A framework for structured modeling of skeletal muscle

The aim of this study is to present a detailed continuum mechanics formulation, and the corresponding algorithms, to predict the deformation of skeletal muscle at different structural levels, starting from the muscle fiber level. The model is used to investigate force production and structural changes during isometric and dynamic contractions of the cat medial gastrocnemius. From a comparison with experimental data obtained in our own laboratories, we conclude that the model faithfully predicts all of the observations pertaining to force production, fascicle length and angle of pennation under various test conditions.     

The multiple roles of titin in muscle contraction and force production

Titin is a filamentous protein spanning the half-sarcomere, with spring-like properties in the I-band region. Various structural, signaling, and mechanical functions have been associated with titin, but not all of these are fully elucidated and accepted in the scientific community. Here, I discuss the primary mechanical functions of titin, including its accepted role in passive force production, stabilization of half-sarcomeres and sarcomeres, and its controversial contribution to residual force enhancement, passive force enhancement, energetics, and work production in shortening muscle. Finally, I provide evidence that titin is a molecular spring whose stiffness changes with muscle activation and actin–myosin-based force production, suggesting a novel model of force production that, aside from actin and myosin, includes titin as a “third contractile” filament. Using this three-filament model of sarcomeres, the stability of (half-) sarcomeres, passive force enhancement, residual force enhancement, and the decrease in metabolic energy during and following eccentric contractions can be explained readily.     

Desmin integrates the three-dimensional mechanical properties of muscles

Striated muscle is a linear motor whose properties have been defined in terms of uniaxial structures. The question addressed here is what contribution is made to the properties of this motor by extramyofilament cytoskeletal structures that are not aligned in parallel with the myofilaments. This question arose from observations that transverse loads increase muscle force production in diaphragm but not in the hindlimb muscle, thereby indicating the presence of structures that couple longitudinal and transverse properties of diaphragmatic muscle. Furthermore, we find that the diaphragms of null mutants for the cytoskeletal protein desmin show 1) significant reductions in coupling between the longitudinal and transverse properties, indicating for the first time a role for a specific protein in integrating the three-dimensional mechanical properties of muscle, 2) significant reductions in the stiffness and viscoelasticity of muscle, and 3) significant increases in tetanic force production. Thus desmin serves a complex mechanical function in diaphragm muscle by contributing both to passive stiffness and viscoelasticity and to modulation of active force production in a three-dimensional structural network. Our finding changes the paradigm of force transmission among cells by placing our understanding of the function of the cytoskeleton in the context of the structural and mechanical complexity of muscles.     

The structure of myosin and its role in energy transduction in muscle

The present understanding of the relationship between the structure of the myosin ATPase and its role in force production for muscle contraction is reviewed. Emphasis is placed on structural transitions in myosin that occur during ATP hydrolysis which may be correlated with force production. Although detailed structural information is presently lacking, numerous spectroscopic and kinetic experiments have indicated that myosin exists in two structural states for each chemical intermediate in the hydrolysis of ATP. Models are discussed which view a transition between these two states as the energy transduction "event" (i.e., force production).     

Entropic elasticity in the generation of muscle force--a theoretical model

A novel simplified structural model of sarcomeric force production in striate muscle is presented. Using some simple assumptions regarding the distribution of myosin spring lengths at different sliding velocities it is possible to derive a very simple expression showing the main components of the experimentally observed force-velocity relationship of muscle: nonlinearity during contraction (Hill, 1938), maximal force production during stretching equal to two times the isometric force (Katz, 1939), yielding at high stretching velocity, slightly concave force-extension relationship during sudden length changes (Ford et al., 1977; Lombardi & Piazzesi, 1990), accurate reproduction of the rate of ATP consumption (Shirakawa et al., 2000; He et al., 2000) and of the extra energy liberation rate (Hill, 1964a). Different assumptions regarding the force-length relationship of individual cross-bridges are explored [linear, power function and worm-like chain (WLC) model based], and it is shown that the best results are obtained if the individual myosin-spring forces are modelled using a WLC model, thus hinting that entropic elasticity could be the main source of force in myosin undergoing the conformational changes associated with the power stroke.     

