The effects of temperature on elastic energy storage and release in a system with a dynamic mechanical advantage latch

Many physiological processes are sensitive to changes in temperature (Bennett, 1984, 1985, 1990; Angilletta et al., 2002; James, 2013). This can pose a challenge to ectothermic organisms that rely on environmental conditions to regulate body temperature and physiological rates. In ectotherms, locomotor modes such as sprint speed, jump distance and swim speed have been shown to be sensitive to changes in temperature (Bennett, 1990; Hertz et al., 1982; Herrel et al., 2007; Navas, 1996; Peplowski and Marsh, 1997; Navas et al., 1999). These temperature-dependent changes in locomotor performance are linked to the strong thermal sensitivity of muscle contractile kinetics (e.g. shortening velocity, rate of force development, and power), and can have consequences for organism survival (Hertz et al., 1982; Putnam and Bennett, 1982; Bennett, 1984; Lutz and Rome, 1996; Peplowski and Marsh, 1997; James, 2013).

Temperature coefficients (Q₁₀) are often used to indicate the rate of change of a variable when temperature is increased by 10°C. A Q₁₀ equal to 1.0 indicates that the variable measured does not change much over a 10°C temperature range. A Q₁₀ equal to 2.0 indicates that the variable measured increases by 2-fold over the range of temperatures measured. A Q₁₀ equal to 0.5 indicates that the variable measured is halved over the range of temperatures measured (Bennett, 1984; Lutz and Rome, 1996; Peplowski and Marsh, 1997; James, 2013). Previous studies have used Q₁₀ to understand how the thermal sensitivity of skeletal muscles informs the thermal sensitivity of locomotor performance (Else and Bennett, 1987; Swoap et al., 1993; Johnson et al., 1996; Peplowski and Marsh, 1997; Navas et al., 1999; Olberding and Deban, 2021).

Some ectothermic organisms use elastic recoil mechanisms to perform high-powered movements that are more robust to changes in temperature than what would be expected based on the Q₁₀ of skeletal muscle (James, 2013; Olberding and Deban, 2021). Muscle contractile velocity, rate of force development or yank (Lin et al., 2019), and power show Q₁₀ values that range between 1.5 and 3.0 (Putnam and Bennett, 1982; Bennett, 1984; Rome, 1990; Peplowski and Marsh, 1997) indicating that with a 10°C change in temperature there is a 2- to 3-fold increase in rate. In contrast, peak velocity, acceleration and power of elastically actuated tongue projection in chameleons shows Q₁₀ values that range from 1.1 to 1.3 across 15–25°C, indicating that tongue projection is relatively insensitive to the change in temperature (Anderson and Deban, 2010). Thermal robustness through elastic recoil has also been observed in salamander tongue projection (Deban and Richardson, 2011; Scales et al., 2017; Deban et al., 2020) and in anuran tongue projection and ballistic mouth opening (Deban and Lappin, 2011; Sandusky and Deban, 2012), suggesting that elastic recoil mechanisms underlie thermally robust behaviors in some ectothermic organisms.

Elastic recoil mechanisms drive some of the fastest and most powerful movements ever recorded in biology (Ilton et al., 2018; Longo et al., 2019; Longo et al., 2021). In animals, these extremely fast and powerful behaviors are the result of finely tuned interactions between muscle, spring (e.g. tendon, aponeurosis, apodeme), latch and projectile (Ilton et al., 2018; Longo et al., 2019). In these systems, the muscle functions as the source of mechanical energy needed to acuate the system, the spring functions to temporarily store said energy, whereas the latch mediates the ability of the system to store and subsequently release mechanical energy loaded into the spring, often at a significantly higher rate than would be possible with direct muscle actuation. While the muscle and spring function are familiar and the structures involved well known, the latch can take many diverse forms. Here, we define a latch as any mechanism (sometimes referred to as a ‘catch’ in the literature) that provides a temporary force that resists motion in the ‘latched’ state while being able to be quickly removed or overcome to mediate the release of energy from a spring (Longo et al., 2019; Divi et al., 2020). Latches may be physical or anatomical structures where contact surfaces resists motion (Larabee et al., 2018). Latches may also result from geometric changes in a system where shifts in geometry can rapidly mediate a shift from storing energy in a spring to releasing that energy to actuate a projectile (Roberts and Marsh, 2003; Steinhardt et al., 2021).

