Is Achilles tendon compliance optimised for maximum muscle efficiency during locomotion?

Tendon elasticity is important for economical locomotion; however it is unknown whether tendon stiffness is appropriate to achieve an optimal efficiency in various muscles. Here we test the hypothesis that the Achilles tendon is of an appropriate stiffness to maximise medial gastrocnemius muscle efficiency during locomotion with different power requirements. To test this hypothesis we used a three element Hill muscle model to determine how muscle fascicles would be required to change length if the series elastic element stiffness is varied, whilst the limb kinematics and muscle properties are held constant. We applied a model of muscle energetics to these data to predict muscle efficiency for a range of stiffness values in both walking and running conditions. We also compared the model results to in vivo data collected using ultrasonography. The muscle model predicted that optimal series elastic element stiffness for maximising efficiency is equal or slightly higher than that of the average Achilles tendon in running and walking, respectively. Although the peak efficiency values for running (26%) and walking (27%) are similar, the range of stiffness values achieving high efficiency in running is much smaller than that during walking. These results suggest that a compliant tendon, such as the Achilles tendon, is required for efficient running. Such a finding is important, because it describes how the stiffness of a tendon may be adapted to achieve optimal efficiency for particular athletic pursuits. The influence of varying tendon stiffness on kinematic performance may, however, play an important role in determining the efficiency of the muscle.     

Bipedal walking and running with spring-like biarticular muscles

Compliant elements in the leg musculoskeletal system appear to be important not only for running but also for walking in human locomotion as shown in the energetics and kinematics studies of spring-mass model. While the spring-mass model assumes a whole leg as a linear spring, it is still not clear how the compliant elements of muscle-tendon systems behave in a human-like segmented leg structure. This study presents a minimalistic model of compliant leg structure that exploits dynamics of biarticular tension springs. In the proposed bipedal model, each leg consists of three leg segments with passive knee and ankle joints that are constrained by four linear tension springs. We found that biarticular arrangements of the springs that correspond to rectus femoris, biceps femoris and gastrocnemius in human legs provide self-stabilizing characteristics for both walking and running gaits. Through the experiments in simulation and a real-world robotic platform, we show how behavioral characteristics of the proposed model agree with basic patterns of human locomotion including joint kinematics and ground reaction force, which could not be explained in the previous models.     

Stretching Your Energetic Budget: How Tendon Compliance Affects the Metabolic Cost of Running

Muscles attach to bones via tendons that stretch and recoil, affecting muscle force generation and metabolic energy consumption. In this study, we investigated the effect of tendon compliance on the metabolic cost of running using a full-body musculoskeletal model with a detailed model of muscle energetics. We performed muscle-driven simulations of running at 2–5 m/s with tendon force–strain curves that produced between 1 and 10% strain when the muscles were developing maximum isometric force. We computed the average metabolic power consumed by each muscle when running at each speed and with each tendon compliance. Average whole-body metabolic power consumption increased as running speed increased, regardless of tendon compliance, and was lowest at each speed when tendon strain reached 2–3% as muscles were developing maximum isometric force. When running at 2 m/s, the soleus muscle consumed less metabolic power at high tendon compliance because the strain of the tendon allowed the muscle fibers to operate nearly isometrically during stance. In contrast, the medial and lateral gastrocnemii consumed less metabolic power at low tendon compliance because less compliant tendons allowed the muscle fibers to operate closer to their optimal lengths during stance. The software and simulations used in this study are freely available at simtk.org and enable examination of muscle energetics with unprecedented detail.     

Winter energetics of Virginia opossums Didelphis virginiana and implications for the species' northern distributional limit

While climatic limitations are widely recognized as primary factors determining the distributions of many species, the physiological link between climate and species' persistence is poorly understood. The Virginia opossum Didelphis virginiana is a species for which winter energetics have been evaluated and a northern geographical limit has been hypothesized. Expansion of opossum populations beyond this limit, however, suggests that a previous winter energetics model requires modification. I update this energetics model by incorporating random foraging success to estimate the probability of opossum survival under varying winter temperature regimes. Estimation of opossum "success" for winters in Amherst, Massachusetts, since 1926 showed that juvenile females, the key breeding component of the population, would survive at a rate high enough to maintain a stable population in only 4 of the 77 yr. The model correctly predicted the fate of 13 of 14 opossums monitored in the Amherst area during the winters of 2000–2003. The current energetics model does not correctly predict autumn weight gain, but it does accurately estimate opossum winter survival. However, the model predicts that opossums should be winter-limited in areas such as Amherst, Massachusetts, where in fact they are well established. This discrepancy may be explained in three ways: weather station data do not adequately reflect available microclimates, opossums show high levels of flexibility in cold-weather foraging behavior, and most likely, humans provide food and shelter that mitigate the effect of winter.     

