VINCENZO LOMBARDI From the Dipartimento di Scienze Fisiologiche, Universita degli Studi di Firenze, Viale G. B. Morgagni, 63, Firenze, Italy

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1 Journal of Physiology (1992), 445, pp With 32 figures Printed in Great Britain TENSION TRANSIENTS DURING STEADY LENGTHENING OF TETANIZED MUSCLE FIBRES OF THE FROG BY GABRIELLA PIAZZESI, FABIO FRANCINI, MARCO LINARI AND VINCENZO LOMBARDI From the Dipartimento di Scienze Fisiologiche, Universita degli Studi di Firenze, Viale G. B. Morgagni, 63, Firenze, Italy (Received 28 February 1991) SUMMARY 1. Steady lengthenings at different velocities ( jum/s per half-sarcomere, temperature C) were imposed on isolated frog muscle fibres at the plateau of the isometric tetanus (tension To) When tension during lengthening had attained a steady value (Ti), which varied from about 15 to about 2 times To depending on lengthening velocity, tension transients were elicited by applying step length changes of different amplitudes. The change in length of a selected segment, close to the end of the fibre connected to the force transducer, was controlled by means of a striation follower. 2. The instantaneous plots of tension versus the length change during the step itself showed that at the high forces developed during steady lengthening, as at the plateau of isometric tetanus, the elasticity of the fibre was almost undamped in the whole range of lengthening velocities used. 3. The tension transient elicited by step length changes imposed in isometric conditions exhibited the characteristic four phases described previously: following the tension change simultaneous with the step (phase 1), there was a quick partial recovery (phase 2, the speed of which increased going from the largest step stretch to the largest step release), a subsequent pause or inversion in recovery (phase 3) and finally a slower approach to the tension before the step (phase 4). 4. In the region of small steps the plot of the extreme tension attained during the step (Ti) versus step amplitude appeared more linear during steady lengthening than in isometric conditions and deviated progressively from linearity with increase in the size of step releases. The amount of instantaneous shortening necessary to drop tension to zero (Y1), measured by the abscissa intercept of the straight line drawn through T1 points for small steps, was about 41 nm per half-sarcomere in isometric conditions and 5-4 nm per half-sarcomere during lengthening at low speed (9,tm/s per half-sarcomere, Ti about 16 To). Taken altogether this indicates, in agreement with previous work, that force enhancement during steady lengthening is due to increase in both number and extension of attached cross-bridges. During lengthening at high speed (8,um/s per half-sarcomere), further enhancement in steady force (T7 about 1-9 To) was accompanied by increase of Y to 6-3 nm per half-sarcomere, MiS 9188

2 66 G. PIAZZESI ASND OTHERS indicating that increase in lengthening velocity exclusively produces increase in cross-bridge extension. 5. During slow lengthening the amplitude and the speed of the quick component (phase 2) of the tension recovery following step releases of small and moderate size (< 6 nm per half-sarcomere) were reduced with respect to isometric conditions, while the amplitude of quick recovery became larger following large releases. The T2 curve, obtained by plotting the tension attained at the end of phase 2 versus step size, intersected the length axis at about 14 4 nm, a value 3-1 nm larger than that of the isometric T2 curve. Phase 3 of the recovery became less distinct and the final approach to the tension before the step (phase 4) became faster. Increase in lengthening speed produced further depression in amplitude and speed of tension recovery during phase 2 and further increase in speed of phase 4. The two phases merged so that it became progressively more difficult to give a quantitative estimate of phase At lengthening velocities above -4,um/s per half-sarcomere the whole recovery was complete within 1-15 ms independent of the step size and was briefer for step stretches than for step releases. The amount of post-step lengthening necessary to attain 95% of recovery of tension (195) was linearly related to step amplitude and independent of lengthening speed. The intercept of this relation on the ordinate, just below 4 nm, represents an estimate of the amplitude of cross-bridge extension during lengthening at high speed. 7. The depression of the quick phase of tension recovery, in transients elicited by step releases of small and moderate size imposed during slow lengthening, is expected because of the larger extension of cross-bridge elastic component. The difference in the shift (13 versus 3-1 nm) produced by steady lengthening at low speed on the intercepts of Ti and T2 curves indicates that cross-bridges are not only extended (about 3 nm) with respect to their average attachment position, but also redistributed towards lower force-generating states. However, with a 1 times increase in speed of steady lengthening mostly further extension of cross-bridges (about 1 nm) will take place. 8. Increase in temperature reduced the effects of slow lengthening on phase 2, while it did not affect the tension transient elicited during fast lengthening. 9. During lengthening at high speed the whole tension transient is dominated by processes occurring during phase 4, i.e. detachment and reattachment of crossbridges. The findings (i) that the time course of recovery is not influenced by temperature; and (ii) that the amount of lengthening necessary, after a given step, to reattain the steady state is constant and independent of the lengthening speed, indicate that during fast lengthening neither the detachment nor the reattachment process represents a rate-limiting step in the cross-bridge turnover; detachment becomes very fast beyond a critical degree of cross-bridge extension and is followed by reattachment that is much faster than attachment in isometric conditions. 1. The tension transients following steps imposed either in isometric conditions or during steady lengthening were satisfactorily simulated with a model of contraction already described in a recent paper by two of the authors of this paper. The model, includes a detachment process in an early stage of the cross-bridge cycle.

3 TENSIOiN TRANSIENTS IN MUSCLE DURING STRETCH 661 INTRODUCTION In a recent paper (Lombardi & Piazzesi, 199) the mechanical characteristics of the response of an active muscle fibre to forcible lengthening (Katz, 1939; Edman, Elzinga & Noble, 1978) were reinvestigated with length control at the level of a selected population of sarcomeres. It was found that the steady force developed during lengthening at velocities below -25-A4,tm/s per half-sarcomere rises sharply with lengthening speed, up to a value about twice that at the plateau of an isometric tetanus (To). Further increase in velocity up to 12,am/s per half-sarcomere produced only minor further increase in steady force. The value of stiffness, measured by the tension response to small step length changes (about 1-5 nm per half-sarcomere), was found to be slightly higher (about 15%) than the isometric value and almost independent of lengthening speed in the whole range of velocities used. These results were explained by assuming that the cross-bridges under steady lengthening are involved in a cycle of attachment and detachment that is different from the normal cycle under isometric conditions. The increased cross-bridge extension, caused by lengthening, reduces the probability that attached crossbridges will complete the normal cycle, so that detachment occurs from an early stage of the power stroke, in a process that becomes very fast beyond a critical amount of extension. According to the Huxley and Simmons theory (1971) of force generation the tension transient elicited by a length step perturbation represents specific events of cross-bridge activity: the tension change simultaneous with the step (phase 1) is related to the change in extension of the elastic component of the cross-bridges; the subsequent quick tension recovery (phase 2) is due to a change in the equilibrium between different force-generating states of the attached cross-bridges, consequent on the change in force per cross-bridge. In this view, the tension transient elicited during steady force response to lengthening would reflect the distribution and extension of cross-bridges before the step. The experiments described in this paper deal with the tension response to step length changes superposed on steady lengthening at different velocities. To eliminate the contribution to the force response of both tendon compliance and inhomogeneity of mechanical characteristics of different sarcomere populations along the fibres, experiments were done with control of the length of a selected segment in fibres chosen for their high degree of homogeneity (Lombardi & Piazzesi, 199). The results concerning slow lengthening (< l,um/s per half-sarcomere) are presented separately from those concerning fast lengthening, to facilitate the description of the events occurring in the two conditions. It will be shown that: (1) during slow lengthening the tension recovery following a length step is characterized by an early phase that changes with velocity of steady lengthening and size of the step, in agreement with the thermodynamic requirements of 'Huxley & Simmons' transitions' between different force-generating states of the attached cross-bridges; (2) during lengthening at high velocity (> -4,um/s per halfsarcomere), the tension recovery is mainly determined by a fast detachment process, occurring beyond a critical amount of cross-bridge extension, followed by a rapid reattachment.

4 662 G. PIAZZESI AND OTHERS Brief reports of preliminary experiments done in collaboration with Francesco Colomo have already been published (Colomo, Lombardi & Piazzesi, 1986, 1989a, b). METHODS Apparatus and procedure Details of the mounting of the fibres and of the recording system have already been described by Lombardi & Piazzesi (199). Single fibres (4-6 mm long) were dissected from the lateral head of the tibialis anterior muscle of the frog (Rana esculenta). Frogs were killed by decapitation, followed by the destruction of the spinal cord. The fibres were mounted horizontally in the experimental trough between a capacitance-gauge force transducer (Huxley & Lombardi, 198) with resonant frequency ranging from 4 to 6 khz (time constant of decay of oscillations when unloaded 3-6,us) and a loudspeaker motor similar to those already described (Cecchi, Colomo & Lombardi, 1976; Ambrogi-Lorenzini, Colomo & Lombardi, 1983). The present version of the motor has an underdamped frequency of 7 khz and can deliver steps complete within about 5,us. In the experiments described in this work step duration was set not shorter than 11,us in order to avoid stimulating longitudinal oscillations in the fibre. The length changes of a fibre segment (-7-2 mm long) were monitored by means of a striation follower similar to that described by Huxley, Lombardi & Peachey (1981 a, b). For details of the striation follower use, see also Cecchi, Colomo, Lombardi & Piazzesi (1987) and Lombardi & Piazzesi (199). The output signal from the striation follower was proportional to the change in sarcomere length within the fibre segment. To minimize the time of propagation of a mechanical perturbation from the sarcomere length detector to the tension transducer, the segment was chosen close to the force transducer end, generally within the third of the fibre closest to the transducer. The very end of the fibre, where sarcomeres are shorter (Huxley & Peachey, 1961), was excluded. The tendon attachment connected to the force transducer was the one that produced the minimum longitudinal and transversal movements of the fibre during contraction. The motor-servo system operated in 'fixed-end' mode when the feedback signal originated from the photodiode sensor signalling the position of the motor lever, or in 'length-clamp' mode when the feedback signal originated from the striation follower output. In agreement with Ford, Huxley & Simmons (1977, p. 452), we found that in length-clamp mode it was much easier to achieve clean responses to fast steps when the velocity component of the feedback signal was derived from the motor position signal. Experimental protocol Fibres were tetanically stimulated at about 4 C (unless specified otherwise) and at a sarcomere length of about 2-1,um. When the tetanus plateau was reached, the control was changed from fixedend to length-clamp mode and the selected fibre segment was maintained isometric or forced to lengthen at a pre-set velocity. The amount of steady lengthening was not larger than 4 nm per half-sarcomere, in order to keep the sarcomere length in the plateau region of the tension-sarcomere length relation. Step changes in length of different amplitudes were imposed in length-clamp mode either at the plateau of isometric tetanus or after about 3 nm per half-sarcomere of lengthening, when tension had attained a steady value. Length steps were usually complete within about 12,us; with this step duration, fibres become slack for steps larger than about 5 and 7 nm per half-sarcomere in isometric and in isovelocity conditions, respectively. For large step releases, in order to minimize the possibility that the segment length-clamp would fail during the slack period, the step duration was increased as necessary. In some preparations, the length step was followed by a few transverse vibrations of the fibre, because the tendon attachments were not perfectly aligned with the longitudinal axis of the fibre. This, together with the presence of a certain degree of obliquity of the striation pattern, caused an artifact in the striation follower signal (see, for instance, the segment length signal for the largest step in Fig. 3A). In some other preparations, during the imposed lengthening, the striation-follower signal was occasionally affected by artifacts due to the presence of irregularities in the striation pattern. In both cases in length-clamp mode there were corresponding disturbances in the movement of the motor lever. When the sarcomere length signal was reliable for the relevant part

5 TEhNVSION TRANSIENTS IN MUSCLE DURING STRETCH of the lengthening, the experiment was made in fixed-end mode. However, since the end compliance affects the tension transient, especially at a low tension level, experiments in fixed-end mode were mainly devoted to analysing the responses to step stretches or small step releases. This is the case in the experiments shown in Figs 3-5. In no case were experiments in fixed-end mode used for the analysis of tension recovery following step releases of different size. Simulation The simulation was made on a Bull GCOS8-DPS9/91 system, Telematici, University of Florence. Centro Servizi Informatici e Definitions 1: length of the fibre or the segment at a sarcomere length of 21,um. V: velocity of ramp length change imposed on the fibre segment (lengthening positive). As: size of a step change in segment length (nm per half-sarcomere). T: tension exerted at a given time after a step length change. To: tension developed at the plateau of isometric tetanus. T1: steady tension attained during lengthening at constant velocity. Tl: extreme tension attained during a step length change. T2: tension attained at the end of phase 2. e1: slope of the straight line fitted to the linear part of the T, curve. e2: slope of the straight line fitted to the linear part of the T2 curve. Y1: amount of instantaneous shortening necessary to drop the force to zero. r: reciprocal of the time necessary to attain 63 % of tension recovery during phase 2. rw: reciprocal of the time necessary to attain 63 % of the whole tension recovery. 663 RESULTS Tension transients during slow lengthening When steady lengthening is imposed on a tetanized fibre, tension rises above the isometric plateau value and, beyond a critical degree of lengthening, attains a steady value that is a function of the lengthening velocity (Lombardi & Piazzesi, 199). Step length changes of different amplitude were applied to tetanized fibres both at the plateau of the isometric tetanus and during lengthening at low speed (-2-1,tm/s per half-sarcomere), at a time when tension had attained a steady value (generally at 3 nm per half-sarcomere of lengthening). The time courses of tension responses following the steps superposed on lengthening at 81,um/s per half-sarcomere are compared with those at the plateau of the isometric tetanus in Figs 1 and 2. The tension transients in isometric conditions show the characteristic four phases already described by Huxley and co-workers (Huxley & Simmons, 1971; Ford et al. 1977): a tension change simultaneous with the step itself (phase 1); a quick partial tension recovery, complete within 2-3 ms (phase 2); a subsequent reduction or reversal of rate of tension recovery lasting a few milliseconds (phase 3) and eventually a much slower recovery to the tension value before the step (phase 4). For large step releases (> 4 nm per half-sarcomere) phase 4 is 95 % complete within about 15 ms. For step stretches tension passes through a minimum (end of phase 2), then rises again (phase 3) up to a maximum and for steps larger than 1-15 nm per half-sarcomere maintains a value above that developed before the step for several hundred milliseconds (Fig. 3A); the larger the size of the step, the higher the tension potentiation during this phase. When the same steps are superposed on the tension response attained after 3 nm

