Temperature dependence of the force-generating process in single fibres from frog skeletal muscle

Abstract

Generation of force and shortening in striated muscle is due to the cyclic interactions of the globular portion (the head) of the myosin molecule, extending from the thick filament, with the actin filament. The work produced in each interaction is due to a conformational change (the working stroke) driven by the hydrolysis of ATP on the catalytic site of the myosin head. However, the precise mechanism and the size of the force and length step generated in one interaction are still under question. Here we reinvestigate the endothermic nature of the force-generating process by precisely determining, in tetanised intact frog muscle fibres under sarcomere length control, the effect of temperature on both isometric force and force response to length changes. We show that raising the temperature: (1) increases the force and the strain of the myosin heads attached in the isometric contraction by the same amount (∼70 %, from 2 to 17 °C); (2) increases the rate of quick force recovery following small length steps (range between −3 and 2 nm (half-sarcomere)⁻¹) with a Q₁₀ (between 2 and 12 °C) of 1.9 (releases) and 2.3 (stretches); (3) does not affect the maximum extent of filament sliding accounted for by the working stroke in the attached heads (10 nm (half-sarcomere)⁻¹). These results indicate that in isometric conditions the structural change leading to force generation in the attached myosin heads can be modulated by temperature at the expense of the structural change responsible for the working stroke that drives filament sliding. The energy stored in the elasticity of the attached myosin heads at the plateau of the isometric tetanus increases with temperature, but even at high temperature this energy is only a fraction of the mechanical energy released by attached heads during filament sliding.

In the contracting muscle the interaction of the globular portion of the myosin molecule (the myosin head) with the actin leads to a conformational change in the myosin head, the working stroke, which is responsible for the generation of the force that pulls the actin filament towards the centre of the sarcomere. The work produced in the interaction is accounted for by the free energy liberated by the ATP hydrolysed in the catalytic site of the myosin head. Biochemical and crystallographic studies (Sutoh et al. 1991; Rayment et al. 1993; Miller et al. 1995; for an extensive review see Cooke, 1997) indicate that the interaction is initiated by the formation of a weak electrostatic bond between the positively charged residues on the surface of the catalytic domain of the myosin head and the negatively charged residues on the actin surface, and is followed by the formation of strong stereospecific bonds that promote the change in the head conformation responsible for the working stroke. Calorimetric studies (Highsmith, 1977; Kodama & Woledge, 1979; Smith et al. 1984) show that ATP hydrolysis and formation of the actomyosin bond are accompanied by an increase in enthalpy, indicating that the process must be driven forward for entropic reasons. This view is supported by mechanical studies in single muscle fibres showing that temperature enhances the force developed per actin-myosin interaction and thus that force generation is an endothermic process occurring in the myosin cross-bridges attached to actin (Ford et al. 1977; Ranatunga & Wylie, 1983; Goldman et al. 1987; Bershitsky & Tsaturyan, 1992, 2002; Davis & Harrington, 1993; Zhao & Kawai, 1994; Davis & Rodgers, 1995; Ranatunga, 1996; Wang & Kawai, 2001). Force can be generated in an entropically driven process by the capture of the energy of the thermal fluctuation of an elastic element in the myosin head with the formation of a strong actin-myosin bond that represents a more favourable state (Cooke, 1997). The higher the temperature the larger the thermal fluctuation and thus the force on the elastic element upon formation of the bond. In terms of a thermal ratchet model the energy released in the execution of the working stroke by the myosin head that drives filament sliding is accounted for by the thermal energy captured upon formation of the bond. However, according to the Huxley & Simmons model of the working stroke (Huxley & Simmons, 1971), the formation of the first bond is followed by one or several strain-dependent state transitions to the most favourable state, which implies a production of work substantially larger than the mechanical energy captured in the first step. This latter model is experimentally based on the characteristics of the force transient elicited by step reductions in length superimposed on the steady isometric force, T₀. The force attained at the end of the step, T₁, reflects the pure elasticity of the attached myosin heads and myofilaments: the size of the step necessary to drop the isometric force to zero, ∼4 nm half-sarcomere (hs)⁻¹, measures the strain on these elements during the isometric contraction. The elastic response is followed by a quick recovery of force (complete within 1–2 ms in frog fibres at 4 °C) to a value, T₂, that, for releases up to 5 nm hs⁻¹, is similar to the original isometric force and then reduces progressively with the increase in the release size, attaining zero for releases of ∼12 nm hs⁻¹. According to Huxley & Simmons (1971): (i) the quick force recovery is the consequence of the synchronous execution of the working stroke in the attached myosin heads, due to the rapid re-equilibration of different force-generating states brought out of the equilibrium by the drop in mechanical energy; (ii) the rate of quick recovery increases with the release size because the mechanical energy in the cross-bridge elasticity is part of the activation energy for the forward state transition, and (iii) the mechanical energy released in one actin-myosin interaction (defined by the profile of the T₂ curve) is several times larger than that stored in the elastic component (defined by the profile of the T₁ curve).

Previous work supports the view that force generation depends on temperature through the same state transitions that are involved in the working stroke elicited by a release: (1) in intact frog fibres the rate of quick recovery following a length step has been found to be largely temperature dependent (Q₁₀ 1.8–2.5, Ford et al. 1977; Piazzesi et al. 1992); (2) in permeabilised (skinned) mammalian fibres an early component of the force rise generated by a step increase in temperature was found to be related to the rapid phase of the force transient elicited by a step release (Goldman et al. 1987; Bershitsky & Tsaturyan, 1992; Davis & Harrington, 1993; Ranatunga, 1999). However, the early phase of the force rise following a temperature jump could be the response not to temperature per se, but rather to a length-release type of effect due to thermal expansion (Davis & Harrington, 1993). A common mechanism for force generation by temperature increase and by step release has been more recently challenged by structural and mechanical data on demembranated fibres (Bershitsky et al. 1997; Tsaturyan et al. 1999; Bershitsky & Tsaturyan, 2002) and by models inspired by those data (Huxley, 2000).

