Muscle Energetics and the Fenn Effect

This study suggests and statistically tests a consistent analytical method for isolating the critical variables governing the Fenn effect in frog sartorius muscle. It demonstrates that when the Hill factor (P₀ - P) is used to normalize the heat of shortening, or the work, or their sum, the enthalpy of shortening, these reduced energy terms are highly linear functions of the time of shortening. Linear correlation of a given form of reduced energy for a given muscle, against the time of shortening yields correlation coefficients of 0.998 or greater. From the regression equations, equations similar in form to the 1938 Hill force-velocity, and 1964 Hill heat of shortening equations, are deduced. The analysis suggests that the efficiency with which the extra energy of shortening is converted into work is nearly constant over most of the range of fractional loads, and it confirms Fenn's observation that the total extra energy of shortening is about 1.3 times the work. Thus it is also consistent with the results of most of the biochemical studies which correlate the extra breakdown of high energy phosphates entirely with the work done, and no component exclusively with the distance shortened. While the analysis successfully identifies the two factors regulating the release of the energy of shortening, time and the fractional load, there was insufficient data in the literature to develop the form of the relation between the size and the geometry of the muscle and the rate of energy release. Therefore until more data are available it seems unjustified to assume that either b/l₀ and a/P₀ (or the equivalent linear regression constants) are independent of these factors.     

Theoretical series elastic element length in Rana pipiens sartorius muscles

Assuming a two component system for the muscle, a series elastic element and a contractile component, the analyses of the isotonic and isometric data points were related to obtain the series elastic stiffness, dP/dl_s, from the relation, See PDF for Equation From the isometric data, dP/dt was obtained and shortening velocity, v, was a result of the isotonic experiments. Substituting (P₀ - P)/T for dP/dt and (P₀ - P)/(P + a) times b for v, dP/dl_s = (P + a) /bT, where P < P₀, and a, b are constants for any lengths l ≤ l₀ (Matsumoto, 1965). If the isometric tension and the shortening velocity are recorded for a given muscle length, l₀, although the series elastic, l_s, and the contractile component, l_c, are changing, the total muscle length, l₀ remains fixed and therefore the time constant, T. Integrating, See PDF for Equation the stress-strain relation for the series elastic element, See PDF for Equation is obtained; l_sc0 - l_s + l_c0where l_co equals the contractile component length for a muscle exerting a tension of P₀. For a given P/P₀, l_s is uniquely determined and must be the same whether on the isotonic or isometric length-tension-time curve. In fact, a locus on one surface curve can be associated with the corresponding locus on the other.     

Autonomic energy conversion. II. An approach to the energetics of muscular contraction

All discussions of muscle energetics concern themselves with the Hill force-velocity relation, which is also the general output relation of a class of self-regulated energy converters and as such contains only a single adjustable parameter —the degree of coupling. It is therefore important to see whether in principle muscle can be included in this class. One requirement is that the muscle should possess a working element characterized by a dissipation function of two terms: mechanical output and chemical input. This has been established by considering the initial steady phase of isotonic and isometric tetanic contraction to represent a stationary state of the fibrils (a considerable body of evidence supports this). Further requirements, which can be justified for the working element, are linearity and incomplete coupling. Thus the chemical input of the muscle may be expected to follow the inverse Hill equation (see Part I). The relatively large changes in activities of reactants which the equation demands could only be controlled by local operation of the regulator, and a scheme is outlined to show how such control may be achieved. Objections to this view recently raised by Wilkie and Woledge rest on at least two important assumptions, the validity of which is questioned: (a) that heat production by processes other than the immediate driving reaction is negligible, which disregards the regulatory mechanism (possibly this involves the calcium pump), and (b) that the affinity of the immediate driving reaction is determined by over-all concentrations. The division of heat production into “shortening heat” and “maintenance heat” or “activation heat” is found to be arbitrary.     