6 664 G. PIAZZESI AND OTHERS V V `1\J o1nm 12 kn/m s-1 ms V V V Fig. 1. Tension responses elicited by length steps imposed at the isometric tetanus plateau (left column) and during lengthening at the velocity of -81 4Um/s per half-sarcomere, when tension had attained a value of 1P5 To (right column). In each frame the upper trace is segment-length change, the lower trace is tension. The dots below the tension traces in

7 TENSION TRANSIENTS IN MUSCLE DURING STRETCH per half-sarcomere of slow lengthening, the transient shows typical modifications (see also Colomo et al. 1989b). The tension change simultaneous with the step is always larger than that at the plateau of isometric tetanus. The amount of tension recovered during the first 2 ms is smaller and slower for relatively small steps and becomes larger for large releases. For step releases larger than about 11 nm per half-sarcomere a quick tension recovery is still clearly recognizable, whereas, in isometric conditions, it has already vanished. In transients elicited during slow lengthening there is no clear evidence for phase 3. The final tension recovery (phase 4) becomes faster than in isometric conditions and is faster for step stretches than for step releases (not shown in Fig. 1 because the ramp ends before the end of recovery). In the fibres for which ramp duration after the step was long enough to allow the measurement of the time course of the whole recovery, tension attained 95 % of the value before the step within 3 ms following a step stretch and within 1 ms following a step release. The reduction in amount and speed of quick recovery and the increase in speed of late recovery, caused by steady lengthening in transients elicited by steps of small and moderate size, become progressively more evident the more the lengthening speed is increased (Figs 4 and 5). The early component of the recovery becomes a progressively smaller fraction of the whole recovery, and the evaluation of the contribution of each phase becomes more difficult. With a step release (Fig. 4), for velocities lower than about -15,m/s per half-sarcomere the tension recovered within the first 2 ms after the step is progressively depressed by the increase in lengthening speed. For velocities larger than about -15,um/s per half-sarcomere the tension recovered within the first 2 ms rises again; the later recovery (phase 4), becoming progressively faster with increase in velocity, starts to influence the early part of tension recovery. Following a step stretch (Fig. 5), the range of velocities for which there is a progressive depression of the early component of recovery is much smaller. Actually, at -2,um/s per half-sarcomere, tension recovered within the first 2 ms has already attained a minimum. Further increase in lengthening speed produces only a rise in velocity of the later recovery, so that the early component is speeded up by the increasing influence of the later component. A quantitative estimation of the level of tension attained at the end of the quick recovery (phase 2) during slow lengthening can be done by using the tangent method (Ford et al. 1977; Colomo et al. 1989b), i.e. by extrapolating the tangent fitted to the later, slower recovery back to the time of the step (Fig. 6A). It was found that for step releases there was a sufficiently large range of velocities (-2-41,um/s per halfsarcomere) for which the tangent could be reliably fitted to a slow and almost linear the two upper frames mark the stimuli. The figure on the left of each row indicates the approximate size of the step in nm per half-sarcomere. The portion of the records between the arrow-heads is under segment length-clamp conditions. The time base between the small bars close to the tension records is 1 times faster than beyond them, in order to provide the appropriate resolution of the fast and slow events. A small shift in the traces, simultaneous with change in speed of time base, originated in A-D conversion in the Nicolet oscilloscope and was used to mark the time in which the trace speed changes. The length step was complete within 13 iss for releases < 5 nm. The step was made slower for releases larger than 5 nm, in isometric conditions, and larger than 6'5 nm, during lengthening. This was done in order to avoid the fibre becoming slack and having the segment length-clamp fail during the early part of phase 2. Fibre and segment length, 5 79 and 93 mm; sarcomere length, 24,um; temperature, 3 9 'C. 665

8 666 G. PIAZZESI ANND OTHERS nm 11 kn/m2 2 ms Fig. 2. Early components of the tension response to step length changes imposed at the isometric tetanus plateau (left column) and during lengthening at -81,tm/s per halfsarcomere (right column). Same records as in Fig. 1 but on a faster time base and with higher vertical sensitivity. The figure on the left of each row indicates the approximate size of the step in nm per half-sarcomere. In each frame from top to bottom the traces

9 TENSION TRANSIENTS IN MUSCLE DURING STRETCH 667 A... ~ ~~ ~ ~ ~ ~ ~ ~ ~ ~... B C b a 2 kn/m2 b 15 nm c~~~~~~~~~~~~~~~~~ 1 s-1 ms Fig. 3. Superposed tension responses (upper traces) to step stretches of different size (lower traces) imposed at the isometric tetanus plateau (A) and during lengthening at 9,um/s per half-sarcomere (B). The dots above the tension traces mark the stimuli. In the tension traces, the starting point for superposition is tension before the step, so that tension traces at rest do not superpose exactly. The three segment-length traces are shifted vertically relative to each other for clarity. The lower-case letters indicate the correspondence between tension and length traces. The size of the steps (nm per halfsarcomere) are 96 (a), 3-12 (b) and 4-6 (c) in A and 94 (a), 1-56 (b) and 2-92 (c) in B. The time base between the small bars close to the tension records is 1 times faster than beyond them. Fibre and segment length, 4-84 and 1-36 mm; segment sarcomere length, 2-9,um; temperature, 5 'C. phase of recovery starting at about 8-1 ms after the time of the step. The measurements of T2 made independently by two of us were in good agreement. For the step releases at velocities > 1,um/s per half-sarcomere and for step stretches at any velocity, it was impossible to find a reliable criterion to fit a tangent to the later phase; as the whole recovery became faster, it made the tension trace more curvilinear at all times and it was not possible to fit a tangent reproducibly. In fact, for step stretches, tension during the whole recovery drops continuously without any clear separation between a quick and a late phase, due to the more marked increase in speed of phase 4. Within 3 ms, regardless of step size, tension always recovers the same value (Fig. 3B). For these reasons, we felt that in any case the recovery following a stretch during slow lengthening was mainly determined by processes related to phase 4, i.e. detachment and reattachment, and we did not attempt any quantification of the quick recovery. Figure 6B shows the results obtained for step releases imposed during lengthening at velocities in the range -2-1,um/s per half-sarcomere. The proportion of tension recovered at the end of phase 2, measured by the ratio (T2 - T)/(IlT - T1), decreases progressively with increase in lengthening velocity at the expense of that recovered during phase 4. show: segment length change, tension response, and resting tension. The poor quality of the length signal at the end of the nm step in isometric conditions is due to the fibre remaining slack under segment length control. This is not the case with the same step during lengthening, because quick recovery is still present.

10 668 G. PIAZZESI AND OTHERS b a.3 a.5 b, a ab,,- āom b.. a a rb.11 b - " bo, b a n a = a af /!M s'21-36 ~~~~~~~~~~~~~~~~~5 kn/ 7 ms Fig. 4. Tension responses to a step release of about 1 5 nm per half-sarcomere applied during lengthening at different velocities (b) superposed on tension response to the same step applied at the plateau of isometric tetanus (a). Starting point for superposition is the extreme tension attained at the end of the step. Figures close to each record indicate the velocity of steady lengthening (,um/s per half-sarcomere). Fibre and segment length, 4-2 and 1P8 mm; sarcomere length, 2-14 /tm; temperature, 5-5 C; To = 175 kn/m2. TABLE 1. Means (± S.E.M., n = 1) of the intercept on the length axis of T1 curve (Y, second column) and the intercept of T2 curve (third column) both in isometric conditions and during slow lengthening (-87 ± 3 um/s). Y T2 curve intercept el e2 T, /TO (nm) (nm) (nm-i) (nm-') Isometric (±-75) (±-29) (±5) (±-7) Slow lengthening 1P (±-43) (±4174) (±286) (±-1) (±.5) Difference (±-144) (+99) Tension before the step (Tl), relative to the isometric tetanic tension, is indicated in the first column. e1 and e2 are the slopes of the straight lines fitted to the linear parts of T, and T2 curves, according to the methods described in the text. The third row shows the mean shifts to the left produced on the respective parameters by steady lengthening. Temperature, 3-9 (±416) C; sarcomere length, 2-1 (± 1) /Sm. Phase 1 T, curve. The plots of the extreme tension attained during the step versus step amplitude (T1 curve), both under isometric conditions and during slow lengthening, are shown by open symbols in Fig. 7 for two different fibres. As has already been shown (Ford et al. 1977; Lombardi & Piazzesi, 199), the T1 curve in isometric conditions () deviates slightly from linearity in such a way that the curve becomes progressively less steep as it proceeds from the largest stretch to the largest release. A good fit to the experimental points could be obtained with a parabola. In this case the intercept of the T1 relation on the length axis, that represents a measure of Y (the amount of instantaneous shortening necessary to drop tension from the isometric

11 TENSION TRANSIENTS IN MUSCLE DURING STRETCH *6 a b b~~~~ 5 ms 7 kn/m2.16 a= ab,=~~~~ -- - a a _J b ~~~~ Fig. 5. Tension responses to a step stretch of about 13 nm per half-sarcomere applied during lengthening at different velocities (b) superposed on the tension response to the same step, applied at the plateau of isometric tetanus (a). Starting point for superposition is the extreme tension attained at the end of the step, Figures close to each record indicate the velocity of steady lengthening (,um/s per half-sarcomere). In the record at -2,um/s per half-sarcomere, the step was applied 7 ms after the start of lengthening, i.e. at 14 nm per half-sarcomere of lengthening when tension and stiffness had not yet attained the maximum value. Fibre and segment length, 5-1 and 1-69 mm; sarcomere length, 2-14,um; temperature, 5-5 C; To = 14 kn/m2..24 A B 1. r v I1 nm 2 kn/m2 T2 -Tl T -Ti.8 t.6 a a 1 s-1 ms.4l..5 O1 Lengthening velocity per half-sarcomere (,um/s) Fig. 6. A, method to measure T2 following a step release (3 7 nm per half-sarcomere) superposed on steady lengthening at low speed. The record belongs to the same series of records shown in Fig. 1. Upper trace, segment length change; lower trace, tension. Arrows, small bars and dots as in Fig. 1. The tangent is fitted to the tension trace starting 8-1 ms after the imposition of the step. B, the proportion of tension recovered at the end of phase 2 ((T2-T )/(T1-1')) in transients elicited during lengthening plotted against lengthening speed. Different symbols refer to different experiments. For each experiment step amplitude and temperature were respectively, 1-5 nm and 5 5 C; V, 1-3 nm and 3.9 C; El, 1-2 nm and 4*7 C. The data points on the ordinate represent the mean value, for each fibre, from several records in isometric conditions.

12 67 G. PIAZZESI AND OTHERS A B &A 'A ' o T6/To ;; P T1/To 1.A *,' AA' T2. T' 4T2TO T21To/," A,,.4 ol ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 'i Step amplitude per half-sarcomere (nm) Step amplitude per half-sarcomere (nm) Fig. 7. T, (open symbols) and T2 (filled symbols) relations obtained with steps imposed at the isometric tetanus plateau (circles) and during steady-force response to lengthening at low speed (triangles) in two different preparations. A, fibre and segment length, 4-68 and 1'8 mm; sarcomere length, 2-14 /sm; temperature, 4-2 C; V = -86 /sm/s. B, same fibre as in Fig. 1; V = -81 /sm/s. T1 and T2 points are plotted relative to the isometric tetanic tension (To). The intercept of T1 relation on the length axis (YI) is estimated by extrapolating the line drawn by eye through the T1 points obtained with small steps (dashed lines), where the relation is approximately linear (Ford et at. 1977; Lombardi & Piazzesi, 199). The dotted lines are the regression lines on the T2 points obtained with step releases larger than 5 nm. The intercepts of Tl and T2 curves on the length axis are listed below: YO T2 curve Fibre T/TO (nm) intercept (nm) A (Isometric 1P P47 A V = -86,um/s 1P (JIsometric B V = -81 /sm/s plateau to zero), depends too heavily on the points for large releases. These points are the most affected by the quick recovery occurring during the step itself (Ford et al. 1977) and require correction. We did not think it worth while to go through such a complicated analysis. We considered it sufficient to estimate Y as the intercept of the T, relation obtained by extrapolating the straight line drawn through the T1 points for small step releases (< 3 nm). The mean value of Y for ten segments from as many fibres was 4-65 nm per half-sarcomere (Table 1). The TI curve obtained when steps were superposed on slow lengthening (A) appears to be linear for a larger range of step amplitudes and steeper than that determined at the isometric tetanus plateau. The intercept on the length axis of the straight line extrapolated from the linear part of the T1 curve is shifted to the left with respect to that in the isometric condition. The mean value of the intercept for the ten fibres was 5 43 nm per half-sarcomere (Table 1). Under the experimental conditions used here, phase 1 is determined mainly by the cross-bridge elasticity (Ford et al. 1977, 1981). The shift of about 1-3 nm per half-

13 TENSION TRANSIENTS IN MUSCLE DURING STRETCH 671 Lfl"'\ o LO 6 LO o ao E i OF C L- a) 9 CJ 2 Lf rl E C a) E ul CY) n a) CY) C> LLU E C4 + DO O O C', n C - a) C, LC) 6 JC) LO C174 O U) O.:. o o -c OC Fig. 8. For legend see p. 672.

