold by a 10 °C increase in temperature. The characteristics of the two fibres are: Rana temporaria, same fibre as in Fig. 1; and Rana esculenta, l_s,o= 2.19 μm, l_f,o= 5.75 mm and A = 6810 μm².

Figure 2. Experimental tracings showing the effect of temperature, and of a delay between end of stretch and release to T_o on phase 2 shortening against T_o.

Oscilloscope traces of phase 2 shortening against T_o measured at 4 °C (top four rows, four fibres, a–d) and at 14 °C (bottom two rows, two fibres, c and d, illustrated on a threefold faster time scale). Phase 2 shortening was measured after the indicated time intervals (Δt_isom= 0, 1, 5, 10, 20, 30, 50 and 300 ms) of fixed-end contraction following the end of a fast ramp stretch. Each column shows phase 2 tracings with similar shape and amplitude. For Δt_isom= 0-50 ms, the thicker, upper tracing of each pair is recorded at the fibre end, whereas the thin, lower record (when available) is recorded simultaneously in a fibre segment. For Δt_isom= 300 ms, the lower, thin record in each pair (‘isometric’) shows, for comparison, the phase 2 shortening measured at the fibre end after release to ∼0.9 T_o from a state of isometric contraction without pre-stretch. Note: (i) the similarity of the tracings recorded at the fibre end and in the fibre segment; (ii) the consistent changes in shape of the phase 2 velocity transient with Δt_isom; (iii) a maximum of phase 2 shortening (column labelled ‘maximum amplitude’) is attained at Δt_isom∼30 ms at 4 °C and at Δt_isom= 5-10 ms at 14 °C; (iv) the tracings at 4 °C are similar to those at 14 °C shown on a threefold faster time scale, indicating that phase 2 is accelerated by temperature with a Q₁₀ of ∼3. The characteristics of the Rana temporaria fibres are: (a) l_s,o= 2.13 μm, l_f,o= 5.37 mm and A = 16 690 μm²; (b) l_s,o= 2.12 μm, l_f,o= 5.00 mm and A = 13 660 μm²; (c) l_s,o= 2.14 μm, l_f,o= 6.30 mm and A = 15 730 μm²; and (d) same fibre as in Fig. 1.

Whereas, after large ramp stretches at 4 °C, one exponential term was sufficient for a fair interpolation of the fast phase of stress relaxation (Colomo et al. 1989; Cavagna, 1993), after the short, fast ramp stretch used in the present study, two exponential terms were found to be necessary at 4 °C in order to account for the initial very rapid tension decay. Therefore, after the fast stretch at 4 °C, the decay of tension T during stress relaxation was fitted (KaleidaGraph 3.5) to the following equation:

(1)

where T₁ is the tension at the end of the ramp stretch and T₂ is the tension at the end of the 300 ms stress relaxation period, t_o is the time at the beginning of stress relaxation, and A_ff, r_ff, A_f, r_f, A_s and r_s are the amplitudes and the rate constants of the very fast, fast and slow phases of stress relaxation, respectively (Fig. 3).

After the fast stretch at 14 °C, a single exponential term was sufficient to interpolate the fast phase of stress relaxation (Fig. 3). After the slow stretch at 4 °C, the fast tension decay was small and could be interpolated more correctly, by a single exponential term, using only the first 100 ms of stress relaxation. Fast tension decay was absent after the slow stretch at 14 °C (Fig. 1). The third term of eqn (1) was used for the interpolation (Fig. 3), but was not taken into account in the present study, which is only concerned with the amplitude and the rate constant of the fast phase of stress relaxation (Table 1). In addition, as shown in Fig. 3, no attempt was made to interpolate the delayed tension increase after the fast tension decay (‘stretch activation’: Pringle, 1949; Rüegg et al. 1970), with an additional exponential term which was assumed to represent cross-bridge attachment by Piazzesi et al. (1997).

Table 1.

Parameters characterizing the effect of a fast and slow stretch at 4 °C, and of a fast stretch at 14 °C on (i) the kinetics of fast tension decay (A_f, A_ff, r_f and r_ff) and (ii) the isometric stress (S_o), the Young's modulus (Y) and the maximum strain (x_max) of the undamped elastic elements during stress relaxation after the fast stretch (used for eqn (3), see text)

	Rana temporaria			Rana esculenta
	4°C		14°C	4°C		14°C
Parameter	Fast stretch (n)	Slow stretch (n)	Fast stretch (n)	Fast stretch (n)	Slow stretch (n)	Fast stretch (n)
Af	0.61 ± 0.07 (10)	0.26 ± 0.10 (6)	0.76 ± 0.00 (2)	0.60 ± 0.03 (13)	0.20 ± 0.00 (2)	0.63 ± 0.05 (3)
rf (ms−1)	0.11 ± 0.03 (10)	0.08 ± 0.04 (6)	0.35 ± 0.07 (2)	0.17 ± 0.03 (13)	0.12 ± 0.09 (2)	0.52 ± 0.02 (3)
Aff	0.25 ± 0.10 (10)	—	—	0.25 ± 0.04 (13)	—	—
rff (ms−1)	0.82 ± 0.25 (10)	—	—	1.39 ± 0.53 (13)	—	—
S0 (kN m−2)	161	—	299	128	—	193
Y (MN m−2)	28.9	—	33.4	30.6	—	33.3
xmax (%)	0.89	—	0.54	0.88	—	0.50

Slow stretch at 14°C is not considered because it has a negligible effect. Values are means ±s.d. A_f, A_ff, and r_f and r_ff are, respectively, the amplitudes and the rate constants of the two exponential terms necessary to interpolate the fast phase of stress relaxation after fast stretch at 4°C. Values are the average of data measured on n fibres during Δt_isom= 300 ms. Only one term is required at 14°C and after slow stretch at 4°C. S_o and Y are calculated from the stress–strain plot of the undamped elastic elements, and x_max is the maximum strain in the plot; the findings are derived from same experiments as those in Fig. 6. Note that: (i) after fast stretches, tension decay is faster in Rana esculenta than in Rana temporaria; and (ii) in both species, r_f increases with temperature with a Q₁₀ of ∼3.