The bending stiffness of shoes is beneficial to running energetics if it does not disturb the natural MTP joint flexion

A local minimum for running energetics has been reported for a specific bending stiffness, implying that shoe stiffness assists in running propulsion. However, the determinant of the metabolic optimum remains unknown. Highly stiff shoes significantly increase the moment arm of the ground reaction force (GRF) and reduce the leverage effect of joint torque at ground push-off. Inspired by previous findings, we hypothesized that the restriction of the natural metatarsophalangeal (MTP) flexion caused by stiffened shoes and the corresponding joint torque changes may reduce the benefit of shoe bending stiffness to running energetics. We proposed the critical stiffness, k_cr, which is defined as the ratio of the MTP joint (MTPJ) torque to the maximal MTPJ flexion angle, as a possible threshold of the elastic benefit of shoe stiffness. 19 subjects participated in a running test while wearing insoles with five different bending stiffness levels. Joint angles, GRFs, and metabolic costs were measured and analyzed as functions of the shoe stiffness. No significant changes were found in the take-off velocity of the center of mass (CoM), but the horizontal ground push-offs were significantly reduced at different shoe stiffness levels, indicating that complementary changes in the lower-limb joint torques were introduced to maintain steady running. Slight increases in the ankle, knee, and hip joint angular impulses were observed at stiffness levels exceeding the critical stiffness, whereas the angular impulse at the MTPJ was significantly reduced. These results indicate that the shoe bending stiffness is beneficial to running energetics if it does not disturb the natural MTPJ flexion.     

Modeling the energetics of 100-m running by using speed curves of world champions.

The present study aims to assess energy demand and supply in 100-m sprint running. A mathematical model was used in which supply has two components, aerobic and anaerobic, and demand has three components, energy required to move forward (C), energy required to overcome air resistance (Caero), and energy required to change kinetic energy (Ckin). Supply and demand were equated by using assumed efficiency of converting metabolic to external work. The mathematical model uses instantaneous velocities registered by the 1997 International Association of Athletics Federations world champions at 100 m in men and women. Supply and demand components obtained in the male champion were (in J/kg) aerobic 30 (5%), anaerobic 607 (95%), C 400 (63%), Caero 83 (13%), Ckin 154 (24%). Comparatively, a model that uses the average velocity of the male and female 100-m champions overestimates Ckin by 37 and 44%, respectively, and underestimates Caero by 14%. We argued that such a model is not appropriate because Ckin and Caero are nonlinear functions of velocity. Neither height nor body mass seems to have any advantage in the energetics of sprint running.     

Metabolic dynamics in skeletal muscle during acute reduction in blood flow and oxygen supply to mitochondria: in-silico studies using a multi-scale, top-down integrated model

Control mechanisms of cellular metabolism and energetics in skeletal muscle that may become evident in response to physiological stresses such as reduction in blood flow and oxygen supply to mitochondria can be quantitatively understood using a multi-scale computational model. The analysis of dynamic responses from such a model can provide insights into mechanisms of metabolic regulation that may not be evident from experimental studies. For the purpose, a physiologically-based, multi-scale computational model of skeletal muscle cellular metabolism and energetics was developed to describe dynamic responses of key chemical species and reaction fluxes to muscle ischemia. The model, which incorporates key transport and metabolic processes and subcellular compartmentalization, is based on dynamic mass balances of 30 chemical species in both capillary blood and tissue cells (cytosol and mitochondria) domains. The reaction fluxes in cytosol and mitochondria are expressed in terms of a general phenomenological Michaelis-Menten equation involving the compartmentalized energy controller ratios ATP/ADP and NADH/NAD(+). The unknown transport and reaction parameters in the model are estimated simultaneously by minimizing the differences between available in vivo experimental data on muscle ischemia and corresponding model outputs in coupled with the resting linear flux balance constraints using a robust, nonlinear, constrained-based, reduced gradient optimization algorithm. With the optimal parameter values, the model is able to simulate dynamic responses to reduced blood flow and oxygen supply to mitochondria associated with muscle ischemia of several key metabolite concentrations and metabolic fluxes in the subcellular cytosolic and mitochondrial compartments, some that can be measured and others that can not be measured with the current experimental techniques. The model can be applied to test complex hypotheses involving dynamic regulation of cellular metabolism and energetics in skeletal muscle during physiological stresses such as ischemia, hypoxia, and exercise.     