14 672 G. PIAZZESI AND OTHERS sarcomere of the intercept of the T1 curve during slow lengthening therefore indicates a corresponding increase in the average extension of attached cross-bridges. On the other hand, the whole Tq curve is steeper during lengthening, as the force before the step is increased by about 63 %, i.e. more than in proportion to the increase in the value of Y, and this indicates a corresponding increase in stiffness. The mean value for the increase in stiffness measured by the slope of the linear part of the T1 relation (el) during lengthening is about 23% (Table 1), a value comparable to that determined with small step releases in a previous paper (Lombardi & Piazzesi, 199, Table 2, column 4). Instantaneous tension-length plots. A limitation of the use of the slope of T1 curve to estimate the stiffness derives from the fact that the step has a finite duration (about 12 /is), and the rate of quick recovery changes with step amplitude (Huxley & Simmons, 1971). Consequently, the proportion of the tension recovery which occurs during the step itself changes with the step size. Moreover, the finding that the quick recovery following a step is slower during lengthening than in isometric conditions might lead to overestimation of the increase in stiffness evaluated by comparing the slope of T1 curves. A more precise characterization of instantaneous stiffness can be obtained by plotting the tension versus the length during the step itself (Ford et al. 1977). In this way, during the early part of each record, there has been negligible time for the quick recovery to affect the tension response. Figure 8 shows the instantaneous force-extension curves determined for steps imposed both under isometric conditions (A, B and C) and during slow lengthening (D, E and F). Tension points during the length step were corrected for the force-transducer inertia, according to the method used by Huxley and co-workers (Ford et al. 1977, Appendix F). In Fig. 8A, tension was plotted against fibre length as measured by the motor position; for the largest releases the points corresponding to the beginning of the step are clearly shifted to the left, indicating a lag of force with respect to the length change. In agreement with Ford et al. (1977), this lag is mainly due to the time of propagation of the length perturbation along the fibre (including tendon attachments) from the motor end to the transducer end. When tension is plotted Fig. 8. Instantaneous tension-length plots during steps of different size imposed at the plateau of the isometric tetanus (A, B and C) and during lengthening at O81 1um/s per half-sarcomere (D, E and F). Tension expressed relative to the isometric tetanic tension is corrected for force-transducer response. With the force transducer used in this work (about 5 times faster than that used by Ford et al. 1977) the tension signal is almost unaffected by the correction. The time interval between points sampled by the Nicolet digital oscilloscope is 1,us; for reasons of clarity some points belonging to the last part of each step are omitted. The tension is plotted against change in fibre length (% lo) imposed by the motor in A and D, and against change in segment length in B, C, E and F. The amplitude of the segment-length steps is listed in graphs B and E. Corresponding steps in graph rows are indicated by using the same symbols. Note that in both isometric and isovelocity conditions of A and D the points corresponding to the beginning of the step releases do not superpose, those belonging to the largest-step releases being shifted to the left with respect to the points belonging to the other releases. The inflexion has almost vanished when tension is plotted against segment-length change (graphs B and E). Correction for resting viscosity makes the points belonging to different step sizes no longer superposable. Same fibre as Fig. 1.

15 TENSION TRANSIENTS IN MUSCLE DURING STRETCH against segment-length change (Fig. 8B), there is no detectable lag, at least within the precision set by the 1,us sampling rate, provided that the segment was chosen in the third of the fibre close to the force transducer end. This behaviour was found in the great majority of the fibres used. Consequently, in order to keep the analysis simple, no correction was made for propagation time of length perturbation along the fibre. TABLE 2. Means (± S.E.M., n = 5) of Y} (second column) measured by the intercepts on the length axis of straight lines drawn through the instantaneous plots of tension versus segment length during steps applied in isometric conditions and during slow lengthening ( um/s). T/TO (nm) (nm-i) Isometric (±8) (±-6) Slow lengthening (±-52) (±-193) (±-8) Difference 1-35 (±+244) el is the average slope of the straight lines. The third row indicates the mean shift of the intercept produced by steady lengthening. Temperature, 4-28 (± 2) C, sarcomere length, (± 1),um. Figure 8B shows clearly that, for most of their courses, the traces belonging to different step sizes superpose so that a straight line can be drawn through the points belonging to small shortening (< 2 nm) without ambiguity. When corrected for the viscosity exhibited by the fibre at rest (Ford et al. 1977; Lombardi & Menchetti, 1984), the traces no longer superposed (Fig. 8C); the points belonging to the largest releases were shifted above those belonging to the smallest releases. This indicates that there is no significant viscosity-induced deviation in the traces from what would be expected on the basis of a pure elasticity. The intercept on the length axis of the straight line drawn by eye (see Ford et al. 1977) through the points for shortening < 2 nm was 3-6 nm per half-sarcomere. This analysis was done on five fibres. The mean value for the intercept on the length axis was 3-73 nm per half-sarcomere, a value 8 % lower than that estimated by the intercept on the length axis of the line extrapolated from the T1 curve (Fig. 7 and Table 1), and very close to that estimated from instantaneous tension-length plots by Ford et al. (1977, 1981). The instantaneous force-extension curves determined during steady lengthening at low velocity are shown in Fig. 8D (length signal from motor position) and E and F (length signal from striation follower). Again in graph D the points corresponding to the early part of larger steps are shifted to the left due to fibre inertia. In the plots of force against segment-length change there is, as in isometric conditions, no detectable deviation from a straight line across the isometric point, and traces belonging to different step sizes superpose, when no correction is made for the resting viscosity. This indicates that the cross-bridge elastic component behaves as a pure elasticity at the high forces developed during steady lengthening, as well as at the isometric plateau. In Fig. 8E the intercept on the length axis of the straight line fitted by eye to the points for shortening < 2 nm is 4-7 nm per half-sarcomere. In the five fibres used the mean value of YO determined from the instantaneous plots was 5 4 nm per half-sarcomere. 22 PHY 445 YO el 673

16 674 G. PIAZZESI AND OTHERS These results are summarized in Table 2. Note that also during slow lengthening, as in isometric conditions, the value of Y obtained with the instantaneous plots is significantly smaller than that obtained with T, curves. On the other hand, with either procedure, the estimates of both the shift of the intercept and the increase in the stiffness produced by steady lengthening are very similar. Phase 2 T2 curve. The level of tension attained at the end of phase 2, TV was measured according to the method already described (Ford et al. 1977; Colomo et al. 1989b; Fig. 6A, this work). The relation between T2 and step amplitude (T2 curve) is shown by the filled symbols in Fig. 7 for both the isometric and the isovelocity conditions. Due to the difficulty in discriminating between phases 2 and 4 in step stretches superposed on lengthening, the present analysis of phase 2 in transients elicited during lengthening is limited to step releases. A '---- %.- -, " Ti1-T -5-./',;'. 1 -~2,,, -T1- T2d (A),pottedagi sep.4 L v.4.2~~~~~~ Fi.roorio o tnsonreovre a te ndofphs 2. (T-)/l-Ti)in1 / Step amplitude per haif-sarcomere (nm) Step amplitude per half-sarcomere (nm) Fig. 9. Proportion of tension recovered at the end of phase 2 ((T2~-T 1T- Th) in transients elicited in isometric conditions () and during steady lengthening at low speed (A), plotted against step amplitude. Lengthening velocities (V) were -86 (A) and -81 (B),um/s. A and B show data from the same fibres as in Fig. 7A and B. The T2 curve in isometric conditions shows the characteristic features described by Ford et al. (1977): the tension recovery at the end of phase 2 is almost complete for small steps, while it becomes progressively smaller for large step releases. Consequently the T2 curve shows a flat part for step releases up to about 5 nm per half-sarcomere, and then, for larger releases, a region of linear decrease with an interception on the length axis at about 11-5 nm per half-sarcomere. During lengthening the tension recovered at the end of phase 2 is reduced for small steps, so that the T2 curve is steeper in this region and becomes more linear. For releases of 4-6 nm per half-sarcomere the curve shows an inflexion and a consequent reduction in slope; eventually, for even larger releases, it shows a constant slope not significantly different from that in isometric conditions. The straight line fitted to the points obtained with releases larger than 5 nm intersects the length axis at about 14-5 nm per half-sarcomere, a value about 3 nm per half-sarcomere to the left of the

17 TENSION TRANSIENTS IN MUSCLE DURING STRETCH 675 interception of the isometric T2 curve. The results obtained in the ten fibres used in this work are summarized in Table 1. Note that the shift of T2 curve intercept is significantly higher than the shift of YO. The effect of steady lengthening on the proportion of tension recovered at the end of phase 2 is more evident in Fig. 9, where the quantity (T2-T1)/(Ti-71), the A B +1-4 C ~~ : T-T2 v ~~1 kn/m2 T T.+- 2 ms D 1.: T-T2-4.8 Tl -T2-1..5: Time (ms) Fig. 1. Left panels, time courses of quick recovery in tension transients elicited in isometric conditions (To = 212 kn/m2, A) and during lengthening at -86,tm/s per halfsarcomere (T1 = 1-55 To, B). Traces superposed starting from tension before the step. Figures close to records indicate the size of the imposed segment-length step (in nm per half-sarcomere). The 5 khz noise (that becomes progressively more evident with increasing size of step) was very probably due to transverse oscillations of the fibre resonating with transverse oscillations in the force transducer lever. Due to some degree of skewness of striations, this produced a spurious signal in the striation-follower output that appears on the force trace in segment length-clamp mode. Right panels, traces from left panels replotted in semi-log scale and superposed after the normalization procedure used by Ford et al. (1977) in order to scale up to the same amplitude. Interpolation has been made between points to eliminate the noise. The normalization procedure cannot be applied to the recovery following the step stretches during lengthening because of the uncertainty in the estimate of T2. Note that in both isometric (C) an isovelocity (D) conditions the speed of tension recovery, indicated by the slope of the curves, is higher for larger step releases. Note also that, for any step size, the slope is lower during lengthening than in the isometric condition. Same fibre as Fig. 7A. recovered tension relative to the initial tension fall, is plotted against the step amplitude for the two fibres of Fig. 7. The curve in isometric conditions shows the same general features as the T2 curve, a flat region for small steps and a region of linear decrease for steps larger than about 5 nm per half-sarcomere. During steady lengthening, for small releases the proportion of tension recovered is smaller than in isometric conditions, but it increases with the size of the release. (T2- T1)/(Ti- T1) is maximum for releases of about 5 nm per half-sarcomere and for larger releases there is a linear decrease with a slope lower than that in isometric conditions. Consequently, for steps larger than about 7 nm per half-sarcomere the fraction of tension recovered 22-2

18 676 G. PIAZZESI AND OTHERS A '3 B '3. r (ms-1) r (ms-1) 2 2. * i Step amplitude per half-sarcomere (nm) Step amplitude per half-sarcomere (nm) Fig. 11. Rate constant (r, as defined in the text) of tension recovery in phase 2 against step amplitude, for the same two fibres as in Fig. 7. Steps superposed either on the isometric tetanus plateau () or on steady lengthening at low speed (*). Lengthening velocities (V) were 86 (A) and 81 (B),um/s. A '3 B 3 r (ms-1) r (ms-1). 2 2 * Step amplitude per half-sarcomere (nm) Step amplitude per half-sarcomere (nm) Fig. 12. Relations between r and step amplitude as in Fig. 11, but with the relation determined during lengthening shifted to the right by 2-8 nm (A) and by 2 77 nm (B) in accordance with the shift in the intercept on the length axis of the corresponding T curves plotted in Fig. 7. Symbols and lengthening velocities as in Fig. 11. becomes larger than in isometric conditions and the intercept on the length axis is shifted by the same amount as the T2 curve intercept. Speed of quick recovery. In isometric conditions the rate of recovery during phase 2 changes with the size and the direction of the step, and this deviation from a simple viscoelastic behaviour supports the idea that the underlying process is an active property of the cross-bridges (Huxley & Simmons, 1971). It is clear from records such

19 TENSION TRANSIENTS IN MUSCLE DURING STRETCH as those in Figs 2, 4 and IOA and B that, during the quick recovery following a small step release, the tension recovery relative to the instantaneous tension change is slower in isovelocity than in isometric conditions. The rate of tension recovery can be characterized by r, the reciprocal of the time necessary to attain 63 % of the tension recovered at the end of phase 2 (T2-T1). In fact the recovery in phase 2 is 3 r (ms-1) vo ~~~~ 1 7 V Velocity per half-sarcomere (,um/s) Fig. 13. Relation between r in tension transient following a small-step release and speed of lengthening or shortening. Data points for lengthening refer to the same records used for data plotted in Fig. 6B. Symbols and step amplitudes as in Fig. 6B. not represented by a single exponential curve, so that r is not the rate constant of an exponential process (Ford et al. 1977). The semi-log plot of tension against time during quick recovery is not linear (Fig. IO C and D), but for any step size, the overall slope of the semi-log plot of normalized tension recovery against time is smaller during steady lengthening (Fig. lod) than in isometric conditions (Fig. 1C). In Fig. 11, r is plotted against step size. For any given step, r is smaller during lengthening (@) than in isometric conditions () so that the whole relation is shifted to the left. In Fig. 12 the experimental points representing r during lengthening (@) are shifted to the right by an amount equal to the difference between intercepts on the length axis of corresponding T2 curves in Fig. 7. It can be seen that, after this manoeuvre, the relation during lengthening does not exactly superpose that in the isometric condition, but lies just above as if the shift had been overestimated. The relation between speed of early recovery and lengthening or shortening velocity is shown in Fig. 13 for the same experiments as in Fig. 6B. It can be seen that the depression of r produced by steady lengthening is proportional to the lengthening velocity in the same narrow range of velocities ( 2-1 sm/s per halfsarcomere) in which there was a marked depressant effect of lengthening on the amount of quick recovery. In Fig. 13, and El refer to two experiments in which the transient was also recorded for steps superposed on steady shortening. In this case, in agreement with previous results (Ford, Huxley & Simmons, 1985), phase 2 is faster than in isometric conditions. Note that there is a slight discontinuity in the relation across the isometric point, the slope being larger in the region of lengthenings.