RESULTS

Digital oscilloscope records of length and tension changes taking place at low (top panels) and high temperature (bottom panels), using fast (left panels) and slow ramp stretches (right panels), are shown in Fig. 1 for one complete experiment on one fibre. The three upper traces in each panel were recorded on a slow time scale, and show the events taking place just before, during and after the ramp stretch. These traces were used to determine the velocity and the amplitude of the ramp stretch and the end of the transient isotonic shortening against T_o (phase 2 shortening, or phase 2). The three lower traces were recorded on a fast time scale, and show the events taking place immediately before, during and after release to T_o. These traces were used to determine the duration of the elastic recoil (phase 1 shortening, or phase 1), the stiffness of the fibre and of the sarcomeres (Mantovani et al. 1999), and the exact initial length of the phase 2 shortening. The amplitude of phase 2 shortening was calculated from the difference between the initial length measured on the fast traces and the final length measured on the slow traces. Sarcomere length changes within the selected segment are shown, when available, below the fibre length changes (thin traces). The left traces in each panel were obtained by releasing the fibre immediately after the stretch, whereas the right traces were obtained by releasing the fibre after the time interval when the amplitude of phase 2 shortening is at a maximum (see below).

The experimental records in Fig. 1 show that the increase in tension taking place during the fast ramp stretch was greater than during the slow ramp stretch and, for a given velocity of stretching, it was greater at 4 °C than at 14 °C. After the fast stretch, the amplitude of phase 2 shortening increased when a time interval of 30 ms at 4 °C, but of only 10 ms at 14 °C, was interposed between the end of the ramp and the isotonic release. These delays correspond to those required, on average, to elicit the maximum increase in the amplitude of phase 2 shortening (Figs 2-4). After the slow stretch, on the other hand, the changes in the amplitude of phase 2 shortening with the delay were less or nil, and the phase 2 amplitude was larger at the lower temperature than at the higher temperature (Fig. 4).

Figure 4. Average data of phase 2 shortening and work against T_o compared with the average trend of fast tension decay.

Mean values (filled circles) of phase 2 shortening against T_o of fibres of Rana temporaria (left panels) and of Rana esculenta (right panels) released to T_o at discrete time intervals, Δt_isom (abscissa) after the end of a fast ramp stretch (two upper panels) and a slow ramp stretch (bottom panels). In each panel, the upper graphs refer to 4 °C and the lower graphs to 14 °C. The mass-specific phase 2 work ordinate has been calculated from the phase 2 shortening ordinate and the average value of the isotonic load during phase 2 (∼T_o) in each group of data. The exponential terms of fast tension decay, r_f and r_ff, are calculated from the average data given in Table 1 (except for the slow stretch at 14 °C where the fast phase of stress relaxation is nil). The vertical arrows indicate a value on the abscissa equal to 5τ, where τ is calculated as the reciprocal of the average rate constant r_f in Table 1 (at 5τ, the amplitude of the exponential term is reduced by 99.3 %). The filled circles and error bars indicate, respectively, the mean and the s.d. of the mean (plotted for convenience on one side only of the symbol) of the number of measurements indicated beside the symbols; the lines through the symbols are traced by eye. The average data confirm the conclusions reached on one fibre in Fig. 3 after a fast stretch, i.e. the simultaneity of fast tension decay and phase 2 increment with a Q₁₀ of ∼3 for both; in all cases, the maximum phase 2 shortening is attained before or at, but not after, the time when the fast tension decay is practically complete (arrows). Note also that, after a slow ramp stretch at 4 °C, when the amplitude of the fast phase of tension decay is less than half that after fast stretches, the increment in phase 2 work during the fast phase of stress relaxation is reduced, but the phase 2 work done immediately after the ramp W_{Δt = 0} is increased. The data are grouped, when applicable, into the following intervals along the abscissa: 0, 0.5-1, 2-3, 5-7, 10, 20, 30, 50, 100 and 300 ms (except for low velocity at 4 °C, where the intervals are 0, 5-10, 20, etc. for Rana temporaria, and 0, 5-10, 20-30, 50, etc. for Rana esculenta). The characteristics of the eight groups of fibres are as follows. Rana temporaria: fast stretch at 4 °C, l_f,o= 5.65 ± 0.68 mm and A = 15 750 ± 6330 μm² (n = 12); slow stretch at 4 °C, l_f,o= 5.84 ± 0.60 mm and A = 15 870 ± 7110 μm² (n = 7); and fast and slow stretch at 14 °C, l_f,o= 5.70 ± 0.85 mm and A = 11 750 ± 5630 μm² (n = 2). Rana esculenta: fast stretch at 4 °C, l_f,o= 5.43 ± 0.41 mm and A = 7590 ± 2490 μm² (n = 13); slow stretch at 4 °C, l_f,o= 5.44 ± 0.43 mm and A = 7690 ± 1980 μm² (n = 7); and fast and slow stretch at 14 °C, l_f,o= 5.70 ± 0.18 mm and A = 9380 ± 2700 μm² (n = 3).

