Can Quick Release Experiments Reveal the Muscle Structure? A Bionic Approach

The goal of this study was to understand the macroscopic mechanical structure and function of biological muscle with respect to its dynamic role in the contraction.A recently published muscle model,deriving the hyperbolic force-velocity relation from first-order mechanical principles,predicts different force-velocity operating points for different load situations.With anew approach,this model could be simplified and thus,transferred into a numerical simulation and a hardware experiment.Two types of quick release experiments were performed in simulation and with the hardware setup,which represent two extreme cases of the contraction dynamics:against a constant force (isotonic) and against an inertial mass.Both experiments revealed hyperbolic or hyperbolic-like force-velocity relations.Interestingly,the analytical model not only predicts these extreme cases,but also additionally all contraction states in between.It was possible to validate these predictions with the numerical model and the hardware experiment.These results prove that the origin of the hyperbolic force-velocity relation can be mechanically explained on a macroscopic level by the dynamical interaction of three mechanical elements.The implications for the interpretation of biological muscle experiments and the realization of muscle-like bionic actuators are discussed.     

Limitations to maximum sprinting speed imposed by muscle mechanical properties

It has been suggested that the force-velocity relationship of skeletal muscle plays a critical limiting role in the maximum speed at which humans can sprint. However, this theory has not been tested directly, and it is possible that other muscle mechanical properties play limiting roles as well. In this study, forward dynamics simulations of human sprinting were generated using a 2D musculoskeletal model actuated by Hill muscle models. The initial simulation results compared favorably to kinetic, kinematic, and electromyographic data recorded from sprinting humans. Muscle mechanical properties were then removed in isolation to quantify their effect on maximum sprinting speed. Removal of the force-velocity, excitation-activation, and force-length relationships increased the maximum speed by 15, 8, and 4%, respectively. Removal of the series elastic force-extension relationship decreased the maximum speed by 26%. Each relationship affected both stride length and stride frequency except for the force-length relationship, which mainly affected stride length. Removal of all muscular properties entirely (optimized joint torques) increased speed (+22%) to a greater extent than the removal of any single contractile property. The results indicate that the force-velocity relationship is indeed the most important contractile property of muscle regarding limits to maximum sprinting speed, but that other muscular properties also play important roles. Interactions between the various muscular properties should be considered when explaining limits to maximal human performance.     

Skeletal muscle performance determined by modulation of number of myosin motors rather than motor force or stroke size

Skeletal muscle can bear a high load at constant length, or shorten rapidly when the load is low. This force-velocity relationship is the primary determinant of muscle performance in vivo. Here we exploited the quasi-crystalline order of myosin II motors in muscle filaments to determine the molecular basis of this relationship by X-ray interference and mechanical measurements on intact single cells. We found that, during muscle shortening at a wide range of velocities, individual myosin motors maintain a force of about 6 pN while pulling an actin filament through a 6 nm stroke, then quickly detach when the motor reaches a critical conformation. Thus we show that the force-velocity relationship is primarily a result of a reduction in the number of motors attached to actin in each filament in proportion to the filament load. These results explain muscle performance and efficiency in terms of the molecular mechanism of the myosin motor.     

Force-velocity test: effect of a heavy-load start on maximal anaerobic power and blood lactate levels

The purpose of this study was to determine the effect of starting the force-velocity test with a heavy load on both maximal anaerobic power and blood lactate concentration. Nine male subjects aged 23.4 +/- 1.3 yr (mean +/- sem) participated in a first force-velocity test (FV1) which had an initial load of 1 kg (classical protocol). Then a week later in a second force-velocity test (FV2) which had an initial load corresponding to maximal power developed during FV1 (W1). The increase in load was of 1 kg for FV1 and FV2. Our results show that during FV2, compared to FV1: 1) maximal anaerobic power developed (W2) is superior to W1 (W1 = 1,165.2 +/- 70.4 W; W2 = 1,278.6 +/- 92.3 W; p less than 0.02); 2) blood lactate concentration after the first load is inferior (p less than 0.001); 3) blood lactate concentration is not significantly different at the peak of power. Thus, starting the force-velocity test with a heavy load allows an increase of maximal anaerobic power until a blood lactate concentration which may be compared to the one obtained during the classic force-velocity test. In conclusion, maximal anaerobic power measured during the force-velocity test seems to depend on protocol used.     