20 678 G. PIAZZESI AND OTHERS Tension transient during fast lengthening When the velocity of steady lengthening is increased above -25-4,um/s per halfsarcomere, it becomes more and more difficult to distinguish a quick phase of nmi 25 kn/m2 I ms Fig. 14. Tension responses to steps of different size imposed during fast lengthening at three different velocities. Figures on top of each column indicate the lengthening speed in,um/s per half-sarcomere. Figures to the right of each row indicate the approximate size of the step in nm per half-sarcomere. In each frame, from top to bottom, traces show segment length change, tension response, resting tension. Fibre and segment length, 4-76 and 1-23 mm; sarcomere length, 2-8,um; temperature, 4-1 C; To = 245 kn/m2. recovery in the tension transient following a step of small or moderate size (< 4 nm per half-sarcomere). This is due to the progression of the effects seen at lower lengthening velocity, i.e. the reduction in amount and speed of phase 2 and the increase in speed of phase 4. Figure 14 shows sample records of tension transients following steps of different sizes superposed on steady lengthening of the fibre segment at three different velocities ( 44, -84, 1-65,tm/s per half-sarcomere). A characteristic feature of the transient at these velocities is that, even for large step releases, tension recovery to

21 A TENSION TRANSIENTS IN MUSCLE DURING STRETCH B kn/m2 2 ms 15 kn/m2 a b c 15 nm a b c 11 nm c Fig. 15. A, superposed tension responses (upper traces) to step stretches of different size (lower traces) delivered during steady lengthening at 7,um/s per half-sarcomere. The dots above the tension trace mark the stimuli. In tension traces the starting point for superposition is the steady tension attained before the step. The three segment-length traces are shifted vertically for clarity. The small letters show the correspondence between tension and length traces. The sizes of the step are -78, 1-59 and 3-11 nm per halfsarcomere for a, b and c respectively. The small bars indicate 1 times magnification in trace speed. Records are from the same experiment as in Fig. 3. B, same records as in A but on a faster time base and with increased vertical sensitivity, to make clearer the effect of step size on tension recovery.. z rz I Step amplitude per half-sarcomere (nm) Fig. 16. T1 curves, determined in isometric conditions () and during lengthening at 9 (A) and 7 (OI) um/s per half-sarcomere. In all curves T1 is made relative to tension at the plateau of isometric tetanus. The straight lines are drawn by eye through the points for small steps. Fibre and segment length 4-9 and 1-4 mm; sarcomere length, 28,um; temperature, 4-8 C. the value before the step is 95 % complete within about 15 ms. In fact, an early, faster phase of recovery is quite distinct, especially for lengthening at moderate velocity (Fig. 14, left and middle columns). At these velocities the tangent method cannot be used reliably to estimate the proportion of tension recovered during the

22 68 G. PIAZZESI AND OTHERS early phase, yet it can be seen that, qualitatively, this proportion increases with the size of the step release. In any case, tension recovery following steps superposed on fast lengthening is mainly determined by a process distinct from phase 2. In fact, the speed of recovery is higher for step stretches than for step releases, just the opposite of what is shown for phase 2 in transients elicited in isometric conditions or during slow lengthening. TABLE 3. Means (± S.E.M., n = 6) of intercepts on the length axis of the linear part of Ti curves (YO, second column) both in isometric conditions and during fast lengthening ( ltm/s). yo el Ti/TO (nm) (nm-i) Isometric (±129) (±8) Fast lengthening (±+47) (±+222) (±-16) Difference (±-155) First column, tension before the step, relative to the isometric tetanic tension. Third column, slopes of the straight lines drawn through the T, points in the range of small releases. The third row shows the shift (± S.E.M.) produced on the intercept by lengthening at high velocity. Temperature, 4-5 (±27) 'C. Sarcomere length, 29 (±2) am. At high lengthening velocity the tension recovery following a step passes through a minimum or a maximum for stretches or releases, respectively. The amount of the undershoot following step stretches increases with the size of the step (Fig. 15). T, curve. Figure 16 shows the T1 curve, obtained from one experiment in which tension transients were elicited either in isometric conditions () or during lengthening at both low (A) and high (OI) velocities. It can be seen that the main effect of increasing lengthening speed is an increase in the shift of the intercept on the length axis of the linear component of T, curve. As usual, the straight line was drawn by eye through the points for small releases (< 3 nm). In this fibre Y1 increased from 5f3 nm per half-sarcomere during lengthening at 9,um/s per half-sarcomere to 6-5 nm per half-sarcomere during lengthening at -7,um/s per half-sarcomere. On the other hand, due to the increased steady tension at the higher velocity (1P91 To at 7,um/s per half-sarcomere compared with 1P62 To at 9,im/s per half-sarcomere), the slope of the curve is almost unaffected by the increase in velocity. This implies that, in accordance with previous results (Lombardi & Piazzesi, 199), stiffness during lengthening is independent of lengthening velocity in this range. Similar results were obtained in six fibres and are summarized in Table 3. Instantaneous tension-length plots. The plots of tension versus segment-length change during the step itself for the same fibre as Fig. 16 are shown in Fig. 17. It is evident that during fast lengthening (steady tension 1-91 TO), as during slow lengthening (steady tension 162 To) and in isometric conditions, the force traces belonging to different step size superpose during most of their course, which indicates that there is no significant viscous component in the force response. Actually in this fibre, especially in isometric conditions (Fig. 17A), there is a small horizontal shift between the traces across zero length, which indicates a lag of the force signal with respect to the segment-length signal. This is in contrast to the conclusions, drawn from plots like those in Fig. 8,

23 TENSION TRANSIENTS IN MUSCLE DURING STRETCH 681 CE, (i o 4.) uq 4.) Co~~~~~c E Q CV) t-z ( o._ r sr-. -._ X) t a73 su) o a)= EC,) -c = 4 O L. C>. 4a C) ) '..== ( ed Qz k1 -E -. E > S - o; N CE, E O (C CE, IL CD + + I < a I o D

24 682 G. PIAZZESI AND OTHERS that the effects of the propagation time between the segment under inspection and the force transducer are beyond the limit of resolution of our records. A degree of deviation comparable to that in Fig. 17 was present in another fibre among those used for this analysis, and again it was more marked for steps imposed at To (isometric conditions) than at higher tensions (during lengthening). The lag was not correlated with the compliance of tendon attachment as estimated by the difference between the value of Y} for the segment and for the fibre (tendon included). It is likely that this lag is related to the mass of the tendon at the transducer end, in relation to the dimensions of the fibre. At the time of the experiments we did not take into account the mass of the tendon, but we measured the dimensions of all fibres. The fibre of Fig. 17 had a cross-sectional area of 4712,tm2, whereas the mean value (±S.E.M.) for cross-sectional area in all the fibres used for the instantaneous tension-length plots (eight) was 678 (±516) /um2. TABLE 4. Means (± S.E.M., n = 5) of Y1 measured by the intercept on the length axis of the straight lines drawn through the instantaneous plot of tension versus segment length during steps applied both in isometric conditions and during fast lengthening ( ,am/s) YO el Ti/<TO(nm) (nm-1) Isometric (± 149) (±-12) Fast lengthening (±48) (±229) (±16) Difference 2 31 (± 136) Two of the five fibres contributed also to data reported in Table 2 for slow lengthenings. First column, tension before the step. Third column, slopes of the straight lines. The third row indicates the mean shift produced on the intercept by lengthening at high velocity. Temperature, 4 3 (± 2) 'C. Sarcomere length, 2-8 (± 2),um. In Fig. 17, the intercepts on the length axis of the straight lines drawn by eye as shown in the graphs, are 4, 486, 622 nm per half-sarcomere, in A, B and C respectively. Table 4 summarizes the results obtained during fast lengthening in five fibres. The estimate of the lengthening-dependent shift of the intercept is similar in Table 4 and Table 3. Comparison with data obtained at low velocity (Table 2) shows that, in accordance with the results given in the previous section, the increase in tension with lengthening speed is related to the shift in the intercept on the length axis, with no change in the slope of the curve and, therefore, in the stiffness. Tension recovery. Tension recovery that followed steps applied at velocity higher than 4,am/s per half-sarcomere appears dependent mainly on processes not related to the size of the step, such as those occurring during phase 4. In phase 4 crossbridges reattain the distribution characteristic of the steady state by detachment and reattachment further along the actin filament (Huxley, 1974). The relations shown in Figs 18 and 19 give further support to the view that the tension transient elicited during fast lengthening is mainly related to cross-bridge detachment and reattachment. Figure 18 shows the dependence of the speed of the entire recovery process following steps of different size on velocity of lengthening. The speed of recovery was characterized by the reciprocal of time necessary to attain 63 % of the whole recovery of tension (rw). For a given step size, rw is linearly related to the velocity of the ramp on which the step is superposed. The slope of the relation obtained for different step sizes is progressively smaller going from the largest step stretch to the largest release.

25 TENSION TRANSIENTS IN MUSCLE DURING STRETCH There was little variation in this relationship between fibres provided that no large inhomogeneity developed with the increase in lengthening speed, and that the experiment was made in length-clamp mode. Figure 19 describes the recovery process in terms of the lengthening necessary to reattain the steady state after the imposed step. The amount of post-step lengthening E <P1.5 ; / K; { S ~~~~~ Lengthening velocity per half-sarcomere (,um/s) Fig. 18. Rate of whole tension recovery (rj) versus speed of steady lengthening for different step sizes, as indicated by the figures (nm per half-sarcomere) close to each relation. Open symbols refer to the same fibre as Fig. 14; filled symbols refer to data pooled from three other fibres. The lines are fitted to the corresponding data points by linear regression analysis, in accordance with the equation rw = a V+ b, where a and b are the slope and the ordinate intercept of the straight line. For and nm steps the regression line is fitted to data pooled from all four experiments. Parameters (±S.D.) of regression equations are listed below: Step amplitude (nm) a (smm-1 x 13) b (ms-1) (±49) -86 (±52) - 1*5-76 (±5) --52 (±56) The intercepts were not significantly different from zero (P > 41 for nm and P > -2 for -15 nm). (195), that has occurred by the time that tension recovery is 95 % complete is plotted against step amplitude for different lengthening velocities. There is a linear relation between 195 and step size with a slope of 71 and an intercept on the ordinate of 3-84 nm per half-sarcomere. This relation is apparently independent of lengthening velocity. To further investigate this point, the data of Fig. 19B were grouped into three classes of velocities (4--7, , P65,um/s per half-sarcomere) and the regression analysis was done separately for each class. Neither the intercepts on the ordinate nor the slopes of the lines for the three classes of velocities differ significantly from each other or from those fitted to the pooled data (parameters given in Fig. 19 legend). However there appears to be a tendency of the regression line to be shifted downward slightly by increasing the lengthening speed.

26 684 G. PIAZZESI AND OTHERS Tension transients at different temperatures Tension transients were elicited at different temperatures to further characterize the nature of the processes underlying the tension recovery during slow and fast lengthening. Figure 2 shows sample records from one experiment in which a step A 16 B 16 /95 (nm) /95 (nm) o44p m s\ 8 to 44, m12 12 ~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~ 8 8 o.44 Mim/s a *4-.7 Mm/s v 1.65 V Step amplitude per half-sarcomere (nm) Step amplitude per half-sarcomere (nm) Fig. 19. Relation between the amount of post-step lengthening necessary to attain 95% of tension recovery (I95) and step amplitude, for the same fibre as Fig. 14 (A) and for four fibres (B). In each graph the regression line is fitted to all data points according to the equation 1l5 = m As + q, where As is the size of the step length change, and m and q are the slope and the ordinate intercept of the straight line. In A, different symbols refer to different velocities of steady lengthening (um/s per half-sarcomere) as listed in the graph. In B different symbols refer to different classes of velocity (4um/s per half-sarcomere) as listed in the graph. The parameters (± S.D.) of the regression analysis done on data grouped for classes of velocity and on the pooled data are listed below. Velocity (,um/s per half-sarcomere) m q (nm) 4-7 -O693 (±32) 4-66 (±182) (±39) (±-195) 1P (±-31) (±-166) Pooled --76 (± 2) 3-84 (±-17) According to the analysis of covariance, there are no significant differences between classes of velocity for both m (P > 25) and q (P > -1). release of 1-2 nm per half-sarcomere was imposed at three different temperatures (4'7, 11., and 17 C). In isometric conditions (first row), at all temperatures, the early recovery is responsible for a large part of the whole recovery. The rise in temperature increases the speed of all the phases of the transient. Values of r for phase 2 were 1-44, 2-17 and 3-8 ms-1 for 4-7, 11 and 17- C respectively. The corresponding temperature coefficient (Qlo) is about 2. Slow lengthening (-7,um/s per half-sarcomere) causes a decrease in the amount of tension recovered in phase 2 that is more pronounced at a low temperature (Fig. 2, second row and Fig. 22A). By comparison with the higher temperatures, it seems that at a low temperature the depression occurs especially at the expense of the slow

27 iso TENSION TRANSIENTS IN MUSCLE DURING STRETCH 4.7 C R 11 OC 17 C G~~~~~~~~~~~~~~ I I -49 I V V V 5 ms Fig. 2. Tension transients elicited by a step release of about 1-2 nm per half-sarcomere applied at three different temperatures (4 7 C, To = 2 kn/m2; 11 C, To = 236 kn/m2; 17 C, To = 271 kn/m2) either in isometric conditions (Iso) or during lengthening at different velocities. The approximate values of lengthening speed are indicated by the figures (#um/s per half-sarcomere) next to the corresponding rows. The small bar indicates the time at which trace speed is reduced 1 times. In some records during lengthening at high velocity the ramp ended before the change in trace speed as indicated by the arrow-heads. The corresponding change in tension response is rounded due to the inability of the command signal to provide a corner sharp enough in relation to the velocity of lengthening. Fibre and segment length, 4-67 and 1-85 mm, sarcomere length, 2-9 Um. I 54 kn/m2 TABLE 5. Rates of quick recovery following step release and step stretch in isometric conditions r r Temperature To step release step stretch (OC) (kn/m2) (ms-i) (ms-i) (±3) (±13) (±-41) (± 75) (±6) (±9) (±-46) (±-11) Q X 2-65 Mean values ( ± S.E.M.) of rates of quick recovery following a step release of 1-16 ( ± 6) nm per half-sarcomere, n = 4, and a step stretch of 1-29 (± 8) nm per half-sarcomere, n = 5, imposed at the plateau of isometric tetanus at two different temperatures as indicated in the first column. The last row shows the corresponding Qlo values. component of phase 2 recovery; however, because of the disappearance of phase 3, it is difficult to decide where phase 2 ends. Consequently, it is possible that the depressant effect consists not only of the reduction of the final tension attained in phase 2 but, in some degree, also consists of the extreme reduction of the rate of the slow component of phase 2. The overall rate of phase 2 is in any case markedly depressed at all temperatures. At 17 C phase 3 is still recognizable, even if it occurs later than in isometric conditions.