The dynamic properties of mammalian skeletal muscle

The dynamic characteristics of the rat gracilis anticus muscle at 17.5°C have been determined by isotonic and isometric loading. For a fixed initial length these characteristics were represented either as a family of length-velocity phase trajectories at various isotonic afterloads or as a series of force-velocity curves at different lengths. An alternate method of viewing these data, the length-external load-velocity phase space, was also generated. When the muscle was allowed to shorten from different initial lengths, the velocity of shortening achieved at a given length was lower for longer initial lengths. The amount of departure was also dependent upon the isotonic load, the greater the load the greater the departure. The departures were not caused by changes in the elastic elements of the muscle or fatigue in the ordinary sense. When the behavior of the muscle was investigated at different frequencies of stimulation, the shortening velocity was a function of the number of stimulating pulses received by the muscle at a given frequency. The shortening velocity of the rat gracilis anticus muscle is, therefore, not only a function of load and length, but also of an additional variable related to the time elapsed from onset of stimulation.     

CONTRACTILITY AND ULTRASTRUCTURE IN GLYCEROL-EXTRACTED MUSCLE FIBERS : I. The Relationship of Contractility to Sarcomere Length

This study was undertaken to determine whether glycerol-extracted rabbit psoas muscle fibers can develop tension and shorten after being stretched to such a length that the primary and secondary filaments no longer overlap. A method was devised to measure the initial sarcomere length and the ATP-induced isotonic shortening in prestretched isolated fibers subjected to a small preload (0.02 to 0.15 P₀). At all degrees of stretch, the fiber was able to shorten (60 to 75 per cent): to a sarcomere length of 0.7 µ when the initial length was 3.7 µ or less, and to an increasing length of 0.9 to 1.8 µ with increasing initial sarcomere length (3.8 to 4.4 µ). At sarcomere lengths of 3.8 to 4.5 µ, overlap of filaments was lost, as verified by electron microscopy. The variation in sarcomere length within individual fibers has been assessed by both light and electron microscopic measurements. In fibers up to 10 mm in length the stretch was evenly distributed along the fiber, and with sarcomere spacings greater than 4 µ there was only a slight chance of finding sarcomeres with filament overlap. These observations are in apparent contradiction to the assumption that an overlap of A and I filaments is necessary for tension generation and shortening.     

Maximum shortening velocity of smooth muscle: zero load-clamp vs. afterloaded method

Previous reports from this laboratory of force-velocity relationships of canine tracheal smooth muscle (TSM) have presented maximum shortening velocities (Vmax) mathematically derived from the linearized transformation of the Hill equation (A. V. Hill, Proc. Roy. Soc. London, Ser. B., 126:136-195, 1938). Recent technical advances enable us to measure Vmax directly using an electromagnetic lever system that can instantaneously clamp to a zero load, thus we compared values of Vmax derived mathematically and those directly measured on the same TSM strips. Derived Vmax values from afterloaded isotonic shortening curves for loads greater than preload were 0.328 +/- 0.021 optimal length (lO)/s and were not significantly different from zero load-clamp measurements of 0.301 +/- 0.022 lO/s from the same (n = 15) muscles. These data indicate that Vmax values mathematically derived for TSM from conventional isotonic afterloaded force-velocity curves are valid estimates of zero load velocity, because they were not significantly different from values obtained by direct measurement using the zero load-clamp technique.     

Mechanical control of the rising phase of contraction of frog skeletal and cardiac muscle

The effect of shortening on contractile activity was studied in experiments in which shortening during the rising phase of an isotonic contraction was suddenly stopped. At the same muscle length and the same time after stimulation the rise in tension was much faster, if preceded by shortening, than during an isometric contraction, demonstrating an increase in contractile activity. In this experiment the rate of tension rise determined in various phases of contraction was proportional to the rate of isotonic shortening at the same time after stimulation. Therefore, the time course of the isotonic rising phase could be derived from the tension rise after shortening. The rate of isotonic shortening was found to be unrelated to the tension generated at various lengths and to correspond closely to the activation process induced by shortening. The length response explains differences between isotonic and isometric contractions with regard to energy release (Fenn effect) and time relations. These results extend previous work which showed that shortening during later phases of a twitch prolongs, while lengthening abbreviates contraction. Thus the length responses, which have been called shortening activation and lengthening deactivation, control activity throughout an isotonic twitch.     

