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Journal of Physiology (2002), 545.1, pp. 145-151
© Copyright 2002 The Physiological Society
DOI: 10.1113/jphysiol.2002.028969

The size and the speed of the working stroke of muscle myosin and its dependence on the force

Gabriella Piazzesi, Leonardo Lucii and Vincenzo Lombardi

	ABSTRACT

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Myosin II is the motor protein that produces force and shortening in muscle by ATP-driven cyclic interactions of its globular portion, the head, with the actin filament. During each interaction the myosin head undergoes a conformational change, the working stroke, which, depending on the mechanical conditions, can generate a force of several piconewtons or an axial displacement of the actin filament toward the centre of the sarcomere of several nanometres. However, the sizes of the elementary force and length steps and their dependence on the mechanical conditions are still under question. Due to the small fraction of the ATPase cycle time myosin II spends attached to actin, single molecule mechanics failed to produce definitive measurements of the individual events. In intact frog muscle fibres, however, myosin II's working stroke can be synchronised in the few milliseconds following a step reduction in either force or length superimposed on the isometric contraction. Here we show that with 150 µs force steps it is possible to separate the elastic response from the subsequent early rapid component of filament sliding due to the working stroke in the attached myosin heads. In this way we determine how the size and the speed of the working stroke depend on the clamped force. The relation between mechanical energy and force provides a molecular basis for muscle efficiency and an estimate of the isometric force exerted by a myosin head.

(Received 20 July 2002; accepted after revision 10 September 2002; first published online 20 September 2002)
Corresponding author V. Lombardi: Dipartimento di Scienze Fisiologiche, Università di Firenze, Viale G. B. Morgagni, 63, I-50134 Firenze, Italy. Email: vincenzo.lombardi{at}unifi.it

	INTRODUCTION

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The interaction between force and motion in muscle is characterised by the well known steady-state relation between force and velocity of filament sliding (the T-V relation; Hill, 1938). Most of mechanical and energetic features of the muscle contracting under different loads can be quantitatively related to the kinetics of a simple two-state model of the myosin cross-bridges acting in parallel in the region of overlap between the myosin and the actin filaments in each half-sarcomere (hs) (Huxley, 1957). The myosin head domain protruding from the thick filament can be either attached to actin or detached and the rate functions for making and breaking the myosin cross-bridges depend on the relative position of the myosin head with respect to the attachment site on actin. However, as first demonstrated by Podolsky (1960), following a step reduction in force from the isometric force developed in a tetanus (T₀), the steady shortening velocity characteristic of the T-V relation is approached through a velocity transient composed of an instantaneous shortening, due to elasticity of myosin cross-bridges and myofilaments, followed by shortening at a velocity much higher than the late steady velocity. According to the Huxley & Simmons (1971) theory for force generation, which implies a multi-step structural change in the attached myosin heads, the early rapid shortening should carry information on the mechanics and kinetics of the working stroke. The difficulty of separating the instantaneous elastic response from the early rapid shortening in the velocity transient (Huxley, 1971) led A. F. Huxley and co-workers to exploit the converse type of experiment, the study of the force transient following step displacement of the filaments (Huxley & Simmons, 1971; Huxley, 1974; Ford et al. 1977). The drop in force simultaneous with the length step is followed by a quick force recovery, complete within 1-2 ms, due to the rapid re-equilibration between different force generating states of the myosin heads that remain attached on the millisecond time scale (Ford et al. 1974; Lombardi et al. 1992). One major advantage of the length step over the force step for studying the kinetics of the working stroke was thought to be the possibility of determining the strain dependence of the rate constants of state transitions in the attached heads directly from the rate of quick force recovery following different step sizes (Huxley & Simmons, 1971). However, it subsequently became clear that ~50 % of the half-sarcomere compliance resides in the myofilaments (Huxley et al. 1994; Wakabayashi et al. 1994; Dobbie et al. 1998; Linari et al. 1998), so allowance must be made for the interfilamentary sliding during the force recovery that slows down its time course.

The isotonic sliding velocity transient following a force step, in contrast, is free from the influence of myofilament compliance. Thus, provided that the mechanical apparatus can discriminate it from the elastic response, the early component of the velocity transient should reflect solely the motion of the attached heads, giving a direct estimate of the size and kinetics of the myosin working stroke and its dependence on the clamped force. With our mechanical apparatus we can deliver force steps complete within 150 µs and record the early rapid component of filament sliding with 1 µs time resolution, showing for the first time how the size and the mechanical energy of the working stroke depend on the force.