28 686 G. PIAZZESI AND OTHERS A B 5 C 1.5 C kn/m2.65 ~ ~~~5ms Fig. 21. Superimposed tension transients elicited by a step stretch of 11 nm per halfsarcomere applied during slow lengthening at two different velocities, indicated by the figures (,tm/s per half-sarcomere) close to the corresponding traces. A, temperature, 5 C, To = 115 kn/m2; B, temperature, 15 C, To = 184 kn/m2. Starting point for superposition is the extreme tension attained during the step. Note that, at the higher temperature, with steady lengthening at -2,um/s per half-sarcomere, change in tension during the step is smaller than at -6,um/s per half-sarcomere, indicating that at very low velocity, stiffness has not yet attained the maximum value. Fibre and segment length, 5 4 and 1 1 mm; sarcomere length; 2-12 /tm. A r (ins-1) T2 -T B Velocity per half-sarcomere (,urmis) Fig. 22. A, proportion of tension attained at the end of quick recovery ((T2- T,)I(T -T)) in transient elicited by a step release of about 1-2 nm per half-sarcomere applied during slow lengthening, plotted versus lengthening speed at three different temperatures ( C); Q, 4-7; V, 1 1; D, 17. Measurements refer to records from the same fibre as Fig. 2. B, open symbols, rate (r) of quick recovery measured in the same records as in A, according to the method already described on page 677. Filled symbols show the relations at 11 C (v) and at 4-7 C (v) shifted to the right along the horizontal axis by 8 and 14,um/s respectively. During lengthening at -49,tm/s per half-sarcomere (Fig. 2, third row), the recovery is 95 % complete within about 15 ms at all temperatures; at the same time an early component is clearly present and, as for the lowest velocity, it is more depressed at a low temperature. Further increase in lengthening speed (last two rows) produces, at all temperatures, the disappearance of the early component and the progressive increase in speed of the late recovery, so that tension time courses become progressively more superposable. Note that in the three records of the last row, tension recoveries appear almost identical. The effect of temperature on the tension transient following a step stretch is similar. In isometric conditions all phases of the transient become faster with rise in

29 TEN,VSION.r TRANSIENTS IN MUTSCLE DURING STRETCH temperature. Values of r for phase 2 following a step stretch of about 1 nm were 97 and 1 6 ms-1 at 5 and 1 5 C, respectively. The corresponding Q1 was 2-4. Table 5 summarizes the results obtained from six fibres in isometric conditions. The depressant effect of lengthening at low speed on the early recovery seems less marked rw (ms-') A 2 A 12 Velocity per half-sarcomere (,um/s) Fig. 23. Relationship between rate of the whole recovery (rw) following a step during fast lengthening and lengthening speed at different temperatures. Filled symbols refer to a step stretch of about 1 1 nm in the same fibre as used for Fig. 21. Open symbols refer to a step release of about 1-2 nm in the same fibre as used for Fig. 2. Temperatures are indicated by ( C):, 5; A, 1-5;, 47; A, 11; CD, 17-. at high temperature (Fig. 21). At 5 C (A) an increase in speed from about -2 to about 6,am/s per half-sarcomere does not produce a depression of the rate of quick recovery, but an increase, probably due to the final recovery becoming faster (see also Fig. 5). At 1 5 C (B), the early part of tension recovery at -6,tm/s superposes on that at 2,tm/s, but, if we take into account the amount of the initial tension drop, the early component of recovery turns out to be more depressed at the higher velocity. An increase in the speed of lengthening above 25-4,tm/s per halfsarcomere makes the tension recovery become identical at all temperatures. Figure 22 shows the relation of amount (A) and rate (B) of early recovery versus lengthening velocity for speeds ranging from -2 to 1,tm/s per half-sarcomere. The rate of recovery at a high temperature is similar to that obtained at a lower temperature if the velocity of lengthening is appropriately increased. In fact, when the relation at 11 C (A) is shifted horizontally to the right by 8,tm/s, and the relation at 4 7 C () is shifted to the right by 14,tm/s, the three relations approximately superpose. Figure 23 shows the relationship between the rate of the entire recovery (r") and lengthening velocity at different temperatures, for the range of speeds ( 4-3,um/s per half-sarcomere) at which we assumed the recovery to be a single process. The

30 688 G. PIAZZESI AND OTHERS relations are again linear as in Fig. 18 and, for a given step size and velocity, data obtained at different temperatures superpose, so that for a given step size there is a single relation at all temperatures. For the step release, in the lowest range of velocities, data taken at a higher temperature actually tend to fall above those taken at a lower temperature. The reason for this may be evident from records in Fig. 2, third row; at moderately high velocities the quick component of recovery is more pronounced at a higher than at a lower temperature, and this would lead to an overestimate of the rate of the entire recovery. DISCUSSION Elasticity of the tetanized fibre during forcible lengthening T, curve When a step length change is superposed on lengthening at low speed ( < 1 jtm/s per half-sarcomere) the tension change simultaneous with the step is about 2% larger than that produced by the same step applied in isometric conditions. Consequently, the slope of the T1 curve is about 2% larger during lengthening. Under the experimental conditions used in this work, and assuming that the quick tension recovery does not affect significantly the extreme tension attained at the end of the step, the slope of the T1 curve measures the stiffness of the sarcomere, which mainly depends on the stiffness of cross-bridges themselves (Ford et al. 1977, 1981). If the cross-bridge stiffness is the same at 1P5-2 To (the levels of force attained during lengthening) as at To, an increase by 2% of the overall slope of T1 curve indicates a corresponding increase in the number of attached cross-bridges. When the slope stiffness of the T, curves is compared in the same region of forces, the increase in stiffness during lengthening is somewhat lower, between 11 and 115 times the isometric value (Lombardi & Piazzesi, 199). During slow lengthening the increase in steady force attained before the step is about 6 %; even if we take the higher value (2 %) for the increase in number of attachments, there must be a substantial increase in force sustained by each cross-bridge. This shows up in the T, curve as a shift to the left of the intercept on the length axis (Y}) by about 13 nm per halfsarcomere, that is YI is about 33 % larger than in isometric conditions. In conclusion, in agreement with previous work (Lombardi & Piazzesi, 199), the rise in force produced by imposed steady lengthening is due mainly to increase in extension of attached cross-bridges and to a lesser extent to rise in number of attachments. Increasing the velocity of lengthening from 1 to f8 /im/s per half-sarcomere does not produce significant changes in stiffness as measured by the slope of the respective T, curves (compare data in Tables 1 and 3), while steady tension increases from about 16 to about 1P9 To. As a consequence, there is a further shift in the intercept of T, curve, with the total shift attaining a value of about 2 2 nm; that is, Y1 becomes about 62 % larger than in the isometric conditions. It is evident, therefore, that the increase in velocity of lengthening produces only further extension in the crossbridge spring, with no further increase in number of attachments. Instantaneous tension-length plots The instantaneous plots of tension against segment-length change during steps imposed in isometric conditions indicate that: (i) there is no detectable lag between

31 TENSION TRANSIENTS IN MUSCLE DURING STRETCH the segment-length perturbation and the tension signal, provided that the segment is close to the force-transducer end and that the mass and the compliance of tendon attachments are small; and (ii) the elasticity of the contractile machinery is almost undamped. The finding that the instantaneous tension response of cross-bridges is purely elastic agrees with the conclusion of previous work (Ford et al. 1977) showing that the viscosity exhibited by the fibre at rest disappears with activation. The straight line drawn through the instantaneous tension-length plots gives an intercept on the length axis of 3-7 nm per half-sarcomere, 8 % lower than that estimated from the T1 curve. This indicates that, with steps complete within 12 Its, quick recovery causes small but still significant truncation of the extreme tension attained in phase 1. Ford et al. (1977, 1981) also found a value of the intercept very close to 3-7 nm per half-sarcomere, though in their work the instantaneous plots necessitated a larger correction for both the inertia of tension transducer and the time of propagation of the perturbation along the fibre. The instantaneous tension-segment length plots during steps superposed on lengthening show the absence of any significant viscosity in the force response during the step also at the high forces developed under steady stretch. During lengthening there is an increase in the slope of these relations and a shift to the left of Y with respect to the isometric conditions. The shift of Y increases with lengthening speed in the same proportion as steady tension attained before the step, so that the slope of the plots is almost the same at low and high velocity (Tables 2 and 4). During lengthening the value of Y estimated from the instantaneous tension-segment length plots is reduced, with respect to that estimated from the T1 curves, by a smaller amount than in isometric conditions. This may be due to a reduction of truncation of T, which in turn is due to a velocity-dependent reduction in the rate of quick recovery (this T1 curve appears to be linear for a larger range of step amplitudes than that determined in isometric conditions). However the estimate of the velocitydependent increase in extension of the elastic component of attached cross-bridges is about the same, for both methods. General considerations on tension recovery following a step The tension recovery following a step change in length imposed at the isometric tetanus plateau shows well-known characteristic features (Ford et al. 1977). The analysis of the quick phase of recovery (phase 2) provides a T2 curve and a relation of rate of quick recovery versus step amplitude which are very similar to those of Ford et al. (1977), with the exception of the value of the intercept of the T2 curve on the length axis, which is significantly smaller in our experiments (Fig. 7, * and Table 1) than in those of Ford et al. (1977, see, for instance, their Fig. 13). This difference cannot be attributed to the different duration of the step, since this does not influence the T2 curve (Ford et al. 1977, p. 472). It could be related to the method used for determining the T2 value with large step' releases, but comparison of tension responses to the largest releases (Figs 1 and 2 in this paper and Figs 11 and 12 in Ford et al. 1977) seems to indicate a genuine difference between the responses obtained in the two preparations. Our results nevertheless confirm those of Ford et al. (1977) as regards the conclusion that there is a limited amount of instantaneous sliding (11-13 nm per half-sarcomere) for which force can be recovered within the first 2 ms after the imposition of the step. The relation between rate of quick recovery and step 689

32 69 G. PIAZZESI AND OTHERS amplitude (Fig. 11) is almost identical to that obtained by Ford et al. (1977) when points from their Fig. 24 (reciprocal of half-time) are made comparable to ours (reciprocal of time to 63%) under the simplifying assumption that recovery is a single exponential and the difference in temperature is taken into account. The tension recovery following length steps superposed on steady lengthening applied to a tetanized muscle fibre shows different characteristics, according to the velocity of lengthening. At very low velocity, below -1,tm/s per half-sarcomere, the tension transient following a step release can be analysed with the same procedure as the tension transients studied by Huxley and co-workers in isometric conditions or during steady shortening (Huxley & Simmons, 1971; Ford et al., 1977, 1981, 1985). Compared to the isometric conditions, the quick recovery (phase 2) following the tension change simultaneous with the step is smaller and slower for step releases of small and moderate size, but becomes larger for large releases; phase 3 reduces or vanishes; the speed of late recovery (phase 4) becomes greater. If the velocity of lengthening is increased above 1,Im/s per half-sarcomere the progression of the effects seen at low velocity makes the whole recovery following the step depend mainly on the processes occurring during phase 4, i.e. detachment and reattachment of cross-bridges: the time for tension to reattain the value before the step becomes progressively shorter, while the amplitude of phase 2 is further reduced. Eventually, at very high velocity, the time for the whole recovery becomes comparable with the time for quick recovery in isometric conditions. There is, in any case, very convincing evidence for the assumption that the recovery in this case is not a phase-2-like process: the tension recovered is independent of the size of step release. We were therefore able to describe quantitatively the quick phase of the transient only at low velocity, while at higher velocity we could not separate phase 2 and made an approximation, considering the whole recovery as a single process. Events occurring at low and high lengthening velocities have therefore been analysed separately. The dependence on lengthening velocity of the processes responsible for the various phases of the transients does not in fact show any discontinuity. The transients at all velocities can be simulated by using the model of cross-bridge reactions alreadv described in a previous paper (Lombardi & Piazzesi, 199), as shown below. Tension recovery following a step during slow lengthening The main effects of lengthening at low speed ( < 1,um/s per half-sarcomere) on tension transient are: (1) reduction of both amplitude and rate of quick recovery (phase 2) for step releases of small and moderate size; (2) increase in the amplitude of quick recovery for large-step releases (> 7 nm per half-sarcomere), with consequent increase in the value of the intercept of T2 curve on the length axis; (3) reduction or disappearance of phase 3; (4) increase in speed of late recovery. Effects (1) and (2) find a straightforward interpretation in terms of Huxley and Simmons' 1971 theory of force generation. The theory postulates that the attached crossbridges exist in different configurations or states, which, in isometric conditions, exert different forces; the transition towards a state generating more force is associated with a gradient of chemical energy, while work is done on the elastic component of the cross-bridge. The rate of transition and the equilibrium between states depend on mechanical conditions, since the work on the cross-bridge elastic

