New paradigms in actomyosin energy transduction: Critical evaluation of non-traditional models for orthophosphate release

ReleaseoftheATPhydrolysisproductortophosphate(Pi)fromtheactivesiteofmyosin is central in chemo-mechanical energy transduction and closely associated with the main force-generating structural change, the power-stroke. Despite intense investigations, the relative timing between Pi-release and the power-stroke remains poorly understood. This hampers in depth understanding of force production by myosin in health and disease and our understanding of myosin-active drugs. Since the 1990s and up to today, models that incorporate the Pi-release either distinctly before or after the power-stroke,inunbranchedkineticschemes,havedominatedtheliterature.However, in recent years, alternative models have emerged to explain apparently contradictory findings.Here,wefirstcompareandcriticallyanalyzethreeinfluentialalternativemod-els proposed previously. These are either characterized by a branched kinetic scheme or by partial uncoupling of Pi-release and the power-stroke. Finally, we suggest critical tests of the models aiming for a unified picture.


INTRODUCTION
Adenosine triphosphate (ATP)-powered generation of force and motion by myosin molecules on actin filaments is central in eukaryotes. [1][2][3] The contractile machinery, consisting of actin, myosin, and accessory proteins, is particularly well-organized in striated muscle (skeletal muscle and heart) with the proteins assembled in μm wide myofibrils that fill the muscle cells ( Figure 1A). The myofibrils consist of series-connected half-sarcomeres which, in turn, are composed of myosin-containing thick filaments and overlapping actin-containing thin filaments in a regular lattice ( Figure 1B). The focus in this paper is F I G U R E 1 Striated muscle as example of actin-myosin contractile system. (A) An isolated myofibril, mounted for force-measurements. [36] Myofibrils fill the muscle cells and are made up of repetitive structures, sarcomeres. (B) Schematic of half-sarcomere with overlapping, thick, myosin-containing filaments, and thin, actin-containing filaments (with regulatory proteins troponin and tropomyosin) attached to Z-line. Accessory proteins, myosin binding protein C (MyBPC) and titin also shown. (C) Myosin molecule with two motor domains (heads or sub-fragment 1; S1). The proteolytic fragment heavy meromyosin (HMM) is also indicated. Note, that the orientation of the myosin molecule is flipped horizontally compared to its orientation in the half-sarcomere in B. (D) Myosin motor domain with key functional elements indicated as well as essential (ELC) and regulatory (RLC) light chains. (E) Schematic drawing of myosin S1 as used in figures below. Panels B-D from [37] reproduced under license CC BY 4.0.
also unclear exactly what happens when externally added Pi-rebinds to myosin. [18,27,[30][31][32][33][34][35] We define a "power-stroke" (or "working stroke"), similar to previous studies (e.g., [11,27,35] ), as a swing of the myosin lever arm ( Figure 1D) generating force and motion. Whereas the full range of the swing is expected to occur in the absence of load, a counteracting load, for example, during an isometric contraction, may translate the swing into elastic straining of the lever arm. Our power-stroke definition is consistent with current views in biochemistry, biophysics, and structural biology (cf. [2,24,38] ). However, it differs from the definition used in some theoretically oriented studies, (e.g., [39][40][41] ) where a "power-stroke" is often contrasted against Brownian ratchet mechanisms. Importantly, our use of the power-stroke term does not mean that we rule out roles played by Brownian ratchet mechanisms, or more generally thermal fluctuations, in generation of force and motion (see further [2,39,40,[42][43][44][45][46][47][48] ). While beyond the main scope of the present paper we discuss these and related issues briefly in the Supporting Information.