Recent work examining the role of the latch in mediating energy flow has shown that the latch substantially influences elastic recoil performance (Ilton et al., 2018; Abbott et al., 2019; Divi et al., 2020). Previous modeling studies have shown that the amount of energy returned decreased with relatively slower latch removal velocities, indicating that the elastic recoil mechanism is sensitive to latch behavior (Ilton et al., 2018; Abbott et al., 2019; Divi et al., 2020; Acharya et al., 2021). However, it is not yet understood whether external factors such as temperature can affect latch performance and how changes to latch dynamics may allow for the contribution of direct muscle power to actuate the system.

Frog jumps are one of the most well-studied examples of elastically actuated movements (Marsh, 1994; Peplowski and Marsh, 1997; Roberts and Marsh, 2003; Azizi and Roberts, 2010, 2012; Astley et al., 2013; Astley and Roberts, 2014; Astley, 2016; Mendoza et al., 2020). Previous studies have shown a temporal decoupling of muscle contraction from joint movement as evidence of elastic energy storage at the ankle joint (Roberts and Marsh, 2003; Azizi and Roberts, 2010; Astley and Roberts, 2012). The elastic recoil mechanism in frog jumps is mediated by a dynamic mechanical advantage latch (a type of geometric latch), where the poor mechanical advantage of the hindlimb extensor muscles and the body's inertia resist hindlimb joint extension early in the jump and allow for energy storage. Once the hindlimb extensor muscles build sufficient force to overcome the latch, the hindlimb joints begin to extend, and the mechanical advantage of the extensor muscles improves rapidly to release energy (Roberts and Marsh, 2003; Olberding et al., 2019; Astley and Roberts, 2014). Results from previous studies suggest that the muscle may continue to shorten as the tendon recoils, providing some evidence that this latching mechanism allows for a combination of elastic and direct muscle energy to contribute to jumps (Roberts and Marsh, 2003; Azizi and Roberts, 2010; Astley and Roberts, 2012; Sutton et al., 2019).

Despite the use of elastic mechanisms, frog jumping performance remains temperature sensitive (Hirano and Rome, 1984; John-Alder et al., 1988; Whitehead et al., 1989; Rome, 1990; Marsh, 1994; Peplowski and Marsh, 1997; Navas et al., 1999; Olberding and Deban, 2021). For example, Hirano and Rome (1984) showed that jump power operated with Q₁₀ of 2.67, jump velocity with a Q₁₀ of 3.33, and jump distance with a Q₁₀ of 1.58 between 15 and 25°C in the leopard frog (Rana pipiens). In this study, we investigated how the latching mechanics of the frog jump mechanism affect their ability to store and release energy across temperature. Here, we aimed to reveal the mechanism that causes frog jumps to be temperature dependent and to determine whether the observed thermal sensitivity is due to differences in energy storage or release. We used an in vitro muscle preparation coupled with an in silico model of a jumper because it enabled us to control muscle performance through direct stimulation, while allowing our muscles to interact with realistic movement dynamics. We hypothesized that the amount of energy stored (i.e. work) would not change with temperature as slowing muscle contractions would only increase the duration and not the amount of elastic energy storage. Additionally, muscle contractile rates are sensitive to temperature; thus, we hypothesized that the amount of energy released (i.e. work) would be temperature dependent because of the additional work that may be contributed by extensor muscles during elastic recoil and joint extension (Bennett, 1984, 1985; Rall and Woledge, 1990; Roberts and Marsh, 2003; Azizi and Roberts, 2010; Astley and Roberts, 2012).

Six similarly sized (mean±s.e.m. mass 100.25±4.23 g) bullfrogs, Rana catesbeiana (Shaw 1802), were purchased from a herpetological vendor (Rana Ranch, Twin Falls, ID, USA). The animals were group housed in glass terraria, maintained at 20–21°C and were fed calcium-enriched crickets ad libitum. Animal husbandry and use were approved by the University of California, Irvine, Animal Care and Use Committee (protocol AUP 20-129).

