The biphasic force-velocity relationship in whole rat skeletal muscle in situ

The Mechanical Properties of in Situ Canine Skeletal Muscle

This study was undertaken to determine if fiber arrangement was responsible for differences in the whole muscle mechanical properties. Experiments were carried out in situ in blood perfused dog skeletal muscles at approximately normal body temperature between 36° and 38°C. The following mechanical relationships were studied using a pneumatic muscle lever to measure Tension (P), length (L) and dP/dt: and dL/dt with a high frequency oscillograph (500–1000 Hz): 1.) Length:Tension; 2.) Force:Velocity; and 3.) Stress:Strain of Series Elastic. Electron microscopy and fiber typing were done as adjunctive studies. Muscles were stimulated by direct nerve stimulation with 0.1msec stimuli at a rate of 1 impulse per second for twitch contractions, or in 200 msec bursts of 100 Hz 0.1 msec stimuli for brief tetanic contractions. The pennate short fibered gastrocnemius plantaris developed 1.0 kg/g of tension during brief tetanic stimulation, at optimal length (Lo) with full stimulus voltage, while the parallel long fibered semitendinosus developed 0.5 kg/g under the same conditions. The Length:Tension relationship for these two muscles was qualitatively similar but quantitatively different. The Force:Velocity relationship (ΔL/L₀ vs. P/P₀) for both muscles were also qualitatively similar and could be described by the previously proposed rectangular hyperbola but a better predicted fit to the observed data could be produced by adding a descending exponential function to the rectangular hyperbola. Unlike previous studies, the Stress:Strain properties of the series elastic component measured by quick release (ΔL/Li vs. ΔP/Po) were linear and gastrocnemius was 25 per cent higher than the semitendinosus. Overall, both muscles were found to have mechanical properties that differed little from the previously reported literature for amphibian, cardiac and small mammalian muscles studied by others in vitro. The major differences that we found were in the shapes of the force:velocity curve of the contractile component, and the Stress:Strain curve of series elastic component. Equations and explanations for these differences are devised and presented.

Is curvature of the force-velocity relationship affected by oxygen availability? Evidence from studies in ex vivo and in situ rat muscles

The power of shortening contractions in skeletal muscle is determined by the force-velocity relationship. Fatigue has been reported to either increase or decrease the force-velocity curvature depending on experimental circumstances. These discrepant findings may be related to experimental differences in oxygen availability. We therefore investigated how the curvature of the force-velocity relationship in soleus and gastrocnemius rat muscles is affected during fatigue, in both an ex vivo setup without an intact blood perfusion and in an in situ setup with an intact blood perfusion. Furthermore, we investigated the effect of reduced oxygen concentrations and reduced diffusion distance on the curvature of the force-velocity relationship in ex vivo muscles, where muscle oxygen uptake relies on diffusion from the incubation medium. Muscles were electrically stimulated to perform repeated shortening contractions and force-velocity curves were determined in rested and fatigued conditions. The curvature increased during fatigue in the soleus muscles (both in situ and ex vivo), and decreased for the gastrocnemius muscles (in situ) or remained unchanged (ex vivo). Furthermore, under ex vivo conditions, neither reduced oxygen concentrations nor reduced diffusion distance conferred any substantial effect on the force-velocity curvature. In contrast, reduced oxygen availability and increased diffusion distance did increase the loss of maximal power during fatigue, mainly due to additional decreases in isometric force. We conclude that oxygen availability does not influence the fatigue-induced changes in force-velocity curvature. Rather, the observed variable fatigue profiles with regard to changes in curvature seem to be linked to the muscle fiber-type composition.

