The myofilament elasticity and its effect on kinetics of force generation by the myosin motor

Archives of Biochemistry and Biophysics xxx (2014) xxx–xxx

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The myofilament elasticity and its effect on kinetics of force generation by the myosin motor Gabriella Piazzesi, Mario Dolfi, Elisabetta Brunello, Luca Fusi 1, Massimo Reconditi, Pasquale Bianco, Marco Linari, Vincenzo Lombardi ⇑ Laboratory of Physiology, Department of Biology, University of Florence, Via G. Sansone 1, 50019 Sesto Fiorentino (FI), Italy

a r t i c l e

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Article history: Received 4 November 2013 and in revised form 5 February 2014 Available online xxxx Keywords: Skeletal muscle Muscle contraction Myofilament elasticity Myosin motor Cross-bridge kinetics

a b s t r a c t The half-sarcomere is the functional unit of striated muscle, in which, according to a ‘‘linear’’ mechanical model, myosin motors are parallel force generators with an average strain s acting between the opposing myosin and actin filaments that behave as a series elastic element with compliance Cf. Thus the definition of the mechanism of force generation by myosin motors in muscle requires integration of the crystallographic model of the working stroke with the mechanical constraints provided by the organization of motors in the half-sarcomere. The relation between half-sarcomere compliance and force (Chs–T) during the development of isometric contraction deviates, at low forces, from that predicted by the linear model, indicating the presence of an elastic element in parallel with the myosin motors, which may influence the estimate of s. A working stroke model, kinetically constrained by the early phase of the isotonic velocity transient following a force step, predicts that the rate of quick force recovery following a length step is reduced to the observed value by a Cf of 12.6 nm/MPa. With this value of Cf, the fit of Chs–T relation during the isometric force rise gives s = 1.8–1.9 nm, similar to the values estimated using the linear model. Ó 2014 Elsevier Inc. All rights reserved.

Introduction Force in the muscle sarcomere is generated by the head (Subfragment 1 or S1) of the motor protein myosin II extending from the thick filament and attaching to the thin actin filament to form cross-bridges that pull the thin filament toward the center of the sarcomere through an ATP driven structural working stroke. Crystallographic studies suggest that the working stroke consists of a 70° tilting of the light chain domain (LCD) of the myosin head about a fulcrum in the actin-attached catalytic domain (CD), corresponding to 11 nm movement at the junction between the LCD and the rod-like subfragment 2 (S2) that connects the cross-bridge to the thick filament (Fig. 1A). However, the extent of the motor movement for force generation during isometric contraction cannot be defined with in vitro studies, as it depends on the stress experienced by the motor while it undergoes the structural transition. The question can be ⇑ Corresponding author. Fax: +39 0554572387. E-mail addresses: gabriella.piazzesi@unifi.it (G. Piazzesi), mario.dolfi@unifi.it (M. Dolfi), elisabetta.brunello@unifi.it (E. Brunello), [email protected] (L. Fusi), massimo.reconditi@unifi.it (M. Reconditi), pasquale.bianco@unifi.it (P. Bianco), marco.linari@unifi.it (M. Linari), vincenzo.lombardi@unifi.it (V. Lombardi). 1 Present address: Randall Division of Cell and Molecular Biophysics, King’s College London, London SE1 1UL, UK.

addressed by determining the elastic characteristics of the halfsarcomere, the functional unit in which myosin motors and myofilaments are organized in a complex series–parallel network. According to the analysis by Ford et al. [1], this complex network can be reduced to a ‘‘linear’’ mechanical model, in which myosin motors are parallel force generators between the opposing myosin and actin filaments that act as elastic elements in series (model 1 in Fig. 1B). The compliance of the actin and myosin filaments has been estimated in frog skeletal muscle (either whole muscle or single fibers) by measuring their force-extension relation with X-ray diffraction [2–6]. From these measurements, the filament compliance (Cf)2 functionally in series with the array of myosin motors could be calculated [1] and was found to be more than 50% of the half-sarcomere compliance (Chs). The average strain of the myosin motors (s) during 2 Abbreviations used: Ccb, compliance of the myosin motor array at T0; Cf, compliance of the myofilaments; Chs, compliance of the half-sarcomere; CP, compliance of the element in parallel to the myosin motor array; e, stiffness of a myosin motor; f, structural transition between two consecutive states of the myosin motor; L, change in length of the half-sarcomere; M1–M5 states of the attached myosin motor; PM, signal from the position of the lever connected to the motor-length trasducer; r2, rate of quick force recovery; s, average motor strain; T, force; T0, plateau force in an isometric tetanus; V2, the initial velocity of the isotonic velocity transient; x, relative position between the myosin motor and the actin filament; y, strain of the motor in a given state; Y, half-sarcomere strain; z, change in axial position of attached motors.

http://dx.doi.org/10.1016/j.abb.2014.02.017 0003-9861/Ó 2014 Elsevier Inc. All rights reserved.

