A model of multicomponent cardiac fibre

In order to simulate the contraction of a cardiac myofibre, a multicomponent fibre model has been developed. This model is composed of a series of segments which are activated in succession. Each segment is represented by the Hill's three component model of the sarcomere. The contractile element behaviour is described by the Huxley's theory and the time dependence agrees with the activation factor proposed by Julian for skeletal muscle, and modified by Wong for cardiac muscle. The two elastic elements have non-linear exponential characteristics. The isometric contraction of the multicomponent fibre has been simulated by means of a computer program. The results show the tension generated by the fibre, the propagation of the contraction along the fibre and the different contribution of each segment depending on its position inside the fibre.     

Optimum power output and structural design of sarcomeres

A model of a "general" sarcomere is presented for the calculation of power output as a function of (i) contraction range, (ii) contraction velocity, (iii) muscle fibre stimulation (active state) and (iv) structural parameters of the sarcomere (i.e. lengths of actin, myosin, and bare zone on myosin, and thickness of the Z-disc). The model is applicable to virtually all types of striated muscle fibres. By computer simulation, particular combinations of actin and myosin lengths were found that maximize the specific power output for particular functional demands, specified in terms of contraction range and contraction velocity. The accuracy of the prediction of the optimum sarcomere design by the model depends on the quality of its input, i.e. the available knowledge of the in vivo spectrum of contraction velocities and sarcomere excursions. Predictions of sarcomere design from model simulations were compared with ultrastructural data from the literature. With the present model, the complete variation in the ratio of myosin length over actin length (from about 1.05 down to 0.65, as observed in insect and vertebrate sarcomeres) can be explained as a series of adaptations for optimum power output from a small to a large contraction range, respectively.     

Relation of muscle cell contraction and calcium distribution in sarcomere

A model of activation of muscle contraction has been proposed. It is based on calcium diffusion and binding to specific regulatory sites in a sarcomere. Calcium ions activate interactions of contractile proteins and thus the generation of force. The model quantifies the relation between calcium released from intracellular stores and the elicited force.     

Cooperative behaviour of the elementary sarcomere units and the cross-bridge step size

We present a model of muscle contraction based on purely physical grounds and modulated by a parameter, k, related to the visco-elastic hindrances of the contractile apparatus. The model predicts a strong cooperation among sarcomere units and proposes that viscous hindrance is a fundamental component of the economy of the contraction. The concept of cross-bridge step size is also discussed and it is concluded that the step size is of various and probably undeterminable length.     

Dynamic analysis of the structure and function of sarcomeres

We attempted to analyze the relationships between the steric structure of the sarcomere and its physiological functions by the use of a sarcomere model of muscle contraction, which includes the geometric arrangement of the thick and thin filaments of the sarcomere, as well as of the cross-bridges and actin sites. Motions of both cross-bridges and myofilaments were considered in terms of our three-state model of the elementary cycle under constraints caused by the steric structure of the sarcomere proposed by Huxley and Brown. Each cross-bridge moves in a molecular potential of our three-state model under the influence of the sliding motions of myofilaments. The sarcomere model described well the tension-velocity relation and isotonic transient processes quantitatively and consistently. In addition, it allowed independence of the no-load shortening velocity upon the overlap of the thick and thin filaments, although the motions of cross-bridges were not independent. Effects of the helical periodicities of the thick and thin filaments and of the number of cross-bridges upon muscle contraction were studied, and the conditions for smooth and efficient contraction of muscle were obtained.     

Optimal design of vertebrate and insect sarcomeres

This paper offers a model for the normalized length-tension relation of a muscle fiber based upon sarcomere design. Comparison with measurements published by Gordon et al. ('66) shows an accurate fit as long as the inhomogeneity of sarcomere length in a single muscle fiber is taken into account. Sequential change of filament length and the length of the cross-bridge-free zone leads the model to suggest that most vertebrate sarcomeres tested match the condition of optimal construction for the output of mechanical energy over a full sarcomere contraction movement. Joint optimization of all three morphometric parameters suggests that a slightly better (0.3%) design is theoretically possible. However, this theoretical sarcomere, optimally designed for the conversion of energy, has a low normalized contraction velocity; it provides a poorer match to the combined functional demands of high energy output and high contraction velocity than the real sarcomeres of vertebrates. The sarcomeres in fish myotomes appear to be built suboptimally for isometric contraction, but built optimally for that shortening velocity generating maximum power. During swimming, these muscles do indeed contract concentrically only. The sarcomeres of insect asynchronous flight muscles contract only slightly. They are not built optimally for maximum output of energy across the full range of contraction encountered in vertebrate sarcomeres, but are built almost optimally for the contraction range that they do in fact employ.     