In this study, we followed methods outlined by Azizi and Roberts (2014) and Mendoza and Azizi (2021). Briefly, frogs were euthanized with a double-pithing protocol. Once death was confirmed, we measured the length of the tibiofibula segment with digital calipers. We isolated the sciatic nerve branch running along the right femur. Then, we exposed the plantaris longus muscle and implanted a sonomicrometry crystal between two muscle fascicles near the muscle origin. A second crystal was implanted ∼8 mm distal to the first and both were secured with 6-0 silk. After instrumentation, we isolated the muscle preparation from the body by detaching the muscle's distal tendon from the plantar fascia and isolating the knee joint (where the muscle originates). The instrumented muscle was secured to a fixed clamp at the knee joint and the distal tendon was threaded through a custom-made clamp. The clamp on the distal tendon was attached to a 50 N servomotor (Aurora Scientific Inc., Ontario, CA, USA). The sciatic nerve was threaded through a custom-made nerve cuff that was connected to a Grass S99D stimulator (Grass Technologies, Warwick, RI, USA), and was used to electrically stimulate the muscle. Finally, the muscle preparation was placed in a bath of circulating anuran Ringer's solution maintained at room temperature (e.g. 20°C) with a temperature controller circulator. The bath was continuously aerated with oxygen.

For each muscle, we determined optimal stimulation voltage by increasing the voltage of twitch contractions by 1 V increments until force stopped increasing with increasing voltage (9–11 V). Next, we determined optimal muscle length (L_o) by characterizing the force–length curve using tetanic fixed-end contractions at variable lengths. L_o was defined as the length at which the muscle produced the highest peak force (P_o). Muscle fascicle length changes were measured with sonomicrometry, and muscle force and muscle–tendon unit (MTU) length were measured with the servomotor. Tetanic stimulation consisted of 0.2 ms pulses at 65 pulses s⁻¹ and durations of 500 ms. Once P_o was determined, muscles were set to the initial length that resulted in the highest force during our isometric tests for all experimental contractions. In most preparations, this value corresponded to an approximate initial length of ∼1.25 L_o.

We calculated in-lever length, L_in, based on the mechanical advantage, MA, required to overcome the gravitational load of the mass and inertia, given a peak force of 0.6P_o and a starting joint angle of 10 deg (see Table 1). The force level of 0.6P_o was selected based on previous muscle fascicle strain data during in vivo jumps showing that muscle fascicles shorten by about 10–15% prior to the movement of the center of mass. This value also allowed the muscles to reach this force level regardless of the slight effects of temperature on maximum isometric force (Olberding and Deban, 2017). We acknowledge that Olberding et al.’s (2019) model contains a calcaneus, and that frogs do not possess this anatomical structure. Again, the goal of the model was not to mimic a frog's anatomy or behavior, but rather to provide us with a controlled framework to compare patterns of energy flow through a system with a MA latch across a range of temperatures.

After characterizing the force–length relationship, we ran the in vitro/in silico experiments across four temperature treatments: 10, 15, 20 and 25°C. In all experiments, the first set of tests were performed at 20°C followed by either sequence 10–15–25–20°C or 25–15–10–20°C. We ran the temperature experiments in this fashion to minimize the potential confounding interactions between muscle fatigue and temperature. The final tests at 20°C were compared with the initial tests at 20°C to make sure that the muscle had not dropped in its force capacity throughout the experiment. The difference in force during the initial and final tests at 20°C was used to examine the effect of treatment order on the muscle's overall performance. Treatment order did not have a significant effect on the trajectory of muscle force throughout the experiments (Student's t-test, P=0.31).

The temperature of the circulating Ringer's solution was manipulated with a temperature controller, and we monitored the solution's temperature with a temperature probe. Once the Ringer's solution reached treatment temperature, we waited 20 min to allow the muscle to reach the experimental temperature (Olberding and Deban, 2017). Then, the muscle was lengthened to L_o, the model and the reactive feedback control loop were initiated, and the muscle was stimulated tetanically as described above. The details of the real-time feedback controller were previously described by Reynaga et al. (2019). Briefly, the servomotor registered the force generated by the muscle and broadcasted the analog signals in the ±10 V range. A peripheral 16-bit sampling analog-to-digital unit on the custom printed circuit board converted this signal to a digital force measurement, which was then passed via the serial peripheral interface bus to the BeagleBone, where it was smoothed with a software-implemented low-pass filter (Reynaga et al., 2019). The smoothed force and starting MA were used to calculate the displacement of the projectile's center of mass. The digital displacement value was then passed from the BeagleBone to a peripheral 16-bit multiplying digital-to-analog converter chip, where it was converted to an analog control signal. The analog control signal was sent to the servomotor, which controlled the position of the muscle lever (MTU length; Fig. 1A). This feedback loop continued until the muscle reached the force threshold. Once the muscle generated sufficient force to overcome the force threshold, the modeled joint began to extend and accelerate the mass. Experiments ended when the joint angle reached 180 deg or when force was equal to zero.

