Force and shortening are generated in the sarcomere, the structural unit of striated muscle, by cyclical, ATP-driven interactions between myosin motors, which extend in two bipolar arrays from the center of the thick filament, and the overlapping thin actin-containing filaments that originate from the sarcomere extremities (Figure 1).

In each interaction, the free energy of the hydrolysis of ATP in the catalytic site of the myosin motor is converted into mechanical work through a structural change (the working stroke) that, according to the crystallographic model (Rayment et al., 1993a; Rayment et al., 1993b), consists of an inter-domain tilting of the lever arm about the catalytic domain firmly attached to the actin site. The working stroke is associated with the sequential release of the hydrolysis products orthophosphate (Pi) and ADP. Calorimetric and protein solution studies have shown that a large enthalpy change is related to Pi release from the quaternary complex actin–myosin-ADP-Pi (AMADPPi), suggesting that the Pi dissociation step is the chemical transition associated with the mechanical working stroke (White and Taylor, 1976; Hibberd et al., 1985; Hibberd and Trentham, 1986).

Photo-liberation of caged Pi in skinned muscle fibers showed that the force during an isometric contraction is reduced by a stepwise increase in [Pi] with a rate that increases asymptotically with [Pi] (Dantzig et al., 1992; Millar and Homsher, 1992; Walker et al., 1992; Homsher et al., 1997), implying that, according to the minimum five-step reaction of Scheme 1, the Pi release (step 4) follows the force-generating transition from a weak AMADPPi to a strong AM'ADPPi state (step 3).

In Reaction Scheme 1, which summarizes the results from solution protein studies and caged compound experiments in skinned fibers, ATP binding to the actomyosin (AM) complex, after the release of the hydrolysis products and the execution of the working stroke, promotes rapid dissociation of myosin from actin (step 1), which is followed by the recovery stroke (the reversal of the working stroke) and ATP hydrolysis (step 2). The detached myosin with the hydrolysis products is in rapid equilibrium with the weakly bound AMADPPi state. The closure of the actin-binding cleft of the myosin catalytic domain (Geeves and Holmes, 2005) forms the strongly attached, stiffness-generating AM'ADPPi state, which is allosterically associated with the structural change that leads to tilting of the lever arm responsible for force generation and filament sliding (step 3). Thus, according to the simplified Scheme 1, step 3 is the combination of two processes, namely, the formation of a strongly bound motor and the generation of force, with kinetics limited by the much slower attachment process. This definition of step 3 is in agreement with the fast kinetics of force generation, as estimated from the quick force recovery following a step release (Huxley and Simmons, 1971; Ford et al., 1977), and it is supported by the evidence that the force increases in proportion to the number of myosin motors during isometric force development by a muscle fiber (Brunello et al., 2006; Caremani et al., 2008). Pi is released without any further contribution to force (step 4, Dantzig et al., 1992). Thus, the sum of the fractional occupancies of the AM'ADPPi and AM'ADP states constitutes the fraction (f) of actin-attached motors that contribute equally to the force and stiffness. The subsequent ADP release (step 5) occurs at a rate that, in the isometric contraction, is low (Nyitrai and Geeves, 2004; Sleep et al., 2005; West et al., 2005). According to Scheme 1, the development of isometric force is rate-limited by both the ATP hydrolysis step (step 2) and the attachment of motors (step 3), while the steady-state flux through the whole cycle (the rate of ATP hydrolysis) is limited by the rate of the ADP release (step 5). ADP release is conformation-dependent and becomes relatively fast only when attached motors preferentially populate the final state of the working stroke, such as during shortening at low load (Nyitrai and Geeves, 2004; Caremani et al., 2015). Thus, in vitro and in situ kinetic studies support Scheme 1 as the unique path through which myosin–actin interaction in muscle hydrolyzes ATP and produces force and power.

This conclusion, however, is questioned on the basis of several pieces of challenging evidence, which are discussed in detail in the next sections. First, most crystallographic models of the myosin motor domain (sub-fragment 1, S1) (Smith and Rayment, 1996; Houdusse et al., 2000; Geeves and Holmes, 2005; Llinas et al., 2015) support the alternative view that Pi is released prior to the working stroke (see also Robert-Paganin et al., 2020; Moretto et al., 2022; Mansson et al., 2023; Rassier and Mansson, 2025). Second, the finding that an increase in [Pi] reduces the isometric force of the Ca²⁺-activated fibers more than the ATPase rate (Cooke and Pate, 1985; Caremani et al., 2008) suggests that an increase in [Pi] increases the probability of an alternative pathway of the chemo-mechanical cycle (Linari et al., 2010; Debold et al., 2013), with premature detachment of motors from a strongly bound force-generating state, followed by the release of the hydrolysis products. Third, the attachment–detachment kinetics that fits the rate of isometric force generation is too slow in relation to the maximum power developed during steady shortening, which suggests that during shortening, an attached motor could complete its ATPase cycle by slipping to the next actin monomer farther from the sarcomere center (Lombardi et al., 1992; Piazzesi and Lombardi, 1995; Caremani et al., 2013; Pertici et al., 2023).