Comparison of linear, hyperbolic and double-hyperbolic models to assess the force-velocity relationship in multi-joint exercises

AbstractThis study assessed the validity of linear, hyperbolic and double-hyperbolic models to fit measured force-velocity (F-V) data in multi-joint exercises and the influence of muscle excitation on the F-V relationship. The force-joint angle and F-V relationships were assessed in 10 cross-training athletes and 14 recreationally resistance-trained subjects in the unilateral leg press (LP) and bilateral bench press (BP) exercises, respectively. A force plate and a linear encoder were installed to register external force and velocity, respectively. Muscle excitation was assessed by surface EMG recording of the quadriceps femoris, biceps femoris and gluteus maximus muscles during the unilateral LP. Linear, Hill's (hyperbolic) and Edman's (double-hyperbolic) equations were fitted to the measured F-V data and compared. Measured F-V data were best fitted by double-hyperbolic models in both exercises (p < 0.05). F-V data deviated from the rectangular hyperbola above a breakpoint located at 90% of measured isometric force (F₀) and from the linearity at ≤45% of F₀ (both p < 0.05). Hyperbolic equations overestimated F₀ values by 13 ± 11% and 6 ± 6% in the LP and BP, respectively (p < 0.05). No differences were found between muscle excitation levels below and above the breakpoint (p > 0.05). Large associations between variables obtained from linear and double-hyperbolic models were noted for F₀, maximum muscle power, and velocity between 25% and 100% of F₀ (r = 0.70-0.99; all p < 0.05). The F-V relationship in multi-joint exercises was double-hyperbolic, which was unrelated with lower muscle excitation levels. However, linear models may be valid to assess F₀, maximal muscle power and velocity between 25% and 100% of F₀.

The effects of inorganic phosphate on muscle force development and energetics: challenges in modelling related to experimental uncertainties

Muscle force and power are developed by myosin cross-bridges, which cyclically attach to actin, undergo a force-generating transition and detach under turnover of ATP. The force-generating transition is intimately associated with release of inorganic phosphate (Pi) but the exact sequence of events in relation to the actual Pi release step is controversial. Details of this process are reflected in the relationships between [Pi] and the developed force and shortening velocity. In order to account for these relationships, models have proposed branched kinetic pathways or loose coupling between biochemical and force-generating transitions. A key hypothesis underlying the present study is that such complexities are not required to explain changes in the force–velocity relationship and ATP turnover rate with altered [Pi]. We therefore set out to test if models without branched kinetic paths and Pi-release occurring before the main force-generating transition can account for effects of varied [Pi] (0.1–25 mM). The models tested, one assuming either linear or non-linear cross-bridge elasticity, account well for critical aspects of muscle contraction at 0.5 mM Pi but their capacity to account for the maximum power output vary. We find that the models, within experimental uncertainties, account for the relationship between [Pi] and isometric force as well as between [Pi] and the velocity of shortening at low loads. However, in apparent contradiction with available experimental findings, the tested models produce an anomalous force–velocity relationship at elevated [Pi] and high loads with more than one possible velocity for a given load. Nevertheless, considering experimental uncertainties and effects of sarcomere non-uniformities, these discrepancies are insufficient to refute the tested models in favour of more complex alternatives.

On the Shape of the Force-Velocity Relationship in Skeletal Muscles: The Linear, the Hyperbolic, and the Double-Hyperbolic