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Fig. 1. Mechanical model of the half-sarcomere. A. Schematic representation of the half-sarcomere in which actin and myosin compliances are lumped into two springs in series with the array of myosin motors arranged in parallel. B. Mechanical models of the elements contributing to the half-sarcomere compliance. Model 1: actin and myosin compliances are represented as a spring with compliance (Cf) in series with the array of parallel elastic elements with a constant strain s, representing the attached myosin motors (two only are shown for simplicity). The compliance of the array is s/T, and is inversely proportional to the level of isometric force exerted by the array that increases linearly with the number of attached motors. Model 2: an elastic element with compliance CP, independent of the isometric force T, is added in parallel with the array of motors.

isometric contraction (force T0) was therefore less than half of the half-sarcomere strain (<2 nm). If the isometric force developed by a myosin motor is 5 pN, as suggested by a series of mechanical, structural and energetic studies on single fibers and by single molecule experiments [6–9], then the stiffness (e) of the individual cross-bridge that links the thick and thin filaments is >(5/2=) 2.5 pN/nm. Here following we show how the mechanical parameters defined in single muscle fibers from frog skeletal muscle constrain the model of force generation and how an independent estimate of the filament compliance, obtained with the analysis of mechanical transients, supports the view that myosin motors in isometric contraction generate force with an average strain <2 nm, due to a distribution of cross-bridge states biased to the beginning of the working stroke. The elastic properties of the half-sarcomere constrain the model of force generation by myosin motors The hypothesis that the isometric force is generated by a mechanism based on the all-or-none 11 nm movement implies that the fraction of attached motors responsible for force is less than (2/ 11=) 0.18, while the remaining fraction of the attached motors (larger than 0.82) remains in a pre-working stroke conformation. However this mechanism seems unlikely on thermodynamic grounds, because the mechanical energy implied in the generation of isometric force by an actin-attached motor (Em = ½ef2, where f is the size of the force generating step) at 4 °C would be larger than 150 zJ, or 40 kBh (where kB is the Boltzmann constant and h is the absolute temperature). This is almost twice the free energy released during the hydrolysis of one ATP molecule (83 zJ, [10]), and the equilibrium constant of the force generating transition would

be too low to account for the isometric force. Moreover an all-ornone 11 nm working stroke implies a dispersion of conformations of the myosin heads attached in the isometric contraction larger than that indicated by X-ray diffraction studies [5,6]. As far as the kinetics of the working stroke is concerned, a foundation stone is represented by the experimental and theoretical work of Huxley and collaborators who first developed a model of force generation based on stepwise structural changes in the actin-attached myosin head controlled by strain-dependent rate functions inferred from the force responses to sudden stretches and releases [11]. In the original model it was assumed that the myosin motors are the only source of compliance in the halfsarcomere, so that, according to the relatively large value of the half-sarcomere strain estimated at that time, the working stroke could be explained by a single state transition in the attached head. Improving the performance of single fiber mechanics, it became evident that the half-sarcomere strain (and thus the motor strain) is at least three times smaller than the sliding accounted for by the working stroke [1,12]. This suggested that the working stroke could be composed of more than one step, though the concept of a tight coupling between mechanical steps and biochemical steps (essentially the release of ATP hydrolysis products orthophosphate and ADP) of the chemo-mechanical cycle was maintained [13]. A further reduction of the myosin motor compliance by a factor of two became evident following X-ray diffraction demonstration that the compliance of the myofilaments could account for more than ½ of the half-sarcomere compliance [2,3]. The progressive reduction in motor compliance was taken into account in an extension of the original model, in which, following previous modeling work [14,15], three states, separated by two structural steps of 5 nm, were assumed [16]. Eventually on the basis of energetic, structural and biochemical studies [4–7,17] the alternative idea emerged that the biochemical steps and the mechanical steps are not tightly coupled [18,19] and the ATP hydrolysis cycle can terminate and the motor detach from actin at an intermediate stage of the 11 nm working stroke. According to this idea, the small average strain of the cross-bridges responsible for the isometric force (s < 2 nm) reflects a limited dispersion of motor conformations characterized by a narrow distribution of LCD angles biased towards the beginning of the working stroke. In other words, the force developed by the halfsarcomere during isometric contraction is the sum of the relatively constant force contributions by the actin-attached myosin heads which have undergone a structural change that is a small fraction (<1/5) of the 11 nm working stroke. During steady shortening against a load
The compliances of the myofilaments and myosin motors according to the linear mechanical model of the half-sarcomere Strong support for the idea that isometric force in a muscle fiber is proportional to the number of attached motors, in which the strain and thus the force is constant, came from sarcomere-level mechanical experiments in both rabbit skinned fibers and frog intact fibers. In these experiments, the relation between half-sarcomere stiffness and isometric force was determined while force changed either with [Ca2+] (skinned fibers, [20]) or during the development of isometric tetanus (intact fibers, [21,22]). Under these conditions it was found that the half-sarcomere strain (Y) increases with force (T) in proportion to the increase of myofilament

