Nonlinear Cross-Bridge Elasticity and Post-Power-Stroke Events in Fast Skeletal Muscle Actomyosin

Biophysical Journal Volume 105 October 2013 1871 1881 1871 Nonlinear Cross-Bridge Elasticity and Post-Power-Stroke Events in Fast Skeletal Muscle Actomyosin Malin Persson, Elina Bengtsson, Lasse ten Siethoff, and Alf Månsson* Department of Chemistry and Biomedical Sciences, Linnaeus University, Kalmar, Sweden ABSTRACT Generation of force and movement by actomyosin cross-bridges is the molecular basis of muscle contraction, but generally accepted ideas about cross-bridge properties have recently been questioned. Of the utmost significance, evidence for nonlinear cross-bridge elasticity has been presented. We here investigate how this and other newly discovered or postulated phenomena would modify cross-bridge operation, with focus on post-power-stroke events. First, as an experimental basis, we present evidence for a hyperbolic [MgATP]-velocity relationship of heavy-meromyosin-propelled actin filaments in the in vitro motility assay using fast rabbit skeletal muscle myosin (28 29 C). As the hyperbolic [MgATP]-velocity relationship was not consistent with interhead cooperativity, we developed a cross-bridge model with independent myosin heads and strain-dependent interstate transition rates. The model, implemented with inclusion of MgATP-independent detachment from the rigor state, as suggested by previous single-molecule mechanics experiments, accounts well for the [MgATP]-velocity relationship if nonlinear cross-bridge elasticity is assumed, but not if linear cross-bridge elasticity is assumed. In addition, a better fit is obtained with load-independent than with load-dependent MgATP-induced detachment rate. We discuss our results in relation to previous data showing a nonhyperbolic [MgATP]-velocity relationship when actin filaments are propelled by myosin subfragment 1 or full-length myosin. We also consider the implications of our results for characterization of the cross-bridge elasticity in the filament lattice of muscle. INTRODUCTION The power output of striated muscle varies with external load according to a characteristic load-velocity relationship (1,2), with molecular origin in the MgATP-driven cycling of an ensemble of actomyosin cross-bridges (2 7). Specifically, the maximum velocity (at zero load) is believed to be directly proportional (7 9) to the rate of detachment of post-power-stroke cross-bridges. These have undergone their force-generating structural change, released inorganic phosphate, and, on average, been brought (by the action of other cross-bridges) into regions where they become negatively strained and resist shortening (exert a drag stroke). The proportionality between the rate of cross-bridge detachment and the unloaded sliding velocity follows directly from the idea (3) that forces due to cross-bridges executing their power-stroke directly balance resisting forces of drag-stroke cross-bridges (see below for details). Now, as the rate of cross-bridge detachment is expected to vary hyperbolically with the MgATP concentration, this would also apply to the sliding velocity (10 12). The basis for an expected hyperbolic behavior is very fast reversible binding of MgATP to the active sites of post-power-stroke crossbridges together with saturation of the active sites at Submitted May 14, 2013, and accepted for publication August 28, 2013. *Correspondence: alf.mansson@lnu.se This is an Open Access article distributed under the terms of the Creative Commons-Attribution Noncommercial License (http://creativecommons. org/licenses/by-nc/2.0/), which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited. Editor: Hideo Higuchi. close-to-physiological MgATP concentrations, further associated with the idea that MgATP binding is obligate for cross-bridge detachment. On these assumptions, the detachment rate is approximately proportional to the MgATP concentration at low [MgATP] (due to a proportional increase in active-site occupation) but asymptotically reaches a constant value when the MgATP concentration is increased to saturating levels. This constant value is determined by the slowest of the individual cross-bridge transitions (MgATPinduced detachment or MgADP dissociation) surrounding MgATP binding in the detachment process. In accordance with the above account, properties of the post-power-stroke cross-bridges are reflected (10 12) in the relationship between the MgATP concentration and the gliding velocity of actin filaments propelled by myosin. In further agreement with the theoretical expectations, the relationship has been found to be well approximated by a hyperbola of the Michaelis-Menten type in skinned skeletal muscle fibers (11 13), myofibrils (14), and a majority of the in vitro motility assay studies (15,16). However, recent results have called into question central ideas about operation of post-power-stroke cross-bridges in striated muscle. First, the shape of the [MgATP]-velocity relationship in recent in vitro motility assay experiments (17,18) with a particularly large number of data points at low [MgATP] deviates significantly from a hyperbola (see also Elangovan et al. (19)). Second, evidence has been presented that crossbridge elasticity may be nonlinear (20) instead of Hookean, with appreciably reduced stiffness in the drag-stroke region. Finally, the existence of velocity dependent interhead cooperativity in skeletal muscle myosin II has been Ó 2013 The Authors 0006-3495/13/10/1871/11 $2.00 http://dx.doi.org/10.1016/j.bpj.2013.08.044

1872 Persson et al. proposed (7,21,22), with expected complex effects on the [MgATP]-velocity relationship. In the latter connection, it is of interest that the bipyridine drug amrinone, which inhibits a strain-dependent ADP-release step (7,22) and affects the load-velocity relationship of skeletal muscle (23,24), increases the tendency for the two partner heads of myosin II to bind simultaneously to actin. Our aims are to address the emerging uncertainties, i.e., to investigate whether a hyperbolic [MgATP]-velocity relationship can be reproduced in the in vitro motility assay and whether the shape of this relationship is compatible with interhead cooperativity and nonlinear cross-bridge elasticity. The experimental basis of the study is carefully controlled in vitro motility assay experiments at different MgATP concentrations, including a large number of data points at low [MgATP], as in Hooft et al. (17) and Baker et al. (18). In these experiments, fluorescence-labeled actin filaments are propelled by heavy meromyosin (HMM) motor fragments (with two motor domains) adsorbed to silanized surfaces (25). This mode of surface adsorption of HMM is very well characterized (26 30), with a dominant fraction of the actin-propelling HMM molecules adsorbed to the surface via the C-terminal part of their subfragment 2 (S2) domain. This would be expected to give elastic properties similar to those of the S2 domain in thick filaments such as those used in the study of Kaya and Higuchi (20). We use amrinone in our studies as a pharmacological tool to enhance possible interhead cooperativity. Independent of conditions and in contrast to previous studies (17,18), our experimental results show that the [MgATP] velocity relationship is very well described by a Michaelis-Menten hyperbola. As we found no sign of interhead cooperativity (see, e.g., Månsson (7) and Albet-Torres et al. (22)), we developed a statistical cross-bridge model with independent myosin heads. This model is based on ideas presented in a previous study (7) but is further constrained by recent experimental findings (20,31 35). Thus, we explicitly included well-defined biochemical states at the end of the power stroke (31). In addition, we made the strain dependence of the MgATP-induced detachment-rate function (k 2 (x)) consistent with results of Capitanio et al. (34). Finally, we introduced a strain-dependent, MgATP-independent detachment (k rigor (x)) from an actomyosin rigor state, as suggested by Nishizaka et al. (32,33) (see also Guo and Guilford (35)). The model presented here faithfully reproduces a range of experimental findings, including loadvelocity data and the hyperbolic [MgATP]-velocity relationship. Indeed, for adequate predictions of the latter relationship, nonlinear cross-bridge elasticity seems to be required, as does the load-independent MgATPinduced detachment rate. We discuss ways to exploit our findings for characterization of post-power-stroke crossbridges in the three-dimensionally ordered filament lattice of muscle. We also consider how this model can be modified to form the basis for general insights into both normal striated muscle function and disease conditions (e.g., cardiomyopathies). MATERIALS AND METHODS Ethics statement All experiments using animal material were approved by the Regional Ethical Committee for Animal experiments in Linköping, Sweden (reference numbers 52-05, 58-08, and 96-11) and performed in accordance with national and European Union legislations. Protein purification Myosin was purified from rabbit fast leg muscle and chymotryptically cleaved to yield HMM (Fig. S8 in the Supporting Material) (36). Actin was from rabbit back muscle. Recombinant CapZ (36,37) was expressed in Escherichia coli, purified on a HiTrap column, biotinylated, and labeled with streptavidin-coated quantum dots (QDot 605, Invitrogen, Carlsbad, CA), as described previously (36,37). In vitro motility assays In vitro motility assays were performed as described previously (36,38) using trimethylchlorosilanized (TMCS) glass surfaces for HMM adsorption, except in control experiments, where the surface was coated with 1% nitrocellulose in amylacetate (Collodion, cat. no. 12620-50, Electron Microscopy Sciences, Hatfield, PA) (25). The HMM incubation concentration was 120 mg/ml, and bovine serum albumin was used as a blocking agent. The HMM concentration used is expected (25,27,39,40) to lead to saturated HMM density on a TMCS-derivatized surface (5000 7000 HMM mm 2 ). All solutions were based on buffer A (10 mm MOPS (3-morpholinopropane-1-sulfonic acid, ph 7.4), 1 mm MgCl 2, and 0.1 mm EGTA). Dilutions and washing steps utilized buffer A supplied with 50 mm KCl and 1 mm dithiothreitol (DTT). Assay solutions (ph 7.4) were based on buffer A with the addition of 115 mm KCl (final concentration), an oxygen scavenger system (3 mg/ml glucose, 20 units/ml glucose oxidase, 430 units/ml catalase, and 10 mm DTT), 2.5 mm creatine phosphate, 56 units/ml creatine kinase, and 0.6% (w/v) methylcellulose. The changes in ionic strength (between 125 and 140 mm) associated with variation in [MgATP] between 0.005 and 3 mm had a negligible effect on velocity, as verified in separate experiments. This finding is consistent with previous experiments on TMCS surfaces (36) at the present HMM incubation concentration. For the amrinone experiments, the assay solutions also contained 1 mm amrinone and 2.5 mm lactic acid (ph 7.4) (22). Mean temperature (measured in a water droplet on top of a flow cell after equilibration) varied between 27.9 and 29.2 C in different experiments. Actin filaments were labeled with rhodamine phalloidin (Invitrogen) or Alexa Fluor 488 phalloidin (Invitrogen) (41) and observed using an epifluorescence microscope (Eclipse TE 300, Nikon, Tokyo, Japan). Image sequences were obtained using an electron-multiplying charge-coupled device (EMCCD) camera (C9100-12, Hamamatsu, Hamamatsu City, Japan) at various frame rates (0.3 10 frames/s). Velocity measurements Filament velocities were measured using automatic tracking of the filament centroid in binary images (42), tracking of the leading end of the filament with the computer mouse (42), or tracking based on Gaussian fits (43) to images of quantum dots attached via CapZ to the trailing end of the filament (nanotracking) (36). When using the two former types of tracking (as done Biophysical Journal 105(8) 1871 1881