Early models [49,50] assumed that the power-stroke was simultaneous with Pi-release. In contrast, more recent models have separated these events, placing Pi-release from the active site either distinctly before (Figure 2A) or distinctly after ( Figure 2B) the power-stroke. [7][8][9][10]13,16,20,26,28,30,51] The idea of Pi-release after the power-stroke is consistent with transient biochemical kinetics studies combined with time-resolved fluorescence resonance energy transfer (FRET) used for detection of the power-stroke. [7] The idea is also in accordance with results evaluating transient mechanical responses to perturbations of length or tension at varied [Pi] in muscle cells and isolated myosin motors. [6,10] However, other data from mechanical experiments using isolated myosin and actin, [52] as well as X-ray crystallography and reverse genetics studies [2,16] argue for Pi-release before the power-stroke. Similar conclusions were reached based on theoretical analyses, [28] implying that Pi-release after the powerstroke would cause reduced maximal shortening velocity (V 0 ) of a muscle and only a minimal change in isometric force upon increased [Pi], contrary to experimental findings. [5,13,23,31] We denote the abovementioned models, with Pi-release from the active site either before or after the power-stroke without any branched pathways or other additional assumptions, as "conventional models". In addition to the problems summarized above and in Table 1, these models also have other limitations. [35] Thus, we found [35] that appreciably faster Pi-release rate than measured experimentally [7,16,[53][54][55] had to be assumed to account for V 0 as well as its [ATP] dependence (see further notes of Table 1). The abovementioned difficulties have led to development of alternative models. Recently, we built on ideas arising from experiments using myosin VI [2,16] to develop one such alternative model [35] ( Figure 2C). It assumes that, whereas Pi leaves the myosin active site before the power-stroke, Pi pauses at external (secondary and tertiary) sites before leaving the myosin head. The binding to these external sites explains the observed delay between the power-stroke and detection of the released Pi in solution [7,53] as well as a relatively slow Pi-release. [7,8,53,56] Moreover, this model also reconciles key contractile effects of varied [Pi], previously viewed as contradictory, including the [Pi]-independence of both V 0 and the power-stroke rate and a monotonous decrease in isometric force with increased [Pi]. Further details on the explanatory potential are given in Table 1 and below.
Despite the explanatory power of the described model with multistep Pi-release [35] ("multistep model" below), outstanding challenges remained. First, whereas we presented evidence for both secondary and tertiary Pi-binding sites in striated muscle myosin II we could not unequivocally demonstrate sufficiently slow Pi-release kinetics to account for the observed rate of Pi-release (with 5-100 ms time delay). [7,16,[53][54][55] Moreover, we did not compare the multistep model to other alternative models, for example, with regard to F I G U R E 2 Mechanokinetic actomyosin cross-bridge models reflecting current ideas for relationship between force-generating power-stroke and the release of Pi from the myosin active site. (A) Model with Pi-release from a Pi-release state (encircled) prior to force-generating power-stroke (from [20] ). (B) Model with Pi-release after power-stroke. (C) Multistep Pi-release model from Moretto et al. [35] with Pi either in the active site (next to ADP in figure) or at a secondary/tertiary site (encircled Pi; full definition in text). Note, the rates along the second and third rows in the scheme are the same. Pi-release state (subscript PiR) is here lumped together with the AMDPt' L start-of-powerstroke state. More details are given in inset within dashed box. (D) Simplified version of loose-coupling model [11] focusing on the Pi-release and its temporal relationship to the power-stroke. It is assumed that Pi can be released from the active site either before or after the power-stroke, however, without existence of secondary/tertiary Pi-binding sites. (E) Branched kinetic model [18,31,32] where Pi-rebinding leads to detachment into a post-power-stroke structural state (lower row). Only metastable states, not transition states, are indicated, but related activation energies are implicit in the rate constants. This follows the tradition when using the T.L. Hill formalism. [49,58,59] Key states are labelled by text boxes, with A: actin, M: myosin; T: ATP, D: ADP and P: inorganic phosphate. Rate constants and equilibrium constants are given by lower case and upper case letters, respectively. PiR, phosphate release state; H, high force; L, low force. The argument, (x), indicates that the rate constant (rate function) varies with the strain of the cross-bridge elastic element. Color coding: Detached states (black) and structurally different actin-attached states (different colors). Free energy diagrams (free energies of metastable states vs. x) are shown for models in A-C in Figure S1. TA B L E 1 Models grouped based on Pi-release before, after the power-stroke or undetermined temporal relationship.