33 TENSION TRANSIENTS IN MUSCLE DURING STRETCH component is part of the activation energy for the transition. A relatively small perturbation in length of an active fibre causes a change in extension of the elastic component of the cross-bridge that is associated with the simultaneous tension change (phase 1 of Huxley & Simmons). The change in the force sustained by all the cross-bridges causes a redistribution between the different attached states, which is responsible for the quick component of tension recovery (phase 2). During steady lengthening the force of attached cross-bridges is increased with respect to the isometric conditions: at a lengthening velocity of 41 sam/s per halfsarcomere the force per cross-bridge is 3-4 % higher as estimated from either the ratio of force to stiffness (Lombardi & Piazzesi, 199) or the shift of Y1 (this work). Due to the higher extension of cross-bridge spring, the rate constants for the forward reactions in the force-generating process are expected to be reduced and the equilibrium between states of the attached cross-bridge is expected to be changed, with a redistribution towards lower force-generating states. Following a small perturbation in length the amplitude and the rate of tension recovery is reduced because the rate of transition to the higher force-generating state remains low. According to Huxley and Simmons' theory, the intercept of the T2 curve on the length axis represents the maximum amount of step shortening for which crossbridges can recover force while attached. During steady lengthening, the intercept of T2 curve is shifted to the left by about 3-1 nm. Therefore the maximum amount of step release which attached cross-bridge can recover is increased accordingly. This indicates that, before the step, the position of the cross-bridge, with respect to the actin site to which it is attached, is shifted in the lengthening direction by 3-1 nm. At the same time, the average extension of the cross-bridge spring during lengthening, as measured by Yo, is increased by only 1-3 nm. The difference between the shift in the position of attached cross-bridges (31 nm) and the actual increase in extension of the cross-bridge spring (1-3 nm) can be accounted for by assuming that attached cross-bridges are redistributed towards lower force-generating states during steady lengthening. It must be noted here that the average extension of the cross-bridge spring depends not only on the state of the attached cross-bridge but also on the position with respect to the actin site to which it is attached. A broader distribution of positions towards the lengthening direction can account for part of the above mentioned difference. Note that both phenomena (redistribution of attached crossbridges towards a lower force-generating state and broadening of spatial distribution) and predicted by the model of contraction presented in the previous paper (Lombardi & Piazzesi, 199, Fig. 14). A large-step release applied during steady lengthening reduces the cross-bridge extension enough to allow cross-bridges to go through the force-generating process which, in this case, can afford a larger amount of sliding. The quantity (T2- T1)/(Ti-Ti) emphasizes the amount of force recovered in relation to the step size (Fig. 9). For step releases larger than 6-7 nm per half-sarcomere, 'lengthening' crossbridges can recover more force, i.e. deliver more energy, than the isometric ones. For step releases smaller than 6 nm per half-sarcomere, the tension recovered at the end of phase 2 is smaller during lengthening than in isometric conditions and the speed of recovery is lower. There could be some uncertainty in the difference in the amplitude of phase 2 recovery between isometric and isovelocity conditions, due to the approximation intrinsic to the tangent method (Fig. 6). An overestimate of the 691

34 692 G. PIAZZESI AND OTHERS amplitude of depression of tension recovered would produce an increase in the estimated speed of recovery. Since the reduction in T2 occurs together with a reduction in speed of recovery, the depressant effect of lengthening on phase 2 cannot be due to this effect. However, a small overestimate of the rate of recovery could be present and could explain the finding that, when the relation between rate of recovery and step size determined during lengthening is shifted to the right by an amount corresponding to the shift of T2 curve intercept, which represents an estimation of cross-bridge extension before the step, the points do not superpose exactly on those determined in isometric conditions, but tend to lie slightly above it (Fig. 12). Tension recovery following step stretches In tension transients following step stretches applied during slow lengthening, we could not reliably separate phase 2 from the whole recovery. In isometric conditions there is a transient rise in force (phase 3) separating early and late recovery, but during slow lengthening tension recovers monotonically to the same value, regardless of the step size (Fig. 3). There is no doubt, however, that the time course of the early part of tension recovery is slower during lengthening (Fig. 5), in accordance with the expected effects of lengthening on phase 2. During steady lengthening, the final value of tension recovered following step stretches is equal to the value before the step, while in isometric conditions tension remains potentiated for several hundred milliseconds and the degree of potentiation is proportional to step size (Fig. 3). Tension potentiation following ramp lengthening, and proportional to the size of the stretch, has been described by Edman et al. (1978). It seems likely that, regardless of stretch velocity, a process of cross-bridge recruitment occurs as a consequence of lengthening, since stiffness has been found to be increased both after steps (Colomo, Lombardi & Piazzesi, 1987) and ramps (Colomo, Lombardi, Menchetti & Piazzesi, 1989) imposed at the plateau of an isometric tetanus. No further potentiation is expected following a step stretch imposed beyond 3 nm of ramp lengthening (the case in our Fig. 3B), since, according to Edman et al. (1978), the amount of lengthening necessary to have the maximum effect is less than 25 nm. Tension recovery following a step during fast lengthening We considered the whole transient elicited during fast lengthening to be determined mainly by processes occurring during phase 4, i.e. cross-bridge detachment and reattachment. This assumption is strongly supported by the finding that, even if the time for the whole recovery becomes comparable to the time for the phase 2 recovery in isometric conditions, the transient exhibits very different characteristics. Its rate, calculated assuming the recovery as a single process, increases going from step releases to step stretches and, for a given step size, is directly related to the velocity of steady lengthening. If the recovery were dominated by processes occurring during phase 4, we would expect that the steady state, after a step, is reattained when all cross-bridges displaced by the step have detached and reattached to a new actin site further along the thin filament. The linear relation between the amount of post-step lengthening necessary to attain 95 % of tension recovery and step size, and the independence of this relation of lengthening speed (Fig. 19), indicate that the condition that determines re-establishment of the steady

35 TENSION TRANSIENTS IN MUtSCLE DL.tRINVG STRETCH state is that the total lengthening, including the step itself, approaches a critical value (just below 4 nm), at which cross-bridges detach quickly. Reattachment of cross-bridges must also be very fast. In fact, if reattachment was a rate-limiting step. the relation in Fig. 19 would have been affected by the velocity of lengthening. A very fast reattachment of cross-bridges detached under strain was postulated previously to explain the finding that stiffness remains high during lengthening at high velocity (Lombardi & Piazzesi, 199). The value of 3 84 nm reported here for the average displacement attained by attached cross-bridges during fast lengthening should indicate that a tenfold increase in lengthening velocity (from about 8,am/s, Table 1, to 1,am/s per halfsarcomere, average value in Fig. 19) produces on cross-bridges further extension of -7-8 nm. Actually, if 3-1 nm is the extension of cross-bridges at -8,tm/s per halfsarcomere, as estimated by the shift in the intercept of the T2 curve (Table 1), the increase in extension with increase in velocity is ( = ) -74 nm. This value is probably slightly too low because the increase in extension of cross-bridge spring estimated by the increase in Y1 for the same range of velocities is much closer to t nm. The presence of the overshoot (for step releases) or the undershoot (for step stretches) produces per se an underestimate of the amount of restretch necessary for tension to recover the steady-state value. A relation similar to that in Fig. 19 has already been described for a fibre in fixedend mode (Colomo et al. 1989b). In that case the intercept of the regression line on the ordinate was significantly lower (3 4 nm) than that reported here. The reason for this discrepancy was an assumption that the value of tension to be recovered is that attained just before the step. This ignored the fact that recovery is complicated by the presence of an undershoot, for step stretches, or an overshoot, for step releases. The slope of the relation of Fig. 19 is significantly smaller than unity. We should expect a slope of unity if the step produced only a shift in position of attached crossbridges, taking them farther from (step release) or bringing them closer to (step stretch) the critical degree of extension at which they detach. A slope less than one should indicate that the amount of post-step lengthening necessary to attain the critical extension is only partly dependent on the size of the step. Several factors can contribute to the reduction of the slope: (1) the presence of the overshoot or the undershoot in tension can influence not only the intercept of the relation (see above) but also its slope. (2) It is possible that a fraction of cross-bridges detach and reattach at any time, independent of the step size (Schoenberg, Brenner, Chalovic, Greene & Eisenberg, 1984); this fraction will contribute to a reduction of the slope of the relation of 195 against step size. The simulation reported below shows that this effect can be explained solely by the overshoot/undershoot. The overshoot of tension during recovery following a step release superposed on fast lengthening can be explained by assuming that some degree of quick recovery, due to Huxley & Simmons transitions, is still present. In this case, there is a rapid redistribution of cross-bridges towards higher force-generating states after the step. If the subsequent recovery towards the distribution characteristic of the steady state during lengthening takes longer than the time necessary to attain the critical extension for rapid detachment, the tension will pass through a maximum before declining to the steady-state value. The undershoot in tension following a step stretch applied during fast lengthening could be due to a transitory reduction in the 693

36 694 G. PIAZZESI AND OTHERS number of attached cross-bridges caused by the sudden increase in detachment rate of cross-bridges extended by the step. A quantitative assessment of these possibilities is described in a later section. Tension transient at different temperatures Isometric Qlo values of 2 and 2-6 for the rate of phase 2 recovery following a step release and stretch respectively are in agreement with the values given previously (Ford et al. 1977). The rate of quick recovery following a step is also very sensitive to temperature during slow lengthening. This gives further support to the procedure used to separate phase 2 from the rest of the recovery. At a given low lengthening velocity, quick tension recovery seems less depressed at a high temperature than at a low one. For a more complete analysis of this point, it might be necessary to take into account that the quick recovery is not a single exponential process (Ford et al. 1977). It seems likely that the various exponential components have a different temperature dependence as well as a different strain dependence. Using the simple analysis of the rate of recovery, the present results are qualitatively consistent with the expectations of Huxley and Simmons' theory. The rise of temperature is expected to reduce the lengthening-dependent depression of quick recovery through a reduction of steadystate extension of cross-bridges for the same lengthening speed, due to the increase in the values of the rate constants of all processes (including detachment and reattachment). An increase in lengthening velocity in proportion to the increase in rate constants would counteract the effects of a rise in temperature and produce an increase in the average extension of cross-bridges. Conversely, lowering the lengthening velocity at the same temperature produces the same effects as increasing the temperature. In fact, the relationship of r with lengthening velocity at different temperatures gives a continuous line when the relationship at a lower temperature is shifted to the right by an appropriate amount along the velocity axis. Change in temperature has almost no effect on speed of recovery following a step imposed during fast lengthening. This finding has important consequences in terms of the kinetics of cross-bridge cycling. Tension recovery in this case depends mainly on detachment and reattachment of cross-bridges. Since detachment and reattachment rates are also expected to rise with temperature, it must be assumed that these rates are in any case very high during fast lengthening, in order not to influence the rate of recovery when temperature is changed. It is well known that the mechanical responses of an active fibre, which depend on the rates of cross-bridge attachment and detachment, such as the rate of tension rise in an isometric tetanus and the tension developed during steady shortening at constant velocity, are strongly dependent on temperature (for an extensive review see Woledge, Curtin & Homsher, 1985). The finding that the time course of tension recovery following a step imposed during fast lengthening is insensitive to temperature gives strong support to the view that in this case attachment and detachment of cross-bridges occur by a different mechanism from attachment and detachment in the isometric condition or during shortening. A different detachment process during lengthening, occurring before completion of the chemical cycle, and therefore without ATP splitting, was first postulated by Huxley (1957, 198) to account for the low energy consumption of active muscle under stretch. Because of the strain-dependent redistribution to a

37 TENk'SION TRANSIENTS IN MUSCLE DLTRING STRETCH low force-generating state, postulated here on the basis of analysis of the tension transients elicited during slow lengthening, detachment must occur from an early (low-force) state of attached cross-bridges. In terms of current theories of the biochemical cycle of the cross-bridges (Huxley, 198, p. 95; Eisenberg, Hill & Chen. 198) this state has the characteristics of a strong-bound state such as those occurring after inorganic phosphate (Pi) dissociation. Model simulation This section describes the results of a computer simulation of the tension transients elicited during steady lengthening at different velocities. The model of cross-bridge cycling and the details of calculations have been discussed previously (Lombardi & Piazzesi, 199). We describe here only those aspects that concern the results of the present work. The model incorporates the Huxley & Simmons theory (1971) and also includes the possibility of detachment of cross-bridges from an early state in the force-generating process. This detachment becomes relevant for critical degrees of cross-bridge compression or extension. Reattachment of cross-bridges in this case is much faster than attachment after the completion of the normal cycle. The scheme of cross-bridge reactions assumed in the model is as follows: DI-- k, --fal_ ka2 A2_ k3 - k4 Dl k-, k 2 /k_ k5 k D2 The scheme includes two detached (DI and D2) and three attached (Al, A2 and A3) states. The force rises progressively with transitions Al -+ A2 -+ A3 due to the increase in the extension of cross-bridge elastic component for each of these transitions. Explanations of the functions expressing the dependence on x (the relative position between the myosin head and the actin site. taken as zero when, with the cross-bridge in Al state, the spring has zero extension) of the rate constants (k's) controlling the transitions are given in the Appendix (see also Lombardi & Piazzesi, 199, p. 162). Some modifications have been introduced in the present version of the model in order to optimize the fit to tension transients following quick releases. The modifications did not affect the steady-state relationships of force and stiffness with lengthening velocity of the steady-state x-distribution of cross-bridge attached states at a given velocity (Fig. 24). Force and stiffness are calculated according to the method given in Lombardi & Piazzesi (199) and are plotted relative to the values at the isometric tetanus plateau. The new assumptions are listed in the Appendix, but their consequences for the simulated responses are described here together with the model properties. In isometric conditions, attachment is controlled by k1, which is relatively low. Equilibrium in reaction 5 is fully in favour of the attached state, so that detachment of cross-bridges occurs only through step 4. The assumptions made for the x dependence of the rate constants controlling the transitions Al -+ A2 -* A3 provide