Crystallographic studies (Smith and Rayment, 1996; Houdusse et al., 2000; Geeves and Holmes, 2005; Llinas et al., 2015) have the unique ability to provide details of the inter-domain movements within the motor domain (S1, Figure 1) that associate the closure of the actin-binding cleft, which is responsible for the formation of the strongly-bound motor, to the catalytic site and then to the converter that controls the orientation of the lever arm (Figure 2). At the catalytic site, modifications in the orientation of the relevant structural elements (P-loop, switch 1, and switch 2) that join the subdomains promote both a decrease in affinity for orthophosphate and a chain of changes in neighboring residues, culminating in a tilt in the converter.

In contrast to this view, Förster resonance energy transfer (FRET) which, unlike crystallography, allows for the direct recording of structural dynamics within the myosin motor, indicates that the working stroke precedes Pi release (Muretta et al., 2015). Similar conclusions have been reached with single-molecule mechanics using a three-bead assay system (Takagi et al., 2004), which were later confirmed with an enhanced time resolution obtained by eliminating the effect of the trap compliance by imposing resistive loads from the very beginning of the interaction (Capitanio et al., 2012; Woody et al., 2019). A critical issue in all the above in vitro mechanical studies is that the loading condition under which the myosin–actin interaction occurs is far from reproducing the physiological load range, while the phenomena under investigation, the working stroke and the release of Pi, have individual unknown load sensitivities as far as their extent and kinetics are concerned. In particular, it must be considered that FRET measurements were achieved in unloaded conditions, and thus, the fractional occupancy of attached motors derived from this study is biased to the state at the end of the working stroke. On the other hand single-molecule measurements showed a working stroke rate that increases and a lifetime of actin-attached motors that decreases with the increase of the resisting load, opposite to the relations expected from the bulk of in situ mechanical and energetic data (Hill, 1938; Huxley and Simmons, 1971; Reconditi et al., 2004; Pertici et al., 2023).

In our experience, the most reliable approach for clarifying the issue of the relative timing and load-dependence of the working stroke and the Pi release is the sarcomere-level mechanics with nanometer-microsecond resolution achieved by integrating fast force transducers (Huxley and Lombardi, 1980) with the striation follower (Huxley et al., 1981), which was first developed for intact fibers isolated from frog skeletal muscle (Lombardi and Piazzesi, 1990) and then applied to single skinned fibers of the mammalian skeletal muscle (Linari et al., 2007; Caremani et al., 2013) (Supplementary Figure 1A). The striation follower is an optoelectronic apparatus that enables the measurement and control of half-sarcomere (hs) length changes in a selected population of sarcomeres, with sub-nanometer precision and 2-μs time resolution. As shown in Figure 3A and Supplementary Figure 1B, a stepwise reduction of half-sarcomere length by a few nanometers (complete in ∼100 μs) superimposed on the isometric contraction elicits a force transient composed of a force change simultaneous with the length step (phase 1) to a value T₁, which depends on elastic properties of the half-sarcomere, followed by a force recovery due to the active properties of the myosin motor. The earliest component is a rapid force recovery (phase 2), which represents the mechanical manifestation of the working stroke of attached motors synchronized by the step release. For releases larger than 6 nm per hs, the force recovered in phase 2 is only a fraction of the force decrease during the elastic response and is followed by a pause in force recovery (phase 3) that precedes the final recovery to the original isometric force T ₀ (phase 4). Phase 3 results from the synchronous attainment of the end of the working stroke by attached motors and their accelerated detachment, whereas phase 4 is explained by the isometric attachment–detachment kinetics. Phase 2 recovery holds the information used for developing a model of the working stroke (Huxley and Simmons, 1971; Ford et al., 1977): the maximum extent of filament sliding for which attached motors can maintain force (estimated by the abscissa intercept of the T ₂ relation) was ∼11 nm (Figure 3C), which is in agreement with the crystallographic model of the working stroke proposed 20 years later (Rayment et al., 1993a). The speed of phase 2 force recovery, r ₂, measured as the reciprocal of the time to recover from T ₁ to T ₁ + 0.63*(T ₂-T ₁), is larger for release than for stretch and increases with the release size (Figure 3D). This dependence of r ₂ on the step size and direction excludes the response of a passive viscoelasticity and holds the constraints for the kinetic model of force generation (Huxley and Simmons, 1971).

