Many animals use a combination of skeletal muscle and elastic structures to amplify power output for fast motions. Among vertebrates, tendons in series with skeletal muscle are often implicated as the primary power-amplifying spring, but muscles contain elastic structures at all levels of organization, from the muscle tendon to the extracellular matrix to elastic proteins within sarcomeres. The present study used ex vivo muscle preparations in combination with high-speed video to quantify power output, as the product of force and velocity, at several levels of muscle organization to determine where power amplification occurs. Dynamic ramp-shortening contractions in isolated frog flexor digitorum superficialis brevis were compared with isotonic power output to identify power amplification within muscle fibers, the muscle belly, free tendon and elements external to the muscle tendon. Energy accounting revealed that artifacts from compliant structures outside of the muscle–tendon unit contributed significant peak instantaneous power. This compliance included deflection of clamped bone that stored and released energy contributing 195.22±33.19 W kg−1 (mean±s.e.m.) to the peak power output. In addition, we found that power detected from within the muscle fascicles for dynamic shortening ramps was 338.78±16.03 W kg−1, or approximately 1.75 times the maximum isotonic power output of 195.23±8.82 W kg−1. Measurements of muscle belly and muscle–tendon unit also demonstrated significant power amplification. These data suggest that intramuscular tissues, as well as bone, have the capacity to store and release energy to amplify whole-muscle power output.

Rapid movements in many biological systems are actuated by active muscles and passive elastic structures. Mantis shrimp can accelerate their appendages fast enough to cavitate water during a predatory strike, and numerous frog species can jump many times their own body length to avoid predators (Astley et al., 2013; Patek and Caldwell, 2005; Peplowski and Marsh, 1997). A commonality of these rapid movements is they occur when active muscles contract slowly to deform passive elastic structures that can recoil at high velocities following energy storage. This process is analogous to the rapid shot of an arrow from a bow; muscles in the upper limb contract and bend the elastic bow that, when released, launches the arrow at much higher velocity than one could achieve by throwing the arrow with limb muscles alone. In animals, such mechanisms allow power outputs that exceed the isotonic force–velocity limit of skeletal muscles, and have been referred to as catapult mechanisms, power amplification or latch-mediated spring actuation (Bennet-Clark, 1975; Longo et al., 2019; Roberts and Azizi, 2011).

An advantage of the bow and arrow analogy is the simplicity of the mechanism. The archer's muscles provide actuation energy, the bow stores and releases elastic energy, and the release of the string by the archer's fingers provides a latch mechanism to govern the process of energy release. The structures providing each role are distinct, and these distinct structures are central to an understanding of how the mechanism works. The analogy might give the impression that identifying power-amplified systems in nature generally proceeds through a process of discovering structures that act as springs or latches. Yet, in many cases, power-amplified systems were first uncovered by a process of ‘black box’ energy accounting, with little or no knowledge of the structures that might serve as springs or latches. The original discovery of power amplification in flea jumps, mantis shrimp strikes and frog leaps resulted from the observation that the motion either occurred in a time period too short for a muscle contraction, or involved a power output that exceeded muscle capacity (Bennet-Clark and Lucey, 1967; Burrows, 1969; Marsh and John-Alder, 1994). Subsequent work identified the structural bases for these mechanisms, though in some cases the exact source of muscle power, spring energy and catch mechanism are still not fully understood.

Jumping in frogs is a system where the structural basis for power amplification is not well understood. Evidence indicates that most of the hindlimb muscles are involved (Peplowski and Marsh, 1997), and direct measurements of muscle length changes in bullfrogs support the idea that tendons are stretched prior to a jump, and thus likely contribute as springs (Astley and Roberts, 2012). Dynamic catch mechanisms that serve to aid in the process of energy storage by effectively resisting motion during a period of energy storage have also been identified (Astley and Roberts, 2014), though a physical latch or catch has not yet been discovered.

Studies of vertebrate jumping have focused on tendon as a spring able to contribute to power amplification by storing and releasing energy produced from muscle contraction. Yet, muscle is intertwined with elastic structures at multiple levels from titin and other proteins within the force generating fibers themselves, to the extracellular matrix that holds muscle fascicles together, all of which have been suggested to contribute to movement during active muscle shortening (Eng et al., 2018; Nishikawa et al., 2012; Roberts et al., 2019; Sleboda and Roberts, 2019). Given intrinsic limits for power generation by the contractile machinery, utilizing the energy storing capabilities of these elastic elements might be advantageous in many circumstances.

The present study seeks to identify contributions to whole-muscle power output from potential springs within muscle using an ex vivo bullfrog isolated muscle preparation. Force and velocity were measured using a muscle motor during dynamic ramp-shortening contractions, similar to what might occur during a frog jump. Muscle power output was quantified and compared with a maximum determined from individual force–velocity curves constructed from isotonic contractions. Power outputs from other levels of muscle organization, including from the muscle fascicles, muscle belly and muscle–tendon unit, were calculated with whole-muscle force and specific measures of velocity obtained from point-tracking landmarks on high-speed videos. If present, springs within a muscle capable of energy storage and recovery could act to amplify the power output at the whole-muscle level. Therefore, we hypothesized that at all the levels of muscle organization measured, including at the level of the muscle fascicles, power output during dynamic contractions with variable force would exceed the maximum power output predicted from isotonic contractions.

Animals

All husbandry and experimental protocols were approved by the Brown University Institutional Animal Care and Use Committee. Six large adult bullfrogs [Lithobates catesbeianus (Shaw 1802)] were purchased from a herpetological vendor (Rana Ranch, ID, USA) and housed in the Brown University Animal Care Facility. Animals inhabited large aquaria that included both terrestrial and aquatic regions, and were kept on a diet of large crickets provided ad libitum.

Ex vivo preparation

The flexor digitorum superficialis brevis (FDSB) muscle was selected as the muscle of study. This selection was motivated by the lack of any central tendon, small size, and the simplicity of its unipennate architecture with nearly all fascicles at the same pennation angle for a given muscle length (Fig. 1). These characteristics facilitated videography, marker implantation and data analysis.

Fig. 1.

Isolated muscle preparation schematic. (A) Schematic of the mounted muscle next to an enlarged image of the muscle belly illustrating pennation angle (θ) about a red muscle fascicle and a red dashed vertical axis. (B) Still image from a video recording, with segments labeled illustrating how length changes of the muscle–tendon unit (LMTU), muscle belly (LBelly) and muscle fascicles (LFasc), and instantaneous length changes about a single point along the distal bone (ΔLBone), were calculated.

Fig. 1.

Isolated muscle preparation schematic. (A) Schematic of the mounted muscle next to an enlarged image of the muscle belly illustrating pennation angle (θ) about a red muscle fascicle and a red dashed vertical axis. (B) Still image from a video recording, with segments labeled illustrating how length changes of the muscle–tendon unit (LMTU), muscle belly (LBelly) and muscle fascicles (LFasc), and instantaneous length changes about a single point along the distal bone (ΔLBone), were calculated.