Myonemal contraction of spirostomum. II. Some mechanical properties of the contractile apparatus

In several respects, notably the high velocity of shortening, Ca²⁺ dependence, and ATP independence, contraction of Spirostomum resembles the spasmonemal mechanism of the peritrich ciliates. In this report further mechanical properties of the contractile apparatus are described that extend this comparison. The velocity-load characteristic is more appropriate to an elastomer than to a muscle where contraction force is load-dependent. Active tension is found to relate linearly to cell length for extensions up to and beyond resting length (l_r), an elastic limit is reached around 1.5 l_r. At resting length this tension, measured by the deformation of a glass microbalance, is similar to that predicted from consideration of the hydrodynamic forces normally resisting shortening. The tension-length relation for the unstimulated (passive) cell is also linear between l_r and the elastic limit, but is displaced from the active tension-length curve and is of reduced stiffness. Kinetic studies suggest that maximum tension and maximum velocity coincide. Calculations are presented that support a model of contraction in Spirostomum in which the myonemes behave as a mechanochemical engine powered directly by the chemical potential of Ca²⁺.     

The Mechanical Properties of Drosophila Jump Muscle Expressing Wild-Type and Embryonic Myosin Isoforms

Transgenic Drosophila are highly useful for structure-function studies of muscle proteins. However, our ability to mechanically analyze transgenically expressed mutant proteins in Drosophila muscles has been limited to the skinned indirect flight muscle preparation. We have developed a new muscle preparation using the Drosophila tergal depressor of the trochanter (TDT or jump) muscle that increases our experimental repertoire to include maximum shortening velocity (V_slack), force-velocity curves and steady-state power generation; experiments not possible using indirect flight muscle fibers. When transgenically expressing its wild-type myosin isoform (Tr-WT) the TDT is equivalent to a very fast vertebrate muscle. TDT has a V_slack equal to 6.1 ± 0.3 ML/s at 15°C, a steep tension-pCa curve, isometric tension of 37 ± 3 mN/mm², and maximum power production at 26% of isometric tension. Transgenically expressing an embryonic myosin isoform in the TDT muscle increased isometric tension 1.4-fold, but decreased V_slack 50% resulting in no significant difference in maximum power production compared to Tr-WT. Drosophila expressing embryonic myosin jumped <50% as far as Tr-WT that, along with comparisons to frog jump muscle studies, suggests fast muscle shortening velocity is relatively more important than high tension generation for Drosophila jumping.     

Comparison of Caplan's Irreversible Thermodynamic Theory of Muscle Contraction with Chemical Data

Recently Caplan (1) applied the concepts of irreversible thermodynamics and cybernetics to contracting muscle and derived Hill's force-velocity relation. Wilkie and Woledge (2) then compared Caplan's theory to chemical rates inferred from heat data and concluded that the theory was not consistent with the data. Caplan defended his theory in later papers (3, 4) but without any direct experimental verifications. As Wilkie and Woledge (2) point out, the rate of phosphorylcreatine (PC) breakdown during steady states of shortening has not been observed because of technical difficulties. In this paper it is shown that the rate equations may be directly integrated with time to obtain relations among actual quantities instead of rates. The validity of this integration is based on experimental evidence which indicates that certain combinations of the transport coefficients are constant with muscle length. These equations are then directly compared to experimental data of Cain, Infante, and Davies (5) with the following conclusions: (a) The measured variations of ΔPC for isotonic contractions are almost exactly as predicted by Caplan's theory. (b) The value of the chemical rate ratio, ν_m/ν_o, obtained from these data was 3.53 which is close to the value of 3 suggested by Caplan (3). (c) The maximum value of the chemical affinity for PC splitting was found to be 10.6 k cal/mole which is as expected from in vitro measurements (2). Because of the excellent agreement between theory and experiment, we conclude that Caplan's theory definitely warrants further investigation.     

Force-velocity shifts with repetitive isometric and isotonic contractions of canine gastrocnemius in situ.

