A rheological motor model for vertebrate skeletal muscle in due consideration of non-linearity

A rheological motor model that satisfies the major mechanical properties of the skeletal muscle is proposed. The model consists of two Maxwell elements and a Voigt element connected in parallel with each other and has a force generator in it. The model well explains the mechanical behavior in quick and slow recovery phases in the isometric contraction of the muscle and achieves a sufficient isotonic shortening speed. The energy liberation of the motor in isotonic contraction is calculated and a mechanism of control is proposed, which operates so as to decrease the dissipated energy by altering the weights of the elastic and viscous constants in Maxwell elements. And thereby it becomes possible for the motor to possess non-linearity in energy liberation and load-velocity relation alike in muscle. The model would be a base model to be utilized for analyzing the kinetics of human macrosystems and/or for modeling the human neuromuscular system of motion control.     

The phenomenological model of muscle contraction with a controller to simulate the excitation–contraction (E–C) coupling

We previously proposed a systematic motor model for muscle with two parallel Maxwell elements and a force generator P. The motor model showed the non-linear behavior of a muscle, such as the force–velocity relation and the force depression and enhancement, by using weight functions. Our newly proposed muscle model is based on the molecular mechanism of myosin cross-bridges. We assume that each parallel Maxwell element represents the mechanical properties of weak and strong binding of the myosin head to actin. Furthermore, we introduce a controller to simulate the excitation–contraction coupling of the muscle. The new muscle model satisfies all the properties obtained in our previous model and reduces the wasted energy of the viscous component to less than 5% of the total energy. The controller enables us to simulate contractions of slow and fast twitch muscles, which are driven by an artificial action potential or a processing electromyography signal despite their same mechanical components. The maximum velocities are calculated to be 3.4L₀ m/s for the fast twitch muscle model and 2.5L₀ m/s for the slow twitch muscle model, where L₀ is the initial length of the muscle model.     

The history dependence of force production in mammalian skeletal muscle following stretch-shortening and shortening-stretch cycles

The purpose of this study was to determine the history dependence of force production during and following stretch-shortening and shortening-stretch cycles in mammalian skeletal muscle. Thirty-three different isometric, stretch, shortening, stretch-shortening and shortening-stretch experiments were preformed in cat soleus (n=8) using previously established methods. Stretch-shortening and shortening-stretch cycles are not commutative with respect to the isometric forces following the length changes. Whereas force depression following shortening is virtually unaffected by previous stretching of the muscle, force enhancement following stretch depends in a dose-dependent manner on the amount of muscle shortening preceding the stretch. The history dependence of isometric force following shortening-stretch cycles can conveniently be modelled using an elastic (compressive and tensile) element that engages at the length of muscle activation. Such an "elastic" mechanism has been proposed by Edman and Tsuchiya (1996) (Edman, K.A. P., Tsuchiya, T., 1996. Strain of passive elements during force enhancement by stretch in frog mucle fibres. Journal of Physiology 490. 1, 191-205) based on experimental observations, and has been implemented theoretically in a rheological model of muscle (Forcinito et al., 1997) (Forcinito, M., Epstein, M., Herzog, W., 1997. Theoretical considerations on myofibril stiffness. Biophysics Journal 72, 1278-1286). The history dependence of isometric force following stretch-shortening cycles appears independent of the stretch preceding the shortening, except perhaps, if stretching occurs at very high speeds (i.e. 6-10 times fibre length per second). The results of this study are hard to reconcile with the two major mechanisms associated with history dependence of force production: sarcomere length non-uniformity (Edman et al., 1993) and stress-induced cross-bridge inhibition (Maréchal and Plaghki, 1979) (Maréchal, G., Plaghki, L., 1979. The deficit of the isometric tetanic tension redeveloped after a relase of frog muscle at a constant velocity. Journal of General Physiology 73, 453-467). It appears that studying the history dependence of force production under more functionally relevant conditions than has been done to date may provide new information that contributes to our understanding of possible mechanisms associated with force depression and force enhancement following muscular length changes.     

Mechanical work as predictor of force enhancement and force depression

The steady-state force following active muscle shortening or stretch differs from the maximum isometric force associated with the final length. This phenomenon proves that the isometric force production is not only dependent on current muscle length and length time derivative, but depends on the preceding contraction history. Isolated extensor digitorum longus and soleus muscles from mice (NMRI strain) were used to investigate the force produced by a muscle, and some parameters hypothetically influencing this history-dependent force modification. The muscles were pre-stimulated at a fixed length, then different stretch/shortening episodes were introduced, whereafter changes of the active force were recorded while the muscles were held isometrically to approach a steady-state force before de-stimulation. The mechanical work during active stretch and shortening was evaluated by integrating the product of force and ramp velocity over the length-varying period. The results show a negative linear correlation between the force modification and the mechanical work produced on or by the muscle, continuous between shortening and stretch. A corresponding modification of the passive force component following each stimulation was also observed. The conclusion is that the isometric force attained after stretch or shortening is well described by an asymptotic force which is determined by the mechanical work.     