The shape of the force-velocity (F-V) relationship has important implications for different aspects of muscle physiology, such as muscle efficiency and fatigue, the understanding of the pathophysiology of several myopathies or the mechanisms of muscle contraction per se, and may be of relevance for other fields, such as the development of robotics and prosthetic applications featuring natural muscle-like properties. However, different opinions regarding the shape of the F-V relationship and the underlying mechanisms exist in the literature. In this review, we summarize relevant evidence on the shape of the F-V relationship obtained over the last century. Studies performed at multiple scales ranging from the sarcomere to the organism level have described the concentric F-V relationship as linear, hyperbolic or double-hyperbolic. While the F-V relationship has most frequently been described as a rectangular hyperbola, a large number of studies have found deviations from the hyperbolic function at both ends of the F-V relation. Indeed, current evidence suggests that the F-V relation in skeletal muscles follows a double-hyperbolic pattern, with a breakpoint located at very high forces/low velocities, which may be a direct consequence of the kinetic properties of myofilament cross-bridge formation. Deviations at low forces/high velocities, by contrast, may be related to a recently discovered, calcium-independent regulatory mechanism of muscle contraction, which may also explain the low metabolic cost of very fast muscle shortening contractions. Controversial results have also been reported regarding the eccentric F-V relationship, with studies in prepared muscle specimens suggesting that maximum eccentric force is substantially greater than isometric force, whereas in vivo studies in humans show only a modest increase, no change, or even a decrease in force in lengthening contractions. This review discusses possible reasons reported in the literature for these discrepant findings, including the testing procedures (familiarization, pre-load condition, and temperature) and a potential neural inhibition at higher lengthening velocities. Finally, some unresolved questions and recommendations for F-V testing in humans are reported at the end of this document.

Myosin Crossbridge, Contractile Unit, and the Mechanism of Contraction in Airway Smooth Muscle: A Mechanical Engineer's Perspective

Muscle contraction is caused by the action of myosin motors within the structural confines of contractile unit arrays. When the force generated by cyclic interactions between myosin crossbridges and actin filaments is greater than the average load shared by the crossbridges, sliding of the actin filaments occurs and the muscle shortens. The shortening velocity as a function of muscle load can be described mathematically by a hyperbola; this characteristic force–velocity relationship stems from stochastic interactions between the crossbridges and the actin filaments. Beyond the actomyosin interaction, there is not yet a unified theory explaining smooth muscle contraction, mainly because the structure of the contractile unit in smooth muscle (akin to the sarcomere in striated muscle) is still undefined. In this review, functional and structural data from airway smooth muscle are analyzed in an engineering approach of quantification and correlation to support a model of the contractile unit with characteristics revealed by mathematical analyses and behavior matched by experimental observation.

Force-velocity relationship during isometric and isotonic fatiguing contractions

Fatiguing contractions change the force-velocity relationship, but assessment of this relationship in fatigue has usually been obtained after isometric contractions. We studied fatigue caused by isometric or isotonic contractions, by assessment of the force-velocity relationship while the contractions maintaining fatigue were continued. This approach allowed determination of the force-velocity relationship during a steady condition of fatigue. We used the in situ rat medial gastrocnemius muscle, a physiologically relevant preparation. Intermittent (1/s) stimulation at 170 Hz for 100 ms resulted in decreased isometric force to ~35% of initial or decreased peak velocity of shortening in dynamic contractions to ~45% of initial. Dynamic contractions resulted in a transient initial increase in velocity, followed by a rapid decline until a reasonably steady level was maintained. Data were fit to the classic Hill equation for determination of the force-velocity relationship. Isometric and dynamic contractions resulted in similar decreases in maximal isometric force and peak power. Only V_max was different between the types of contraction ( P < 0.005) with greater decrease in V_max during isotonic contractions to 171.7 ± 7.3 mm/s than during isometric contractions to 208.8 mm/s. Curvature indicated by a/Po (constants from fit to Hill equation) changed from 0.45 ± 0.04 to 0.71 ± 0.11 during isometric contractions and from 0.51 ± 0.04 to 0.85 ± 0.18 during isotonic contractions. Recovery was incomplete 45 min after stopping the intermittent contractions. At this time, recovery of low-frequency isometric force was substantially less after isometric contractions, implicating force during intermittent contractions as a determining factor with this measure of fatigue. NEW & NOTEWORTHY The force-velocity relationship was captured while fatigue was maintained at a constant level during isometric and dynamic contractions. The curvature of the force-velocity relationship was less curved during fatigue than prefatigued, but within 45 min this recovered. Low-frequency fatigue persisted with greater depression of low-frequency force after isometric contractions, possibly because of higher force contractions during intermittent contractions.

