An energy 'sources' and 'fractions' approach to the mechanical energy expenditure problem--II. Movement of the multi-link chain model

Movement of a multi-link chain model is treated using the 'sources' and 'fractions' concept introduced earlier (part I of this series, J. Biomechanics 19, 287-293). 'The energy balance equations', i.e. the relations between the powers of the sources and the time rate of change of the total energy of the links are obtained. It is shown that the action of joint forces can promote energy transfers between links and energy transformations of the energy fractions of the links but it cannot change the total energy of the system. A formula for the mechanical energy expenditure for the control moments is deduced. 'External' and 'internal' energy balance equations are derived. 'External' energy is the energy of the general center of mass, 'internal' energy is the energy of the links in their motion relative to the general center of mass. It is shown that 'external' and 'internal' work depend on each other and their sum is not equal to the mechanical energy expenditure which occurs during movement of the body. This is because of the possibility of some source powers to change the external and internal energy of the system simultaneously out of phase with each other.     

An energy 'sources' and 'fractions' approach to the mechanical energy expenditure problem--III. Mechanical energy expenditure reduction during one link motion

Mechanical energy economy and transformation during one link motion are analyzed on the basis of the theory developed in the previous publications (parts I and II of this series, J. Biomechanics 19, 287-300). The 'compensation coefficient' characterizing mechanical energy economy is introduced. The attempts to estimate MEE using only energy curves and neglecting the powers of real sources of energy implicitly lead to replacement of real force and moment systems by the systems reduced to the centers of mass. But such an unintentional substitution of imaginary sources for real ones, specifically, the reduction of forces acting on the link to the equivalent system, changes estimates of mechanical energy expenditure (MEE). That is why the methods of calculating MEE economy based on the determination of so-called 'quasi-mechanical' work (the sum of the kinetic and potential energy increases per one cycle of motion) are not correct. There are two mechanisms to reduce the MEE using the antiphase fluctuations (corresponding to energy transformations) of the (a) rotational and translational fractions of the total energy (at the expense of the F-sources); (b) potential and kinetic energies (at the expense of the mg-source).     

Mechanical energy analysis identifies compensatory strategies in disabled elders' gait

Current concepts in disablement emphasize the importance of identifying mobility impairments in aging humans to enable timely intervention and, ultimately, prevent disability. Because mobility impairments are likely to result in compensatory movement strategies, recognizing and understanding those strategies may be critical in designing effective interventions for preventing disability. We sought to determine if mechanical energy methods are useful for identifying and understanding lower extremity compensatory movement strategies due to disabilities. Aleshinski's method was used to compute mechanical energy expenditure (MEE) and mechanical energy compensation (MEC) for the sagittal plane stance leg and low-back joints of healthy elders (HE) and disabled elders (DE) during preferred speed and paced (120 steps min(-1)) gait. DE subjects expended less ankle energy in late-stance and more low-back energy in mid-stance than did the HE subjects. When controlling for walking speed, the difference in ankle MEE disappeared, but mid-stance hip MEE was significantly higher for the DE subjects. Despite increased hip and low-back MEE, the DE subjects compensated hip and low-back muscles greater then HE subjects by increasing energy transferred into the pelvis, particularly when walking faster than their self-selected speed. Increased energy transfers into the pelvis during mid-stance may be a strategy used to assist in advancing and controlling the contralateral limb's swing phase. Increased trunk energy, however, may compromise dynamic stability and increase the risk of falling. We conclude that mechanical energy methods are useful for identifying and understanding compensatory movement strategies in elders with disabilities.     

A quantitative assessment of the "tendon" action of two-joint muscles]

Two-joint muscles are able to transmit mechanical energy between the links of the body having no common joint ("tendon action" of the muscles). It is proposed to calculate difference between control moment power in a joint and the sum of powers developed by all muscles serving this joint in order to determine the direction and rate of mechanical energy transfer through the two-joint muscles. It was shown that in the shock-absorbing phase of support in running two-joint muscles the energy transfers from distal to proximal links (from foot to thigh, and from shank to pelvis), in take-off phase-from proximal links to distal ones (from pelvis to shank, and from thigh to foot).     