Autonomic energy conversion. I. The input relation: phenomenological and mechanistic considerations

The differences between completely and incompletely coupled linear energy converters are discussed using suitable electrochemical cells as examples. The output relation for the canonically simplest class of self-regulated incompletely coupled linear energy converters has been shown to be identical to the Hill force-velocity characteristic for muscle. The corresponding input relation (the “inverse” Hill equation) is now derived by two independent methods. The first method is a direct transformation of the output relation through the phenomenological equations of the converter; Onsager symmetry has no influence on the result. The second method makes use of a model system, a hydroelectric device with a regulator mechanism which depends only on the operational limits of the converter (an electro-osmosis cell operated in reverse) and on the load. The inverse Hill equation is shown to be the simplest solution of the regulator equation. An interesting and testable series of relations between input and output parameters arises from the two forms of the Hill equation. For optimal regulation the input should not be greatly different in the two limiting stationary states (level flow and static head). The output power will then be nearly maximal over a considerable range of load resistance, peak output being obtained at close to peak efficiency.     

Relative shortening velocity in locomotor muscles: turkey ankle extensors operate at low V/V(max)

The force-velocity properties of skeletal muscle have an important influence on locomotor performance. All skeletal muscles produce less force the faster they shorten and typically develop maximal power at velocities of approximately 30% of maximum shortening velocity (V(max)). We used direct measurements of muscle mechanical function in two ankle extensor muscles of wild turkeys to test the hypothesis that during level running muscles operate at velocities that favor force rather than power. Sonomicrometer measurements of muscle length, tendon strain-gauge measurements of muscle force, and bipolar electromyographs were taken as animals ran over a range of speeds and inclines. These measurements were integrated with previously measured values of muscle V(max) for these muscles to calculate relative shortening velocity (V/V(max)). At all speeds for level running the V/V(max) values of the lateral gastrocnemius and the peroneus longus were low (<0.05), corresponding to the region of the force-velocity relationship where the muscles were capable of producing 90% of peak isometric force but only 35% of peak isotonic power. V/V(max) increased in response to the demand for mechanical power with increases in running incline and decreased to negative values to absorb energy during downhill running. Measurements of integrated electromyograph activity indicated that the volume of muscle required to produce a given force increased from level to uphill running. This observation is consistent with the idea that V/V(max) is an important determinant of locomotor cost because it affects the volume of muscle that must be recruited to support body weight.     

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.     

Crank inertial load has little effect on steady-state pedaling coordination

Inertial load can affect the control of a dynamic system whenever parts of the system are accelerated ordeclerated. During steady-state pedating, because within-cycle variations in crank angular acceleration still exist, the amount of crank inertia present (which varies widely with road-riding gear ratio) may affect the within-cycle coordination of muscles. However, the effect of inertial load on steady-state pedaling coordination is almos always assumed to be negligible, since the net mechanical energy per cycle developed by muscles only depends on the constant cadence and workload. This study tests the hypothesis that under steady-state conditions, the net joint torques produced by muscles at the hip, knee, and ankle are unaffected by crank inertial load. To perform the investigation, we constructed a pedaling apparatus which could emulate the low inertial load of a standard ergometer or the high inertial load of a road bicycle in high gear. Crank angle and bilateral pedal force and angle data were collected from ten subjects instructed to pedal steadily (i.e. constant speed across cycles) and smoothly (i.e. constant speed within a cycle) against both inertias at a constant workload. Virtually no statistically significant changes were found in the net hip and knee muscle joint torques calculated from an inverse dynamics analysis. Though the net ankle muscle joint torque, as well as the one- and two-legged crank torque, showed statistically significant increases at the higher inertia, the changes were small. In contrast, large statistically significant reductions were found in crank kinematic variability both within a cycle and between cycles (i.e. cadence), primarily because a larger inertial load means a slower crank dynamic response. Nonetheless, the reduction in cadence variability was somewhat attenuated by a large statistically significant increase in one-legged crank torque variability. We suggest, therefore, that muscle coordination during steady-state pedaling is largely unaffected, though less well regulated, when crank inertial load is increased.     