A study of the muscle force waveform using a population stochastic model of skeletal muscle

A population stochastic model based on the differing properties and the independent activation of motor units is used to describe the production of force in the contracting skeletal muscle. Detailed force predictions of the model concerning a hand muscle are obtained by computer simulation. General features of the force signal are established analyticaly on the basis of the general properties of the neuromuscular system which the population model takes into account. The results show that the asynchronous activity of motor units and the distribution of their filtering and firing properties at various levels of muscle contraction are esponsible, at least partially, for the main features of the muscle force waveform, including tremor.     

Modelling concentric contraction of muscle using an improved cross-bridge model.

Despite its overwhelming acceptance in muscle research, the cross-bridge theory does not account for all phenomena observed during muscular contractions. A phenomenon which has received much attention in the biomechanics literature, but has evaded convincing explanation and is not accounted for in the formulation of the classic cross-bridge theory, is the persistent aftereffects of muscular length changes on force production. For example, following muscle shortening, the isometric force of a muscle is depressed for a long time period ( > 5 s) compared to the corresponding isometric force following no length change. In the present study, the classic cross-bridge model was modified in two ways in an attempt to account for the force depressions following muscle shortening. First, the steady-state force depressions following shortening were described by a single scalar variable: the work performed by the muscle during shortening; and second, the dynamic, history-dependent cross-bridge properties were described using a fading memory function. The proposed model was developed and tested for shortening of the cat soleus at constant speeds ranging from 4 to 32 mm/s, for shortening at changing speeds, and for shortening of different magnitudes ranging from 2 to 10 mm. The history-dependent forces during shortening and the steady-state force depressions following shortening were well captured with the modified cross-bridge model. The present model contains two mathematically simple adaptations to the classic cross-bridge model, and is the first such model to account for the long-lasting force depressions following muscle shortening using a single scalar variable.     

EMG-force modeling using parallel cascade identification

Measuring force production in muscles is important for many applications such as gait analysis, medical rehabilitation, and human-machine interaction. Substantial research has focused on finding signal processing and modeling techniques which give accurate estimates of muscle force from the surface-recorded electromyogram (EMG). The proposed methods often do not capture both the nonlinearities and dynamic components of the EMG-force relation. In this study, parallel cascade identification (PCI) is used as a dynamic estimation tool to map surface EMG recordings from upper-arm muscles to the induced force at the wrist. PCI mapping involves generating a parallel connection of a series of linear dynamic and nonlinear static blocks. The PCI model parameters were initialized to obtain the best force prediction. A comparison between PCI and a previously published Hill-based orthogonalization scheme, that captures physiological behaviour of the muscles, has shown 44% improvement in force prediction by PCI (averaged over all subjects in relative-mean-square sense). The improved performance is attributed to the structural capability of PCI to capture nonlinear dynamic effects in the generated force.     

Activation of smooth muscle in the airway wall, force production, and airway narrowing.

Airway narrowing depends on smooth muscle force production and muscle shortening, but the structural and geometric properties exhibited by individual generations of the bronchial tree largely determine the extent and characteristics of airway narrowing. Properties of major importance include the nature and integrity of the epithelium, the structural and mechanical properties of the airway wall, as well as airway diameter. The influence of these properties on airway narrowing measured as flow or flow resistance in large and small diameter segments of airways from pig lung is described using a novel preparation, the perfused bronchial segment.     

The kinetics relating calcium and force in skeletal muscle.