In all ex vivo steps, the muscle was bathed in amphibian Ringer's solution (115 mmol l−1 NaCl, 2.5 mmol l−1 KCl, 1.0 mmol l−1 MgSO4, 20 mmol l−1 imidazole, 1.8 mmol l−1 CaCl2, 11 mmol l−1 glucose, pH 7.9) and aerated with 100% oxygen.

Six large adult bullfrogs (L. catesbeianus) were anesthetized with inhaled isoflurane and euthanized with a double pithing protocol. FDSB muscles (n=6, 0.392±0.019 g, mean±s.e.m. mass) were isolated from the lower hindlimbs. The insertion of each muscle was severed to free it from digits IV and V. The proximal origin on the tibiofibula was kept intact as its attachment to the talus and calcaneum was freed. The tibiofibula was placed in a custom clamp, and the remaining distal tendon of the FDSB was tied to a stainless-steel chain using silk suture ranging in size from 2-0 to 4-0.

Under a Zeiss dissecting scope (Stemi 2000-C), the muscle was bathed in oxygenated Ringer's solution and several black polyethylene microspheres ranging from 355 to 425 µm in diameter were inserted along superficial muscle fascicles. Markers were implanted near both ends of a single fascicle where it bordered an aponeurosis to quantify muscle fascicle length and pennation angle (Fig. 1). To avoid puncture damage to the muscle, a blunt custom-made tool was used for implantation and the number of markers implanted was kept to a minimum.

Following marker implantation, the tibiofibula clamp was secured to a rigid mount and the chain was attached to a servo-controlled muscle motor (310 B-LR, Aurora Scientific Inc., Ontario, CA, USA). This preparation was enclosed in a rectangular clear plastic housing filled with oxygenated amphibian Ringer's solution. Care was taken to ensure that the muscle, silk thread and steel chain were oriented vertically directly underneath the lever of the muscle motor. Field stimulation was provided by two thin sheets of platinum foil (0.025 mm) placed on either side of the isolated muscle. The electrodes were connected to a high current stimulus isolation unit (USIO, Hugo Sachs Elektronik, Germany) which was controlled by a stimulator (Grass S48, A-M Systems, Sequim, WA, USA) to provide field stimulation to the muscle. The temperature of the Ringer's solution was monitored and maintained throughout the time course of the experiment (mean±s.e.m., 23.1±0.1°C).

Data collection

Motor signals were collected at 1000 Hz via a PowerLab data acquisition system and LabChart 7.2.2 software (ADInstruments). Supramaximal stimulation voltage was determined by increasing the voltage of isometric twitch contractions until twitch force no longer increased; 150% of this voltage was used for the remaining duration of the experiment (∼10–15 V). A series of isometric twitch contractions at different lengths was used to construct a length–tension curve with optimum muscle length (L0) defined as the length at peak active force. The muscle was then tetanically stimulated to determine its maximum active force (P0) for a duration of 250 ms. This step was repeated at the end of each experiment to record any decline in muscle force production owing to fatigue. All post-experimental values for P0 were within 5% of pre-experimental values, indicating negligible fatigue. Each tetanic stimulation was performed using 0.2 ms pulses at 100 Hz. Next, a series of six isotonic shortening contractions was performed at approximately 10%, 20%, 30%, 40%, 60% and 80% of P0, along with an isometric contraction representing 100% P0 to characterize the force–velocity relationship. These data were fit by a modified Hill equation that included a linear component:
formula
(1)
where V is the velocity of shortening, P/P0 is force as a fraction of maximum isometric force, B and C are constants with dimensions of velocity, and A is a dimensionless constant (Marsh and Bennett, 1986). Predicted isotonic power was obtained from the product of force and velocity for each muscle.

For dynamic contractions, linear shortening ramp signals were input to the motor length controller via custom-made scripts in Igor Pro 7 software (WaveMetrics, Lake Oswego, OR, USA). Linear ramps were chosen as a simple approximation for the shortening pattern a frog hindlimb muscle might undergo during a jump, and a range of ramp speeds were chosen (60, 90, 120 and 160 mm s−1) that were similar and faster than values reported in the literature for jumping (∼2–5 L0 s−1) (Astley and Roberts, 2012; Azizi and Roberts, 2010; Lutz and Rome, 1994; Roberts and Marsh, 2003), and covered the range of speeds predicted from simulations (Sawicki et al., 2015a). As the focus of the study was to dissect out contributions to muscle power output, we did not attempt to directly imitate in vivo muscle shortening patterns during a jump, but note that the pattern of fascicle length change we observed was qualitatively similar to that observed during jumping in ranid frogs (Astley and Roberts, 2012; Azizi and Roberts, 2010). The shortening ramp was administered after the muscle was stimulated for 100 ms, allowing the muscle to reach 80–90% P0. This delay to allow for muscle pre-activation is similar to values reported in the literature for other species of frogs (Marsh, 2022). The excursion of the ramp was set to obtain 30–40% fascicle shortening (Azizi and Roberts, 2010) and the duration of the ramp was tuned based on the duration of the tetanic stimulus. It is notable that the abrupt shift from the motor holding the muscle isometric to shortening following a delay provided a ‘quick release’ of force that is likely more abrupt than the transition that occurs in vivo (Astley and Roberts, 2012). Each individual FDSB muscle was subjected to both isotonic contractions and dynamic ramp contractions.

A high-speed video camera (Flare 12M180MCX, IO Industries, London, Ontario, Canada) connected to a digital video recorder (CORE2 DVR Express, IO Industries) recorded all contractions at 500 frames s−1. Following data collection, the mass of the muscle was recorded, and the muscle was reviewed under a dissecting scope to select markers that remained in their original position and were well placed along a fascicle for point tracking.

Data analysis

Marker 2-D positions were tracked from grayscale video using XMALab 1.5.3-1.5.5 (Knörlein et al., 2016). Data processing was performed in MATLAB R2020b (The MathWorks Inc., Natick, MA, USA), Igor Pro 7 software (WaveMetrics, Lake Oswego, OR, USA) and RStudio (RStudio Team, Boston, MA, USA). For any smoothing, Reinsch's smoothing spline as implemented in Igor Pro 7 software was used (Reinsch, 1967). Because a goal of the study was to calculate power at multiple levels of organization, tracked marker coordinates were used to calculate length and velocity for the muscle fascicles, muscle belly, muscle–tendon unit and bone. Length and velocity at the level of the whole system were determined from the output of the muscle servo system. Fascicle length was measured as the linear distance between two fascicle markers (Fig. 1B), which was assumed to be representative of all fascicles. Pennation angle was calculated as the angle of the measured fascicle relative to the vertical axis (Fig. 1A). Muscle belly length was calculated as the length of the muscle without the free tendon but including the aponeurosis (Fig. 1B). Muscle–tendon unit length was measured as the length of the muscle plus free tendon (Fig. 1B). The velocity of the motor was used to calculate total system power. To determine the contribution of bone deflection to overall power output, which includes contributions from any compliant equipment, such as clamping hardware, displacement of the bone was measured from manual tracking of a single point of high contrast on the distal tip directly underneath the motor and where muscle attachment occurred (Fig. 1B). No external marker was placed at this location; natural high contrast landmarks were opportunistically chosen.