Predictions and ecperimental tests of a visco-elastic muscle model using elastic and inertial loads

A simple, linear visco-elastic model of muscle is described which contains five parameters: a series and a parallel elasticity, a viscosity, and a magnitude and rate constant for the decay of the active state. The effects of adding springs in series with a muscle are predicted. The responses to random stimulus trains can be used to evaluate the parameters of the model. The effects of applying inertial loads to the muscle can also be predicted. These predictions are in good agreement with experimental observations on plantaris muscle of the cat. For example, damped oscillations of the predicted frequencies can be observed for various inertial loads. The gain of the frequency response falls off sharply (as the fourth power of frequency) at higher frequencies. However, responses to lower frequency signals, including most of the frequencies important for cyclic movements, are only slightly affected by a wide variation in inertial load.Graduate student of the Medical Research Council of Canada.Formerly a Post-doctoral Fellow of the Muscular Dystrophy Association of Canada.     

Stiffness-distortion sarcomere model for muscle simulation.

A relatively simple method is presented for incorporating cross-bridge mechanisms into a muscle model. The method is based on representing force in a half sarcomere as the product of the stiffness of all parallel cross bridges and their average distortion. Differential equations for sarcomeric stiffness are derived from a three-state kinetic scheme for the cross-bridge cycle. Differential equations for average distortion are derived from a distortional balance that accounts for distortion entering and leaving due to cross-bridge cycling and for distortion imposed by shearing motion between thick and thin filaments. The distortion equations are unique and enable sarcomere mechanodynamics to be described by only a few ordinary differential equations. Model predictions of small-amplitude step and sinusoidal responses agreed well with previously described experimental results and allowed unique interpretations to be made of various response components. Similarly good results were obtained for model reproductions of force-velocity and large-amplitude step and ramp responses. The model allowed reasonable predictions of contractile behavior by taking into account what is understood to be basic muscle contractile mechanisms.     

A comparison of the mechanical behavior of the cat soleus muscle with a distribution-moment model

A state-variable model for skeletal muscle, termed the "Distribution-Moment Model," is derived from A. F. Huxley's 1957 model of molecular contraction dynamics. The state variables are the muscle stretch and the three lowest-order moments of the bond-distribution function (which represent, respectively, the contractile tissue stiffness, the muscle force, and the elastic energy stored in the contractile tissue). The rate equations of the model are solved under various conditions, and compared to experimental results for the cat soleus muscle subjected to constant stimulation. The model predicts several observed effects, including yielding of the muscle force in constant velocity stretches, different "force-velocity relations" in isotonic and isovelocity experiments, and a decrease of peak force below the isometric level in small-amplitude sinusoidal stretches. Chemical energy and heat rates predicted by the model are also presented.     

Effects of aging on force, velocity, and power in the elbow flexors of males

The effect of aging on muscular power development was investigated by determining the force-velocity relationship. The muscle cross-sectional area (CSA) was estimated by the thickness of the elbow flexors. The subjects were 19 elderly males aged 69.1+/-3.7 years old (G-70 group), 15 middle-aged males aged 50.9+/-3.5 years old (G-50), and 19 young males aged 21.2+/-1.3 years old (G-20). The G-70 group had the slowest shortening velocities under various load conditions, resulting in the lowest force-velocity relationship. The maximum values for force (Fmax), velocity (Vmax), power (Pmax), dynamic constants (a, b), and the a/Fmax ratio were determined using Hill's equation. The a/Fmax ratio determines the degree of concavity in the force-velocity curve. The a/Fmax ratio was greatest in G-70, followed by those in G-50 and G-20, while the maximum values for force (Fmax), velocity (Vmax), and power (Pmax) were significantly lower in G-70 than in the other groups. Fmax and Pmax per CSA were lowest in G-70, and Vmax per unit muscle length was also lowest in G-70 as compared to the other age groups. The ratio of G-70/G-20 was greatest in Pmax (69.6%), followed by Fmax (75.3%) and Vmax (83.4%). However, there were no significant differences in CSA among the 3 age groups. Our findings suggest that muscle force and shortening velocity may decline gradually in the process of aging attributed to declining muscle function rather than CSA.     