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Fig. 2. Half-sarcomere strain and compliance during tetanus rise in fibers from R. temporaria (A and B and triangles in E) and R. esculenta (C and D and circles in E). A, time course of force (T, thick line), stiffness (e, triangles) and half-sarcomere shortening (nm, thin line) during the development of an isometric tetanus in a muscle fiber from R. temporaria. Force and stiffness are relative to the tetanic plateau values T0 and e0, respectively. Zero time is the start of stimulation. The insets show force (relative to T0) and length change per half-sarcomere (nm) during 4 kHz sinusoidal length changes imposed 35 and 280 ms after the start of stimulation when force was 0.25 T0 and T0. Stiffness was measured as the ratio between the in-phase component of force and half-sarcomere length responses to oscillations. From Fig. 1A and B of Brunello et al. [21]. B. Halfsarcomere strain (Y, triangles) plotted against relative force; the slope of the straight line fitted to these points, 2.6 ± 0.1 nm/T0, estimates the filament compliance and its ordinate intercept, 1.8 ± 0.1 nm, the average strain of the motors. Values are mean ± SD. From Fig. 1D of Brunello et al. [21]. C. Force (upper trace) and half-sarcomere length change (lower trace) during the development of an isometric tetanus in a muscle fiber from R. esculenta with superimposed 4 kHz length oscillations (2 nm hs1 peak to peak) applied starting from 0.2 T0. Stiffness was calculated in correspondence of the vertical bars above the force trace. The insets show force response (upper trace) and length oscillations (lower trace) on a faster time scale at 0.3 T0 and T0, as indicated by the arrows. From Fig. 3A of Fusi et al. [22]. D. Half-sarcomere strain (circles) plotted against force relative to T0 from the fiber in C. The continuous line is the linear regression on filled circles (slope 1.77 ± 0.05 nm/T0 and ordinate intercept 1.83 ± 0.04 nm). From Fig. 3B of Fusi et al. [22] (modified adding the point collected at 0.2 T0). E. Half-sarcomere strain data from panels B (triangles, R. temporaria) and D (circles, R. esculenta) plotted against force in SI units. The dashed line is the linear regression to the strain-force data for forces larger than 50 kPa: slope, 11.9 ± 0.3 nm/MPa, ordinate intercept, 1.82 ± 0.03 nm. F. Half-sarcomere compliance–force relation. Same data as in panel E. Dashed line: relation expected from the linear model 1. Continuous line: all data fitted with model 2, after fixing filament compliance to 12.6 nm/MPa as explained in the text.

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(triangles data from R. temporaria, circles data from R. esculenta). It can be seen that data from both species superimpose and a unique regression line can be fitted for T > 50 kPa, while the downward deviation appears only below a given threshold stress on the filaments, which in these experiments occurs only in fibers from R. esculenta, as they have a 30% lower T0. The fit of the linear part of the Y–T relation obtained by merging data from the two frog species gives Cf = 11.9 ± 0.3 nm/MPa and s = 1.82 ± 0.03 nm.

strain, according to the linear mechanical model of Fig. 1B, where Y is given by:

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with s the average strain of the array of myosin motors. Experiments were done in single intact fibers from frog skeletal muscle (at 4 °C and 2.15 lm sarcomere length) electrically stimulated to obtain fused tetanic contractions. As shown in Fig. 2B and D, the half-sarcomere strain-force (Y–T) relation, determined by superimposing 4 kHz sinusoidal oscillations on the force development during an isometric tetanus (Fig. 2A and C), was found to be linear for forces larger than 0.3 T0, with a slope and an ordinate intercept that, according to Eq. (1), estimated Cf and s respectively. Actually, while the estimates of s from the ordinate intercept is not significantly different in fibers from Rana temporaria (1.8 ± 0.1 nm, Fig. 2B from Brunello et al. [21]) and from Rana esculenta (1.76 ± 0.05 nm, from Fusi et al. [22], Fig. 2D and two more fibers), the slope of the regression line that estimates Cf is 2.6 ± 0.1 nm/T0 in R. temporaria and 1.91 ± 0.07 nm/T0 in R. esculenta. Moreover, while the relation in R. temporaria is linear over the whole range of force studied (from 0.25 T0 to T0), in R. esculenta the relation deviates from linearity at forces <0.4 T0, as shown by the point at 0.2 T0, which lies below the regression line fitted to filled circles. As first observed by Barclay et al. [10], the apparent contradictions are resolved by taking into account the actual force exerted on the filaments, that is calculating Cf in SI units. The value of T0 was 217 ± 28 kPa in the fibers from R. temporaria, while it was 152 ± 8 kPa in the fibers of R. esculenta. The corresponding Y–T relations in SI units are merged in Fig. 2E

