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

Design Principles and Benefits of Spatially Explicit Models of Myofilament Function

Spatially explicit models of muscle contraction include fine-scale details about the spatial, kinetic, and/or mechanical properties of the biological processes being represented within the model network. Over the past 25 years, this has primarily consisted of a set of mathematical and computational algorithms representing myosin cross-bridge activity, Ca2+-activation of contraction, and ensemble force production within a half-sarcomere representation of the myofilament network. Herein we discuss basic design principles associated with creating spatially explicit models of myofilament function, as well as model assumptions underlying model development. A brief overview of computational approaches is introduced. Opportunities for new model directions that could investigate coupled regulatory pathways between the thick-filament and thin-filaments are also presented. Given the modular design and flexibility associated with spatially explicit models, we highlight some advantages of this approach compared to other model formulations.

Advancing clinical translation of cardiac biomechanics models: a comprehensive review, applications and future pathways

Cardiac mechanics models are developed to represent a high level of detail, including refined anatomies, accurate cell mechanics models, and platforms to link microscale physiology to whole-organ function. However, cardiac biomechanics models still have limited clinical translation. In this review, we provide a picture of cardiac mechanics models, focusing on their clinical translation. We review the main experimental and clinical data used in cardiac models, as well as the steps followed in the literature to generate anatomical meshes ready for simulations. We describe the main models in active and passive mechanics and the different lumped parameter models to represent the circulatory system. Lastly, we provide a summary of the state-of-the-art in terms of ventricular, atrial, and four-chamber cardiac biomechanics models. We discuss the steps that may facilitate clinical translation of the biomechanics models we describe. A well-established software to simulate cardiac biomechanics is lacking, with all available platforms involving different levels of documentation, learning curves, accessibility, and cost. Furthermore, there is no regulatory framework that clearly outlines the verification and validation requirements a model has to satisfy in order to be reliably used in applications. Finally, better integration with increasingly rich clinical and/or experimental datasets as well as machine learning techniques to reduce computational costs might increase model reliability at feasible resources. Cardiac biomechanics models provide excellent opportunities to be integrated into clinical workflows, but more refinement and careful validation against clinical data are needed to improve their credibility. In addition, in each context of use, model complexity must be balanced with the associated high computational cost of running these models.

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.

Molecular mechanisms of muscle contraction: A historical perspective

Studies of muscle structure and function can be traced to at least 2,000 years ago. However, the modern era of muscle contraction mechanisms started in the 1950s with the classic works by AF Huxley and HE Huxley, both born in the United Kingdom, but not related and working independently. HE Huxley was the first to suggest that muscle contraction occurred through the sliding of two sets of filamentous structures (actin or thin filaments and myosin or thick filaments). AF Huxley then developed a biologically inspired mathematical model suggesting a possible molecular mechanism of how this sliding of actin and myosin might take place. This model then evolved from a two-state to a multi-state model of myosin-actin interactions, and from one that suggested a linear motor causing the sliding to a rotating motor. This model, the cross-bridge model of muscle contraction, is still widely used in biomechanics, and even the more sophisticated cross-bridge models of today still contain many of the features originally proposed by AF Huxley. In 2002, we discovered a hitherto unknown property of muscle contraction that suggested the involvement of passive structures in active force production, the so-called passive force enhancement. It was quickly revealed that this passive force enhancement was caused by the filamentous protein titin, and the three-filament (actin, myosin, and titin) sarcomere model of muscle contraction evolved. There are many suggestions of how these three proteins interact to cause contraction and produce active force, and one such suggestion is described here, but the molecular details of this proposed mechanism still need careful evaluation.

Using Multiscale Simulations as a Tool to Interpret Equatorial X-ray Fiber Diffraction Patterns from Skeletal Muscle

Synchrotron small-angle X-ray diffraction is the method of choice for nm-scale structural studies of striated muscle under physiological conditions and on millisecond time scales. The lack of generally applicable computational tools for modeling X-ray diffraction patterns from intact muscles has been a significant barrier to exploiting the full potential of this technique. Here, we report a novel "forward problem" approach using the spatially explicit computational simulation platform MUSICO to predict equatorial small-angle X-ray diffraction patterns and the force output simultaneously from resting and isometrically contracting rat skeletal muscle that can be compared to experimental data. The simulation generates families of thick-thin filament repeating units, each with their individually predicted occupancies of different populations of active and inactive myosin heads that can be used to generate 2D-projected electron density models based on known Protein Data Bank structures. We show how, by adjusting only a few selected parameters, we can achieve a good correspondence between experimental and predicted X-ray intensities. The developments presented here demonstrate the feasibility of combining X-ray diffraction and spatially explicit modeling to form a powerful hypothesis-generating tool that can be used to motivate experiments that can reveal emergent properties of muscle.

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.

