Probing the limits to muscle-powered accelerations: lessons from jumping bullfrogs
Biology Department, Northeastern University, 414 Mugar, 360 Huntington Ave, Boston, MA 02115, USA
* Author for correspondence at present address: Oregon State University, Zoology Department, 3029 Cordley Hall, Corvallis, OR 97331, USA (e-mail: robertst{at}bcc.orst.edu)
Accepted 10 April 2003
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Summary |
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These results demonstrate that an elastic element can increase the work output in a muscle-powered acceleration. Elastic elements uncouple muscle fiber shortening velocity from body movement to allow the muscle fibers to operate at slower shortening velocities and higher force outputs. A variable muscle mechanical advantage improves the effectiveness of elastic energy storage and recovery by providing an inertial catch mechanism. These results can explain the high power outputs observed in jumping frogs. More generally, our model suggests how the function of non-muscular elements of the musculoskeletal system enhances performance in muscle-powered accelerations.
Key words: locomotion, muscle work, muscle power, jumping, frog, Rana catesbeiana, elastic, tendon, acceleration
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Introduction |
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The supramaximal powers observed during jumping probably result from the
rapid release of strain energy from elastic elements (Marsh,
1994,
1999
). Elastic structures can
operate as muscle power amplifiers because they are not bound by the
constraints on shortening velocity that limit power output of muscle
contractile elements (Hill,
1950a
; Alexander,
1988
). Yet, how and when these elastic elements are stretched to
store the energy of muscular contraction in frog jumping is unclear.
Furthermore, although the hypothesis that elastic elements provide the high
power outputs of a jump is reasonable, this hypothesis has not been supported
by direct measurements of muscle function.
Nature's Olympian jumpers are insects that utilize a catapult-like
mechanism to amplify muscle power for jumping. Fleas, click beetles and
locusts contract their muscles to load elastic elements in their limbs prior
to initiating a jump (Bennet-Clark and
Lucey, 1967; Evans,
1972
; Bennet-Clark,
1975
). A physical or muscular catch mechanism provides the
resistance necessary to allow the preloading of elastic elements by muscular
contraction. The release of the catch mechanism triggers the explosive release
of elastic strain energy and spectacularly high jump power production. These
catapult mechanisms produce extraordinary jumping performance because they
solve the problem of the mismatch between muscle contractile behavior and the
behavior of an accelerating body. Muscles do the most work when they contract
slowly, due to the forcevelocity relationship, yet jumping involves a
very rapid movement. By separating in time the performance of muscular work
from the application of mechanical work to the body, the catapult mechanism
overcomes intrinsic constraints of skeletal muscle function
(Bennet-Clark and Lucey, 1967
;
Bennet-Clark, 1977
). Without a
catch mechanism, it is unclear how and when muscular energy might be loaded
into elastic elements and whether the temporal redistribution of muscle work
can lead to a performance advantage for jumping.
We undertook the present study to determine whether frogs could produce the
high power outputs observed during jumping in the absence of a physical catch
mechanism for elastic energy storage. Models of jumping have suggested a small
benefit of elastic storage and recovery in jumps without a catch mechanism
(Alexander, 1995;
Bobbert, 2001
), although the
power amplification is expected to be smaller. We hypothesized that the action
of elastic elements in jumping frogs allows muscles to operate, on average, at
slower shortening velocities and higher work outputs. To test this hypothesis,
we used a combined empirical and modeling approach. Our model of a
muscle-powered acceleration consisted of a muscle actuator with typical
contractile properties, operating to accelerate an inertial load. We modeled
single contractions with and without an in-series tendon and under conditions
of variable effective mechanical advantage (EMA) between the muscle and the
load. This simple model reproduced the complex mechanical behavior that
results from the interaction of muscletendon contractile properties and
the inertial behavior of an accelerated load. We predicted that the highest
accelerations would occur when the muscle operated in series with a compliant
tendon. Our empirical measurements consisted of synchronous high-speed video
and sonomicrometry measurements in jumping frogs. The pattern of force,
velocity and power applied to the body was calculated from the high-speed
video recordings and compared with the same parameters measured for the body
in the simulated acceleration. Sonomicrometry measurements in the plantaris
muscle, an ankle extensor with a large tendon, allowed us to compare the
shortening pattern of the frog muscle with the shortening pattern of the
muscle in the simulated acceleration. We predicted that the observed pattern
of movement of the body and muscle in the jumping frog would be most closely
matched by the configuration of the simulated acceleration that produced the
most mechanical work.