However, the presence of a significant compliance in the myofilaments, functionally in series with that of the motor array in each half-sarcomere (Huxley et al., 1994; Wakabayashi et al., 1994), complicates the use of the rate of quick force recovery r ₂ as the constraint for modeling the working stroke kinetics (Linari et al., 2009; Piazzesi et al., 2014). The rationale is illustrated with a simplified mechanical model of the half-sarcomere (Supplementary Figure 2), in which the distributed compliance of actin and myosin filaments is represented by an elastic element in series with the array of myosin motors. Under these conditions, force recovery occurs while motors move in the shortening direction to increase stress in the filaments, and its rate r ₂ decreases in proportion to filament compliance. The availability of a capacitance force transducer with a resonant frequency of 50 kHz (Huxley and Lombardi, 1980) and, thus, an adequately fast force clamp enabled an alternative protocol in which the velocity transient was recorded following stepwise decreases in force superimposed on the isometric steady force (Figure 3B; Piazzesi et al., 2002). The elastic shortening in phase 1 (L ₁) simultaneous with the force step is followed by a rapid phase-2 shortening (due to the synchronization of the working stroke in the attached motors) with an extent (L ₂ - L ₁, where L ₂ is the hs shortening attained at the end of phase 2) that is larger and attained earlier at lower force. L ₂ (the sum of the elastic strain and the working stroke) for a decrease in force to zero coincides with the abscissa intercept of the T ₂ relation (Figure 3C). Phase-2 shortening is followed by a pause (phase 3, due to synchronized detachment of motors) and then a steady shortening (phase 4) at a lower velocity that is characteristic of the detachment–attachment kinetics underpinning the force–velocity relation of the contracting muscle. The phases that follow the elastic (phase 1) response occur under force clamp and, thus, are not affected by the compliance of myofilaments. Under this condition, the dependence on force of the speed V ₂ estimated by the tangent to the initial part of phase-2 shortening (Figure 3B) directly measures the load dependence of the speed of the working stroke (Figure 3E), providing the constraints for a mechanical–kinetic model of the myosin working stroke in situ (Figures 4A–C).

For simplicity, the model considers only the mechanical and kinetic properties of the attached myosin motors that determine phase 2 of the responses to length and force steps; thus, the kinetic scheme is based on Huxley–Simmons 1971 model (Huxley and Simmons, 1971) and does not include further steps of the actin–myosin interaction, such as the attachment and detachment of motors. In this study, we provide a synthetic description of the model, which is described in detail by Piazzesi et al. (2014). The stiffness of the myosin motor is 3 pN/nm, and this sets the slope of the parabolas representing the free energy profile of attached states and, thus, the size (2.75 nm) and number (4) of the force-generating steps necessary to fit the transition and equilibrium kinetics of the working stroke. The rate functions of the transitions between the five structural states marking the progression of the working stroke (Figure 4D) were selected by the model simulation to achieve the best fit of the phase-2 velocity transient and its dependence on load (Figures 4E, F). Then, the same kinetic scheme was applied to simulate the relation between the rate of phase-2 force transient (r ₂) and the size of the step release (Figure 4G, filled triangles). The simulated relation was shifted upward with respect to the observed relation (open circles), with the simulated r ₂ already four times higher for 1.5 nm release. The difference between the observed and simulated relations was the consequence of neglecting the depressant effect of filament compliance on the rate of force generation by the motor array in the model. Assuming that the filament compliance was the only adjustable parameter, it was found that the same kinetic scheme can fit the observed relation when an equivalent filament compliance of ∼2 nm/T ₀ per hs is assumed (filled circles), a value in fairly good agreement with the filament compliance estimated with mechanical and X-ray diffraction experiments on the same preparation (Brunello et al., 2014).

The simplified kinetic scheme does not imply detachment/attachment; thus, the simulation of the velocity transients ends with the equilibrium distribution of attached states at each force, which underpins working stroke amplitude that reduces with the load, mostly due to the reduced contribution of the recoil of elastic elements (Figure 4E). The reduction in the observed working stroke amplitude with increase in the load is larger (Figure 3B) because the completion of the working stroke is progressively truncated by the ensuing synchronized detachment in phase 3.

These conclusions were straightforwardly confirmed in their structural counterpart, achieved by using the same mechanical protocol (Figure 5A) in combination with time-resolved X-ray diffraction from synchrotron light (Supplementary Figure 1) (Reconditi et al., 2004).

X-ray diffraction by itself lacks the phase information required to directly define the change in shape of the diffracting unit, but the size and direction of the movement can be recovered with sub-nanometer precision by exploiting the X-ray interference between the two bipolar arrays of motors in each sarcomere. Following the decrease in force, the changes in the interference fine structure of the so-called M3 reflection originating from the 14.5-nm axial periodicity of myosin motors (Figure 5B) indicate the movement of attached motors toward the center of the sarcomere in phase 2, coherent with the execution of the working stroke, and their movement away from the center at the onset of phase 3, associated with motor detachment at the end of the working stroke followed by re-attachment farther from the sarcomere center (Figure 5C). The structural data (Figure 5D, right panel) match the striation follower signal (left panel), indicating a larger and faster working stroke with decreasing load.

A reliable method for in situ recording of the working stroke is the prerequisite for investigating its sensitivity to [Pi]. By applying the striation follower technology and the fast force clamp protocols to skinned fibers from rabbit psoas (Caremani et al., 2013), it has been shown that increase in [Pi] to 10 mM (i) reduces the isometric force by 35%, without a significant effect on the unloaded shortening velocity (V ₀) (see also Cooke et al., 1988), (ii) does not affect the rate constant of the working stroke, as measured by the speed of phase-2 shortening, and (iii) shortens phase-3 pause and accelerates the subsequent transition to the final steady shortening velocity (Figure 6).