Titin force in muscle cells alters lattice order, thick and thin filament protein formation

Skeletal muscle force production is increased at longer compared to shorter muscle lengths because of length-dependent priming of thick filament proteins in the contractile unit before contraction. Using small-angle X-ray diffraction in combination with a mouse model that specifically cleaves the stretch-sensitive titin protein, we found that titin cleavage diminished the length-dependent priming of the thick filament. Strikingly, a titin-sensitive, length-dependent priming was also present in thin filaments, which seems only possible via bridge proteins between thick and thin filaments in resting muscle, potentially myosin-binding protein C. We further show that these bridges can be forcibly ruptured via high-speed stretches. Our results advance a paradigm shift to the fundamental regulation of length-dependent priming, with titin as the key driver.

Mitochondrial network configuration influences sarcomere and myosin filament structure in striated muscles

Sustained muscle contraction occurs through interactions between actin and myosin filaments within sarcomeres and requires a constant supply of adenosine triphosphate (ATP) from nearby mitochondria. However, it remains unclear how different physical configurations between sarcomeres and mitochondria alter the energetic support for contractile function. Here, we show that sarcomere cross-sectional area (CSA) varies along its length in a cell type-dependent manner where the reduction in Z-disk CSA relative to the sarcomere center is closely coordinated with mitochondrial network configuration in flies, mice, and humans. Further, we find myosin filaments near the sarcomere periphery are curved relative to interior filaments with greater curvature for filaments near mitochondria compared to sarcoplasmic reticulum. Finally, we demonstrate variable myosin filament lattice spacing between filament ends and filament centers in a cell type-dependent manner. These data suggest both sarcomere structure and myofilament interactions are influenced by the location and orientation of mitochondria within muscle cells.

Small Angle X-ray Diffraction as a Tool for Structural Characterization of Muscle Disease

Small angle X-ray fiber diffraction is the method of choice for obtaining molecular level structural information from striated muscle fibers under hydrated physiological conditions. For many decades this technique had been used primarily for investigating basic biophysical questions regarding muscle contraction and regulation and its use confined to a relatively small group of expert practitioners. Over the last 20 years, however, X-ray diffraction has emerged as an important tool for investigating the structural consequences of cardiac and skeletal myopathies. In this review we show how simple and straightforward measurements, accessible to non-experts, can be used to extract biophysical parameters that can help explain and characterize the physiology and pathology of a given experimental system. We provide a comprehensive guide to the range of the kinds of measurements that can be made and illustrate how they have been used to provide insights into the structural basis of pathology in a comprehensive review of the literature. We also show how these kinds of measurements can inform current controversies and indicate some future directions.

Effect of Myosin Isoforms on Cardiac Muscle Twitch of Mice, Rats and Humans

To understand how pathology-induced changes in contractile protein isoforms modulate cardiac muscle function, it is necessary to quantify the temporal-mechanical properties of contractions that occur under various conditions. Pathological responses are much easier to study in animal model systems than in humans, but extrapolation between species presents numerous challenges. Employing computational approaches can help elucidate relationships that are difficult to test experimentally by translating the observations from rats and mice, as model organisms, to the human heart. Here, we use the spatially explicit MUSICO platform to model twitch contractions from rodent and human trabeculae collected in a single laboratory. This approach allowed us to identify the variations in kinetic characteristics of α- and β-myosin isoforms across species and to quantify their effect on cardiac muscle contractile responses. The simulations showed how the twitch transient varied with the ratio of the two myosin isoforms. Particularly, the rate of tension rise was proportional to the fraction of α-myosin present, while the β-isoform dominated the rate of relaxation unless α-myosin was >50%. Moreover, both the myosin isoform and the Ca2+ transient contributed to the twitch tension transient, allowing two levels of regulation of twitch contraction.

FiberSim: A flexible open-source model of myofilament-level contraction

FiberSim is a flexible open-source model of myofilament-level contraction. The code uses a spatially explicit technique, meaning that it tracks the position and status of each contractile molecule within the lattice framework. This allows the model to simulate some of the mechanical effects modulated by myosin-binding protein C, as well as the dose dependence of myotropes and the effects of varying isoform expression levels. This paper provides a short introduction to FiberSim and presents simulations of tension-pCa curves with and without regulation of thick and thin filament activation by myosin-binding protein C. A myotrope dose-dependent response as well as slack/re-stretch maneuvers to assess rates of tension recovery are also presented. The software was designed to be flexible (the user can define their own model and/or protocol) and computationally efficient (simulations can be performed on a regular laptop). We hope that other investigators will use FiberSim to explore myofilament level mechanisms and to accelerate research focusing on the contractile properties of sarcomeres.

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.