Nonlinear Actomyosin Elasticity 1873 in most experiments), we adjusted the frame rate (in the range 0.3 10 s 1 ) to the sliding velocity (15,42), as guided by tracking in pilot experiments using several different frame rates. This was important, because frame rates that are too low lead to underestimations of velocity due to truncation of the filament paths (15), whereas those that are too high cause overestimations of velocity (15,36,42) due to various sources of noise. In both versions, the algorithm searched for the 10 frames in a sequence where the coefficient of variation (CV) of the frame-to-frame velocity is lowest. The velocity data represent the mean value for several filaments of such 10-frame averages with CV < 0.5. In the nanotracking procedure, we could use a frame rate of 5 s 1 even at very low sliding velocities without noticeable errors (36). The algorithms were implemented in MATLAB version 8.0 (The MathWorks, Natick, MA). The manual algorithm was used for measuring velocities at various filament lengths for velocity-length plots and at R0.25 mm MgATP for [MgATP]-velocity relationships when only filaments longer than 2 mm were studied. Actin filament lengths were estimated from the filament intensity (Fig. S9 (39)). Data from three different HMM preparations were pooled, since there was no statistically significant difference between these data (Fig. S10). Data for each preparation were obtained on different experimental dates, without significant differences. Model simulations The model predictions were derived by numerically solving the appropriate system of differential equations (Eqs. S2 S7) and then calculating observable parameters for steady-state shortening using Eqs. S25 S27. The differential equations were solved using the Runge-Kutta-Fehlberg method, which is suitable for stiff differential equations. The algorithm was implemented in Simnon software (version 1.3; SSPA, Gothenburg, Sweden). To obtain [MgATP]-velocity relationships, velocity was iteratively adjusted for each given MgATP concentration to give an integrated motor force equal to zero (for [MgATP]-velocity relationships without external loads) or equal to an externally imposed load. Statistical analysis Curve fittings and one-way analysis of variance were implemented in GraphPad Prism v. 6.0 (GraphPad Software, San Diego, CA). Some Michaelis-Menten curve fitting was constrained to fit mean values only, without taking into account the scatter of individual data points. THEORY Several recent results from transient and steady-state kinetics studies of actomyosin (AM) in solution (44 46) are consistent with the reaction mechanism in Scheme 1. The lower and upper rows in Scheme 1 represent myosin (M) and actomyosin (AM) states, respectively, with substrate (MgATP, denoted ATP) or products (MgADP and inorganic phosphate, denoted ADP and P i, respectively) at the active site of myosin. The quantities K 1, K 3, etc., are equilibrium constants, whereas k 2, k 4, etc., are rate constants (compare Albet-Torres et al. (22) and Nyitrai et al. (31)). The gray highlighting indicates an AM*ADP state with a closed nucleotide binding pocket. Negative strain working on the lever arm is expected to accelerate the opening of the nucleotide pocket (with rate constant k 5 ) before MgADP leaves with rate constant k 6. The mean residence time in the AM*ADP state is prolonged in the presence of the drug amrinone (22), which may enhance interhead cooperative effects in skeletal muscle myosin II. Assuming a velocity-independent myosin power-stroke distance (h) and elastic properties of actomyosin crossbridges (3), it is expected (12) that sliding velocity is proportional to the cross-bridge detachment rate constant. This follows from force balance considerations. Thus, during in vitro sliding F þ ¼ F, where F þ is the average positive force due to the fraction of cross-bridges that promote shortening and F is a negative force due to the remaining crossbridges that have been brought into the drag-stroke region to resist shortening. The forces can be written as and F F þ ¼ k c c h (1a) ¼ k c x ; (1b) where k c and k c represent stiffness of the cross-bridge elasticity with average positive strain, c h (where c is a constant, 0 < c % 1) and average negative strain, <x >, respectively. The numerical value of the latter is given by x v f ¼ v f t on ð½mgatpšþ ¼ k diss ð½mgatpšþ ; (2) where v f is the in vitro sliding velocity and t on ([MgATP]) and k diss ([MgATP]) are the [MgATP]-dependent average waiting time and rate constant, respectively, for detachment of post-power-stroke cross-bridges. Now, making use of Eqs. 1a 2, and setting F þ ¼ F, it follows that the sliding velocity is proportional to the detachment-rate constant: v f ¼ k c ch ðt on ð½mgatpšþ k c Þ ¼ k cchk diss ð½mgatpšþ : kc Assuming that k c c/k c ¼ 1, the following hyperbolic relationship between velocity and [MgATP] is expected (see Introduction; (10 12)): v f ¼ hk diss ð½mgatpšþ ¼ hksat diss ½MgATPŠ K M þ½mgatpš ; (3) where kdiss sat is the value of k diss ([MgATP]) at saturating [MgATP]. The quantity K M is the MgATP concentration at which 50% of the post-power stroke cross-bridges have MgATP at their active site. Whereas K M is a function of K 1, k 2, k 5, and k 6 in Scheme 1 (12), kdiss sat is determined by k 2, k 5, and k 6 or, if the magnitude is very different for these separate rate constants, by the slowest of them (12,31,47). Biophysical Journal 105(8) 1871 1881