The kinetics relating Ca2+ transients and muscle force were examined using data obtained with the photoprotein aequorin in skeletal muscles of the rat, barnacle, and frog. These data were fitted by various models using nonlinear methods for minimizing the least mean square errors. Models in which Ca2+ binding to troponin was rate limiting for force production did not produce good agreement with the observed data, except for a small twitch of the barnacle muscle. Models in which cross-bridge kinetics were rate limiting also did not produce good agreement with the observed data, unless the detachment rate constant was allowed to increase sharply on the falling phase of tension production. Increasing the number of cross-bridge states did not dramatically improve the agreement between predicted and observed force. We conclude that the dynamic relationship between Ca2+ transients and force production in intact muscle fibers under physiological conditions can be approximated by a model in which (a) two Ca2+ ions bind rapidly to each troponin molecule, (b) force production is limited by the rate of formation of tightly bound cross-bridges, and (c) the rate of cross-bridge detachment increases rapidly once tension begins to decline and free Ca2+ levels have fallen to low values after the last stimulus. Such a model can account not only for the pattern of force production during a twitch and tetanus, but also the complex, nonlinear pattern of summation which is observed during an unfused tetanus at intermediate rates of stimulation.     

The Three Filament Model of Skeletal Muscle Stability and Force Production

Ever since the 1950s, muscle force regulation has been associated with the cross-bridge interactions between the two contractile filaments, actin and myosin. This gave rise to what is referred to as the "two-filament sarcomere model". This model does not predict eccentric muscle contractions well, produces instability of myosin alignment and force production on the descending limb of the force-length relationship, and cannot account for the vastly decreased ATP requirements of actively stretched muscles. Over the past decade, we and others, identified that a third myofilament, titin, plays an important role in stabilizing the sarcomere and the myosin filament. Here, we demonstrate additionally how titin is an active participant in muscle force regulation by changing its stiffness in an activation/force dependent manner and by binding to actin, thereby adjusting its free spring length. Therefore, we propose that skeletal muscle force regulation is based on a three filament model that includes titin, rather than a two filament model consisting only of actin and myosin filaments.     

Increased Residual Force Enhancement in Older Adults Is Associated with a Maintenance of Eccentric Strength

Despite an age-related loss of voluntary isometric and concentric strength, muscle strength is well maintained during lengthening muscle actions (i.e., eccentric strength) in old age. Additionally, in younger adults during lengthening of an activated skeletal muscle, the force level observed following the stretch is greater than the isometric force at the same muscle length. This feature is termed residual force enhancement (RFE) and is believed to be a combination of active and passive components of the contractile apparatus. The purpose of this study was to provide an initial assessment of RFE in older adults and utilize aging as a muscle model to explore RFE in a system in which isometric force production is compromised, but structural mechanisms of eccentric strength are well-maintained. Therefore, we hypothesised that older adults will experience greater RFE compared with young adults. Following a reference maximal voluntary isometric contraction (MVC) of the dorsiflexors in 10 young (26.1±2.7y) and 10 old (76.0±6.5y) men, an active stretch was performed at 15°/s over a 30° ankle joint excursion ending at the same muscle length as the reference MVCs (40° of plantar flexion). Any additional torque compared with the reference MVC therefore represented RFE. In older men RFE was ∼2.5 times greater compared to young. The passive component of force enhancement contributed ∼37% and ∼20% to total force enhancement, in old and young respectively. The positive association (R ² = 0.57) between maintained eccentric strength in old age and RFE indicates age-related mechanisms responsible for the maintenance of eccentric strength likely contributed to the observed elevated RFE. Additionally, as indicated by the greater passive force enhancement, these mechanisms may be related to increased muscle series elastic stiffness in old age.     

Mysteries of muscle contraction

According to the cross-bridge theory, the steady-state isometric force of a muscle is given by the amount of actin-myosin filament overlap. However, it has been known for more than half a century that steady-state forces depend crucially on contractile history. Here, we examine history-dependent steady-state force production in view of the cross-bridge theory, available experimental evidence, and existing explanations for this phenomenon. This is done on various structural levels, ranging from the intact muscle to the myofibrillar and isolated contractile protein level, so that advantages and limitations of the various preparations can be fully exploited and overcome. Based on experimental evidence, we conclude that steady-state force following active muscle stretching is enhanced, and this enhancement has a passive and an active component. The active component is associated with the cross-bridge kinetics, and the passive component is associated with a calcium-dependent increase in titin stiffness.     

