Towards a unified theory of muscle contraction. I: foundations

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.

A novel kinetic model to demonstrate the independent effects of ATP and ADP/Pi concentrations on sarcomere function

Understanding muscle contraction mechanisms is a standing challenge, and one of the approaches has been to create models of the sarcomere-the basic contractile unit of striated muscle. While these models have been successful in elucidating many aspects of muscle contraction, they fall short in explaining the energetics of functional phenomena, such as rigor, and in particular, their dependence on the concentrations of the biomolecules involved in the cross-bridge cycle. Our hypothesis posits that the stochastic time delay between ATP adsorption and ADP/Pi release in the cross-bridge cycle necessitates a modeling approach where the rates of these two reaction steps are controlled by two independent parts of the total free energy change of the hydrolysis reaction. To test this hypothesis, we built a two-filament, stochastic-mechanical half-sarcomere model that separates the energetic roles of ATP and ADP/Pi in the cross-bridge cycle's free energy landscape. Our results clearly demonstrate that there is a nontrivial dependence of the cross-bridge cycle's kinetics on the independent concentrations of ATP, ADP, and Pi. The simplicity of the proposed model allows for analytical solutions of the more basic systems, which provide novel insight into the dominant mechanisms driving some of the experimentally observed contractile phenomena.

Force and kinetics of fast and slow muscle myosin determined with a synthetic sarcomere-like nanomachine

Myosin II is the muscle molecular motor that works in two bipolar arrays in each thick filament of the striated (skeletal and cardiac) muscle, converting the chemical energy into steady force and shortening by cyclic ATP-driven interactions with the nearby actin filaments. Different isoforms of the myosin motor in the skeletal muscles account for the different functional requirements of the slow muscles (primarily responsible for the posture) and fast muscles (responsible for voluntary movements). To clarify the molecular basis of the differences, here the isoform-dependent mechanokinetic parameters underpinning the force of slow and fast muscles are defined with a unidimensional synthetic nanomachine powered by pure myosin isoforms from either slow or fast rabbit skeletal muscle. Data fitting with a stochastic model provides a self-consistent estimate of all the mechanokinetic properties of the motor ensemble including the motor force, the fraction of actin-attached motors and the rate of transition through the attachment-detachment cycle. The achievements in this paper set the stage for any future study on the emergent mechanokinetic properties of an ensemble of myosin molecules either engineered or purified from mutant animal models or human biopsies.

Basic science methods for the characterization of variants of uncertain significance in hypertrophic cardiomyopathy

With the advent of next-generation whole genome sequencing, many variants of uncertain significance (VUS) have been identified in individuals suffering from inheritable hypertrophic cardiomyopathy (HCM). Unfortunately, this classification of a genetic variant results in ambiguity in interpretation, risk stratification, and clinical practice. Here, we aim to review some basic science methods to gain a more accurate characterization of VUS in HCM. Currently, many genomic data-based computational methods have been developed and validated against each other to provide a robust set of resources for researchers. With the continual improvement in computing speed and accuracy, in silico molecular dynamic simulations can also be applied in mutational studies and provide valuable mechanistic insights. In addition, high throughput in vitro screening can provide more biologically meaningful insights into the structural and functional effects of VUS. Lastly, multi-level mathematical modeling can predict how the mutations could cause clinically significant organ-level dysfunction. We discuss emerging technologies that will aid in better VUS characterization and offer a possible basic science workflow for exploring the pathogenicity of VUS in HCM. Although the focus of this mini review was on HCM, these basic science methods can be applied to research in dilated cardiomyopathy (DCM), restrictive cardiomyopathy (RCM), arrhythmogenic cardiomyopathy (ACM), or other genetic cardiomyopathies.

Matching Mechanics and Energetics of Muscle Contraction Suggests Unconventional Chemomechanical Coupling during the Actin-Myosin Interaction

The mechanical performances of the vertebrate skeletal muscle during isometric and isotonic contractions are interfaced with the corresponding energy consumptions to define the coupling between mechanical and biochemical steps in the myosin-actin energy transduction cycle. The analysis is extended to a simplified synthetic nanomachine in which eight HMM molecules purified from fast mammalian skeletal muscle are brought to interact with an actin filament in the presence of 2 mM ATP, to assess the emergent properties of a minimum number of motors working in ensemble without the effects of both the higher hierarchical levels of striated muscle organization and other sarcomeric, regulatory and cytoskeleton proteins. A three-state model of myosin-actin interaction is able to predict the known relationships between energetics and transient and steady-state mechanical properties of fast skeletal muscle either in vivo or in vitro only under the assumption that during shortening a myosin motor can interact with two actin sites during one ATP hydrolysis cycle. Implementation of the molecular details of the model should be achieved by exploiting kinetic and structural constraints present in the transients elicited by stepwise perturbations in length or force superimposed on the isometric contraction.

Varying thin filament activation in the framework of the Huxley'57 model

Muscle contraction is triggered by the activation of the actin sites of the thin filament by calcium ions. It results that the thin filament activation level varies over time. Moreover, this activation process is also used as a regulation mechanism of the developed force. Our objective is to build a model of varying actin site activation level within the classical Huxley'57 two-state framework. This new model is obtained as an enhancement of a previously proposed formulation of the varying thick filament activation within the same framework. We assume that the state of an actin site depends on whether it is activated and whether it forms a cross-bridge with the associated myosin head, which results in four possible states. The transitions between the actin site states are controlled by the global actin sites activation level and the dynamics of these transitions is coupled with the attachment-detachment process. A preliminary calibration of the model with experimental twitch contraction data obtained at varying sarcomere lengths is performed.

A thermodynamically consistent monte carlo cross-bridge model with a trapping mechanism reveals the role of stretch activation in heart pumping

Changes in intracellular calcium concentrations regulate heart beats. However, the decline in the left ventricular pressure during early diastole is much sharper than that of the Ca²⁺ transient, resulting in a rapid supply of blood to the left ventricle during the diastole. At the tissue level, cardiac muscles have a distinct characteristic, known as stretch activation, similar to the function of insect flight muscles. Stretch activation, which is a delayed increase in force following a rapid muscle length increase, has been thought to be related to autonomous control in these muscles. In this numerical simulation study, we introduced a molecular mechanism of stretch activation and investigated the role of this mechanism in the pumping function of the heart, using the previously developed coupling multiple-step active stiffness integration scheme for a Monte Carlo (MC) cross-bridge model and a bi-ventricular finite element model. In the MC cross-bridge model, we introduced a mechanism for trapping the myosin molecule in its post-power stroke state. We then determined the rate constants of transitions for trapping and escaping in a thermodynamically consistent manner. Based on our numerical analysis, we draw the following conclusions regarding the stretch activation mechanism: (i) the delayed force becomes larger than the original isometric force because the population of trapped myosin molecules and their average force increase after stretching; (ii) the delayed force has a duration of more than a few seconds owing to a fairly small rate constant of escape from the trapped state. For the role of stretch activation in heart pumping, we draw the following conclusions: (iii) for the regions in which the contraction force decreases earlier than the neighboring region in the end-systole phase, the trapped myosin molecules prevent further lengthening of the myocytes, which then prevents further shortening of neighboring myocytes; (iv) as a result, the contraction forces are sustained longer, resulting in a larger blood ejection, and their degeneration is synchronized.