1874 Persson et al. We argue that limited processivity, with two sequential head actions on average (7) at certain MgATP concentrations, corresponds to an apparently larger step length (7) (see Supporting Material), giving an [MgATP]-dependent step length (Fig. S1 and Eq. S1). This would introduce systematic deviations from a hyperbola of the [MgATP]-velocity relationship (Eq. S1). Moreover, the drug amrinone would be expected to increase the degree of processivity (16,23,24,48) due to effects on interhead synchronization (7,22), as evidence for double-headed attachment of HMM to actin was found in the presence of 1 mm amrinone but not in its absence (22). For a more detailed representation of actomyosin crossbridge function, we adapted the cross-bridge model of Månsson (7) (for details, see Figs. S2 S7, Eqs. S2 S7, and accompanying text in the Supporting Material) for mammalian muscle at high temperature and constrained the model further by including the AMADP, AM, and AMATP states of Scheme 1, with rate functions defined based on experimental results in the literature ((31,33,34); Eqs. S8 S24). In this model, referred to below as the statistical model, the free energy of each cross-bridge state depends on the distance (x) from the myosin head-tail junction to the nearest binding site on the actin filament, where x ¼ 0 nm is defined as the distance with zero force in the AMADP state. The minimum free energy of each state corresponds to the free energy in solution (3,4,49), and for Hookean cross-bridge elasticity, the free energy varies with x according to a parabola (Fig. S2). The relationship between the model states, each existing for a distribution of strains, is indicated in Scheme 2, together with relevant interstate transitions. The states in Scheme 2 are analogous (but not identical, due to strain dependence) to those in Scheme 1, except in the case of state D, which corresponds to the detached states, MATP and MADPP i, in rapid equilibrium with weakly bound states. In contrast to Scheme 1, we have included an ATP-independent detachment step (governed by k rigor (x); Eq. S23) in some versions of the statistical model. This was done to accommodate experimentally observed forcedependent detachment from the AM state based on experiments in the absence of ATP (32,33,35). It is important to further constrain the model in this way on the basis of experimental data, particularly because the AM state becomes appreciably populated at negative x during shortening at low MgATP concentrations (see below). It is important to note that there is negligible detachment by this pathway at positive x due to a negligible population of the AM state. The reason no further ATP-independent detachment processes have been included, e.g., from AMADP and AM*ADP states, is explained below and in the Supporting Results and Discussion. The model is a single-site model, i.e., each myosin head is assumed to reach only one attachment site on the actin filament (49) and the neighboring sites are separated by 36 nm. We (7) and others (50,51) have previously considered the effects of this approximation. In the Supporting Material, we further consider the 36 nm spacing in simulations to test the importance of full equilibration of subspecies of the D state between neighboring sites during rapid shortening. The results of these simulations support the lumping together of all detached states into a single state, D, as well as the validity of the single-site model in other regards. It is important to note that we consider only low and constant concentrations of inorganic phosphate (<1 mm), which allows us to neglect the effects (17,52,53) of high phosphate concentrations that affect the [MgATP]-velocity curves by incompletely elucidated mechanisms. Unlike other transitions in Scheme 2, the transitions between the AMADP, AM, and AMATP states were assumed to be strain-independent and governed by the rate and equilibrium constants k 6, k 6, and K 1, respectively (Scheme 1). The strain independence is consistent with the structural similarity of these states, e.g., an open nucleotide pocket, similar angle of the lever arm, and free-energy minima for the same value of x. Detailed parameter values (Table S1), the relationship between free energies and rate functions (Eqs. S8 S10), and detailed x dependence of rate functions (Figs. S6 and S7 and Eqs. S12 S24) are given in the Supporting Material and in Fig. 1. Here, k diss 0 (x, [MgATP]) is the composite transition rate constant for the transition from the AM ADP(x) state to the D(x) state and k diss (x, [MgATP]) is the composite detachment rate function (depending on k 5 (x), k 6, K 1, and k 2 (x) and [MgATP]) from the AM*ADP(x) to D(x) via the AMADP(x), AM(x), and AMATP(x) states (Eqs. S20 S22). Incorporation of nonlinear cross-bridge elasticity into the model is implemented by setting the stiffness of negatively strained cross-bridges, k c, to 5% of the stiffness of positively strained cross-bridges, k c (Figs. S3 S5). The effects of this intervention and removal of strain dependence of MgATP-induced detachment (setting the Bell critical strain, x crit, to 0 nm) are illustrated in Fig. 1, B, D, and F. In particular, it can be seen that nonlinear cross-bridge elasticity appreciably reduces the rate of detachment (k rigor (x)) from the AM state for negative strains. RESULTS AND DISCUSSION We performed in vitro motility assays at different MgATP concentrations to gain insight into the properties of postpower-stroke cross-bridges. First, we performed experiments using either nitrocellulose-coated or TMCS-derivatized glass surfaces for HMM adsorption (Fig. S11). These experiments confirmed earlier findings (25,30) of higher HMM-driven actin filament velocity on TMCS but otherwise gave similar [MgATP]-velocity relationships for the two surfaces. We Biophysical Journal 105(8) 1871 1881

Nonlinear Actomyosin Elasticity 1875 FIGURE 1 Rate functions for cross-bridge transitions from the end of the power stroke (the AM*ADP state) to the detached state, D. (A, C, and E) Standard conditions with k c ¼ k c ¼ 2.8 pn/nm and Bell parameter (Eq. S24) x crit ¼ 0.6 nm but k rigor (x) according to Nishizaka et al. (33). (B, D, and F) Nonlinear cross-bridge elasticity (k c ¼ 0.05k c ) and strainindependent MgATP-induced detachment (x crit ¼ 0). Concentrations of MgATP were 5 mm (A and B), 0.1 mm MgATP (C and D), and 0.01 mm MgATP (E and F). Red, k 2 (x); blue, k 5 (x); orange, k rigor (x); black solid line, composite rate function, k diss, for transitions from the AM*ADP to the D state (MATP/MADPP i ); black dashed line, k 0 diss, for transitions from the AMADP to the D state. The maximum value of any rate function set to 20,000 s 1 for practical reasons if not otherwise stated. duty ratio, i.e., the ratio between the time (t on ) that a myosin head spends strongly attached to actin and the ATP-turnover time (t cycle z 32 ms) (55). Finally, r is the number of myosin heads/mm interacting with the actin filament, a function of the density of myosin heads that are appropriately oriented and sufficiently close to the filament to allow interactions. To a first approximation (see Theory), the quantity t on, and hence the duty ratio, may be assumed to vary with the MgATP concentration (compare Pate et al. (8) and Walcott et al. (56) ast on ¼ h/v f ([MgATP]), where h is the myosin step length (8 nm (20)) and v f ([MgATP]) is the velocity as a function of [MgATP]. In accordance with this behavior and Eq. 4, we found that at [MgATP] R0.25 mm, the velocity increased with filament length, saturating at lengths >2 mm (Fig. 2). No length dependence was observed at lower [MgATP]. For the TMCS surface, r was found to be 91 mm 1 (Fig. S11). This corresponds reasonably well to the value of 130 mm 1 calculated for a total surface density of ~6000 HMM molecules/mm 2 (27,39,40) on the assumptions that 1), HMM molecules with two active heads that are adsorbed within a 30-nm-wide band with the actin filament in the center reach binding sites on actin; 2), 70% of the myosin heads are fully active and not surface immobilized (40); and 3), ~50% of the HMM molecules are correctly oriented relative to the actin polarity (Fig. S5). Interpretation of in vitro motility data in terms of average cross-bridge properties, obtained from numerical solution of the master equations (Eqs. S2 S7), requires that a therefore used TMCS-derivatized surfaces in all experiments below. Furthermore, blocking actin, often employed to block MgATP-insensitive heads (15), was omitted as motivated by a high fraction of motile filaments (0.7 or higher) in the absence of blocking actin and a lack of effect of blocking actin on the [MgATP]-velocity relationship (Fig. S12). The sliding velocity (v f ) varies with the number of myosin heads potentially available for interaction with the actin filament, i.e., with surface adsorption sufficiently close to the filament and in appropriate orientation (54): v f ¼ v N f 1 ð1 fþ rl ; (4) where v f N is the sliding velocity for infinitely long filaments, l is the filament length in micrometers, and f is the FIGURE 2 Velocity versus filament length at different [MgATP]. Data at 1 mm MgATP fitted by Eq. 4 after fixing v N f to the average velocity at filament lengths R2 mm and the duty ratio to a value calculated from t on and t cycle, as described in the text. For other MgATP concentrations, no fits were obtained. Instead, Eq. 4 was plotted using the duty ratio, and v N f was determined as described in the text and the parameter value r obtained from fitting Eq. 4 to the data at 1 mm MgATP. Biophysical Journal 105(8) 1871 1881