Can a rheological muscle model predict force depression/enhancement?

A new phenomenological model of activated muscle is presented. The model is based on a combination of a contractile element, an elastic element that engages upon activation, a linear dashpot and a linear spring. Analytical solutions for a few selected experiments are provided. This model is able to reproduce the response of cat soleus muscle to ramp shortening and stretching and, unlike standard Hill-type models, computations are stable on the descending limb of the force–length relation and force enhancement (depression) following stretching (shortening) is predicted correctly. In its linear version, the model is consistent with a linear force–velocity law, which in this model is a consequence rather than a fundamental characteristic of the material. Results show that the mechanical response of activated muscle can be mimicked by a viscoelastic system. Conceptual differences between this model and standard Hill-type models are analyzed and the advantages of the present model are discussed.     

Muscular force production after concentric contraction

The steady-state force following active shortening does not reach the maximum isometric force associated with the final length. Isolated extensor digitorum longus and soleus muscles from mice (NMRI strain) were used to investigate the force produced by a muscle, and some parameters hypothetically influencing this shortening-induced force depression. The muscles were pre-stimulated at fixed length, shortened and then held isometrically to give maximum post-shortening forces, before de-stimulation. The shortening magnitude was 0.18, 0.36 or 0.72mm (about 2-7% of optimal length), time of shortening was chosen as 0.03, 0.06 and 0.12s, and final length as +0.72, 0 and -0.72mm, related to optimal length. The mechanical work during active shortening was evaluated by integrating the product of force and shortening velocity over the shortening period. The results show a positive correlation between the force depression and the mechanical work, whereas the force depression was not correlated to the velocity of shortening. Depression of the passive force component was also observed following all stimulations. Experiments show that the fully stimulated redevelopment of isometric force following concentric contraction follows a time function similar to the creation of force when isometric muscle is initially stimulated. The conclusion is that the isometric force development after active shortening can be well described by an asymptotic force which is decided by the produced work, and the initial isometric time constant.     

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.     

Effects of shortening on stretch-induced force enhancement in single skeletal muscle fibers

The main purpose of this study was to evaluate the effects of shortening on the stretch-induced force enhancement in single muscle fibers, and indirectly test the hypothesis that force enhancement may be associated with the engagement of a passive element upon activation. Fibers were placed on the descending limb of the force-length relationship, and stretch and shortening contractions were performed. Fibers underwent two sets of shortening-stretch cycles. First, fibers were shortened by a fixed amplitude and speed (10% fiber length, and at 40% fiber length/s), and then were stretched (10% fiber length, and at 40% fiber length/s) immediately following shortening, or 500 or 1000 ms following the shortening. Second, fibers were shortened by varying amounts (5%, 10% and 15% fiber length) and at a constant speed (40% fiber length/s) immediately preceding a given fiber stretch (10% fiber length, and at 40% fiber length/s). When stretching was immediately preceded by shortening, force enhancement was decreased proportionally with the shortening magnitude. When intervals were introduced between shortening and stretch, the effects of shortening on the stretch-induced force enhancement became less prominent. We concluded that, in contrast to published suggestions, shortening affects the stretch-induced force enhancement in an amplitude-dependent manner in single fibers, as it does in whole muscles, but this effect is diminished by increasing the time period between the shortening and stretch phases.     

Cross-bridge mechanism of residual force enhancement after stretching in a skeletal muscle

A muscle model that uses a modified Langevin equation with actomyosin potentials was used to describe the residual force enhancement after active stretching. Considering that the new model uses cross-bridge theory to describe the residual force enhancement, it is different from other models that use passive stretching elements. Residual force enhancement was simulated using a half sarcomere comprising 100 myosin molecules. In this paper, impulse is defined as the integral of an excess force from the steady isometric force over the time interval for which a stretch is applied. The impulse was calculated from the force response due to fast and slow muscle stretches to demonstrate the viscoelastic property of the cross-bridges. A cross-bridge mechanism was proposed as a way to describe the residual force enhancement on the basis of the impulse results with reference to the compliance of the actin filament. It was assumed that the period of the actin potential increased by 0.5% and the amplitude of the potential decreased by 0.5% when the half sarcomere was stretched by 10%. The residual force enhancement after 21.0% sarcomere stretching was 6.9% of the maximum isometric force of the muscle; this value was due to the increase in the number of cross-bridges.     