The changes in shape of the phase 2 velocity transient, which took place after different time intervals following the end of a fast ramp stretch (Δt_isom= 0-300 ms), are illustrated in Fig. 2 for four fibres of Rana temporaria. In all except the right-hand column, the thick trace shows the fibre length change and the thin trace (when available) shows the simultaneous sarcomere length change within the selected segment. In the right-hand column, the thin trace shows phase 2 shortening after release from a state of isometric contraction to ∼0.9 T_o for comparison with release to T_o after Δt_isom= 300 ms.

In three of the four fibres in Fig. 2, the sarcomeres lengthened, as expected, during Δt_isom, when the force fell and the tendons shortened in the fixed-end mode of contraction. On the other hand, in fibre b of Fig. 2, the sarcomeres shortened during Δt_isom, indicating that the sarcomeres within the segment were stronger than average. For this reason, in subsequent analysis, the amplitude of phase 2 shortening was measured only at the fibre end where it represented the average sarcomere behaviour (Cavagna et al. 1994).

When release took place immediately after the end of the fast stretch, phase 2 showed an oscillatory character (left column of Fig. 2). The oscillation was more marked (as shown by a positive slope at the beginning of phase 2) at 4 °C than at 14 °C and in Rana temporaria than in Rana esculenta (data not shown). As the time from the perturbation caused by the stretch increased, the oscillation was damped out and the amplitude of phase 2 increased. This process was markedly accelerated by temperature (note the threefold difference in the time scales for the 10 °C difference in temperature in Fig. 2) so that the maximum phase 2 amplitude was attained 20-30 ms after the end of the stretch at 4 °C and 5-10 ms after the end of the stretch at 14 °C (see column labelled ‘maximum amplitude’). With a further increase of Δt_isom, the amplitude of phase 2 decreased. After a 300 ms delay, phase 2 shortening against T_o resembled in shape and size the well-known phase 2 velocity transient which takes place when the force is suddenly reduced below T_o from an isometric contraction without previous stretching.

The shape of the phase 2 velocity transient at a given temperature is remarkably similar in different fibres for a given time delay after the end of the ramp (columns in Fig. 2). However, at the same temperature, some fibres exhibited a different duration of phase 2: for example, fibres a and b in Fig. 2 show a longer transient than fibres c and d. Phase 2 duration was measured after the fast ramp stretch for Δt_isom= 0-50 ms, i.e. before phase 2 shortening resolved into a single, short step (as occurred after Δt_isom= 300 ms in Fig. 2; see also Fig. 4 of Cavagna et al. 1986). The average phase 2 durations in R. temporaria were: 77.7 ± 30.5 ms (n = 66) at 4 °C and 30.9 ± 8.0 ms (n = 11) at 14 °C, resulting in a Q₁₀ of 2.5. The average phase 2 durations in Rana esculenta were: 57.9 ± 12.6 ms (n = 86) at 4 °C and 15.3 ± 5.5 ms (n = 21) at 14 °C, resulting in a Q₁₀ of 3.8. Note that in Rana temporaria: (1) phase 2 was on average slower (1.3-fold slower, P < 0.001, at 4 °C, and 2.0-fold, P < 0.001, at 14 °C); and (2) the Q₁₀ was one-third less than in Rana esculenta.

The evolution in the amplitude of phase 2 shortening taking place after different time intervals following the end of a fast ramp stretch (Δt_isom= 0-50 ms) is compared in Fig. 3 with the kinetics of the fast phase of stress relaxation which took place when the fibre was held isometric (fixed-end) for the same time period. Results obtained on one fibre of Rana temporaria and one fibre of Rana esculenta are compared for contractions at 4 and 14 °C. The insets show records of the tension decay over a period of 300 ms after the end of the stretch, which includes part of the slow phase of stress relaxation. The oscilloscope records of tension decay during the first 50 ms are shown in the upper graph of each panel (dotted trace), along with the curve fit (circles) and the exponential terms calculated as explained in Methods above (eqn (1)). It appears that: (1) phase 2 shortening attained a maximum value of 5.0-6.0 nm per half-sarcomere at 4 °C near the end of the fast phase of stress relaxation; (2) the time intervals necessary to attain both the maximum phase 2 shortening and the end of the fast phase of stress relaxation decreased from ∼30 ms at 4 °C to ∼10 ms at 14 °C; and (3) the amplitude of phase 2 shortening taking place immediately after release (Δt_isom= 0 on the abscissa) was greater in Rana esculenta than in Rana temporaria.

The average amplitude of phase 2 shortening and work done against T_o are shown in Fig. 4, together with the average exponential terms of the fast phase of stress relaxation (Table 1), as a function of the time interval between the end of the ramp and the release to T_o for both fast and slow velocities of stretching, at 4 °C and 14 °C, for Rana temporaria and Rana esculenta. For the fast stretch, the average values confirm the results enumerated above for one fibre of each species (Fig. 3). For the slow stretch at 4 °C, the changes in amplitude of phase 2 shortening for Δt_isom= 0-30 ms were smaller than after the fast stretch and not significant: this was accompanied by a smaller amplitude of the fast phase of stress relaxation. For the slow stretch at 14 °C, both the changes in amplitude of phase 2 shortening and the fast phase of stress relaxation were nil.

Average values of sarcomere and fibre stiffness measured during phase 1 elastic recoil are shown in Fig. 5 as a function of Δt_isom for both fast and slow velocities of stretching, at 4 and 14 °C, in Rana temporaria and Rana esculenta. As expected, the stiffness was higher in the sarcomeres than in the fibre, which includes tendon compliance, but both showed the same trend with Δt_isom. Immediately after the fast ramp stretch, the stiffness increased above the isometric value and then decreased during stress relaxation towards the isometric value within ∼50 ms at 4 °C and 10-30 ms at 14 °C. After the slow ramp stretch, the stiffness changes were negligible (data not shown at 14 °C).