After experimentation, the muscle was detached from its origin at the knee joint and the distal free tendon was removed. Muscle mass, fascicle length and pennation angle were measured and used to calculate muscle stress using a known muscle density of 1.06 g cm⁻² (Mendez and Keys, 1960). Muscle stress was calculated to ensure that the muscle preparations were of good quality (i.e. within physiological range, ∼20 N cm⁻²; Table 1) (Roberts et al., 2011; Mendoza and Azizi, 2021).

All data were processed in Igor Pro software (Wavemetrics, Lake Oswego, OR, USA). To interrogate the role of a dynamic MA latch in mediating energy flow, we partitioned our data into three phases (Fig. 1B). The first loading phase (L1) was defined as the time from the beginning of muscle stimulation until a threshold force (0.6P_o) was reached. During this phase, the latch was engaged, and the force required to accelerate the projectile had not been reached. The second loading phase (L2) was defined as the time from the muscle reaching the threshold force until the muscle reached its peak force. During this phase, the inertia of the projectile delays unlatching, providing additional time for energy to be loaded into elastic elements. Given that the model limb was not extending, the MTU was maintained at a constant length so any shortening of the muscle fascicles would result in an equal stretch of the elastic elements (Fig. 2A). Finally, the unloading phase (U3) was defined as the time at which the muscle reached peak force until the model took-off from the ground (Fig. 1B). In the unloading phase, the latch was removed, and this allowed the elastic elements to recoil while the muscle fascicles continued to do work during joint extension, simultaneously.

The mechanical work (energy) associated with each of the three phases was calculated by plotting muscle fascicle length against muscle force during the loading phase and calculating the area under this curve (Fig. 3). We interrogated the unloading phase further by parsing out the relative contributions of tendon recoil work and direct muscle fascicle work (i.e. during joint extension). To do this, we measured total MTU work by plotting muscle force against MTU length during the unloading phase. The area under this curve was the total MTU work done during the unloading phase. To calculate tendon recoil work, we subtracted unloading muscle fascicle work from total MTU work. All measurements of work were converted to muscle mass-specific work by dividing by plantaris longus muscle mass.

We calculated tendon efficiency by taking the ratio of tendon recoil work to loading muscle fascicle work (L1 and L2). Additionally, we calculated the relative contributions of direct muscle fascicle work and tendon recoil work during joint extension, by taking the ratio of tendon recoil work to unloading muscle fascicle work. Lastly, we measured the duration of both the loading and unloading phase. We used this to calculate muscle fascicle power and tendon power.

(Deban and Lappin, 2011; Deban and Richardson, 2011; Anderson and Deban, 2012; Scales et al., 2017; Olberding and Deban, 2017; Olberding et al., 2018). We reported Q₁₀ of durations as inverse Q₁₀ values (e.g. 1/Q₁₀) to express them as rates (Deban and Lappin, 2011). The P-values for the regression coefficients for all tests were adjusted using the Benjamini–Hochberg procedure to control for false discovery (Benjamini and Hochberg, 1995). Temperature coefficients were significantly different from 1.0 if the P-value for the regression coefficients was less than the adjusted alpha.