Most of the literature, in contrast to the above conclusion, reports a direct effect of Pi on the force generation process. This finding can be explained considering that those studies lack the resolution to discriminate between motor attachment and force generation (for instance, the process 2πb identified with sinusoidal analysis by Kawai and Halvorson, 1991). A study on the effect of [Pi] on the rate of the working stroke, which was determined in the same preparation using the quick force recovery elicited by length steps (Ranatunga et al., 2002), produced contradictory results: increasing [Pi] did not affect the rate of quick force recovery following a step release, whereas it increased the rate following a step stretch, mainly at the expense of its latter component. These results, however, can be justified by taking into account that in the quick force recovery following a step stretch, there is a contamination of the reversal of the working stroke by rapid detachment–attachment. With double-step protocols in single frog muscle fibers, it has been demonstrated that (i) the quick force recovery from a step release up to 6 nm per hs is complete within 2 ms and is almost fully accounted for by the synchronous execution of the working stroke because the detachment–attachment responsible for the rapid repriming of the working stroke is three times slower (Lombardi et al., 1992); (ii) the recovery from a stretch of 4 nm (a size comparable to that of Ranatunga et al., 2002), instead, takes ∼20 ms, and only 1/3 of it is explained by the synchronous reversal of the working stroke while the remaining part is due to the ensuing detachment–attachment that, in the force transient following a stretch, merges with the reversal within phase 2 (Piazzesi et al., 1997). Thus, the sensitivity to [Pi] of the later part of the recovery from a stretch is explained by the Pi-dependent acceleration of detachment in phase 3 of the velocity transient (Caremani et al., 2013).

In summary, the analysis of the velocity transients shows that the rate of the working stroke is independent of [Pi] and solely depends on the load and that the increase in [Pi] accelerates the phase-3 detachment–attachment (Caremani et al., 2013). These conclusions constrain the modeling of the chemically and structurally explicit cycle implemented from Reaction Scheme 1 (Reaction Scheme 4 presented later) to simulate both transient (phases 2 and 3) and steady-state (phase 4) shortening responses following the decrease in force below T ₀ (Caremani et al., 2013), which is in agreement with the original idea of Huxley and Simmons (1971) that the progression through the working stroke is controlled by rate functions that solely depend on the strain under which the transition from a lower to a higher force-generating state occurs. At the same time, the release of Pi from the catalytic site of the myosin motor can occur at any stage of the structural transitions. In this way, the release of Pi is an orthogonal process to the working stroke, and its rate does not directly depend on the motor strain, which is compelling for a chemical step. At the same time, as detailed in the section describing the properties of Reaction Scheme 4, the rate of Pi release can be made conformation-dependent, in which case it increases as the motor progresses through the working stroke.

Addition of Pi to the solution induces a decrease in the Ca²⁺-activated steady isometric force (T ₀) of a skinned fiber (Figure 7A) (Cooke and Pate, 1985; Kawai and Halvorson, 1991; Dantzig et al., 1992; Caremani et al., 2008) and, to a lesser extent, a decrease in the isometric ATPase rate (Figure 7B) (Webb et al., 1986; Bowater and Sleep, 1988; Cooke et al., 1988; Potma et al., 1995; Potma and Stienen, 1996). In 10 mM Pi, T ₀ is reduced to ½ the value in control solution (with 0.7 mM–1 mM of contaminating Pi Pate and Cooke, 1989a; Millar and Homsher, 1990), while the ATPase rate is reduced by less than 20%.

In terms of a conventional chemo-mechanical cycle in which Pi release occurs with the working stroke (Pate and Cooke, 1989a; Pate and Cooke, 1989b), the apparent contradiction was explained by hypothesizing that an increase in [Pi] shifts attached motors toward a state generating lower force without any effects on the ATP turnover. However, this idea was contradicted by evidence that the Pi-dependent reduction in force is accompanied by a proportional reduction in the number of attached motors (Caremani et al., 2008), a finding anticipated by work using sinusoidal analysis (Kawai and Halvorson, 1991), even though the conclusion in that case appeared questionable because of both the limited time resolution and the absence of analysis of the effects of filament compliance. A Pi-dependent reduction in the number of attached motors in isometric contraction without a comparable reduction in the ATP turnover indicates that the flux of myosin motors through the attachment/force-generating step (step 3 in Reaction Scheme 1) is larger than the flux through the conventional cycle estimated through steps 5 and 1, demonstrating that the conventional model cannot explain the reduced effect of [Pi] on the ATPase rate with respect to that on the force. The problem is solved by assuming a branched pathway that allows the myosin in the AM'ADPPi force-generating state to detach from actin before the release of ADP and Pi and then quickly release the hydrolysis products through an unconventional cycle (see Reaction Scheme 2; Linari et al., 2010). This conclusion was later supported by the results and modeling of the effect of Pi on the in vitro mechanics of a mini-ensemble of myosin motors (Debold et al., 2013). Notably, to be effective, the release of hydrolysis products from the AM'ADPPi state must be assumed irreversible; so, for the mass action, an increase in Pi promotes only the flux through the AMADPPi state.