Figure 5. Sarcomere and fibre stiffness compared with fast tension decay and phase 2 shortening against T_o.

In each panel, the upper graph shows the sarcomere stiffness (•—•) and the fibre stiffness (○- - -○) during stress relaxation, whereas the lower graphs show, for comparison, the phase 2 shortening and the fast tension decay illustrated in Fig. 4. Stiffness is normalized by dividing the Young's modulus, measured during release (phase 1), by the isometric stress (S_o), measured before each ramp. The normalized stiffness measured during release from a state of isometric contraction to ∼0.9 T_o is indicated by the horizontal lines and corresponds to: at 4 °C, 161 ± 10 (n = 8) for the sarcomeres and 110 ± 13 (n = 11) for the fibre in Rana temporaria, 180 ± 34 (n = 9) for the sarcomeres and 133 ± 21 (n = 12) for the fibre in Rana esculenta; and at 14 °C, 100 ± 18 (n = 2) for the sarcomeres and 86 ± 20 (n = 2) for the fibre in Rana temporaria, and 119 ± 8 (n = 3) for the sarcomeres and 94 ± 13 (n = 3) for the fibre in Rana esculenta. As expected, sarcomere stiffness is greater than fibre stiffness, which includes the tendon compliance. Note that, after fast stretches: (i) stiffness decreases towards the isometric value during the fast phase of stress relaxation; (ii) when tension decay is accelerated by an increase in temperature, the stiffness decay is also accelerated similarly; (iii) the increase in phase 2 shortening is accompanied by a decrease in stiffness indicating that it is not due to an increase in the number of cross-bridges; and (iv) after slow stretches, when the fast phase of stress relaxation is small (4 °C) or nil (14 °C, data not shown), the stiffness changes are not significant. Data are derived from the same experiments as in Fig. 4; the number of data comprising the average shown by the open circles (fibre end) is the same as in Fig. 4 (except for one average), whereas, for the filled circles (sarcomeres), the numbers near the symbols indicate the number of usable striation follower records. Lines through the symbols are traced by eye.

DISCUSSION

The aim of the present study was to investigate the mechanism of work enhancement induced by stretching in tetanized muscle fibres, using ramp stretches of small amplitude imposed at two very different speeds and temperatures. Work enhancement was tested by applying to the fibre, after the stretch, a load equal to the maximum isometric force developed during a tetanus on the plateau of the force-length relation, T_o, i.e. a load against which no shortening can occur without previous stretching. Release to T_o took place either immediately after the end of the stretch, or after different time intervals of isometric contraction and stress relaxation, Δt_isom, introduced between the end of the stretch and the release to T_o.

The work enhancement induced by previous stretching was measured as the positive work done, after release, during a transient shortening against T_o (phase 2 shortening; Cavagna et al. 1986, 1994). Since the force applied to the fibre during phase 2 shortening is held constant, the phase 2 work done against T_o cannot derive from the recoil of the undamped elastic elements and must derive from a damped mechanism. This damped mechanism could be: (1) the release of energy stored during stretching; and/or (2) energy set free de novo by a greater number of cross-bridges recruited because of stretching. These two possibilities will be considered in the first part of the Discussion. The structures responsible for energy storage and recovery will be considered in the second part of the Discussion.

The phase 2 work done against T_o immediately after the stretch, W_{Δt = 0}, must derive from processes occurring during the stretch (or during the isotonic phase itself). On the other hand, an increment in phase 2 work done after the stretch, ΔW_out= W - W_{Δt = 0}, must derive from processes taking place during the period of stress relaxation (or during the isotonic phase). A simple mechanism explaining ΔW_out could be the transfer of elastic energy, taking place during tension decay, from undamped elements within the tendons and sarcomeres to damped elements within the sarcomere context.

When the force falls during stress relaxation, the undamped elastic elements shorten and, since the fibre is held isometric (fixed-end mode), elastic work is done on the sarcomeres which, on average, must lengthen, possibly storing mechanical energy. In addition, even in an isometric or a shortening sarcomere, an energy transfer may occur during stress relaxation from the undamped to the damped element of each cross-bridge (Huxley & Simmons, 1971). However, it is possible that the work done during phase 2 shortening against T_o may derive from other sources. For example, as mentioned above, it could derive from a greater number of attached cross-bridges recruited during or after the ramp stretch. In this case, ΔW_out may not be related to the elastic energy transfer taking place during stress relaxation and could even exceed the elastic work done on and within the sarcomeres. Information on these possibilities can be obtained by comparing ΔW_out with the elastic energy set free by the undamped elastic elements during stress relaxation. This comparison was made using the records obtained after fast stretches only because stress relaxation and work enhancement are negligible after slow stretches (Fig. 1).

Energy balance during stress relaxation

The elastic energy released by the undamped elastic elements corresponds to a work input into the sarcomere context, W_in, which was calculated at different time intervals of stress relaxation, Δt_isom, as:

(2)

The first term is the average force measured from the force-length relation of the undamped elastic elements: T_max is the force attained at the end of the ramp, and T is the force measured just before release, at the end of Δt_isom. The second term is the shortening of the undamped elastic elements during Δt_isom, measured as the elastic recoil of the fibre, ΔL_max, occurring when it is released to T_o immediately after the ramp (i.e. before stress relaxation: Δt_isom= 0) minus the elastic recoil, ΔL, occurring when it is released after an isometric delay (i.e. after stress relaxation: Δt_isom > 0). The elastic recoil is measured at the fibre end, and therefore, includes the recoil of tendons and connections. The fibre mass, M, is calculated from the relaxed fibre length at the end of the ramp, l_f,o, the fibre cross-sectional area, A, and the fibre density, δ (assumed equal to that of water).