Decrease in light diffraction intensity of contracting muscle fibres

Single fibres from the semitendinosus muscle of frog were illuminated normally with a He–Ne laser. The intensity transient and fine structure pattern of light diffracted from the fibre undergoing isometric twitches were measured. During fibre shortening, the intensity decreased rapidly and the fine structure pattern preserved its shape and moved swiftly away from the undiffracted laser beam. The fine structure patterns of the contracting and resting fibre were nearly identical. The ratio of intensities of the contracting and resting fibre of the same sarcomere length was determined as a function of the time elapsed after fibre stimulation. The time-resolved intensity ratio increased with sarcomere length and became unity when sarcomere length was between 3.5 m and 3.7 m. A diffraction theory based on the sarcomere unit was developed. It contained a parameter describing the strength of filament interaction. The comparison between the theory and data shows that the initial intensity drop during contraction is primarily due to filament interactions. At a later stage of contraction, sarcomere disorder becomes the major component causing the intensity to decrease. Diffraction models which use the Debye-Waller formalism to explain the intensity decrease are discussed. The sarcomere-unit diffraction model is applied to previously reported intensity measurements from active fibres.     

Muscle active force-length curve explained by an electrophysical model of interfilament spacing

The active isometric force-length relation (FLR) of striated muscle sarcomeres is central to understanding and modeling muscle function. The mechanistic basis of the descending arm of the FLR is well explained by the decreasing thin:thick filament overlap that occurs at long sarcomere lengths. The mechanistic basis of the ascending arm of the FLR (the decrease in force that occurs at short sarcomere lengths), alternatively, has never been well explained. Because muscle is a constant-volume system, interfilament lattice distances must increase as sarcomere length shortens. This increase would decrease thin and thick-filament electrostatic interactions independently of thin:thick filament overlap. To examine this effect, we present here a fundamental, physics-based model of the sarcomere that includes filament molecular properties, calcium binding, sarcomere geometry including both thin:thick filament overlap and interfilament radial distance, and electrostatics. The model gives extremely good fits to existing FLR data from a large number of different muscles across their entire range of measured activity levels, with the optimized parameter values in all cases lying within anatomically and physically reasonable ranges. A local first-order sensitivity analysis (varying individual parameters while holding the values of all others constant) shows that model output is most sensitive to a subset of model parameters, most of which are related to sarcomere geometry, with model output being most sensitive to interfilament radial distance. This conclusion is supported by re-running the fits with only this parameter subset being allowed to vary, which increases fit errors only moderately. These results show that the model well reproduces existing experimental data, and indicate that changes in interfilament spacing play as central a role as changes in filament overlap in determining the FLR, particularly on its ascending arm.     

Model of calcium diffusion, binding and membrane transport in the sarcomere of frog skeletal muscle

A model of calcium distribution in the sarcomere during activation of contraction was developed. It allows for diffusion and binding of calcium ions to various sarcoplasmic binding sites in the three dimensional spatial coordinate system. The model was used to analyze the influence of kinetic characteristic of binding processes on the temporal and spatial distribution of calcium in the sarcomere during activation of contraction by the action potential and by rectangular depolarizing pulses. The hypothesis concerning the calcium release control in the membrane of terminal cisternae was tested.     

A mathematical model of the sarcomere mechanics of cross-striated muscle with consideration of the stretch and twist of actin filaments

A mathematical model of sarcomere mechanics, which takes into account the elongation of actin and myosin filaments and also twisting of the actin filaments in the sarcomere of striated muscle during contraction is presented. The model accounts for the experimentally observed phenomena of the stretch and twist of the actin filaments due to strong binding of myosin heads and the pulling force. Some model parameters were estimated from published experimental data. The results of modeling show that the twist of actin filaments can play a substantial role in the mechanical responses of contracting muscle fibers to step changes of their length.     

