Actomyosin based contraction: one mechanokinetic model from single molecules to muscle?

Mechanistic insights into effects of the cardiac myosin activator omecamtiv mecarbil from mechanokinetic modelling

Introduction

Small molecular compounds that affect the force, and motion-generating actin-myosin interaction in the heart have emerged as alternatives to treat or alleviate symptoms in severe debilitating conditions, such as cardiomyopathies and heart failure. Omecamtiv mecarbil (OM) is such a compound developed to enhance cardiac contraction. In addition to potential therapeutic use, its effects may help to elucidate myosin energy transduction mechanisms in health and disease and add insights into how the molecular properties govern contraction of large myosin ensembles in cardiac cells. Despite intense studies, the effects of OM are still incompletely understood.

Methods

Here we take an in silico approach to elucidate the issue. First, we modify a model, previously used in studies of skeletal muscle, with molecular parameter values for human ventricular β-myosin to make it useful for studies of both myosin mutations and drugs. Repeated tests lead to at a set of parameter values that allow faithful reproduction of range of functional variables of cardiac myocytes. We then apply the model to studies of OM.

Results and discussion

The results suggest that major effects of OM such as large reduction of the maximum velocity with more limited effects on maximum isometric force and slowed actin-activated ATPase can be accounted for by two key molecular effects. These encompass a reduced difference in binding free energy between the pre- and post-power-stroke states and greatly increased activation energy for the lever arm swing during the power-stroke. Better quantitative agreement, e.g., isometric force minimally changed from the control value by OM is achieved by additional changes in model parameter values previously suggested by studies of isolated proteins.

Mechanisms of myosin II force generation: insights from novel experimental techniques and approaches

Myosin II is a molecular motor that converts chemical energy derived from ATP hydrolysis into mechanical work. Myosin II isoforms are responsible for muscle contraction and a range of cell functions relying on the development of force and motion. When the motor attaches to actin, ATP is hydrolyzed and inorganic phosphate (Pi) and ADP are released from its active site. These reactions are coordinated with changes in the structure of myosin, promoting the so-called "power stroke" that causes the sliding of actin filaments. The general features of the myosin-actin interactions are well accepted, but there are critical issues that remain poorly understood, mostly due to technological limitations. In recent years, there has been a significant advance in structural, biochemical, and mechanical methods that have advanced the field considerably. New modeling approaches have also allowed researchers to understand actomyosin interactions at different levels of analysis. This paper reviews recent studies looking into the interaction between myosin II and actin filaments, which leads to power stroke and force generation. It reviews studies conducted with single myosin molecules, myosins working in filaments, muscle sarcomeres, myofibrils, and fibers. It also reviews the mathematical models that have been used to understand the mechanics of myosin II in approaches focusing on single molecules to ensembles. Finally, it includes brief sections on translational aspects, how changes in the myosin motor by mutations and/or posttranslational modifications may cause detrimental effects in diseases and aging, among other conditions, and how myosin II has become an emerging drug target.

Probing actin-activated ATP turnover kinetics of human cardiac myosin II by single molecule fluorescence

Mechanistic insights into myosin II energy transduction in striated muscle in health and disease would benefit from functional studies of a wide range of point-mutants. This approach is, however, hampered by the slow turnaround of myosin II expression that usually relies on adenoviruses for gene transfer. A recently developed virus-free method is more time effective but would yield too small amounts of myosin for standard biochemical analyses. However, if the fluorescent adenosine triphosphate (ATP) and single molecule (sm) total internal reflection fluorescence microscopy previously used to analyze basal ATP turnover by myosin alone, can be expanded to actin-activated ATP turnover, it would appreciably reduce the required amount of myosin. To that end, we here describe zero-length cross-linking of human cardiac myosin II motor fragments (sub-fragment 1 long [S1L]) to surface-immobilized actin filaments in a configuration with maintained actin-activated ATP turnover. After optimizing the analysis of sm fluorescence events, we show that the amount of myosin produced from C2C12 cells in one 60 mm cell culture plate is sufficient to obtain both the basal myosin ATP turnover rate and the maximum actin-activated rate constant (k cat). Our analysis of many single binding events of fluorescent ATP to many S1L motor fragments revealed processes reflecting basal and actin-activated ATPase, but also a third exponential process consistent with non-specific ATP-binding outside the active site.