In cases where microspheres did not start and end at the limits of the muscle or muscle fascicle, data representing distance of a muscle or muscle fascicle were multiplied by a scaling factor. For instance, after measuring frames from each video, if the distance between two markers was found to be 6 mm representing a fascicle that was 8 mm in length, the marker-to-marker distance over time was multiplied by 4/3. If correction values were required, corrected values were checked at both long and short fascicle lengths for agreement.

Force was always recorded from the muscle servo system. Each of the measured components (muscle fascicles, muscle belly, muscle–tendon unit, silk thread, steel chain and distal bone) were assumed to operate effectively in series and, therefore, all experience the force detected at the muscle motor. This assumption relies on the fact that although the transmission force within any of these units maybe be spatially complex (e.g. the transmission of force between aponeurosis and fibers; Bossuyt et al., 2023; Kawakami and Lieber, 2000), force must be equivalent at the motor and a unit (e.g. muscle belly) when there is no alternative pathway for force transmission. This assumption holds for muscle fascicles as a unit, as there is no alternative pathway for force from distal aponeurosis to proximal aponeurosis. One caveat with regard to measured fascicle power is that our analysis did not account for the fact that fascicle are off-axis to the degree to which they are pennate. In theory, because not all of the force muscle fascicles produce is along the vertical axis, they experience slightly higher forces along the fiber axis (the axis for which we measured length) than is measured along the muscle line of action. In order to account for this, a cosine correction would be used to approximate the amount of force lost. However, in the case of the present study, the pennation angles are low when peak power is reached (6.08±1.69 deg) and would provide negligible correction values, and correcting for pennation angle would increase muscle fascicle power output. Therefore, muscle fascicle force was not corrected for pennation angle in order to provide a conservative estimate for fascicle force and power.

Power was calculated as the product of force from the muscle motor and the velocity measured from the high-speed video tracking as described above, except for the whole-system power, which was calculated from motor-measured velocity. Power amplification was determined by comparing the power output of dynamic shortening contractions with the maximum power output determined from isotonic contractions for each muscle.

Statistical tests

To test whether power output differs between each level of organization (muscle fascicles, muscle belly, muscle–tendon unit, motor) for the fastest ramp shortening contraction (160 mm s−1), a one-way repeated-measures ANOVA was performed. A natural logarithmic transformation was applied to these data to fit a normal distribution. Assumptions of data normality, homoscedasticity and sphericity were assessed using Shapiro–Wilk, Levene's and Mauchly's tests, respectively. A post hoc analysis was performed using Student's t-tests with a Bonferroni correction. The relationship between shortening ramp velocity (x) and power output (y) for each level of organization (muscle fascicles, muscle belly, muscle–tendon unit, motor) was determined using linear regressions. Both statistical analyses were computed using RStudio and the level of significance was set at P<0.05.

The relationship between force, velocity and power for frog FDSB muscle was quantified from a series of isotonic contractions (Fig. 2). Peak isometric stress averaged between the six individual muscle preparations was 19.53±1.14 N cm−2 (mean±s.e.m.) (Table 1). Isotonic force and velocity data were well fit by a Hill equation modified to include a linear component (Marsh and Bennett, 1986). Peak isotonic power was 195.23±8.82 W kg−1 on average (Table 1).

Fig. 2.

The isotonic force, velocity and power relationship. Individual normalized force and velocity data (black) are well fit by a modified Hill equation (gray) (Marsh and Bennett, 1986). Results from each individual muscle are plotted as different shapes (n=6), and power (blue) was calculated from the modified Hill equation.

Fig. 2.

The isotonic force, velocity and power relationship. Individual normalized force and velocity data (black) are well fit by a modified Hill equation (gray) (Marsh and Bennett, 1986). Results from each individual muscle are plotted as different shapes (n=6), and power (blue) was calculated from the modified Hill equation.

Table 1.

Key contractile and architectural properties of bullfrog flexor digitorum superficialis brevis muscle (n=6)

Key contractile and architectural properties of bullfrog flexor digitorum superficialis brevis muscle (n=6)
Key contractile and architectural properties of bullfrog flexor digitorum superficialis brevis muscle (n=6)

Measurements of length changes at multiple locations in the preparation allowed for the calculation of power output at several levels of organization. The simplest measure of power, and the one most commonly used in muscle preparations, is the that measured by the muscle motor (Askew and Marsh, 1997; Brooks et al., 1990; Nelson et al., 2004). Significant power amplification was observed at the level of motor measurements, with peak instantaneous power outputs as high as 1353.67±51.15 W kg−1 for 160 mm s−1 shortening ramps, or 6.96±0.23 times peak isotonic power output (Figs 3, 4, Table 2). This power amplification was for the fastest shortening ramp, 160 mm s−1, and occurred at the onset of ramp shortening (Fig. 3). Power output from the motor was significantly higher when compared with all other levels of muscle organization (the muscle fascicles, the muscle belly and the muscle–tendon unit) (P<0.001; Fig. 4).

Fig. 3.

Contractile time series data. Data from a single representative individual (n=1) 60 mm s−1 ramp shortening contraction is plotted against time in milliseconds. Length and power of each measured level of muscle organization are shown, along with force measured from the muscle motor (gray). The red horizontal line indicates the measured maximum isotonic power.

Fig. 3.

Contractile time series data. Data from a single representative individual (n=1) 60 mm s−1 ramp shortening contraction is plotted against time in milliseconds. Length and power of each measured level of muscle organization are shown, along with force measured from the muscle motor (gray). The red horizontal line indicates the measured maximum isotonic power.

Fig. 4.

Power from potential springs at multiple levels of organization. (A) Power (W kg−1) and power amplification (power relative to isotonic max.) are shown for a ramp shortening contraction of 160 mm s−1. (B) An illustration highlights potential sources of elastic energy storage at multiple levels of organization. Opaque dashed boxes on the bar plot are estimates meant to highlight differences between bars potentially due to elastic elements at each level; elastic elements are named within these boxes as possible contributors (Apo, aponeurosis). The red horizontal line indicates the measured maximum isotonic power and the results are shown for the one-way repeated-measures ANOVA (*P<0.05, ***P<0.001; n=6).

Fig. 4.