As expected, an increase in temperature resulted in a decrease in the duration of all phases of the experiments (Fig. 2A–C). Total loading time (L1 and L2) was longer than unloading time (U3). The duration of the first phase of loading (L1) decreased with increasing temperature, with the temperature coefficients differing significantly from 1.00 (Fig. 2B). Specifically, 10–20°C had a 1/Q₁₀ of 1.31 (P=0.005), 15–25°C had a 1/Q₁₀ of 1.07 (P=0.012), and 10–25°C had a 1/Q₁₀ of 1.19 (P=0.006; Fig. 2B). The duration of the second phase of loading (L2) decreased with increasing temperature, with the temperature coefficients differing significantly from 1.00 for temperature ranges of 10–20°C and 10–25°C (Fig. 2C). Specifically, 10–20°C had a 1/Q₁₀ of 1.14 (P=0.005), 15–25°C had a 1/Q₁₀ of 1.02 (P=0.192), and 10–25°C had a 1/Q₁₀ of 1.08 (P=0.015; Fig. 2C). Unloading time (U3) did not have temperature coefficients that were significantly different from 1.00 across all three temperature ranges. Specifically, the temperature coefficient at 10–20°C was 1/Q₁₀=1.07 (P=0.035), 10–25°C was 1/Q₁₀=1.04 (P=0.037), and 15–25°C was 1/Q₁₀=1.00 (P=0.427; Fig. 2C).

We found that peak force increased with increasing temperature and peaked at 20°C (Fig. 2E). The Q₁₀ for peak force across 10–20°C and 10–25°C was 1.2 and 1.1, respectively, both significantly different from 1.0 (P=0.00 and P=0.005, respectively). The Q₁₀ for the temperature range 15–25°C was 1.00 (P=0.326; Fig. 2E).

Instantaneous force–length plots were used to quantify the flow of mechanical energy during in vitro/in silico contractions (Fig. 3). Muscle work generated during the first phase of loading (L1) did not vary with temperature (Fig. 3A,B and 4A). The Q₁₀ for L1 muscle work did not significantly differ from 1.0 and was 1.0 for 10–20°C (P=0.765), 1.0 for 10–25°C (P=0.919), and 1.1 for 15–25°C (P=0.302; Fig. 4A). However, muscle work generated during the second phase of loading (L2) did vary with temperature over the 10–20°C and 10–25°C temperature ranges (Figs 3A,B and 4B). The Q₁₀ for muscle work during L2 was 2.5 for 10–20°C (P=0.005), 1.6 for 10–25°C (P=0.014), and 1.4 for 15–25°C (P=0.296; Fig. 4B). Tendon recoil work increased with increasing temperature (Fig. 4C). The temperature range 10–20°C had a Q₁₀ of 1.5, which was significantly different from 1.0 (P=0.003). The temperature range 10–25°C and 15–25°C had Q₁₀ values equal to 1.2 (P=0.036) and 1.0 (P=0.316), respectively, which were not significantly different from 1.0 (Fig. 4C). Unloading muscle work increased significantly with increasing temperature (Figs 3B,C and 4D). The Q₁₀ for unloading muscle work across all temperature ranges significantly differed from 1.0 and was 2.2 for 10–20°C (P=0.0004), 1.7 for 10–25°C (P=0.003), and 1.6 for 15–25°C (P=0.052; Fig. 4C).

Muscle power during the first phase of loading (L1) increased with increasing temperature (Fig. 5A). The temperature coefficient was significantly different from 1.0 for the temperature range 10–20°C (Q₁₀=2.9, P=0.002), 10–25°C (Q₁₀=1.9, P=0.005; Fig. 5A), and 15–25°C (Q₁₀=1.6, P=0.022; Fig. 5A). Muscle power during the second phase of loading (L2) increased with increasing temperature (Fig. 5B). The temperature coefficient was significantly different from 1.0 for the temperature range 10–20°C (Q₁₀=3.8, P=0.001) and for the temperature range 10–25°C (Q₁₀=1.9, P=0.005; Fig. 5A). The temperature coefficient was not significantly different from 1.0 for the temperature range 15–25°C (Q₁₀=1.5, P=0.267; Fig. 5B). Tendon recoil power increased with increasing temperature (Fig. 5C). The temperature range 10–20°C had a Q₁₀ of 2.1 (P=0.007) and the temperature range 10–25°C had a Q₁₀ of 1.5 (P=0.016) and both were significantly different from 1.0 (Fig. 5C). The temperature range 15–25°C had a Q₁₀ of 1.0 (P=0.985) and was not significantly different from 1.0 (Fig. 5C). Unloading muscle power increased with increasing temperature (Fig. 5D). The Q₁₀ was significantly different from 1.0 for temperature range 10–20°C (Q₁₀=3.2, P=0.0004) and 10–25°C (Q₁₀=2.1, P=0.002; Fig. 5D). The Q₁₀ was not significantly different from 1.0 for the temperature range 15–25°C (Q₁₀=1.6, P=0.071; Fig. 5D).