The force-velocity (F-V) relationships of canine gastrocnemius-plantaris muscles at optimal muscle length in situ were studied before and after 10 min of repetitive isometric or isotonic tetanic contractions induced by electrical stimulation of the sciatic nerve (200-ms trains, 50 impulses/s, 1 contraction/s). F-V relationships and maximal velocity of shortening (Vmax) were determined by curve fitting with the Hill equation. Mean Vmax before fatigue was 3.8 +/- 0.2 (SE) average fiber lengths/s; mean maximal isometric tension (Po) was 508 +/- 15 g/g. With a significant decrease of force development during isometric contractions (-27 +/- 4%, P < 0.01, n = 5), Vmax was unchanged. However, with repetitive isotonic contractions at a low load (P/Po = 0.25, n = 5), a significant decrease in Vmax was observed (-21 +/- 2%, P < 0.01), whereas Po was unchanged. Isotonic contractions at an intermediate load (P/Po = 0.5, n = 4) resulted in significant decreases in both Vmax (-26 +/- 6%, P < 0.05) and Po (-12 +/- 2%, P < 0.01). These results show that repeated contractions of canine skeletal muscle produce specific changes in the F-V relationship that are dependent on the type of contractions being performed and indicate that decreases in other contractile properties, such as velocity development and shortening, can occur independently of changes in isometric tension.     

Energetics of Isometric and Isotonic Twitches in Toad Sartorius

Contractile energetics have been studied in twitches of toad sartorius muscle at 6-7°C. Isometric and isotonic energy production has been measured and plotted against a wide range of developed tensions and tension-time integrals. These parameters were varied by altering the isotonic load or by changing the preset isometric length. The isometric tension-independent heat was 1.12 ±0.18 (SD) mcal/g. The isometric heat coefficient Pl₀/H was 12.0 ±1.4 in muscles having twitch to tetanus ratios ranging from 0.4 to 0.6. Isometric enthalpy increased monotonically with tension or tension-time integral but the correlation between isometric heat and these parameters was poor. Isotonic enthalpy consumption was always higher than isometric enthalpy for any given tension or tension-time integral; however, isotonic heat production was consistently less than isometric heat production. The isotonic heat for the highest load (3 g) was not significantly different from the isometric tension-independent heat. Thus isotonic heat production first decreased and then increased with increasing tension or tension-time integral. In the discussion it is shown that the results conflict with all current interpretations of muscle energetics.     

A Phenomenological Theory of Muscular Contraction: II. Generalized Length Variations

In part I of this series, the theory of irreversible thermodynamics was applied to the sliding filament model to obtain rate equations for a contracting muscle at the in situ length l_o. In this paper we extend the theory to include length variations derived from the sliding filament model of contracting muscle using the work of Gordon, Huxley, and Julian (1). Accepting the validity of Hill's forcevelocity relation (2) at the in situ length, we show that Hill's equation is valid for any length provided that the values of the parameters, a, b, and V_m vary with length as derived herein. The predicted variation with length of the velocity for a lightly loaded isotonic contraction is shown to agree well with that measured by Gordon, Huxley, and Julian (1). Chemical rates are derived as functions of length using parameters that can be obtained experimentally.     

Contractile properties of the shortening rat diaphragm in vitro

Diaphragmatic fatigue has been defined in terms of the failure of the muscle to continue to generate a given level of tension. Appropriate shortening of the diaphragm is, however, just as important for adequate ventilation. In this study we have examined in vitro the contractile properties of the rat diaphragm under afterloaded isotonic conditions and the effect of fatigue on the ability of the diaphragm to shorten. Shortening of the muscle strips was found to depend on size of afterload, frequency of stimulation, duration of stimulation, and initial length of the muscle. The afterloaded isotonic length-tension relationship coincided with the relationship between length and active isometric tension only for relatively small afterloads. Fatigue of the muscle strips, induced by isometric or afterloaded isotonic contractions, was associated with a decline in the extent of shortening as well as a decrease in active isometric tension. Ability to shorten and ability to develop isometric tension did not decrease to the same extent under all conditions. We conclude that active shortening, as well as active isometric tension, is decreased by muscular fatigue and that changes in these properties can be different depending on experimental conditions. The results suggest that the definition of diaphragmatic fatigue should be expanded to include the ability of the muscle to shorten by an appropriate amount. The results also suggest that measurement of isometric performance may not provide a complete estimate of the overall performance of the fatigued diaphragm.     