Notably, the branched pathway introduced in Reaction Scheme 2, while enhancing the ATP consumption at any [Pi] fitting the observed relation (red line in Figure 7B), maintains the property of Reaction Scheme 1. It fits the dependence on [Pi] of either the rate of isometric force development (k _TR, Figure 7C; filled symbols, experiment; continuous blue line, Reaction Scheme 1; continuous red line Reaction Scheme 2) or the rate of force reduction elicited by a stepwise increase in [Pi] produced by photo-liberation of caged Pi during the isometric contraction (k _Pi, Figure 7C; open symbols experiment; dashed blue line Reaction Scheme 1; dashed red line Reaction Scheme 2) equally well. This is because both the schemes assume that force generation and Pi release occur as a two-step process, with the first step (force generation, corresponding to step 3 in Reaction Scheme 1) and the second step (the release of Pi, corresponding to step 4 in Reaction Scheme 1) being sufficiently fast (>500/s) to be considered in rapid equilibrium, with a K ∼10 mM (Dantzig et al., 1992; Homsher et al., 1997).

The first evidence of a loose coupling between the mechanical and biochemical cycle of actin–myosin interactions during shortening was the finding that the working stroke on the attached myosin motors could be reprimed much faster than expected from the rate of the ATP hydrolysis (Lombardi et al., 1992). With double-step releases, each <5 nm per half sarcomere, superimposed on the isometric contraction (Figure 9), it was demonstrated that, if delivered within 1–2 ms following the first release, the second step release elicits a quick recovery that corresponds to that elicited by the sum of the two releases. Meanwhile, the quick recovery from a second step release imposed progressively later, increases and recovers the value expected from a single-step release with a time constant of ∼6 ms (at 4 °C), which is at least four times faster than expected from the kinetics of the attachment step, underpinning the rate of isometric force development from 0 (time constant ∼26 ms; Figure 3A, B in Linari et al. (2015)).

Structural evidence of the fast repriming of the working stroke was obtained shortly afterward using X-ray diffraction (Irving et al., 1992), exploiting changes in the intensity of the third-order myosin-based meridional reflection to track the changes in the orientation of the lever arm of the myosin motor. In isometric contraction, the motor lever is oriented nearly perpendicular to the filament axis. During the quick force recovery after the step release, it tilts toward the center of the sarcomere, marking the synchronous execution of the working stroke, and then, within the next 15 ms, it recovers the isometric conformation in parallel with the recovery of the ability to generate a second working stroke, which is much faster than expected from the rate of the isometric force development.

The problem that the rate of motor attachment responsible for the transition from 0 to T ₀ in isometric conditions is too low to account for the rate of the working stroke regeneration during shortening is further exacerbated by the finding that the kinetics of the attachment step underpinning the development of isometric force is too low to account for the maximum power developed during isotonic contractions against loads of approximately 1/3 T ₀ (Pertici et al., 2023). On the other hand, the increase in the rate constant for the motor attachment required to fit the maximum power would imply an increase in the rate of force development to values larger than those observed. A way to keep the rate of isometric force development as low as required by that observed, preserving the power of the shortening contraction, is to assume a specific geometrical hindrance in the isometric condition consequent to the mismatch between actin and myosin periodicities (Marcucci et al., 2021). In this case, however, the fit of the observed power produced during shortening implies that the ATP hydrolysis rate (or the rate of energy liberation) increases by 8–9 times with respect to the isometric rate. This is a major drawback of the model of Marcucci et al. (2021) because the observed rate of energy liberation during an isotonic contraction developing maximum power is much lower, ranging from 2 (mammalian muscle) to 4 (frog muscle) times larger the isometric energy rate (Pertici et al., 2023, and references therein).

As shown by the model of Marcucci et al. (2021), any model that attempts to simultaneously simulate the limited rate of isometric force development and the power during isotonic shortening by increasing the apparent rate of attachment during shortening under the assumption that more motors become available with shortening (Chen and Brenner, 1993) finds its limit in an exaggerated increase in the rate of energy consumption for the maximum power relative to the isometric rate. This same limitation also applies to the Huxley and Tideswell (1997) model, in which the first attached motor of a myosin dimer during shortening facilitates the attachment of the partner motor. Eventually, the hypothesis that the reduced rate of isometric force development can be explained by mechanosensing-based thick filament activation (Linari et al., 2015), masking the effect of a rate of attachment that is high enough to account for the power developed during isotonic contraction, is contradicted on the energetic basis. In fact, in this case, the underlying ATP hydrolysis rate during steady isometric contraction would be fivefold higher than that observed (Marcucci and Reggiani, 2016).