Effect of Active Lengthening and Shortening on Small-Angle X-ray Reflections in Skinned Skeletal Muscle Fibres

Our purpose was to use small-angle X-ray diffraction to investigate the structural changes within sarcomeres at steady-state isometric contraction following active lengthening and shortening, compared to purely isometric contractions performed at the same final lengths. We examined force, stiffness, and the 1,0 and 1,1 equatorial and M3 and M6 meridional reflections in skinned rabbit psoas bundles, at steady-state isometric contraction following active lengthening to a sarcomere length of 3.0 µm (15.4% initial bundle length at 7.7% bundle length/s), and active shortening to a sarcomere length of 2.6 µm (15.4% bundle length at 7.7% bundle length/s), and during purely isometric reference contractions at the corresponding sarcomere lengths. Compared to the reference contraction, the isometric contraction after active lengthening was associated with an increase in force (i.e., residual force enhancement) and M3 spacing, no change in stiffness and the intensity ratio I_1,1/I_1,0, and decreased lattice spacing and M3 intensity. Compared to the reference contraction, the isometric contraction after active shortening resulted in decreased force, stiffness, I_1,1/I_1,0, M3 and M6 spacings, and M3 intensity. This suggests that residual force enhancement is achieved without an increase in the proportion of attached cross-bridges, and that force depression is accompanied by a decrease in the proportion of attached cross-bridges. Furthermore, the steady-state isometric contraction following active lengthening and shortening is accompanied by an increase in cross-bridge dispersion and/or a change in the cross-bridge conformation compared to the reference contractions.

The Synergic Role of Actomyosin Architecture and Biased Detachment in Muscle Energetics: Insights in Cross Bridge Mechanism Beyond the Lever-Arm Swing

Muscle energetics reflects the ability of myosin motors to convert chemical energy into mechanical energy. How this process takes place remains one of the most elusive questions in the field. Here, we combined experimental measurements of in vitro sliding velocity based on DNA-origami built filaments carrying myosins with different lever arm length and Monte Carlo simulations based on a model which accounts for three basic components: (i) the geometrical hindrance, (ii) the mechano-sensing mechanism, and (iii) the biased kinetics for stretched or compressed motors. The model simulations showed that the geometrical hindrance due to acto-myosin spatial mismatching and the preferential detachment of compressed motors are synergic in generating the rapid increase in the ATP-ase rate from isometric to moderate velocities of contraction, thus acting as an energy-conservation strategy in muscle contraction. The velocity measurements on a DNA-origami filament that preserves the motors’ distribution showed that geometrical hindrance and biased detachment generate a non-zero sliding velocity even without rotation of the myosin lever-arm, which is widely recognized as the basic event in muscle contraction. Because biased detachment is a mechanism for the rectification of thermal fluctuations, in the Brownian-ratchet framework, we predict that it requires a non-negligible amount of energy to preserve the second law of thermodynamics. Taken together, our theoretical and experimental results elucidate less considered components in the chemo-mechanical energy transduction in muscle.

Multiscale modeling of twitch contractions in cardiac trabeculae

Understanding the dynamics of a cardiac muscle twitch contraction is complex because it requires a detailed understanding of the kinetic processes of the Ca2+ transient, thin-filament activation, and the myosin-actin cross-bridge chemomechanical cycle. Each of these steps has been well defined individually, but understanding how all three of the processes operate in combination is a far more complex problem. Computational modeling has the potential to provide detailed insight into each of these processes, how the dynamics of each process affect the complexity of contractile behavior, and how perturbations such as mutations in sarcomere proteins affect the complex interactions of all of these processes. The mechanisms involved in relaxation of tension during a cardiac twitch have been particularly difficult to discern due to nonhomogeneous sarcomere lengthening during relaxation. Here we use the multiscale MUSICO platform to model rat trabecular twitches. Validation of computational models is dependent on being able to simulate different experimental datasets, but there has been a paucity of data that can provide all of the required parameters in a single experiment, such as simultaneous measurements of force, intracellular Ca2+ transients, and sarcomere length dynamics. In this study, we used data from different studies collected under similar experimental conditions to provide information for all the required parameters. Our simulations established that twitches either in an isometric sarcomere or in fixed-length, multiple-sarcomere trabeculae replicate the experimental observations if models incorporate a length-tension relationship for the nonlinear series elasticity of muscle preparations and a scheme for thick-filament regulation. The thick-filament regulation assumes an off state in which myosin heads are parked onto the thick-filament backbone and are unable to interact with actin, a state analogous to the super-relaxed state. Including these two mechanisms provided simulations that accurately predict twitch contractions over a range of different conditions.

Biophysically detailed mathematical models of multiscale cardiac active mechanics

We propose four novel mathematical models, describing the microscopic mechanisms of force generation in the cardiac muscle tissue, which are suitable for multiscale numerical simulations of cardiac electromechanics. Such models are based on a biophysically accurate representation of the regulatory and contractile proteins in the sarcomeres. Our models, unlike most of the sarcomere dynamics models that are available in the literature and that feature a comparable richness of detail, do not require the time-consuming Monte Carlo method for their numerical approximation. Conversely, the models that we propose only require the solution of a system of PDEs and/or ODEs (the most reduced of the four only involving 20 ODEs), thus entailing a significant computational efficiency. By focusing on the two models that feature the best trade-off between detail of description and identifiability of parameters, we propose a pipeline to calibrate such parameters starting from experimental measurements available in literature. Thanks to this pipeline, we calibrate these models for room-temperature rat and for body-temperature human cells. We show, by means of numerical simulations, that the proposed models correctly predict the main features of force generation, including the steady-state force-calcium and force-length relationships, the length-dependent prolongation of twitches and increase of peak force, the force-velocity relationship. Moreover, they correctly reproduce the Frank-Starling effect, when employed in multiscale 3D numerical simulation of cardiac electromechanics.