Effects of manipulating tetanic calcium on the curvature of the force-velocity relationship in isolated rat soleus muscles

AIM

In dynamically contracting muscles, increased curvature of the force-velocity relationship contributes to the loss of power during fatigue. It has been proposed that fatigue-induced reduction in [Ca⁺⁺ ]_i causes this increased curvature. However, earlier studies on single fibres have been conducted at low temperatures. Here, we investigated the hypothesis that curvature is increased by reductions in tetanic [Ca⁺⁺ ]_i in isolated skeletal muscle at near-physiological temperatures.

METHODS

Rat soleus muscles were stimulated at 60 Hz in standard Krebs-Ringer buffer, and contraction force and velocity were measured. Tetanic [Ca⁺⁺ ]_i was in some experiments either lowered by addition of 10 μmol/L dantrolene or use of submaximal stimulation (30 Hz) or increased by addition of 2 mmol/L caffeine. Force-velocity curves were constructed by fitting shortening velocity at different loading forces to the Hill equation. Curvature was determined as the ratio a/F₀ with increased curvature reflecting decreased a/F₀ .

RESULTS

Compared to control levels, lowering tetanic [Ca⁺⁺ ]_i with dantrolene or reduced stimulation frequency decreased the curvature slightly as judged from increase in a/F₀ of 13 ± 1% (P = < .001) and 20 ± 2% (P = < .001) respectively. In contrast, increasing tetanic [Ca⁺⁺ ]_i with caffeine increased the curvature (a/F₀ decreased by 17 ± 1%; P = < .001).

CONCLUSION

Contrary to our hypothesis, interventions that reduced tetanic [Ca⁺⁺ ]_i caused a decrease in curvature, while increasing tetanic [Ca⁺⁺ ]_i increased the curvature. These results reject a simple causal relation between [Ca⁺⁺ ]_i and curvature of the force-velocity relation during fatigue.

A multiscale chemo-electro-mechanical skeletal muscle model to analyze muscle contraction and force generation for different muscle fiber arrangements

The presented chemo-electro-mechanical skeletal muscle model relies on a continuum-mechanical formulation describing the muscle's deformation and force generation on the macroscopic muscle level. Unlike other three-dimensional models, the description of the activation-induced behavior of the mechanical model is entirely based on chemo-electro-mechanical principles on the microscopic sarcomere level. Yet, the multiscale model reproduces key characteristics of skeletal muscles such as experimental force-length and force-velocity data on the macroscopic whole muscle level. The paper presents the methodological approaches required to obtain such a multiscale model, and demonstrates the feasibility of using such a model to analyze differences in the mechanical behavior of parallel-fibered muscles, in which the muscle fibers either span the entire length of the fascicles or terminate intrafascicularly. The presented results reveal that muscles, in which the fibers span the entire length of the fascicles, show lower peak forces, more dispersed twitches and fusion of twitches at lower stimulation frequencies. In detail, the model predicted twitch rise times of 38.2 and 17.2 ms for a 12 cm long muscle, in which the fibers span the entire length of the fascicles and with twelve fiber compartments in series, respectively. Further, the twelve-compartment model predicted peak twitch forces that were 19% higher than in the single-compartment model. The analysis of sarcomere lengths during fixed-end single twitch contractions at optimal length predicts rather small sarcomere length changes. The observed lengths range from 75 to 111% of the optimal sarcomere length, which corresponds to a region with maximum filament overlap. This result suggests that stability issues resulting from activation-induced stretches of non-activated sarcomeres are unlikely in muscles with passive forces appearing at short muscle length.

The force-velocity relationship at negative loads (assisted shortening) studied in isolated, intact muscle fibres of the frog

AIM

The study was undertaken to explore the force-velocity relationship under conditions where the myofilament system is subjected to an external force that serves as a negative load and assists the shortening movement.