Biomechanics of a muscular hydrostat: a model of lapping by a reptilian tongue

We have developed a quantitative model of an example of a muscular hydrostat, a reptilian tongue, and have used this model to study a functional movement, protrusion and retrusion, a form of lapping. The model tongue consists of a longitudinal muscle that shortens the tongue when it contracts, and a circumferential muscle wrapped around the longitudinal muscle that lengthens the tongue when it contracts. The anatomy of the model tongue and the pattern of activation of its muscles are based on studies of the tongue of the lizard Tupinambis nigropunctatus (Smith 1984). The mechanics of pressure vessels were used to derive a relationship between the forces in the two muscles. Muscle force production was modelled as the product of length/tension properties, force/velocity properties, and activation due to neural inputs (incorporating both recruitment and firing period). Passive forces were modeled as a force in parallel with the longitudinal muscle. Muscle activation dynamics were modeled as a first order low pass filter. When the model tongue is short, the two muscles can lengthen or shorten it with comparable forces, but as it lengthens, the force that the circumferential muscle can exert drops precipitously. When the tongue is long, it can neither be very stiff, nor can it generate much force. The model also reproduces the kinematics of lapping movements actually observed in Tupinambis.     

Elastic Energy Storage and Radial Forces in the Myofilament Lattice Depend on Sarcomere Length

We most often consider muscle as a motor generating force in the direction of shortening, but less often consider its roles as a spring or a brake. Here we develop a fully three-dimensional spatially explicit model of muscle to isolate the locations of forces and energies that are difficult to separate experimentally. We show the strain energy in the thick and thin filaments is less than one third the strain energy in attached cross-bridges. This result suggests the cross-bridges act as springs, storing energy within muscle in addition to generating the force which powers muscle. Comparing model estimates of energy consumed to elastic energy stored, we show that the ratio of these two properties changes with sarcomere length. The model predicts storage of a greater fraction of energy at short sarcomere lengths, suggesting a mechanism by which muscle function shifts as force production declines, from motor to spring. Additionally, we investigate the force that muscle produces in the radial or transverse direction, orthogonal to the direction of shortening. We confirm prior experimental estimates that place radial forces on the same order of magnitude as axial forces, although we find that radial forces and axial forces vary differently with changes in sarcomere length.     

A coupled framework of in situ and in silico analysis reveals the role of lateral force transmission in force production in volumetric muscle loss injuries

Volumetric muscle loss injuries (VML) are challenging to treat because of the variability in wound location. Regenerative medicine offers promising alternative treatments, but there is little understanding of the correlation between magnitude of VML injuries and corresponding functional deficits that must be addressed. There is a need for a tool that can elucidate the relationship between VML injury and force loss, as well as the impact on specific mechanisms responsible for force production. The purpose of this study was to develop a novel coupled framework of in situ and in silico methods to more precisely understand the relationship between injury location and force production deficits. We created a three-dimensional finite-element model of the pennate latissimus dorsi (LD) muscle in the rat and validated the model experimentally. We found that the model’s prediction (2.6 N/g Model I, 2.1 N/g Model V) compared favorably to in situ testing of isometric force generation of the injured rat LD muscle (2.8 ± 0.3 N/g Experimental I, 2.0 ± 0.2 N/g Experimental V). Further model analysis revealed that the contribution from lateral and longitudinal force transmission to the total force varied with injury location and led to a greater understanding of the mechanisms responsible for VML-related force deficits. In the future, the coupled computational and experimental framework can be used to inform development of preclinical VML injury models that better recapitulate the spectrum of VML injuries observed in affected patients, and the mechanistic insight can accelerate the creation of improved regenerative therapeutics for VML injuries.     