Active Force Generation in Cardiac Muscle Cells: Mathematical Modeling and Numerical Simulation of the Actin-Myosin Interaction

Cardiac in silico numerical simulations are based on mathematical models describing the physical processes involved in the heart function. In this review paper, we critically survey biophysically-detailed mathematical models describing the subcellular mechanisms behind the generation of active force, that is the process by which the chemical energy of ATP (adenosine triphosphate) is transformed into mechanical work, thus making the muscle tissue contract. While presenting these models, that feature different levels of biophysical detail, we analyze the trade-off between the accuracy in the description of the subcellular mechanisms and the number of parameters that need to be estimated from experiments. Then, we focus on a generalized version of the classic Huxley model, built on the basis of models available in the literature, that is able to reproduce the main experimental characterizations associated to the time scales typical of a heartbeat-such as the force-velocity relationship and the tissue stiffness in response to small steps-featuring only four independent parameters. Finally, we show how those parameters can be calibrated starting from macroscopic measurements available from experiments.

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.

Straightening Out the Elasticity of Myosin Cross-Bridges

In a contracting muscle, myosin cross-bridges extending from thick filaments pull the interdigitating thin (actin-containing) filaments during cyclical ATP-driven interactions toward the center of the sarcomere, the structural unit of striated muscle. Cross-bridge attachments in the sarcomere have been reported to exhibit a similar stiffness under both positive and negative forces. However, in vitro measurements on filaments with a sparse complement of heads detected a decrease of the cross-bridge stiffness at negative forces attributed to the buckling of the subfragment 2 tail portion. Here, we review some old and new data that confirm that cross-bridge stiffness is nearly linear in the muscle filament lattice. The implications of high myosin stiffness at positive and negative strains are considered in muscle fibers and in nonmuscle intracellular cargo transport.

Estimation of Forces on Actin Filaments in Living Muscle from X-ray Diffraction Patterns and Mechanical Data

Many biological processes are triggered or driven by mechanical forces in the cytoskeletal network, but these transducing forces have rarely been assessed. Striated muscle, with its well-organized structure provides an opportunity to assess intracellular forces using small-angle X-ray fiber diffraction. We present a new methodology using Monte Carlo simulations of muscle contraction in an explicit 3D sarcomere lattice to predict the fiber deformations and length changes along thin filaments during contraction. Comparison of predicted diffraction patterns to experimental meridional X-ray reflection profiles allows assessment of the stepwise changes in intermonomer spacings and forces in the myofilaments within living muscle cells. These changes along the filament length reflect the effect of forces from randomly attached crossbridges. This approach enables correlation of the molecular events, such as the current number of attached crossbridges and the distributions of crossbridge forces to macroscopic measurements of force and length changes during muscle contraction. In addition, assessments of fluctuations in local forces in the myofilaments may reveal how variations in the filament forces acting on signaling proteins in the sarcomere M-bands and Z-discs modulate gene expression, protein synthesis and degradation, and as well to mechanisms of adaptation of muscle in response to changes in mechanical loading.

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.

Disrupted mechanobiology links the molecular and cellular phenotypes in familial dilated cardiomyopathy

Familial dilated cardiomyopathy (DCM) is a leading cause of sudden cardiac death and a major indicator for heart transplant. The disease is frequently caused by mutations of sarcomeric proteins; however, it is not well understood how these molecular mutations lead to alterations in cellular organization and contractility. To address this critical gap in our knowledge, we studied the molecular and cellular consequences of a DCM mutation in troponin-T, ΔK210. We determined the molecular mechanism of ΔK210 and used computational modeling to predict that the mutation should reduce the force per sarcomere. In mutant cardiomyocytes, we found that ΔK210 not only reduces contractility but also causes cellular hypertrophy and impairs cardiomyocytes' ability to adapt to changes in substrate stiffness (e.g., heart tissue fibrosis that occurs with aging and disease). These results help link the molecular and cellular phenotypes and implicate alterations in mechanosensing as an important factor in the development of DCM.

Computational Tool to Study Perturbations in Muscle Regulation and Its Application to Heart Disease

Striated muscle contraction occurs when myosin thick filaments bind to thin filaments in the sarcomere and generate pulling forces. This process is regulated by calcium, and it can be perturbed by pathological conditions (e.g., myopathies), physiological adaptations (e.g., β-adrenergic stimulation), and pharmacological interventions. Therefore, it is important to have a methodology to robustly determine the impact of these perturbations and statistically evaluate their effects. Here, we present an approach to measure the equilibrium constants that govern muscle activation, estimate uncertainty in these parameters, and statistically test the effects of perturbations. We provide a MATLAB-based computational tool for these analyses, along with easy-to-follow tutorials that make this approach accessible. The hypothesis testing and error estimation approaches described here are broadly applicable, and the provided tools work with other types of data, including cellular measurements. To demonstrate the utility of the approach, we apply it to elucidate the biophysical mechanism of a mutation that causes familial hypertrophic cardiomyopathy. This approach is generally useful for studying muscle diseases and therapeutic interventions that target muscle contraction.

Insights into Actin-Myosin Interactions within Muscle from 3D Electron Microscopy

Much has been learned about the interaction between myosin and actin through biochemistry, in vitro motility assays and cryo-electron microscopy (cryoEM) of F-actin, decorated with myosin heads. Comparatively less is known about actin-myosin interactions within the filament lattice of muscle, where myosin heads function as independent force generators and thus most measurements report an average signal from multiple biochemical and mechanical states. All of the 3D imaging by electron microscopy (EM) that has revealed the interplay of the regular array of actin subunits and myosin heads within the filament lattice has been accomplished using the flight muscle of the large water bug Lethocerus sp. The Lethocerus flight muscle possesses a particularly favorable filament arrangement that enables all the myosin cross-bridges contacting the actin filament to be visualized in a thin section. This review covers the history of this effort and the progress toward visualizing the complex set of conformational changes that myosin heads make when binding to actin in several static states, as well as the fast frozen actively contracting muscle. The efforts have revealed a consistent pattern of changes to the myosin head structures as determined by X-ray crystallography needed to explain the structure of the different actomyosin interactions observed in situ.

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.

A new myofilament contraction model with ATP consumption for ventricular cell model

A new contraction model of cardiac muscle was developed by combining previously described biochemical and biophysical models. The biochemical component of the new contraction model represents events in the presence of Ca2+-crossbridge attachment and power stroke following inorganic phosphate release, detachment evoked by the replacement of ADP by ATP, ATP hydrolysis, and recovery stroke. The biophysical component focuses on Ca2+ activation and force (F b) development assuming an equivalent crossbridge. The new model faithfully incorporates the major characteristics of the biochemical and biophysical models, such as F b activation by transient Ca2+ ([Ca2+]-F b), [Ca2+]-ATP hydrolysis relations, sarcomere length-F b, and F b recovery after jumps in length under the isometric mode and upon sarcomere shortening after a rapid release of mechanical load under the isotonic mode together with the load-velocity relationship. ATP consumption was obtained for all responses. When incorporated in a ventricular cell model, the contraction model was found to share approximately 60% of the total ATP usage in the cell model.

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.