Modeling thick filament activation suggests a molecular basis for force depression

Multiscale models aiming to connect muscle's molecular and cellular function have been difficult to develop, in part due to a lack of self-consistent multiscale data. To address this gap, we measured the force response from single, skinned rabbit psoas muscle fibers to ramp shortenings and step stretches performed on the plateau region of the force-length relationship. We isolated myosin from the same muscles and, under similar conditions, performed single-molecule and ensemble measurements of myosin's ATP-dependent interaction with actin using laser trapping and in vitro motility assays. We fit the fiber data by developing a partial differential equation model that includes thick filament activation, whereby an increase in force on the thick filament pulls myosin out of an inhibited state. The model also includes a series elastic element and a parallel elastic element. This parallel elastic element models a titin-actin interaction proposed to account for the increase in isometric force after stretch (residual force enhancement). By optimizing the model fit to a subset of our fiber measurements, we specified seven unknown parameters. The model then successfully predicted the remainder of our fiber measurements and also our molecular measurements from the laser trap and in vitro motility. The success of the model suggests that our multiscale data are self-consistent and can serve as a testbed for other multiscale models. Moreover, the model captures the decrease in isometric force observed in our muscle fibers after active shortening (force depression), suggesting a molecular mechanism for force depression, whereby a parallel elastic element combines with thick filament activation to decrease the number of cycling cross-bridges.

Modeling Thick Filament Activation Suggests a Molecular Basis for Force Depression

Multiscale models aiming to connect muscle's molecular and cellular function have been difficult to develop, in part, due to a lack of self-consistent multiscale data. To address this gap, we measured the force response from single skinned rabbit psoas muscle fibers to ramp shortenings and step stretches performed on the plateau region of the force-length relationship. We isolated myosin from the same muscles and, under similar conditions, performed single molecule and ensemble measurements of myosin's ATP-dependent interaction with actin using laser trapping and in vitro motility assays. We fit the fiber data by developing a partial differential equation model that includes thick filament activation, whereby an increase in force on the thick filament pulls myosin out of an inhibited state. The model also includes a series elastic element and a parallel elastic element. This parallel elastic element models a titin-actin interaction proposed to account for the increase in isometric force following stretch (residual force enhancement). By optimizing the model fit to a subset of our fiber measurements, we specified seven unknown parameters. The model then successfully predicted the remainder of our fiber measurements and also our molecular measurements from the laser trap and in vitro motility. The success of the model suggests that our multiscale data are self-consistent and can serve as a testbed for other multiscale models. Moreover, the model captures the decrease in isometric force observed in our muscle fibers after active shortening (force depression), suggesting a molecular mechanism for force depression, whereby a parallel elastic element combines with thick filament activation to decrease the number of cycling cross-bridges.

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

Release of the ATP hydrolysis product ortophosphate (Pi) from the active site of myosin 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, in unbranched kinetic schemes, have dominated the literature. However, in recent years, alternative models have emerged to explain apparently contradictory findings. Here, we first compare and critically analyze three influential alternative models 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.

Skeletal muscle structure, physiology, and function

Contractions of skeletal muscles provide the stability and power for all body movements. Consequently, any impairment in skeletal muscle function results in some degree of instability or immobility. Factors that influence skeletal muscle structure and function are therefore of great interest scientifically and clinically. Injury, neuromuscular disease, and old age are among the factors that commonly contribute to impairments in skeletal muscle function. The goal of this chapter is to summarize the fundamentals of skeletal muscle structure and function to provide foundational knowledge for this Handbook volume. We examine the molecular interactions that provide the basis for the generation of force and movement, discuss mechanisms of the regulation of contraction at the level of myofibers, and introduce concepts of the activation and control of muscle function in vivo. Where appropriate, the chapter updates the emerging science that will increase understanding of muscle function.