Polarized light observations on striated muscle contraction in a mite

Contraction of individual sarcomeres within the living mite Tarsonemus sp. was observed by polarized light microscopy. In unflattened animals the usual range of contraction was such that the minimum sarcomere length approximated the length of the A region, and the maximum sarcomere length was about twice the length of the A region. The central sarcomeres of the dorsal metapodosomal muscles were observed in detail. The A band length increased slightly with increasing sarcomere length since the regression of I region length on sarcomere length had an average slope of 0.91. When the A band length in a sarcomere which was shortening was compared with the length when the same sarcomere lengthened, no significant difference was seen. The A band of each sarcomere seemed to act as a not too rigid limit to further shortening; this agreed with the reversible shortening of a muscle in which the A band had been experimentally shortened. An H region was visible at long sarcomere lengths and was not visible at short sarcomere lengths, even when the muscle was actively shortening. The rate of change of H region length with sarcomere length suggested that I filament length may increase as sarcomere length increases. Despite this effect and the small increase in A length with sarcomere length, the results are considered to be consistent with a model in which shortening occurs by the relative movement of A and I filaments, with little or no change in length of either set of filaments. Sarcomere shortening was clearly associated with an increase in the retardation of the A region.     

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 sliding filament model of muscle contraction. III. Stability analysis and sinusoidal perturbations

A theoretical model of the energy transducing part of a single sarcomere, which has previously accounted for most of the steady state energetic and dynamical properties of striated muscle, is subjected to stability analysis. The steady states of isotonic and isometric contraction turn out to be stable and the magnitudes of the characteristic roots allow accurate reduction of the original seven state theory to a four state theory. Both versions possess a complex pair of roots giving a damped oscillatory character to the contraction velocity, the tension generated and the populations of the crossbridge states.     

ANALYSIS OF MUSCLE CONTRACTION BY ULTRAVIOLET MICROBEAM DISRUPTION OF SARCOMERE STRUCTURE

In an effort to differentiate between the sliding filament theory for muscle contraction and alternative views which propose attachment between actin and myosin filaments at or across the H zone, rabbit psoas myofibrils were irradiated in various areas of the sarcomere with an ultraviolet microbeam. Irradiation of the I band appears to destroy the actin filaments; in vitro irradiation of F actin causes an irreversible depolymerization of the protein. Irradiation of the A band disorients the myosin but causes no apparent loss of dry mass. These effects are maximal at the wavelength of maximum absorption of the proteins involved. Actin filaments, released at the Z line of a sarcomere, are seen to slide into the A band on addition of ATP. Irradiation of a full A band prevents contraction, whereas irradiation of two-thirds of the A band, leaving a lateral edge intact, permits contraction at the non-irradiated edge. Thus contraction can occur in what is in essence only one-third of a sarcomere, eliminating any necessity for postulated H zone connections. These observations are in complete accord with the classical sliding filament theory but incompatible with either the contralateral filament hypothesis or the actin folding model for muscle contraction.     

Distribution of calcium during contraction and relaxation of crayfish skeletal muscle fibre.

A model of activation of muscle contraction has been applied to the crayfish isolated skeletal muscle fibre. The model is based on calcium diffusion and binding to specific regulatory sites in a sarcomere. Calcium ions activate interactions of contractile proteins and thus the generation of force. The model quantifies the relation between calcium released from intracellular stores and force elicited. Experimental tension records from isolated crayfish skeletal muscle fibres under voltage clamp conditions are analyzed. Model parameters were determined either via approximation of the onset of tension by the model solution or from the model based relations between the tension maximum, and depolarizing pulse length and amplitude. This allowed to determine time changes of free and bound calcium distribution in the sarcomere and the calcium release from terminal cisternae. The steady state calcium concentration at terminal cisternae showed S-shaped voltage dependence with saturation below approx. 10 mumol/l at positive membrane potentials.     