Mechanical parameters of the molecular motor myosin II determined in permeabilised fibres from slow and fast skeletal muscles of the rabbit

KEY POINTS

The different performance of slow and fast muscles is mainly attributed to diversity of the myosin heavy chain (MHC) isoform expressed within them. In this study fast sarcomere-level mechanics has been applied to Ca²⁺ -activated single permeabilised fibres isolated from soleus (containing the slow myosin isoform) and psoas (containing the fast myosin isoform) muscles of rabbit for a comparative definition of the mechano-kinetics of force generation by slow and fast myosin isoforms in situ. The stiffness and the force of the slow myosin isoform are three times smaller than those of the fast isoform, suggesting that the stiffness of the myosin motor is a determinant of the isoform-dependent functional diversity between skeletal muscles. These results open the question of the mechanism that can reconcile the reduced performance of the slow MHC with the higher efficiency of the slow muscle.

ABSTRACT

The skeletal muscle exhibits large functional differences depending on the myosin heavy chain (MHC) isoform expressed in its molecular motor, myosin II. The differences in the mechanical features of force generation by myosin isoforms were investigated in situ by using fast sarcomere-level mechanical methods in permeabilised fibres (sarcomere length 2.4 μm, temperature 12°C, 4% dextran T-500) from slow (soleus, containing the MHC-1 isoform) and fast (psoas, containing the MHC-2X isoform) skeletal muscle of the rabbit. The stiffness of the half-sarcomere was determined at the plateau of Ca²⁺ -activated isometric contractions and in rigor and analysed with a model that accounted for the filament compliance to estimate the stiffness of the myosin motor (ε). ε was 0.56 ± 0.04 and 1.70 ± 0.37 pN nm^-1 for the slow and fast isoform, respectively, while the average strain per attached motor (s₀ ) was similar (∼3.3 nm) in both isoforms. Consequently the force per motor (F₀  = εs₀ ) was three times smaller in the slow isoform than in the fast isoform (1.89 ± 0.43 versus 5.35 ± 1.51 pN). The fraction of actin-attached motors responsible for maximum isometric force at saturating Ca²⁺ (T_0,4.5 ) was 0.47 ± 0.09 in soleus fibres, 70% larger than that in psoas fibres (0.29 ± 0.08), so that F₀ in slow fibres was decreased by only 53%. The lower stiffness and force of the slow myosin isoform open the question of the molecular basis of the higher efficiency of slow muscle with respect to fast muscle.

Robust mechanobiological behavior emerges in heterogeneous myosin systems

Biological complexity presents challenges for understanding natural phenomenon and engineering new technologies, particularly in systems with molecular heterogeneity. Such complexity is present in myosin motor protein systems, and computational modeling is essential for determining how collective myosin interactions produce emergent system behavior. We develop a computational approach for altering myosin isoform parameters and their collective organization, and support predictions with in vitro experiments of motility assays with α-actinins as molecular force sensors. The computational approach models variations in single myosin molecular structure, system organization, and force stimuli to predict system behavior for filament velocity, energy consumption, and robustness. Robustness is the range of forces where a filament is expected to have continuous velocity and depends on used myosin system energy. Myosin systems are shown to have highly nonlinear behavior across force conditions that may be exploited at a systems level by combining slow and fast myosin isoforms heterogeneously. Results suggest some heterogeneous systems have lower energy use near stall conditions and greater energy consumption when unloaded, therefore promoting robustness. These heterogeneous system capabilities are unique in comparison with homogenous systems and potentially advantageous for high performance bionanotechnologies. Findings open doors at the intersections of mechanics and biology, particularly for understanding and treating myosin-related diseases and developing approaches for motor molecule-based technologies.

Modeling the Actin.myosin ATPase Cross-Bridge Cycle for Skeletal and Cardiac Muscle Myosin Isoforms

Modeling the complete actin.myosin ATPase cycle has always been limited by the lack of experimental data concerning key steps of the cycle, because these steps can only be defined at very low ionic strength. Here, using human β-cardiac myosin-S1, we combine published data from transient and steady-state kinetics to model a minimal eight-state ATPase cycle. The model illustrates the occupancy of each intermediate around the cycle and how the occupancy is altered by changes in actin concentration for [actin] = 1-20K_m. The cycle can be used to predict the maximal velocity of contraction (by motility assay or sarcomeric shortening) at different actin concentrations (which is consistent with experimental velocity data) and predict the effect of a 5 pN load on a single motor. The same exercise was repeated for human α-cardiac myosin S1 and rabbit fast skeletal muscle S1. The data illustrates how the motor domain properties can alter the ATPase cycle and hence the occupancy of the key states in the cycle. These in turn alter the predicted mechanical response of the myosin independent of other factors present in a sarcomere, such as filament stiffness and regulatory proteins. We also explore the potential of this modeling approach for the study of mutations in human β-cardiac myosin using the hypertrophic myopathy mutation R453C. Our modeling, using the transient kinetic data, predicts mechanical properties of the motor that are compatible with the single-molecule study. The modeling approach may therefore be of wide use for predicting the properties of myosin mutations.

A simple model of cardiac muscle for multiscale simulation: Passive mechanics, crossbridge kinetics and calcium regulation

A simple model of cardiac muscle was designed for multiscale simulation of heart mechanics. Relaxed cardiac muscle was described as a transversally isotropic hyperelastic material. Active tension caused by actin-myosin crossbridges depends on the ensemble averaged strain of myosin heads bound to actin. Calcium activation was modeled by Ca²⁺ binding to troponin-C. To account for the dependence of troponin affinity for Ca²⁺ on myosin heads strongly bound to actin, the kinetics of troponin binding to Ca²⁺ in the overlap zone of the thin and thick filaments and outside it were separated. The changes in the length of these zones during muscle shortening or lengthening were accounted for explicitly. Simplified version of the model contains only 5 ordinary differential equations (ODE). Model parameters were estimated from a limited set of experiments with skeletal and cardiac muscle. Simulations have shown that model reproduces qualitatively a number of experimental observations: steady-state force-velocity and stiffness-velocity relations; mechanical responses to step changes in muscle length or load; steep Ca²⁺-tension relationship and its dependence on sarcomere length tension (the Frank-Starling mechanism); tension, shortening and Ca²⁺-transients in twitch isometric and isotonic contractions, tension development and redevelopment upon fast change in Ca²⁺ concentration or muscle release followed by re-stretch. We believe that the model can be effectively used for modeling contraction and relaxation of the heart.