Model
Reproducing V 0 versus [Pi] at pH 7 [23,30] Reproducing rate of power-stroke versus [Pi] (e.g., [6,10] ) Predicts re-synthesis of ATP upon Pi-binding [32,33] Main limitation Pi-release from active site before power-stroke (a) Figure 2A ( [20]20]; see also [28] ) a Yes No Yes -"No" in one column and requirement for higher Pi-release rate than measured to account for maximum velocity (b) Figure 2C (multi-step model; [34] ) b Yes Yes Yes -Rate of Pi-release from secondary and tertiary sites not known Pi-release from active site after power-stroke (a) Figure 2B (e.g., [9,10] ) a No Yes Yes -"No" in one column and requirement for higher Pi-release rate than measured to account for maximum velocity (b) Figure 2E (branched model; [27,30] Figure 2D (loose coupling model [11])d -Unclear if model predicts as slow Pi-release as observed experimentally a The results in the table assume very fast Pi-release rate (10 000 s -1 ) from myosin (and the active site). Pi-release rate as found experimentally (100 s -1 or less) [7,16,45,46] would predict too low V 0 . b The results were obtained with a Pi-release rate from myosin (100 s -1 ) similar to that found experimentally but transfer of Pi from the active site to the secondary site is fast. c In a paper [31] were this model is described in quantitative detail, the cross-bridge attachment, Pi-release step and power-stroke are lumped together, preventing the Pi-bound state from affecting V 0 . However, being recently [27] assumed to have Pi-release after the power-stroke it seems impossible for this model to account for V 0 unless the Pi-release rate is appreciably higher than 1000 s -1 . d The Pi-release rate from the active site varies between different mechanical states from <100 s -1 in pre-power-stroke states to 5000 s -1 in the last postpower-stroke state. This accounts for V 0 but there are other uncertainties (see last column and text) conceptual similarities and/or if they would be as successful in selfconsistently accounting for a range of experimental data. This includes one model [18,27,30,31] with a branched kinetic pathway (denoted "branched model" below) where Pi-release is assumed to occur after the power-stroke but where the cross-bridges may detach from a postpower-stroke state upon Pi-re-binding. A third alternative model [11,57] assumes loose coupling ("loose-coupling model" below) between the power-stroke and Pi-release (as well as ADP release) from the active site, where either the power-stroke or the Pi-release may occur first.
The latter model has similarities to the multistep model [35] but does not include external Pi-binding sites and, unlike the multistep model, Pi may leave the active site either before or after the power-stroke.
Here, we first review distinguishing features and explanatory potential of the three alternative models, proposed previously. [11,18,27,31,35,57] Our focus is on the temporal relationship between the power-stroke and Pi-release from the active site and from the myosin head (noting that these may differ). In this context, we also consider effects of Pi added from the solution. Then, we aim to objectively point out key strengths and weaknesses of the models and consider testable predictions. Figure 2 describes current models that relate the processes of Pi-release (yellow highlight in Figure 2) and Pi-rebinding to the power-stroke (green highlight in Figure 2). For the multistep model ( Figure 2C), the Pi-release, indicated by yellow highlight, refers to Pirelease from external sites. In all other models in Figure 2, the Pi in yellow highlight indicates Pi-release from the active site.

DETAILS OF CURRENT MODELS FOR PI-RELEASE AND RE-BINDING
Mechanokinetic models, like those in Figure 2, incorporate key molecular properties (strain-dependent actomyosin interaction kinetics, myosin elasticity, and coarse grain structure) into kinetic schemes to allow quantitative predictions of contractile function. [49,[58][59][60][61][62][63] The strain dependence is expressed in terms of a variable x, reflecting the distance, along the actin and myosin filaments, between a myosin head and the nearest binding site on actin. This variable is also explicitly related to the strain of the cross-bridge elastic element, with different relationships for different biochemical cross-bridge states (see below).
The models in Figure 2C-E, represent the alternative models in focus here: the multistep model [35] ( Figure 2C), the loose coupling model (full kinetic detail in [11,57] ; see also [21] ; Figure 2D) and the branched model [27,31,32] ( Figure 2E). All models are grouped in Table 1 based on whether Pi is released from the active site before or after the power-stroke (or neither).