1876 Persson et al. sufficiently large number of cross-bridges interact with the filament (56). This condition should be fulfilled in the flat (saturated) region of the velocity-length relationship (Fig. 2). Therefore, at MgATP concentrations R0.25 mm, we only included filaments >2 mm long in the analyses below. The [MgATP]-velocity relationship (at close to physiological ionic strength, ~130 140 mm, and temperature, 28 29 C) is well fitted (Fig. 3, A and B) by a hyperbolic equation (r 2 z 0.998) with V max z 15.28 5 0.28 mm/s (mean 5 SE) and K M z 0.389 5 0.023 mm. Similar good fits (r 2 z 0.994) were also obtained in the presence of 1 mm amrinone (Fig. 3, gray) with V max z 9.93 5 0.27 mm/s and K M z 0.268 5 0.026 mm. The good hyperbolic fits are also clear from the linear doublereciprocal plots in Fig. 3 C. In addition, the mean residuals (Fig. 3 D) from three individual nonlinear regression fits (Fig. S10) using three different HMM preparations (Fig. S8) do not differ significantly from zero. Similar results were found in the presence of 1 mm amrinone. The results in Fig. 3, with very good hyperbolic fits and FIGURE 3 [MgATP]-dependence of HMM propelled actin filament velocities. (A) Velocities in the absence (black) and presence (gray) of 1 mm amrinone. Experimental data are represented by solid circles (mean 5 SE, n ¼ 6 flow cells in the absence of amrinone and n ¼ 3in the presence of amrinone unless otherwise stated). The mean temperature was 28.2 29.2 C. Solid lines represent the Michaelis-Menten fits to experimental data. Dotted lines indicate the 95% confidence limits. See text for parameter values of V max and K M obtained in these fits. (B) Data in A limited to low [MgATP]. (C) Data in A as a double reciprocal plot. Solid lines (shown with 95% confidence limits) were obtained by linear regression, where the slope is equal to K M.(D) Velocity residuals from the Michaelis-Menten fits for 0 0.5mM MgATP (mean 5 95% confidence interval for three different HMM preparations; see Fig. S10) in the presence and absence of amrinone and for the model (Fig. S1, dashed line) with assumed interhead interactions. (Inset) Velocity residuals in the range 0 0.1 mm MgATP. lack of systematic deviations from a hyperbola, in both the presence and absence of amrinone, fail to corroborate the existence of interhead cooperativity (Fig. 3 D, dashed line; based on Fig. S1). The V max value was slightly higher than that observed in previous in vitro motility assay studies using HMM (15,57) and similar to those in skinned (58) and intact (59) skeletal muscle fibers of fast mammalian muscle. The K M value for the [MgATP]-velocity relationship was higher than in previous studies that used muscle cells (60), myofibrils (14), and in vitro motility assays (13,15,16). The higher K M value compared to those of cells and myofibrils can most likely be attributed due to the usually low temperature (generally <15 C) in mechanical studies of cells. Accordingly, it was mentioned (10) that the K M for rabbit psoas muscle increased from ~0.15 mm to ~0.6 mm when temperature increased from 10 to 35 C. The low K M value in previous in vitro motility assay studies may be attributed to differences in ionic strength (15) and/or myosin preparation (species or myosin/hmm/subfragment 1 (S1)) (13,18) compared to those used here. Appreciable differences between S1 and HMM in kinetics and elastic properties (affecting both K M and V max ) have been demonstrated (20,22,61,62) and myosin has been suggested to impose external load via interaction of its light meromyosin tail part with actin (57). The latter effect would reduce both K M and V max (compare Greenberg et al. (63); Fig. S13). However, the values of these parameters are also influenced by the degree of linearity of the cross-bridge elasticity (see below). Whereas a hyperbolic [MgATP]-velocity relationship has generally been observed (11 16) in the in vitro motility assay, recent results from studies that used fulllength myosin or S1 (17 19) and an unusually large number of data points at low [MgATP] exhibited appreciable deviations from a hyperbola. The possible reason for this is considered below. As we found no evidence for sequential operation of the two myosin heads, we investigated whether the [MgATP]- velocity data could be accounted for by a statistical crossbridge model that assumes independent myosin heads. As in earlier work (6,7,10), we first assumed linear cross-bridge elasticity and a strain-dependent rate of MgATP-induced detachment (x crit ¼ 0.6 nm in Eq. S24). We also ignored MgATP-independent detachment (33) (from the AM state; k rigor ¼ 0). Using this standard parameter set (Table S1), we obtained model predictions in good agreement with the experimental findings (Fig. 4 A, green solid symbols). However, if MgATP-independent detachment from the AM state (32,33) is taken into account, as seems reasonable (see reasoning in Theory section and Supporting Material), model predictions (Fig. 4 A, open symbols) deviate appreciably from the hyperbolic shape. This is particularly clear from the double-reciprocal plot in Fig. 4 B. In terms of the model, the deviation is attributed to the dominance of MgATP-independent detachment at low [MgATP] (Fig. 1, Biophysical Journal 105(8) 1871 1881

Nonlinear Actomyosin Elasticity 1877 FIGURE 4 Modeling of experimental data using different versions of the statistical cross-bridge model. (A) Michaelis-Menten hyperbolic fits to the experimental data from Fig. 3 A (black) and simulated data (green circles). The standard parameter set (including strain-dependent MgATP-induced detachment and linear cross-bridge elasticity) with (open green symbols) and without (solid green symbols) MgATP-independent detachment (k rigor (x)) from the rigor (AM) state. The dashed and solid lines represent Michaelis-Menten fits for each parameter set. (Inset) Details at low [MgATP]. (B) Double reciprocal plots of the data in A and C. Regression lines are fitted to all data. (C) Michaelis-Menten fits to the experimental data (black and gray; solid symbols and lines have the same meaning as in A) and the simulated data (open symbols and dashed lines), all with nonlinear cross-bridge elasticity (k c ¼ 0.05k c ) and MgATP-independent detachment, (k rigor (x)). Open blue squares, no amrinone, strain independence of MgATPinduced detachment (x crit ¼ 0 nm). Open gray squares, 1 mm amrinone, strain independence of MgATP-induced detachment (x crit ¼ 0 nm). Open blue circles, no amrinone, strain dependence of MgATP-induced detachment (x crit ¼ 0.6 nm). (D) Details of data in C at low [MgATP]. A, C, and E) that reduces the MgATP dependency of velocity (see also Fig. S14, cross-bridge distributions). One may argue that there are also other types of ATPindependent detachment process, e.g., force-induced detachment from the AMADP and AM*ADP states with rate functions similar to that for ATP-induced detachment from the AM state (Fig. 1). However, such detachment is highly unlikely in the drag-stroke region during steady-state actin filament sliding due to negligible population of the AMADP and AM*ADP states at relevant strains (Fig. S14). Whereas detachment from the AM*ADP state at high positive cross-bridge strains cannot be fully excluded, it is of very limited importance, as verified by control simulations (Fig. S15). Also detachment by reversal of the attachment step (from the AMADPP i to the D state) is negligible. Thus, during steady-state actomyosin motor action, the transition from the AMADPP i to the AMADP state drains the AMADPP i state, because the average x-value of attached cross-bridges is reduced by filament sliding (compare Fig. S14). For detailed arguments on alternative detachment processes, see the Supporting Material. One straightforward way to eliminate the nonhyperbolic deviation of the [MgATP]-velocity relationship seen with an MgATP-independent detachment step is to introduce nonlinear cross-bridge elasticity, as Kaya and Higuchi did (20). The latter modification (see Fig. 4 C, blue open symbols) appreciably reduces the highly load-dependent (33) rate function k rigor (x) at moderately negative strains (Fig. 1, B, D, and F) (see also Fig. S14) and therefore restores a nearly hyperbolic shape of the [MgATP]-velocity relationship. However, the maximum velocity was slightly higher than that found experimentally. The latter finding can be understood as follows. It is clear that several crossbridge states in the drag-stroke region become increasingly populated (Fig. S14) with nonlinear compliance due to reduction in k 2 (x) (Fig. 1) (not shown for x crit ¼ 0.6 nm), k rigor (x), and k 5 (x). However, it is important to note that this does not lead to increased drag force because the drag force is proportional not only to the number of negatively strained cross-bridges and their average strain but also to the cross-bridge stiffness. Because the latter was reduced to 5% in the drag-stroke region for the nonlinear compared to the linear case, the increase in the number of attached cross-bridges and their average negative strain in the drag-stroke region in the nonlinear case was insufficient to lower the maximum velocity to the experimental value (Fig. 4 C). This discrepancy was eliminated (Fig. 4, C and D, blue open squares) by further reducing the overall cross-bridge detachment rate in the drag-stroke region. This was most conveniently achieved by reducing x crit (Eq. S24) from 0.6 to 0 nm (Fig. 1, B, D, and F), thereby completely removing the load dependence of the MgATP-induced detachment rate, k 2 (x), making it constant and equal to its lowest value, k 2 (0). This intervention made the overall detachment rate, Biophysical Journal 105(8) 1871 1881