Force-velocity relationship and biochemical-to-mechanical energy conversion by the sarcomere

The intracellular control mechanism leading to the well-known linear relationship between energy consumption by the sarcomere and the generated mechanical energy is analyzed here by coupling calcium kinetics with cross-bridge cycling. A key element in the control of the biochemical-to-mechanical energy conversion is the effect of filament sliding velocity on cross-bridge cycling. Our earlier studies have established the existence of a negative mechanical feedback mechanism whereby the rate of cross-bridge turnover from the strong, force-generating conformation to the weak, non-force-generating conformation is a linear function of the filament sliding velocity. This feedback allows the analytic derivation of the experimentally established Hill's equation for the force-velocity relationship. Moreover, it allows us to derive the transient length response to load clamps and the transient force response to sarcomere shortening at constant velocity. The results are in agreement with experimental studies. The mechanical feedback regulates the generated power, maintains the linear relationship between energy liberated by the actomyosin-ATPase and the generated mechanical energy, and determines the efficiency of biochemical-to-mechanical energy conversion. The mechanical feedback defines three elements of the mechanical energy: 1) external work done; 2) pseudopotential energy, required for cross-bridge recruitment; and 3) energy dissipation caused by the viscoelastic property of the cross bridge. The last two elements dissipate as heat.     

An energy 'sources' and 'fractions' approach to the mechanical energy expenditure problem--V. The mechanical energy expenditure reduction during motion of the multi-link system

Mechanical energy economy during motion of the multi-link system is analyzed on the basis of the theory developed in the previous publications (parts I-IV of this series, J. Biomechanics 19, 287-309). The compensation coefficients for the F- and M-sources and also the absolute compensation coefficient reflecting the mechanical energy economy due to four possible resources are introduced. These resources are the antiphase fluctuations of (I) each link's total energy fractions involving energy transformations between (1) rotational and translational fractions by F-sources, (2) kinetic and potential fractions by mg-source; (II) the links' total energies involving energy transfers between (3) links by F-sources, (4) links by M-sources. The conditions of mechanical energy economy, particularly due to M-sources, are analyzed.     

Mechanical energy expenditure on human movement and the anthropomorphic mechanism]

Mechanical energy expenditures of the man and anthropomorphic locomotion machine during movement are compared theoretically. Sources of the mechanical energy affecting movement of human's lower extremity are modelled by 8 muscles, 3 of which are the two-joint muscles. The model of the lower extremity of anthropomorphic locomotion machine is moved by joint moments. It was shown that in the same movement the model of the human lower extremity can spend less mechanical energy than that of the model of the anthropomorphic locomotion machine. It is caused by the presence of two-joint muscles in the first model. Such an economy of mechanical energy expenditures realized by the two-joint muscle is possible at simultaneous execution of three conditions: 1) signs of the muscle powers, which are produced by that muscle at both joints, are opposite; 2) moments produced by that muscle at each of both joints have the same direction with the joint moments at these joints; 3) one-joint antagonistic muscles are not active. An expression which makes it possible to estimate the mechanical energy savings by the two-joint muscles during humans' movement was developed.     

Determinants of increased energy cost of submaximal exercise in obese subjects

The energy cost of submaximal cycling exercises is studied in 23 obese (OS) and 13 lean control (LS) subjects at 1) a constant pedaling frequency (60 rpm) and at various work loads [external work loads (Wmec) up to 100 W] for one group of OS and LS, and at 2) constant Wmec (brake free and 60 or 70 W) and various frequencies (38-70 rpm) for a second group of OS and LS. The total energy expenditure (WO2) is calculated from O2 consumption (VO2) measured in both conditions and is compared with anthropometric data. The results show that at rest or at the same Wmec, WO2 is always greater in OS than in LS. At rest the quotients of WO2 over body surface area are not significantly different. At work the difference in WO2 cannot be explained by the muscular mechanical efficiency, which is not statistically different in OS (26 +/- 7.8%) and LS (25 +/- 4.6%). The calculated increase in the work of breathing of OS can account only for 5-15% of the energy overexpenditure. The energy cost of leg movement is estimated in brake-free cycling trials; it is significantly greater in OS than in LS (118 J compared with 68 J/pedal stroke), but when divided by leg volume the figures are not different (9.2 compared with 8.5 J X dm-3 X pedal stroke-1). Leg moving may account for approximately 60-70% of the energy cost of moderate exercise in cycling OS. The remaining difference in WO2 between OS and LS (20-30%) may be explained by an increase in muscular postural activity related to the lack of physical training of OS.     