Atomistic simulation approach to a continuum description of self-assembled beta-sheet filaments

We investigated the supramolecular structure and continuum mechanical properties of a beta-sheet nanofiber comprised of a self-assembling peptide ac-[RARADADA]2-am using computer simulations. The supramolecular structure was determined by constructing candidate filaments with dimensions compatible with those observed in atomic force microscopy and selecting the most stable ones after running molecular dynamics simulations on each of them. Four structures with different backbone hydrogen-bonding patterns were identified to be similarly stable. We then quantified the continuum mechanical properties of these identified structures by running three independent simulations: thermal motion analysis, normal mode analysis, and steered molecular dynamics. Within the range of deformations investigated, the filament showed linear elasticity in transverse directions with an estimated persistence length of 1.2-4.8 microm. Although side-chain interactions govern the propensity and energetics of filament self-assembly, we found that backbone hydrogen-bonding interactions are the primary determinant of filament elasticity, as demonstrated by its effective thickness, which is smaller than that estimated by atomic force microscopy or from the molecular geometry, as well as by the similar bending stiffness of a model filament without charged side chains. The generality of our approach suggests that it should be applicable to developing continuum elastic ribbon models of other beta-sheet filaments and amyloid fibrils.     

Air resistance and its influence on the biomechanics and energetics of sprinting at sea level and at altitude

Following an examination of the processes by which chemical energy is converted into useful work during running, a mathematical model of the energetics of sprinting is constructed. This is used in conjunction with a careful analysis of Olympic records, in particular those obtained in the 1968 Games at Mexico City, to determine the magnitude of the rate of working against air resistance during running. It is established that times in the 100 m, 200 m and 400 m events at the Mexico Olympics were approximately 1.7% lower than they would otherwise have been if the races had been run at sea level. This information is used to deduce that the external work done per unit time against air resistance is about 7.5-9% of the total power output of a sprinter, running at maximum speed at sea level. These figures compare well with the value of 7.8% obtained independently by Davies (J. appl. Physiol 48, 702-709, 1980). The analysis provides evidence that a linear relation exists between running speed and the rate of degradation of mechanical energy into thermal energy up to the highest sprinting speeds attainable. The maximum power generated by a sprinter is approximately 3 kW.     

Effects of altered stride frequency and contact time on leg-spring behavior in human running

Many studies have demonstrated that contact time is a key factor affecting both the energetics and mechanics of running. The purpose of the present study was to further explore the relationships between contact time (t(c)), step frequency (f) and leg stiffness (k(leg)) in human running. Since f is a compound parameter, depending on both contact and aerial time, the specific goal of this study was to independently vary f and t(c) and to investigate their respective effects on spring-mass characteristics during running, seeking to determine if the changes in k(leg) observed when running at different f are mainly due to inherent changes in t(c). We compared three types of constant 3.33 m s(-1) running conditions in 10 male subjects: normal running at the subject's freely chosen f, running with decreased and increased f, and decreased and increased t(c) at the imposed freely chosen f. The data from the varied f trials showed that the variation of t(c) was strongly correlated to that of k(leg) (r(2)=0.90), and the variation of f was also significantly correlated to that of k(leg) (r(2)=0.47). Further, changes in t(c) obtained in various t(c) conditions were significantly correlated to changes in k(leg) (r(2)=0.96). These results confirm that leg stiffness was significantly influenced by step frequency variations during constant speed running, as earlier demonstrated, but our more novel finding is that compared to step frequency, the effect of contact time variations appears to be a stronger and more direct determinant of k(leg). Indeed, 90-96% of the variance in k(leg) can be explained by contact time, whether this latter parameter is directly controlled, or indirectly controlled through its close relationship with step frequency. In conclusion, from the comparison of two experimental procedures, i.e. imposing various step frequency conditions vs. asking subjects to intentionally vary contact time at their freely chosen step frequency, it appears that changes in leg stiffness are mainly related to changes in contact time, rather than to those in step frequency. Step frequency appears to be an indirect factor influencing leg stiffness, through its effect on contact time, which could be considered a major determinant of this spring-mass characteristic of human running.     

A unified theory for the energy cost of legged locomotion

Small animals are remarkably efficient climbers but comparatively poor runners, a well-established phenomenon in locomotor energetics that drives size-related differences in locomotor ecology yet remains poorly understood. Here, I derive the energy cost of legged locomotion from two complementary components of muscle metabolism, Activation–Relaxation and Cross-bridge cycling. A mathematical model incorporating these costs explains observed patterns of locomotor cost both within and between species, across a broad range of animals (insects to ungulates), for a wide range of substrate slopes including level running and vertical climbing. This ARC model unifies work- and force-based models for locomotor cost and integrates whole-organism locomotor cost with cellular muscle physiology, creating a predictive framework for investigating evolutionary and ecological pressures shaping limb design and ranging behaviour.     