The integrated function of muscles and tendons during locomotion

The mechanical roles of tendon and muscle contractile elements during locomotion are often considered independently, but functionally they are tightly integrated. Tendons can enhance muscle performance for a wide range of locomotor activities because muscle-tendon units shorten and lengthen at velocities that would be mechanically unfavorable for muscle fibers functioning alone. During activities that require little net mechanical power output, such as steady-speed running, tendons reduce muscular work by storing and recovering cyclic changes in the mechanical energy of the body. Tendon stretch and recoil not only reduces muscular work, but also allows muscle fibers to operate nearly isometrically, where, due to the force-velocity relation, skeletal muscle fibers develop high forces. Elastic energy storage and recovery in tendons may also provide a key mechanism to enable individual muscles to alter their mechanical function, from isometric force-producers during steady speed running to actively shortening power-producers during high-power activities like acceleration or uphill running. Evidence from studies of muscle contraction and limb dynamics in turkeys suggests that during running accelerations work is transferred directly from muscle to tendon as tendon stretch early in the step is powered by muscle shortening. The energy stored in the tendon is later released to help power the increase in energy of the body. These tendon length changes redistribute muscle power, enabling contractile elements to shorten at relatively constant velocities and power outputs, independent of the pattern of flexion/extension at a joint. Tendon elastic energy storage and recovery extends the functional range of muscles by uncoupling the pattern of muscle fiber shortening from the pattern of movement of the body.     

Limit to steady-state aerobic power of skeletal muscles

Like any other kind of cell, muscle cells produce energy by oxidizing the fuel substrate that they absorb together with the needed oxygen from the surroundings. Oxidation occurs entirely within the cell. It means that the reactants and products of reaction must at some time be dissolved in the cell’s cytosol. If a cell operates at steady state, its cytosol composition remains constant. Therefore, the cytosol in a muscle that produces work at steady state must contain a constant amount of fuel, oxygen, and product of reaction dissolved in it. The greater the power produced, the higher the concentration of these solutes. There is a limit, however, to the maximum amount of solutes that the cytosol can contain without damaging the cell. General thermodynamic arguments, which are reviewed in this paper, help relate this limit to the dehydration and overhydration limits of the cell. The present analysis shows that the same limits entail a limit to the maximum power that a muscle can produce at steady state. This limit depends on the composition of the fuel mixture used by the muscle. The analysis also determines the number of fuel carbon atoms that must be oxidized in parallel within a cell to produce a given power. It may well happen that a muscle cannot reach the maximum attainable power because it cannot activate all the parallel oxidation paths that are needed to produce it. This may be due to a series of reasons ranging from health issues to a lack of training. The paper shows how the methods of indirect calorimetry can provide all the experimental data needed to determine the actual number of parallel oxidation paths that at steady state must be active in a muscle in a given exercise. A diagram relating muscle power to the number of parallel oxidation paths and fuel composition is finally presented. It provides a means to assess the power capacity of animal muscles and can be applied to evaluate their fitness, stamina, margins for improvement, and athletic potential.     

Heavily loaded flight and limits to the maximum size of dragonflies (Anisoptera) and griffenflies (Meganisoptera)

An original hypothesis is presented that the maximum mass and size of living anisopteran dragonflies are constrained by a physiological performance limit: the wing muscle power required to permit reproductively successful males to carry heavier females in the so‐called ‘wheel position’ in flight. It is proposed that the same limit cannot have applied to all fossil Odonatoptera. As the physiology of the giant Carboniferous griffenfly Namurotypus sippeli precludes flight in the wheel position, it did not need to carry any substantial load aside from exogenous aerial prey. Based on its thorax dimensions, it is argued that Namurotypus flew with a relatively low maximum specific muscle power output in comparison with living Anisoptera. The extinction of some families of large Mesozoic Odonatoptera may have been exacerbated by competition with smaller (stem‐) Anisoptera that evolved higher specific power outputs and superior flight performance similar to living Anisoptera. To investigate the credibility of this flight‐performance size‐limit hypothesis and its consequences, an analysis of the scaling of the required flight power and available muscle power is presented using allometric relations. It is found that for living Anisoptera and fossil Odonatoptera, there are different limiting sizes, above which the required specific flight power would exceed the available muscle specific power. These limits are directly related to maximum load‐carrying capacity and the atmospheric air density at the habitual altitude. It is suggested that the largest living species of Petaluridae, Petalura ingentissima, is close to the proposed Anisoptera size limit at current near‐sea‐level air density conditions.     