ALTERNATIVE MODEL 1: MULTISTEP ORTHOPHOSPHATE RELEASE
The multistep model [35] has foundations in structural and reverse genetics studies of Llinas et al., [16] applied primarily to myosin VI. Their central result was evidence for the so-called Pi-release state, the state from which Pi is expected to leave myosin. [16] This state, that had not been previously observed, is characterized by stereo-specific weak, actin-attachment that differs from the initial actin-attached state by shift of the nucleotide-interacting switch-II to an open position, being the basis for actin-activation of Pi-release, that is, the opening of the back door is triggered by actin binding. We recently incorporated the Pi-release state into a mechanokinetic model for myosin II operation, [19,20] allowing us to model the mechanism of action of the small molecular myosin II inhibitor blebbistatin. [20] Later, structural modelling ( Figure 3A) corroborated the Pi-release properties of a similar state in myosin II. [35] Of further importance for the development of the multistep model, [35] Llinas et al. [16] found evidence for a secondary Pi binding site outside the active site in myosin VI, consistent with previous suggestions. [28] Using molecular modelling [35] ( Figure 3B), we demonstrated a similar secondary Pi-binding site in myosin II. Additionally, we found evidence for tertiary Pi-binding sites, possibly related to a positive electrostatic surface potential ( Figure 3C). The latter evidence is based on single molecule studies showing that increased [Pi] competitively inhibited non-specific binding of fluorescent ATP to myosin. [35] The tertiary sites were not explicitly considered by Llinas et al., [16] but gained strong support from the single molecule competition assay. [35] The latter results from myosin II suggest overlap with non-specific ATP binding sites and positively charged residues ( Figure 3C-D). Altogether, there is strong evidence [2,16,66,67] that Pi, after its release from the active site, is guided via the so-called back door ( Figure 3A) to bulk solution. Additionally, the evidence for Pibinding sites external to the active site in close proximity to the back door, makes it likely that Pi pauses at these sites before leaving the myosin head. The explanatory power of this multistep model is substantial (e.g., Table 1; Introduction), accounting for a range of contractile phenomena that appear contradictory when interpreted using conventional models. [35] On the one hand, this includes the [Pi] independence of V 0 and the monotonous decrease in isometric force with increased [Pi]. On the other hand, the model also accounts for the slow Pirelease [7,16,53,54] and the simultaneous lack of effects of altered [Pi] on mechanical transients of muscle and isolated actomyosin. [6,10,27] It is of interest to consider these characteristics in relation to optical tweezers data suggesting fast Pi-dependent detachment rates of early states of the actin-myosin complex. [10] These data were, together with other findings, taken to support the idea of Pi-release after the power-stroke. [10] However, Robert-Paganin et al. [2] argued that the finding is consistent with opening of the Pi-release tunnel before the power-stroke, that is, that Pi leaves the active site before the stroke, as in the multistep model. Other strengths of the multistep model [35] include evidence from both experiments and modelling [16,35] for the existence of secondary as well as tertiary Pi-binding sites. A limitation is that it is unclear if the kinetics of Pi-binding and subsequent release from secondary and tertiary sites is sufficiently slow to account for the delay (5-100 ms; varying between myosin isoforms) [7,16,53,54] of Pi-appearance in solution.

ALTERNATIVE MODEL 2: THE LOOSE COUPLING MODEL
The loose coupling model of Caremani et al. [11,57] (see also [21] ) was developed to account for the effects of varied [Pi] on transient and steady-state effects of changes in load on isometrically contracting muscle fibers. The general principles are illustrated in Figure 2D with details in Figure 4. [11,57] The key features of the model are: (i) the power-stroke has three sub-strokes corresponding to one pre-powerstroke state (upper row in Figure 4A) and three post-power-stroke states (M 2 -M 4 in Figure 4A) with strain-dependent transition rates in between, (ii) each sub-stroke occurs with the same rate whether both ADP and Pi or only ADP is at the active site, (iii) the release of Pi (step 4 in Figure 4A) occurs before release of ADP (step 5 in Figure  The myosin II surface structure similar to that in C (PDB: 5N6A) but showing positively charged amino acids (blue) and ATP-binding areas (yellow) identified using the program ATPint [70] (for further details see [71] ). Positively charged amino acids that overlap or are very close to identified ATP binding regions are labeled in boxes. Dashed circle indicates opening of back door as in C. Panels A-C reproduced from [35] under license CC BY 4.0. Panel D rendered using VMD (v. 1.9.3; http://www.ks.uiuc.edu/Research/vmd/. [72] the actin filament while it is at an intermediate state of its biochemical and structural (force-generating) cycle ( Figure 4B and steps 8-9 in Figure 4A), and (vi) myosin may undergo unconventional detachment from actin with the hydrolysis products still at the active site (step 7 in Figure 4A), followed by a fast release of the products and binding of a new ATP. The latter feature was originally introduced, [73] to explain a smaller reduction with increased [Pi] in ATP turnover rate than in tension development during isometric contraction. [34,74,75] The loose coupling model, similar to the multistep model, does not assume a fixed temporal relationship between the power-stroke With lack of the external Pi-binding sites in the loose coupling model, the Pi-release from the active site must be sufficiently slow to account for the slow appearance of Pi in solution [7,16,53,54] because this effect cannot be attributed to binding outside the active site. At the same time, the Pi-release rate must be sufficiently fast to be compatible with V 0 of muscle. Finally, contrary to the ideas of Houdusse, Sweeney and co-workers, [2,16] Pi release from the active site cannot be a necessary condition for occurrence of the sub-strokes in the loose coupling model.