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The influence of a non-linear element on the mechanical parameters of the half-sarcomere The conclusions based on the linear mechanical model of the half-sarcomere (model 1 in Fig. 1B) have been challenged by taking the downward concavity of the half-sarcomere Y–T relation at forces <50 kPa (Fig. 2E) as an evidence for the presence of an elastic element with constant stiffness in parallel with the force-generating motors [23] (model 2 in Fig. 1B). This parallel elasticity was attributed to weakly-bound myosin heads, and implied that the calculation of the motor compliance using model 1 introduced a 40% underestimate of s, and a corresponding overestimate of Cf. However the study was conducted by using as length signal the position of the lever (PM) connected to the motor controlling the length of the fiber (tendons included), assuming that the length change at the level of the sarcomeres was a constant fraction of PM, that is assuming that the tendon compliance behaves just like the half-sarcomere compliance in the whole range of forces investigated. The conclusions based on stiffness measurements on the

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Fig. 3. Force and velocity transients. A. Early components of force transients elicited by steps of different size imposed at the steady state of the isometric contraction (T0). Upper trace is sarcomere length change (L), the straight horizontal lines are baselines for the force response to 2.5 nm (a) and 4.6 nm (b) step release. T1 and T2 indicate respectively the force attained at the end of phase 1 (1) and phase 2 (2). From Fig. 3A of Piazzesi and Lombardi [15]. B. Rate of early force recovery (r2) versus step amplitude. r2 is estimated as the reciprocal of time necessary to attain 63% of the T2 recovery. The lines are obtained by fitting the r2 data points with an exponential. From Fig. 3C of Piazzesi and Lombardi [15]. C. Isotonic velocity transients following a step in force superimposed on the isometric tetanic force (T0). Upper trace: stepwise drop to a force T = 0.5 T0; middle trace: force baseline; lower traces: change in half-sarcomere length (L) corresponding to steps to the forces indicated by the figures on the right. The figures on the left of the 0.1 T0 trace identify the phases of the transient. From Fig. 1 of Piazzesi et al. [45]. D. Relation between initial shortening velocity of phase 2 (V2) and force (T/T0). The continuous line is an exponential fit to data. From Fig. 3A of Piazzesi et al. [45].

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C hs ¼ C f þ s  C P =ðs þ C P  TÞ

ð2Þ

where CP is the constant compliance of the parallel element. The fit fails to give consistent results because there are too many free parameters (s, Cf, and CP) and there is only one point in the region (T < 50 kPa) where the effect of the element responsible for the non-linearity is more relevant. The problem is solved by getting an independent estimate of Cf from the analysis, described in the following sections, of the effect of filament compliance on the kinetics of quick force recovery elicited by step releases. With Cf estimated in this way (12.6 nm/MPa), the fit of Eq. (2) to the whole set of half-sarcomere compliance – force data in Fig. 2F (continuous line), gives CP = 311 ± 55 nm/MPa and s = 1.88 ± 0.04 nm, which is slightly, but not significantly (P > 20%) higher than s estimated by fitting model 1 to data for T > 50 kPa. The compliance of the motor array at T0 (Ccb) in SI units is (1.88/0.217=) 8.7 nm/MPa in R. temporaria and (1.93/0.152=) 12.4 nm/MPa in R. esculenta. Thus Ccb is 40% larger in R. esculenta than in R. temporaria, in agreement with the reported difference in the number of myosin motors at T0 [22,27], but both Ccb values are 30 times smaller than CP. Such a large value of the compliance of the parallel elastic element with respect to that of the motor array at T0 explains why the effect of the parallel element becomes evident only at low forces, where its compliance is comparable to that of the motors. As far as the nature of the parallel element we do not have enough elements for its identification. It could be due to the filamentous protein titin [28–30], as well as to weakly bound cross-bridges [23]; but it could also be due to the transient presence, at the start of activation, of links between actin and myosin filaments attributable to myosin binding protein C [31–36].