The tendon recoil work to unloading muscle work ratio did not significantly change with increasing temperature (Fig. 6A). Q₁₀ values for the temperature ranges 10–20°C (Q₁₀=0.7; P=0.018), 10–25°C (Q₁₀=0.8; P=0.120) and 15–25°C (Q₁₀=0.7; P=0.217) were not significantly different from 1.0 (Fig. 6A). Efficiency did not change with temperature (Fig. 6B). Average efficiency was approximately (mean±s.e.m.) 44.69±4.47% across all temperature treatments. Q₁₀ values for the temperature ranges 10–20°C (Q₁₀=1.0; P=0.845), 10–25°C (Q₁₀=1.1; P=0.581) and 15–25°C (Q₁₀=0.9; P=0.739) were not significantly different from 1.0 (Fig. 6B).

Thermal robustness achieved through spring actuation has been demonstrated in some ectothermic organisms (Anderson and Deban, 2010; Deban and Lappin, 2011; Deban and Richardson, 2011; Deban et al., 2020; Olberding and Deban, 2021). However, it appears that the degree of insensitivity to changes in temperature varies across systems. Jumping in frogs has been shown to be relatively more sensitive to changes in temperature than other systems (e.g. chameleons) despite documented use of spring actuation (Hirano and Rome, 1984; Azizi and Roberts, 2010; Astley and Roberts, 2012). This suggests that the specific mechanics of latch-mediated spring actuation (LaMSA) are critical for producing movements that are thermally robust. In this study, we investigated how the latching mechanics mediated energy flow in a jumper with a dynamic MA latch. We hypothesized that loading muscle fascicle work would not differ across temperature treatment because unlatching would not occur until the muscle reached the force threshold and lowering the temperature would only increase the time required to reach such a threshold. Moreover, we predicted that unloading muscle fascicle work would differ across temperature treatment because of temperature effects on muscle contractile rates (Bennett, 1984, 1985; Rall and Woledge, 1990). We did in fact find that the mechanical work the muscle generated during the first phase of loading was not affected by temperature (Fig. 4A). However, during the second phase, the loading time available for additional energy storage was determined by the inertia of the mass (projectile) and warm muscles generated force more quickly and better utilized this phase (Fig. 4B). This finding is consistent with theoretical analyses of relating spring dynamics and inertial loads, and highlights the significance of the rate of force generation (via changes in temperature or fiber type recruitment) in maximizing the time available for energy storage (Galantis and Woledge, 2003; Richards and Sawicki, 2012). We found that tendon recoil work showed temperature dependence that reflected the work pattern observed during the loading phase (Fig. 4B). Furthermore, we found that unloading muscle fascicle work showed strong temperature dependence which supported the hypothesis (Fig. 4D). Our results suggest that a dynamic mechanical advantage latch cannot fully decouple muscle contraction from joint motion, allowing for temperature effects to affect motion. Movements that are actuated by a combination of elastic recoil and direct muscle actuation will not display the thermal robustness observed in other LaMSA systems (Anderson and Deban, 2010; Deban and Lappin, 2011; Deban and Richardson, 2011; Deban et al., 2020; Olberding and Deban, 2021). Together, our results indicate that temporal decoupling of muscle contraction from movement is critical for thermally robust movements (e.g. chameleon tongue projection; Wainwright and Bennett, 1992; Anderson and Deban, 2010).

Previous studies have shown that the plantaris longus muscle continues to shorten during limb extension in frog jumps (Roberts and Marsh, 2003; Azizi and Roberts, 2010; Astley and Roberts, 2012), yet the relative importance of this contribution remained unknown because in vivo measurements of muscle force are difficult to acquire in frogs (but see Richards and Biewener, 2007; Moo et al., 2017). In our study, we found that muscle fascicle work during the unloading phase showed strong thermal dependence across all temperature ranges (Fig. 4D). We found that unloading muscle fascicle work operated with Q₁₀ values of 2.2, 1.6 and 1.7 across the temperature ranges of 10–20°C, 15–25°C and 10–25°C, respectively. At the coldest treatment, the muscle was able to contribute on average about 3.7 J kg⁻¹ muscle mass, while at the warmest temperature, the muscle contributed on average approximately 7.8 J kg⁻¹ muscle mass (Fig. 4C). At warmer treatments, the amount of work contributed by the muscle fascicles during the unloading phase was on a par with that returned by tendon recoil (Figs 4C,D and 6A). Our results showed that the work contributions of muscle during the unloading phase are as important as those of tendon recoil and highlight the hybrid nature of this jump mechanism (Sutton et al., 2019; Olberding et al., 2019).