Insights into Muscle Contraction Derived from the Effects of Small-Molecular Actomyosin-Modulating Compounds

Bottom-up mechanokinetic models predict ensemble function of actin and myosin based on parameter values derived from studies using isolated proteins. To be generally useful, e.g., to analyze disease effects, such models must also be able to predict ensemble function when actomyosin interaction kinetics are modified differently from normal. Here, we test this capability for a model recently shown to predict several physiological phenomena along with the effects of the small molecular compound blebbistatin. We demonstrate that this model also qualitatively predicts effects of other well-characterized drugs as well as varied concentrations of MgATP. However, the effects of one compound, amrinone, are not well accounted for quantitatively. We therefore systematically varied key model parameters to address this issue, leading to the increased amplitude of the second sub-stroke of the power stroke from 1 nm to 2.2 nm, an unchanged first sub-stroke (5.3−5.5 nm), and an effective cross-bridge attachment rate that more than doubled. In addition to better accounting for the effects of amrinone, the modified model also accounts well for normal physiological ensemble function. Moreover, a Monte Carlo simulation-based version of the model was used to evaluate force−velocity data from small myosin ensembles. We discuss our findings in relation to key aspects of actin−myosin operation mechanisms causing a non-hyperbolic shape of the force−velocity relationship at high loads. We also discuss remaining limitations of the model, including uncertainty of whether the cross-bridge elasticity is linear or not, the capability to account for contractile properties of very small actomyosin ensembles (<20 myosin heads), and the mechanism for requirements of a higher cross-bridge attachment rate during shortening compared to during isometric contraction.

Multistep orthophosphate release tunes actomyosin energy transduction

Muscle contraction and a range of critical cellular functions rely on force-producing interactions between myosin motors and actin filaments, powered by turnover of adenosine triphosphate (ATP). The relationship between release of the ATP hydrolysis product ortophosphate (Pi) from the myosin active site and the force-generating structural change, the power-stroke, remains enigmatic despite its central role in energy transduction. Here, we present a model with multistep Pi-release that unifies current conflicting views while also revealing additional complexities of potential functional importance. The model is based on our evidence from kinetics, molecular modelling and single molecule fluorescence studies of Pi binding outside the active site. It is also consistent with high-speed atomic force microscopy movies of single myosin II molecules without Pi at the active site, showing consecutive snapshots of pre- and post-power stroke conformations. In addition to revealing critical features of energy transduction by actomyosin, the results suggest enzymatic mechanisms of potentially general relevance.

Release of the ATP hydrolysis product orthophosphate (Pi) from the myosin active site is central in force generation but is poorly understood. Here, Moretto et al. present evidence for multistep Pi-release reconciling apparently contradictory results.

A reverse stroke characterizes the force generation of cardiac myofilaments, leading to an understanding of heart function

Changes in the molecular properties of cardiac myosin strongly affect the interactions of myosin with actin that result in cardiac contraction and relaxation. However, it remains unclear how myosin molecules work together in cardiac myofilaments and which properties of the individual myosin molecules impact force production to drive cardiac contractility. Here, we measured the force production of cardiac myofilaments using optical tweezers. The measurements revealed that stepwise force generation was associated with a higher frequency of backward steps at lower loads and higher stall forces than those of fast skeletal myofilaments. To understand these unique collective behaviors of cardiac myosin, the dynamic responses of single cardiac and fast skeletal myosin molecules, interacting with actin filaments, were evaluated under load. The cardiac myosin molecules switched among three distinct conformational positions, ranging from pre- to post-power stroke positions, in 1 mM ADP and 0 to 10 mM phosphate solution. In contrast to cardiac myosin, fast skeletal myosin stayed primarily in the post-power stroke position, suggesting that cardiac myosin executes the reverse stroke more frequently than fast skeletal myosin. To elucidate how the reverse stroke affects the force production of myofilaments and possibly heart function, a simulation model was developed that combines the results from the single-molecule and myofilament experiments. The results of this model suggest that the reversal of the cardiac myosin power stroke may be key to characterizing the force output of cardiac myosin ensembles and possibly to facilitating heart contractions.