The Sliding Filament Theory Since Andrew Huxley: Multiscale and Multidisciplinary Muscle Research

Two groundbreaking papers published in 1954 laid out the theory of the mechanism of muscle contraction based on force-generating interactions between myofilaments in the sarcomere that cause filaments to slide past one another during muscle contraction. The succeeding decades of research in muscle physiology have revealed a unifying interest: to understand the multiscale processes-from atom to organ-that govern muscle function. Such an understanding would have profound consequences for a vast array of applications, from developing new biomimetic technologies to treating heart disease. However, connecting structural and functional properties that are relevant at one spatiotemporal scale to those that are relevant at other scales remains a great challenge. Through a lens of multiscale dynamics, we review in this article current and historical research in muscle physiology sparked by the sliding filament theory.

Force-velocity and tension transient measurements from Drosophila jump muscle reveal the necessity of both weakly-bound cross-bridges and series elasticity in models of muscle contraction

Muscle contraction is a fundamental biological process where molecular interactions between the myosin molecular motor and actin filaments result in contraction of a whole muscle, a process spanning size scales differing in eight orders of magnitude. Since unique behavior is observed at every scale in between these two extremes, to fully understand muscle function it is vital to develop multi-scale models. Based on simulations of classic measurements of muscle heat generation as a function of work, and shortening rate as a function of applied force, we hypothesize that a model based on molecular measurements must be modified to include a weakly-bound interaction between myosin and actin in order to fit measurements at the muscle fiber or whole muscle scales. This hypothesis is further supported by the model's need for a weakly-bound state in order to qualitatively reproduce the force response that occurs when a muscle fiber is rapidly stretched a small distance. We tested this hypothesis by measuring steady-state force as a function of shortening velocity, and the force transient caused by a rapid length step in Drosophila jump muscle fibers. Then, by performing global parameter optimization, we quantitatively compared the predictions of two mathematical models, one lacking a weakly-bound state and one with a weakly-bound state, to these measurements. Both models could reproduce our force-velocity measurements, but only the model with a weakly-bound state could reproduce our force transient measurements. However, neither model could concurrently fit both measurements. We find that only a model that includes weakly-bound cross-bridges with force-dependent detachment and an elastic element in series with the cross-bridges is able to fit both of our measurements. This result suggests that the force response after stretch is not a reflection of distinct steps in the cross-bridge cycle, but rather arises from the interaction of cross-bridges with a series elastic element. Additionally, the model suggests that the curvature of the force-velocity relationship arises from a combination of the force-dependence of weakly- and strongly-bound cross-bridges. Overall, this work presents a minimal cross-bridge model that has predictive power at the fiber level.

Actin bundle architecture and mechanics regulate myosin II force generation

The actin cytoskeleton is a soft, structural material that underlies biological processes such as cell division, motility, and cargo transport. The cross-linked actin filaments self-organize into a myriad of architectures, from disordered meshworks to ordered bundles, which are hypothesized to control the actomyosin force generation that regulates cell migration, shape, and adhesion. Here, we use fluorescence microscopy and simulations to investigate how actin bundle architectures with varying polarity, spacing, and rigidity impact myosin II dynamics and force generation. Microscopy reveals that mixed-polarity bundles formed by rigid cross-linkers support slow, bidirectional myosin II filament motion, punctuated by periods of stalled motion. Simulations reveal that these locations of stalled myosin motion correspond to sustained, high forces in regions of balanced actin filament polarity. By contrast, mixed-polarity bundles formed by compliant, large cross-linkers support fast, bidirectional motion with no traps. Simulations indicate that trap duration is directly related to force magnitude and that the observed increased velocity corresponds to lower forces resulting from both the increased bundle compliance and filament spacing. Our results indicate that the microstructures of actin assemblies regulate the dynamics and magnitude of myosin II forces, highlighting the importance of architecture and mechanics in regulating forces in biological materials.

The spatial distribution of thin filament activation influences force development and myosin activity in computational models of muscle contraction

Striated muscle contraction is initiated by Ca²⁺ binding to, and activating, thin filament regulatory units (RU) within the sarcomere, which then allows myosin cross-bridges from the opposing thick filament to bind actin and generate force. The amount of overlap between the filaments dictates how many potential cross-bridges are capable of binding, and thus how force is generated by the sarcomere. Myopathies and atrophy can impair muscle function by limiting cross-bridge interactions between the filaments, which can occur when the length of the thin filament is reduced or when RU function is disrupted. To investigate how variations in thin filament length and RU density affect ensemble cross-bridge behavior and force production, we simulated muscle contraction using a spatially explicit computational model of the half-sarcomere. Thin filament RUs were disabled either uniformly from the pointed end of the filament (to model shorter thin filament length) or randomly throughout the length of the half-sarcomere. Both uniform and random RU 'knockout' schemes decreased overall force generation during maximal and submaximal activation. The random knockout scheme also led to decreased calcium sensitivity and cooperativity of the force-pCa relationship. We also found that the rate of force development slowed with the random RU knockout, compared to the uniform RU knockout or conditions of normal RU activation. These findings imply that the relationship between RU density and force production within the sarcomere involves more complex coordination than simply the raw number of RUs available for myosin cross-bridge binding, and that the spatial pattern in which activatable RU are distributed throughout the sarcomere influences the dynamics of force production.