As mentioned above, W_in is the decrease in the elastic potential energy, stored during stretching within the undamped elastic elements, and is equal to the increase in the total energy available for storage in damped elements within the sarcomere context during stress relaxation. For this reason, W_in is measured as a positive number in Fig. 6, increasing from an initial value equal to zero when Δt_isom= 0. W_in increases as the force falls and the undamped elastic elements shorten during Δt_isom. This increase in work is plotted in Fig. 6 (filled circles) as a function of the fall in force relative to T_o, i.e. (T_max - T)/T_o=ΔT/T_o, taking place during stress relaxation from Δt_isom= 0.5 ms to Δt_isom= 300 ms (as indicated by the numbers near the filled squares). Note that ΔT/T_o= 0 on the abscissa corresponds to the high force value attained at the end of the ramp stretch, before the beginning of the stress relaxation period.

Figure 6. Energy balance during stress relaxation after a fast ramp stretch.

Left panels, Rana temporaria; right panels, Rana esculenta; upper panels, 4 °C; lower panels, 14 °C. The filled circles in each panel indicate the mechanical energy, W_in, set free within the sarcomere context during stress relaxation of the undamped elastic elements. W_in is calculated using eqn (2) for each tetanus, and averaged for equal or similar values of Δt_isom, as indicated by the numbers near the squares. The open circles indicate the increase, during stress relaxation, in the mass-specific phase 2 work done against T_o, ΔW_out, calculated for the same tetani using eqn (6). The filled squares indicate ΔW_out - W_in, calculated as eqn (6) minus eqn (2). All data are plotted as a function of the fall in force relative to T_o. The continuous and dotted lines are calculated as described in the text; the dashed line is traced by eye. Note that, during the fast phase of stress relaxation (Δt_isom= 0-30 ms at 4 °C and 0-10 ms at 14 °C, corresponding to ΔT/T_o∼0-1 and ∼0-0.5, respectively, on the abscissa), the increase of W_in is accompanied by a proportional increase of ΔW_out, suggesting that some of the elastic energy delivered during stress relaxation is stored and recovered by damped structures within the sarcomere context. Data are derived from the same experiments as in Fig. 4 (fast stretches only and excluding Δt_isom= 0; the number of data comprising the averages are the same as in Fig. 4, except for Rana temporaria at 4 °C where they are fewer because, in one experiment, the record for Δt_isom= 0 was missing, and as a consequence, work measurements could not be made).

The filled circles in Fig. 6 are W_in measurements made according eqn (2) for each tetanus, and averaged for equal or similar values of Δt_isom as indicated. The continuous and dotted lines through the filled circles are calculated as follows. The stress, S = T/A, developed just before each release to T_o, has been plotted as a function of the subsequent fibre recoil during phase 1, x =ΔL/l_f,o, for all the data obtained in each fibre type and temperature after the fast stretch and interpolated with a linear relation:

(3)

where S_o is the isometric stress and Y is the Young's modulus. The values of S_o and Y are given in Table 1 for the two fibre types and temperatures. Integrating eqn (3) during the fall in force:

(4)

where V is the fibre volume, x_max (Table 1) is the maximum strain of the undamped elastic elements attained in the experiments with Δt_isom= 0, and x is the strain after each period of stress relaxation, Δt_isom. The fall in stress during stress relaxation is ΔS = Y (x_max - x). Substituting in eqn (4) and rearranging, one obtains:

(5)

which can be used to interpolate the data in Fig. 6, since ΔS/S_o=ΔT/T_o. The continuous line through the filled circles is the least squares fit of all the experimental data made according to an equation of the form of eqn (5) (Kaleidagraph General curve fit), whereas the dotted line is the predicted trend calculated from eqn (5) using the constants in Table 1.

The total energy input in the sarcomere context, W_in, which represents the total energy available for storage in the damped elements, is to be compared with the increase in the total energy output by the damped mechanism taking place after release against T_o. This increase, ΔW_out, results from the average strain of the damped elements within or between the sarcomeres, most of which lengthen, but some of which shorten during stress relaxation (see Fig. 2, fibre b). It must be pointed out that, in the shortening sarcomeres, ΔW_out cannot be due to structures in parallel with them, and must be due to structures within the sarcomere itself, such as the damped elements of the cross-bridges lengthening at the expense of undamped elements in series with them (Huxley & Simmons, 1971). The middle curve in each panel of Fig. 6 (open circles) is the increment in the mass-specific phase 2 work, ΔW_out, measured after different times of stress relaxation, Δt_isom, as:

(6)

where Sh and Sh_{t = 0} are the phase 2 shortening measured after each Δt_isom, and immediately after stretching, respectively, and T_iso∼T_o is the average force during phase 2 shortening (see Methods).

The continuous line through the open circles is calculated assuming that, during the fast phase of stress relaxation, when ΔW_out increases to a maximum (Fig. 4), the increase in phase 2 work is a constant fraction of the elastic energy delivered by the undamped elastic elements, W_in. Over the range of Δt_isom where phase 2 work increases, the ratio ΔW_out/W_in was measured for each tetanus and its average value was multiplied by the continuous line through the filled circles to obtain the continuous line through ΔW_out (open circles). The agreement between calculated and measured values of ΔW_out shows that, during the fast phase of stress relaxation, the increase in phase 2 work is a constant fraction of the elastic energy delivered by the undamped elastic elements, W_in. After the fast phase of stress relaxation, W_in continues to increase whereas ΔW_out decreases (dashed lines in Fig. 6).