Power from potential springs at multiple levels of organization. (A) Power (W kg−1) and power amplification (power relative to isotonic max.) are shown for a ramp shortening contraction of 160 mm s−1. (B) An illustration highlights potential sources of elastic energy storage at multiple levels of organization. Opaque dashed boxes on the bar plot are estimates meant to highlight differences between bars potentially due to elastic elements at each level; elastic elements are named within these boxes as possible contributors (Apo, aponeurosis). The red horizontal line indicates the measured maximum isotonic power and the results are shown for the one-way repeated-measures ANOVA (*P<0.05, ***P<0.001; n=6).

Table 2.

Peak power measured at multiple levels of organization

Peak power measured at multiple levels of organization
Peak power measured at multiple levels of organization

Power output measured from the product of video-tracked muscle–tendon unit velocity and force from the motor was significantly lower than power measured at the muscle motor, with peak instantaneous values of 670.51±28.05 W kg−1 for 160 mm s−1 ramp shortening, or about half that detected at the muscle motor (P<0.001; Fig. 4). Power output from the muscle–tendon unit was significantly higher when compared with power output from the muscle fascicles and the muscle belly (P<0.001; Fig. 4). The significant difference between the muscle–tendon unit and the motor results from elastic deformation measured at the muscle motor that occurs outside of the muscle–tendon unit. Compliance in any of the structures shown in Fig. 4B that are outside of the muscle–tendon unit can allow for length changes under load and, ultimately, power amplification. Displacement of one of these structures, the bone, was measured by video. The point velocity of the tibiofibula at the muscle insertion point on the distal bone corresponded to a mean peak power output of 195.22±33.19 W kg−1 during 160 mm s−1 shortening ramps (Fig. 3, Table 2). The remaining compliance, as indicated by the difference between motor and muscle–tendon unit power, is indicated by a box with dashed lines in Fig. 4A.

Measurements from video of the muscle–tendon unit allowed further dissection of the contributors to power amplification. Muscle belly peak instantaneous power was calculated from the product of velocity of the muscle–tendon unit excluding the tendon and motor force as 452.46±26.09 W kg−1 during 160 mm s−1 shortening ramps (Fig. 3, Table 2). Power output from the muscle belly was significantly lower when compared with that from the motor and the muscle–tendon unit (P<0.001; Fig. 4), and significantly higher than that from the muscle fascicles (P=0.016; Fig. 4). Power of the muscle fascicles was calculated from the product of velocity of a single muscle fascicle and motor force as 338.78±16.03 W kg−1 during 160 mm s−1 shortening ramps (Fig. 3, Table 2). A control isotonic contraction held at 20% P0 for each individual never exceeded the measured maximum isotonic power (Table 2). Power output increased significantly with increasing shortening ramp velocity from 60 to 160 mm s−1 for the motor (y=260+6.9x, R2=0.90, P<0.0001), the muscle–tendon unit (y=160+3.2x, R2=0.85, P<0.0001), the muscle belly (y=220+1.6x, R2=0.60, P<0.0001) and the muscle fascicles (y=180+1x, R2=0.57, P<0.0001) (Fig. 5).

Fig. 5.

Power output increases with increasing ramp shortening velocity. Power is plotted in W kg−1 over the velocity of shortening ramps input from a control signal to the muscle motor. All four different shortening conditions are shown. Standard error of the mean is represented by the error bars (n=6). The red horizontal line indicates the measured maximum isotonic power.

Fig. 5.

Power output increases with increasing ramp shortening velocity. Power is plotted in W kg−1 over the velocity of shortening ramps input from a control signal to the muscle motor. All four different shortening conditions are shown. Standard error of the mean is represented by the error bars (n=6). The red horizontal line indicates the measured maximum isotonic power.

The results of our study show significant power amplification in a muscle–tendon unit preparation, with measurements from a muscle motor showing maximum powers exceeding muscle capacity by nearly 7-fold (Fig. 4). Here, isotonic muscle capacity is defined as the maximum power developed in a series of isotonic contractions, and power amplification is identified when power output exceeds this value in non-isotonic contractions. Other studies of isolated muscle have similarly shown muscle power amplification in dynamic contractions meant to mimic those in vivo (Lappin et al., 2006; Sawicki et al., 2015b). Both Sawicki et al. (2015b) and Lappin et al. (2006) showed power amplification in anuran skeletal muscle, demonstrating how the high power outputs occurring in vivo (Marsh and John-Alder, 1994; Peplowski and Marsh, 1997) can be reproduced at the muscle level. These studies, as well as many in vivo studies, have in common that power amplification was identified by the mismatch between maximum isotonic muscle power and power observed, with limited insight into the structures or processes underlying this amplification. The goal of the present study was to measure power at multiple scales to dissect where in the system power amplification occurred. We found power amplification at all measured scales, including within the muscle fibers, pointing towards multiple structures and processes contributing to the development of supramaximal muscle power in dynamic contractions.

Power from system compliance

Video tracking of muscle landmarks allowed us to quantify the contributions to power amplification by elastic structures outside of the muscle–tendon unit. These springs include the bone, steel chain, steel wire, clamps and the silk thread that all reside outside of the muscle–tendon unit and therefore represent compliant artifacts of the isolated preparation. Strikingly, these external elastic structures contributed about half of the measured peak instantaneous power output detected at the muscle motor (Fig. 4, Table 2). Because the methods of the present study allowed for the displacement of the muscle–tendon unit to be quantified from video recordings, less care was taken to reduce equipment compliance, and as a result, compliance was likely higher than in other isolated muscle studies. We measured from fixed-end contractions 0.22±0.02 mm N−1 of system compliance. Some studies report equipment compliance and accordingly make corrections to length and speed measured at the muscle motor (Ettema et al., 1990; Lou et al., 1999). Lou and coworkers (1999) reported that 25% of the compliance in a dogfish muscle preparation rested outside of muscle. Although we expect that the magnitude of artifact caused by equipment compliance is higher than typical in our preparation, the significant power amplification observed highlights the potential for even small amounts of compliance to contribute to measurements of power in dynamic, non-isotonic contractions.

Although most of the system compliance results from elasticity in the equipment, we found that bending in the bone to which the muscle origin was attached made a significant contribution: 195.22±33.19 W kg−1. This power amplification resulted from the rapid displacement in the tibiofibula as force declined rapidly during ramp contractions (Fig. 4, Table 2). The relatively large standard error of bone power output may be a result of the different lengths of bone that were attached to the clamp for each trial. During data collection it was assumed that the bone was not compliant enough to store meaningful elastic strain energy, so care was not taken to limit or consistently place the same amount of bone in-series with the preparation. The present method of clamping the tibiofibula orthogonal to the direction of force generation may not necessarily predict what this bone does in vivo, but does add to existing evidence that bone can act as a spring under the right conditions (Jenkins et al., 1988). This idea draws parallels with some invertebrates, such as mantis shrimp, whose rigid supportive exoskeleton can function as a spring to store and release energy, amplifying power output beyond that predicted from isotonic muscle contractions (Patek et al., 2004; Zack et al., 2009).