In this work, we reinvestigate the nature of the endothermic process leading to force generation in the myosin head by precisely determining, in single intact fibres from frog muscle, the effects of temperature on both isometric force and force response to length changes. The results show that the temperature modulates the isometric force through the same state transition as that promoted by length changes. However, the working stroke elicited by temperature is only a fraction of the maximal working stroke elicited by shortening, which is 10 nm, independent of temperature. This gives further support to the Huxley & Simmons (1971) multistate transition model of the working stroke.

Methods

Fibre preparation and mounting

Single fibres (4–6 mm long) were dissected from the lateral head of the tibialis anterior muscle of Rana esculenta after killing by decapitation and pithing (following the official regulation of the European Community Council, Directive 86/609/EEC and conforming with the UK Animals Scientific Procedures Act, 1986). Tendon attachments were carefully trimmed and clamped in T-shaped aluminium foil clips (Ford et al. 1977) to minimise end compliance and to obtain optimal alignment between the fibres and the transducer hooks.

Fibres were horizontally mounted in a thermoregulated trough between the lever arms of a capacitance gauge force transducer (resonant frequency 35–50 kHz; Huxley & Lombardi, 1980) and a loudspeaker motor servo-system (Lombardi & Piazzesi, 1990). The physiological solution bathing the fibre (mm:115 NaCl, 2.5 KCl, 1.8 CaCl₂, 3 phosphate buffer, at pH 7.1) was set at the desired temperature by means of a servo-controlled thermoelectric module. Fibre measurements (sarcomere length, fibre length and cross-sectional area, CSA) were carried out under a microscope with a × 40 water immersion objective and a × 25 eyepiece. Trains of stimuli of alternate polarity to elicit fused tetani were delivered by means of platinum plate electrodes 4 mm apart.

Mechanical apparatus and data acquisition

A striation follower (Huxley et al. 1981) was used to either record or control the length of a population of sarcomeres in a 1–2 mm fibre segment, close to the force transducer end to minimize the effect of propagation time of mechanical perturbation. The force transducer, the loudspeaker motor and the striation follower have been already described in detail (Lombardi & Piazzesi, 1990; Piazzesi et al. 1992). Experiments were carried out both in length-clamp mode (the feedback signal for the loudspeaker motor is from the striation follower) and in fixed-end mode (the feedback signal is from the position of the loudspeaker motor hook). Tension transients were elicited in length-clamp mode by superimposing step length changes (range from +3 to −12 nm hs⁻¹, negative sign for releases) on the isometric tetanus. The loudspeaker motor used in these experiments was able to deliver steps complete in about 50 μs; however, to avoid stimulating longitudinal oscillations in the fibre, the duration of step was set to 110 μs. Force-velocity relations were determined in fixed-end mode by imposing, on the isometric tetanus, ∼50 nm hs⁻¹ shortenings and lengthenings at constant velocities.

The outputs from the tension transducer, the striation follower and the loudspeaker motor were recorded on a digital oscilloscope (Nicolet Pro20). In the experiments on tension transients a multifunction I/O board (PCI-6110E, National Instruments) and a program written with LabVIEW (National Instruments) were used for signal recording. Acquisition rate was 200 kHz to resolve the elastic response and the early phases of quick force recovery following a length step.

Experimental protocol

To determine the relation of isometric tetanic force and stiffness versus temperature, for each fibre typically two cycles of several series of tetani (one series for each temperature) were recorded. The lowest temperature used was 2 °C, since in muscle fibres from Rana esculenta the excitation may fail at temperatures lower than 2 °C. In the first cycle five series of tetani referred to five different temperatures 2, 5, 10, 17 and 24 °C. The order of the series was random and was different from that of the second cycle. Each series in the first cycle consisted of four fixed-end tetani, without superimposed steps, for a total of 20 tetani. The duration of tetanic stimulation was reduced as temperature was increased (Table 1, first row) in accordance with the temperature-dependent increase in rate of development of tetanic force (see Fig. 1). Following the change in temperature the first tetanus could be preceded by a larger time interval than that set between tetani within the same series and this resulted in a T₀ value higher than in the following tetani of the same series. For this reason only the last three tetani of each series were used for the statistics. For each temperature the frequency of electrical stimulation and the time interval between consecutive tetani were selected in preliminary experiments. Frequency was set to the minimum value capable of producing complete fusion in force (Table 1, second row); the time interval between tetani was set to the minimum value able to maintain constant T₀ in the last three tetani of the series (Table 1, third row). In the second cycle, which concerned series of tetani at all the above temperatures except 24 °C, a step in sarcomere length was superimposed on the isometric tetani, starting from the second tetanus, to elicit the tension transient. The step was delivered as soon as the force had attained the isometric plateau value (Table 1, fourth row). The length control system was switched from fixed-end mode to sarcomere length-clamp mode 30 ms before the step and switched back to fixed-end at the time of the last stimulus. At each temperature the number of tetani was between five and eight to select the best gain for the amplifiers in the length-clamp loop. We completed the protocol in five fibres, recording the force transient for three step amplitudes (−2.5, −1.3 and +1.3 nm hs⁻¹, negative sign for releases) at all four temperatures before the fibre showed sign of run-down (more than 5 % drop in T₀ with respect to the value during the first cycle at the same temperature).

Table 1.

Parameters of the experimental protocol

	2°C	5°C	10°C	17°C	24°C
Stimulus duration (ms)	500	400	300	200	100
Stimulus frequency (Hz)	10–15	15–25	35–50	70–90	120
Time interval between tetani (min)	4	3	3	2	2
Step time (ms after stimulus start)	350–400	250–300	150–250	100–150	—

Figure 1. Effect of temperature on the development of the tetanic force and on the force transient following length steps.

Force responses (upper traces in each panel) to step changes of half-sarcomere (hs) length (lower traces) at four temperatures (in °C, indicated by numbers in A). The size of the steps were 1.33 nm hs⁻¹ (A), −1.28 nm hs⁻¹ (B) and −2.5 nm hs⁻¹ (C). The earlier part of the records, separated from the latter part by the interruption in the traces, refers to the development of the tetanus and is 200 times slower than the latter part, so as to have adequate resolution for the phenomenon under analysis. This gives evidence of the temperature dependence of latency time, rate of force development during the rise of the tetanus and steady force attained at the tetanus plateau. The interruptions of the traces are progressively larger from the lowest to the highest temperature, so that the steps are graphically synchronised and it is easier to compare the evolution of the force transients at the different temperatures. The force traces at the four temperatures are superimposed on the same baseline. The length traces are superimposed starting from the value just before the length step. This illustrates the consistency of the steps imposed in length-clamp mode at the four different temperatures, in contrast to the shortening of the sarcomeres during the development of the tetanus in fixed-end mode, which may change from one tetanus to another by a few nanometres per hs. The small bars under the force traces indicate the time of the first stimulus. Fibre length: 6.29 mm, sarcomere length: 2.14 μm, cross-sectional area (CSA): 14 300 μm².