Despite the abovementioned differences the loose-coupling model seems to give similar predictions as the multistep model [35] in sev-eral respects. This includes [Pi] independence of both the power-stroke rate and V 0 . [11] Presumably, the loose-coupling model also predicts a monotonous reduction in isometric force with increased [Pi] although this was not explicitly tested for a range of Pi-concentrations. However, it is not clear that the model can explain the findings that Pi-release from actomyosin in solution is appreciably slower than the powerstroke. [7] In the loose-coupling model, the Pi-release rate is taken as > 1000 s −1 for states towards the end of the power-stroke (presumably to account for the high V 0 ). These states (e.g., M 4 in Figure 4A) are likely to be heavily populated in solution experiments in which no elastic strain prevents the power-stroke. Therefore, one would expect the loose-coupling model [11] to predict a negligible delay between the power-stroke and the Pi-release, contrary to observations by Muretta et al. [7] To summarize, a challenge related to the loose coupling model is whether it can simultaneously account for the slow Pi-release rate observed in biochemical studies and the high V 0 observed in muscle and in vitro motility assays. A question that also must be answered is if the power-stroke can occur with Pi in the active site.

ALTERNATIVE MODEL 3: BRANCHED MODEL
The branched model [18,27,[30][31][32] is characterized by: (i) a branched kinetic pathway with Pi-binding to a post-power-stroke, actin-attached cross-bridge state, leading to detachment of the myosin head into a state with the lever arm in the post-power-stroke position ( Figure 2E).
(ii) Pi binding to the active site in the rigor (AM) state, competing with ATP binding to the active site. In early versions of the model the initial Pi-release (before any Pi-rebinding) was lumped together with myosin-actin attachment and the power-stroke. However, in a recent version [27] this Pi-release from the active site was believed to occur after the power-stroke (cf. Figure 2E) which seems logical considering that Pi-re-binding to the active site is assumed to occur in the post-power-stroke state.
The main discriminating feature between this model, the multistep model [35] and most other models, is the branched pathway (i). The other discriminating feature, the competitive inhibition by Pi of ATP binding to the active site in the rigor state (ii), is not unique to this model (see e.g., [35,76] ) and we will not consider it further as it is not directly relevant for the Pi-release mechanism. Regarding the branching, a similar pathway (with somewhat different properties) was considered for isometric contraction by Linari et al., [73] using a simple kinetic scheme. In the more complete mechanokinetic analysis of that group, [11,57] giving the loose coupling model, the branched pathway only plays a secondary role. We therefore focus on the version put forward by Debold, Walcott and co-workers. [18,27,31,32] The main motivations behind the branched model were to account for two findings that are difficult to explain using conventional models [18] : First, that increased [Pi] reduces isometric force development appreciably more than the ATP turnover rate during isometric contraction and second, that the gliding velocity in the in vitro motility assay is increased with increased [Pi] at low pH (< 7). With regard to the first of these findings, a serial kinetic scheme, without consideration of cross-bridge elasticity (e.g., no x-dependence of rate constants in Figure 2A-B) was found to predict [73] a larger reduction of the isometric ATP turnover rate than of isometric force upon increased [Pi].
This effect is in striking contrast to what is seen experimentally with small reduction in the isometric ATP turnover rate compared to isometric force with increased [Pi]. This discrepancy could be accounted for by introducing a branched pathway as in Figure 2E. [73] The second motivation for the branched model was that the gliding velocity in the in vitro motility assay increases in response to increased [Pi] at low pH. The authors [31] proposed that Pi-binding to the post-power-stroke AMADP state, followed by detachment into a post-power-stroke state, would explain this result. They relied on findings [77] that the AMADP state is prolonged compared to the AM (rigor) state at low pH and high [ATP], making the lifetime of the AMADP state more important in limiting the detachment rate and thereby the maximum actin gliding velocity. This would explain that Piinduced detachment from the AMADP state has greater effect on the velocity [31] at low pH. A subsequent detailed mechanokinetic version of the branched model seems to also account for a range of additional effects of increased [Pi]. [32] However, that version [32] lumped myosin-actin attachment, Pi-release and power-stroke together into one transition, preventing assessment of effects of low Pi-release rate on V 0 . [7,16,53,54] If the Pi-release step (with the experimental rate) is explicitly included, it seems clear that this model would predict appreciably lower V 0 than found experimentally, in similarity to other models with Pi-release from the active site after the power-stroke (see notes of Table 1).