Muscle mechanical work requirements during normal walking: the energetic cost of raising the body's center-of-mass is significant

Inverted pendulum models of walking predict that little muscle work is required for the exchange of body potential and kinetic energy in single-limb support. External power during walking (product of the measured ground reaction force and body center-of-mass (COM) velocity) is often analyzed to deduce net work output or mechanical energetic cost by muscles. Based on external power analyses and inverted pendulum theory, it has been suggested that a primary mechanical energetic cost may be associated with the mechanical work required to redirect the COM motion at the step-to-step transition. However, these models do not capture the multi-muscle, multi-segmental properties of walking, co-excitation of muscles to coordinate segmental energetic flow, and simultaneous production of positive and negative muscle work. In this study, a muscle-actuated forward dynamic simulation of walking was used to assess whether: (1). potential and kinetic energy of the body are exchanged with little muscle work; (2). external mechanical power can estimate the mechanical energetic cost for muscles; and (3.) the net work output and the mechanical energetic cost for muscles occurs mostly in double support. We found that the net work output by muscles cannot be estimated from external power and was the highest when the COM moved upward in early single-limb support even though kinetic and potential energy were exchanged, and muscle mechanical (and most likely metabolic) energetic cost is dominated not only by the need to redirect the COM in double support but also by the need to raise the COM in single support.     

An energy 'sources' and 'fractions' approach to the mechanical energy expenditure problem--I. Basic concepts, description of the model, analysis of a one-link system movement

Different methods of determining mechanical energy expenditure for human movement are used in scientific research. However, the validity of some of these methods is open to question. The concept of 'sources' and 'fractions' of mechanical energy is introduced in this paper. Power phenomena in moving a one-link system are analyzed through the use of 'energy balance equations'. They represent the interrelationships between the powers of the 'sources' and the time rate of change the mechanical energy 'fractions'. Two ways of minimizing mechanical energy expenditure are discussed. They correspond to 'whip-type' and 'pendulum-type' motion.     

Mechanical energy states during running

Changes in total mechanical work and its partitioning into different energy states (kinetic, potential and rotational) during a step cycle of running were investigated on six well trained athletes who ran at the test speeds of 40, 60, 80, and 100% (9.3 +/- 0.3 m/s) of maximum. Cinematographic techniques were utilized to calculate the mechanical energy states as described by Norman et al. (1976), using a 13 segment mechanical model of a runner as the basis for the computations. The data showed that both the kinetic and rotational energy increased parabolically but the potential energy decreased linearly with increases in running velocity. The calculated power of the positive work phase increased quadratically with running speed. During the phase when the runner was in contact with the ground, the applied calculations gave similar increases for the positive and negative works, and the power ratio (Wneg/Wpos) stayed the same at all measured speeds. Therefore, it is likely that the method used to calculate the various mechanical energy states did not reflect accurately enough the physiological energy costs at higher running speeds. It may, however, be quite acceptable for estimating the mechanical energy states during walking and slow running, in which case the role of negative work is less and consequently the storage and utilization of elastic energy is small.     

Role of extracellular glutamic acids in the stability and energy landscape of bacteriorhodopsin

Bacteriorhodopsin (BR), a specialized nanomachine, converts light energy into a proton gradient to power Halobacterium salinarum. In this work, we analyze the mechanical stability of a BR triple mutant in which three key extracellular residues, Glu⁹, Glu¹⁹⁴, and Glu²⁰⁴, were mutated simultaneously to Gln. These three Glu residues are involved in a network of hydrogen bonds, in cation binding, and form part of the proton release pathway of BR. Changes in these features and the robust photocycle dynamics of wild-type (WT) BR are apparent when the three extracellular Glu residues are mutated to Gln. It is speculated that such functional changes of proteins go hand in hand with changes in their mechanical properties. Here, we apply single-molecule dynamic force spectroscopy to investigate how the Glu to Gln mutations change interactions, reaction pathways, and the energy barriers of the structural regions of WT BR. The altered heights and positions of individual energy barriers unravel the changes in the mechanical and the unfolding kinetic properties of the secondary structures of WT BR. These changes in the mechanical unfolding energy landscape cause the proton pump to choose unfolding pathways differently. We suggest that, in a similar manner, the changed mechanical properties of mutated BR alter the functional energy landscape favoring different reaction pathways in the light-induced proton pumping mechanism.     