1878 Persson et al. k diss (x, [MgATP]) (from state AM*ADP to state D) strainindependent in the drag-stroke region but strain-sensitive with a negative slope for positive x values. This (Fig. 1, B, D, and F) is in fair agreement with results from recent single-molecule experiments where the constant k1 (34) presumably corresponds to our k diss (x, [MgATP]), most likely with a contribution from k rigor (x) at micromolar [MgATP]. In the model version with linear cross-bridge elasticity, on the other hand, load dependence of k 2 (x) (and hence of k diss (x, [MgATP])) was essential to prevent the high drag forces seen with the 20-fold-higher stiffness at negative x. It is of interest to note that the effects of 1 mm amrinone on the [MgATP]-velocity relationship (reduced V max and K M ) were faithfully reproduced by the model Fig. 4, C and D, gray symbols and lines) on the basis of our previously found mechanism, i.e., increased stability of the AM*ADP state relative to the AMADP state (lowered difference between the free-energy minima of these states ðdg 12 Þ from 1.7 to 0.1 k B T).Whereas the nonhyperbolic deviation of the [MgATP]-velocity relationship (Fig. 4, A and B) is in conflict with the experimental results presented here, the deviation is similar to that found in previous in vitro motility assay experiments using full-length myosin or myosin subfragment 1 (17 19). This is interesting, because linear cross-bridge elasticity is indeed expected in the case with subfragment 1, according to the evidence of Kaya and Higuchi (20) that the nonlinear elasticity is due to bending of the S2 domain for negatively strained crossbridges (Figs. S3 S5). Whether the S2 domain is prevented from bending in nitrocellulose-adsorbed full-length myosin, giving linear cross-bridge elasticity and explaining the nonhyperbolic deviation (17,18) in this case, remains to be clarified. On the other hand, TMCS-adsorbed HMM is likely to exhibit a nonlinearity similar to that found by Kaya and Higuchi (20). This follows from the flexible surface adsorption of HMM via the subfragment-2 part (27) (see details in the Supporting Material). Whereas we think that the above arguments are convincing and clear, at least for S1 and HMM on TMCS, it will ultimately be important to directly investigate the elastic characteristic of the different combinations of surfaces and motor fragments using single-molecule mechanical studies. The validity of the model simulations is supported by the solid theoretical and experimental foundations of model parameters and states determined in independent experimental studies. The only parameter values of relevance for the [MgATP]-velocity relationship that was obtained based on fitting the model to the in vitro motility assay results described here (V max and K M in the presence and absence of amrinone) were those describing the transition between the AM*ADP and AMADP states. All other parameter values of relevance were directly derived from independent experimental results (20,31,33,34,64) modified only within the experimental uncertainties of those experiments. Parameter values related to pre-power-stroke states were also based on experimental data, but in a less stringent way (see the Supporting Material). However, we show in Fig. S15 that a wide range of changes in these parameter values have a negligible effect on the [MgATP]-velocity relationship. When the parameter values, as obtained above, were used in model simulations, the model faithfully reproduced the [MgATP]-velocity relationship in the presence and absence of amrinone (Fig. 4) and the effects of load (Fig. S13) and varied [MgADP] (Fig. S16) without further parameter adjustments. In addition, it is important to note that the load-velocity relationship (Fig. S17) including characteristic effects of amrinone (23,24) was accounted for. The model presented here benefits greatly from recent, experimental results that provide a detailed characterization of the cross-bridge elasticity, power-stroke distance (20), and individual transition rates between model states (31,34,65), including their strain dependence (33,34). Indeed, the experimental characterization approaches a level of sophistication that will soon allow a close-to-unique definition of the model parameter space. Repriming of the power stroke after a length step (66) and the discrepancy between high power output at intermediate loads, on the one hand, and a relatively low rate of rise of force during an isometric tetanus on the other, are currently incompletely understood phenomena. It has been proposed (7,67) that they can be attributed to interhead cooperativity with sequential action of the two heads of myosin II. Our failure to detect evidence for such cooperativity could indicate other causes; for instance, pre-power-stroke crossbridges may have differing properties during shortening and in isometric contraction (68 70). Alternatively, interhead cooperativity may only exist in a three-dimensionally ordered filament lattice (21) and/or in the presence of external loads rather than internal loads (low [MgATP]). These are areas for future study. CONCLUSIONS AND PERSPECTIVES Our data are consistent with the idea that actin propelling heavy meromyosin motor fragments in the in vitro motility assay exhibit nonlinear elasticity similar to that of actomyosin cross-bridges in synthetic myosin filaments (20). The results suggest that detailed studies of the [MgATP]-velocity relationship (including a large number of data points at low [MgATP]) in skinned fibers or myofibrils also should give clues as to whether the cross-bridge elasticity is linear or nonlinear in the three-dimensionally ordered myofilament lattice. The appreciable effect of the nonlinear cross-bridge elasticity on the [MgATP]-velocity relationship could mean that aspects of the energy economy of muscle, e.g., in fatigue, may be evolutionary driving forces for development of nonlinear cross-bridge elasticity. Further developed statistical cross-bridge models, where pre-power-stroke Biophysical Journal 105(8) 1871 1881

Nonlinear Actomyosin Elasticity 1879 events and cooperative activation (71) are also considered in greater detail, would be useful for elucidating this issue. Currently, limited experimental insight hampers such model developments, e.g., uncertainties about the temporal relationship between the phosphate release step and the main force-generating power-stroke event (70,72 77). This model is of a classical type, developed from our previous models (7,22) and based on breakthrough ideas (2,3,78,79) that were later formalized in great detail (4,49). Models of this type, or closely related to this type, have been expanded in several respects and used quite extensively (6,7,22,50,56,80,81). It is therefore of appreciable interest that the introduction of nonlinear crossbridge elasticity does not compromise the capability of a model of this type to account for contractile phenomena such as [MgATP]-velocity and load-velocity relationships. This version of the classical models should therefore form a solid basis for further development to elucidate how kinetic and elastic properties on the single-motor level determine the contractile behavior of a motor ensemble. This may contribute to deciphering of mechanisms whereby point mutations in contractile protein genes lead to disturbed function on the ensemble level (7), as seen in severely debilitating sarcomere myopathies (82 84) of striated muscle. These diseases include cardiomyopathies, the leading cause of sudden cardiac death in otherwise healthy young individuals. A large fraction of these disease states are attributed to point mutations in the cardiac myosin (also expressed in slow skeletal muscle). That possible disease mechanisms may be revealed by modeling is illustrated by previous studies (7,81,85) suggesting that certain combinations of parameter values lead to mechanical sarcomere instabilities and that small changes in model parameters (e.g., due to point mutations) may be sufficient to introduce instabilities with the potential to compromise cellular integrity (7). SUPPORTING MATERIAL Seventeen figures, one table, references (86 93) and additional supplemental information are available at http://www.biophysj.org/biophysj/ supplemental/s0006-3495(13)00991-0. A.M. and M.P. designed this study; M.P., A.M., L.t.S., and E.B. performed the experiments and simulations; M.P., A.M., and E.B. analyzed the data; and A.M., M.P., and E.B. wrote the article. 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Supplementary Material: Nonlinear cross-bridge elasticity and post-powerstroke events in fast skeletal muscle actomyosin by Malin Persson, Elina Bengtsson, Lasse ten Siethoff, Alf Månsson Department of Chemistry and Biomedical sciences, Linnaeus University, SE-391 82 Kalmar, Sweden 1