Three-Dimensional Muscle Architecture and Comprehensive Dynamic Properties of Rabbit Gastrocnemius,Plantaris and Soleus: Input for Simulation Studies

The vastly increasing number of neuro-muscular simulation studies (with increasing numbers of muscles used per simulation) is in sharp contrast to a narrow database of necessary muscle parameters. Simulation results depend heavily on rough parameter estimates often obtained by scaling of one muscle parameter set. However, in vivo muscles differ in their individual properties and architecture. Here we provide a comprehensive dataset of dynamic (n = 6 per muscle) and geometric (three-dimensional architecture, n = 3 per muscle) muscle properties of the rabbit calf muscles gastrocnemius, plantaris, and soleus. For completeness we provide the dynamic muscle properties for further important shank muscles (flexor digitorum longus, extensor digitorum longus, and tibialis anterior; n = 1 per muscle). Maximum shortening velocity (normalized to optimal fiber length) of the gastrocnemius is about twice that of soleus, while plantaris showed an intermediate value. The force-velocity relation is similar for gastrocnemius and plantaris but is much more bent for the soleus. Although the muscles vary greatly in their three-dimensional architecture their mean pennation angle and normalized force-length relationships are almost similar. Forces of the muscles were enhanced in the isometric phase following stretching and were depressed following shortening compared to the corresponding isometric forces. While the enhancement was independent of the ramp velocity, the depression was inversely related to the ramp velocity. The lowest effect strength for soleus supports the idea that these effects adapt to muscle function. The careful acquisition of typical dynamical parameters (e.g. force-length and force-velocity relations, force elongation relations of passive components), enhancement and depression effects, and 3D muscle architecture of calf muscles provides valuable comprehensive datasets for e.g. simulations with neuro-muscular models, development of more realistic muscle models, or simulation of muscle packages.     

Some proposals in cardiac muscle mechanics and energetics

Based on A. V. Hill's three-component model, mechanical properties of the contractile element (CE), such as velocity and tension, were determined as muscle shortening and loads in quick-release or afterloaded isotonic contraction. The method is applicable for studying cardiac mechanics, to obtain force-velocity data of the same CE length at varous afterloads. Analysis of the energetics of cardiac muscle was based on simulation studies of cardiac mechanics (Wong 1971, 1972). By proper derivation, the conventional contractile element work (CEW) was found to be a minor energy determinant. The tension-time integral and tension-independent heat (Ricchiuti and Gibbs, 1965) represent energy utilization for activation and maintenance of tension, the primary energy determinant.     

Improvements to Hoang et al.'s method for measuring passive length–tension properties of human gastrocnemius muscle in vivo

While the passive mechanical properties of a musculo-articular complex can be determined using the relationship between the articular angle and the passive torque developed in resistance to motion, the properties of different structures of the musculo-articular complex cannot be easily assessed. Recently, an elegant method has been proposed to estimate the passive length–tension properties of gastrocnemius muscle–tendon unit (Hoang et al., 2005). In the present paper, two improvements of this method are proposed to decrease the number of parameters required to assess the passive length–tension relationship from 9 to 2. Furthermore, these two parameters have physical meaning as they represent a passive muscle–tendon stiffness index (α) and the muscle–tendon slack length (l₀). α and l₀ are relevant clinical parameters to study the chronic effects of aging, training protocols or neuromuscular pathologies on the passive mechanical properties of the muscle–tendon unit. Eight healthy subjects performed passive loading/unloading cycles at 5°/s with knee angle at 6 knee angles to assess the torque–angle relationships and to apply the modified method. Numerical optimization was used to minimize the squared error between the experimental and the modeled relationships. The experiment was performed twice to assess the reliability of α and l₀ across days. The results showed that the reliability of the two parameters was good (α: ICC=0.82, SEM=6.1 m^?1, CV=6.3% and l₀: ICC=0.83, SEM=0.29 cm, CV=0.9%). Using a sensitivity analysis, it was shown that the numerical solution was unique. Overall, the findings may provide increased interest in the method proposed by Hoang et al. (2005).     