larger releases (compare a and b force responses in Fig. 3A) and this is a fundamental constraint in the Huxley and Simmons theory of force generation [11], because it indicates that the mechanical energy involved in the state transition is part of the free energy barrier, so that the transition rate constant increases if the drop in mechanical energy is larger. The strain dependent kinetics of the process is shown by plotting the rate of the quick force recovery (r2), measured by the reciprocal of the time to recover from T1 to T1 + 0.63  (T2 – T1), against the step size (Fig. 3B, from Piazzesi and Lombardi [15]). r2 increases roughly exponentially from 0.43 ms1 for the 1.8 nm stretch to 2.04 ms1 for the 3.2 nm release. The finding that a large proportion of the compliance of the half-sarcomere (>50%) is in the actin and myosin filaments ([2–4,42]; this work) made evident the limits of the length step protocol to determine the kinetics of the myosin working stroke. In fact, the presence of a significant myofilament compliance in series with the myosin motors slows down the rate of force recovery for any given step release imposed on the half-sarcomere, because (i) the drop in motor strain induced by the release is substantially less than that expected from the size of the half-sarcomere length step, and (ii) the myosin motors move in the direction of half-sarcomere shortening during force recovery to reload the filament compliance. The alternative protocol, first introduced by Podolsky [43], consists in recording the isotonic velocity transient in response to a stepwise reduction in force and avoids the limits discussed above, because any series elasticity in the fiber maintains a constant length after the step. However, the force step protocol was not pursued because it was found difficult to clamp

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fiber (tendon included) would be wrong if the tendon compliance, as it seems likely, increases more than half-sarcomere compliance at low forces [24–26]. The Chs–T relation obtained from the same experiments as in Fig. 2E is reported in Fig. 2F. It can be seen that the point (open circle) below 50 kPa is shifted downward with respect to the relation predicted by the linear model (dashed line, from the linear fit in Fig. 2E), because in this range of forces the progressive reduction of the stiffness of the motor array allows the presence of the parallel element responsible for the non-linearity to be revealed. According to model 2 (Fig. 1B), the whole range of Chs–T data in Fig. 2F can be fitted with the equation:

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Huxley and collaborators [1,11,12,37] have shown that the kinetics of the myosin working stroke can be studied in situ by recording the early phases of the transient responses to step perturbations in length or force controlled at the level of the half-sarcomere. Later experiments, including double step protocols [38] and time-resolved X-ray diffraction on single frog fibers [4,39– 41] have demonstrated that during the first few milliseconds following the step there is no substantial detachment and reattachment of myosin motors. Under these conditions the mechanical and structural responses elicited by the step are due to the motors that were attached before the step. As shown in Fig. 3A, a stepwise reduction in half-sarcomere length superimposed on the isometric tetanus produces a drop to a force T1 synchronous with the length step (phase 1 of the transient), due to the elasticity of myosin motors and myofilaments, followed by a quick force recovery to a force T2 (phase 2), which is the mechanical manifestation of the working stroke in the myosin motors synchronized by the drop of the mechanical component of the free energy difference between subsequent states. The quick force recovery is faster for

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the force in the few hundreds microseconds following the step, until the introduction of a 50 kHz force transducer [44] and the refinement of the procedures for the minimization of the compliance and non-linearity of tendon attachments [45]. As shown in Fig. 3C, a stepwise drop in force superimposed on the isometric tetanus produces an elastic shortening of amplitude L1 (phase 1) followed by an early rapid shortening (phase 2), which occurs under isotonic conditions and therefore records directly the inter-filamentary sliding induced by the synchronous execution of the working stroke by the attached myosin motors. The velocity next decreases to a minimum value (phase 3) and then increases to attain the steady value (phase 4), characteristic of the force–velocity relation, one order of magnitude lower than that in phase 2 [45]. Phases 3 and 4 are due to motor detachment/attachment and thus are not considered here. The speed and the amplitude of phase 2 shortening increase with the reduction of the clamped force. However the amplitude of the shortening attained at the end of phase 2, which represents the amplitude of the working stroke, is also influenced by the ensuing detachment, the rate of which increases with reduction of the force [4,19]. A reliable parameter for estimating the kinetics of phase 2 was found to be the initial shortening speed (V2) measured by the tangent to the initial part of phase 2 shortening. V2 increases from 0.9 lm/s per half-sarcomere for 0.81 T0 to 12.0 lm/s per half-sarcomere for 0.10 T0 (Fig. 3D, from [45]).