SUPPLEMENTARY THEORY On possible sequential action of the two myosin heads If there is velocity dependent, limited processivity e.g., with two (on average) sequential actions of the two heads in a pair (1,2), a deviation from a hyperbolic [MgATP]-velocity relationship (Eq. 3 in main paper) may be expected, corresponding to apparently larger step length, h at MgATP concentrations (and velocities) that are optimal for sequential actions of the two heads (1). In our analysis we therefore simulated the limited processivity by assuming a percentage increase of the power-stroke distance as a function of velocity according to recent predictions (1). This led to a velocity dependent multiplicator, M(v) (Fig. 1), for the power-stroke distance in Eq. 1. This velocity dependent multiplicator was transformed into a [MgATP]-dependent multiplicator by solving the equation: (S1) where M(v f ) was taken as a piecewise linear function according to the inset of Fig. 1 and h and K M were approximated from experimental data. k 1.8 M([MgATP]) 2.0 1.6 1.8 1.4 1.6 1.4 1.2 1.2 1.0 0.0 0.5 1.0 1.5 Velocity fraction of 1 mm 1.0 0.0 0.2 0.4 0.6 0.8 1.0 [MgATP] (mm) M(v) FIGURE S1 [MgATP] dependent multiplicator, M([MgATP]). This MgATP dependent multiplicator was obtained by solving equation S1 and using a velocity dependent multiplicator (M(v)) as shown in the inset. The latter was approximated from Månsson (7). Use of an increased apparent power-stroke distance to simulate processivity implies that the attachment of the leading head causes the trailing head to detach rapidly, thereby spending negligible time in the drag stroke. Such immediate detachment also seems essential to account for the re-priming of the power-stroke (3) and also to increase the power output during shortening at intermediate velocity (4). Whereas the exact mechanism is unclear there may be effects of strain caused by the azimuthal displacement of the neighboring myosin binding sites along the actin filament. In analogy with the idea of increased processivity at low MgATP concentrations, myosin V, modified to loose processivity under physiological conditions, regained this processivity by lowering [ATP] to the µm range (5). 2

Statistical model with strain-dependent rates where each head is an independent forcegenerator General The present statistical model for muscle contraction is based on the states depicted in Scheme 2 in the main paper. The steady-state properties of the model are obtained from the state probabilities amdp(x), am*d(x), amd(x), am(x), amt(x) and d(x) for the states AMADPP i, AM*ADP, AMADP, AM, AMATP and D, respectively. These state probabilities are obtained by numerically (using the Runge-Kutta Fehlberg method) solving the following system of non-linear ordinary differential equations (S2 S7; cf. (4,6)): damdp = -(k a (x) d(x) + k -4 (x) am*d(x) (k -a (x) + k 4 (x)) amdp(x))/v dx (S2) dam* d = -(k 4 (x) amdp(x) + k -5 (x) amd(x) (k -4 (x) + k 5 (x)) am*d(x))/v dx (S3) damd dx = -(k 5 (x) am*d(x) + k -6 [MgADP] am (x) (k -5 (x) + k 6 ) amd(x))/v (S4) dam = dx (S5) damt = -(k +1 [MgATP] am (x) (k -1 + k 2 (x) )amt (x))/v dx (S6) dd dx = -(k 2 (x) amt(x) + k -a (x) amdp(x)+ k rigor (x)am(x) k a (x)) d(x))/v (S7) The rate functions (rate constants varying with x) for transitions between states are defined in Scheme 2 in the main paper. The velocity, v, is negative for shortening i.e., sliding with the actin filament pointed end at front. The analytical expression for the rate functions are given below. Note that the transition governed by the rate function k 4 (x) is associated with release of inorganic phosphate (P i ) and the main power-stroke. Convesely, the reverse rate function k - 4(x) is associated with P i -rebinding and reversal of the power-stroke. Here, the P i dependence of the latter function could be neglected due to our assumption of constant P i -concentration. Adaptation of previous model - overview The model based on the kinetic Scheme 2 in the main paper was developed from that of Månsson (1). The latter model gave excellent fits to the force-velocity curves, the time course of an isometric tetanus and the Huxley & Simmons tension transients (7,8) in frog muscle fibres in the temperature range 2 4 o C. The model (1), was here modified to account for force-velocity data of mammalian muscle at high temperature. Furthermore, we introduced the AMADP, AM and AMATP cross-bridge states at the end of the power-stroke using inter-state transition rates from (9). Finally, the detachment rate function was changed, now varying with cross-bridge strain according to the Bell relationship (as suggested in (10) and (11)) instead of the empiric relationship used previously (1) Adaptation of free energy profiles One characteristic effect of increases in temperature as suggested by experimental results (12) is increased free energy difference between pre-force attached states and the main forcegenerating state with minimal effects on cross-bridge stiffness and the number of attached cross-bridges. The effect of increased temperature on the free energy profiles was modeled by increasing (compared to (1)) the difference in minimum free energy (G min ) between the states AMADPP i and AM*ADP and AM*ADP and AMADP from 10 to 14 k B T and from 0.6 k B T to 3

1.7 k B T, respectively (k B : Boltzmann constant, T: absolute temperature). The magnitudes of the differences in free energy minima between states are motivated below. When cross-bridge elasticity is Hookean, the free energy profiles are symmetrical and parabolic (Fig. S2) according to the following relationship: G state (x)= G state min + k c (x-x 0 ) 2 /2 (S8) where G state (x) and G state min give the x-dependence and minimal value of the free energy of the actomyosin cross-bridge in a given state. Further, the magnitude of the variable x gives the distance between the available myosin binding site on actin (here limited to one available site)(1) and a reference point on the myosin filament. The value x=0 nm is defined as the position where the force developed in the AMADP state is zero (see further below). Finally, k c is the stiffness (2.8 pn/nm in the case of linear cross-bridge elasticity) of each individual cross-bridge and x 0 is the x-position where the free energy in the given state attains its minimum. Forward and reverse rate constant k ij (x) and k ji (x) between states i and j are related as follows: k ij (x)/k ji (x)=exp((g i (x)-g j (x))/k B T) (S9) for thermodynamic consistency. The free-energy diagrams used in analayses assuming linear cross-bridge elasticity are illustrated in Fig. S2 for 5 mm MgATP (Fig. S2A) and 0.01 mm MgATP (Fig. S2B). The free energy profiles are ultimately constrained by the total free energy of ATP hydrolysis ΔG ATP. This quantity (expressed per molecule) is given by: ΔG ATP = ΔG ATP o k B T (S10) where ΔG ATP o is the standard free energy of ATP hydrolysis (~13.1 k B T)(13), k B is the Bolzmann constant and T is absolute temperature. Further, [MgATP], [MgADP] and [P i ] are the Molar concentrations of MgATP, MgADP and inorganic phosphate (P i ). We used creatine-kinase creatine phosphate (CK/CP) for ATP regeneration from ADP thereby keeping the MgATP/MgADP ratio constant with low [MgADP]. It is therefore assumed here that [MgADP] is 10 µm, somewhat lower than the physiological rest value (13) whereas the inorganic phosphate concentration is around 0.5 mm (14). Free energy profiles and non-linear cross-bridge elasticity In some simulations we used non-linear cross-bridge elasticity (Fig. S3) as proposed recently (15). In these cases the free energy profiles were altered away from the parabolic shape (to shapes in Fig. S4) The appearance of non-linearities like those of Kaya et al. (15) are likely also in our in vitro motility assay using HMM. Thus, for HMM molecules adsorbed to trimethylchlorosilane (TMCS) derivatized surfaces the coiled-coil subfragment-2 domain (S2) is extending from the surface (16) and may therefore buckle under compressive forces or the bending elasticity of S2 may constitute the most compliant element between the actin filament and the surface (cf. Fig. S5). The buckling force F b is given by inserting the persistence length (L P = 130 nm(17)) and length (L~55 nm (17)) of S2 into the relationship (18): 20.2 (S11) 4