Our results bear directly on the more general problem of how muscles might function optimally to accelerate loads. Although acceleration is one of the most common functions of skeletal muscle, the pattern of muscle contraction that might be expected to provide maximal mechanical work output during acceleration is relatively unexplored. During acceleration, the load on the contracting muscle continuously changes because, as the mass accelerates, the force produced by the muscles changes due to forcevelocity and lengthtension effects. The changing muscle force in turn affects the acceleration of the load. Understanding the mechanisms for optimizing performance during accelerations requires an increased knowledge of the nature of the reciprocal interactions between the changing load, the properties of the muscle and the musculoskeletal structures that link the contracting muscle fibers to the load.
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Materials and methods |
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Sonomicrometry and electromyography
Sonomicrometer crystals and electromyographic (EMG) electrodes were
surgically implanted to measure length change and activity, respectively, in
the plantaris muscle of six animals. Details of transducer implantation and
surgical procedures are similar to those presented previously by Olson and
Marsh (1998). Animals were
anesthetized by immersion in a bath of MS-222. When the animals had reached a
surgical plane of anesthesia, a small incision was made in the skin caudal to
the iliosacral joints at approximately the midpoint of the urostyle.
Electrodes were routed subcutaneously from this position to the point of
implantation.
Small (1 mm) sonomicrometer crystals (Triton Technology, San Diego, CA,
USA) were used to measure fascicle segment lengths in the plantaris longus
muscle. The plantaris is a pinnate muscle that acts primarily as an ankle
extensor. The tendinous origin of the plantaris muscle is complex, with one
portion attaching to the tibiofibula near the knee and two other portions
crossing the knee. Thus, some of the muscle may act with a knee flexor moment.
Sonomicrometer crystals were aligned along fascicles that were visible and
could be traced from the more superficial aponeurosis to the deep aponeurosis
near the point of origin of the muscle. A small incision was made in the
muscle fascia between visible muscle fascicles. Sonomicrometer crystals were
inserted into this space, and fine 6-0 silk suture was used to secure small
stainless steel crystal holders to the surface of the muscle. Care was taken
to minimize the depth of these sutures and to minimize damage to muscle
fibers. Crystals were implanted 10-15 mm apart. A sonomicrometer (Model 120;
Triton Technology) was used to measure length changes from the sonomicrometer
crystals. The individual pairs of sonomicrometer crystals were calibrated
before implantation and corrections entered for the offset error due to the
holders and the epoxy lens (Olson and
Marsh, 1998).
Bipolar electromyographic electrodes were constructed from 0.076
mm-diameter Teflon-coated stainless steel wire. Wire ends were bared over
approximately 1 mm, and the wires were twisted into the `simple double hook'
configuration (Loeb and Gans,
1986). The electrodes were inserted in the region of length
measurement using a 25-gauge hypodermic needle. The location of the EMG
electrodes was verified in dissection after the completion of the
measurements. EMG recordings were made with WPI DAM-50 amplifiers operating
with a low-pass filter of 3 kHz and a high-pass filter of 10 Hz.
Frogs were allowed to recover from surgery for one day, and measurements were taken for 2-3 days following surgery. Jumping measurements were made in an enclosed jumping area approximately 40 cm wide by 2 m long. Lightweight, 1.5 m-long leads were connected to the recording transducers using small multi-pin connectors (Microtech, Boothwyn, PA, USA). The total mass of these leads was less than 2% of the mass of the frog, and care was taken to ensure that leads moved freely and did not interfere with jumping. Sonomicrometer and EMG signals were recorded at 4000 samples per second using a MacAdios 12-bit A/D board (GW Instruments, Somerville, MA, USA) in a Macintosh computer. Sonomicrometer signals were filtered in software (Superscope II; GW Instruments) with a 60 Hz smoothing filter. EMG signals were filtered with a 200 Hz high-pass FIR filter. For sonomicrometer measurements, the length of the muscle prior to the jump was used as the resting length, Lo, of the muscle. Muscle velocity was calculated from the differentiated length signal. Velocity traces were filtered in software using a 130 Hz smoothing filter (Superscope II).
Measurements of center of mass dynamics
Animals were videotaped with an NAC 200 high-speed video camera operating
at 500 frames s-1. Animals were videotaped in lateral view, and
only those jumps that occurred in the sagittal plane were used for analysis.