Complete T₂ relations, with releases up to 12 nm hs⁻¹, were determined at only two temperatures, 2 and 10 °C, in another six fibres. In fact, for releases larger than 4 nm hs⁻¹ the fibre becomes slack during the phase 1 response and this makes it difficult to obtain clean force responses under sarcomere length-clamp conditions. This problem was reduced by increasing the duration of the step according to the increase in its amplitude, but the number of trials and the time spent achieving satisfactory records for each step size above 4 nm hs⁻¹ made it unlikely that we obtained complete T₂ curves from the same fibre for more than two temperatures before the fibre run-down. The force attained at the end of quick recovery (T₂) was estimated according to the tangent method already described (Ford et al. 1977; Piazzesi et al. 1992).

The relation between force and velocity of shortening and lengthening (T–V relation) was determined at three different temperatures (5, 11 and 17 °C) in a separate set of experiments.

Results

Effect of temperature on the isometric force and stiffness

The force and stiffness developed at the plateau of the isometric tetanus were determined at different temperatures in the ranges 2–24 °C and 2–17 °C, respectively. Figure 1 illustrates a typical experiment. In each panel the records at four different temperatures (2, 5, 10 and 17 °C) of the force (upper frame) and the length change in the population of the sarcomeres under striation follower inspection (lower frame) are superimposed. The earlier part of the records refers to the development of the tetanic force in fixed-end conditions up to the isometric plateau value T₀, and the latter part of the records, running at a speed 200 times higher and graphically separated from the first part by an interruption in the traces, refers to the force transient following a step imposed under sarcomere length-clamp conditions. It is evident from direct observation of the records, that, while the force attained at the tetanus plateau progressively increases with temperature, attaining at 17 °C a value ∼70 % larger than at 2 °C, the change in force simultaneous with the length step is roughly the same at the different temperatures. The relation of the isometric tetanic force versus the temperature (range 2–24 °C) is reported in Fig. 2. Force rises from 140 ± 3 kPa (mean ± s.e.m., five fibres) at 2 °C to 258 ± 3 kPa at 24 °C. Even though the increase in force with the rise in temperature decreases at higher temperatures, there is no sign of saturation up to 24 °C.

Figure 2. Temperature dependence of the isometric tetanic force.

Data are the mean values from five fibres at five temperatures (2, 5, 10, 17 and 24 °C). Error bars (± s.e.m.) are within the size of the symbols. The two different scales on the ordinate refer to kPa (left) and to values relative to T₀ at 2 °C (T₀/T_0,2, right).

Stiffness is measured by the ratio of the change in force during phase 1 of the force transient over the corresponding change in half-sarcomere length. It has been shown by Ford et al. (1977) that, depending on the time taken for the step to be complete, the quick recovery of force during phase 2 of the force transient produces a truncation of the tension attained at the end of the step (T₁), and thus an underestimate of the stiffness. The speed of the quick force recovery and thus the truncation effect are larger the larger the step release, and this explains the progressive upward deviation from linearity of the relation between T₁ and step amplitude, and the corresponding decrease in the stiffness (estimated by the slope of the relation) with the increase in the size of the release. Since the speed of quick force recovery is markedly temperature dependent (Ford et al. 1977; Piazzesi et al. 1992), different degrees of truncation are expected at different temperatures, even with the same step size, and this would influence the measure of the effect of temperature on stiffness. To avoid the truncation effect, the stiffness must be estimated by the slope of the relation of the change in force (T) versus the change in half-sarcomere length (Δl) during the step itself (Ford et al. 1977). The plots for two different temperatures (2 °C, open symbols, and 10 °C, filled symbols) from a typical experiment are shown in Fig. 3A. The values at the intersection with the ordinate are the isometric tetanic forces just before the step. In accordance with previous work (Ford et al. 1977; Piazzesi et al. 1992), at each temperature the plots for different step sizes superimpose, showing that the response in phase 1 is purely elastic. The Young modulus (E = (T/CSA) × (hs length/Δl)) reported in Fig. 3B for the four different temperatures is the mean ± s.e.m. from five fibres in the range of Δl from +2 to −3 nm hs⁻¹. The value at 2 °C, 42.7 ± 3.4 MPa, is not significantly different from the values at any other temperature. In accordance with this result, the relation of the abscissa intercept of the instantaneous T:Δl plots (Y₀) versus T₀ at different temperatures (T_0,t, Fig. 3C) can be fitted by a linear regression equation Y₀ = bT_0,t + a, where b = 0.025 ± 0.002 nm hs⁻¹ kPa⁻¹, with a not significantly different from zero (a = 0.002 ± 0.394 nm hs⁻¹). Thus changes in T₀ are accompanied by proportional changes in strain of the two main structures contributing to the half-sarcomere elasticity, the myofilaments and the attached myosin heads (Linari et al. 1998) (this conclusion would be wrong only if the two mechanical components of the half-sarcomere had non-linear elasticities that together would generate a linear elasticity). Assuming that the overall linear elasticity of the half-sarcomere provides the evidence that its main components have linear characteristics, the elastic strain in myosin heads (Y_h) and myofilaments (Y_f) at T₀ at each temperature can be calculated from the observed Y₀, since we know the fractional contribution of cross-bridges and myofilaments to total half-sarcomere compliance at 4 °C in fibres from Rana esculenta: cross-bridges 0.6, myofilaments 0.4 (Linari et al. 1998; Dobbie et al. 1998). As reported in Table 2, Y_h varies from 2.09 nm at 2 °C to 3.49 nm at 17 °C. Extrapolating the relation in Fig. 3C for temperatures higher than 17 °C, at 24 °C, where T_0,24/T_0,2 = 1.84, Y_h is 3.85 nm.