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.

The effect of variable troponin C mutation thin filament incorporation on cardiac muscle twitch contractions

One of the complexities of understanding the pathology of familial forms of cardiac diseases is the level of mutation incorporation in sarcomeres. Computational models of the sarcomere that are spatially explicit offer an approach to study aspects of mutational incorporation into myofilaments that are more challenging to get at experimentally. We studied two well characterized mutations of cardiac TnC, L48Q and I61Q, that decrease or increase the release rate of Ca²⁺ from cTnC, k_-Ca, resulting in HCM and DCM respectively [1]. Expression of these mutations in transgenic mice was used to provide experimental data for incorporation of 30 and 50% (respectively) into sarcomeres. Here we demonstrate that fixed length twitch contractions of trabeculae from mice containing mutant differ from WT; L48Q trabeculae have slower relaxation while I61Q trabeculae have markedly reduced peak tension. Using our multiscale modelling approach [2] we were able to describe the tension transients of WT mouse myocardium. Tension transients for the mutant cTnCs were simulated with changes in k_-Ca, measured experimentally for each cTnC mutant in whole troponin complex, a change in the affinity of cTnC for cTnI, and a reduction in the number of detached crossbridges available for binding. A major advantage of the multiscale explicit 3-D model is that it predicts the effects of variable mutation incorporation, and the effects of variations in mutation distribution within thin filaments in sarcomeres. Such effects are currently impossible to explore experimentally. We explored random and clustered distributions of mutant cTnCs in thin filaments, as well as distributions of individual thin filaments with only WT or mutant cTnCs present. The effects of variable amounts of incorporation and non-random distribution of mutant cTnCs are more marked for I61Q than L48Q cTnC. We conclude that this approach can be effective for study on mutations in multiple proteins of the sarcomere. SUMMARY: A challenge in experimental studies of diseases is accounting for the effect of variable mutation incorporation into myofilaments. Here we use a spatially explicit computational approach, informed by experimental data from transgenic mice expressing one of two mutations in cardiac Troponin C that increase or decrease calcium sensitivity. We demonstrate that the model can accurately describe twitch contractions for the data and go on to explore the effect of variable mutant incorporation and localization on simulated cardiac muscle twitches.

The Transient Mechanics of Muscle Require Only a Single Force-Producing Cross-Bridge State and a 100 Å Working Stroke

Simple Summary

With modern increased computational power, newly developed computer programs can be used to simulate how muscle contracts. Here, we created, in silico, a “virtual” muscle that includes modelled myosin cross-bridges, and, using statistical mechanical methods, we calculated the macroscopic response of the muscle during contraction and as a result of applied transients. Good fits to many experimental observations were obtained with this simple model with one attached force-producing state and using a single cross-bridge step size of 100 Å.

Abstract

An informative probe of myosin cross-bridge behaviour in active muscle is a mechanical transient experiment where, for example, a fully active muscle initially held at constant length is suddenly shortened to a new fixed length, providing a force transient, or has its load suddenly reduced, providing a length transient. We describe the simplest cross-bridge mechanical cycle we could find to model these transients. We show using the statistical mechanics of 50,000 cross-bridges that a simple cycle with two actin-attached cross-bridge states, one producing no force and the other producing force, will explain much of what has been observed experimentally, and we discuss the implications of this modelling for our understanding of how muscle works. We show that this same simple model will explain, reasonably well, the isotonic mechanical and X-ray transients under different loads observed by Reconditi et al. (2004, Nature 428, 578) and that there is no need to invoke different cross-bridge step sizes under these different conditions; a step size of 100 Å works well for all loads. We do not claim that this model provides a total mechanical explanation of how muscle works. However, we do suggest that only if there are other observations that cannot be explained by this simple model should something more complicated be considered.

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.

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.

Titin: A Tunable Spring in Active Muscle

Muscle has conventionally been viewed as a motor that converts chemical to kinetic energy in series with a passive spring, but new insights emerge when muscle is viewed as a composite material whose elastic elements are tuned by activation. New evidence demonstrates that calcium-dependent binding of N2A titin to actin increases titin stiffness in active skeletal muscles, which explains many long-standing enigmas of muscle physiology.

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.