There are other issues with the branched model. First, the lower effect of varied [Pi] on the isometric ATP turnover rate than on isometric force is not unique to models with branched pathways. The effect is an inherent component of a range of other mechanokinetic models [19,28,50] that explicitly include effects of the cross-bridge elasticity (e.g., strain-dependence of rate constants). As outlined above, the strain dependence can be described by a variable x. If this variable is defined as the strain of the elastic element in the post-power-stroke cross-bridge state, the strain in the pre-power-stroke state is equal to x-h if h is the power-stroke size and if there is only one powerstroke (cf. [62] ). Assuming such a simple model, cross-bridge attachment into the pre-power-stroke state occurs primarily for large x (around x = h) whereas, cross-bridge detachment (including ADP release and ATP re-binding) from the post-power-stroke state primarily occurs at low x. The cross-bridges that carry highest force are those in the post-power-stroke state at high x, together with cross-bridges in their pre-power-stroke state with high strain already at attachment. These high-force cross-bridges are, however, least prone to undergo detachment by ADP release and ATP binding (due to strain-dependence of rate constants) but most prone to undergo detachment by reversal of the power-stroke as a result of Pi-rebinding (cf. [2,19,78,79] ). In isometric contraction, this predicts that increased [Pi] alters the steady-state cross-bridge distribution as illustrated in Figure 5 for the model in [19] (similar to Figure 2A) Similar predictions that a significant portion of the isometric force is maintained by pre-power-stroke cross-bridges follows from theoretical arguments [80] and seem compatible with electron paramagnetic resonance results from muscle cells. [81] Regarding the [Pi]-effects on velocity at low pH, there are also limitations that need to be considered. First, to the best of our knowledge, the increased velocity in response to increased [Pi] at low pH has only been demonstrated experimentally using in vitro experiments on isolated proteins. [31] These results may differ from those obtained in muscle fiber experiments (cf. [63,[82][83][84] ). Accordingly, V 0 in muscle cells, Modified from [19] under license CC BY 4.0.
in contrast to the in vitro motility assay data in, [31] seem to be little affected by altered [Pi] at low pH. [17,85] Indeed, some studies have even found that increased [Pi] and lowered pH are synergistic in reducing velocity in muscle fibers. [86] These issues need to be further elucidated.
Even if the experimental [Pi]-pH velocity relationship is consistent with the branched kinetic model, it has been found [33,34] that Pi-rebinding to myosin II leads to re-synthesis of ATP. This is consistent with a reversal of the attachment step and the ATP hydrolysis step on the myosin active site (e.g., models in Figure 2A-D) rather than detachment into a post-power-stroke state as in the branched model ( Figure 2E).
Effects of an S217A mutation in myosin V (S236A in myosin II and S203A in myosin VI) have been taken as support for the branched model, combined with initial Pi release after the power-stroke. Forgacs et al. [87] found an appreciable slowing of the Pi-release step in myosin V with the S217A mutation as did Llinas et al. [16] for the corresponding mutation in myosin VI and myosin II. Scott et al. [27] used optical trapping to study the effects of the myosin V S217A mutation on the power-stroke in single molecules. They found that, both in the presence and absence of 30 mM Pi, wild-type myosin V produced a fast power-stroke (≥500 s −1 ) without detectable delay following actin binding, but with the 25% longest binding events eliminated by added Pi. Further, they found that the S217A myosin V construct generated a power-stroke following binding to actin that was similar in rate and size as for wild-type myosin V. All these findings were taken as evidence [27] that myosin V generates its power-stroke with Pi in its active site and that Pi-re-binding leads to detachment into a post-power-stroke state as proposed in the branched model. However, these results warrant interpretation within the framework of differ-ent models. First, even if the S217A mutation prevents (or slows) the entrance from the active site into the back door, leading to slower Pirelease, it is not clear if the Pi would stay in the active site for longer time due to this effect. Instead, Pi may be trapped in an intermediate position like the secondary site in the multistep Pi-release model [35] .