The relationship between force depression following shortening and mechanical work in skeletal muscle

Force depression following muscle shortening was investigated in cat soleus (n=6) at 37 degrees C for a variety of contractile conditions with the aim to test the hypotheses that force depression was independent of the speed of shortening and was directly related to the mechanical work produced by the muscle during shortening. Force depression was similar for tests in which the mechanical work performed by the muscle was similar, independent of the speed of shortening (range of speeds: 4-256mm/s). On the other hand, force depression varied significantly at a given speed of shortening but different amounts of mechanical work, supporting the hypothesis that force depression was not speed - but work dependent. The variations in the mechanical work produced by the muscle during shortening accounted for 87-96% of the variance observed in the force depression following shortening further supporting the idea that the single scalar variable work accounts for most of the observed loss in isometric force after shortening. The results of the present study are also in agreement with the notion that the mechanism underlying force depression might be associated with an inhibition of cross-bridge attachments in the overlap zone formed during the shortening phase, as proposed previously (Herzog and Leonard, 1997. Journal of Bimechanics 30 (9), 865-872; Maréchal and Plaghki, 1979.     

Rapid cytoskeletal response of epithelial cells to force generation by type IV pili

Many bacterial pathogens interfere with cellular functions including phagocytosis and barrier integrity. The human pathogen Neissieria gonorrhoeae generates grappling hooks for adhesion, spreading, and induction of signal cascades that lead to formation cortical plaques containing f-actin and ezrin. It is unclear whether high mechanical forces generated by type IV pili (T4P) are a direct signal that leads to cytoskeletal rearrangements and at which time scale the cytoskeletal response occurs. Here we used laser tweezers to mimic type IV pilus mediated force generation by T4P-coated beads on the order of 100 pN. We found that actin-EGFP and ezrin-EGFP accumulated below pilus-coated beads when force was applied. Within 2 min, accumulation significantly exceeded controls without force or without pili, demonstrating that T4P-generated force rapidly induces accumulation of plaque proteins. This finding adds mechanical force to the many strategies by which bacteria modulate the host cell cytoskeleton.     

Variation of muscle stiffness with force at increasing speeds of shortening

Single frog skeletal muscle fibers were attached to a servo motor and force transducer by knotting the tendons to pieces of wire at the fiber insertions. Small amplitude, high frequency sinusoidal length changes were then applied during tetani while fibers contracted both isometrically and isotonically at various constant velocities. The amplitude of the resulting force oscillation provides a relative measure of muscle stiffness. It is shown from an analysis of the transient force responses observed after sudden changes in muscle length applied both at full and reduced overlap and during the rising phase of short tetani that these responses can be explained on the basis of varying numbers of cross bridges attached at the time of the length step. Therefore, the stiffness measured by the high frequency length oscillation method is taken to be directly proportional to the number of cross bridges attached to thin filament sites. It is found that muscle stiffness measured in this way falls with increasing shortening velocity, but not as rapidly as the force. The results suggest that at the maximum velocity of shortening, when the external force is zero, muscle stiffness is still substantial. The findings are interpreted in terms of a specific model for muscle contraction in which the maximum velocity of shortening under zero external load arises when a force balance is attained between attached cross bridges some of which are aiding and others opposing shortening. Other interpretations of these results are also discussed.     

Residual force enhancement in myofibrils and sarcomeres

Residual force enhancement has been observed following active stretch of skeletal muscles and single fibres. However, there has been intense debate whether force enhancement is a sarcomeric property, or is associated with sarcomere length instability and the associated development of non-uniformities. Here, we studied force enhancement for the first time in isolated myofibrils (n=18) that, owing to the strict in series arrangement, allowed for evaluation of this property in individual sarcomeres (n=79). We found consistent force enhancement following stretch in all myofibrils and each sarcomere, and forces in the enhanced state typically exceeded the isometric forces on the plateau of the force-length relationship. Measurements were made on the plateau and the descending limb of the force-length relationship and revealed gross sarcomere length non-uniformities prior to and following active myofibril stretching, but in contrast to previous accounts, revealed that sarcomere lengths were perfectly stable under these experimental conditions. We conclude that force enhancement is a sarcomeric property that does not depend on sarcomere length instability, that force enhancement varies greatly for different sarcomeres within the same myofibril and that sarcomeres with vastly different amounts of actin-myosin overlap produce the same isometric steady-state forces. This last finding was not explained by differences in the amount of contractile proteins within sarcomeres, vastly different passive properties of individual sarcomeres or (half-) sarcomere length instabilities, suggesting that the basic mechanical properties of muscles, such as force enhancement, force depression and creep, which have traditionally been associated with sarcomere instabilities and the corresponding dynamic redistribution of sarcomere lengths, are not caused by such instabilities, but rather seem to be inherent properties of the mechanisms of contraction.     