Power from intra-muscular and tendon springs

The ability for tendon to function as a spring during animal movement and muscle contraction is well established. Tendon is often assumed to be the primary spring storing and releasing elastic strain energy in movements ranging from single bout jumps to cyclic locomotion potentially amplifying, conserving or attenuating power (Alexander and Vernon, 1975; Hedrick et al., 2004; Peplowski and Marsh, 1997; Roberts and Azizi, 2011). In the present study, the power attributed to the free tendon alone was relatively small. This small contribution is likely due to the short length of the free tendon in series with the preparation; most of the free distal tendon was removed in dissection.

Power output from the level of the muscle belly also exceeded what was predicted from isotonic contractions. Within the muscle belly, both tendon aponeuroses and collagenous extracellular matrix have the potential to store and recover elastic energy and contribute to power amplification. The position of the aponeurosis allows for biaxial loading, both parallel and perpendicular to the muscle line of action, where it can act as an effective spring in each direction (Arellano et al., 2019; Azizi et al., 2009; Scott and Loeb, 1995), and a role of aponeurosis in energy storage and recovery has been identified in several systems (Eng and Roberts, 2018; Lichtwark et al., 2007; Roberts et al., 1997).

The muscle extracellular matrix also has the potential to contribute to power amplification via elastic stretch and recoil of the dense network of collagen fibers (Eng et al., 2018; Roberts et al., 2019). The extracellular matrix transmits forces from muscle contractile elements (Huijing et al., 1998) and thus bears load during contraction. Recent studies have suggested that the extracellular matrix can be loaded from contractile force transmitted through incompressible intramuscular fluid and influence both passive and active muscle tension (Sleboda and Roberts, 2017, 2019). Eng et al. (2018) proposed that in pennate muscles, force developed in muscle fibers orthogonal to the muscle line of action could pressurize intramuscular fluid and load the extracellular matrix in tension, thereby resisting the off-axis forces and acting as a spring. The results of the present study support the hypothesis that the extracellular matrix, loaded by pressurized intramuscular fluid, may act as a spring during dynamic contractions where it can store and release energy amplifying muscle power beyond the isotonic prediction. Although we expect that the spring-like action of the extracellular matrix is a contributor to power amplification, our approach does not allow determination of the relative contributions of aponeurosis versus extracellular matrix to muscle belly power. Our approach also could not detect interactions between elastic elements at separate levels of organization which may exist.

Power amplification within muscle fibers

Interestingly, a significant portion of the power amplification within the muscle–tendon unit was detected within the muscle fascicles themselves (Fig. 4, Table 2). Video tracking allowed for the measurement of fascicle velocity for both isotonic and ramp-shortening contractions, and the product of this velocity and motor force was used to measure power output. Power at the level of the muscle fascicles exceeded the maximum predicted muscle power determined from isotonic contraction during dynamic shortening ramps by as much as 1.7-fold (Fig. 4, Table 2). The mechanisms and structures underlying this power amplification are unclear at this point. Below, several possible contributors are discussed, including elastic elements within or external to sarcomeres, as well as possible differences in cross-bridge dynamics between isotonic and non-isotonic contractions.

Elasticity within contractile elements has the potential to contribute transiently to power output in force-varying contractions. Early experiments exploring tension transients during quick releases in muscle fibers measured compliance in both cross-bridges and myofilament backbones, determining that ∼50% of sarcomere compliance resides within the cross-bridges and ∼50% resides within the backbones of the thick (myosin) and thin (actin) filaments (Higuchi et al., 1995; Huxley et al., 1994; Wakabayashi et al., 1994). Although the range of fiber strain over which a single cross-bridge can contribute elastically is limited to about 0.4% (Ford et al., 1977; Huxley and Simmons, 1971), presumably cross-bridges and filaments could contribute elastically over a large range under conditions of cross-bridge cycling. Beyond primary myofilaments, an extensive scaffold of constituent proteins within skeletal muscle, e.g. intermediate filaments, has the potential to contribute to elastic storage and recovery (Henderson et al., 2018; Lazarides, 1980; Linari et al., 2003; Wang et al., 1993).

It has been suggested that the giant intrasarcomeric protein titin plays a role in active and passive muscle force production via its spring-like action (Dutta et al., 2018; Fukutani et al., 2021; Herzog, 2018; Monroy et al., 2017; Nishikawa et al., 2012; Prado et al., 2005). The spring-like behavior of titin is well established (Kellermayer et al., 1997; Linke, 2000; Wang et al., 1993), and evidence suggests that titin is stiffened under conditions of muscle activation and can contribute to elastic energy storage when active muscle is stretched (Dutta et al., 2018; Hessel et al., 2017; Powers et al., 2014). Lappin and colleagues (2006) found very high power outputs (9600 W kg−1) in an in situ toad jaw muscle preparation and speculated that spring-like behavior of titin might contribute (Lappin et al., 2006). A recent study showed significant power amplification in single fibers during stretch–shortening cycles, with stretches at the highest velocities producing power outputs during subsequent shortening that were approximately 3-fold greater than maximum power output predicted from isotonic contractions (Tomalka et al., 2021). The spring-like behavior of titin could contribute to power amplification observed at the fiber level in the present study, but it is important to note that most models for titin suggest elastic strain energy is stored during pre-stretch of a stretch–shortening cycle (Fukutani and Herzog, 2020; Powers et al., 2014; Tomalka et al., 2021).

We identifed power amplification as a mismatch between power measured in dynamic contractions (varying force) and the maximum determined from isotonic contractions. Although this mismatch is most commonly explained by contributions of elastic elements, it is also possible that the performance of the contractile machinery differs between dynamic and isotonic contractions. One study reported that the relationship between force and velocity measured during a continuously changing load was different than when measured isotonically and varied based on the compliance of the non-steady-state load (Iwamoto et al., 1990). Studies of enhanced muscle force production after stretch have, in some cases, explained this phenomena by a potentiation of cross-bridge action, either due to increased force per cross-bridge (Cavagna et al., 1985) or increased number of bound cross-bridges (Fusi et al., 2010).

It is important to note that although we consider above several possible explanations for the high power outputs measured at the fiber level in our study, these results do not shed new light on the question of whether titin, cross-bridges or transverse elastic elements contribute to force or power enhancement at the fiber level. This debate has been reviewed elsewhere (Granzier, 2010; Herzog et al., 2006; Minozzo and de Lira, 2013). Also, the studies cited above focus largely on muscle behavior following an active stretch. Our contraction protocol did not include an active stretch, and thus mechanistic inferences from stretch–shortening contractions may, or may not, be essential to understanding the present results.