the myosin II motors, because the early rapid shortening in phase 2 depends only on the strain-dependent kinetics of the myosin working stroke [7,19,45]. Thus the first step is to select, with the model simulation, the rate equations of the state transitions that best fit the early velocity transient and its dependence on the load, and then apply the same kinetic scheme to the simulation of the quick force recovery during phase 2 of the force transient, to test the effect of filament compliance on the kinetics of force generation. In turn, fitting the force transient with the kinetic scheme selected for the velocity transient simulation implies that the only adjustable parameter is the filament compliance, which can be compared with the estimates obtained from stiffness measurements. The model concerns only the mechanical and kinetic properties of the attached myosin motors that determine the early phases of the responses to length and force steps, thus the kinetic scheme is based on Huxley–Simmons model [11] (see Appendix) and does not include further steps of the actin–myosin interaction, like attachment and detachment of motors. The simulation is constrained by the half-sarcomere mechanical parameters obtained from studies on fibers from R. esculenta at 4 °C. During isometric contraction the average strain in the attached myosin motors s is 1.7 nm and the force per motor 5 pN [7,27], thus the stiffness of the myosin motors (e) is 2.9 pN/nm. As discussed in several previous studies [10,14–16,18,46–48] and detailed in the Appendix, the value of e constrains the size of the structural step, f, between successive states of the working stroke: with e = 2.9 pN/nm, f is 2.75 nm and the number of steps is 4 in agreement with Linari et al. [46]. Since the simplified kinetic scheme does not imply detachment/ attachment, the simulation of the velocity transients (Fig. 4A) ends with the equilibrium distribution of attached states at each

Model simulation of the transient responses to step perturbations in length and force; the force transient as a tool to estimate the filament compliance The isotonic velocity transient provides quite stringent constraints for a mechanical–kinetic model of the in situ action of

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Fig. 5. Model simulation of force transients. A. Simulated force transient following a step release of 1 nm assuming Cf = 0 nm/T0. Upper trace, length change taken by the motors (z), which in this case corresponds to the imposed length step. B. Simulated force transient following a step release of the same total amplitude of 1 nm, assuming a Cf = 2 nm/T0. In this case z at the end of the step is reduced to 0.45 nm. C. Time course of force recovery from A and B after normalization by T2  T1. D. Filled symbols and continuous lines: simulated relations between rate of early force recovery and length change in the absence of filament compliance, triangles, and in the presence of 1.7 nm/T0 (squares), 2 nm/T0 (filled circles), 2.3 nm/T0 (diamonds) filament compliance. Open circles are experimental data from Fig. 3B.

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G. Piazzesi et al. / Archives of Biochemistry and Biophysics xxx (2014) xxx–xxx

clamped force. The initial velocity of shortening following the force step is measured in the simulated response with the same tangent method as that used for the experimental records and plotted versus T in Fig. 4B (filled circles connected by continuous line). The set of rate equations selected to simulate the isotonic velocity transient is applied without any modification to predict the quick force recovery following length steps. The size of the length step is kept small enough to justify the assumption that the force recovery is not influenced by detachment/attachment of motors. Moreover, the simulation is applied only to responses to step releases, given the evidence that, following a stretch, rapid attachment of myosin motors to actin occurs in the submillisecond time scale [49]. The first step in the force transient simulation is to assume that the contribution of filament compliance is zero, in which case the amplitude of the length step L corresponds to the displacement of the isometric distribution of attached motors (z), and the subsequent force recovery occurs without any further change in z. For a step release of 1 nm, the force drops to 0.41 T0, in agreement with a stiffness of the attached motor population of (1/1.7=) 0.59 T0/nm, and then nearly recovers the isometric value within 1 ms (Fig. 5A). The rate of force recovery (r2) is quantified using the same method as for the experimental records, namely by determining the reciprocal of the time to 63% of force recovery. The recovery is faster for larger releases, as shown by the relation between r2 and L (triangles joined by the continuous line in Fig. 5D). It can be seen also that the r2–L relation simulated assuming zero filament compliance is shifted upward with respect to the observed r2–L relation (open circles) by a factor which increases from 3 to 5 by increasing the size of the release. The next step in the simulation is to identify which is the value of filament compliance that provides the best fit of the experimental r2–L relation. During isometric contraction, at the steady force T0, the filaments are under constant strain and thus the distribution of the attached motors is the same as that with zero filament compliance (see Appendix, Fig. A1C). When the isometric contraction is perturbed by a step release of size L, which brings the force to T1, the displacement z induced on the distribution of attached motors is a fraction of L (for a description of the calculation see Appendix). In the simulation shown in Fig. 5, the total compliance of the half-sarcomere (3.7 nm/T0) is composed of a 55% (2 nm) contribution from the myofilaments and 45% from the motor array (1.7 nm). Thus for a step release L = 1 nm, z = 0.45 nm. During force recovery from T1 to T2, z continues to increase because the increase in force implies a movement of the myosin motors in the shortening direction to lengthen the filaments. If T2 = T0, z at the end of the force recovery becomes as large as the imposed length step. Shortening during the force recovery slows down the force recovery rate, as shown in Fig. 5C by superimposing the time course of force responses from A and B normalized for the amount of force recovery (T2  T1). The effect is larger with larger filament compliance; filled symbols joined by continuous lines in Fig. 5D show the simulated r2–L relation with different contributions of Cf: 1.7 nm/T0 (squares), 2.0 nm/T0 (circles) and 2.3 nm/T0 (diamonds). It can be seen that the experimental r2–L relation (open circles) is best fitted with Cf = 2 nm/T0. Considering that T0 in these experiments on fibers from R. esculenta ranged from 152 kPa [22] to 165 kPa [7], the estimate of Cf in SI units obtained with this procedure is restricted to a range of values from 13.2 to 12.1 nm/MPa, in good agreement with the estimates obtained from mechanical and X-ray diffraction studies. The average value, 12.6 nm/MPa is used to constrain Cf in the fit of the Chs–T relation of Fig. 2F (continuous line).