Figure 3. Temperature dependence of half-sarcomere stiffness and strain.

A, instantaneous force-length (T:Δl) relations at 2 °C (open symbols) and 10 °C (filled symbols). Different symbols refer to different step sizes between +3 and −4 nm hs⁻¹. Fibre length: 6.01 mm, sarcomere length: 2.16 μm, CSA: 8200 μm². B, mean values (±s.e.m., 5 fibres) of Young modulus (E) at the different isometric forces corresponding to the four temperatures (T_0,t) obtained by instantaneous T:Δl relations as those in A. C, plot of Y₀, the abscissa intercept of T:Δl plots, against T_0,t (mean and s.e.m. from the same five fibres as in B). Continuous line is the linear regression on the average data points, with slope 0.025 ± 0.002 nm hs⁻¹T₀⁻¹ and ordinate intercept 0.002 ± 0.394 nm hs⁻¹.

Table 2.

Dependence on temperature of isometric tetanic force, strain in the attached myosin heads (Y_h), fraction of the attached myosin heads in the force-generating state (n₂/(n₁+n₂)) and free energy difference (ΔG) between the non-force- and force-generating states

Temperature (°C)	T0/T0,2	Yh (nm)	n2/(n1+n2)	ΔG (J × 10−21)
2	1	2.09	0.50	−0.03
5	1.18	2.47	0.59	−1.35
10	1.42	2.97	0.72	−3.55
17	1.67	3.49	0.84	−6.64

Isometric force values are made relative to the force at 2°C (T₀/T_0,2), Y_h is calculated from the relation in Fig. 3C, n₂/(n₁+n₂) and ΔG are calculated from eqn (4) in the text.

Thus the temperature-dependent increase in T₀ is accounted for by a corresponding increase in the average force per attached head without significant change in the fraction of attached, stiffness-generating, heads. This finding suggests that during a steady isometric contraction the temperature-dependent shift in the equilibrium distribution among different force-generating states of attached myosin heads affects the flux from attached to detached states and the flux from detached to attached states to the same extent. Since it is known that the rate of ATPase increases with temperature, this finding indicates that the fraction of cycle time spent by myosin heads in the attached states remains constant and that the main determinant of the fraction of attached myosin heads in the isometric contraction is the steric correspondence between myosin heads and actin sites.

Effect of temperature on the quick force recovery following a length step

Figure 4 shows T₁ relations (filled symbols) and T₂ relations (open symbols) from a typical experiment at 2 °C (circles) and 10 °C (triangles). T₂ relations show a downward concavity in the region of small releases and then, for releases larger than 6 nm hs⁻¹, a linear drop with the step size. Dashed lines are the 1st order regression equations fitted to the linear part of T₂ curves (see also Piazzesi et al. 1992). While the T₂ relations run almost parallel in the region of small releases, in the linear region the slope of the relation at 10 °C is larger than that at 2 °C. The intercept on the length axis of the regression lines fitted to the linear part of the T₂ relation (dashed lines) represents the maximum sliding accounted for by the phase 2 process (L₀). As shown in the table in Fig. 4, in the six fibres examined, L₀ is 11.3 ± 0.3 nm hs⁻¹ (mean ± s.e.m.) at 2 °C and 11.8 ± 0.4 nm hs⁻¹ at 10 °C.

Figure 4. Effect of temperature on T₂ relations.

T₁ (filled symbols) and T₂ (open symbols) relations at 2.1 °C (circles) and 10.0 °C (triangles). Force is relative to the isometric tetanic value at 2 °C. Dashed lines are the linear regressions on T₂ points for step releases larger than 6 nm hs⁻¹. Fibre length, 5.17 mm; sarcomere length, 2.11 μm; CSA, 6500 μm². The table reports the average values (±s.e.m.) in 6 fibres for T₀ and L₀ (the abscissa intercept of T₂ relation) at the two temperatures.

The kinetic characteristics of quick force recovery following small length steps (from +2 to −3 nm hs⁻¹) were determined at the same four temperatures as stiffness, 2, 5, 10 and 17 °C (see Fig. 1) in five fibres. The range of step amplitudes was limited to minimise the contribution of rapid detachment/attachment of myosin heads to quick force recovery (Lombardi et al. 1992; Piazzesi et al. 1997). The rate of quick recovery (r) was measured by the reciprocal of the time taken for the force to recover from T₁ to T₁ + 0.63(T₂ − T₁). r (mean ± s.e.m., five fibres) at the four temperatures is plotted against the step amplitude in Fig. 5A. Length steps are grouped in three classes: +1.3 nm (range +1 to +1.5), −1.3 nm (range −1.1 to −1.4) and −2.5 nm (range −2.1 to −2.9). At each temperature r reduces exponentially going from the largest release to the stretch (Huxley & Simmons, 1971; Ford et al. 1977). For any step size, r increases with the rise in temperature, without sign of saturation at the highest temperature tested (17 °C). In Fig. 5B the temperature dependence of the speed of recovery for any step amplitude is shown by plotting lnr versus the reciprocal of the absolute temperature (1/θ). The symbols refer to the same experimental data as in Fig. 5A. The slopes of the relations show that the temperature dependence of r increases going from the largest release to the stretch. Corresponding Q₁₀ values in the range 2–12 °C are 1.92 for −2.5 nm, 1.99 for −1.3 nm and 2.30 for + 1.3 nm. This is in agreement with previous work: Ford et al. (1977) report 1.85 and 2–2.5 for releases and stretches respectively; Piazzesi et al. (1992) report 2 and 2.65 for releases and stretches respectively. In contrast, in skinned fibres the speed of quick tension recovery following a length step does not change with temperature (Bershitsky & Tsaturyan, 2002), but in this preparation it is difficult to record the rapid phase of the tension transient under sarcomere length control (Goldman et al. 1987; Linari et al. 1993).

Figure 5. Effect of temperature on the speed of quick force recovery.