Myosin and gelsolin cooperate in actin filament severing and actomyosin motor activity

Actin is a major intracellular protein with key functions in cellular motility, signaling, and structural rearrangements. Its dynamic behavior, such as polymerization and depolymerization of actin filaments in response to intracellular and extracellular cues, is regulated by an abundance of actin binding proteins. Out of these, gelsolin is one of the most potent for filament severing. However, myosin motor activity also fragments actin filaments through motor-induced forces, suggesting that these two proteins could cooperate to regulate filament dynamics and motility. To test this idea, we used an in vitro motility assay, where actin filaments are propelled by surface-adsorbed heavy meromyosin (HMM) motor fragments. This allows studies of both motility and filament dynamics using isolated proteins. Gelsolin, at both nanomolar and micromolar Ca²⁺ concentration, appreciably enhanced actin filament severing caused by HMM-induced forces at 1 mM MgATP, an effect that was increased at higher HMM motor density. This finding is consistent with cooperativity between actin filament severing by myosin-induced forces and by gelsolin. We also observed reduced sliding velocity of the HMM-propelled filaments in the presence of gelsolin, providing further support of myosin-gelsolin cooperativity. Total internal reflection fluorescence microscopy–based single molecule studies corroborated that the velocity reduction was a direct effect of gelsolin binding to the filament and revealed different filament severing pattern of stationary and HMM propelled filaments. Overall, the results corroborate cooperative effects between gelsolin-induced alterations in the actin filaments and changes due to myosin motor activity leading to enhanced F-actin severing of possible physiological relevance.

Hypothesis: Single Actomyosin Properties Account for Ensemble Behavior in Active Muscle Shortening and Isometric Contraction

Muscle contraction results from cyclic interactions between myosin II motors and actin with two sets of proteins organized in overlapping thick and thin filaments, respectively, in a nearly crystalline lattice in a muscle sarcomere. However, a sarcomere contains a huge number of other proteins, some with important roles in muscle contraction. In particular, these include thin filament proteins, troponin and tropomyosin; thick filament proteins, myosin binding protein C; and the elastic protein, titin, that connects the thin and thick filaments. Furthermore, the order and 3D organization of the myofilament lattice may be important per se for contractile function. It is possible to model muscle contraction based on actin and myosin alone with properties derived in studies using single molecules and biochemical solution kinetics. It is also possible to reproduce several features of muscle contraction in experiments using only isolated actin and myosin, arguing against the importance of order and accessory proteins. Therefore, in this paper, it is hypothesized that “single molecule actomyosin properties account for the contractile properties of a half sarcomere during shortening and isometric contraction at almost saturating Ca concentrations”. In this paper, existing evidence for and against this hypothesis is reviewed and new modeling results to support the arguments are presented. Finally, further experimental tests are proposed, which if they corroborate, at least approximately, the hypothesis, should significantly benefit future effective analysis of a range of experimental studies, as well as drug discovery efforts.

Hierarchical modeling of force generation in cardiac muscle

Performing physiologically relevant simulations of the beating heart in clinical context requires to develop detailed models of the microscale force generation process. These models, however, may reveal difficult to implement in practice due to their high computational costs and complex calibration. We propose a hierarchy of three interconnected muscle contraction models-from the more refined to the more simplified-that are rigorously and systematically related to each other, offering a way to select, for a specific application, the model that yields a good trade-off between physiological fidelity, computational cost and calibration complexity. The three model families are compared to the same set of experimental data to systematically assess what physiological indicators can be reproduced or not and how these indicators constrain the model parameters. Finally, we discuss the applicability of these models for heart simulation.