Indeed, the corresponding serine residues in myosin VI [16] and myosin II [35] do not only communicate with the active site but also participate in Pi-coordination in the secondary Pi-binding site outside the active site. Accordingly, the effect of the S217A mutation on Pi-release, previously [2,16] led to the conclusion that Pi must leave the active site to enable the power-stroke.

Multistep model
Despite firm evidence for external Pi-binding sites, it is neither clear that the Pi leaving the active site actually pauses at these external sites nor that Pi-binding kinetics is sufficiently slow to account for the observed Pi-release rate. [7,16,53,54] If the off rate from the external sites is faster than the overall Pi-release rate, it would suggest that Pi-dissociation from the active site is rate limiting, arguing against the multistep model. Indeed, such a finding would also argue against the idea that pausing of Pi at external sites is of any importance at all. Possibly, the latter idea may also be tested by using fluorescent nucleotides (e.g., ε-aza-ATP [88] ) or fluorescent amino acids/probes (cf. [22] ) provided that binding of Pi to external sites affects the local environment of the fluorophore in question.
We proposed [35] that the kinetics of Pi-binding at external sites (and indeed any such binding) may be tested by mutations of these sites that either change the binding kinetics or disrupt the sites. We specifically proposed that the double mutation R243E/E466R in cardiac myosin II could be useful [89] and preferred over the single R243E mutation (disrupting the secondary Pi-site) because the double mutation is expected to preserve other aspects of actomyosin function. [89] We have found that both the R243E and R243E/E466R mutations are properly transfected into C2C12 cells by a new virus free method [90] that would allow convenient screening of a wide range of mutants. Both mutants also express well ( Figure S2). We will next perform basic functional characterization followed by quantification of the Pi-release rate constant. The advantage of using cardiac myosin II compared to Dictyostelium myosin is that the R243 residue is also a hotspot for disease causing mutations, for example, in cardiomyopathies. [91,92] The multistep model predicts that secondary Pi-binding accounts for a major fraction of the delay of up to 100 ms before Pi appears in solution after the power-stroke (data from bovine cardiac βmyosin [55] ). Therefore, we expect that the R243E/E466R mutation, disrupting the secondary site, would appreciably increase the Pidissociation rate from 10 s −1 or somewhat higher in wild-type human β-myosin to possibly 1000 s −1 .
Recently, [35] we raised the possibility that Pi-binding to the tertiary sites may contribute significantly to the observed delay of Pi-appearance in solution. This should be testable using a similar strategy, that is, quantification of the Pi-release rate of β-myosin constructs with tertiary site mutations designed to prevent Pi-binding. However, the location of the tertiary sites is less well-defined. On the one hand, we noted a positive surface electrostatic potential around the opening of the back door for Pi release ( Figure 3C). On the other hand, we found that the Pi binding outcompeted almost all non-specific Alexa647-ATP binding to the surface of the myosin head. The possibility that theAlexa647-ATP binding sites include the positively charged regions around the back door opening is analyzed for bovine cardiac β-myosin in Figure 3D. Here, we indicate all positively charged amino acids (blue) on the myosin surface close to the back door opening together with residues (yellow) involved in non-specific ATP binding according to the program ATPint [70] ; see also [71] ). Specifically, we label positively charged amino acids in Figure 3D that are parts of the non-specific ATP-binding sites or in close proximity to these. Such positively charged amino acids would be interesting candidates for mutations to disrupt tertiary Pi-binding sites. Importantly, these residues, letter coded in Figure 3D, are conserved between bovine ( Figure 3) and human cardiac β-myosin (sequence alignment in ClustalW).
After mutation of secondary and putative tertiary Pi-binding sites, according to the above strategy, and further strategies used in similar studies of another molecule, [93] we will characterize the general functionality, followed by detailed studies of the Pi-release.