MyBP-C: one protein to govern them all

The heart is an extraordinarily versatile pump, finely tuned to respond to a multitude of demands. Given the heart pumps without rest for decades its efficiency is particularly relevant. Although many proteins in the heart are essential for viability, the non-essential components can attract numerous mutations which can cause disease, possibly through alterations in pumping efficiency. Of these, myosin binding protein C is strongly over-represented with ~ 40% of all known mutations in hypertrophic cardiomyopathy. Therefore, a complete understanding of its molecular function in the cardiac sarcomere is warranted. In this review, we revisit contemporary and classical literature to clarify both the current standing of this fast-moving field and frame future unresolved questions. To date, much effort has been directed at understanding MyBP-C function on either thick or thin filaments. Here we aim to focus questions on how MyBP-C functions at a molecular level in the context of both the thick and thin filaments together. A concept that emerges is MyBP-C acts to govern interactions on two levels; controlling myosin access to the thin filament by sequestration on the thick filament, and controlling the activation state and access of myosin to its binding sites on the thin filament. Such affects are achieved through directed interactions mediated by phosphorylation (of MyBP-C and other sarcomeric components) and calcium.

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.

High-speed AFM reveals subsecond dynamics of cardiac thin filaments upon Ca2+ activation and heavy meromyosin binding

The advent of high-speed atomic force microscopy (HS-AFM) changed the field of biology considerably. HS-AFM is the only method where in situ dynamics of biological samples and imaging can be coupled with a spatial resolution of 1 to 5 nm in the horizontal direction. Unlike electron or cryo-electron microscopy, HS-AFM does not require fixation or freezing of the samples, and has the ability to derive kinetic parameters by recording the live movements of single-molecule dynamics. In this paper, we used HS-AFM to investigate directly the mechanisms of cardiac muscle activation. We visualized the muscle regulatory tropomyosin–troponin complex movements during activation by calcium or myosin (motor that drives contraction), and the structural transitions that happen during these events.

High-speed atomic force microscopy (HS-AFM) can be used to study dynamic processes with real-time imaging of molecules within 1- to 5-nm spatial resolution. In the current study, we evaluated the 3-state model of activation of cardiac thin filaments (cTFs) isolated as a complex and deposited on a mica-supported lipid bilayer. We studied this complex for dynamic conformational changes 1) at low and high [Ca²⁺] (pCa 9.0 and 4.5), and 2) upon myosin binding to the cTF in the nucleotide-free state or in the presence of ATP. HS-AFM was used to directly visualize the tropomyosin–troponin complex and Ca²⁺-induced tropomyosin movements accompanied by structural transitions of actin monomers within cTFs. Our data show that cTFs at relaxing or activating conditions are not ultimately in a blocked or activated state, respectively, but rather the combination of states with a prevalence that is dependent on the [Ca²⁺] and the presence of weakly or strongly bound myosin. The weakly and strongly bound myosin induce similar changes in the structure of cTFs as confirmed by the local dynamical displacement of individual tropomyosin strands in the center of a regulatory unit of cTF at the relaxed and activation conditions. The displacement of tropomyosin at the relaxed conditions had never been visualized directly and explains the ability of myosin binding to TF at the relaxed conditions. Based on the ratios of nonactivated and activated segments within cTFs, we proposed a mechanism of tropomyosin switching from different states that includes both weakly and strongly bound myosin.

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.

Elastic domains of giant proteins in striated muscle: Modeling compliance with rulers

Chase examines a study using the MUSICO model of striated muscle to evaluate the function of giant elastic proteins titin and nebulin.

Nebulin and titin modulate cross-bridge cycling and length-dependent calcium sensitivity

Various mutations in the structural proteins nebulin and titin that are present in human disease are known to affect the contractility of striated muscle. Loss of nebulin is associated with reduced actin filament length and impairment of myosin binding to actin, whereas titin is thought to regulate muscle passive elasticity and is likely involved in length-dependent activation. Here, we sought to assess the modulation of muscle function by these sarcomeric proteins by using the computational platform muscle simulation code (MUSICO) to quantitatively separate the effects of structural changes, kinetics of cross-bridge cycling, and calcium sensitivity of the thin filaments. The simulations show that variation in thin filament length cannot by itself account for experimental observations of the contractility in nebulin-deficient muscle, but instead must be accompanied by a decreased myosin binding rate. Additionally, to match the observed calcium sensitivity, the rate of TnI detachment from actin needed to be increased. Simulations for cardiac muscle provided quantitative estimates of the effects of different titin-based passive elasticities on muscle force and activation in response to changes in sarcomere length and interfilament lattice spacing. Predicted force-pCa relations showed a decrease in both active tension and sensitivity to calcium with a decrease in passive tension and sarcomere length. We conclude that this behavior is caused by partial redistribution of the muscle load between active muscle force and titin-dependent passive force, and also by redistribution of stretch along the thin filament, which together modulate the release of TnI from actin. These data help advance understanding of how nebulin and titin mutations affect muscle function.