The effects of inorganic phosphate on muscle force development and energetics: challenges in modelling related to experimental uncertainties

Muscle force and power are developed by myosin cross-bridges, which cyclically attach to actin, undergo a force-generating transition and detach under turnover of ATP. The force-generating transition is intimately associated with release of inorganic phosphate (Pi) but the exact sequence of events in relation to the actual Pi release step is controversial. Details of this process are reflected in the relationships between [Pi] and the developed force and shortening velocity. In order to account for these relationships, models have proposed branched kinetic pathways or loose coupling between biochemical and force-generating transitions. A key hypothesis underlying the present study is that such complexities are not required to explain changes in the force–velocity relationship and ATP turnover rate with altered [Pi]. We therefore set out to test if models without branched kinetic paths and Pi-release occurring before the main force-generating transition can account for effects of varied [Pi] (0.1–25 mM). The models tested, one assuming either linear or non-linear cross-bridge elasticity, account well for critical aspects of muscle contraction at 0.5 mM Pi but their capacity to account for the maximum power output vary. We find that the models, within experimental uncertainties, account for the relationship between [Pi] and isometric force as well as between [Pi] and the velocity of shortening at low loads. However, in apparent contradiction with available experimental findings, the tested models produce an anomalous force–velocity relationship at elevated [Pi] and high loads with more than one possible velocity for a given load. Nevertheless, considering experimental uncertainties and effects of sarcomere non-uniformities, these discrepancies are insufficient to refute the tested models in favour of more complex alternatives.

Regeneration of Assembled, Molecular-Motor-Based Bionanodevices

The guided gliding of cytoskeletal filaments, driven by biomolecular motors on nano/microstructured chips, enables novel applications in biosensing and biocomputation. However, expensive and time-consuming chip production hampers the developments. It is therefore important to establish protocols to regenerate the chips, preferably without the need to dismantle the assembled microfluidic devices which contain the structured chips. We here describe a novel method toward this end. Specifically, we use the small, nonselective proteolytic enzyme, proteinase K to cleave all surface-adsorbed proteins, including myosin and kinesin motors. Subsequently, we apply a detergent (5% SDS or 0.05% Triton X100) to remove the protein remnants. After this procedure, fresh motor proteins and filaments can be added for new experiments. Both, silanized glass surfaces for actin-myosin motility and pure glass surfaces for microtubule-kinesin motility were repeatedly regenerated using this approach. Moreover, we demonstrate the applicability of the method for the regeneration of nano/microstructured silicon-based chips with selectively functionalized areas for supporting or suppressing gliding motility for both motor systems. The results substantiate the versatility and a promising broad use of the method for regenerating a wide range of protein-based nano/microdevices.

Blebbistatin Effects Expose Hidden Secrets in the Force-Generating Cycle of Actin and Myosin

Cyclic interactions between myosin II motors and actin filaments driven by ATP turnover underlie muscle contraction and have key roles in the motility of nonmuscle cells. A remaining enigma in the understanding of this interaction is the relationship between the force-generating structural change and the release of the ATP-hydrolysis product, inorganic phosphate (Pi), from the active site of myosin. Here, we use the small molecular compound blebbistatin to probe otherwise hidden states and transitions in this process. Different hypotheses for the Pi release mechanism are tested by interpreting experimental results from in vitro motility assays and isolated muscle fibers in terms of mechanokinetic actomyosin models. The data fit with ideas that actomyosin force generation is preceded by Pi release, which in turn is preceded by two serial transitions after/coincident with cross-bridge attachment. Blebbistatin changes the rate limitation of the cycle from the first to the second of these transitions, uncovering functional roles of an otherwise short-lived pre-power stroke state that has been implicated by structural data.