7

Conclusions  The definition of the mechanism of force generation by myosin motors in muscle requires integration of the crystallographic model of the working stroke with the elastic characteristics of the functional unit of muscle, the half-sarcomere. According to a ‘‘linear’’ mechanical model, constrained by a bulk of mechanical and X-ray diffraction evidence, myosin motors in each half-sarcomere are parallel force generators between the opposing myosin and actin filaments acting as a series elastic element with a compliance (Cf) that accounts for more than one half of the half-sarcomere compliance.  The relation between half-sarcomere compliance and force (Chs–T relation) during the development of an isometric contraction indicates the presence of an elastic element in parallel with the myosin motors. In turn the compliance of this element (CP) influences the mechanical estimates of Cf and of the average strain (s) of force-generating myosin motors.  The early phases (phase 1 and 2) of the transient following a step in length or force superimposed on isometric contraction reveal the elastic and kinetic properties of the actin-attached myosin motors. The early rapid phase of the velocity transient (phase 2) following a stepwise drop in force occurs in isotonic conditions and thus is not influenced by the myofilament compliance and is a direct measure of the strain-dependent kinetics of the working stroke.  A mechanical-kinetic model of the working stroke constrained by the phase 2 velocity transients (single fibers of R. esculenta at 4 °C) predicts that the rate (r2) of the quick force recovery following a step release in the absence of filament compliance would be 4 times higher than that measured from the quick force recovery following a step release, and is reduced to the observed value by a Cf of 2 nm/T0. In SI units this corresponds to 12.6 nm/MPa, in quite good agreement with values reported in previous mechanical and X-ray diffraction studies.  This value of Cf from the kinetic model is used as a constraint to fit the Chs–T relation determined during the rise of isometric contraction with a mechanical model that includes an elastic element in parallel with the myosin motors. The resulting estimate of the compliance of the parallel element CP is 311 ± 55 nm/MPa and the average strain in the motors s is 1.88 ± 0.04 nm. These results demonstrate that the average strain of motors during isometric contraction is not significantly different from the value obtained from the linear model.  An average strain of the force generating motors <2 nm results from the weighted mean of the first three of a five-state working stroke model, in which the step size of the transition between consecutive states is 2.75 nm.

Appendix Model simulation The simulation is based on an earlier version [46], in which the Huxley and Simmons model of force generation was integrated with filament compliance, and is adapted to the half-sarcomere mechanical parameters obtained from studies on fibers from R. esculenta at 4 °C, as reported in Figs. 2B and 3. The stiffness of the myosin motor is e = 2.9 pN/nm and this sets the slope of the parabolas representing the free energy profile of attached states and thus the number of force-generating steps necessary to fit the transition and equilibrium kinetics of the working stroke [7,15,16,47,48]. Once the horizontal distance between minima of neighboring

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8

G. Piazzesi et al. / Archives of Biochemistry and Biophysics xxx (2014) xxx–xxx

states, f (the step size), is set to 2.75 nm, the maximum working stroke of 11 nm is preserved if the number of states is five (M1–M5, Fig. A1A). The difference in the minimum of free energy between neighboring states (DG0) is the same for all the states and is 12.5 zJ, so that the free energy parabola of the next state intersects that of the preceding state at its minimum, and the maximum mechanical energy of the working stroke is (12.5  4=) 50 zJ. According to the principle of nearest-neighbor interaction, attachment to the actin monomer (approximately 5.5 nm in diameter) of a myosin motor in the first of the five states (M1) occurs for a range of x from –2.75 to 2.75 nm, where x is the relative axial position between the motor and the actin monomer, and is zero for the position of the center of the distribution of attachments of the motors in the M1 state. At the same x, the strain depends on the conformation, increasing by 2.75 nm for each state transition M1–M2, M2–M3, M3–M4 or M4–M5. Therefore, at any position x, the total strain on the motor in the Mi state is given by yi = (fi + x), where i is 0, 1, 2, 3, 4 for the states M1, M2, M3, M4, M5, respectively and the average strain of the motors s is the weighted mean of the strain in each state.