A, relation between the rate of quick recovery, r, calculated as the reciprocal of the time necessary to recover 63 % of the difference between T₂ and T₁, and the size of the length step at 2 (circles), 5 (squares), 10 (triangles) and 17 (diamonds) °C. Values are reported as means ±s.e.m. from the same 5 fibres as in Fig. 2. Error bars are omitted when smaller than symbol size. Continuous lines are the exponential fits on pooled data according to the equation r =r₀/2(1 + exp(-al), where r₀ is the intercept on the ordinate and l is the imposed length change. B, Arrhenius plot of rates of quick recovery for the three step sizes in A: −2.5 nm hs⁻¹, squares; −1.3 nm hs⁻¹, circles; 1.3 nm hs⁻¹, triangles. The temperature axis also shows the scale in °C.

Effect of temperature on the force response to steady lengthening

The increase in steady force in response to constant velocity lengthening imposed on an active fibre has been shown to depend on the force developed by attached myosin heads in the original isometric condition (Piazzesi et al. 1994): when the isometric force is depressed by an increase in the osmotic pressure of the bathing solution, the increase in steady force in response to lengthening is larger than in the normal solution. The explanation is that when the equilibrium distribution of different attached states in the isometric condition is shifted toward the beginning of the working stroke by the osmotic intervention, the stretch-induced backward redistribution of attached states (Piazzesi et al. 1992) is reduced, producing a smaller compensation for the effect of lengthening on the strain of the attached heads. Thus, an independent way to test whether the temperature-dependent increase in isometric force is due to a shift in the equilibrium between the states of the myosin heads that are under mechanical control is to record the force response to steady lengthening at different temperatures.

As previously demonstrated (Colomo et al. 1988; Lombardi & Piazzesi, 1990), during constant velocity stretches a steady force is attained beyond ∼20 nm hs⁻¹ of lengthening as far as the imposed lengthening is homogeneously redistributed among the sarcomeres; otherwise, when intersarcomere inhomogeneity develops, force continues to creep up during the whole lengthening. Data reported here are from five fibres selected for intersarcomere homogeneity of lengthening speeds at the three temperatures tested (5, 11 and 17 °C). Figure 6 shows the results from one experiment. The steady force attained during constant velocity shortening and lengthening is plotted against velocity (T–V relation). Force is expressed both normalised per CSA (kPa, Fig. 6A) and normalised for the isometric tetanic force at the same temperature (T/T₀, Fig. 6B). It can be seen that the lengthening-dependent increase in steady force above the isometric value is smaller the higher the temperature. The first evidence of this phenomenon has already been given in whole frog muscle (Flitney & Hirst, 1978). The velocity of lengthening at which the increase in steady force becomes almost insensitive to further increase in velocity (the phenomenon described as ‘give’ by Katz, 1939) is progressively higher the higher the temperature. As shown in Fig. 6B, the values of lengthening velocity necessary to attain the force for ‘give’ (T_G) are < 1 μm s⁻¹ at 5 °C, just above 2 μm s⁻¹ at 11 °C and ∼4 μm s⁻¹ at 17 °C. This agrees with the previous finding (Colomo et al. 1987) that at 11 °C, to obtain the same force-lengthening relation as at 5 °C, the lengthening speed must be increased 2–3 times. The table in Fig. 6 reports the mean values (± s.e.m.) of T_G relative to T₀ at the corresponding temperature for five fibres tested at the three temperatures. T_G/T₀ progressively decreases as the temperature increases, indicating that the higher the temperature, the larger the stretch-induced backward redistribution between attached states that partially absorbs the effect of lengthening.

Figure 6. Effects of temperature on force-velocity relation.

A, relation between shortening (negative values) and lengthening (positive values) velocity and force at 4.9 °C (triangles), 11 °C (squares) and 17 °C (diamonds). Note that the force attained at each lengthening velocity is almost the same, independent of the temperature. B, same relations as in A with force expressed relative to T₀ at the corresponding temperature. Fibre length: 4.7 mm, sarcomere length: 2.13 μm, CSA: 8000 μm². The table reports means and s.e.m. (5 fibres) of the lengthening velocity at which the force starts to ‘give’ (V_G) and the force for ‘give’ relative to T₀ at the corresponding temperature (T_G/T₀). The asterisk indicates that the higher temperature has been used in only 3 fibres.

Discussion

Thermodynamic implications of the temperature dependence of isometric force

The force developed in the isometric tetanus increases by 70 % rising the temperature from 2 to 17 °C, while the half-sarcomere stiffness remains constant. Thus the number of attached myosin heads is unaffected by temperature and the rise in force is fully explained by the rise in the force generated per attached myosin head. Under these conditions temperature controls the equilibrium between non-force and force-generating states of the attached myosin heads. Assuming the simplest case of two states A1 (with zero force) and A2 (with force F₀), the force exerted in isometric conditions in the half-sarcomere depends on the equilibrium of the reaction:

where k+ and k− are the forward and reverse rate constants. Using n₁ and n₂ to indicate the fractions of attached heads in the states A1 and A2 and taking (n₁ + n₂) as constant (according to stiffness results), the isometric force at temperature t, T_0,t, is F₀n₂ and the maximum isometric force, T_o,max, is F₀(n₁ + n₂). Hence, T_0,t/T_o,max = n₂/(n₁ + n₂). According to the van't Hoff equation (see also Goldman et al. 1987), relating the equilibrium constant K to the free-energy change, ΔG:

(1)

where k_b is the Boltzmann constant and θ is the absolute temperature.

Equation (1) can be written as:

(2)

or

(3)

where ΔH is the enthalpy change and ΔS is the entropy change. Equation (3) expresses the dependence of T₀ on temperature and can be rearranged to the form:

(4)

Figure 7 shows the same T₀ points as Fig. 2 re-plotted versus the reciprocal of θ (filled circles) and eqn (4) fitted to the points (continuous line). The fit gives the estimates of T_o,max (278.6 ± 8.2 kPa), ΔH ((121.3 ± 11.3) × 10⁻²¹ J) and ΔS ((0.441 ± 0.044) × 10⁻²¹ J K⁻¹). It must be noted that Wang & Kawai (2001), using the sinusoidal perturbation technique to investigate the kinetics of actin-myosin interaction in psoas skinned fibres, estimated a ΔH of ∼100 × 10⁻²¹ J for the endothermic force-generating step.

Figure 7. Relation between force and reciprocal of absolute temperature.