	METHODS

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Fibre preparation and mechanical apparatus

Frogs (Rana esculenta) were killed by decapitation and pithing, following the official regulations of the European Community Council (Directive 86/609/EEC), and conforming with the UK Animals (Scientific procedures) Act 1986. Single fibres dissected from the tibialis anterior muscle were mounted, by means of aluminium clips clamped to the tendons, between the lever arms of a capacitance force transducer with frequency resonance 50 kHz (Huxley & Lombardi, 1980) and a loudspeaker motor (see Lombardi & Piazzesi, 1990, and references therein) at sarcomere length ~2.1 µm. The physiological solution (115 mM NaCl, 2.5 mM KCl, 1.8 mM CaCl₂, 3 mM phosphate buffer at pH 7.1) bathing the fibre was kept at 4.0 ± 0.1 °C, by means of a servo-controlled thermoelectric module. A striation follower (Huxley et al. 1981) continuously recorded the sarcomere length changes in a 1-1.5 mm fibre segment close to the force transducer end (to minimise the delay between length and force responses due to propagation of the mechanical perturbation along the fibre) with a time resolution of 1 µs and a sensitivity of 100 mV nm^-1 hs^-1. The loudspeaker motor was servo-controlled using as feedback signal the output from either the position sensor on the loudspeaker lever (fixed-end mode) or the force transducer (force clamp mode).

Experimental protocol

Fibres were stimulated by means of platinum plate electrodes with pulse trains of current at the optimal frequency (10-20 Hz) to produce fused tetani of 1 s duration. When the isometric tetanic force had attained the plateau value (T₀), 300 ms after the first stimulus, the control was shifted from fixed-end mode to force clamp mode and 20 ms later a step in force, complete within 150 µs, was imposed by using as a command signal the output of an integrated circuit that generated steps to preset fractions of T₀. When the isotonic shortening attained 40 nm hs^-1, the control was shifted back to fixed-end mode to terminate the tetanus in isometric conditions at the new length. In force clamp mode only the direct signal of force was used in the feedback, while the velocity signal was taken from the motor lever position sensor. At any clamped force, several trials were necessary to adjust the gains of direct, velocity and lag amplifiers to optimise the 150 µs step. Force, motor position and sarcomere length changes were recorded at both 5 µs and 1 ms sampling intervals with a digital oscilloscope (Nicolet ProSystem 20) and analysed with a PC running dedicated software.

	RESULTS

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When the force of a contracting muscle fibre is changed in a stepwise manner from the isometric tetanic value (T₀, 155 ± 12 kN m^-2, mean ± S.E.M., 5 fibres) to a value T lower than T₀, the half-sarcomere shortens by a few nanometres during the force step (Fig. 1A and B). This is phase 1 of the velocity transient and reflects the undamped elastic properties of the attached myosin heads and the filaments. The following isotonic phase of the velocity transient consists in a rapid shortening (phase 2), which is followed by a period of shortening at reduced speed (phase 3) and eventually evolves into shortening at a steady velocity V (phase 4). As clearly seen in Fig. 1A and B, the lower the value of T the larger the shortening in phase 1, and the faster the shortening in phase 2; in fact phase 2 takes from more than 10 ms at 0.8T₀ to less than 1 ms at 0.1T₀. Phase 3 is also briefer at lower values of T. The transition to the steady velocity may be delayed by one small velocity oscillation, which is faster the lower the value of T. In accordance with the corresponding phases of the force transient following a length step (Huxley, 1974), phase 2 is related to the synchronous execution of the working stroke in the attached heads and phases 3 and 4 to detachment and attachment. In Fig. 1C the relevant parameters of the early components of the velocity transient are shown: L₁, the length attained at the end of the force step, measures the instantaneous compliance of the half-sarcomere; L₂, the length attained at the end of phase 2 of the velocity transient, is the sum of the elastic response and working stroke response in the attached heads. At any value of T, the shortening velocity decreases progressively during phase 2. The responses to step reductions in force to values larger than 0.8T₀ are not used in this analysis because they were recorded only occasionally. The transient elicited at 0.9T₀, apart the slower evolution, did not show the pronounced oscillatory behaviour found in previous work (Edman & Curtin, 2001).