The data in Fig. 6 show that the elastic energy input to the sarcomere context, W_in, is always larger than the incremental phase 2 work done against T_o during stress relaxation, ΔW_out. Therefore, the mechanical energy input is always sufficient to account for the increment in the mechanical energy output. A causal link between W_in and ΔW_out after fast stretches is strongly suggested by the following findings: (1) the duration of the fast phase of stress relaxation and the time necessary to attain the maximum phase 2 work are similar (Fig. 4); (2) both times decrease by the same amount when temperature is increased from 4 to 14 °C (Fig. 4); and (3) for each experimental condition, ΔW_out/W_in is constant (Fig. 6).

Assuming that no processes other than the mechanical energy transfer from the undamped to the damped elements during stress relaxation are contributing to ΔW_out, this would derive from a balance between the energy input W_in and an energy loss, ΔW_out - W_in, which has been calculated as eqn (6) minus eqn (2), and is indicated by the filled squares in Fig. 6 (the continuous curve through the squares is obtained by subtracting the continuous line through the filled circles from that through the open circles). This energy loss is possibly due to detachment of strained cross-bridges and, particularly after large ramp stretches, it may be due to stress-relaxation of passive visco-elastic elements recruited during stretching (Cavagna, 1993; Cavagna et al. 1994).

Stiffness changes during stress relaxation

Since sarcomere stiffness is an increasing function of the number of attached cross-bridges (Mijailovich et al. 1996), an increase in stiffness during fast tension decay would support the hypothesis that the concomitant increase in phase 2 work (ΔW_out) is due to an increased number of attached cross-bridges. On the other hand, a decrease in sarcomere stiffness during fast tension decay would not be compatible with that hypothesis.

It is known that the beginning of a ramp stretch, or a step stretch, causes an abrupt increase in sarcomere stiffness above the value measured during release from a state of isometric contraction; if the ramp stretch is continued, the stiffness reverts towards the isometric value (Bressler et al. 1988; Sugi & Tsuchiya, 1988; Morgan et al. 1996; Mantovani et al. 1999). The initial increase in stiffness seems too fast to be attributed entirely to the formation of new cross-bridges. More likely, since the initial increase in stiffness is proportional to the isometric stiffness existing prior the stretch (Mantovani et al. 1999), it is a consequence of the distortion of already attached cross-bridges as suggested by Sugi & Tsuchiya (1988).

During stress relaxation, the stiffness falls towards the isometric value simultaneously with the fall in tension (Colomo et al. 1989), suggesting that the increase in stiffness induced by stretching is bound to the increase in tension. The liaison between stretch-induced stiffness and tension is substantiated by the present study which shows that, when the fast phase of tension decay is accelerated by an increase in temperature, the stiffness decay is accelerated similarly, and that, when the velocity of stretching is too slow to cause an appreciable increase in tension, the increase in stiffness is also reduced (Fig. 5). The finding that stiffness decreases during the fast phase of stress relaxation whereas phase 2 shortening increases (Fig. 5) indicates that ΔW_out is not due to an increased number of attached cross-bridges induced by the ramp stretch. In fact, a greater number of attached cross-bridges would cause an increase, not a decrease in stiffness.

It must be pointed out that stiffness was measured during phase 1 and not during phase 2. In principle, it is possible that cross-bridge attachment occurs after the transition into the isotonic phase and that these newly attached cross-bridges could cause the described shortening against T_o. However, in this case, cross-bridge attachment induced by release to T_o should change with Δt_isom in such a way as to simulate the energy transfer mechanism described above. In particular, cross-bridge attachment immediately after the stretch should be small, whereas cross-bridge attachment at the end of the fast phase of stress relaxation should be maximum. This last possibility seems unlikely because: (1) the velocity of phase 2 shortening is maximal immediately after release (see column ‘maximum amplitude’ in Fig. 2) just when cross-bridge attachment induced by release to T_o should begin; and (2) cross-bridge attachment immediately after release should compensate for cross-bridge detachment (indicated by the fall in stiffness) taking place during the preceding phase of stress relaxation.

The results considered above strongly suggest instead that the enhanced work capability induced by stretching is due to energy storage in damped elements within the sarcomere context. In order to determine if these damped elements reside in the cross-bridges or in passive visco-elastic elements, the following issues will be considered next: (1) the rate of tension decay and of ΔW_out production during the fast phase of stress-relaxation; (2) the effect of temperature on tension recovery and phase 2 shortening; (3) the phase 2 duration; and (4) the amount of phase 2 work done against T_o.

Rate of stress relaxation and ΔW_out production

After a steady-state condition is attained by the end of a large ramp stretch, the tension fall during stress-relaxation was fitted with reasonable accuracy by the sum of two exponential terms describing the fast and the slow phases of tension decay (Colomo et al. 1989; Cavagna, 1993). A rise in temperature markedly increased the rate constant of the fast phase (Q₁₀= 2.1), whereas it had little effect on the rate constant of the slow phase (Q₁₀= 1.2) (Cavagna, 1993). Stress relaxation of passive visco-elastic elements would hardly be accelerated by a rise in temperature as was the fast phase of stress relaxation. The more likely explanation is that given by Huxley & Simmons (1971) for the fast tension recovery (stress relaxation) after step stretches: the transfer of energy from the undamped to the damped structure within the cross-bridges. In fact, the rate constant (0.12 ms⁻¹ at 4 °C) and the Q₁₀ found by Cavagna (1993) for the fast phase were similar to those reported by, respectively, Huxley & Simmons (1971; ∼0.13 ms⁻¹ in their Fig. 4, after the largest step stretch used at the same temperature) and Ford et al. (1977; Q₁₀= 2.0-2.5).