A

100

Free energy (zJ)

M5

M4

M3

M2

M1

80 60 40 20 0 -15

-10

-5

0

5

x (nm)

-1

Rate constant (s )

B

40000 30000 20000 10000 2000

4

2

3

1

0 -15

State occupancy

C

-10

-5

0

5

x (nm) 1.0 M1 M2 M3

0.8 0.6 0.4 0.2 0.0 -4

-2

0

2

4

x (nm) Fig. A1. Relevant parameters of the kinetic model. A. Free energy diagrams of the myosin motor states (M1–M5) as a function of the relative position (x) between the myosin head and the actin monomer. x is set to zero for the minimum free energy of an attached motor in state M1. The thick line marks the axial distribution of motors during the isometric contraction (range 2.75 to + 2.75 nm). B. Functions expressing the x-dependence of the rate constants for the forward (continuous lines) and the backward (dashed lines) transitions. The figures indicate the order of the transition. C. Occupancy of the various states as a function of x during the isometric contraction.

The forward and reverse rate constants (ki(x) and k–i(x), where i = 1, 4) for each of the four transitions are related to the change in free energy between neighboring states (DG(x)) through the equation:

ki ðxÞ  ¼e ki ðxÞ

DGðxÞ kb h

ðA1Þ

The x-dependence of the rate functions for the forward transition between two consecutive states is expressed according to the formalism introduced by Slawnych et al. [50] as an implementation of Huxley and Simmons formalism [11] (Fig. A1B):

ki ðxÞ ¼ Ei 

e



efðxxmi Þ kb h



1þe

efðxxmi Þ

ðA2Þ

kb h

where ei is the maximum value of the rate for the transition i and xm is the midway position between two consecutive states. The rate function for the reverse transition is calculated from Eq. (A1). Since e, f and DG0 are constrained as explained above, the only free parameters in the simulation are the Ei values for the four forward transitions and the first step in the simulation is to select the sets of values that best fit the time course of the early rapid shortening and its dependence on the clamped force. The four Ei values selected are reported in Fig. A1B. In isometric contraction the relevant state occupancy is limited to the first two states (Fig. A1A and C), due to the high stiffness of the motor. Significant occupancy of the other states occurs only when transitions through the working stroke are allowed by dropping the mechanical energy barrier represented by the isometric stress in the motors, with either a length or load release [7]. The algorithm for the simulation of the force transient in Piazzesi and Lombardi [15] was implemented so that the isotonic condition was applied following the stepwise displacement of the isometric distribution of the attached motors by the amount corresponding to the desired drop in force. The velocity V2 is estimated, as in the experimental records, by the slope of the tangent to the initial part of the simulated shortening. The set of rate equations selected to simulate the isotonic velocity transient is applied without any modification to the simulation of quick force recovery following length steps. The first step in the force transient simulation is to assume that the contribution of filament compliance is zero, in which case the amplitude of the length step L corresponds to the displacement of the isometric distribution of attached motors (z) and the subsequent force recovery occurs without any further change in z. The next step in the simulation is to assume various amounts of filament compliance. During isometric contraction, at the steady force T0, the filaments are under constant strain and thus the distribution of the attached motors is the same as that with zero filament compliance (Fig. A1C). After a step release of size L, which brings the force to T1, the displacement of the distribution of attached motors (z) is given by the equation: z1 = z0 + [L  Cf(T1  T0)]. During force recovery from T1 to T2, z is calculated at time intervals Dt of 10 ls with the equation zt = z(t  Dt) + Cf(T(t  Dt) – Tt). The rate of quick recovery is estimated on the simulated force transient with the same method as for the experimental records, that is calculating the reciprocal of the time to recover from T1 to T1 + 0.63  (T2  T1). References [1] L.E. Ford, A.F. Huxley, R.M. Simmons, J. Physiol. 311 (1981) 219–249. [2] H.E. Huxley, A. Stewart, H. Sosa, T. Irving, Biophys. J. 67 (1994) 2411–2421. [3] K. Wakabayashi, Y. Sugimoto, H. Tanaka, Y. Ueno, Y. Takezawa, Y. Amemiya, Biophys. J. 67 (1994) 2422–2435.

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