Filled circles are the same data as open circles in Fig. 2. The line is the fit to experimental points according to the equation T₀=T_o,max(1 + exp(ΔH/k_bθ−ΔS/k_b). The temperature axis also shows the scale in °C.

An important consequence of our analysis is that, given the values for ΔH and ΔS, the change in the entropic factor, θΔS, becomes almost equal to the change in enthalpy at 2 °C. Therefore at this temperature ΔG = 0, the attached heads are populating the two states in the same proportion and the isometric force is 0.5 T_o,max. As temperature increases, θΔS increases, determining the drop in ΔG that promotes the shift in the equilibrium distribution toward the force-generating state (Table 2).

In mammalian muscle fibres the temperature for half-maximum isometric force is ∼10 °C (Ranatunga & Wylie, 1983; Coupland et al. 2001). A difference of ∼8 °C between frog and mammalian muscles agrees with the consideration that the rates of the temperature-sensitive reactions of force generation scale with the working temperature of the muscle and not with the absolute temperature (Davis & Harrington, 1993).

Model simulation of the temperature dependence of isometric force and quick force recovery

The speed of quick force recovery following small length steps superimposed on the isometric contraction increases with the temperature, in agreement with previous results on intact frog fibres under the same sarcomere length-clamp conditions (Ford et al. 1977; Piazzesi et al. 1992). The temperature dependence of the speed of quick force recovery indicates that temperature affects the strain-dependent rate constants of state transitions responsible for the restoration of the isometric force perturbed by a length step. To test whether this occurs by the same mechanism that determines the temperature dependence of the isometric force generated by the attached myosin heads, we used a kinetic-mechanical model of myosin actin interaction (Piazzesi & Lombardi, 1995) that incorporates the Huxley & Simmons (1971) theory of force generation. In the simulation reported here we limited the reaction cycle to the two transitions between the three attached states (A1 ⇌²A2 ⇌³A3), considering that: (i) as shown in this work, the fraction of attached myosin heads during the isometric contraction is not affected by temperature, and (ii) the size of the length step (-2.5 to +1.5 nm) is small enough to assume that the quick force recovery is only due to the rapid re-equilibration between different force-generating states of attached heads, without substantial attachment/detachment (Lombardi et al. 1992; Piazzesi et al. 1997).

In the original model, all the half-sarcomere compliance (∼4 nm T₀⁻¹ at 4 °C) was attributed to the elasticity of the myosin cross-bridges. Actually the finding that ∼50 % of the half-sarcomere compliance resides in the actin and myosin filaments (Huxley et al. 1994; Wakabayashi et al. 1994; Dobbie et al. 1998) would reduce the compliance of myosin cross-bridges to ∼2 nm T₀⁻¹. For the purpose of this simulation, it is possible to maintain the whole 4 nm T₀⁻¹ compliance in the myosin cross-bridges, since during the quick force recovery the heads undergo the state transitions against a series compliance given by both cross-bridge elasticity and myofilament elasticity. Consequently, as shown in Fig. 8A, the dependence of the free-energy of the three states on x (the relative position of the actin site with respect to the myosin head) is the same as in Fig. 1A in Piazzesi & Lombardi (1995). In isometric conditions cross-bridges are assumed to be uniformly distributed between −1.6 and 3.9 nm. The forward and reverse rate constants for each of the two transitions (k_i(x) and k_-i(x), where i = 2, 3) are related to the change in free-energy between neighbouring states (ΔG(x)) through the equation:

(5)

Figure 8. Model simulation of temperature effects.

A, parabolic free-energy profiles of the three states of the attached myosin heads, A1, A2 and A3, as a function of x (the relative axial position between the myosin head and the actin site, taken as zero when the cross-bridge in A1 state exerts zero force) at 2 °C (continuous line) and 10 °C (dashed line). The free-energy profile of A1 state is the same at both temperatures and is defined setting G_min= 40 × 10⁻²¹ J. Bold traces indicate the free-energy distribution of attached myosin heads at the isometric plateau: the force is larger at 10 °C than at 2 °C, because at 10 °C in the range of x between 1 and 2.5 nm, the attached heads prefer the A2 state, which exhibits a steeper free-energy curve. B, relation of the rate of quick force recovery (r) versus the size of the length step calculated with the model (symbols) compared with the observed relation (continuous lines from Fig. 5A).

The equations expressing the dependence on x of the forward rate constants for the two transitions are:

(6)

(7)

where e is the stiffness of the elastic component of myosin cross-bridges (0.7 pN nm⁻¹), z is the change in length of the elastic component (4.5 nm) during either transition A1 ⇌ A2 or A2 ⇌ A3, and B is the maximum value assumed by k₂ and k₃ for x < −z/2 nm and −3z/2 nm, respectively. The reverse rate constants are calculated from eqn (5) after choosing the appropriate values for the equation expressing the forward rate constant. For a detailed explanation of the assumptions and the definitions of parameters, see Piazzesi & Lombardi (1995).

Simulation at 2 °C

First, the equilibrium distribution of attached heads in the isometric condition at 2 °C is obtained by setting the difference between the minima of the consecutive free-energy parabolas, ΔG_min = 10.5 × 10⁻²¹ J (Fig. 8A continuous lines). In this way the fraction of myosin heads in the A1 state (that by approximation can be assumed as the non-force-generating state) is similar to the fraction in the force-generating states, A2 and A3 (Table 3). Then, at the same temperature, the speed of quick force recovery following a step and its dependence on the step amplitude are simulated by setting B = 7000 s⁻¹ (Fig. 8B, circles).

Table 3.

Dependence on temperature of the relevant parameters of the three-state model and of the calculated fractional occupancy of the three states

			Fractional occupancy
Temperature (°C)	ΔGmin (J × 10−21)	B (s−1)	A1	A2	A3
2	−10.5	7000	0.50	0.47	0.03
5	−12.6	15 000	0.40	0.54	0.06
10	−16.0	41 000	0.25	0.61	0.14
17	−18.6	103 000	0.16	0.61	0.23

ΔG_min is the difference of the minima of two consecutive free energy curves and B is the maximum value assumed by the forward rate constants in eqns (6) and (7).