This gives F b 4 pn. However, even at minimal negative strains, the bending stiffness of S2 may dominate the elastic influence of the acto-hmm complex on motility (cf. (15)). Thus, inserting the values of L P and the S2-length into the equation for stiffness of a cantilevered beam (18) gives a value of 0.01 pn/nm (see also Kaya and Higuchi (15)). This is several orders of magnitude less than myosin head stiffness of 2-3 pn/nm (19,20). In our simulations we used 0.14 pn/nm for the cross-bridge stiffness in the drag-stroke region. This is similar to the stiffness values found by Kaya and Highuchi for negative strains in the range -10-0 nm. ATP ADP ADP P i ATP ADP P i ADP ADP ADP FIGURE S2. Free energy diagrams illustrating x-dependence for different states in Scheme 2 in main paper. A. 5 mm MgATP. Schematic structure shown as insets, indicating each state with tentative lever arm position and degree of lever arm bending (corresponding to tentative elastic element). B. 0.01 mm MgATP. 5

FIGURE S3. Cross-bridge forceextension curves illustrating simplified implementation of non-linear crossbridge elasticity. Full line: Hookean crossbridge compliance. Dashed line and solid circles: Non-linear cross-bridge compliance with stiffness 2.8 pn/nm (0.7k B T nm -2 ) at positive strains (k c ) and 0.14 pn/nm (k c - = 0.05k c ) at negative strains. FIGURE S4. Free energy diagrams with non-linear cross-bridge elasticity. The force-extension curve of the cross-bridge elasticity is similar to that illustrated in Fig. S3 with cross-bridge stiffness for positive x-values of 2.8 pn/nm but with 0.14 pn/nm for negative x-values (as in Fig. S3). The minima of the free energies for each state are similar to those in Fig. S2A and occur at the same x-values as in that figure. v A B - power-stroke - + S2 F + S1 drag-stroke lever arm FIGURE S5. Schematic illustration of elastic elements in the myosin head. A. Two correctly oriented surface adsorbed HMM molecules (only one head, subfragment 1 [S1], shown for each molecule) interacting with an actin filament (plus and minus ends indicated). The left-most head exerts its power-stroke with a strained lever-arm domain exerting force in the shortening direction (upper arrow) on the filament. The rightmost head is in the dragstroke (resisting shortening) with bending of subfragment 2 (S2). B. Incorrectly oriented HMM molecule with bending/buckling of S2 during the power-stroke. 6

Interestingly, similar arguments to those above, about bending and buckling of S2, suggest an inefficient power-stroke of incorrectly oriented HMM-molecules, i.e., those that are adsorbed to the surface in a way that their motor domains point towards the actin filament minus end (Fig. S5B). This is consistent with the low velocities of actin filaments that are transported away from the midline of a bipolar myosin filament (21) Moreover, there is experimental evidence that incorrectly oriented myosins do not impose a load on the correctly oriented ones, i.e., when the two types of heads operate simultaneously the velocity is determined by the fast ones (21). In view of these arguments we assume that the motile behavior in the in vitro motility assay is largely unaffected by the presence of incorrectly oriented HMM molecules. This idea is consistent with findings that the sliding velocity in our experiments is close to that found in muscle fibers (~13 µm/s at 25 o C) (22). Adaptation of attachment rate function Full Ca 2+ activation of the thin filaments is assumed in the model and, as motivated previously (1), the attachment rate function varies with x according to a Gaussian function centered close to the free energy minimum of the AMADPP i state. Here, we focus on steadystate shortening and, instead of implementing a velocity dependent attachment rate to account for both shortening and isometric rate of rise of force (1) we used a high value of the attachment rate constant at all loads. However, in order to fit data at high temperature, rather than frog data at low temperature, the maximum value of the Gaussian was set to 600 s -1 (> 5- fold higher than in previous work(1)). The attachment rate function is illustrated in Fig. S6 together with its reversal. The rate function (in s -1 ) of the attachment process is given by: k a kc ( x ( x0 μ)) ( x) = exp( Δ Gd0 /(2 k BT)) exp( ) (S12) 2ε k T B Here, k c, is the cross-bridge stiffness (0.7 k B T nm -2 = 2. 8 pn nm -1 ). The numerical values of the parameters μ and ε are given in Table S1. The amplitude and width of the Gaussian function is influenced by the cross-bridge stiffness and the free energy difference Δ G d 0 between the D and AMADPP i states, respectively. The rate function (in s -1 ) for reversal of the attachment process is given by: k a ( x ) min( k a' ( x ), 20000 ) = where kc( x x ) ' ( x ) k ( x )exp( G /( k T ))exp( 0 2 k a = a Δ d0 B ) (S13) 2k BT Here, x 0, is the position of the free energy minimum for the AMADPP i state. If k a ( x ) > 20 000 s -1 it was set to 20 000 s -1 to eliminate problems with instability in the numerical calculations. This simplifying assumption had negligible effect on the outcome of the calculations. The rate function k -a (x) is approximately centered around x=x0 where the force in the AMADPPi state is zero. This is in approximate agreement with the force dependence of the related constant k 3 in Capitanio et al. (10) that was symmetrical around 0 pn. 2 7

FIGURE S6. Rate function for cross-bridge attachment to (k a (x)) and detachment from (k -a (x)) the AMADPP i state. Applies to situation with constant cross-bridge stiffness, k c =2.8 pn/nm. Rate functions for transitions between attached states The shape of the rate functions for transitions between attached states (Fig. S7) were chosen following principles in (4) and (2) on basis of the original ideas of Huxley & Simmons (8) to account for T 1 - and T 2 -curves in transient muscle mechanics. The maximum rates of these transitions were considerably increased to accommodate the high temperature and recent results (23). The rate function ( k 4 ( x )) for transition from the AMADPP i to the AM*ADP state is given by: k 4 ( x ) = min( k 4' ( x ), 20000 ) (S14) where k c( ( x x )x x x ) k ' ( x ) k exp( G /( k T ))exp( 2 2 0 1 0 + 2 1 4 = 4a Δ 01 B ) (S15) 2 k T Here, k 4a is a constant (Table S1), Δ G 01 is the difference in minimal free energy between the AMADPP i and AM*ADP states and x 1 is the position of the free energy minimum of the AM*ADP state. This equation applies to conditions with the same stiffness, k c, in the AMADPP i and AM*ADP states. If the two states have different stiffness, as may occur with non-linear filament compliance where stiffness is lower for negatively strained heads, the equation is changed as described in (6). The rate function k- 4 (x) is given by: k c( ( x x )x x x ) k ( x ) k ( x )exp( G /( k T ))exp( 2 2 0 1 0 + 2 1 4 = 4 Δ 01 B ) (S16) 2 k T The rate function for transition from the AM*ADP to the AMADP state ( ( x ) ) is given by: k 5 ( x ) = min( k 5' ( x ), 20000 ) (S17) B B k 5 where k ' ( x ) = k exp( Δ G 5 5a 12 /( k B kc( ( x x )x x x ) T ))exp( 2 1 2 1 + 2 2 ) 2 k T B 2 (S18) 8