The unlatching mechanics affected the tendon recoil efficiency and introduced thermal sensitivity. In our experiments, we found that elastic recoil operated with an average efficiency of 44.69% across all temperatures (Fig. 6B). Efficiency is substantially lower than previous reports of approximately 90% efficiency in tendon when measured during relatively slow cyclical tensile conditions (e.g. Ker, 1981). This low efficiency is likely due to energy dissipation during unlatching. Abbott et al. (2019) modeled an elastic recoil system with an antagonist muscle as a latch and showed that unlatching velocity (i.e. muscle relaxation rate) was critical for determining whether power would be amplified or attenuated. They found that the fastest unlatching resulted in substantial power amplification and the slowest unlatching resulted in power attenuation. Furthermore, Divi et al. (2020) modeled latches with different latch removal velocities and showed that slower unlatching resulted in increased control of projectile launch at the cost of efficiency. While our study does not examine unlatching velocity, our estimates of efficiency suggest that there may be an emphasis on control of jump trajectory during actuation that may result in a tradeoff with efficiency. The studies mentioned above support our findings and collectively suggest that unlatching duration may serve as a form of control on output performance at the incurred cost of energy loss (Hyun et al., 2023). Furthermore, although integration of the two phases resulted in temperature-sensitive movements, it may be of importance for control. In the frog jumping mechanism, muscles loaded work into elastic elements while the latch was engaged and continued to do work during unlatching and limb extension. Continuous contribution of work by the muscle throughout the jump suggests that frogs may have the ability to control the performance and directionality of the jump after unlatching. In fact, the ability to impart neural control during take-off may be an important feature of systems using geometric latches (Sutton and Burrows, 2008) and may not be available to systems with faster latches (Sutton and Burrows, 2010). The ability to control a movement in real time after unlatching would likely not be possible in organisms where the muscle only contributes energy during energy storage, and where the latch temporally decouples energy storage from energy return (Roberts, 2019). Thus, latches like the anuran latch may result in reduced efficiency and robustness to environmental perturbations (e.g. temperature), but they afford the organism greater control of energy release and movement during the actuation phase.

Thermal robustness through elastic recoil is observed in chameleon and salamander tongue projection, and ballistic mouth opening in frogs and toads, but less so in frog or house cricket jumping (Hirano and Rome, 1984; Anderson and Deban, 2010; Deban and Richardson, 2011; Deban and Lappin, 2011; Scales et al., 2017; Olberding and Deban, 2021; Deban and Anderson, 2021). Deban and Anderson (2021) showed that jumping performance in house crickets was relatively more temperature sensitive than jumping in fleas and other insects despite use of elastic recoil mechanisms. The authors suggest that this could be due to additional muscle contributions during the takeoff phase, or dealing with high loads, which are known to result in temperature-dependent work outputs (Olberding and Deban, 2017). In our study, the required MA to overcome the latch was calculated with peak muscle force equal to 0.6P_o to reduce the load experienced by the muscle and fatigue; therefore, it is not likely that there was an interaction effect of high load and temperature. In our study, continuous contribution of work by the muscle fascicles and the unlatching mechanics resulted in the integration of the loading and unloading phases. This continuity between the two phases allowed for transmission of muscle thermal sensitivity into the energy stored and energy returned by elastic structures. The strongest temperature effects were observed during the unloading phase where the muscle contributed work during joint extension. Thus, while frog jump performance is relatively more insensitive to changes in temperature than underlying muscle contractile properties, frog jump performance is relatively more sensitive to changes in temperature than other thermally robust systems (e.g. chameleon tongue projection). Our work suggests that organisms that use elastic recoil mechanisms will perform temperature-independent movements only when the latch temporally decouples muscle contraction from energy release (i.e. no additional contribution by the muscle during actuation).