Conclusions

Skeletal muscle power output that exceeds the maximum value calculated from isotonic contractions has been documented repeatedly, yet in many cases the exact source or mechanism amplifying power output is not fully understood. Often, this power amplification is thought to be a result of the storage and release of elastic strain energy in large tendons. Yet, skeletal muscle is a three-dimensional and hierarchical structure that operates within and around elastic elements that influence its mechanical output. We found that subunits of a muscle–tendon preparation all contributed to power amplification, including the fibers, muscle belly, tendon and bone. Additionally, we found that a large amount of the power detected was the result of artifacts from an overly compliant setup, providing a cautionary reminder to muscle physiologists. Although the exact structures and mechanisms underlying power amplification in each of these subunits is unclear, the results demonstrate that many different elements at multiple scales can contribute to increasing muscle performance beyond peak isotonic power.

We gratefully thank Richard Marsh for helpful discussion that helped improve the paper, as well as Erika Tavares for proofreading, and Maria Contreras and Mary Kate O'Donnell for assistance with data collection.

Author contributions

Conceptualization: J.C.P., T.J.R.; Methodology: J.C.P., T.J.R.; Formal analysis: J.C.P.; Investigation: J.C.P., T.J.R.; Resources: T.J.R.; Data curation: J.C.P.; Writing - original draft: J.C.P.; Writing - review & editing: J.C.P., T.J.R.; Visualization: J.C.P.; Supervision: T.J.R.; Funding acquisition: T.J.R.

Funding

This project was supported by National Science Foundation grant 1832795 and National Institutes of Health grant AR55295. Deposited in PMC for release after 12 months.

Data availability

The data used in this study are available from the Dryad Digital Repository: https://doi.org/10.5061/dryad.3tx95x6nh.