Simulation of the effects of increase of temperature

According to the conclusion in the previous section that raising the temperature shifts the equilibrium distribution of attached myosin heads toward the force-generating state by increasing the free-energy drop involved in the state transition (Table 2), the temperature dependence of T₀ was simulated by shifting downward the parabolic free-energy curves of both A2 and A3 states by the amount necessary to fit the increase of T₀ relative to T₀ at 2 °C (Table 3 and, for the 10 °C case, dashed lines in Fig. 8A). At the same time, the temperature-dependent increase of the speed of quick recovery (Fig. 5) implies that the increase in free-energy drop between states is accompanied by an increase in the sum of the forward and reverse rate constants, (k₂ + k_–2) and (k₃ + k_–3). Upward shifts of the relation of r versus step size similar to those observed can be reproduced by increasing B in eqns (6) and (7). However, as the temperature increases, the slope of the simulated relation (symbols in Fig. 8B) becomes progressively larger than that of the observed relation (continuous lines in Fig. 8B). This is particularly evident at 17 °C (diamonds), where the value of B that is able to shift the curve by the amount necessary to fit r for the −1.3 nm point produces an underestimate and an overestimate of r for the +1.3 nm point and the −2.5 nm point, respectively. The discrepancy is likely to be related to the combined effect of the high slope of the x dependence of the forward rate constant and the reduced number (2) of state transitions. For any given temperature and x, the transition A3 ⇌ A2 is slower than the transition A2 ⇌ A1 and at high temperatures, in contrast with low temperatures, the fractional occupancy of the A3 state is large enough (Table 3) to give a marked contribution to the quick recovery elicited by a stretch. This explanation of the discrepancy suggests that a working stroke model with only two transitions is a rough approximation to the real process that is likely to be based on a continuous structural transition between the myosin head conformations at the beginning and at the end of the working stroke.

From the values reported in Table 3 we can see that temperature-dependent changes in ΔG_min are not exactly linearly related to temperature, as would be expected from the linear dependence of free-energy change ΔG on the entropic factor θΔS (Table 2 and Fig. 7). In this respect it must be considered that we assume the same stiffness (and thus the same slope of free-energy parabolas) for all the three attached states and the same step size ΔG_min between consecutive parabolas (Piazzesi & Lombardi, 1995). With more free parameters the isometric force could be reproduced imposing that changes in ΔG_min are linearly dependent on temperature. This however would imply ad hoc modifications of the original model that would weaken the strength of the simulation in demonstrating which are the relevant parameters that control the effect of temperature.

Temperature and length steps probe a unique mechanism of force generation

One important result of these experiments is that the temperature-dependent increase in isometric force is accompanied by a proportional increase in the extension of the half-sarcomere elasticity (Y₀, Fig. 3C), while the increase in the abscissa intercept of the T₂ relation (L₀), measuring the extent of filament sliding accounted for by working stroke, is less than proportional (Fig. 4). The increase in Y₀ is distributed between all structural components of the half-sarcomere, in relation to their fractional contributions to the overall compliance. With 40 % of the half-sarcomere compliance in the myofilaments (Linari et al. 1998), at 2 °C Y_f is (3.48 nm × 0.4 =) 1.39 nm. The expected change in myofilament extension for a 42 % increase in isometric force between 2 and 10 °C is (1.39 nm × 0.42 =) 0.58 nm, which is almost identical to the increase in L₀ in the table of Fig. 4. Thus, when the increased strain in the myofilaments is taken into account, the abscissa intercept of the T₂ relation is the same, independent of temperature. This implies that the increased strain of myosin cross-bridges with the isometric force between 2 and 10 °C, 0.9 nm (Table 2), occurs at the expense of the same working stroke mechanism that is responsible for filament sliding.

The maximum extent of filament sliding due to myosin heads attached at the isometric plateau (L_w) is given by the sum of the working stroke necessary to load the head elasticity to T₀ (Piazzesi et al. 1997, 2002a), which corresponds to Y_h and the working stroke elicited by shortening, L_s. Thus L_w = Y_h + L_s. L_w can also be calculated as (L₀ − Y_f), and is ∼10 nm at both 2 and 10 °C (table in Fig. 4 and Table 2). However, at the higher temperature the sliding occurs at higher forces. In this respect the working stroke implies a larger work and thus a larger efficiency at higher temperature.

Independent evidence that temperature controls the isometric force through the same state transition controlled by mechanical conditions is given by the effects of temperature on the steady force developed in response to lengthening (Fig. 6). At the higher temperature the increase in force by lengthening is smaller and this is explained if the equilibrium distribution between states in the isometric condition is further ahead, so that the backward shift of the equilibrium induced by lengthening (Lombardi & Piazzesi, 1990; Piazzesi et al. 1992, 1997) can be larger than that at the lower temperature and produce a larger attenuation of the increase in strain of attached heads.

The finding that the speed of quick force recovery following a length step increases with the increase in temperature (Fig. 5) further supports the view that both temperature and mechanical conditions act on the same force-generating process. This evidence is missing in recent experiments on skinned fibres (Bershitsky & Tsaturyan, 2002). The discrepancy between skinned and intact fibres leaves open the question of possible limits in the resolution of fast mechanical events in skinned fibres (see later in this section). However, X-ray diffraction data from partially cross-linked skinned fibres (Bershitsky et al. 1997; Tsaturyan et al. 1999) show that the force rise following a temperature jump is accompanied by an intensification of the 1st actin layer line. This finding, together with the absence of temperature dependence of the rate of quick recovery following a release, was interpreted as evidence that the force rise induced by temperature is based on a different process than the working stroke elicited by a length step. In fact the intensity of the 1st actin layer line should reflect both azimuthal and axial movement of the attached heads to acquire a conformation more strictly related to the actin symmetry. X-ray interference between the two arrays of myosin heads in each thick filament (Linari et al. 2000) shows that an increase in temperature produces an axial movement of the myosin heads toward the centre of the sarcomere (Reconditi et al. 2002), as when heads step forward in the working stroke (Piazzesi et al. 2002b). In this respect the intensification of the 1st actin layer line by temperature jump may indicate that the motion of the myosin heads during the working stroke has an azimuthal as well as an axial component (Corrie et al. 1999). According to our conclusion that temperature and shortening influence the same structural transition in the attached myosin heads, a step increase in temperature in isometric conditions should produce a redistribution of the attached heads toward higher force-generating states, as occurs following a step release. However this is not evident in the existing work in skinned fibres: most of the rise in force following a step increase in temperature is one order of magnitude slower than the quick tension recovery following a length step (Goldman et al. 1987; Bershitsky & Tsaturyan, 1992, 2002; Davis & Harrington, 1993; Davis & Rodgers, 1995; Ranatunga, 1996). Even in the most recent work (Bershitsky & Tsaturyan, 2002) where a laser diffractometer is used to clamp sarcomere length, there is no evidence of a significant 1000 s⁻¹ component in the force response. An explanation could be that laser diffraction from a skinned fibre does not provide a feedback signal adequate to length clamp a homogeneous population of sarcomeres with the gain required by the rapid component of the force response. If this is the case it could explain why, in the same work (Bershitsky & Tsaturyan, 2002), the rate of quick force recovery following a length step fails to show temperature dependence.