METHODS

The experiments were carried out on single muscle fibres isolated from the anterior tibialis muscle of Rana temporaria. The fibres, being operated under load-clamp control, were released to shorten during tetanic stimulation at sarcomere lengths where the fibres carried different degrees of passive tension. The shortening thus occurred while the sarcomeres were subjected to a force that may be characterized as a 'negative load', that is, a force assisting the shortening movement.

RESULTS

The force-velocity relationship below zero load was found to be a smooth continuation of the force-velocity curve recorded at positive loads with the shortening velocity increasing steeply at loads below zero. A negative load amounting to merely 10% of the isometric force, thus raised the shortening velocity to a level two to three times higher than V0 , the velocity recorded at zero load.

CONCLUSIONS

The results provide evidence that, even in the presence of a longitudinal compressive force, the speed of shortening of the muscle fibre is determined by the cycling rate of the interacting cross-bridges. The force-velocity relationship at negative loads may play a relevant part during fast movements of striated muscle as pointed out in the discussion.

Hill's equation of muscle performance and its hidden insight on molecular mechanisms

Muscles shorten faster against light loads than they do against heavy loads. The hyperbolic equation first used by A.V. Hill over seven decades ago to illustrate the relationship between shortening velocity and load is still the predominant method used to characterize muscle performance, even though it has been regarded as purely empirical and lacking precision in predicting velocities at high and low loads. Popularity of the Hill equation has been sustained perhaps because of historical reasons, but its simplicity is certainly attractive. The descriptive nature of the equation does not diminish its role as a useful tool in our quest to understand animal locomotion and optimal design of muscle-powered devices like bicycles. In this Review, an analysis is presented to illustrate the connection between the historic Hill equation and the kinetics of myosin cross-bridge cycle based on the latest findings on myosin motor interaction with actin filaments within the structural confines of a sarcomere. In light of the new data and perspective, some previous studies of force-velocity relations of muscle are revisited to further our understanding of muscle mechanics and the underlying biochemical events, specifically how extracellular and intracellular environment, protein isoform expression, and posttranslational modification of contractile and regulatory proteins change the interaction between myosin and actin that in turn alter muscle force, shortening velocity, and the relationship between them.

A 3D electro-mechanical continuum model for simulating skeletal muscle contraction

A thermodynamically consistent three-dimensional electro-mechanical continuum model for simulating skeletal muscle contraction is presented. Active and passive responses are accounted for by means of a decoupled strain energy function into passive and active contributions. The active force is obtained as the maximum tetanic force penalized by two functions that consider the external stimulus frequency and the overlap between actin and myosin filaments. Passive response is modelled by a transversely isotropic strain energy function. The robustness of the model is analyzed by means of finite element simulations that reproduce the one-dimensional isometric, concentric and eccentric contractions in a simplified model of a muscle. The model has also been implemented to reproduce isometric and concentric contractions on a three-dimensional finite element model of the rat tibialis anterior (TA) muscle. The finite element model was obtained from magnetic resonance imaging and the preferential directions associated with the collagen and muscular fibres were considered. The proposed model was able to reproduce the observed experimental response of the active force generated by the isolated rat TA muscle during isometric and concentric contractions. In addition, the predicted force-velocity relationship is in good agreement with experimental data reported for the fast-twitch extensor digitorum longus (e.d.l) muscle of male rats.