Muscle work is biased toward energy generation over dissipation in non-level running

This study tested the hypothesis that skeletal muscles generate more mechanical energy in gait tasks that raise the center of mass compared to the mechanical energy they dissipate in gait tasks that lower the center of mass despite equivalent changes in total mechanical energy. Thirteen adults ran on a 10° decline and incline surface at a constant average velocity. Three-dimensional (3D) joint powers were calculated from ground force and 3D kinematic data using inverse dynamics. Joint work was calculated from the power curves and assumed to be due to skeletal muscle–tendon actuators. External work was calculated from the kinematics of the pelvis through the gait cycle. Incline vs. decline running was characterized with smaller ground forces that operated over longer lever arms causing larger joint torques and work from these torques. Total lower extremity joint work was 28% greater in incline vs. decline running (1.32 vs. −1.03 J/kg m, p<0.001). Total lower extremity joint work comprised 86% and 71% of the total external work in incline (1.53 J/kg m) and decline running (−1.45 J/kg m), which themselves were not significantly different (p<0.180). We conjectured that the larger ground forces in decline vs. incline running caused larger accelerations of all body tissues and initiated a greater energy-dissipating response in these tissues compared to their response in incline running. The runners actively lowered themselves less during decline stance and descended farther as projectiles than they lifted themselves during incline stance and ascended as projectiles. These data indicated that despite larger ground forces in decline running, the reduced displacement during downhill stance phases limited the work done by muscle contraction in decline compared to incline running.     

Membrane electricity as a convertible energy currency for the cell

The role of transmembrane electric potential difference (delta psi) in mitochondria, chloroplasts, and bacteria has been considered. Since the electric capacitance of membranes is much lower than the pH buffer capacitance of water phases, delta psi proves to be the primary form of energy produced by generators of electrochemical H+ potential difference (delta mu-H). There are 11 distinct types of delta mu-H-generating systems in coupling membranes, involved in respiratory and light-dependent electron and proton transfer, as well as in ATP and PP1 hydrolysis and synthesis. Bacteriorhodopsin is the simplest delta mu-H generator. However, even in this case, the molecular mechanism of delta psi production remains obscure. Many types of work can be supported by delta mu-H with no ATP involved so that delta mu-H proves to be not only a transient intermediate of oxidative and photosynthetic phosphorylation but also a convertible energy currency for the cell. Among the delta mu-H-supported activities, mechanical work was recently demonstrated. It can be exemplified by the motility systems of (i) flagellar bacteria and (ii) blud--green algae. As was found in multicellular cyanobacteria, delta mu-H can be used for a power transmission over distances as long as 1 mm. It seems to be probable that in large cells of eukaryotes (e.g., in muscle fibers) giant mitochondria may serve as power-transmitting structures. Na+--K+ gradients can be used to stabilize delta mu-H in bacteria. It is suggested that the primary function of unequal distribution of these cations between the microbial cell and the medium is delta mu-H buffering.     

Power in sports: A literature review on the application,assumptions, and terminology of mechanical power in sport research

The quantification of mechanical power can provide valuable insight into athlete performance because it is the mechanical principle of the rate at which the athlete does work or transfers energy to complete a movement task. Estimates of power are usually limited by the capabilities of measurement systems, resulting in the use of simplified power models. This review provides a systematic overview of the studies on mechanical power in sports, discussing the application and estimation of mechanical power, the consequences of simplifications, and the terminology. The mechanical power balance consists of five parts, where joint power is equal to the sum of kinetic power, gravitational power, environmental power, and frictional power. Structuring literature based on these power components shows that simplifications in models are done on four levels, single vs multibody models, instantaneous power (IN) versus change in energy (EN), the dimensions of a model (1D, 2D, 3D), and neglecting parts of the mechanical power balance. Quantifying the consequences of simplification of power models has only been done for running, and shows differences ranging from 10% up to 250% compared to joint power models. Furthermore, inconsistency and imprecision were found in the determination of joint power, resulting from inverse dynamics methods, incorporation of translational joint powers, partitioning in negative and positive work, and power flow between segments. Most inconsistency in terminology was found in the definition and application of ‘external’ and ‘internal’ work and power. Sport research would benefit from structuring the research on mechanical power in sports and quantifying the result of simplifications in mechanical power estimations.     

A comparison of muscular mechanical energy expenditure and internal work in cycling

The hypothesis that the sum of the absolute changes in mechanical energy (internal work) is correlated with the muscular mechanical energy expenditure (MMEE) was tested using two elliptical chainrings, one that reduced and one that increased the internal work (compared to circular). Upper and lower bounds were put on the extra MMEE (work done by net joint torques in excess of the external work) with respect to the effect of intercompensation between joint torques due to biarticular muscles. This was done by having two measures of MMEE, one that allowed no intercompensation and one that allowed complete intercompensation between joints spanned by biarticular muscles. Energy analysis showed no correlation between internal work and the two measures of MMEE. When compared to circular, the chainring that reduced internal work increased MMEE, and phases of increased crank velocity associated with the elliptical shape resulted in increased power absorbed by the upstroke leg as it was accelerated against gravity. The resulting negative work necessitated additional positive work. Thus, the hypothesis that the internal work is correlated with MMEE was found to be invalid, and the total mechanical work done cannot be estimated by summing the internal and external work. Changes in the dynamics of cycling caused by a non-circular chainring may affect performance and must be considered during the non-circular chainring design process.     