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Figure 1. Isotonic velocity transients following a step in force superimposed on the isometric tetanic force Upper traces in A, step changes in force to values of T normalised to T0; lower traces in A, change in sarcomere length. B, same length responses as in A at a lower trace speed, showing all phases of the velocity transient. Unlike the shortening in response to force decreases, the lengthening following a force increase has a much slower time course. In both panels dotted lines show the 0.3T0 velocity transient simulated using the Piazzesi & Lombardi (1995) kinetic model. C, early components of the length response (lower trace) to a force step to 0.6T0 (upper trace) to show the method for estimating L1 and L2. With the increase in step size, L1 would be progressively overestimated, due to both the time taken for the step to be complete (~150 µs) and the increase in speed of phase 2. To account for this we measured L1 by extrapolating back to the half-time of the force step (t1/2, see inset) the tangent to the initial part of phase 2 response. L2 is measured by extrapolating the ordinate intercept of the tangent to the phase 3 response back to t1/2, to suppress the small contribution of phase 3. Fibre cross-sectional area, 6100 µm2; T0, 176 kN m-2; sarcomere length, 2.12 µm; temperature, 4.0 °C.

The relation between L₁ and T (Fig. 2A, ) is linear and the ordinate intercept, expressing the compliance of the undamped elasticity of filaments and attached myosin heads, is 3.73 ± 0.03 nm hs^-1, the same as that obtained from force transient experiments (Piazzesi & Lombardi, 1995 and Fig. 4). The relation between L₂ and T () is also linear in the range of forces for which it has been determined (< 0.8T₀), showing that the sliding distance accounted for by the contribution of both the undamped elasticity and the myosin working stroke increases with the decrease in T from ~5 nm hs^-1 at 0.8T₀ up to ~10 nm hs^-1 at 0.1T₀. The maximum sliding distance, attained with zero force, is 10.67 ± 0.07 nm hs^-1 (the intercept on the ordinate of a straight line fitted to the L₂ points). This value is also similar to the corresponding value estimated from the intercept of the T₂ curve in the force transient experiments (Piazzesi & Lombardi, 1995 and Fig. 4). The slope of the L₂ relation is larger than the slope of the L₁ relation so that, even after subtraction of the contribution of the undamped elasticity in the half-sarcomere, L_T (= L₂ - L₁), the amount of filament sliding accounted for by the working stroke elicited by the decrease in force, is larger for smaller values of T (Fig. 2B, ). L_T increases from 4.0 ± 0.5 nm hs^-1 at T = 0.8T₀ to 6.7 ± 0.9 nm hs^-1 at T = 0.1T₀.

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Figure 2. Extent of half-sarcomere shortening and mechanical energy during the working stroke Data are the means (± S.E.M.) from the five fibres used in these experiments. A, shortening simultaneous with the force step (L1, ) and total shortening at the end of the phase 2 of the velocity transient (L2, ) in relation to the force attained at the end of the step (T/T0). B, dependence on T/T0 of the isotonic filament sliding during the working stroke (LT, ) and of the mechanical energy ().

The overall mechanical energy of the working stroke in the myosin heads is:

E = E_iso + E_T, (1)

where E_iso is the energy stored by the elasticity during the isometric force generation, and E_T is the work delivered during phase 2 isotonic shortening. For myosin heads that attach at the isometric tetanus plateau, E_iso is a constant quantity that implies only the work to strain their own compliance (2.3 nm hs^-1 T₀^-1) (Dobbie et al. 1998; Linari et al. 1998), while E_T varies depending on both the clamped force and the amount of filament sliding. According to these definitions:

E_iso = 2.3 nm hs^-1 1/2T₀

and E_T = L_T T.

From the value of T₀ per cross-sectional area obtained in the five fibres used in this work it can be calculated that E increases from (0.28 ± 0.02) 10^-3 J m^-2 hs^-1 at 0.1T₀ to (0.68 ± 0.07) 10^-3 J m^-2 hs^-1 at 0.8T₀ (Fig. 2B, ).

The initial rate of rapid shortening following the force step (V₂), which is measured by the slope of the tangent to the length trace at the beginning of phase 2 (inset of Fig. 1C), is due to the synchronisation, promoted by the force drop, of the working stroke in the population of attached heads; the decay of this rate is the result of the progressive exhaustion of the population undergoing this process. As T reduces, V₂ increases (Fig. 3A, ) in a way that is similar to that observed for the steady-state shortening velocity (V, Fig. 3A, ). However, at each value of T, V₂, the sliding velocity due to the working stroke of the heads attached in the original isometric conditions, is eight times higher than V, the sliding velocity related to the rate of the actin-myosin ATPase cycle. For instance, at T = 0.1T₀, V₂ is 12 µm s^-1 hs^-1 and V is 1.5 µm s^-1 hs^-1. Also the shape of the phase 2 force-power relation (Fig. 3B, ) is similar to that at the steady state (Fig. 3B, ), both showing a maximum at about 0.4T₀. However, in phase 2 the maximum power is ~0.33 W m^-2 hs^-1, and in phase 4 it is ~0.04 W m^-2 hs^-1.