The sarcomere length-tension relation in skeletal muscle

Tension development during isometric tetani in single fibers of frog semitendinosus muscle occurs in three phases: (a) in initial fast-rise phase; (b) a slow-rise phase; and (c) a plateau, which lasts greater than 10 s. The slow-rise phase has previously been assumed to rise out of a progressive increase of sarcomere length dispersion along the fiber (Gordon et al. 1966. J. Physiol. [Lond.]. 184:143--169;184:170-- 192). Consequently, the "true" tetanic tension has been considered to be the one existing before the onset of the slow-rise phase; this is obtained by extrapolating the slowly rising tension back to the start of the tetanus. In the study by Gordon et al. (1966. J. Physiol. [Lond.] 184:170--192), as well as in the present study, the relation between this extrapolated tension and sarcomere length gave the familiar linear descending limb of the length-tension relation. We tested the assumption that the slow rise of tension was due to a progressive increase in sarcomere length dispersion. During the fast rise, the slow rise, and the plateau of tension, the sarcomere length dispersion at any area along the muscle was less than 4% of the average sarcomere length. Therefore, a progressive increase of sarcomere length dispersion during contraction appears unable to account for the slow rise of tetanic tension. A sarcomere length-tension relation was constructed from the levels of tension and sarcomere length measured during the plateau. Tension was independent of sarcomere length between 1.9 and 2.6 microgram, and declined to 50% maximal at 3.4 microgram. This result is difficult to reconcile with the cross-bridge model of force generation.     

Sarcomere Mechanics in Capillary Endothelial Cells

Tension generation in endothelial cells of the aorta, spleen, and eye occurs in actin stress fibers, and is necessary for normal cell function. Sarcomeres are the tension-generating units of actin stress fibers in endothelial cells. How sarcomeres generate and maintain tension in stress fibers is not well understood. Using femtosecond laser ablation, we severed living stress fibers and measured sarcomere contraction under zero tension. The length of the sarcomere decreased in two phases: an instantaneous initial response, followed by a slower change in length attributed to myosin activity. The latter phase ceased abruptly after a minimum sarcomere length was reached, suggesting a rigid resistance that prevents further contraction. Furthermore, severed, contracted stress fibers did not relax when treated with myosin inhibitors, indicating that contracted stress fibers do not store elastic potential energy. These novel measurements combined with modeling suggest that myosin-generated forces in adjacent sarcomeres are directly in balance, and argue against sarcomere models with springlike elements in parallel with myosin contractile elements. We propose a new model for tension generation in the sarcomere, which provides a mechanistic interpretation for our observations and previous observations of inhomogeneous sarcomere contraction and apparent stress fiber viscoelastic behavior.     

THE RELATIONSHIP BETWEEN MYOFILAMENT PACKING DENSITY AND SARCOMERE LENGTH IN FROG STRIATED MUSCLE

In cross-sections of single fibers from the frog semitendinosus muscle the number of thick myofilaments per unit area (packing density) is a direct function of the sarcomere length. Our data, derived from electron microscopic studies, fit well with other data derived from in vivo, low-angle X-ray diffraction studies of whole semitendinosus muscles. The data are consistent with the assumption that the sarcomere of a fibril maintains a constant volume during changes in sarcomere length. The myofilament lattice, therefore, expands as the sarcomere shortens. Since the distance between adjacent myofilaments is an inverse square root function of sarcomere length, the interaction of the thick and the thin myofilaments during sarcomere shortening may occur over distances which increase 70 A or more. The "expanding-sarcomere, sliding-filament" model of sarcomere shortening is discussed in terms of the current concepts of muscle architecture and contraction.     

Sarcomere Length Nonuniformity and Force Regulation in Myofibrils and Sarcomeres

The smallest contractile unit in striated muscles is the sarcomere. Although some of the classic features of contraction assume a uniform behavior of sarcomeres within myofibrils, the occurrence of sarcomere length nonuniformities has been well recognized for years, but it is yet not well understood. In the past years, there has been a great advance in experiments using isolated myofibrils and sarcomeres that has allowed scientists to directly evaluate sarcomere length nonuniformity. This review will focus on studies conducted with these preparations to develop the hypotheses that 1) force production in myofibrils is largely altered and regulated by intersarcomere dynamics and that 2) the mechanical work of one sarcomere in a myofibril is transmitted to other sarcomeres in series. We evaluated studies looking into myofibril activation, relaxation, and force changes produced during activation. We conclude that force production in myofibrils is largely regulated by intersarcomere dynamics, which arises from the cooperative work of the contractile and elastic elements within a myofibril.