Smeared multiscale finite element model for electrophysiology and ionic transport in biological tissue

Basic functions of living organisms are governed by the nervous system through bidirectional signals transmitted from the brain to neural networks. These signals are similar to electrical waves. In electrophysiology the goal is to study the electrical properties of biological cells and tissues, and the transmission of signals. From a physics perspective, there exists a field of electrical potential within the living body, the nervous system, extracellular space and cells. Electrophysiological problems can be investigated experimentally and also theoretically by developing appropriate mathematical or computational models. Due to the enormous complexity of biological systems, it would be almost impossible to establish a detailed computational model of the electrical field, even for only a single organ (e.g. heart), including the entirety of cells comprising the neural network. In order to make computational models feasible for practical applications, we here introduce the concept of smeared fields, which represents a generalization of the previously formulated multiscale smeared methodology for mass transport in blood vessels, lymph, and tissue. We demonstrate the accuracy of the smeared finite element computational models for the electric field in numerical examples. The electrical field is further coupled with ionic mass transport within tissue composed of interstitial spaces extracellularly and by cytoplasm and organelles intracellularly. The proposed methodology, which couples electrophysiology and molecular ionic transport, is applicable to a variety of biological systems.

Single cardiac ventricular myosins are autonomous motors

Myosin transduces ATP free energy into mechanical work in muscle. Cardiac muscle has dynamically wide-ranging power demands on the motor as the muscle changes modes in a heartbeat from relaxation, via auxotonic shortening, to isometric contraction. The cardiac power output modulation mechanism is explored in vitro by assessing single cardiac myosin step-size selection versus load. Transgenic mice express human ventricular essential light chain (ELC) in wild- type (WT), or hypertrophic cardiomyopathy-linked mutant forms, A57G or E143K, in a background of mouse α-cardiac myosin heavy chain. Ensemble motility and single myosin mechanical characteristics are consistent with an A57G that impairs ELC N-terminus actin binding and an E143K that impairs lever-arm stability, while both species down-shift average step-size with increasing load. Cardiac myosin in vivo down-shifts velocity/force ratio with increasing load by changed unitary step-size selections. Here, the loaded in vitro single myosin assay indicates quantitative complementarity with the in vivo mechanism. Both have two embedded regulatory transitions, one inhibiting ADP release and a second novel mechanism inhibiting actin detachment via strain on the actin-bound ELC N-terminus. Competing regulators filter unitary step-size selection to control force-velocity modulation without myosin integration into muscle. Cardiac myosin is muscle in a molecule.

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

Models of muscle contraction often assume that a myosin motor interacts with areas of one to three distinct binding sites along an actin filament. Månsson shows that computational models employing different numbers of sites return similar predictions, suggesting one-site models are often sufficient.

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 stochastic simulation of skeletal muscle calcium transients in a structurally realistic sarcomere model using MCell

Skeletal muscle contraction is initiated when an action potential triggers the release of Ca2+ into the sarcomere in a process referred to as excitation-contraction coupling. The speed and scale of this process makes direct observation very challenging and invasive. To determine how the concentration of Ca2+ changes within the myofibril during a single activation, several simulation models have been developed. These models follow a common pattern; divide the half sarcomere into a series of compartments, then use ordinary differential equations to solve reactions occurring within and between the compartments. To further develop this type of simulation, we have created a realistic structural model of a skeletal muscle myofibrillar half-sarcomere using MCell software that incorporates the myofilament lattice structure. Using this simulation model, we were successful in reproducing the averaged calcium transient during a single activation consistent with both the experimental and previous simulation results. In addition, our simulation demonstrated that the inclusion of the myofilament lattice within our model produced an asymmetric distribution of Ca2+, with more Ca2+ accumulating near the Z-disk and less Ca2+ reaching the m-line. This asymmetric distribution of Ca2+ is also apparent when we examine how the Ca2+ are bound to the troponin-C proteins along the actin filaments. Our simulation model also allowed us to produce advanced visualizations of this process, including two simulation animations, allowing us to view Ca2+ release, diffusion, binding and uptake within the myofibrillar half-sarcomere.

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.

A short history of the development of mathematical models of cardiac mechanics

Cardiac mechanics plays a crucial role in atrial and ventricular function, in the regulation of growth and remodelling, in the progression of disease, and the response to treatment. The spatial scale of the critical mechanisms ranges from nm (molecules) to cm (hearts) with the fastest events occurring in milliseconds (molecular events) and the slowest requiring months (growth and remodelling). Due to its complexity and importance, cardiac mechanics has been studied extensively both experimentally and through mathematical models and simulation. Models of cardiac mechanics evolved from seminal studies in skeletal muscle, and developed into cardiac specific, species specific, human specific and finally patient specific calculations. These models provide a formal framework to link multiple experimental assays recorded over nearly 100 years into a single unified representation of cardiac function. This review first provides a summary of the proteins, physiology and anatomy involved in the generation of cardiac pump function. We then describe the evolution of models of cardiac mechanics starting with the early theoretical frameworks describing the link between sarcomeres and muscle contraction, transitioning through myosin-level models to calcium-driven systems, and ending with whole heart patient-specific models.