Compliance Accelerates Relaxation in Muscle by Allowing Myosin Heads to Move Relative to Actin

The mechanisms that limit the speed at which striated muscles relax are poorly understood. This work presents, to our knowledge, novel simulations that show that the time course of relaxation is accelerated by interfilamentary movement resulting from series compliance; force drops faster when myosin heads move relative to actin during relaxation. This insight was obtained by using cross-bridge distribution techniques to simulate the mechanical behavior of half-sarcomeres that were connected in series with springs of varying stiffness. (The springs mimic the combined effects of half-sarcomere heterogeneity and muscle's series elastic component.) Half-sarcomeres that shortened by >∼10 nm when they were activated subsequently relaxed with a biphasic profile; force initially declined slowly and approximately linearly before collapsing with a fast exponential time course. Stretches imposed during the linear phase quickened relaxation, while shortening movements prolonged the time course. These predictions are consistent with data from experiments performed by many other groups using single muscle fibers and isolated myofibrils. When half-sarcomeres were linked to stiff springs (so that they did not shorten appreciably during the simulations), force relaxed with a slow exponential time course and did not show biphasic behavior. Together, these results suggest that fast relaxation of striated muscle is an emergent property that reflects multiscale interactions within the muscle architecture. The nonlinear behavior during relaxation reflects perturbations to the dynamic coupling of regulated binding sites and cycling myosin heads that are induced by interfilamentary movement.

Three-dimensional stochastic model of actin-myosin binding in the sarcomere lattice

The effect of molecule tethering in three-dimensional (3-D) space on bimolecular binding kinetics is rarely addressed and only occasionally incorporated into models of cell motility. The simplest system that can quantitatively determine this effect is the 3-D sarcomere lattice of the striated muscle, where tethered myosin in thick filaments can only bind to a relatively small number of available sites on the actin filament, positioned within a limited range of thermal movement of the myosin head. Here we implement spatially explicit actomyosin interactions into the multiscale Monte Carlo platform MUSICO, specifically defining how geometrical constraints on tethered myosins can modulate state transition rates in the actomyosin cycle. The simulations provide the distribution of myosin bound to sites on actin, ensure conservation of the number of interacting myosins and actin monomers, and most importantly, the departure in behavior of tethered myosin molecules from unconstrained myosin interactions with actin. In addition, MUSICO determines the number of cross-bridges in each actomyosin cycle state, the force and number of attached cross-bridges per myosin filament, the range of cross-bridge forces and accounts for energy consumption. At the macroscopic scale, MUSICO simulations show large differences in predicted force-velocity curves and in the response during early force recovery phase after a step change in length comparing to the two simplest mass action kinetic models. The origin of these differences is rooted in the different fluxes of myosin binding and corresponding instantaneous cross-bridge distributions and quantitatively reflects a major flaw of the mathematical description in all mass action kinetic models. Consequently, this new approach shows that accurate recapitulation of experimental data requires significantly different binding rates, number of actomyosin states, and cross-bridge elasticity than typically used in mass action kinetic models to correctly describe the biochemical reactions of tethered molecules and their interaction energetics.

Including Thermal Fluctuations in Actomyosin Stable States Increases the Predicted Force per Motor and Macroscopic Efficiency in Muscle Modelling

Muscle contractions are generated by cyclical interactions of myosin heads with actin filaments to form the actomyosin complex. To simulate actomyosin complex stable states, mathematical models usually define an energy landscape with a corresponding number of wells. The jumps between these wells are defined through rate constants. Almost all previous models assign these wells an infinite sharpness by imposing a relatively simple expression for the detailed balance, i.e., the ratio of the rate constants depends exponentially on the sole myosin elastic energy. Physically, this assumption corresponds to neglecting thermal fluctuations in the actomyosin complex stable states. By comparing three mathematical models, we examine the extent to which this hypothesis affects muscle model predictions at the single cross-bridge, single fiber, and organ levels in a ceteris paribus analysis. We show that including fluctuations in stable states allows the lever arm of the myosin to easily and dynamically explore all possible minima in the energy landscape, generating several backward and forward jumps between states during the lifetime of the actomyosin complex, whereas the infinitely sharp minima case is characterized by fewer jumps between states. Moreover, the analysis predicts that thermal fluctuations enable a more efficient contraction mechanism, in which a higher force is sustained by fewer attached cross-bridges.

Mathematical models are of fundamental importance in the quantitative verification of biological hypotheses. Muscle contraction models assume the existence of several stable states for the myosin head, whereas the transition rates between states are defined to fit experimental data. The ratio of the forward and backward rates is linked to the ratio of the probabilities of being in one or other stable state at equilibrium through a detailed balance condition. A commonly used assumption leads to a relatively simple expression for this balance condition that depends only on the values of the energy at the minima and not on the minima shape. Mathematically, this hypothesis corresponds to infinite sharpness at these minima; physically, it neglects the small thermal fluctuations within actomyosin stable states. In this work, we compare this classical approach with a model that includes thermal fluctuations within wide minima, and quantitatively assess how much this hypothesis affects the model outcomes at the single molecule, single fiber, and whole heart levels. It is shown that, using parameters compatible with known behavior in muscle mechanics, relaxing the infinitely sharp minima hypothesis improves the predicted force generation and efficiency at the macroscopic level.

Ventricular fiber optimization utilizing the branching structure

In this paper, we propose an algorithm that optimizes the ventricular fiber structure of the human heart. A number of histological studies and diffusion tensor magnetic resonance imaging analyses have revealed that the myocardial fiber forms a right-handed helix at the endocardium. However, the fiber formation changes its orientation as a function of transmural depth, becoming a left-handed helix at the epicardium. To determine how nature can construct such a structure, which obtains surprising pumping performance, we introduce macroscopic modeling of the branching structure of cardiac myocytes in our finite element ventricular model and utilize this in an optimization process. We put a set of multidirectional fibers around a central fiber orientation at each point of the ventricle walls and simulate heartbeats by generating contraction forces along each of these directions. We examine two optimization processes using the workloads or impulses measured in these directions to update the central fiber orientation. Both processes improve the pumping performance towards an optimal value within several tens of heartbeats, starting from an almost-flat fiber orientation. However, compared with the workload optimization, the impulse optimization produces better agreement with experimental studies on transmural changes of fiber helix angle, streamline patterns of characteristic helical structures, and temporal changes in strain. Furthermore, the impulse optimization is robust under geometrical changes of the heart and tends to homogenize various mechanical factors such as the stretch and stretch rate along the fiber orientation, the contraction force, and energy consumption. Copyright © 2015 John Wiley & Sons, Ltd.

Statistical mechanics of the Huxley-Simmons model

The chemomechanical model of Huxley and Simmons (HS) [A. F. Huxley and R. M. Simmons, Nature 233, 533 (1971)NATUAS0028-083610.1038/233533a0] provides a paradigmatic description of mechanically induced collective conformational changes relevant in a variety of biological contexts, from muscles power stroke and hair cell gating to integrin binding and hairpin unzipping. We develop a statistical mechanical perspective on the HS model by exploiting a formal analogy with a paramagnetic Ising model. We first study the equilibrium HS model with a finite number of elements and compute explicitly its mechanical and thermal properties. To model kinetics, we derive a master equation and solve it for several loading protocols. The developed formalism is applicable to a broad range of allosteric systems with mean-field interactions.