Topography and mechanical properties of single molecules of type I collagen using atomic force microscopy

Although the mechanical behavior of tendon and bone has been studied for decades, there is still relatively little understanding of the molecular basis for their specific properties. Thus, despite consisting structurally of the same type I collagen, bones and tendons have evolved to fulfill quite different functions in living organisms. In an attempt to understand the links between the mechanical properties of these collageneous structures at the macro- and nanoscale, we studied trimeric type I tropocollagen molecules by atomic force microscopy, both topologically and by force spectroscopy. High-resolution imaging demonstrated a mean (+/- SD) contour length of (287 +/- 35) nm and height of (0.21 +/- 0.03) nm. Submolecular features, namely the coil-pitch of the molecule, were also observed, appearing as a repeat pattern along the length of the molecule, with a length of approximately 8 nm that is comparable to the theoretical value. Using force spectroscopy, we established the stretching pattern of the molecule, where both the mechanical response of the molecule and pull-off peak are convoluted in a single feature. By interpreting this response with a wormlike chain model, we extracted the value of the effective contour length of the molecule at (202 +/- 5) nm. This value was smaller than that given by direct measurement, suggesting that the entire molecule was not being stretched during the force measurements; this is likely to be related to the absence of covalent binding between probe, sample, and substrate in our experimental procedure.     

Contribution of actin filaments and microtubules to quasi-in situ tensile properties and internal force balance of cultured smooth muscle cells on a substrate

The effects of actin filaments (AFs) and microtubules (MTs) on quasi-in situ tensile properties and intracellular force balance were studied in cultured rat aortic smooth muscle cells (SMCs). A SMC cultured on substrates was held using a pair of micropipettes, gradually detached from the substrate while maintaining in situ cell shape and cytoskeletal integrity, and then stretched up to approximately 15% and unloaded three times at the rate of 1 mum every 5 s. Cell stiffness was approximately 20 nN per percent strain in the untreated case and decreased by approximately 65% and approximately 30% following AF and MT disruption, respectively. MT augmentation did not affect cell stiffness significantly. The roles of AFs and MTs in resisting cell stretching and shortening were assessed using the area retraction of the cell upon noninvasive detachment from thermoresponsive gelatin-coated dishes. The retraction was approximately 40% in untreated cells, while in AF-disrupted cells it was <20%. The retraction increased by approximately 50% and decreased by approximately 30% following MT disruption and augmentation, respectively, suggesting that MTs resist intercellular tension generated by AFs. Three-dimensional measurements of cell morphology using confocal microscopy revealed that the cell volume remained unchanged following drug treatment. A concomitant increase in cell height and decrease in cell area was observed following AF disruption and MT augmentation. In contrast, MT disruption significantly reduced the cell height. These results indicate that both AFs and MTs play crucial roles in maintaining whole cell mechanical properties of SMCs, and that while AFs act as an internal tension generator, MTs act as a tension reducer, and these contribute to intracellular force balance three dimensionally.     

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.     

Shortening-induced force depression is primarily caused by cross-bridges in strongly bound states

The steady-state isometric force following active muscle shortening is smaller than the corresponding force obtained for purely isometric contractions. This so-called residual force depression has been observed consistently for more than half a century, however its mechanism remains a matter of scientific debate. [Maréchal, G., Plaghki, L., 1979. The deficit of the isometric tetanic tension redeveloped after a release of frog muscle at a constant velocity. J. Gen. Physiol. 73, 453–467] suggested that force depression might be caused by alterations in the cross-bridge kinetics following muscle shortening, but there is no research studying force depression systematically for altered cross-bridge kinetic conditions. The purpose of this study was to investigate if force depression affects so-called weakly and strongly bound cross-bridges to the same degree. In order to achieve this aim, we modified the ratio of weakly to strongly bound cross-bridges with 2,3-butanedione monoxime (BDM) in single frog fibers. BDM inhibits the formation of strongly bound cross-bridges in a dose-dependent manner, thus the ratio of weakly to strongly bound cross-bridges could be altered in a systematic way. We found that the absolute amount of force depression was decreased by 50% while the relative amount was decreased by 12% in BDM exposed fibers compared to fibers in normal Ringer's solution. Furthermore, force depression was accompanied by a decrease in stiffness that was much greater in normal compared to BDM exposed fibers, leading to the conclusion that force depression was caused by an inhibition of cross-bridge attachment following fiber shortening and that this inhibition primarily affected cross-bridges in the strongly bound states.     

Computer simulation of movement-generating cross-bridges.

A stochastic computational method was developed to study properties of cross-bridge models for muscle contraction, by following the time history of individual cross-bridge model of Andrew Huxley (1957) and a modified two-state model with more realistic behavior during steady stretching are used as examples. The method can readily compute steady-state force during shortening and stretching and force-transients following rapid changes in length. Computations of velocity with a steady load and of velocity transients are more sensitive to the randomness inherent in the stochastic method.