Analytical solution to isometric mechanogram of hill’s model of striated muscle

The two-element muscle model considered consists of a contractile element defined by a hyperbolic force-velocity relation connected in series with an “exponential spring”. Differential equations for the isometrically developed force during a tetanic contraction and the corresponding contractile element shortening velocity are derived and their stability is investigated. Analytical solutions to both equations are obtained. Two numerical examples are given, the second chosen to illustrate pressure-induced hypertrophy of a cardiac muscle.     

Active contractions of ureteral segments

The first step in the analysis of the biomechanics of any organ is to obtain its constitutive equation. In pursuit of a constitutive equation describing the peristalsis of the ureter, we measured the relationship between the length of the muscle, the velocity of contraction, and the active tension development of isolated ureter segments. The results of length-tension measurements (giving the maximum tension developed in isometric contraction of a ureter segment of specific length) were similar to those obtained by previous investigators and reflected the behavior of length-tension relationship for other smooth muscles. Two aspects of the force-velocity relationship of the ureter were examined: the effect of releasing the ureter at different times after stimulation, and that at different levels of afterload. Measurements were analyzed using the hyperbolic Hill's equation in the form T/T0 = (1-v/v0) (l + cv/v0)-1 where v is the velocity of contraction, v0 is the velocity of contraction when T = 0, T is the tension in the muscle after release, T0 is the tension in the muscle immediately prior to release, and c is the dimensionless constant. The results of force-velocity measurements showed that the so-called "maximum" velocity v0, is the largest if the tension is released at a time of contraction, early in the rise portion of the contraction cycle. Further, if tension is released from an isometric contraction at a fixed time in the rise portion of the contraction cycle, the largest value of v0 is obtained when the muscle length is in the range of 0.85-0.90 Lmax. Interestingly, the in vivo length of the ureter lies also in this range, 0.85-0.90 Lmax.     

The mechanics of mouse skeletal muscle when shortening during relaxation

The dynamic properties of relaxing skeletal muscle have not been well characterised but are important for understanding muscle function during terrestrial locomotion, during which a considerable fraction of muscle work output can be produced during relaxation. The purpose of this study was to characterise the force-velocity properties of mouse skeletal muscle during relaxation. Experiments were performed in vitro (21 degrees C) using bundles of fibres from mouse soleus and EDL muscles. Isovelocity shortening was applied to muscles during relaxation following short tetanic contractions. Using data from different contractions with different shortening velocities, curves relating force output to shortening velocity were constructed at intervals during relaxation. The velocity component included contributions from shortening of both series elastic component (SEC) and contractile component (CC) because force output was not constant. Early in relaxation force-velocity relationships were linear but became progressively more curved as relaxation progressed. Force-velocity curves late in relaxation had the same curvature as those for the CC in fully activated muscles but V(max) was reduced to approximately 50% of the value in fully activated muscles. These results were the same for slow- and fast-twitch muscles and for relaxation following maximal tetani and brief, sub-maximal tetani. The measured series elastic compliance was used to partition shortening velocity between SEC and CC. The curvature of the CC force-velocity relationship was constant during relaxation. The SEC accounted for most of the shortening and work output during relaxation and its power output during relaxation exceeded the maximum CC power output. It is proposed that unloading the CC, without any change in its overall length, accelerated cross-bridge detachment when shortening was applied during relaxation.     

Viscosity as an inseparable partner of muscle contraction

A simple model is presented where, by an iterative procedure, the forces delivered by the power strokes are summed up to overcome the load. The system is moderated by the viscous hindrance. The model reproduces the features of muscle contraction as defined by the data of He et al. [1997. ATPase kinetics on activation of permeabilized isometric fibres from rabbit and frog muscle: a real time phosphate assay. J. Physiol. 501, 125-148] on rabbit psoas muscle fibres. According to the model power strokes are random. Energy summation take place if the subsequent power stroke occurs before the energy delivered by the previous power stroke is completely used. In order the sarcomere to be competent to contract initial driving force must reach a threshold whose value increases with the load. The step size of the power stroke decreases with the increase of the load. The viscous regime is simulated by the equation, where 1/k measures the viscous hindrance of the system. The relationship between water activity, viscosity and stiffness is discussed. It is concluded that the three parameters vary cyclically and that when water activity decreases (sarcomere shortening, cross-bridge attachment) viscosity and stiffness increase.