Ultra-fast force-clamp spectroscopy data on the interaction between skeletal muscle myosin and actin

Ultrafast force-clamp spectroscopy is a single molecule technique based on laser tweezers with sub-millisecond and sub-nanometer resolution. The technique has been successfully applied to investigate the rapid conformational changes that occur when a myosin II motor from skeletal muscle interacts with an actin filament. Here, we share data on the kinetics of such interaction and experimental records collected under different forces [1]. The data can be valuable for researchers interested in the mechanosensitive properties of myosin II, both from an experimental and modeling point of view. The data is related to the research article “ultrafast force-clamp spectroscopy of single molecules reveals load dependence of myosin working stroke” [2].

Comparing models with one versus multiple myosin-binding sites per actin target zone: The power of simplicity

Mechanokinetic statistical models describe the mechanisms of muscle contraction on the basis of the average behavior of a large ensemble of actin-myosin motors. Such models often assume that myosin II motor domains bind to regularly spaced, discrete target zones along the actin-based thin filaments and develop force in a series of strain-dependent transitions under the turnover of ATP. The simplest models assume that there is just one myosin-binding site per target zone and a uniform spatial distribution of the myosin motor domains in relation to each site. However, most of the recently developed models assume three myosin-binding sites per target zone, and some models include a spatially explicit 3-D treatment of the myofilament lattice and thereby of the geometry of the actin-myosin contact points. Here, I show that the predictions for steady-state contractile behavior of muscle are very similar whether one or three myosin-binding sites per target zone is assumed, provided that the model responses are appropriately scaled to the number of sites. Comparison of the model predictions for isometrically contracting mammalian muscle cells suggests that each target zone contains three or more myosin-binding sites. Finally, I discuss the strengths and weaknesses of one-site spatially inexplicit models in relation to three-site models, including those that take into account the detailed 3-D geometry of the myofilament lattice. The results of this study suggest that single-site models, with reduced computational cost compared with multisite models, are useful for several purposes, e.g., facilitated molecular mechanistic insights.

Nonlinear Actomyosin Elasticity in Muscle?

Cyclic interactions between myosin II motor domains and actin filaments that are powered by turnover of ATP underlie muscle contraction and have key roles in motility of nonmuscle cells. The elastic characteristics of actin-myosin cross-bridges are central in the force-generating process, and disturbances in these properties may lead to disease. Although the prevailing paradigm is that the cross-bridge elasticity is linear (Hookean), recent single-molecule studies suggest otherwise. Despite convincing evidence for substantial nonlinearity of the cross-bridge elasticity in the single-molecule work, this finding has had limited influence on muscle physiology and physiology of other ordered cellular actin-myosin ensembles. Here, we use a biophysical modeling approach to close the gap between single molecules and physiology. The model is used for analysis of available experimental results in the light of possible nonlinearity of the cross-bridge elasticity. We consider results obtained both under rigor conditions (in the absence of ATP) and during active muscle contraction. Our results suggest that a wide range of experimental findings from mechanical experiments on muscle cells are consistent with nonlinear actin-myosin elasticity similar to that previously found in single molecules. Indeed, the introduction of nonlinear cross-bridge elasticity into the model improves the reproduction of key experimental results and eliminates the need for force dependence of the ATP-induced detachment rate, consistent with observations in other single-molecule studies. The findings have significant implications for the understanding of key features of actin-myosin-based production of force and motion in living cells, particularly in muscle, and for the interpretation of experimental results that rely on stiffness measurements on cells or myofibrils.

Controlled Surface Silanization for Actin-Myosin Based Nanodevices and Biocompatibility of New Polymer Resists