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Figure 3. Force dependence of the early and the steady-state shortening velocity and power output Data from the same fibres as in Fig. 2. A, relation between force (T/T0) and shortening velocity (µm hs-1) during the first part of phase 2 () and during phase 4 (). B, relation between force and power during early phase 2 () and during phase 4 ().

We tested the idea that phase 2 of the velocity transient is due to the working stroke in the attached heads by simulating the response to a force step with the mechanical-kinetic model that Piazzesi & Lombardi (1995) developed from the Huxley & Simmons (1971) theory to explain both the force transient following single or multiple length steps (Lombardi et al. 1992) and the steady state T-V relation. In the model the working stroke consists of two successive transitions between attached states of the myosin head: A₁-A₂ and A₂-A₃. The algorithm was implemented so that the isotonic condition was applied following the force step in order to simulate the velocity transient.

As shown by dotted lines in Fig. 1A and B for a force step to 0.3T₀, the whole velocity transient, including the small delayed oscillation, is qualitatively well reproduced by the model, except for the fact that the initial velocity of phase 2 is slower than observed. This is because the rate functions for the transitions A₁-A₂ and A₂-A₃ were selected for the best fit to the quick force recovery in length step experiments, assuming that the observed ~4 nm hs^-1 T₀^-1 compliance was all in the myosin cross-bridges (Piazzesi & Lombardi, 1995). Since ~50 % of the half-sarcomere compliance is in the myofilaments (Huxley et al. 1994; Wakabayashi et al. 1994; Dobbie et al. 1998; Linari et al. 1998), the compliance of myosin heads is in fact ~2 nm hs^-1 T₀^-1 and the length dependence of the rate functions should be much steeper (Huxley & Tideswell, 1996). Thus the model underestimates the kinetic features of the myosin working stroke, such as V₂, though it qualitatively reproduces the mechanical features such as the relation between L_T and force and its intercept on the length axis (6.5 nm hs^-1).

	DISCUSSION

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The size of the working stroke

One main achievement of this work is the demonstration that the sliding distance accounted for by work-producing myosin heads can vary in relation to force ( in Fig. 2B). This conclusion is only possible with adequately time-resolved measurements of the isotonic velocity transients and is not predictable from Huxley's length step experiments (Huxley & Simmons, 1971; Ford et al. 1977), where the full stroke capability can be elicited only with large releases. In fact, the quick force recovery following a length step is due to a fraction of the myosin stroke capability that varies depending on the size of the step, since the isometric condition following the step makes possible a new equilibrium distribution that is farther from the end of the working stroke the smaller the step. With force steps, the clamp to a force below the isometric capability allows the myosin heads to complete the stroke transition, whatever the step size. At higher forces the transition takes longer (>10 ms at 0.8T₀; Fig. 1B) and the probability that the myosin head detaches before completing the working stroke increases: the observed sliding distance during the isotonic phase ( in Fig. 2B) shows a maximum of ~7 nm for T = 0 and reduces to 4 nm at 0.8T₀. This feature is reproduced by models which assume a ~11 nm working stroke based on state transitions of the attached heads (Piazzesi & Lombardi, 1995) and thus does not contradict the idea of a large conformational change in the myosin head domain, such as that seen in crystallographic studies (Rayment et al. 1993; Geeves & Holmes, 1999).

Another striking result of this work is that, as shown in Fig. 4, the L₁ () and L₂ () relations obtained from the isotonic velocity transients almost perfectly superimpose on the corresponding T₁ () and T₂ () relations obtained from the force transients in the same frog species (Piazzesi & Lombardi, 1995). The identity of the relations between force and length for the phase 1 elastic response is straightforward and proves the reliability of our fast force clamp method. The identity for phase 2 working stroke response provides original experimental evidence of the validity of the mechanical-kinetic model based on the assumption of strain-dependent rate constants of state transition in the attached myosin heads integrated with the possibility of detachment of myosin heads from actin at a moderate rate. Under these conditions the sliding distance accounted for by the working stroke reduces for both small force steps and small length steps because detachment at an intermediate stage of the transition becomes relevant when either (1) following a small force step the state transition is slow, or (2) following a small shortening step the state transition does not exhaust the stroke capability.