X-ray diffraction from nonuniformly stretched helical molecules

The fibrous proteins in living cells are exposed to mechanical forces interacting with other subcellular structures. X-ray fiber diffraction is often used to assess deformation and movement of these proteins, but the analysis has been limited to the theory for fibrous molecular systems that exhibit helical symmetry. However, this approach cannot adequately interpret X-ray data from fibrous protein assemblies where the local strain varies along the fiber length owing to interactions of its molecular constituents with their binding partners. To resolve this problem a theoretical formulism has been developed for predicting the diffraction from individual helical molecular structures nonuniformly strained along their lengths. This represents a critical first step towards modeling complex dynamical systems consisting of multiple helical structures using spatially explicit, multi-scale Monte Carlo simulations where predictions are compared with experimental data in a 'forward' process to iteratively generate ever more realistic models. Here the effects of nonuniform strains and the helix length on the resulting magnitude and phase of diffraction patterns are quantitatively assessed. Examples of the predicted diffraction patterns of nonuniformly deformed double-stranded DNA and actin filaments in contracting muscle are presented to demonstrate the feasibly of this theoretical approach.

Mechanics of Vascular Smooth Muscle

Vascular smooth muscle (VSM; see Table 1 for a list of abbreviations) is a heterogeneous biomaterial comprised of cells and extracellular matrix. By surrounding tubes of endothelial cells, VSM forms a regulated network, the vasculature, through which oxygenated blood supplies specialized organs, permitting the development of large multicellular organisms. VSM cells, the engine of the vasculature, house a set of regulated nanomotors that permit rapid stress-development, sustained stress-maintenance and vessel constriction. Viscoelastic materials within, surrounding and attached to VSM cells, comprised largely of polymeric proteins with complex mechanical characteristics, assist the engine with countering loads imposed by the heart pump, and with control of relengthening after constriction. The complexity of this smart material can be reduced by classical mechanical studies combined with circuit modeling using spring and dashpot elements. Evaluation of the mechanical characteristics of VSM requires a more complete understanding of the mechanics and regulation of its biochemical parts, and ultimately, an understanding of how these parts work together to form the machinery of the vascular tree. Current molecular studies provide detailed mechanical data about single polymeric molecules, revealing viscoelasticity and plasticity at the protein domain level, the unique biological slip-catch bond, and a regulated two-step actomyosin power stroke. At the tissue level, new insight into acutely dynamic stress-strain behavior reveals smooth muscle to exhibit adaptive plasticity. At its core, physiology aims to describe the complex interactions of molecular systems, clarifying structure-function relationships and regulation of biological machines. The intent of this review is to provide a comprehensive presentation of one biomachine, VSM.

Poorly understood aspects of striated muscle contraction

Muscle contraction results from cyclic interactions between the contractile proteins myosin and actin, driven by the turnover of adenosine triphosphate (ATP). Despite intense studies, several molecular events in the contraction process are poorly understood, including the relationship between force-generation and phosphate-release in the ATP-turnover. Different aspects of the force-generating transition are reflected in the changes in tension development by muscle cells, myofibrils and single molecules upon changes in temperature, altered phosphate concentration, or length perturbations. It has been notoriously difficult to explain all these events within a given theoretical framework and to unequivocally correlate observed events with the atomic structures of the myosin motor. Other incompletely understood issues include the role of the two heads of myosin II and structural changes in the actin filaments as well as the importance of the three-dimensional order. We here review these issues in relation to controversies regarding basic physiological properties of striated muscle. We also briefly consider actomyosin mutation effects in cardiac and skeletal muscle function and the possibility to treat these defects by drugs.

Dynamic coupling of regulated binding sites and cycling myosin heads in striated muscle

Myosim, a computer model of muscle contraction, includes molecular-level effects and incorporates dynamic coupling of myosin heads and binding sites.

In an activated muscle, binding sites on the thin filament and myosin heads switch frequently between different states. Because the status of the binding sites influences the status of the heads, and vice versa, the binding sites and myosin heads are dynamically coupled. The functional consequences of this coupling were investigated using MyoSim, a new computer model of muscle. MyoSim extends existing models based on Huxley-type distribution techniques by incorporating Ca²⁺ activation and cooperative effects. It can also simulate arbitrary cross-bridge schemes set by the researcher. Initial calculations investigated the effects of altering the relative speeds of binding-site and cross-bridge kinetics, and of manipulating cooperative processes. Subsequent tests fitted simulated force records to experimental data recorded using permeabilized myocardial preparations. These calculations suggest that the rate of force development at maximum activation is limited by myosin cycling kinetics, whereas the rate at lower levels of activation is limited by how quickly binding sites become available. Additional tests investigated the behavior of transiently activated cells by driving simulations with experimentally recorded Ca²⁺ signals. The unloaded shortening profile of a twitching myocyte could be reproduced using a model with two myosin states, cooperative activation, and strain-dependent kinetics. Collectively, these results demonstrate that dynamic coupling of binding sites and myosin heads is important for contractile function.

Power-stroke-driven actomyosin contractility

In ratchet-based models describing actomyosin contraction the activity is usually associated with actin binding potential while the power-stroke mechanism, residing inside myosin heads, is viewed as passive. To show that contraction can be propelled directly through a conformational change, we propose an alternative model where the power stroke is the only active mechanism. The asymmetry, ensuring directional motion, resides in steric interaction between the externally driven power-stroke element and the passive nonpolar actin filament. The proposed model can reproduce all four discrete states of the minimal actomyosin catalytic cycle even though it is formulated in terms of continuous Langevin dynamics. We build a conceptual bridge between processive and nonprocessive molecular motors by demonstrating that not only the former but also the latter can use structural transformation as the main driving force.

Hill's equation of muscle performance and its hidden insight on molecular mechanisms

Muscles shorten faster against light loads than they do against heavy loads. The hyperbolic equation first used by A.V. Hill over seven decades ago to illustrate the relationship between shortening velocity and load is still the predominant method used to characterize muscle performance, even though it has been regarded as purely empirical and lacking precision in predicting velocities at high and low loads. Popularity of the Hill equation has been sustained perhaps because of historical reasons, but its simplicity is certainly attractive. The descriptive nature of the equation does not diminish its role as a useful tool in our quest to understand animal locomotion and optimal design of muscle-powered devices like bicycles. In this Review, an analysis is presented to illustrate the connection between the historic Hill equation and the kinetics of myosin cross-bridge cycle based on the latest findings on myosin motor interaction with actin filaments within the structural confines of a sarcomere. In light of the new data and perspective, some previous studies of force-velocity relations of muscle are revisited to further our understanding of muscle mechanics and the underlying biochemical events, specifically how extracellular and intracellular environment, protein isoform expression, and posttranslational modification of contractile and regulatory proteins change the interaction between myosin and actin that in turn alter muscle force, shortening velocity, and the relationship between them.