Alexander
,
R. M. N.
and
Vernon
,
A.
(
1975
).
The mechanics of hopping by kangaroos (Macropodidae)
.
J. Zool.
177
,
265
-
303
.
Arellano
,
C. J.
,
Konow
,
N.
,
Gidmark
,
N. J.
and
Roberts
,
T. J.
(
2019
).
Evidence of a tunable biological spring: elastic energy storage in aponeuroses varies with transverse strain in vivo
.
Proc. R. Soc. B Biol. Sci.
286
,
20182764
.
Askew
,
G. N.
and
Marsh
,
R. L.
(
1997
).
The effects of length trajectory on the mechanical power output of mouse skeletal muscles
.
J. Exp. Biol.
200
,
3119
-
3131
.
Astley
,
H. C.
and
Roberts
,
T. J.
(
2012
).
Evidence for a vertebrate catapult: elastic energy storage in the plantaris tendon during frog jumping
.
Biol. Lett.
8
,
386
-
389
.
Astley
,
H. C.
and
Roberts
,
T. J.
(
2014
).
The mechanics of elastic loading and recoil in anuran jumping
.
J. Exp. Biol.
217
,
4372
-
4378
.
Astley
,
H. C.
,
Abbott
,
E. M.
,
Azizi
,
E.
,
Marsh
,
R. L.
and
Roberts
,
T. J.
(
2013
).
Chasing maximal performance: a cautionary tale from the celebrated jumping frogs of Calaveras County
.
J. Exp. Biol.
216
,
3947
-
3953
.
Azizi
,
E.
and
Roberts
,
T. J.
(
2010
).
Muscle performance during frog jumping: influence of elasticity on muscle operating lengths
.
Proc. R. Soc. B Biol. Sci.
277
,
1523
-
1530
.
Azizi
,
E.
,
Halenda
,
G. M.
and
Roberts
,
T. J.
(
2009
).
Mechanical properties of the gastrocnemius aponeurosis in wild turkeys
.
Integr. Comp. Biol.
49
,
51
-
58
.
Bennet Clark
,
H. C.
(
1975
).
The energetics of the jump of the locust Schistocerca gregaria
.
J. Exp. Biol.
63
,
53
-
83
.
Bennet-Clark
,
H. C.
and
Lucey
,
E. C.
(
1967
).
The jump of the flea: a study of the energetics and a model of the mechanism
.
J. Exp. Biol.
47
,
59
-
67
.
Bossuyt
,
F. M.
,
Abramovic
,
S.
,
Leonard
,
T.
,
Sawatsky
,
A.
,
Smith
,
C. R.
,
Taylor
,
W. R.
,
Michael Scott
,
W.
and
Herzog
,
W.
(
2023
).
The non-intuitive, in vivo behavior of aponeuroses in a unipennate muscle
.
J. Biomech.
147
,
111430
.
Brooks
,
S. V.
,
Faulkner
,
J. A.
and
McCubbrey
,
D. A.
(
1990
).
Power outputs of slow and fast skeletal muscles of mice
.
J. Appl. Physiol.
68
,
1282
-
1285
.
Burrows
,
M.
(
1969
).
The mechanics and neural control of the prey capture strike in the mantid shrimps Squilla and Hemisquilla
.
Z. Vgl. Physiol.
62
,
361
-
381
.
Cavagna
,
G. A.
,
Mazzanti
,
M.
,
Heglund
,
N. C.
and
Citterio
,
G.
(
1985
).
Storage and release of mechanical energy by active muscle: a non-elastic mechanism?
J. Exp. Biol.
115
,
79
-
87
.
Dutta
,
S.
,
Tsiros
,
C.
,
Sundar
,
S. L.
,
Athar
,
H.
,
Moore
,
J.
,
Nelson
,
B.
,
Gage
,
M. J.
and
Nishikawa
,
K.
(
2018
).
Calcium increases titin N2A binding to F-actin and regulated thin filaments
.
Sci. Rep.
8
,
14575
.
Eng
,
C. M.
and
Roberts
,
T. J.
(
2018
).
Aponeurosis influences the relationship between muscle gearing and force
.
J. Appl. Physiol.
125
,
513
-
519
.
Eng
,
C. M.
,
Azizi
,
E.
and
Roberts
,
T. J.
(
2018
).
Structural determinants of muscle gearing during dynamic contractions
.
Integr. Comp. Biol.
58
,
207
-
218
.
Ettema
,
G. J. C.
,
Huijing
,
P. A.
,
Van Ingen Schenau
,
G. J.
and
De Haan
,
A.
(
1990
).
Effects of prestretch at the onset of stimulation on mechanical work output of rat medical gastrocnemius muscle–tendon complex
.
J. Exp. Biol.
152
,
333
-
351
.
Ford
,
L. E.
,
Huxley
,
A. F.
and
Simmons
,
R. M.
(
1977
).
Tension responses to sudden length change in stimulated frog muscle fibres near slack length
.
J. Physiol.
269
,
441
-
515
.
Fukutani
,
A.
and
Herzog
,
W.
(
2020
).
Differences in stretch–shortening cycle and residual force enhancement between muscles
.
J. Biomech.
112
,
110040
.
Fukutani
,
A.
,
Isaka
,
T.
and
Herzog
,
W.
(
2021
).
Evidence for muscle cell-based mechanisms of enhanced performance in stretch-shortening cycle in skeletal muscle
.
Front. Physiol.
11
,
609553
.
Fusi
,
L.
,
Reconditi
,
M.
,
Linari
,
M.
,
Brunello
,
E.
,
Elangovan
,
R.
,
Lombardi
,
V.
and
Piazzesi
,
G.
(
2010
).
The mechanism of the resistance to stretch of isometrically contracting single muscle fibres
.
J. Physiol.
588
,
495
-
510
.
Gans
,
C.
(
1982
).
Fiber architecture and muscle function
.
Exerc. Sport Sci. Rev.
10
,
160
-
207
.
Granzier
,
H. L.
(
2010
).
Activation and stretch-induced passive force enhancement: are you pulling my chain? Focus on ‘Regulation of muscle force in the absence of actin-myosin-based cross-bridge interaction
’.
Am. J. Physiol. Cell Physiol.
299
,
C11
-
C13
.
Hedrick
,
T. L.
,
Usherwood
,
J. R.
and
Biewener
,
A. A.
(
2004
).
Wing inertia and whole-body acceleration: an analysis of instantaneous aerodynamic force production in cockatiels (Nymphicus hollandicus) flying across a range of speeds
.
J. Exp. Biol.
207
,
1689
-
1702
.
Henderson
,
C. A.
,
Gomez
,
C. G.
,
Novak
,
S. M.
,
Mi-Mi
,
L.
and
Gregorio
,
C. C.
(
2018
).
Overview of the muscle cytoskeleton
.
Compr. Physiol.
7
,
891
-
944
.
Herzog
,
W.
(
2018
).
The multiple roles of titin in muscle contraction and force production
.
Biophys. Rev.
10
,
1187
-
1199
.
Herzog
,
W.
,
Lee
,
E. J.
and
Rassier
,
D. E.
(
2006
).
Residual force enhancement in skeletal muscle
.
J. Physiol.
574
,
635
-
642
.
Hessel
,
A. L.
,
Lindstedt
,
S. L.
and
Nishikawa
,
K. C.
(
2017
).
Physiological mechanisms of eccentric contraction and its applications: a role for the giant titin protein
.
Front. Physiol.
8
,
70
.
Higuchi
,
H.
,
Yanagida
,
T.
and
Goldman
,
Y. E.
(
1995
).
Compliance of thin filaments in skinned fibers of rabbit skeletal muscle
.
Biophys. J.
69
,
1000
-
1010
.
Huijing
,
P. A.
,
Baan
,
G. C.
and
Rebel
,
G. T.
(
1998
).
Non-myotendinous force transmission in rat extensor digitorum longus muscle
.
J. Exp. Biol.
201
,
682
-
691
.
Huxley
,
A. F.
and
Simmons
,
R. M.
(
1971
).
Proposed mechanism of force generation in striated muscle
.
Nature
233
,
533
-
538
.
Huxley
,
H. E.
,
Stewart
,
A.
,
Sosa
,
H.
and
Irving
,
T.
(
1994
).
X-ray diffraction measurements of the extensibility of actin and myosin filaments in contracting muscle
.
Biophys. J.
67
,
2411
-
2421
.
Iwamoto
,
H.
,
Sugaya
,
R.
and
Sugi
,
H.
(
1990
).
Force–velocity relation of frog skeletal muscle fibres shortening under continuously changing load
.
J. Physiol.
422
,
185
-
202
.
Jenkins
,
F. A.
,
Dial
,
K. P.
and
Goslow
,
G. E.
(
1988
).
A cineradiographic analysis of bird flight: the wishbone in starlings is a spring
.
Science
241
,
1495
-
1498
.
Kawakami
,
Y.
and
Lieber
,
R. L.
(
2000
).
Interaction between series compliance and sarcomere kinetics determines internal sarcomere shortening during fixed-end contraction
.
J. Biomech.
33
,
1249
-
1255
.
Kellermayer
,
M. S. Z.
,
Smith
,
S. B.
,
Granzier
,
H. L.
and
Bustamante
,
C.
(
1997
).
Folding-unfolding transitions in single titin molecules characterized with laser tweezers
.
Science
276
,
1112
-
1116
.
Knörlein
,
B. J.
,
Baier
,
D. B.
,
Gatesy
,
S. M.
,
Laurence-Chasen
,
J. D.
and
Brainerd
,
E. L.
(
2016
).