Rensch's rule in large herbivorous mammals derived from metabolic scaling

Rensch's rule, which states that the magnitude of sexual size dimorphism tends to increase with increasing body size, has evolved independently in three lineages of large herbivorous mammals: bovids (antelopes), cervids (deer), and macropodids (kangaroos). This pattern can be explained by a model that combines allometry, life-history theory, and energetics. The key features are that female group size increases with increasing body size and that males have evolved under sexual selection to grow large enough to control these groups of females. The model predicts relationships among body size and female group size, male and female age at first breeding, death and growth rates, and energy allocation of males to produce body mass and weapons. Model predictions are well supported by data for these megaherbivores. The model suggests hypotheses for why some other sexually dimorphic taxa, such as primates and pinnipeds (seals and sea lions), do or do not conform to Rensh's rule.     

Can cycle power predict sprint running performance?

A major criticism of present models of the energetics and mechanics of sprint running concerns the application of estimates of parameters which seem to be adapted from measurements of running during actual competitions. This study presents a model which does not perpetuate this solecism. Using data obtained during supra-maximal cycle ergometer tests of highly trained athletes, the kinetics of the anaerobic and aerobic pathways were modelled. Internal power wasted in the acceleration and deceleration of body limbs and the power necessary to overcome air friction was calculated from data in the literature. Assuming a mechanical efficiency as found during submaximal cycling, a power equation was constructed which also included the power necessary to accelerate the body at the start of movement. The differential equation thus obtained was solved through simulation. The model appeared to predict realistic times at 100 m (10.47 s), 200 m (19.63 s) and 400 m (42.99 s) distances. By comparison with other methods it is argued that power equations of locomotion should include the concept of mechanical efficiency.     

Ground reaction forces during downhill and uphill running

We investigated the normal and parallel ground reaction forces during downhill and uphill running. Our rationale was that these force data would aid in the understanding of hill running injuries and energetics. Based on a simple spring-mass model, we hypothesized that the normal force peaks, both impact and active, would increase during downhill running and decrease during uphill running. We anticipated that the parallel braking force peaks would increase during downhill running and the parallel propulsive force peaks would increase during uphill running. But, we could not predict the magnitude of these changes. Five male and five female subjects ran at 3m/s on a force treadmill mounted on the level and on 3 degrees, 6 degrees, and 9 degrees wedges. During downhill running, normal impact force peaks and parallel braking force peaks were larger compared to the level. At -9 degrees, the normal impact force peaks increased by 54%, and the parallel braking force peaks increased by 73%. During uphill running, normal impact force peaks were smaller and parallel propulsive force peaks were larger compared to the level. At +9 degrees, normal impact force peaks were absent, and parallel propulsive peaks increased by 75%. Neither downhill nor uphill running affected normal active force peaks. Combined with previous biomechanics studies, our normal impact force data suggest that downhill running substantially increases the probability of overuse running injury. Our parallel force data provide insight into past energetic studies, which show that the metabolic cost increases during downhill running at steep angles.     

Segment interactions within the swing leg during unloaded and loaded running

In this study, designed to determine the effect of lower extremity inertia manipulation on joint kinetics and segment energetics during the swing phase, 15 male distance runners were filmed as they performed treadmill running (3.35 m s-1) under five load conditions: no added load and loads of 0.25 kg and 0.50 kg added to each thigh or each foot. Results of this study demonstrated that the energetics of the lower extremity movements during the swing phase of the running cycle were dominated by mechanical energy transfers between adjacent segments attributed to the joint reaction forces, which acted to redistribute mechanical energy within the system. These contributions were considerably greater than those of the net joint moments, which primarily reflected muscular generation and dissipation of mechanical energy. Lower extremity loading caused little change in the movement pattern of the swing leg. However, increases in the joint reaction forces and net moments and in the amount of work done and the energy transfer attributed to the reaction forces and moments were observed, but were limited to the joints proximal to the location of the added load. These results were consistent with the increased aerobic demand associated with increases in lower extremity inertia that have been reported elsewhere and also have implications for the manner in which the neuromuscular system controls the motion of the legs during running.     