The Reaction Scheme 3 (Figure 10) (Pertici et al., 2018) has the characteristics of a model that reconciles both the rapid regeneration of the working stroke determined in the transient regime (Lombardi et al., 1992) and the maximum power determined in the steady-state regime with the constraints that (i) the rate of isometric force development is four times slower than the rate of the working stroke regeneration following a step release and (ii) the rate of energy liberation in the isotonic contraction at the maximum power is no more than four times larger than that in the isometric contraction.

The demonstration, summarized in Figure 11 and Table 1, is given for the performance of the frog skeletal muscle at 4 °C–5 °C (Pertici et al., 2023), but, as shown in the same paper, it is equally valid for the fast mammalian skeletal muscle.

The three-state mechano-kinetic model of the actin–myosin interaction in Figure 10 has already been described in detail (Pertici et al., 2018; Pertici et al., 2020). The way in which the most relevant energetic features of the model are constrained by literature data is summarized here. In isometric contraction, the rate-limiting step in the cycle is detachment from the high-force generating state A2: the rate of ATP splitting per myosin motor (φ) is the minimum under the isometric conditions (φ₀). During steady shortening, φ increases as the rate of motor detachment increases due to the accelerated execution of the working stroke. φ for the maximum power (φ _Pmax) is higher than φ₀ by a factor of four (Barclay et al., 2010). Under this condition, the curvature of the F–V relation and the resulting maximum power can be reproduced only by assuming the integral version of Reaction Scheme 3 (3a), in which, during shortening, the attached myosin motors can rapidly regenerate the working stroke during the same ATPase cycle by slipping to the next actin monomer farther from the center of the sarcomere (Lombardi et al., 1992; Piazzesi and Lombardi, 1995) (step “slip” in Figure 10) and undergoing A1′–A2′ state equilibration according to the strain dependency of step-2 kinetics (Huxley and Simmons, 1971). Detachment from either A1′ or A2’ (step 3′) implies ATP hydrolysis.

All the relevant emergent properties of the array of motors in the half-thick filament (htf) of fast skeletal muscle are reproduced by Reaction Scheme 3a. In Figure 11, the F–V relation (A), where F is the force per htf, is shown, and the P–F relations (B), simulated with Reaction Scheme 3a (green lines), are superimposed on the experimental data (black lines). The corresponding simulated φ–V relation, where φ is calculated by the flux through step 1, is shown in Figure 11C (green line). The relevant parameters are shown in the first line of Table 1. Notably, the observed P _max is simulated with φ (φ _Pmax), which is four times larger than the value for the isometric force F ₀ (φ₀).

The limits of the conventional mechano-chemical cycle in which the ATP hydrolysis is completed within a single actin–myosin interaction are summarized in the simulations of Reaction Scheme 3b and c, where the slipping transition is removed. If no other adjustments are introduced (Reaction Scheme 3b), F ₀ and φ₀ are not substantially affected, while P _max is reduced to less than ½ (violet line in Figure 11B) without a marked change in φ _Pmax (violet arrows in Figure 11C; Table 1). With Reaction Scheme 3c, the mechanical performance of fast skeletal muscle (brown lines in Figures 11A, B) is recovered in the absence of slipping with ad hoc increases in the relevant rate constants for attachment and detachment (for the details of the kinetic assumptions, see Pertici et al., 2023). The consequences are that while φ₀ can be maintained in substantial agreement with the experimental value, φ _Pmax (brown arrow in Figure 11C; Table 1) increases by ∼10-fold with respect to φ₀.

The mechanical parameter that also in isometric conditions is sensitive to the kinetic differences between Reaction Scheme 3a and c is the rate at which force reaches the steady-state isometric value F ₀. This is expected from the consideration that the time course of force increase depends on the rate constants governing attachment (step 1) and force generation (step 2) and, specifically, in this model, on step 1, which is the rate limiting step in the process. To avoid the influence of the time required for thick filament activation on initial force development (Linari et al., 2015), the selected reference parameter is the force redevelopment following a 5% rapid shortening, which can reduce and maintain the isometric force at 0 (Figure 11D, black trace). The rise time t _r (the time from 10% to 90% of F ₀), estimated from the exponential fit to the recorded trace, is 52 ms, and it is perfectly reproduced by the simulation with Reaction Scheme 3a (Figure 11D, green trace, and first row in Table 1). In contrast, t _r is reduced to less than ½ in the simulation with Reaction Scheme 3c (Figure 11D, brown trace, and third row in Table 1), demonstrating that Reaction Scheme 3c fails to predict the observed low rate of the attachment-force development. Notably, t _r simulated by Reaction Scheme 3b (Figure 11D, violet trace, and second row in Table 1) fits the observed value. This is because Reaction Scheme 3b shares the same kinetic scheme as Reaction Scheme 3a, apart from the slipping transition, which is effective only for the performance of the shortening muscle.

Reaction Scheme 4 (Figure 12) (Caremani et al., 2013; Caremani et al., 2015) integrates the properties of Reaction Schemes 1–3, described in the previous sections, into a chemically and structurally explicit model that retains the predictive properties of the previous schemes and adds the possibility to both simulate the velocity transients elicited by step perturbations in force and their modulation by increase in [Pi] and identify the biochemical states that are involved in unconventional pathways such as early detachment from a force-generating state or, during shortening, slippage to the next actin monomer away from the sarcomere center.