Here, k 5a is a constant (Table S1), Δ G 12 is the difference in minimal free energy between the AM*ADP and the AMADP states, and x 2 is the position of the free energy minimum of the AMADP state. The rate function k -5 (x) is given by: k k ( x) = k ( x)exp( ΔG /( k T ))exp( 5 5 12 B c (2( x 1 x2) x x 2k T B 2 1 2 2 + x ) ) (S19) This equation, as well as Eq. S18, applies to conditions with the same stiffness, k c, in the AM*ADP and the AMADP states. If the two states have different stiffness, e.g., with nonlinear filament compliance, the equations are changed as described in (6). FIGURE S7. Rate functions for cross-bridge transitions between states AMADPP i and AM*ADP (k 4 (x) and k -4 (x)) and AM*ADP and AMADP (k 5 (x) and k -5 (x)). Applies to constant cross-bridge stiffness, k c =2.8 pn/nm. Details on detachment rate function Unlike in the previous paper (1) the AMADP, AM and AMATP states were here treated separately and strain-independent inter-state transitions were used according to measurements in solution (9). Using the temperature and ionic strength dependence, the following values were obtained: k 2 1500 s -1, K 1 1.7 mm -1 and k 6 4000 s -1. In the actual model calculations we increased k 6 to 5000 s -1 as the estimate of Nyitrai et al. (9) was a lower estimate. Now, the overall transition rate constant, k diss for transition from the AMADP state to the dissociated actin and MATP states can be approximated by: = + + (S20) where only k 2 may be x-dependent (see below and main paper). This relationship is readily rewritten as (now including the arguments x, and [MgATP]):, (S20A) The overall cross-bridge detachment rate (k diss (x, [MgATP])), from the AM*ADP to the D state can be estimated as follows: k diss (x, [MgATP]),, (S21) 9

Here we take into account reversibility of reactions by assuming either rapid equilibria (governed by K 1 ; see above) or neglecting reversible transitions (MgADP-rebinding and the transition governed by k -5 (x)). Estimation of the overall detachment rate function without the simplifying assumptions in Eq. S21 was achieved as follows: k diss (x, [MgATP]),,,,, (S22) Estimates of the overall detachment rate function using either Eq. S21 or S22 are illustrated in main Fig. 1. If k diss (x, ) > 20 000 s -1, according to the above relationships, it was set to 20 000 s -1 (as with other rate functions). In addition to detachment as considered above, ATP independent detachment from the rigor state (the AM state) may take place according to a rate function as follows: = k 0 exp (S23) Here, the quantities k rigor (0) and x rigor were found by Nishizaka (11) to be 0.016 s -1 and 2.7 nm, respectively (see also (24)). A recent optical tweezers study of myosin-subfragment-1(s1)-actin interactions (10) showed very limited strain-dependence in the drag stroke region of a rate function (k 1 in (10), corresponding largely to our k diss (x, )). It was therefore reasonable to approximate the rate function k 2 (x) with a constant (k 2 (x) k 2 (0); Fig. 1 B, D, F) also making the composite detachment rate, k diss (x, ), x-independent. We do not, a priori, expect any difference in load dependence between S1, used by Capitanio et al. (10), and HMM used here, as the load-sensitivity is most likely attributed to structural effects of load on the motor domain itself. However, with linear cross-bridge stiffness, it was difficult to fit the high velocity of HMM propelled actin filaments at high [MgATP](considerably higher than for S1) without such a strain-dependence (unpublished model studies). Therefore, we here hold the possibility open that ATP-induced acto-hmm detachment, in contrast to acto-s1 detachment, is strain dependent as follows: k 0 exp (S24) with a Bell critical strain (x crit ) up to 0.6 nm. The value of k 2 (0) used here was taken from (9). The numerical values for various model parameters are summarized in Table S1. Simulations of steady-state muscle contraction From the state probabilities, that were obtained by solving equations S2-S7, a number of observables can be calculated for each chosen velocity as described below (where x is in nm). 1. Average force <F>; (pn) per attached cross-bridge <F> = k c ( amdp( x ) ( x x ) + am* d( x ) ( x x ) + amd( x ) ( x x ) + am( x ) ( x x ) + amt( x ) ( x x ))dx where k c is in pn nm -1 and x is in nm. 0 ( amdp( x ) + am* d( x ) + amd( x ) + am( x ) + amt( x ))dx 1 2 2 2 (S25) 2. Number of attached cross-bridges (NC) vs velocity (v) normalized to the number attached during isometric contraction 10

NC = ( amdp( x,v ) + am* d( x,v ) + amd( x,v ) + am( x,v ) + amt( x,v ))dx ( amdp( x, 0 ) + am* d( x, 0 ) + amd( x, 0 ) + am( x, 0 ) + amt( x, 0 ))dx (S26) 3. Average strain (<S>; nm) per attached cross-bridge <S> = <F>/2k c (S27) where the denominator represents the cross-bridge stiffness expressed in pn nm -1. 11

Table S1. Parameter values a in standard rate functions used for standard optimized simulations in the main paper. Parameters Range (directly from Parameter from Parameters experiments and/or Månsson used here from fitting model to Source (1) data) x 0 (AMADPP i ) 7. 35 nm 7. 35 nm 7-10 nm (15,25) x 1 (AM*ADP) 1 nm 1 nm 1.0-1.3 nm (26) x 2 (AM*ADP) 0 nm 0 nm NA NA 2-4 k B T (e.g., to fit Δ G d 0 T 4 k (D- AMADPP i ) B T 3.5 k B T 1 /T 2 -cruves and free energy of AM-ADP- (1,13,27) P i -state) Δ G 01 (AMADPP i AM*ADP) Δ G 12 (AM*ADP- AMADP) 10 k B T 14 k B T Δ G 01 + Δ G 12 : 18-20 k B T b (13,18) 0.6 k B T 1.7 k B T See above (13,18) Δ G ATP 20 k B T 13.1 + ln ([MgATP]/ Free energy of ATPhydrolysis ([MgADP][Pi]) k B T (13) μ 0.5 nm 0.5 nm Empirical (1) ε 1.7 nm 1.7 nm Empricial (1) k 4a 250 s -1 6000 s -1 ~6000 s -1 (23) k 5a 300 s -1 2000 s -1 Obtained in fit c Obtained in fit c k a 24 s -1 100 s -1 d Maximum power output Parameters for detachment rate functions below x crit NA 0.6 nm < 0.2 nm (10) k 6 NA 5000 s -1 >3500 s -1 (9) k -6 NA 14 290 mm -1 s -1 >10 000 mm -1 s -1 (9) [MgADP] NA 0.01 mm <0.03 mm (physiol. rest value) (13) [Pi] NA 0.5 mm ~ 0.5 mm (14) K 1 NA 1.7 mm -1 1.7 mm -1 (9) k +1 NA 14 000 mm -1 s -1 >10 000 mm -1 s -1 e (9) k -1 NA 8 400 s -1 k -1 =k +1 K 1 NA k 2 (0) NA 1530 s -1 1530 s -1 (9) NA: Not applicable a See Supplementary Discussions for further motivation of parameter values. b Data was also well fitted here (see below) by increasing Δ G 01 up to 16 thereby increasing Δ G 01 + Δ G 12 to 17.7 k B T c Fit of force velocity relationship d Attachment rate function chosen to fit maximum power output. e From the assumption of a diffusion limited rate constant > 10 000 mm -1 s -1 (14 000 mm -1 s -1 ) 12

SUPPLEMENTARY METHODS Figures S8-S10 are considered in the Materials and Methods of the main paper FIGURE S8. SDS-PAGE of myosin and HMM preparations used in the present study visualized by Coomassie staining. A. Myosin A and HMM A, 12% SDS-PAGE Tris- Glycine. Left lane: Molecular weight (Mwt) marker: Precision Plus Protein Standard (BioRad). Lanes 1 and 2: myosin and HMM preparation, respectively. B Myosin B and HMM B, 12% SDS-PAGE Tris-Glycine. Similar lanes as in A. Mwt marker: PageRuler (Fermentas). C. Myosin C and HMM C, 4-20% SDS-PAGE Tris-Glycine (NOVEX, Invitrogen). Similar lanes as in A and B. Mwt marker: SeeBlue Plus2 (Invitrogen.) The double lines in A at low molecular weights are artefacts from scanning (shadow) and not present on the gel (cf. the double line for marker at 10kDa). FIGURE S9. Measured filament length vs filament intensity at 5 frames/s. The filament length was obtained from intensity data as described previously (28) using the slope of the regression line (full line) for lengths >2 v f fr -1 where v f is the sliding velocity and fr -1 is the imaging frame rate FIGUR S10. [MgATP]-velocity data plots for the individual three myosin preparations (and related HMMpreparations) used for the summary plots in the main Fig. 3. The three HMM preparations correspond to the gels in Fig. S8. 15 10 5 HMM A HMM B HMM C 0 0 1 2 3 [MgATP] mm 13