Validation of XMALab software for marker-based XROMM
.
J. Exp. Biol.
219
,
3701
-
3711
.
Lappin
,
A. K.
,
Monroy
,
J. A.
,
Pilarski
,
J. Q.
,
Zepnewski
,
E. D.
,
Pierotti
,
D. J.
and
Nishikawa
,
K. C.
(
2006
).
Storage and recovery of elastic potential energy powers ballistic prey capture in toads
.
J. Exp. Biol.
209
,
2535
-
2553
.
Lazarides
,
E.
(
1980
).
Intermediate filaments as mechanical integrators of cellular space
.
Nature
283
,
249
-
255
.
Lichtwark
,
G. A.
,
Bougoulias
,
K.
and
Wilson
,
A. M.
(
2007
).
Muscle fascicle and series elastic element length changes along the length of the human gastrocnemius during walking and running
.
J. Biomech.
40
,
157
-
164
.
Linari
,
M.
,
Woledge
,
R. C.
and
Curtin
,
N. A.
(
2003
).
Energy storage during stretch of active single fibres from frog skeletal muscle
.
J. Physiol.
548
,
461
-
474
.
Linke
,
W. A.
(
2000
).
Titin elasticity in the context of the sarcomere: force and extensibility measurements on single myofibrils
.
Adv. Exp. Med. Biol.
481
,
179
-
206
.
Longo
,
S. J.
,
Cox
,
S. M.
,
Azizi
,
E.
,
Ilton
,
M.
,
Olberding
,
J. P.
,
St Pierre
,
R.
and
Patek
,
S. N.
(
2019
).
Beyond power amplification: latch-mediated spring actuation is an emerging framework for the study of diverse elastic systems
.
J. Exp. Biol.
222
,
1
-
10
.
Lou
,
F.
,
Curtin
,
N. A.
and
Woledge
,
R. C.
(
1999
).
Elastic energy storage and release in white muscle from dogfish Scyliorhinus canicula
.
J. Exp. Biol.
202
,
135
-
142
.
Lutz
,
G. J.
and
Rome
,
L. C.
(
1994
).
Jumping: design frog system
.
Science
263
,
370
-
372
.
Marsh
,
R. L.
(
2022
).
Muscle preactivation and the limits of muscle power output during jumping in the Cuban tree frog Osteopilus septentrionalis
.
J. Exp. Biol.
225
,
jeb244525
.
Marsh
,
B. Y. R. L.
and
Bennett
,
A. F.
(
1986
).
Thermal dependence of contractile properties of skeletal muscle from the lizard Sceloporus occidentalis with comments on methods for fitting and comparing force–velocity curves
.
J. Exp. Biol.
126
,
63
-
77
.
Marsh
,
R. L.
and
John-Alder
,
H. B.
(
1994
).
Jumping performance of hylid frogs measured with high-speed cine film
.
J. Exp. Biol.
188
,
131
-
141
.
Minozzo
,
F. C.
and
De Lira
,
C. A. B.
(
2013
).
Muscle residual force enhancement: a brief review
.
Clinics
68
,
269
-
274
.
Monroy
,
J. A.
,
Powers
,
K. L.
,
Pace
,
C. M.
,
Uyeno
,
T.
and
Nishikawa
,
K. C.
(
2017
).
Effects of activation on the elastic properties of intact soleus muscles with a deletion in titin
.
J. Exp. Biol.
220
,
828
-
836
.
Nelson
,
F. E.
,
Gabaldón
,
A. M.
and
Roberts
,
T. J.
(
2004
).
Force–velocity properties of two avian hindlimb muscles
.
Comp. Biochem. Physiol. A Mol. Integr. Physiol.
137
,
711
-
721
.
Nishikawa
,
K. C.
,
Monroy
,
J. A.
,
Uyeno
,
T. E.
,
Yeo
,
S. H.
,
Pai
,
D. K.
and
Lindstedt
,
S. L.
(
2012
).
Is titin a ‘winding filament’? A new twist on muscle contraction
.
Proc. R. Soc. B Biol. Sci.
279
,
981
-
990
.
Patek
,
S. N.
and
Caldwell
,
R. L.
(
2005
).
Extreme impact and cavitation forces of a biological hammer: strike forces of the peacock mantis shrimp Odontodactylus scyllarus
.
J. Exp. Biol.
208
,
3655
-
3664
.
Patek
,
S. N.
,
Korff
,
W. L.
and
Caldwell
,
R. L.
(
2004
).
Deadly strike mechsanism of a mantis shrimp
.
Nature
428
,
819
-
820
.
Peplowski
,
M. M.
and
Marsh
,
R. L.
(
1997
).
Work and power output in the hindlimb muscles of Cuban tree frogs Osteopilus septentrionalis during jumping
.
J. Exp. Biol.
200
,
2861
-
2870
.
Powell
,
P. L.
,
Roy
,
R. R.
,
Kanim
,
P.
,
Bello
,
M. A.
and
Edgerton
,
V. R.
(
1984
).
Predictability of skeletal muscle tension from architectural determinations in guinea pig hindlimbs
.
J. Appl. Physiol. Respir. Environ. Exerc. Physiol.
57
,
1715
-
1721
.
Powers
,
K.
,
Schappacher-Tilp
,
G.
,
Jinha
,
A.
,
Leonard
,
T.
,
Nishikawa
,
K.
and
Herzog
,
W.
(
2014
).
Titin force is enhanced in actively stretched skeletal muscle
.
J. Exp. Biol.
217
,
3629
-
3636
.
Prado
,
L. G.
,
Makarenko
,
I.
,
Andresen
,
C.
,
Krüger
,
M.
,
Opitz
,
C. A.
and
Linke
,
W. A.
(
2005
).
Isoform diversity of giant proteins in relation to passive and active contractile properties of rabbit skeletal muscles
.
J. Gen. Physiol.
126
,
461
-
480
.
Reinsch
,
C. H.
(
1967
).
Smoothing by spline functions
.
Numer. Math.
10
,
177
-
183
.
Roberts
,
T. J.
and
Azizi
,
E.
(
2011
).
Flexible mechanisms: the diverse roles of biological springs in vertebrate movement
.
J. Exp. Biol.
214
,
353
-
361
.
Roberts
,
T. J.
and
Marsh
,
R. L.
(
2003
).
Probing the limits to muscle-powered accelerations: lessons from jumping bullfrogs
.
J. Exp. Biol.
206
,
2567
-
2580
.
Roberts
,
T. J.
,
Marsh
,
R. L.
,
Weyand
,
P. G.
and
Taylor
,
C. R.
(
1997
).
Muscular force in running turkeys: the economy of minimizing work
.
Science
275
,
1113
-
1115
.
Roberts
,
T. J.
,
Eng
,
C. M.
,
Sleboda
,
D. A.
,
Holt
,
N. C.
,
Brainerd
,
E. L.
,
Stover
,
K. K.
,
Marsh
,
R. L.
and
Azizi
,
E.
(
2019
).
The multi-scale, three-dimensional nature of skeletal muscle contraction
.
Physiology
34
,
402
-
408
.
Sawicki
,
G. S.
,
Robertson
,
B. D.
,
Azizi
,
E.
and
Roberts
,
T. J.
(
2015a
).
Timing matters: tuning the mechanics of a muscle-tendon unit by adjusting stimulation phase during cyclic contractions
.
J. Exp. Biol.
218
,
3150
-
3159
.
Sawicki
,
G. S.
,
Sheppard
,
P.
and
Roberts
,
T. J.
(
2015b
).
Power amplification in an isolated muscle–tendon unit is load dependent
.
J. Exp. Biol.
218
,
3700
-
3709
.
Scott
,
S. H.
and
Loeb
,
G. E.
(
1995
).
Mechanical properties of aponeurosis and tendon of the cat soleus muscle during whole-muscle isometric contractions
.
J. Morphol.
224
,
73
-
86
.
Sleboda
,
D. A.
and
Roberts
,
T. J.
(
2017
).
Incompressible fluid plays a mechanical role in the development of passive muscle tension
.
Biol. Lett
.
13
,
20160630
.
Sleboda
,
D. A.
and
Roberts
,
T. J.
(
2019
).
Internal fluid pressure influences muscle contractile force
.
Proc. Natl. Acad. Sci. USA
117
,
1772
-
1778
.
Tomalka
,
A.
,
Weidner
,
S.
,
Hahn
,
D.
,
Seiberl
,
W.
and
Siebert
,
T.
(
2021
).
Power amplification increases with contraction velocity during stretch–shortening cycles of skinned muscle fibers
.
Front. Physiol.
12
,
644981
.
Wakabayashi
,
K.
,
Sugimoto
,
Y.
,
Tanaka
,
H.
,
Ueno
,
Y.
,
Takezawa
,
Y.
and
Amemiya
,
Y.
(
1994
).
X-ray diffraction evidence for the extensibility of actin and myosin filaments during muscle contraction
.
Biophys. J.
67
,
2422
-
2435
.
Wang
,
K.
,
Mccarter
,
R.
,
Wright
,
J.
,
Beverly
,
J.
and
Ramirez-Mitchell
,
R.
(
1993
).
Viscoelasticity of the sarcomere matrix of skeletal muscles. The titin–myosin composite filament is a dual-stage molecular spring
.
Biophys. J.
64
,
1161
-
1177
.
Zack
,
T. I.
,
Claverie
,
T.
and
Patek
,
S. N.
(
2009
).
Elastic energy storage in the mantis shrimp's fast predatory strike
.
J. Exp. Biol.
212
,
4002
-
4009
.

Competing interests

The authors declare no competing or financial interests.