38 696 G. PIAZZESI AND OTHERS that isometric cross-bridges are mainly distributed between Al and A2 and that forward reaction 3 is the rate-limiting step in cross-bridge cycling. Steady lengthening further reduces the probability of cross-bridges going through the complete cycle and causes a redistribution of cross-bridges towards the lowest force-generating state Relative tension () B Relative stiffness (A) E -g a.2 E. * AD C 1.6 D~~~~~~~~~~~~~~C - z E c n 1t \ n ~~~~~~.1 c D E C.2.2- ~ ) CD~~~~~~~..~~~~~~~~~ A2 E U. x (nm) - % o ~~~~~~~~ 1 UL C Fig. 24. A, simulated relations of steady-state force (O) and stiffness (/\ versus lengthening velocity. Velocity is plotted in the ordinate in conformity with the previous paper (Lombardi &; Piazzesi, 199, Fig. 13). Force and stiffness are made relative to the isometric plateau values. The other panels show the distribution of attached cross-bridges in relation to x, calculated either at B, the isometric tetanus plateau or C, during steadyforce response to lengthening at -78 and D, -78,um/s per half-sarcomere. The ordinate gives the number of attached cross-bridges per nm as fraction of the total number. (Al). Detachment from this state (through step 5) becomes very fast beyond a critical amount of extension, and is followed by rapid reattachment (Lombardi & Piazzesi, I199). Shortening produces a redistribution of cross-bridges toward A3 and a consequent increase in rate of detachment and reattachment through steps 4 and 1. If ATP

39 TENSION TRANSIENTS IN MUSCLE DURING STRETCH splitting is associated with detachment after completion of the cycle there will be a consequent increase in ATP utilization during shortening, in accordance with experimental results (Curtin, Gilbert, Kretzschmar & Wilkie, 1974; Ferenezi. Homsher & Trentham, 1984). To make the normal cross-bridge cycle efficient during shortening, detachment has to occur mainly when A3 cross-bridges exert negative force (see also Huxley, 1957). This condition has been met more completely in the present version of the model, by keeping the value of k4 low for x > -9 nm. This modification also prevents a significant detachment of A3 cross-bridges after a step release, before phase 2 is completed. In order to optimize the fit to tension transients following step releases, the possibility of detachment from an early state of the attached cross-bridges (Al and A2), followed by fast reattachment, has to be assumed for negative values of x as well as for positive values. This is provided by the functions giving the values of k-, and k6 for x <. Indeed, if we assume that the only detachment and reattachment processes during shortening are through steps 4 and 1, the simulated transient following relatively large step releases imposed in isometric conditions has a large and long-standing depression during phase 3 which is not observed experimentally. Detachment from an early state of the attached cross-bridges, becoming significant during fast shortening, has been postulated by Huxley (1973) to explain the finding that energy consumption (Hill, 1964) and ATP splitting (Kushmerick & Davies, 1969) do not increase with shortening velocity at high speed. Our hypothesis of detachment occurring at an early stage of the cycle differs from that of Huxley; we assume the presence of a detachment process toward a state that implies rapid reattachment (steps 5 and 6), whereas in Huxley's hypothesis detachment is simply the reversal of the attachment reaction (step 1 in our scheme) and this implies a reattachment rate that is relatively low. Note, in this respect, that detachment from an early attached state in our scheme populates the state D2 which is the same as that populated by the detachment process occurring during lengthening, beyond a critical amount of cross-bridge strain. This identity is postulated to fit the kinetics of the mechanical results, but its equivalent in biochemical terms has yet to be found. During steady lengthening at low speed (-78,um/s per half-sarcomere, Fig. 24C) the model predicts a large redistribution of attached cross-bridges to the Al state with broadening toward positive values of x, in good quantitative agreement with the conclusions drawn from the analysis of transients (see pp ). The value of x for the median of attached cross-bridges (all states) is shifted to the right by about 3 nm, compared with isometric conditions. Increase in lengthening speed up to -78,tm/s per half-sarcomere (Fig. 24D) produces further broadening of the distribution and a further shift of the median by about 1 nm. Figure 25 compares model predictions for tension transients elicited by steps of different size delivered at the plateau of the isometric tetanus with those elicited during steady-force response to lengthening at low speed (-78,m/s per halfsarcomere). The main features of the experimental transients in isometric conditions as well as the modifications associated with steady lengthening are clearly reproduced by the model. The speed of quick recovery increases from the step stretch (upper trace) to the largest step release (lowest trace). Compared to the isometric case a given step of moderate size imposed during slow lengthening produces a transient 697

40 698 G. PIAZZESI AND OTHERS which differs in three ways: the speed and the amount of quick recovery are reduced and phase 3 disappears (see, for instance, -5). For large step releases (-9), quick recovery is larger during lengthening and phase 3 again becomes evident. The steady-state tension attained during lengthening is lower in the simulation (see below and Lombardi & Piazzesi, 199, p. 162) than in our experimental results. A 2- r TITo n43-.. loo-d a a I I a A I a I Time (ms) 16 2 B 2. T/To Time (ms) Fig. 25. Superposed simulated tension responses to step length changes of different size imposed either at A, the isometric tetanus plateau or B, during lengthening at 78,um/s per half-sarcomere. Tension (T) is relative to the isometric plateau value (TO). The size of the step (nm per half-sarcomere) is indicated by the different symbols as listed in the upper panel. The steps were completed in zero time; for reasons of clarity the corresponding tension changes take 1 #us in the drawings. Negative values of T/To are omitted. Beyond this, the simulated responses fit the experimental ones with two exceptions (Fig. 26): (1) the very early component of recovery from a step stretch is too slow; (2) T2 recovery for moderate step releases (5-9 nm in isometric conditions and 5-12 nm during slow lengthening) is larger than in the experiment. The first discrepancy is reminiscent of the failure by Ford et al. (1977) to fit the experimental

41 A TENSION TRANSIENTS IN MUSCLE DURING STRETCH TITO uwm.mwzz_mumuzmw T/7o = _~ = _ T/TO TITo Time (ms) 4 8 B 1.6.8~ TITO TITo TITo Time (ms) Fig. 26. Comparison of the tension transients obtained experimentally (lines refer to some of the records of Fig. 1) with those simulated by the model (symbols) either A, in isometric conditions or B, during slow lengthening (78 um/s per half-sarcomere, model and -81,um/s per half-sarcomere, experiment). Sizes of the steps (nm per half-sarcomere) indicated by the figure close to each record. In B all the tension values of the simulated transients are multiplied by a factor of 114, the ratio of experimental over simulated T,/TO. responses to step stretches with a series of four exponentials (see Figs 23 and 28 of their paper). The second discrepancy can be explained by a rapid detachment of a fraction of cross-bridges soon after a step release. This reduces the force recovered for moderate releases, but not for the very large ones. Rapid detachment by the time

42 7 G. PIAZZESI AND OTHERS phase 2 is complete has already been hypothesized (Ford, Huxley & Simmons, 1974; Huxley, Simmons, Faruqi, Kress, Bordas & Koch, 1983). Rapid detachment after a step cannot explain the first discrepancy because stiffness measured shortly after a step is reduced by releases (Ford et al. 1974; Cecchi, Griffiths & Taylor, 1986), but A B 2. Co 4) C D T2 -Tl Tj -T, r (ms-1) Step amplitude per half-sarcomere (nm) Fig. 27. A, T1 (open symbols) and T2 (filled symbols) curves determined on the simulated responses obtained either in isometric conditions (circles) or during steady lengthening at 78 um/s per half-sarcomere (T1 = 1-32 To, triangles). All tension values are made relative to To. Lines are drawn by joining the points; the intercepts on the length axis of T2 curves are obtained by extrapolating the straight lines drawn through the points belonging to the largest releases. B, the points show experimental T, (open symbols) and T2 (filled symbols) relations (same experiment as Fig. 7B). Lines for isometric curves are the simulated relations as in panel A; lines for isovelocity curves (78,m/s) are obtained by multiplying the corresponding points in panel A by 1-14, the ratio of the experimental Tl/To to the simulated Tl/To. The lower panels show the relations of both C, the fraction of tension recovered at the end of phase 2 ((T2-Tl7)/(T1 - T1)) and D, the rate of quick recovery (r) versus step amplitude, determined from the simulated responses in isometric conditions (circles) and during slow lengthening (-78,um/s; triangles). increased by stretches (Colomo et al. 1987). Note that during slow lengthening an early component of the tension transient, faster than that expected on the basis of the Huxley and Simmons model, is present also for small releases. In Fig. 26B the discrepancy between the superposed traces is similar in and -2-3 traces. Apparently, in phase 2 recovery from a length step applied to the active sarcomere

43 TENSION TRANSIENTS IN MUSCLE DURING STRETCH under stretch, there is a component that has the characteristics of a linear viscoelasticity (Lombardi & Piazzesi, 1989). The characteristics of phase 2 of the simulated tension transients are quantified in Fig. 27. The procedure for measuring T2 and the rate of quick recovery of the responses was the same as that used for the experimental records (p. 665 and Fig. 6A). Figure 27A shows the T1 (open symbols) and T2 (filled symbols) curves determined either in isometric conditions (circles) or during steady lengthening at -78,um/s per half-sarcomere (triangles). The intercepts on the length axis of T1 curves were 3-6 and 4-6 nm per half-sarcomere in isometric and isovelocity conditions respectively. The intercepts of T2 curves were respectively 1-9 and 14-2 nm per half-sarcomere. The slopes of T, curves are similar in both cases since, as already reported (Lombardi & Piazzesi, 199, p. 162), the model is not able to reproduce the increase in stiffness found experimentally. In fact, no allowance has been made for an increase in the number of sites available for cross-bridge attachment that is likely to occur during lengthening, compared to isometric conditions, as a consequence of the difference in periodicity between sites along the myosin and actin filaments (Huxley & Brown, 1967). Consequently, in the simulated responses, the increase in steady tension during lengthening (Tj = 1-32 To at -78 #tm/s per half-sarcomere) was almost completely due to increased extension of attached cross-bridges. The shifts of T1 and T2 curves intercepts on the length axis found experimentally (Tables 1 and 2) are accurately predicted. In Fig. 27B, as well as in Fig. 26B, all the simulated values of tension during lengthening are multiplied by a factor of 1-14 (the ratio between TlTo in the experiment and T1lTo in the simulation). After this manoeuvre, apart from the above mentioned discrepancy for T2 level with moderate releases, simulated T1 and T2 curves show a reasonably good agreement with the experimental ones. Good agreement with experimental relations is also obtained from the simulated transient by plotting either the fraction of tension recovered at the end of phase 2 (Fig. 27 C) or the rate of recovery (Fig. 27D) versus step amplitude. The changes in x-distribution of attached states of cross-bridges accompanying the tension transient following a step release of 5 nm (triangles in Fig. 25) either in isometric conditions or during slow lengthening are shown in Fig. 28. In isometric conditions (Fig. 28, left column), at 4 ms (1-9 times the half-time of quick recovery) cross-bridges are redistributed towards A2 and A3 states, with very little change in number of attachments (S = -98 So, where So is the number of attached crossbridges at the plateau of isometric tetanus and S is the number of attached crossbridges at a given time after the step). At 6 ms (middle of phase 3), there is some detachment from A3 state, but also substantial detachment from A2 state followed by rapid reattachment and force generation. This process is accompanied by very little change in force and by a reduction of the number of attached cross-bridges (S = -84 SO). At 2 ms (phase 4) cross-bridges are again approaching the steady-state distribution, but the number of attachments is still low (S = -85 SO) due to the relatively slow attachment of cross-bridges detached from A3. During slow lengthening (Fig. 28, right column), the Al state is the most populated one, so that quick recovery following a step release is due in large part to 71

44 72 E c. cm C'a - C._ C LL u-uu Isometric t= ms S/So = I.' I :..! G. PIAZZESI AND OTHERS t=.4 ms S/SO = I _ t= 6 ms S/SO = t= 2 ms S/SO = x (nm) E c en', c).q C) 11 Cu LL im/s t= ms S/SO = : İ t= 6 ms S/SO = t= 2 ms S/SO = x (nm) Fig. 28. The x-distribution of attached cross-bridges at various times (t) after a step release of 5 nm imposed either in isometric conditions (left column) or during steady lengthening at -78,um/s per half-sarcomere (right column). Continuous line, Al state; dotted line, A2 state; dashed line, A3 state. The distribution at t = ms is obtained by shifting to the left by 5 nm the steady-state distribution. The ordinate gives the number of attached cross-bridges per nm as fraction of the total number. The areas under each curve, representing the fraction of attached cross-bridges in each state, are summed to give S/SO, the stiffness at any time relative to that at the isometric tetanus plateau.