Among the many LaMSA systems studied to date, no system has shown a capacity to augment spring actuation with additional muscular work after the spring actuation has begun. While at first glance it would seem favorable for systems to augment mechanical energy output during the actuation phase by continuing to generate muscle work, this ‘hybrid’ actuation operates with some important constraints. It is likely that the duration of the take-off phase of a jumping frog is largely determined by the rate of energy release by tendons constraining the time available for muscle contributions. This would suggest that the power output of the muscle must remain high enough to contribute a substantial amount of mechanical work during a limited period of time. Therefore, it is reasonable to assume that some of the fastest LaMSA systems operating with a more idealized latch simply do not provide muscles enough time to contribute significantly once unlatching has occurred (Divi et al., 2020). This would suggest that in systems using hybrid actuation, maintenance of muscle power may be favored by natural selection whereas idealized systems may move to muscles specialized for high force production (Longo et al., 2019).

In this study, we examined the effects of temperature on elastic energy storage and return in a system with a dynamic mechanical advantage latch. We found that continuous muscle contributions and the unlatching mechanics in this system allowed for integration of energy storage and energy release that resulted in temperature dependence. While the results here demonstrate the role of unlatching mechanics in mediating energy flow, there are limitations to our approach. First, we used an isolated muscle preparation coupled to a model of a jumper (Olberding et al., 2019) to understand the effects of temperature on an elastic recoil system with a dynamic MA latch. We used limb morphology and muscle mass to scale the jumper model to each frog. Our approach assumes that the muscle physiology of the plantaris longus is representative of hip and other hindlimb extensor muscles. The plantaris longus is a biarticulate muscle that flexes the knee and primarily extends the ankle (Olson and Marsh, 1998). It has bipennate architecture and in bullfrogs it has a large aponeurosis sheet wrapping around the muscle belly, which is distinct from some of the hip and hindlimb muscles. Yet, previous studies examining the properties of the plantaris longus muscle suggest that its contractile behavior may be consistent with other hindlimb muscles involved in jumping. For example, Astley (2016) and Mendoza et al. (2020) showed that the underlying muscle properties of the plantaris longus and semimembranosus (parallel fibered hip extensor and knee flexor) are similar across several species of frogs from diverse microhabitats. Additionally, Deban and Lappin (2011) and Olberding et al. (2018) showed that the contractile properties of muscles used in elastic recoil mechanisms are consistent with those of typical skeletal muscle. Furthermore, in vivo studies in jumping bullfrogs showed that several muscles spanning the hip and hindlimb have similar activation patterns to the plantaris longus during jump takeoff, suggesting that they have similar functional roles during a jump (Olson and Marsh, 1998). Another limitation to our approach is that we simplified neural control to supramaximal stimulation, despite significant modulation of activation during locomotion (Gillis and Biewener, 2000). Future studies are necessary to understand whether the patterns observed here reflect in vivo patterns in frog jumps at variable temperatures. The patterns observed in our study are confounded by the limitations outlined above, yet they provide testable hypotheses for future studies examining the temperature effects on elastic energy storage and return in frog jumping in vivo.

Frog jumping performance is known to be relatively more sensitive to changes in temperature than other movements driven by elastic recoil. In this study, we investigated what aspects of the jump mechanism contributed to temperature sensitivity by examining the role of latching mechanics in mediating energy storage and release. We found that continuous muscle contributions and the mechanics of a dynamic mechanical advantage latch resulted in thermal sensitivity of energy storage and energy return. Furthermore, we found that hindlimb muscle plays a substantial role in actuating jumps in addition to the recoil of elastic structures. Finally, we propose that actuation through elastic recoil and direct muscle contributions results in some thermal sensitivity but allows for greater control and modulation of the jump in real time.

We would like to thank Dr Alexander Duman and Adrien Arias for support during experimentation. We thank Azizi lab members for help with animal husbandry and for providing feedback on earlier versions of the manuscript. We are grateful for numerous discussions about elastic systems with Greg Sutton, Sheila Patek, Mark Ilton and Suzanne Cox. Lastly, we thank Dr Natalie Holt and Dr Matt McHenry for constructive feedback during all stages of this project. All sections in this paper are reproduced from the PhD thesis of E.M. (Mendoza, 2023).