In a non-steady-state condition, such as after the short fast ramp stretches used in the present study, an additional exponential term of smaller amplitude characterized by a much higher rate constant is required to adequately fit the data at 4 °C (r_ff= 0.8-1.4 ms⁻¹ in Table 1, corresponding to τ= 1.25-0.71 ms). The velocity of this small amplitude, very fast component is further increased after ∼100 μs step stretches (Piazzesi et al. 1997). However, the rate constant of the dominant exponential term of fast tension decay at 4 °C is similar after large ramp stretches (0.12 ms⁻¹ in Cavagna, 1993), after the short ramp stretches used here (0.08-0.11 ms⁻¹ in Rana temporaria and 0.12-0.17 ms⁻¹ in Rana esculenta, Table 1) and after large step stretches (∼0.13 ms⁻¹ measured from Fig. 4 of Huxley & Simmons, 1971; and 0.12 ms⁻¹ in Table 1 of Piazzesi et al. 1997, corresponding to τ_d= 8.2 ms). It is worth noting that the value of the rate constant of this dominant exponential term approaches that predicted by the theory of Huxley & Simmons (1971, ∼0.2 ms⁻¹).

The finding that the time course of the fast tension decay mirrors the increase in ΔW_out both at 4 and 14 °C strongly suggests, as discussed above, a conservation of some of the mechanical energy stored during stretching in the undamped elastic elements by a damped mechanism within the sarcomeres. The similarity of the dominant rate constant of this process with that predicted by the theory of Huxley & Simmons (1971), and the high dependence on temperature further strengthens the idea that the fast tension decay is due to readjustment in length between undamped and damped structures associated with the cross-bridges, and that the subsequent ΔW_out is the release of energy stored in this way within the damped structures. This interpretation of fast tension decay is consistent with the strong correlation found between the kinetics of the force transients following a stretch (fast tension decay and subsequent ‘stretch activation’) and the isoforms of the myosin heavy chains, which suggests that the heads of various myosin heavy chain isoforms exhibit different kinetic properties in their transfer between attached states on actin (Galler et al. 1994, 1996).

The above interpretation of the experimental data differs from the hypothesis that the same component of tension decay (τ_d= 8.2 ms at ∼4 °C) represents instead the detachment of strained cross-bridges, and that the reversal of the power stroke is only responsible for the very fast components of stress relaxation (Piazzesi et al. 1997). This hypothesis would not explain the increase in ΔW_out that occurs simultaneously with cross-bridge detachment. In addition, according to this hypothesis, the rate constant for the reversal of the power stroke at 4 °C (∼1.9 ms⁻¹, according to Table 1 of Piazzesi et al. 1997) would be about tenfold greater than that predicted by the theory of Huxley & Simmons (1971), which results in the well-known relationship between load and velocity, with velocity approaching a low constant value at high loads (Fig. 4 of Huxley & Simmons, 1971). This relationship, based upon asymmetric energy barriers, is in general agreement with the trend of the experimental data (Ford et al. 1977). Furthermore, the ratchet model, leading to the same load-velocity relationship (Fig. 46–2 of Feynman, 1966) is widely used to describe possible mechanisms of molecular motors in the Brownian domain (e.g. Magnasco, 1993; Astumian, 1997).

Effect of temperature

The large effect of temperature on the duration of phase 2 shortening against T_o reported in a previous study on Rana temporaria (Cavagna et al. 1994) is confirmed by the present experiments (Fig. 1 and 2, Q₁₀= 2.5 in Rana temporaria and 3.8 in Rana esculenta) which, in addition, show that the duration of the post-stretch energy transfer, during which tension decreases and ΔW_out increases, is affected by temperature in a similar way (Fig. 3 and 4, Q₁₀∼ 3.0). This suggests that both the charging process (tension decay) and the discharging process (phase 2 shortening against T_o) have a common, temperature-dependent mechanism. It is worth recalling in this respect that, in Rana esculenta where the tension decay is faster than in Rana temporaria (Table 1), the phase 2 shortening is also of shorter duration than in Rana temporaria.

The large effect of temperature on both the charging and the discharging processes is more easily explained by the transfer between cross-bridge states than by stretch and recoil of passive visco-elastic elements. However, Edman & Tsuchiya (1996) suggested that phase 2 shortening against T_o derives from the recoil of passive visco-elastic elements and explained the large effect of temperature by assuming that the velocity of elastic recoil is determined by the velocity of shortening of contractile elements acting in parallel.

Is phase 2 duration too long to be sustained by cross-bridges?

The phase 2 velocity transient against T_o (Fig. 2) is slower and larger than that inferred from the duration and the amplitude of phase 2 tension transients measured after step releases superposed during ramp stretches (Piazzesi et al. 1992). The discrepancy, pointed-out by Edman & Tsuchiya (1996), is due to the different experimental protocols used by Cavagna et al. (1986, 1994 and the present study) and Piazzesi et al. (1992).

Even if the time scale of a tension transient is not simply related to that of a velocity transient (Ford et al. 1977), both rely on the same underlying mechanism. Therefore, tension transients are illustrated in Fig. 7 to compare the different results obtained using the two different protocols. The protocol used here and in our previous studies can be simulated by a ramp, a shortening step and a period of isometric contraction (Ramp-Step-Hold (RSH), left-hand side of Fig. 7), whereas the protocol used by Piazzesi et al. (1992) is a ramp, a shortening step and a continuation of the ramp (Ramp-Step-Ramp (RSR), right-hand side of Fig. 7).