Force-velocity constants in smooth muscle: afterloaded isotonic and quick-release methods

In using pharmacologic stimuli, force-velocity (FV) curves are usually obtained by the method of quick release (QR) and redevelopment of shortening at peak tetanic tension; the advantage of the method being that the active state is at maximum. However, the QR may itself reduce the intensity of the active state and result in reduced values of FV constants. We tested this by delineating FV curves in canine tracheal smooth muscle using both conventional afterloaded isotonic contractions (ALI), and redevelopment of shortening after QR methods. For both these studies a supramaximal tetanizing electrical stimulus was used. The analysis of 11 experiments revealed that the latter method resulted in statistically significant reductions of all FV constants except for Po (maximum isometric tetanic tension). The means and standard errors for the sets of constants for the ALI and QR, respectively, are as follows: Vmax (maximum velocity of shortening) = 0.275 lo (optimal muscle length)/s +/- 0.024 (SE), and 0.216 lo/s + 0.023; a (hyperbolic constant with units of force) = 294 g/cm2 +/- 35 and 236 g/cm2 +/- 32; b (hyperbolic constant with units of velocity) = 0.059 lo +/- 0.004 and 0.039 lo/s +/- 0.005; a/Po = 0.214 +/- 0.028 and 0.182 +/- 0.026; and Po = 1.362 kg/cm2 +/- 0.106 and 1.294 kg/cm2 +/- 0.097. These data clearly show that the quick-release method for measuring force-velocity relationships in canine smooth muscle results in significant underestimates of muscle shortening properties.     

Unloaded speed of shortening in voltage-clamped intact skeletal muscle fibers from wt, mdx, and transgenic minidystrophin mice using a novel high-speed acquisition system

Skeletal muscle unloaded shortening has been indirectly determined in the past. Here, we present a novel high-speed optical tracking technique that allows recording of unloaded shortening in single intact, voltage-clamped mammalian skeletal muscle fibers with 2-ms time resolution. L-type Ca²⁺ currents were simultaneously recorded. The time course of shortening was biexponential: a fast initial phase, τ₁, and a slower successive phase, τ_2, with activation energies of 59 kJ/mol and 47 kJ/mol. Maximum unloaded shortening speed, v_u,max, was faster than that derived using other techniques, e.g., ∼14.0 L₀ s⁻¹ at 30°C. Our technique also allowed direct determination of shortening acceleration. We applied our technique to single fibers from C57 wild-type, dystrophic mdx, and minidystrophin-expressing mice to test whether unloaded shortening was affected in the pathophysiological mechanism of Duchenne muscular dystrophy. v_u,max and a_u,max values were not significantly different in the three strains, whereas τ₁ and τ₂ were increased in mdx fibers. The results were complemented by myosin heavy and light chain (MLC) determinations that showed the same myosin heavy chain IIA profiles in the interossei muscles from the different strains. In mdx muscle, MLC-1f was significantly increased and MLC-2f and MLC-3f somewhat reduced. Fast initial active shortening seems almost unaffected in mdx muscle.     

Modeling force-velocity relation in skeletal muscle isotonic contraction using an artificial neural network

The aim of this study is to design an artificial neural network (ANN) to model force-velocity relation in skeletal muscle isotonic contraction. We obtained the data set, including physiological and morphometric parameters, by myography and morphometric measurements on frog gastrocnemius muscle. Then, we designed a multilayer perceptron ANN, the inputs of which are muscle volume, muscle optimum length, tendon length, preload, and afterload. The output of the ANN is contraction velocity. The experimental data were divided randomly into two parts. The first part was used to train the ANN. In order to validate the model, the second part of experimental data, which was not used in training, was employed to the ANN and then, its output was compared with Hill model and the experimental data. The behavior of ANN in high forces was more similar to experimental data, but in low forces the Hill model had better results. Furthermore, extrapolation of ANN performance showed that our model is more or less able to simulate eccentric contraction. Our results indicate that ANNs represent a powerful tool to capture some essential features of muscle isotonic contraction.