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Figure 4. Comparison of L1 and L2 relations from force step experiments and T1 and T2 relations from length step experiments Filled symbols are L1 (circles) and L2 (diamonds) data points as in Fig. 2A, with the sign of the length change reversed and the axes rotated by 90 deg to superimpose on length step data; open symbols are T1 (circles) and T2 (diamonds) data points in Fig. 3B of Piazzesi & Lombardi (1995).

Energetics of the working stroke

The mechanical energy liberated with the working stroke ( in Fig. 2B) increases by 2.5 times from 0.1T₀ to 0.8T₀, indicating a corresponding increase in the efficiency of energy conversion. This is expected on thermodynamic grounds since at high forces the state transition in the attached heads occurs closer to the equilibrium between chemical energy that drives the heads towards the end of the working stroke and the opposing mechanical energy barrier. The finding reflects the macroscopic efficiency of muscle that exhibits a maximum value of 0.6 when muscle shortens under a high load (see Woledge et al. 1985, pages 265-267). E in eqn (1) and Fig 2B () can be normalised for the isometric force, giving:

E_r = E/T₀ = 1.15 nm hs^-1 + (L_T T/T₀), (2)

which varies from 1.82 nm hs^-1 at 0.1T₀ to 4.35 nm hs^-1 at 0.8T₀. The units for length change reduce from nm hs^-1 to nm when the relation is calculated for a single myosin head from the array of heads working in parallel in the half-sarcomere. In this case:

E_r = e/F₀, (3)

where e and F₀ are the mechanical energy and the isometric force of a myosin head. F₀ can be calculated if we know E_r and e. According to the experiments reported here, a myosin head works with the highest efficiency at 0.8T₀. Assuming a tight chemo-mechanical coupling, one actin-myosin interaction at 0.8T₀ liberates mechanical energy equivalent to 0.6 of the free energy made available from ATP hydrolysis (83 10^-21 J, Cooke, 1997). Thus at 0.8T₀, F₀ = (50 10^-21 J/4.35 nm =) 11.5 pN. This value is consistent with a low fraction (< 0.2) of myosin heads attached during the isometric contraction, as indicated by spectroscopic (Cooke et al. 1982; Corrie et al. 1999) and X-ray diffraction (Huxley et al. 1982) structural studies, and is higher than the values obtained with single molecule techniques (Finer et al. 1994; Ishijima et al. 1994; Molloy et al. 1995), which suffer from the limitation that the force of individual interactions is recorded through a large series compliance.

Note that, if, as reported by Barclay (1998), the maximum efficiency is 0.45, F₀ becomes 8.6 pN. If we choose an average value of 10 pN for F₀ and consider that during the isometric contraction the attached myosin heads account for 50 % of the 4 nm hs^-1 compliance (Dobbie et al. 1998), the stiffness of a myosin head (s) is ~(10 pN/2 nm =) 5 pN nm^-1. This high value of stiffness is consistent with a relatively small thermal fluctuation of the myosin heads before attachment to actin. Namely at 4 °C the r.m.s. fluctuation ((kT/s)^0.5, where k is the Boltzmann constant, 1.38 10^-23 J K^-1, and T is the temperature in K) is 0.9 nm. Accordingly, the probability of generating a 7 nm working distance by thermal fluctuation is too low. These considerations give further support to the Huxley & Simmons multi-step model of the working stroke rather than thermal ratchet models.

We can conclude that (i) under isotonic conditions the working stroke in myosin II from frog skeletal muscle at 4 °C accounts for a filament sliding that decreases from a maximum of ~7 nm at T = 0 to a minimum of 4 nm at T = 0.8T₀, while the time for the completion of the working stroke increases from < 1 ms (T = 0.1T₀) to > 10 ms (T = 0.8T₀); (ii) the mechanical energy and thus the efficiency of energy conversion increases monotonically with the force and is maximal at 0.8T₀; (iii) the isometric force per attached head is ~10 pN.

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Acknowledgements

We thank Professor A. F. Huxley for helpful criticism, Professor M. Irving for comments on the manuscript and Mr A. Aiazzi and Mr M. Dolfi for mechanical and electronic assistance. This work was supported by CNR, MURST and Telethon (Italy).

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