Cardiac and skeletal actin substrates uniquely tune cardiac myosin strain-dependent mechanics

Cardiac ventricular myosin (βmys) translates actin by transducing ATP free energy into mechanical work during muscle contraction. Unitary βmys translation of actin is the step-size. In vitro and in vivo βmys regulates contractile force and velocity autonomously by remixing three different step-sizes with adaptive stepping frequencies. Cardiac and skeletal actin isoforms have a specific 1 : 4 stoichiometry in normal adult human ventriculum. Human adults with inheritable hypertrophic cardiomyopathy (HCM) upregulate skeletal actin in ventriculum probably compensating the diseased muscle's inability to meet demand by adjusting βmys force-velocity characteristics. βmys force-velocity characteristics were compared for skeletal versus cardiac actin substrates using ensemble in vitro motility and single myosin assays. Two competing myosin strain-sensitive mechanisms regulate step-size choices dividing single βmys mechanics into low- and high-force regimes. The actin isoforms alter myosin strain-sensitive regulation such that onset of the high-force regime, where a short step-size is a large or major contributor, is offset to higher loads probably by the unique cardiac essential light chain (ELC) N-terminus/cardiac actin contact at Glu6/Ser358. It modifies βmys force-velocity by stabilizing the ELC N-terminus/cardiac actin association. Uneven onset of the high-force regime for skeletal versus cardiac actin modulates force-velocity characteristics as skeletal/cardiac actin fractional content increases in diseased muscle.

Acidosis affects muscle contraction by slowing the rates myosin attaches to and detaches from actin

The loss of muscle force and power during fatigue from intense contractile activity is associated with, and likely caused by, elevated levels of phosphate ([Formula: see text]) and hydrogen ions (decreased pH). To understand how these deficits in muscle performance occur at the molecular level, we used direct measurements of mini-ensembles of myosin generating force in the laser trap assay at pH 7.4 and 6.5. The data are consistent with a mechanochemical model in which a decrease in pH reduces myosin's detachment from actin (by slowing ADP release), increases non-productive myosin binding (by detached myosin rebinding without a powerstroke), and reduces myosin's attachment to actin (by slowing the weak-to-strong binding transition). Additional support of this mechanism is found by incorporating it into a branched pathway model for the effects of [Formula: see text] on myosin's interaction with actin. Including pH-dependence in one additional parameter (acceleration of [Formula: see text]-induced detachment), the model reproduces experimental measurements at high and low pH, and variable [Formula: see text], from the single molecule to large ensemble levels. Furthermore, when scaled up, the model predicts force-velocity relationships that are consistent with muscle fiber measurements. The model suggests that reducing pH has two opposing effects, a decrease in attachment favoring a decrease in muscle force and a decrease in detachment favoring an increase in muscle force. Depending on experimental details, the addition of [Formula: see text] can strengthen one or the other effect, resulting in either synergistic or antagonistic effects. This detailed molecular description suggests a molecular basis for contractile failure during muscle fatigue.

Nebulin stiffens the thin filament and augments cross-bridge interaction in skeletal muscle

Nebulin is a giant sarcomeric protein that spans along the actin filament in skeletal muscle, from the Z-disk to near the thin filament pointed end. Mutations in nebulin cause muscle weakness in nemaline myopathy patients, suggesting that nebulin plays important roles in force generation, yet little is known about nebulin's influence on thin filament structure and function. Here, we used small-angle X-ray diffraction and compared intact muscle deficient in nebulin (using a conditional nebulin-knockout, Neb cKO) with control (Ctrl) muscle. When muscles were activated, the spacing of the actin subunit repeat (27 Å) increased in both genotypes; when converted to thin filament stiffness, the obtained value was 30 pN/nm in Ctrl muscle and 10 pN/nm in Neb cKO muscle; that is, the thin filament was approximately threefold stiffer when nebulin was present. In contrast, the thick filament stiffness was not different between the genotypes. A significantly shorter left-handed (59 Å) thin filament helical pitch was found in passive and contracting Neb cKO muscles, as well as impaired tropomyosin and troponin movement. Additionally, a reduced myosin mass transfer toward the thin filament in contracting Neb cKO muscle was found, suggesting reduced cross-bridge interaction. We conclude that nebulin is critically important for physiological force levels, as it greatly stiffens the skeletal muscle thin filament and contributes to thin filament activation and cross-bridge recruitment.