Interactions and trade-offs among physiological determinants of performance and reproductive success

How an animal performs in its natural environment ultimately plays a key role in its reproductive success. While a number of studies have investigated how selection acts on performance-related traits, far fewer studies have examined the mechanisms responsible for variation in performance. Among mechanisms, variable morphology has received the most attention. Although physiological traits have received less attention, they are intrinsically related to performance and ultimately to reproductive success. We present a framework whereby investigators can link some basic physiological functions with organismal performance and ultimately with reproductive success. We propose that performance and ultimately reproductive success are strongly influenced by hormones, immune functions, and energetics. We further argue that no physiological function can be considered in isolation and thus our model emphasizes interactions and trade-offs both within each physiological function as well as among them. Some of the most commonly studied trade-offs are between reproduction and immune functions, with energetics as one of the key common currencies for these trade-offs. From an evolutionary perspective, the largest gaps in our knowledge lie in how these interactions and trade-offs influence reproductive success. We believe that a full understanding of how hormones, immune functions, and energetics influence performance traits related to reproduction and, ultimately, lifetime reproductive success requires recognition of the complex relationships, interactions, and trade-offs among these processes.     

Factor Xa active site substrate specificity with substrate phage display and computational molecular modeling

Structural origin of substrate-enzyme recognition remains incompletely understood. In the model enzyme system of serine protease, canonical anti-parallel beta-structure substrate-enzyme complex is the predominant hypothesis for the substrate-enzyme interaction at the atomic level. We used factor Xa (fXa), a key serine protease of the coagulation system, as a model enzyme to test the canonical conformation hypothesis. More than 160 fXa-cleavable substrate phage variants were experimentally selected from three designed substrate phage display libraries. These substrate phage variants were sequenced and their specificities to the model enzyme were quantified with quantitative enzyme-linked immunosorbent assay for substrate phage-enzyme reaction kinetics. At least three substrate-enzyme recognition modes emerged from the experimental data as necessary to account for the sequence-dependent specificity of the model enzyme. Computational molecular models were constructed, with both energetics and pharmacophore criteria, for the substrate-enzyme complexes of several of the representative substrate peptide sequences. In contrast to the canonical conformation hypothesis, the binding modes of the substrates to the model enzyme varied according to the substrate peptide sequence, indicating that an ensemble of binding modes underlay the observed specificity of the model serine protease.     

Similarity in multilegged locomotion: Bouncing like a monopode

Despite impressive variation in leg number, length, position and type of skeleton, similarities of legged, pedestrian locomotion exist in energetics, gait, stride frequency and ground-reaction force. Analysis of data available in the literature showed that a bouncing, spring-mass, monopode model can approximate the energetics and dynamics of trotting, running, and hopping in animals as diverse as cockroaches, quail and kangaroos. From an animal's mechanical-energy fluctuation and ground-reaction force, we calculated the compression of a virtual monopode's leg and its stiffness. Comparison of dimensionless parameters revealed that locomotor dynamics depend on gait and leg number and not on body mass. Relative stiffness per leg was similar for all animals and appears to be a very conservative quantity in the design of legged locomotor systems. Differences in the general dynamics of gait are based largely on the number of legs acting simultaneously to determine the total stiffness of the system. Four- and six-legged trotters had a greater whole body stiffness than two-legged runners operating their systems at about the same relative speed. The greater whole body stiffness in trotters resulted in a smaller compression of the virtual leg and a higher natural frequency and stride frequency.     

Force measurements during running on different instrumented treadmills

One method to determine the forces produced during running is to conduct extensive kinematic and kinetic analysis. These analyses can be performed by having an individual perform repeated over-ground running trials or simply run continuously on an instrumented treadmill. The forces produced during over-ground running may not be the same as the forces during treadmill running and these differences could be attributed to a number of factors, including the design of the instrumented treadmill. The purpose of this paper was to determine whether there are differences in force measurements on different instrumented treadmill setups in comparison to over-ground running and to correct for any of these differences using a theoretical model. 11 participants ran on three different treadmills and performed over-ground running at 2.7, 3.6, and 4.5 m/s. Ground reaction forces were measured via force plates and an instrumented pressure insole. We found that the magnitude of the vertical ground reaction force differed between the three treadmills and over-ground running. The difference in ground reaction forces estimated by the pressure insole and the treadmill-force-plate system or instrumented treadmill can be explained by a three degree of freedom mechanical model of a person running on a treadmill and this model could potentially be used to correct for errors in force measurement from instrumented treadmills. The model included a force plate, a treadmill, and a wobbling mass with varying natural frequencies and damping characteristics, and constant masses. These findings provide researchers a method to correct forces from an instrumented treadmill set-up to determine a close approximation of the actual forces experienced by a participant during treadmill running.