A comparison of no-flow and low-flow ischemia in the rat heart: an energetic study

Heat production under no-flow ischemia (ISCH) and under hypoperfusion (HYP) conditions was measured in single isovolumetric contractions of perfused rat ventricles at 25 degrees C. Resting heat production (Hr) and resting pressure decreased when the perfusion rate was reduced from 6 to 1.5 mL min(-1) or lower flows (HYP) and by ISCH. Maximal developed pressure (P) decreased to 29% and 20% of control by HYP at 0.8 mL min(-1) and ISCH, respectively. The tension-independent heat (TIH) fraction attributed to Ca2+-binding, measured during single contractions, decreased under HYP with an increase in the ratio between the maximum relaxation rate and P (-P/P ratio). The TIH fractions (attributed to Ca2+ binding and Ca2+ removal processes) decreased under ISCH. The long duration TIH fraction associated with Ca2+-dependent mitochondrial activity disappeared at flow rates of 1.5 mL min(-1) or lower. The ratio between the tension-dependent energy release and P was decreased by ISCH but not by HYP, indicating that under ISCH there was an improvement in contractile economy, but this was not modified by HYP. Overall, the results indicate that no-flow and low-flow ischemias are energetically different models. While the contractile failure under HYP seems to be related to a decrease in myofilament Ca2+ sensitivity, under ISCH it appears to be related to decreased cytosolic Ca2+ availability combined with a more noticeable effect on a fraction of energy that has been attributed to mitochondrial activity. Furthermore, mechanical and energetic responses of both models (i.e., ISCH and HYP) found in the present work were not the same as those previously observed in severe hypoxia so that all these models should not be used indistinctly.     

The sodium cycle--a new type of bacterial energetics

The literature data and experimental results of the author's laboratory on the role of Na+ in bacterial energetics are reviewed. It was shown that certain bacterial species utilize the transmembrane difference of Na+ electrochemical potentials (delta mu Na+) as a convertible membrane-linked form of energy. The membranes of such bacteria were found to contain delta mu Na+ generators (e. g., decarboxylases of some carboxylic acids of NADH-menaquinone reductase). It was shown that delta mu Na+ formed by these generators may support all the three main types of work of the bacterial cell, i. e., chemical (ATP synthesis), osmotic (substrate accumulation) and mechanical (motility).     

The surface energy of water: the largest but forgotten source of energy in biological systems

Many functional proteins perform mechanical, structural or chemical work. Such proteins often use the energy from the hydrolysis of adenosine triphosphate (ATP). The role of ATP as an energy source and its production by metabolism was established in the middle of the twentieth century and replaced glycolysis as the focus of study. Before this time the surface energy of water, quantified in the middle of the nineteenth century, had been visualized as an important source of biological energy. Experimental and theoretical work has shown that the internal work done by this energy source may greatly exceed the energy derived from metabolism. Although the energy from ATP usually does the work external to the body, even this may be supplemented by the surface energy of water to give greater efficiency. The consideration of the principles by which proteins might employ this larger source of energy to do work is germane at this time.     

Membrane-linked energy transductions. Bioenergetic functions of sodium: H+ is not unique as a coupling ion

The concept is developed according to which Na+, like H+, can play the role of a coupling ion in energy-transducing biomembranes. This idea is based on observations that (i) Na+ can be extruded from the cell by primary pumps (Na-motive NADH-quinone reductase, decarboxylase or ATPase), and (ii) the downhill Na+ flux into the cell can be coupled with the performance of all the three types of membrane-linked work i.e. chemical (ATP synthesis), osmotic (accumulation of solutes) and mechanical (motility). Marine alkalotolerant Vibrio alginolyticus represents the first example of such a complete sodium cycle pattern. Simplified versions of the sodium cycle or some of its constituents are found in the cytoplasmic membrane of a great variety of taxa including anaerobic, aerobic and photosynthetic bacteria, cyanobacteria and animals; this fact indicates that Na+ energetics should be regarded as a common case, rather than a rare exception applied to some natural niches only.