Loose coupling model
Testable predictions of the loose coupling model include: (i) Pi must be allowed to leave the active site either before or after the power-stroke, (ii) the secondary and tertiary Pi-binding sites play no role in delaying the Pi-appearance in solution, and (iii) the rates of Pi-release from the pre-and the three post-power-stroke states must be such that both V 0 and the delay of Pi-appearance in solution after the power-stroke are accounted for. The first of these predictions (i) has been partly tested in high-speed atomic force microscopy (hs-AFM) recordings [35] and single molecule force measurements, [52] suggesting that the powerstroke can occur without Pi in the active site. What remains to be shown is that the reverse is also true, that is, that the power-stroke can occur with Pi in the active site. If Pi release from the active site is very fast, it may be challenging to test the latter possibility. However, the loose coupling model assumes rather slow Pi-release from the active site, suggesting that testing may be feasible. Testing the second prediction (ii) that the secondary and tertiary sites do not contribute to delayed appearance of Pi in solution was considered above. Thus, if mutations of the secondary and tertiary sites do not modify the delayed Pi-appearance in bulk solution, the remaining possibility seems to be that the delay is due to slow dissociation from the active site as in the loose coupling model. However, in relation to the third testable prediction (iii), it then also remains to be shown that both such a slow Pi-appearance and a high V 0 can be explained by the same set of model parameter values using a mechanokinetic model (as in [11] ).

Branched model
Most importantly, the classical findings [33,34] that ATP is resynthesized upon Pi-binding need to be explained within the framework of the branched model. These results suggest that Pi-rebinding leads to reversal of the original Pi-release and attachment steps, rather than to detachment via a branched pathway. In such models (those in Figures 2 except the branched model in Figure 2E), one would expect a nucleotide dwell time distribution in single molecule studies [71,76,94] with more slow events upon increased [Pi]. This is because some of the crossbridges that are detached through reversal of the Pi-release and the attachment steps may eventually go through several such back-andforth transitions. The effect would be expected to be most prominent in conditions with already prolonged dwell-times in low-force states (e.g., AMD L in Figure 2) that are particularly highly populated under isometric conditions. Similar effects of increased [Pi] are not expected in the branched pathway if the Pi-re-binding and subsequent product release rates (lower row of Figure 2E) are as fast as assumed in a recent version of the model [32] (see also [73] ). Evidence suggesting more slow ATP dwell-times in an isometric single molecule assay upon increased [Pi], was provided by Amrute-Nayak et al. ( [76] ). A recently modified version [71] of their isometric single molecule ATPase assay is likely better at detecting long events due to removal of artifacts that tend to bias the dwell-times towards short events. Therefore, in order to better distinguish between branched and unbranched models, it would be of interest to repeat the previous experiments, [76] to more definitely corroborate the idea that increased [Pi] increases the fraction of long fluorescent nucleotide dwell-times. It is important to note, in this connection, that both the branched model and the multistep Pi-release model would account for loss of the longest actin-binding events upon increased [Pi] in optical tweezers studies of myosin V. [27] Thus, the longest actin-binding events may equally likely be eliminated by detachment into a post-power-stroke state and by reversal of the cross-bridge attachment. Only in the latter case, however, would the ATP dwell-times be expected to increase as explained above.

General tests
A further way to evaluate effects of Pi-release and binding events would be to monitor effects of the release of caged Pi from wild type and mutated myosin in real-time using either optical tweezers Pi release. It has the potential to study structural changes related to Pi-binding; we hope to soon bypass time limitations by applying a single-line scanning method. This approach enables image scanning, while the AFM cantilever probe is at a fixed x-y position. It allows monitoring of the height fluctuations under the cantilever probe in the z-direction, producing single-line height data (x, z, t) with Angstrom spatial and microsecond (μs) temporal resolution. [95] CONCLUSIONS This review has focused on the timing of the power-stroke and Pirelease from the active site and from the myosin head. We described and classified (Table 1) three alternative models that have previously been proposed for the temporal relationship between the powerstroke on the one hand, and Pi-release and Pi-rebinding to myosin on the other. We also highlighted strengths and weaknesses of the different models and discussed approaches for further testing. Gaining a complete understanding of the Pi-release mechanism, and its relation to the power-stroke, is crucial, not only for full insight into the fundamental mechanisms of energy transduction by actin and myosin. It is also increasingly important because this mechanism is targeted by several small molecular myosin-active compounds with therapeutic potential [55,[96][97][98][99] in cancer, [100] heart failure, [101] cardiomyopathies, [102,103] and skeletal muscle disorders. [104] We hope that experimental tests proposed here may be helpful.