Nonlinear cross-bridge elasticity and post-power-stroke events in fast skeletal muscle actomyosin

Generation of force and movement by actomyosin cross-bridges is the molecular basis of muscle contraction, but generally accepted ideas about cross-bridge properties have recently been questioned. Of the utmost significance, evidence for nonlinear cross-bridge elasticity has been presented. We here investigate how this and other newly discovered or postulated phenomena would modify cross-bridge operation, with focus on post-power-stroke events. First, as an experimental basis, we present evidence for a hyperbolic [MgATP]-velocity relationship of heavy-meromyosin-propelled actin filaments in the in vitro motility assay using fast rabbit skeletal muscle myosin (28-29°C). As the hyperbolic [MgATP]-velocity relationship was not consistent with interhead cooperativity, we developed a cross-bridge model with independent myosin heads and strain-dependent interstate transition rates. The model, implemented with inclusion of MgATP-independent detachment from the rigor state, as suggested by previous single-molecule mechanics experiments, accounts well for the [MgATP]-velocity relationship if nonlinear cross-bridge elasticity is assumed, but not if linear cross-bridge elasticity is assumed. In addition, a better fit is obtained with load-independent than with load-dependent MgATP-induced detachment rate. We discuss our results in relation to previous data showing a nonhyperbolic [MgATP]-velocity relationship when actin filaments are propelled by myosin subfragment 1 or full-length myosin. We also consider the implications of our results for characterization of the cross-bridge elasticity in the filament lattice of muscle.

The working stroke of the myosin II motor in muscle is not tightly coupled to release of orthophosphate from its active site

Skeletal muscle shortens faster against a lower load. This force-velocity relationship is the fundamental determinant of muscle performance in vivo and is due to ATP-driven working strokes of myosin II motors, during their cyclic interactions with the actin filament in each half-sarcomere. Crystallographic studies suggest that the working stroke is associated with the release of phosphate (Pi) and consists of 70 deg tilting of a light-chain domain that connects the catalytic domain of the myosin motor to the myosin tail and filament. However, the coupling of the working stroke with Pi release is still an unsolved question. Using nanometre-microsecond mechanics on skinned muscle fibres, we impose stepwise drops in force on an otherwise isometric contraction and record the isotonic velocity transient, to measure the mechanical manifestation of the working stroke of myosin motors and the rate of its regeneration in relation to the half-sarcomere load and [Pi]. We show that the rate constant of the working stroke is unaffected by [Pi], while the subsequent transition to steady velocity shortening is accelerated. We propose a new chemo-mechanical model that reproduces the transient and steady state responses by assuming that: (i) the release of Pi from the catalytic site of a myosin motor can occur at any stage of the working stroke, and (ii) a myosin motor, in an intermediate state of the working stroke, can slip to the next actin monomer during filament sliding. This model explains the efficient action of muscle molecular motors working as an ensemble in the half-sarcomere.

A cross-bridge cycle with two tension-generating steps simulates skeletal muscle mechanics

We examined whether cross-bridge cycle models with one or two tension-generating steps can account for the force-velocity relation of and tension response to length steps of frog skeletal muscle. Transition-state theory defined the strain dependence of the rate constants. The filament stiffness was non-Hookean. Models were refined against experimental data by simulated annealing and downhill simplex runs. Models with one tension-generating step were rejected, as they had a low efficiency and fitted the experimental data relatively poorly. The best model with two tension-generating steps (stroke distances 5.6 and 4.6 nm) and a cross-bridge stiffness of 1.7 pN/nm gave a good account of the experimental data. The two tensing steps allow an efficiency of up to 38% during shortening. In an isometric contraction, 54.7% of the attached heads were in a pre-tension-generating state, 44.5% of the attached heads had undergone the first tension-generating step, and only 0.8% had undergone both tension-generating steps; they bore 34%, 64%, and 2%, respectively, of the isometric tension. During slow shortening, the second tensing step made a greater contribution. During lengthening, up to 93% of the attached heads were in a pre-tension-generating state yet bore elevated tension by being dragged to high strains before detaching.

Muscle as a metamaterial operating near a critical point

The passive mechanical response of skeletal muscles at fast time scales is dominated by long range interactions inducing cooperative behavior without breaking the detailed balance. This leads to such unusual "material properties" as negative equilibrium stiffness and different behavior in force and displacement controlled loading conditions. Our fitting of experimental data suggests that "muscle material" is finely tuned to perform close to a critical point which explains large fluctuations observed in muscles close to the stall force.

Erratum to: Identification of functional differences between recombinant human α and β cardiac myosin motors

The myosin isoform composition of the heart is dynamic in health and disease and has been shown to affect contractile velocity and force generation. While different mammalian species express different proportions of α and β myosin heavy chain, healthy human heart ventricles express these isoforms in a ratio of about 1:9 (α:β) while failing human ventricles express no detectable α-myosin. We report here fast-kinetic analysis of recombinant human α and β myosin heavy chain motor domains. This represents the first such analysis of any human muscle myosin motor and the first of α-myosin from any species. Our findings reveal substantial isoform differences in individual kinetic parameters, overall contractile character, and predicted cycle times. For these parameters, α-subfragment 1 (S1) is far more similar to adult fast skeletal muscle myosin isoforms than to the slow β isoform despite 91% sequence identity between the motor domains of α- and β-myosin. Among the features that differentiate α- from β-S1: the ATP hydrolysis step of α-S1 is ~ten-fold faster than β-S1, α-S1 exhibits ~five-fold weaker actin affinity than β-S1, and actin·α-S1 exhibits rapid ADP release, which is >ten-fold faster than ADP release for β-S1. Overall, the cycle times are ten-fold faster for α-S1 but the portion of time each myosin spends tightly bound to actin (the duty ratio) is similar. Sequence analysis points to regions that might underlie the basis for this finding.

From single molecule fluctuations to muscle contraction: a Brownian model of A.F. Huxley's hypotheses

Muscular force generation in response to external stimuli is the result of thermally fluctuating, cyclical interactions between myosin and actin, which together form the actomyosin complex. Normally, these fluctuations are modelled using transition rate functions that are based on muscle fiber behaviour, in a phenomenological fashion. However, such a basis reduces the predictive power of these models. As an alternative, we propose a model which uses direct single molecule observations of actomyosin fluctuations reported in the literature. We precisely estimate the actomyosin potential bias and use diffusion theory to obtain a brownian ratchet model that reproduces the complete cross-bridge cycle. The model is validated by simulating several macroscopic experimental conditions, while its interpretation is compatible with two different force-generating scenarios.