The following is a summary of the Reaction Scheme 4. The two-step reaction responsible for force generation and release of Pi in Reaction Scheme 1 (steps 3 and 4, respectively), predicting the hyperbolic dependence of k _Pi (but also of k _TR) on [Pi] (Figure 7C), is reproduced in Reaction Scheme 4 (Figure 12) by steps 3 and 4 (and 4′). The early detachment from the force-generating AM'ADPPi state and the subsequent rapid release of the hydrolysis products in Reaction Scheme 2 (steps 6 and 7, respectively), which are required to explain the reduced effect of the increase of [Pi] on the ATPase rate (Figure 7B), are reproduced in Reaction Scheme 4 by steps 6 (and 6′) and 7 (and 7′), respectively. The slipping of an actin-attached motor during shortening to the next actin monomer farther from the sarcomere centre within the same ATP hydrolysis cycle (step “slip” in Reaction Scheme 3a, Figure 11), required for a rapid regeneration of the working stroke (Figure 9) and for the observed maximum power (Figure 11B) with a limited increase in the rate of energy liberation (Figure 11C) during steady shortening, is reproduced in Reaction Scheme 4 by steps 8 (slipping from the AMADPPi states) and 9 (slipping from the AMADP states).

The biochemically different force-generating states of Reaction Scheme 1 that are significantly populated in physiological conditions (AMADPPi and AMADP) in Reaction Scheme 4 exist in several conformations depending on the progression through the working stroke. Strong binding of M to an actin monomer A (step 3), caused by the closure of the actin-binding cleft (Geeves and Holmes, 2005), implies the formation of the first, AM₁ADPP_i, of several states generating progressively higher force. The kinetic, mechanical, and energetic features constraining the simulation of the working stroke have already been described in detail for the intact frog muscle fiber (Figure 4). In that case, the stiffness of the motor was 3 pN/nm so that the size of the force-generating transition, corresponding to the progressive swinging of the lever arm, was set to 2.75 nm (Piazzesi et al., 2014), and the number of transitions for an 11-nm working stroke was four (Figure 4D). In the case of the mammalian-skinned fiber, the smaller stiffness of the motor (≤1.7 pN/nm; Linari et al., 2007) allowed the assumption of a larger size of the transition (3.1 nm) and only three force-generating transitions, namely, M₁→ M₂, M₂→ M₃, and M₃→ M₄. According to the principle of the nearest-neighbor interaction, attachment to the actin monomer (with approximately 5.5-nm axial separation between monomers along a strand of the actin helix) occurs for a range of x from −2.75 nm to 2.75 nm, where x is the relative axial position between the motor and A, and is 0 for the position of the center of distribution of attachments of the motors in the M₁ state (Figure 13A). AM₁ADPPi undergoes the first transitions leading to the strained conformation responsible for the isometric force (Figure 13A).

Both the release of P_i (step 4) and the release of ADP (step 5) can occur from any of the four attached states, that is P_i release and and ADP release are orthogonal to the working stroke state transitions. The most important consequences for the aims of this report are as follows: first, the working stroke kinetics is independent of the concentration of Pi, as shown from the data in Figure 6D; second, the kinetics of Pi release can be made to depend on the conformation of the motor, increasing with the progression of the motor through the working stroke from M₁ to M₄. Notably, this kinetic feature is excluded in all the reaction schemes in which Pi-release coincides (Pate and Cooke, 1989b) or is in series (Dantzig et al., 1992; Homsher et al., 1997; Ranatunga, 1999; Smith, 2014; Stehle and Tesi, 2017; Offer and Ranatunga, 2020) with the force-generating transition. The rate functions of the reaction scheme are given in Supplementary Table 1. Further details on the calculation procedure can be found in Caremani et al. (2013) and Caremani et al. (2015).

Reaction Scheme 4 (Figure 12) retains all the properties of the three preceding schemes, and most of the mechanical and energetic outputs, observed for mammalian skinned fibers both without and with added P_i, are reported by Caremani et al. (2013). Therefore, its predictive power is demonstrated in detail here, mainly for the transient mechanical properties not simulated in the previous sections. In addition, with Reaction Scheme 4, as already demonstrated with the simplified Reaction Scheme 3 (Figures 11A, B), suppressing the possibility for an attached motor to slip to the next actin monomer farther from sarcomere centre during shortening increases the curvature of the T–V relation (Figure 13D, violet line) and reduces the maximum power (E, violet line) with respect to the experimental data (circles).

Again, as shown in the Reaction Scheme 4, the rate of transition from 0 to the steady-state isometric force is not affected by the suppression of the slipping possibility (Figure 14A; black trace, experimental record; green and violet lines, force rise simulated with Reaction Scheme 4 with and without steps 8–9, respectively).