However, the absence of a quick state redistribution in the force-generating heads following a step in temperature may be real. This is the case if the thermal fluctuation is effective in determining the equilibrium distribution of the attached heads only at an early stage of the force-generating process. In this respect it must be noted that skinned fibre studies on the chemo-mechanical transduction by myosin support the view that force generation occurs at an early stage of the ATPase cycle, before phosphate release (Fortune et al. 1991; Kawai & Halvorson, 1991; Dantzig et al. 1992; Ranatunga, 1999). If the force of the myosin cross-bridge can no longer be influenced by temperature once the endothermic process has been terminated by the formation of a favourable bond, the rise in force following the temperature jump would record the time course of the substitution of the cross-bridges attached before the step with new cross-bridges. This process is ∼10 times slower than the quick force recovery process, as shown by the rate (100 s⁻¹) of the repriming of the ability to elicit a second complete working stroke following a conditioning step release (Lombardi et al. 1992; Piazzesi et al. 1997). This idea provides a key to interpreting skinned fibre results showing that the most temperature-sensitive process, identified with the endothermic force-generating process, is ∼10 times slower than Huxley & Simmons' (1971) quick force recovery. In fact in skinned fibres, in the absence of an adequately fast sarcomere length clamp, the 1000 s⁻¹ working stroke elicited by a step release cannot produce a substantial force recovery (Linari et al. 1993) and most of the recovery occurs during the subsequent slower phase (phase 2 slow in Davis & coworkers' terminology; Davis & Harrington, 1993; Davis & Rodgers, 1995), which kinetically corresponds to the relatively flat phase 3 of the tension transient in Huxley & Simmons' (1971) terminology and to the 100 s⁻¹ repriming process in double step experiments (Lombardi et al. 1992). Both Davis' phase 2 slow following a step release and the rate of force rise in response to a temperature jump have marked temperature sensitivity (Q₁₀ ∼3; Davis & Harrington, 1993; Bershitsky & Tsaturyan, 2002). A direct test of the temperature dependence of the repriming process with the double step technique has still to be performed.

Bershitsky & Tsaturyan (2002) demonstrated, combining temperature jump and length steps, that the length step does not produce effects on the response to a temperature jump by the time the response is complete. Scaling for the different working temperature of frog and mammalian muscles, these results can be explained by the idea above that the force response to a temperature jump records the ∼100 s⁻¹ process of detachment/attachment of myosin heads. This interpretation of the force response to temperature jump does not contradict the evidence that the number of attached heads does not change with temperature, as shown both in temperature jump experiments (Bershitsky & Tsaturyan, 2002) and in this work. In fact, the 100 s⁻¹ detachment/attachment process implies substitution of the original heads with heads attached afresh without a substantial change in the number of attachments (Lombardi et al. 1992). Note that this proposed interpretation of the temperature jump data implies that the process is not necessarily endothermic and that the actual endothermic process of force generation is one order of magnitude faster.

Nature of the force-generating process

Provided that the number of attached myosin heads remains constant over the range of temperatures used, the relation between isometric force and temperature (Fig. 7) indicates that the rise in temperature shifts the equilibrium distribution of the myosin cross-bridges toward higher force-generating states by an entropically driven process, which leads to a progressively more favourable state of the myosin-actin complex. The higher the temperature, the larger the thermal energy captured in the elastic element upon formation of the bond and thus the force generated by the new state. The increase in the entropic factor accounts for the larger drop in free-energy between states in Fig. 8A, the enthalpy increase remaining constant.

A forward shift in the equilibrium between states can be obtained at constant temperature after reducing the mechanical energy with a step release and this is predicted according to the Huxley & Simmons (1971) model by the strain dependence of rate constants for state transition. The rate of quick force recovery following a release increases with the size of the release, indicating that the equilibrium is under mechanical control by the increase in the forward rate constant and not by reduction in backward rate constant. This evidence, which is incorporated in the profile of the activation energy by choosing the strain dependence of rate constants as in eqns (6) and (7), implies that most of the mechanical energy needed to complete the transition is involved in attaining the transition state, which is therefore mechanically closer to the state ahead (Howard, 2001). Under these conditions it becomes evident that thermal fluctuation is a mechanism for populating the high force-generating state in isometric conditions and that the force-generating process acts as a thermal ratchet, so that the higher the temperature the higher the mechanical energy in the elastic element when it is locked by the formation of a favourable bond (Cooke, 1997). On the other hand the 10 nm motion elicited in the attached myosin heads by shortening is more than twice that induced by temperature and stored in the head elasticity (just below 4 nm at 24 °C). This means that the completion of the working stroke implies a multi-state transition that is possible only when the mechanical energy barrier is kept low because filament sliding is permitted and allows a structurally determined maximum sliding distance of ∼10 nm. This is reproduced in Fig. 8A: due to the temperature-dependent downward shift in the free-energy curves, the force developed in isometric conditions is higher, but the maximum sliding distance does not change since the abscissas of the minima of the parabolas remain constant. In conclusion, the results of this work show that in isometric conditions the structural change leading to force generation in the attached myosin heads can be modulated by temperature at the expense of the structural change responsible for the working stroke that drives filament sliding. The energy stored in the elasticity of the attached heads at the plateau of the isometric tetanus increases with temperature, but, even at high temperature, it is a fraction of the mechanical energy released by attached heads during filament sliding.