Identification of functional differences between recombinant human α and β cardiac myosin motors

The myosin isoform composition of the heart is dynamic in health and disease and has been shown to affect contractile velocity and force generation. While different mammalian species express different proportions of α and β myosin heavy chain, healthy human heart ventricles express these isoforms in a ratio of about 1:9 (α:β) while failing human ventricles express no detectable α-myosin. We report here fast-kinetic analysis of recombinant human α and β myosin heavy chain motor domains. This represents the first such analysis of any human muscle myosin motor and the first of α-myosin from any species. Our findings reveal substantial isoform differences in individual kinetic parameters, overall contractile character, and predicted cycle times. For these parameters, α-subfragment 1 (S1) is far more similar to adult fast skeletal muscle myosin isoforms than to the slow β isoform despite 91% sequence identity between the motor domains of α- and β-myosin. Among the features that differentiate α- from β-S1: the ATP hydrolysis step of α-S1 is ~ten-fold faster than β-S1, α-S1 exhibits ~five-fold weaker actin affinity than β-S1, and actin·α-S1 exhibits rapid ADP release, which is >ten-fold faster than ADP release for β-S1. Overall, the cycle times are ten-fold faster for α-S1 but the portion of time each myosin spends tightly bound to actin (the duty ratio) is similar. Sequence analysis points to regions that might underlie the basis for this finding.

Structural changes in isometrically contracting insect flight muscle trapped following a mechanical perturbation

The application of rapidly applied length steps to actively contracting muscle is a classic method for synchronizing the response of myosin cross-bridges so that the average response of the ensemble can be measured. Alternatively, electron tomography (ET) is a technique that can report the structure of the individual members of the ensemble. We probed the structure of active myosin motors (cross-bridges) by applying 0.5% changes in length (either a stretch or a release) within 2 ms to isometrically contracting insect flight muscle (IFM) fibers followed after 5–6 ms by rapid freezing against a liquid helium cooled copper mirror. ET of freeze-substituted fibers, embedded and thin-sectioned, provides 3-D cross-bridge images, sorted by multivariate data analysis into ∼40 classes, distinct in average structure, population size and lattice distribution. Individual actin subunits are resolved facilitating quasi-atomic modeling of each class average to determine its binding strength (weak or strong) to actin. ∼98% of strong-binding acto-myosin attachments present after a length perturbation are confined to “target zones” of only two actin subunits located exactly midway between successive troponin complexes along each long-pitch helical repeat of actin. Significant changes in the types, distribution and structure of actin-myosin attachments occurred in a manner consistent with the mechanical transients. Most dramatic is near disappearance, after either length perturbation, of a class of weak-binding cross-bridges, attached within the target zone, that are highly likely to be precursors of strong-binding cross-bridges. These weak-binding cross-bridges were originally observed in isometrically contracting IFM. Their disappearance following a quick stretch or release can be explained by a recent kinetic model for muscle contraction, as behaviour consistent with their identification as precursors of strong-binding cross-bridges. The results provide a detailed model for contraction in IFM that may be applicable to contraction in other types of muscle.

Quasiperiodic distribution of rigor cross-bridges along a reconstituted thin filament in a skeletal myofibril

Electron microscopy has shown that cross-bridges (CBs) are formed at the target zone that is periodically distributed on the thin filament in striated muscle. Here, by manipulating a single bead-tailed actin filament with optical tweezers, we measured the unbinding events of rigor CBs one by one on the surface of the A-band in rabbit skeletal myofibrils. We found that the spacings between adjacent CBs were not always the same, and instead were 36, 72, or 108 nm. Tropomyosin and troponin did not affect the CB spacing except for a relative increase in the appearance of longer spacing in the presence of Ca(2+). In addition, in an in vitro assay where myosin molecules were randomly distributed, were obtained the same spacing, i.e., a multiple of 36 nm. These results indicate that the one-dimensional distribution of CBs matches with the 36-nm half pitch of a long helical structure of actin filaments. A stereospecific model composed of three actin protomers per target zone was shown to explain the experimental results. Additionally, the unbinding force (i.e., the binding affinity) of CBs for the reconstituted thin filaments was found to be larger and smaller relative to that for actin filaments with and without Ca(2+), respectively.

Impact of myocyte strain on cardiac myofilament activation

When cardiac myocytes are stretched by a longitudinal strain, they develop proportionally more active force at a given sub-maximal Ca(2+) concentration than they did at the shorter length. This is known as length-dependent activation. It is one of the most important contributors to the Frank-Starling relationship, a critical part of normal cardiovascular function. Despite intense research efforts, the mechanistic basis of the Frank-Starling relationship remains unclear. Potential mechanisms involving myofibrillar lattice spacing, titin-based effects, and cooperative activation have all been proposed. This review summarizes some of these mechanisms and discusses two additional potential theories that reflect the effects of localized strains that occur within and between half-sarcomeres. The main conclusion is that the Frank-Starling relationship is probably the integrated result of many interacting molecular mechanisms. Multiscale computational modeling may therefore provide the best way of determining the key processes that underlie length-dependent activation and their relative strengths.

Emerging complex pathways of the actomyosin powerstroke

Actomyosin powers muscle contraction and various cellular activities, including cell division, differentiation, intracellular transport and sensory functions. Despite their crucial roles, key aspects of force generation have remained elusive. To perform efficient force generation, the powerstroke must occur while myosin is bound to actin. Paradoxically, this process must be initiated when myosin is in a very low actin-affinity state. Recent results shed light on a kinetic pathway selection mechanism whereby the actin-induced activation of the swing of myosin's lever enables efficient mechanical functioning. Structural elements and biochemical principles involved in this mechanism are conserved among various NTPase-effector (e.g. kinesin-microtubule, G protein exchange factor and kinase-scaffold protein) systems that perform chemomechanical or signal transduction.

Axial and radial forces of cross-bridges depend on lattice spacing

Nearly all mechanochemical models of the cross-bridge treat myosin as a simple linear spring arranged parallel to the contractile filaments. These single-spring models cannot account for the radial force that muscle generates (orthogonal to the long axis of the myofilaments) or the effects of changes in filament lattice spacing. We describe a more complex myosin cross-bridge model that uses multiple springs to replicate myosin's force-generating power stroke and account for the effects of lattice spacing and radial force. The four springs which comprise this model (the 4sXB) correspond to the mechanically relevant portions of myosin's structure. As occurs in vivo, the 4sXB's state-transition kinetics and force-production dynamics vary with lattice spacing. Additionally, we describe a simpler two-spring cross-bridge (2sXB) model which produces results similar to those of the 4sXB model. Unlike the 4sXB model, the 2sXB model requires no iterative techniques, making it more computationally efficient. The rate at which both multi-spring cross-bridges bind and generate force decreases as lattice spacing grows. The axial force generated by each cross-bridge as it undergoes a power stroke increases as lattice spacing grows. The radial force that a cross-bridge produces as it undergoes a power stroke varies from expansive to compressive as lattice spacing increases. Importantly, these results mirror those for intact, contracting muscle force production.