Molecular motor-based nanodevices require organized cytoskeletal filament guiding along motility-promoting tracks, confined by motility-inhibiting walls. One way to enhance motility quality on the tracks, particularly in terms of filament velocity but also the fraction of motile filaments, is to optimize the surface hydrophobicity. We have investigated the potential to achieve this for the actin-myosin II motor system on trimethylchlorosilane (TMCS)-derivatized SiO₂ surfaces to be used as channel floors in nanodevices. We have also investigated the ability to supress motility on two new polymer resists, TU7 (for nanoimprint lithography) and CSAR 62 (for electron beam and deep UV lithography), to be used as channel walls. We developed a chemical-vapor deposition tool for silanizing SiO₂ surfaces in a controlled environment to achieve different surface hydrophobicities (measured by water contact angle). In contrast to previous work, we were able to fabricate a wide range of contact angles by varying the silanization time and chamber pressure using only one type of silane. This resulted in a significant improvement of the silanization procedure, producing a predictable contact angle on the surface and thereby predictable quality of the heavy meromyosin (HMM)-driven actin motility with regard to velocity. We observed a high degree of correlation between the filament sliding velocity and contact angle in the range 10-86°, expanding the previously studied range. We found that the sliding velocity on TU7 surfaces was superior to that on CSAR 62 surfaces despite similar contact angles. In addition, we were able to suppress the motility on both TU7 and CSAR 62 by plasma oxygen treatment before silanization. These results are discussed in relation to previously proposed surface adsorption mechanisms of HMM and their relationship to the water contact angle. Additionally, the results are considered for the development of actin-myosin based nanodevices with superior performance with respect to actin-myosin functionality.

Do Actomyosin Single-Molecule Mechanics Data Predict Mechanics of Contracting Muscle?

In muscle, but not in single-molecule mechanics studies, actin, myosin and accessory proteins are incorporated into a highly ordered myofilament lattice. In view of this difference we compare results from single-molecule studies and muscle mechanics and analyze to what degree data from the two types of studies agree with each other. There is reasonable correspondence in estimates of the cross-bridge power-stroke distance (7–13 nm), cross-bridge stiffness (~2 pN/nm) and average isometric force per cross-bridge (6–9 pN). Furthermore, models defined on the basis of single-molecule mechanics and solution biochemistry give good fits to experimental data from muscle. This suggests that the ordered myofilament lattice, accessory proteins and emergent effects of the sarcomere organization have only minor modulatory roles. However, such factors may be of greater importance under e.g., disease conditions. We also identify areas where single-molecule and muscle data are conflicting: (1) whether force generation is an Eyring or Kramers process with just one major power-stroke or several sub-strokes; (2) whether the myofilaments and the cross-bridges have Hookean or non-linear elasticity; (3) if individual myosin heads slip between actin sites under certain conditions, e.g., in lengthening; or (4) if the two heads of myosin cooperate.

Kinetic coupling of phosphate release, force generation and rate-limiting steps in the cross-bridge cycle

A basic goal in muscle research is to understand how the cyclic ATPase activity of cross-bridges is converted into mechanical force. A direct approach to study the chemo-mechanical coupling between P_i release and the force-generating step is provided by the kinetics of force response induced by a rapid change in [P_i]. Classical studies on fibres using caged-P_i discovered that rapid increases in [P_i] induce fast force decays dependent on final [P_i] whose kinetics were interpreted to probe a fast force-generating step prior to P_i release. However, this hypothesis was called into question by studies on skeletal and cardiac myofibrils subjected to P_i jumps in both directions (increases and decreases in [P_i]) which revealed that rapid decreases in [P_i] trigger force rises with slow kinetics, similar to those of calcium-induced force development and mechanically-induced force redevelopment at the same [P_i]. A possible explanation for this discrepancy came from imaging of individual sarcomeres in cardiac myofibrils, showing that the fast force decay upon increase in [P_i] results from so-called sarcomere 'give'. The slow force rise upon decrease in [P_i] was found to better reflect overall sarcomeres cross-bridge kinetics and its [P_i] dependence, suggesting that the force generation coupled to P_i release cannot be separated from the rate-limiting transition. The reasons for the different conclusions achieved in fibre and myofibril studies are re-examined as the recent findings on cardiac myofibrils have fundamental consequences for the coupling between P_i release, rate-limiting steps and force generation. The implications from P_i-induced force kinetics of myofibrils are discussed in combination with historical and recent models of the cross-bridge cycle.