Considerations on the history dependence of muscle contraction.

When a skeletal muscle that is actively producing force is shortened or stretched, the resulting steady-state isometric force after the dynamic phase is smaller or greater, respectively, than the purely isometric force obtained at the corresponding final length. The cross-bridge model of muscle contraction does not readily explain this history dependence of force production. The most accepted proposal to explain both, force depression after shortening and force enhancement after stretch, is a nonuniform behavior of sarcomeres that develops during and after length changes. This hypothesis is based on the idea of instability of sarcomere lengths on the descending limb of the force-length relationship. However, recent evidence suggests that skeletal muscles may be stable over the entire range of active force production, including the descending limb of the force-length relationship. The purpose of this review was to critically evaluate hypotheses aimed at explaining the history dependence of force production and to provide some novel insight into the possible mechanisms underlying these phenomena. It is concluded that the sarcomere nonuniformity hypothesis cannot always explain the total force enhancement observed after stretch and likely does not cause all of the force depression after shortening. There is evidence that force depression after shortening is associated with a reduction in the proportion of attached cross bridges, which, in turn, might be related to a stress-induced inhibition of cross-bridge attachment in the myofilament overlap zone. Furthermore, we suggest that force enhancement is not associated with instability of sarcomeres on the descending limb of the force-length relationship and that force enhancement has an active and a passive component. Force depression after shortening and force enhancement after stretch are likely to have different origins.     

A dynamical model of muscle activation, fatigue, and recovery

A dynamical model is presented as a framework for muscle activation, fatigue, and recovery. By describing the effects of muscle fatigue and recovery in terms of two phenomenological parameters (F, R), we develop a set of dynamical equations to describe the behavior of muscles as a group of motor units activated by voluntary effort. This model provides a macroscopic view for understanding biophysical mechanisms of voluntary drive, fatigue effect, and recovery in stimulating, limiting, and modulating the force output from muscles. The model is investigated under the condition in which brain effort is assumed to be constant. Experimental validation of the model is performed by fitting force data measured from healthy human subjects during a 3-min sustained maximal voluntary handgrip contraction. The experimental results confirm a theoretical inference from the model regarding the possibility of maximal muscle force production, and suggest that only 97% of the true maximal force can be reached under maximal voluntary effort, assuming that all motor units can be recruited voluntarily. The effects of different motor unit types, time-dependent brain effort, sources of artifacts, and other factors that could affect the model are discussed. The applications of the model are also discussed.     

New insights into the passive force enhancement in skeletal muscles

The steady-state isometric force following active stretching of a muscle is always greater than the steady-state isometric force obtained in a purely isometric contraction at the same length. This phenomenon has been termed "residual force enhancement" and it is associated with an active and a passive component. The origin of these components remains a matter of scientific debate. The purpose of this work was to test the hypothesis that the passive component of the residual force enhancement is caused by a passive structural element. In order to achieve this purpose, single fibers (n=6) from the lumbrical muscles of frog (Rana pipiens) were isolated and attached to a force transducer and a motor that could produce computer-controlled length changes. The passive force enhancement was assessed for three experimental conditions: in a normal Ringer's solution, and after the addition of 5 and 15mM 2,3-butanedione monoxime (BDM) which inhibits force production in a dose-dependent manner. If our hypothesis was correct, one would expect the passive force enhancement to be unaffected following BDM application. However, we found that increasing concentrations of BDM decreased the isometric forces, increased the normalized residual force enhancement, and most importantly for this study, increased the passive force enhancement. Furthermore, BDM decreased the rate of force relaxation after deactivation following active stretching of fibers, passive stretching in the Ringer's and BDM conditions produced the same passive force-sarcomere length relationship, and passive force enhancement required activation and force production. These results led to the conclusion that the passive force enhancement cannot be caused by a structural component exclusively as had been assumed up to date, but must be associated, directly or indirectly, with cross-bridge attachments upon activation and the associated active force.