Procedures for rat in situ skeletal muscle contractile properties

There are many circumstances where it is desirable to obtain the contractile response of skeletal muscle under physiological circumstances: normal circulation, intact whole muscle, at body temperature. This includes the study of contractile responses like posttetanic potentiation, staircase and fatigue. Furthermore, the consequences of disease, disuse, injury, training and drug treatment can be of interest. This video demonstrates appropriate procedures to set up and use this valuable muscle preparation. To set up this preparation, the animal must be anesthetized, and the medial gastrocnemius muscle is surgically isolated, with the origin intact. Care must be taken to maintain the blood and nerve supplies. A long section of the sciatic nerve is cleared of connective tissue, and severed proximally. All branches of the distal stump that do not innervate the medial gastrocnemius muscle are severed. The distal nerve stump is inserted into a cuff lined with stainless steel stimulating wires. The calcaneus is severed, leaving a small piece of bone still attached to the Achilles tendon. Sonometric crystals and/or electrodes for electromyography can be inserted. Immobilization by metal probes in the femur and tibia prevents movement of the muscle origin. The Achilles tendon is attached to the force transducer and the loosened skin is pulled up at the sides to form a container that is filled with warmed paraffin oil. The oil distributes heat evenly and minimizes evaporative heat loss. A heat lamp is directed on the muscle, and the muscle and rat are allowed to warm up to 37°C. While it is warming, maximal voltage and optimal length can be determined. These are important initial conditions for any experiment on intact whole muscle. The experiment may include determination of standard contractile properties, like the force-frequency relationship, force-length relationship, and force-velocity relationship. With care in surgical isolation, immobilization of the origin of the muscle and alignment of the muscle-tendon unit with the force transducer, and proper data analysis, high quality measurements can be obtained with this muscle preparation.

Calculation of muscle maximal shortening velocity by extrapolation of the force-velocity relationship: afterloaded versus isotonic release contractions

The maximal shortening velocity of a muscle (V(max)) provides a link between its macroscopic properties and the underlying biochemical reactions and is altered in some diseases. Two methods that are widely used for determining V(max) are afterloaded and isotonic release contractions. To determine whether these two methods give equivalent results, we calculated V(max) in 9 intact single fibres from the lumbrical muscles of the frog Xenopus laevis (9.5-15.5 °C, stimulation frequency 35-70 Hz). The data were modelled using a 3-state cross-bridge model in which the states were inactive, detached, and attached. Afterloaded contractions gave lower predictions of Vmax than did isotonic release contractions in all 9 fibres (3.20 ± 0.84 versus 4.11 ± 1.08 lengths per second, respectively; means ± SD, p = 0.001) and underestimated unloaded shortening velocity measured with the slack test by an average of 29% (p = 0.001, n = 6). Excellent model predictions could be obtained by assuming that activation is inhibited by shortening. We conclude that under the experimental conditions used in this study, afterloaded and isotonic release contractions do not give equivalent results. When a change in the V(max) measured with afterloaded contractions is observed in diseased muscle, it is important to consider that this may reflect differences in either activation kinetics or cross-bridge cycling rates.

Characteristics of tetanic force produced by the sternomastoid muscle of the rat

The sternomastoid (SM) muscle plays an important role in supporting breathing. It also has unique anatomical advantages that allow its wide use in head and neck tissue reconstruction and muscle reinnervation. However, little is known about its contractile properties. The experiments were run on rats and designed to determine in vivo the relationship between muscle force (active muscle contraction to electrical stimulation) with passive tension (passive force changing muscle length) and two parameters (intensity and frequency) of electrical stimulation. The threshold current for initiating noticeable muscle contraction was 0.03 mA. Maximal muscle force (0.94 N) was produced by using moderate muscle length/tension (28 mm/0.08 N), 0.2 mA stimulation current, and 150 Hz stimulation frequency. These data are important not only to better understand the contractile properties of the rat SM muscle, but also to provide normative values which are critical to reliably assess the extent of functional recovery following muscle reinnervation.

Contractile performance of striated muscle

The single muscle fiber preparation provides an excellent tool for studying the mechanical behaviour of the contractile system at sarcomere level. The present article gives an overview of studies based on intact single fibers from frog and mouse skeletal muscle. The following aspects of muscle function are treated: (1) The length-tension relationship. (2) The biphasic force-velocity relationship. (3) The maximum speed of shortening, its independence of sarcomere length and degree of activation. (4) Force enhancement during stretch, its relation to sarcomere length and myofilament lattice width. (5) Residual force enhancement after stretch. (6) Force reduction after loaded shortening. (7) Deactivation by active shortening. (8) Differences in kinetic properties along individual muscle fibers.