Figure 7. Effect of two different experimental protocols on the phase 2 tension transient after a shortening step imposed during lengthening.

This figure shows one of some experiments made (in addition to those described in Methods) to clarify discussion about phase 2 duration. The upper graphs show, as a function of time, the length changes imposed at the fibre end by the motor in length feed-back mode. In the left tracing, the fibre is subjected to a large ramp stretch at 1.66 fibre lengths s⁻¹, followed by a fast shortening step (50 μm complete in 220 μs) and by a period of isometric (fixed-end) contraction (Ramp-Step-Hold (RSH), a condition similar to that of the present study where force instead of length is kept constant after the step). In the right tracing, the fibre is subjected to the same ramp and step, but the ramp is continued after the step (Ramp-Step-Ramp (RSR), as in the experimental protocol of Piazzesi et al. 1992). The lower graphs show the force exerted by the fibre before and after the step; the inset on the right shows on an expanded time scale the force and length changes just before and after the step during tension recovery in the RSR protocol. Note that: (i) the shortening step is given when the force is almost steady during the ramp at ∼1.8 T_o; (ii) the recovery of tension after the step attains ∼1.2 T_o on the left (RSH protocol) and ∼1.8 T_o on the right (RSR protocol); and (iii) the tension recovery is complete in a time approximately tenfold shorter in the RSR protocol. The characteristics of the fibre are: Rana temporaria, l_f,o= 6.4 mm, A = 10580 μm² (3.7 °C).

In the RSH protocol, the tension increases after the step to ∼1.2 T_o, indicating that some damped elements, charged during stretching, are capable of raising the force above the isometric value, T_o. In the RSR protocol, the duration of the tension transient following the same shortening step is tenfold shorter than in the RSH protocol and the tension overshoots slightly above the value maintained during the ramp stretch before the step; the shape and the duration of the tension transient (right-hand inset of Fig. 7) are similar to those shown by Piazzesi et al. (1992) in their Fig. 14. The accelerated tension recovery in the RSR protocol is to be expected since, after the shortening step, tension recovery is due not only to the transition of the cross-bridges towards states of lower potential energy (as in the RSH protocol), but also to the lengthening of the undamped elements by the continued ramp stretch. Phase 2 cannot proceed as in RSH because the rise in tension due to lengthening by the motor accelerates the attainment of an equilibrium at a higher energy level. The result appears as an enhanced phase 4 (cross-bridge detachment and attachment) at the expense of phase 2 (Piazzesi et al. 1992).

In conclusion, neither the duration nor the amplitude of phase 2 shortening, as measured with the RSR protocol, can be compared with those measured by Cavagna et al. (1986, 1994) who used an experimental procedure more similar to the RSH protocol, which results in a longer-duration transient and a larger phase 2 amplitude.

Is the amount of phase 2 work compatible with the cross-bridge working stroke?

The maximum phase 2 shortening against T_o measured in the present study after a short ramp stretch is about 5.5 nm per half-sarcomere (Fig. 4). After a four- to fivefold larger ramp stretch, phase 2 shortening attained on average 8 nm per half-sarcomere (Cavagna et al. 1994) and, in one good mirror experiment on one fibre, it attained 10 nm per half-sarcomere (Cavagna et al. 1986, Fig. 3). All together, these values of shortening against T_o correspond to a work output of 1-1.5 J kg⁻¹ (Fig. 4). This amount of work is compatible with the work done by a cross-bridge during a working stroke. Taking a maximum of 1.5 J kg⁻¹, or 1.5 mJ m⁻² per half-sarcomere, and a total number of myosin heads of ∼15 × 10¹⁶ m⁻² per half-sarcomere, of which ∼30 % are generating force in an isometric contraction (Cooke, 1997), the maximum work done by each myosin head would be: 1.5 mJ m⁻² per half-sarcomere/5 × 10¹⁶ m⁻² per half-sarcomere = 30 × 10⁻²¹J. This additional amount of work done against T_o is about one half of the work thought to be done during a single actomyosin interaction without previous stretching (Woledge et al. 1985; Cooke, 1997).

Conclusions

The present study provides evidence showing that, when a steady state is not attained by the end of a ramp stretch, an appreciable fraction of the energy stored in the undamped elastic elements during stretching is conserved during the fast phase of stress relaxation by a transfer into damped elements within the sarcomeres through a temperature-dependent mechanism. The Q₁₀ and the dominant rate constant of this energy transfer are consistent with the model proposed by Huxley & Simmons (1971).

The geometrical constraints of the lever arm model of the power stroke (∼5 nm per half-sarcomere; e.g. Cooke, 1997) would appear to be unable to accommodate both the maximum phase 2 shortening against T_o and the large stretch amplitude required to attain this maximum, a lengthening distance greater than the generally accepted maximum distance over which the cross-bridges can remain attached to actin. However, some mechanisms that may overcome these constraints in vivo cannot be excluded. These mechanisms may involve: (1) the cooperative action of the two myosin heads (Huxley & Tideswell, 1997); (2) two myosin steps per working stroke (Veigel et al. 1999); and (3) multiple stepping by a single myosin head (Kitamura et al. 1999). Conformational changes may also occur within the actin filament where cross-bridges are attached (Yanagida et al. 1984).

Apart from the exact mechanism involved, the present experiments show that the physiological function of the undamped elastic elements in muscle is that of a buffer capable not only of delivering mechanical energy when the velocity of shortening is too high to be followed by the contractile machinery, but also of temporarily conserving mechanical energy when the velocity of stretching is too high for direct energy storage within the contractile machinery.