Reaction Scheme 4 is unique in allowing simulation of the force and velocity transients elicited in response to a step in length or load that synchronizes the execution of the working stroke and its dependence on mechanical and biochemical conditions. As already discussed, the presence of a significant compliance in the myofilaments, functionally in series with the motor array in each half-sarcomere, complicates the use of the rate of quick force recovery as a constraint for modeling the working stroke kinetics (Piazzesi et al., 2014). Instead, phase-2 shortening elicited by a stepwise decrease in force (Figures 3B, 5D) directly provides information on the load dependence of the rate and the size of the working stroke, and the following phase-3 pause provides information on the kinetics of the events leading to working stroke regeneration. Reaction Scheme 4 can satisfactorily fit the isotonic velocity transient (Figure 14B: black, experiment; green, model), providing a direct demonstration of the role of slipping in the rapid regeneration of the working stroke, which is the condition for the maximization of the power during the more physiological steady shortening. Following the removal of steps 8–9, the phase-3 duration increases (Figure 14B, violet), delaying the transition to the steady shortening velocity of phase 4, with consequences on the steady-state (physiological) performance, such as the increase in the curvature of the force–velocity relation and the reduction in the power output (violet lines in Figures 13D, E). As already discussed for the simplified Reaction Scheme 3, this drawback, intrinsic to any conventional chemo-mechanical cycle, cannot be remedied by increasing the rate constant for the attachment step, because of the limit imposed to the kinetics of this step by the observed rate of isometric force development (Figures 11D, 14A).

A consequence of the idea that the release of Pi is orthogonal to the working stroke transitions is that only the kinetics of the working stroke is strain-dependent. Instead, Pi-release kinetics is sensitive to the structural state of the attached motor (conformation-dependent) and, thus, indirectly connected to the mechanical conditions by the speed of the progression through the different conformations (M₁–M₄). Under these conditions, the model satisfies the constraints that the speed of the working stroke (the rate of phase-2 rapid shortening, r ₂) elicited by a stepwise force decrease (i) increases with the size of the step (Figure 14C: filled symbols: black, experiment; green, model) and (ii) is not affected by the increase in [Pi] (Figure 14C; open symbols: black, experiment; green, model).

With smaller steps, i.e., at higher loads, the working stroke is slower (Figures 3B, 5D), which increases the probability that the other reactions taking over in the phase-3 pause (early detachment with the hydrolysis products in the catalytic site (step 6) and slipping to the next actin monomer (steps 8 and 9)) truncate the working stroke and lead to its regeneration. An increase in [Pi] alone also truncates the working stroke and accelerates phase-3 pause for the same relative decrease in force (Figures 6A–C, E). Notably, this effect is accompanied by an increase in both k _Pi and k _TR (Figure 7C), revealing that the involved kinetic steps are not restricted to the unconventional early detachment but also include the conventional cycle.

Reaction Scheme 4 can discriminate two different mechanisms leading to the accelerated working stroke regeneration during shortening: (i) at low Pi, the operating mechanism is based on the possibility that, for critical negative values of x, the force-generating motors, mainly those in the AMADP state, slip to the next Z-ward actin monomer (step 9), reducing phase-3 pause (Figure 14B, green) and, in this way, increasing the sliding distance per one ATPase cycle (see Figure 7A in Caremani et al., 2013), without any effect on k _TR (Figure 14A); (ii) the operating mechanism induced by the increase in Pi (Figure 14D: green, no added Pi; orange, 10 mM Pi) is based on the acceleration of the whole attachment-force generation and detachment cycle, as revealed by a corresponding increase in k _TR (Figure 14E; circles: experiment, line: model simulation), as a consequence of the increase in the second-order rate constant of the reversal of Pi release. Under these conditions, the model predicts that, in isometric contraction and at high loads, an increase in [Pi] increases the rate of ATP hydrolysis per motor (Figure 14F), which is the tension cost.

Conventional models in which attachment-force generation is strain-dependent can predict the finding from Pi jump experiments that the rate of the Pi transient is larger during isovelocity shortening than under isometric conditions (Figure 8A) and increases with an increase in shortening velocity (Figure 8B). However, those models cannot predict that during isovelocity shortening, the rate of Pi transient increases with an increase in final [Pi], as in the isometric contraction (Figure 8C). This is because in all those models, the Pi release is a step in series with the working stroke and thus can be defined by only one rate constant. In Reaction Scheme 4, Pi release is orthogonal to the working stroke, and its kinetics depends on the effects that the progression of the working stroke exerts on the catalytic site. Assuming that the rate of Pi release (k ₄) increases with the transition of motor conformation from M₁ to M₄ (Figure 15, see also Supplementary Material) is the sole condition that allows the rate of Pi transient to depend on either the velocity in isovelocity contractions (Figure 14G) or the final [Pi] both in isometric contraction (Figure 14H, green continuous line) and during shortening (green dashed line), as observed (circles, filled during isometric contraction, and open during shortening). If k ₄ is given a convenient constant value (100 s^-1), the simulation still predicts k _Pi to depend on the final [Pi] in isometric contraction (orange continuous line) but not during shortening (orange dashed line).