45 TENSION TRANSIENTS IN MUSCLE DURING STRETCH 73 A 2.4 B T/1o T/To Time (ms) Time (ms) Fig. 29. Superposed simulated tension responses to step length changes of different size applied during steady lengthening at A, -78 and B, 1 25,m/s per half-sarcomere. Tension (T) is relative to the isometric plateau value (To). The size of the step is indicated (in nm) by: V, + 2; El, + 1;, -2; A, -5. The steps were completed in zero time; for reasons of clarity the corresponding tension changes take 1 lus in the drawings /95 (nm) 5 `4> ) Step amplitude per half-sarcomere (nm) Fig. 3. Relation of the amount of post-step lengthening necessary to attain 95 % of the maximum of tension recovered versus step size, determined from simulated responses obtained during steady lengthening at two different velocities: -78 (A) and 1-24 (),um/s per half-sarcomere. The line is drawn according to the regression equation: 195 = -4 As , where As is the size of the step length change. Note that the intercept on the ordinate is very close to the value found experimentally, while the slope of the relation is significantly lower. the transition Al -+ A2 and, in accordance with the reduction of phase 3, the subsequent phase is characterized by only minor detachment from A3 state (see distribution at 6 ms). At 2 ms, cross-bridges have been re-extended by only 156 nm, so that the distribution is still in favour of A2 state. All the phases are characterized

46 74 G. PIAZZESI AND OTHERS by very little decrease in number of attachments, due to the reduced detachment from A3. Figure 29 shows the simulated tension recovery following steps of different sizes superposed on steady lengthening at -78 (A) and 1-25,tm/s (B). At these velocities the whole recovery is complete within 1 ms and is briefer going from the largest step release to the largest step stretch. For a given size of step, the speed of recovery is higher at the higher velocity. An undershoot and an overshoot in tension are present in the transients following a step stretch and release respectively. They appear to be sharper and briefer than in the experimental records and are followed by a few oscillatory cycles which become progressively more marked with increase in lengthening speed. The smoother appearance of the experimental records may be related to inhomogeneity in the lengthening between different sarcomeres in series along the fibre (Lombardi & Piazzesi, 199). In Fig. 3 the amount of post-step lengthening (195), necessary to attain 95 % of the maximum tension recovered, is plotted against step amplitude for the two velocities used in Fig. 29. As found experimentally, the relation is roughly linear and not significantly affected by the change in velocity. The x-distribution of attached states of cross-bridges which underlie tension recovery following either a step stretch (+ 2 nm, inverted triangles in Fig. 29) or a step release (-5 nm, triangles in Fig. 29) superposed on lengthening at high speed (1P25 jtm/s per half-sarcomere) are shown in Fig. 31. At this velocity attached crossbridges exist mainly in the Al state and their distribution broadens towards positive values of x. The distribution of cross-bridges at any time during the transient following a step is characterized by very little change in total number of attachments. At 4 ms after the step stretch, tension is half recovered, while detachment of Al cross-bridges has occurred only at large positive values of x, followed by rapid reattachment at x around 1 nm. At 4-4 ms, when the undershoot is at its maximum, reattached cross-bridges have not yet reattained the steady-state distribution; there is a deficit of the Al state at a value of x around 8 nm (these cross-bridges were at [8- (125 x 4 4) + 2] = 5 nm just before the step stretch and the ones not attached had no chance to attach due to the sudden shift to the right by 2 nm), and an accumulation of Al cross-bridges at values of x around 5 nm, as a consequence of the abrupt increase in rate of detachment just after the step. Tension recovery following a step release is characterized by an early transition of the least-extended fraction of Al cross-bridges towards A2 (see the x-distribution at 4 ms). The overshoot in force with respect to the steady-state value arises mainly as a consequence of the residual increased number of cross-bridges in the A2 state (see the x-distribution at 7-2 ms) by the time steady lengthening has progressed toward the point where the x- distribution regains the steady state. Note that the presence of both undershoot (after a stretch) and overshoot (after a release) causes an underestimate of the lengthening necessary to reattain the true steady-state distribution. Therefore the slope of the relation 195 against step size is less than unity. The slope is expected to be lower in the model than in the experiment since the overshoot and the undershoot are sharper and briefer.

47 TENSION TRANSIENTS IN MUSCLE DURING STRETCH nm t= ms S/SO = nm t= ms S/SO = E a -2 1._,.15 Co na).1 a) t!.5 t=.4 ms S/So = 1-6 E.2 C a) ) l.15. Co r.5 t=.4 ms S/SO = 1.24 a L. u.t t= 4.4 ms S/so = 1.19 I.) t= 7.2 ms SS/SO = x (nm)., x (nm) Fig. 31. The x-distribution of attached cross-bridges at various times (t) after a step stretch of 2 nm (left column) and a step release of 5 nm (right column), imposed during steady lengthening at 1-25 /tm/s per half-sarcomere (T1 = 1-89 To). Continuous line, Al state; dotted line, A2 state; dashed line, A3 state. The distribution at t = ms is obtained by shifting the steady-state distribution along the x-axis by an amount corresponding to the step size. The ordinate gives the number of attached cross-bridges/nm as fraction of the total number. The bottom graphs refer, in left column, to the distribution at the peak of the undershoot and, in right column, to the distribution at the peak of the overshoot. 23 PHY 445

48 76 G. PIAZZESI AND OTHERS Thermodynamic implications The hypothesis that cross-bridges under stretch are detached by a different mechanism than in isometric conditions and without the hydrolysis of ATP was first formulated by Huxley (1957, 198) to explain the finding that, during stretch of active muscle, in spite of the increase in cross-bridge turnover, energy consumption is reduced (Hill, 1938; Abbott, Aubert & Hill, 1951). The modifications of the tension transient during steady lengthening give experimental evidence for that hypothesis, indicating that attached cross-bridges under stretch are redistributed toward an early stage of the force-generating process. Forcible detachment at this stage occurs beyond a critical degree of cross-bridge extension toward a detached state that exhibits a much faster reattachment rate compared to cross-bridges detaching at the end of power stroke. As emphasized in a previous paper (Lombardi & Piazzesi, 199, pp. 164 and 166) the progressive increase, with lengthening velocity, in force exerted by the cross-bridges in the Al state (the state with the highest chemical energy) provides a straightforward explanation for the endothermic process occurring during lengthening at high speed (Woledge et al. 1985). According to our model, detachment from Al and A2 states before the completion of the normal cycle followed by rapid reattachment has to occur also during shortening, in order that the simulated responses fit the part of tension transient following phase 2. This detachment process differs from the reverse reaction of the attachment process postulated by Huxley (1973) in order to explain the limited increase of both energy production (Hill, 1964) and ATP splitting rate (Kushmerick & Davies, 1969) with shortening at high velocity. In fact, whereas in Huxley's hypothesis detachment from an early stage is provided by the reversal of the attachment reaction, in our hypothesis detachment from A2 implies that crossbridges can go through part of the power-stroke (step 2), detach without completing the cycle and rapidly reattach in a state which can still produce work. A direct test of the presence of a relatively rapid detachment and reattachment of cross-bridges, following phase 2 of the tension transient elicited by a step release imposed in isometric conditions, did not come within the scope of this work; it requires a direct investigation of the processes occurring during phase 3 of the transient. The creditability of the hypothesis that cross-bridges can execute fast cycles of detachment, force generation and reattachment depends on how strictly the hydrolysis of one molecule of ATP is associated with a single cross-bridge cycle. As already mentioned previously (Lombardi & Piazzesi, 199, p. 168), the first aim of this type of simulation was to find a kinetic scheme that fitted the mechanical results. In order to predict the rate of quick recovery found experimentally, c (cross-bridge stiffness) and z (change in extension of cross-bridge spring associated with the transition between attached states) had to be given values (see Appendix) which imply very low free energy associated with each transition compared with the energy associated with ATP hydrolysis. APPENDIX The definitions of symbols used in the Appendix are: Al, A2 and A3: attached cross-bridge states.

49 TENSION TRANSIENTS IN MUSCLE DURING STRETCH 77 DI and D2: detached cross-bridge states. x: relative position of the actin site with respect to the myosin head to which it is attached; x = when, with the cross-bridge in the Al state, extension of cross-bridge elastic component is zero. 15 E 1, 15 1 cac x (nm) 2 3 7, 2 - cr ~ CZ) - k4 L-5, k6-1 x (nm) 1 Cfl CO) 4-8 co c' 4 cr a) 4_ - k2 -- k k3... /C x (nm) Fig. 32. Functions expressing the dependence of the rate constants on x, the longitudinal distance between the cross-bridges and the actin sites. k'5, not shown, has the same x dependence as k', but the figures on the scale must be multiplied by Rate constants are expressed in s-i or s-1 nm-1 (see text). z: change in length of cross-bridge elastic component during either transition Al-A2 or A2-A3. z is assumed to be 4-5 nm. c: stiffness of the cross-bridge elastic component, assumed to be '7 pn/nm. b: Boltzmann constant = 1P385 x 1-2 (pn nm)/k. : absolute temperature = 277'16 K. 23-2

50 78 G. PIAZZESI AND OTHERS The scheme of cross-bridge reactions is: D1k_ -A1 k2 A 2 -A k3 :LtA -i A3 k5 k-5 k_ / ka3 4D1 D2 The corresponding set of differential equations are the same as in Lombardi & Piazzesi (199). The new set of functions expressing the x dependence of the rate constants for the transitions between various states is listed below. The corresponding plots are shown in Fig. 32. Fractions of cross-bridges in Al, A2 and A3 states were calculated at x-points separated by an interval (A x) of -5 nm from each other. Values of k were considered constant within this interval. -1 k1(s-1 nm-1 = kl/ax = { exp (- (e/2) (x -1)2/b) SxS -I<-1 x > 3 k1(s-) = { exp ((- (e/2) (x- 1)2/b) + ((e/2) X2/b - 329)) k2(s-1) - { 7 exp (- (e/2) (2x z + z2)/b) 17 k_2(s exp (-(e/2) (-2xz+Z2)/b) 4 k3(s-1) - { 7 exp (- (e/2) (2 (x + z) z + z2)/bo) k_3(s-1) 7 f exp (- (e/2) (-2 (x+z) z+z2)/b) 4 24 k4(s-1) = -2(x + 8) -2(x- 1) 4 k(s-l nm-1) = k5/ax { 315 exp (-e/2) (x- 1)2/6) (x+ 1) k_5and k6(s-') = exp (-8 (x - 45) x sxs3 x <-l,x> 3 x < -z/2 x < -z/2 x S -z/2 x > -z/2 X < -32 2z X S<- 1Z X >-32Z x < -2-2 S x S < x S-1-1SIx 3 x < -1,x > 3 x <-2-2 < x S -1-1x <x <xs 15-5 x> 15-5

51 TENSION TRANSIENTS IN MUSCLE DURING STRETCH k' and k1 express the value for attachment rate constants independent of the value of interval Ax. Equations for kl, k-1, k2, k12, k3 and k_3 imply parabolic free-energy curves for the attached states. The main modifications with respect to Lombardi & Piazzesi (199) are the following: (1) k4, the rate constant of detachment at the end of the power-stroke has a relatively low value in the whole range of x > -9 nm, and becomes very high for x <-9 nm, where A3 cross-bridges exert negative force. This provides better efficiency for the action of the cross-bridges that complete the cycle and avoids the occurrence of a significant detachment from A3 by the time the quick recovery is complete. (2) z, the extension of cross-bridge spring for each transition Al-1 A2 -* A3, has been reduced from 5 to 4'5 nm. (3) e, the stiffness of the cross-bridge, has been increased from 5 to 7 pn/nm. If the average extension of the cross-bridge spring at the plateau of an isometric tetanus is 3-6 nm, it follows that the force per cross-bridge is 7 x 3-6 = 2-52 pn, larger than the value assumed in the previous simulation, but well within the limits of current estimates. (4) xo, the centre of the Gaussian curve used to express the values of the attachment rates k1 and k5, was shifted from 1-5 to 1- nm. Modifications (2), (3) and (4) alter the x dependence of rate constant for the transitions between different forcegenerating states of attached cross-bridges and give better fits to both T2 curves and relations between rate of quick recovery and step size. We are grateful to Professor Sir Andrew F. Huxley for criticism and suggestions during the course of this work and for reading the manuscript. We also wish to thank Dr Yale E. Goldman and Dr Hideo Higuchi for useful discussion and suggestions on thermodynamic consistency of the simulation, Mr A. Aiazzi and Mr M. Dolfi for skilled technical assistance and Mr A. Vannucchi for the preparation of illustrations. This work was supported by grant from the Italian M.U.R.S.T. REFERENCES ABBOTT, B. C., AUBERT, X. M. & HILL, A. V. (1951). The absorption of work by a muscle stretched during a single twitch or a short tetanus. Proceedings of the Royal Society B 139, AMBROGI-LORENZINI, C., COLOMO, F. & LOMBARDI, V. (1983). Development of force-velocity relation, stiffness and isometric tension in frog single muscle fibres. Journal of Muscle Research and Cell Motility 4, CECCHI, G., COLOMO, F. & LOMBARDI, V. (1976). A loudspeaker servo system for determination of mechanical characteristics of isolated muscle fibres. Bollettino della Societa Italiana di Biologia Sperimentale 52, CECCHI, G., COLOMO, F., LOMBARDI, V. & PIAZZESI, G. (1987). Stiffness of frog muscle fibres during rise of tension and relaxation in fixed-end or length-clamped tetani. Pflugers Archiv 49, CECCHI, G., GRIFFITHS, P. J. & TAYLOR, S. (1986). Stiffness and force in activated frog skeletal muscle fibers. Biophysical Journal 49, COLOMO, F., LOMBARDI, V., MENCHETTI, G. & PIAZZESI, G. (1989). The recovery of isometric tension after steady lengthening in tetanized fibres isolated from frog muscle. Journal of Physiology 415, 13P. COLOMO, F., LOMBARDI, V. & PIAZZESI, G. (1986). Recovery of tension after step-length changes applied to tetanized isolated frog muscle fibres during constant velocity stretches. Journal of Physiology 381, 87P. 79

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