Force-Dependent Recruitment from the Myosin Off State Contributes to Length-Dependent Activation

Cardiac muscle develops more force when it is activated at longer lengths. The concentration of Ca²⁺ required to develop half-maximal force also decreases. These effects are known as length-dependent activation and are thought to play critical roles in the Frank-Starling relationship and cardiovascular homeostasis. The molecular mechanisms underpinning length-dependent activation remain unclear, but recent experiments suggest that they may include recruitment of myosin heads from the off (sometimes called super-relaxed) state. This manuscript presents a mathematical model of muscle contraction that was developed to investigate this hypothesis. Myosin heads in the model transitioned between an off state (that could not interact with actin), an on state (that could bind to actin), and a single attached state. Simulations were fitted to experimental data using multidimensional parameter optimization. Statistical analysis showed that a model in which the rate of the off-to-on transition increased linearly with force reproduced the length-dependent behavior of chemically permeabilized myocardium better than a model with a constant off-to-on transition rate (F-test, p < 0.001). This result suggests that the thick-filament transitions are modulated by force. Additional calculations showed that the model incorporating a mechanosensitive thick filament could also reproduce twitch responses measured in a trabecula stretched to different lengths. A final set of simulations was then used to test the model. These calculations predicted how reducing passive stiffness would impact the length dependence of the calcium sensitivity of contractile force. The prediction (a 60% reduction in ΔpCa₅₀) mimicked the 58% reduction in ΔpCa₅₀ in myocardium from rats that expressed a giant isoform of titin and had low resting tension. Together, these computational results suggest that force-dependent recruitment of myosin heads from the thick-filament off state contributes to length-dependent activation and the Frank-Starling relationship.

Effects of cross-bridge compliance on the force-velocity relationship and muscle power output

Muscles produce force and power by utilizing chemical energy through ATP hydrolysis. During concentric contractions (shortening), muscles generate less force compared to isometric contractions, but consume greater amounts of energy as shortening velocity increases. Conversely, more force is generated and less energy is consumed during eccentric muscle contractions (lengthening). This relationship between force, energy use, and the velocity of contraction has important implications for understanding muscle efficiency, but the molecular mechanisms underlying this behavior remain poorly understood. Here we used spatially-explicit, multi-filament models of Ca2+-regulated force production within a half-sarcomere to simulate how force production, energy utilization, and the number of bound cross-bridges are affected by dynamic changes in sarcomere length. These computational simulations show that cross-bridge binding increased during slow-velocity concentric and eccentric contractions, compared to isometric contractions. Over the full ranges of velocities that we simulated, cross-bridge cycling and energy utilization (i.e. ATPase rates) increased during shortening, and decreased during lengthening. These findings are consistent with the Fenn effect, but arise from a complicated relationship between velocity-dependent cross-bridge recruitment and cross-bridge cycling kinetics. We also investigated how force production, power output, and energy utilization varied with cross-bridge and myofilament compliance, which is impossible to address under typical experimental conditions. These important simulations show that increasing cross-bridge compliance resulted in greater cross-bridge binding and ATPase activity, but less force was generated per cross-bridge and throughout the sarcomere. These data indicate that the efficiency of force production decreases in a velocity-dependent manner, and that this behavior is sensitive to cross-bridge compliance. In contrast, significant effects of myofilament compliance on force production were only observed during isometric contractions, suggesting that changes in myofilament compliance may not influence power output during non-isometric contractions as greatly as changes in cross-bridge compliance. These findings advance our understanding of how cross-bridge and myofilament properties underlie velocity-dependent changes in contractile efficiency during muscle movement.

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.

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.

In vitro actin motility velocity varies linearly with the number of myosin impellers

Cardiac myosin is the motor powering the heart. It moves actin with 3 step-size varieties generated by torque from the myosin heavy chain lever-arm rotation under the influence of myosin essential light chain whose N-terminal extension binds actin. Proposed mechanisms adapting myosin mechanochemical characteristics on the fly sometimes involve modulation of step-size selection probability via motor strain sensitivity. Strain following the power stroke, hypothetically imposed by the finite actin detachment rate 1/t_on, is shown to have no effect on unloaded velocity when multiple myosins are simultaneously strongly actin bound in an in vitro motility assay. Actin filaments slide ∼2 native step-sizes while more than 1 myosin strongly binds actin probably ruling out an actin detachment limited model for imposing strain. It suggests that single myosin estimates for t_on are too large, not applicable to the ensemble situation, or both. Parallel motility data quantitation involving instantaneous particle velocities (frame velocity) and actin filament track averaged velocities (track velocity) give an estimate of the random walk step-size, δ. Comparing δ for slow and fast motility components suggests the higher speed component has cardiac myosin upshifting to longer steps. Variable step-size characteristics imply cardiac myosin maintains a velocity dynamic range not involving strain.