Video measurements were synchronized to sonomicrometry and EMG measurements by
means of a square-wave signal that appeared on the videotape at the onset of
computer data acquisition. Video recordings were digitized into computer and
analyzed using NIH Image software. The point of entry of the sonomicrometer
leads was used as a marker to estimate the movements of the center of mass.
This point is a good estimate of the location of the center of mass
(Marsh and John-Alder, 1994;
Hirsch, 1931
). These video
measurements do not account for the distal movement of the center of mass as
the legs extend during a jump. We feel that this approximation is appropriate
for the following reasons. First, the movement of the center of mass is
relatively small. Hirsch
(1931
) estimated that the
center of mass remained in the middle two-thirds of the urostyle in Rana
ridibunda. Second, most of the movement of the center of mass occurs in
the first two-thirds of the jump, because the ankle is the last joint to
extend and the foot is quite light. Third, kinematic data may actually
underestimate peak power late in the jump due to the effects of smoothing.
Thus, the important aspects of the frog jump, with force, velocity and power
all peaking late in the jump, are accurately reproduced by our data.
Estimates of the position of the center of mass were used to calculate the
force acting on the body and the center of mass velocity and power. Position
data were smoothed using a smoothing spline interpolation in the software
application Igor (Wavemetrics, Lake Oswego, OR, USA). Data were interpolated
to a wave of 1000 points (from approximately 100) with a standard deviation of
0.0015 m. This method is similar to the cubic spline algorithm recommended by
Walker (1998) for calculating
accelerations from position data. Smoothed position data were differentiated
to calculate center of mass velocity, and these data were differentiated to
calculate force in the horizontal (fore aft) and vertical planes
(Marsh and John-Alder, 1994
).
Power was calculated from the rate of change of the sum of horizontal and
vertical kinetic energies and potential energy.
Overview of the model
A computer-aided engineering application, Working Model (Knowledge
Revolution, Redwood City, CA, USA), was used to design a model that simulated
the key mechanical features of a muscletendon-powered acceleration. The
model simulated a muscletendon unit that operated across a lever system
to move a mass (Fig. 1). The
muscle actuator had forcevelocity, lengthtension and activation
properties, and the spring was modeled as a simple linear Hookean spring.
Between the muscle and the load to be accelerated was a gearbox through which
the effective mechanical advantage (EMA) could be adjusted throughout the
contraction to simulate a change in the muscle's leverage against the load. To
operate the simulation, the muscle was activated and allowed to shorten over
30% strain. The velocity, power and acceleration of the load were determined
only by the properties of the muscle actuator; no other controls over force or
velocity were included. These features together modeled a jumping frog as a
single muscle operating across a single joint to accelerate the body mass.
This model does not address multi-joint coordination of muscle forces or the
effects of limb inertia, or potential variation in muscle activation. The
muscle model does address the key features of the dynamic interaction between
muscle forcevelocity properties, elasticity, muscle mechanical
advantage and the dynamics of an accelerating load. A preliminary version of
this model was described by Marsh
(1999). The model and
documentation are available online
(http://jeb.biologists.org).
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Model muscle and load
The muscle actuator was constructed from several actuators operating in
parallel to produce the proper activation timing, forcevelocity and
lengthtension properties during a contraction. Central to the muscle
actuator was the forcevelocity actuator, which developed force
Ffv in proportion to its shortening velocity following a
simple Hill-type equation:
![]() | (1) |
To achieve muscle activation and lengthtension behavior, actuators
were stacked in parallel to act in opposition to the forcevelocity
actuator. We assumed that muscle activation increased linearly over the first
20 ms of muscle contraction. This effect was achieved by an actuator that
resisted the forcevelocity actuator during the first 20 ms of
contraction with force Fact that decreased linearly with
time (t):
![]() | (2) |
Lengthtension properties were modeled by stacking actuators in
parallel with the forcevelocity actuator. Two actuators
(FLT1 and FLT2) were used to
model the length tension effect:
![]() | (3) |
![]() | (4) |
![]() | (5) |
The dimensions and contractile properties of the model muscle were chosen
to represent the musculature of the hindlimb of a bullfrog as a single muscle.
The maximal shortening velocity (Vmax), maximal stress
(Po) and curvature of the forcevelocity curve were
determined from the forcevelocity properties measured in isolated
sartorius of Rana catesbeiana
(Marsh, 1994). A
Q10 of 2 for maximal shortening velocity was used to calculate a
Vmax of 9 L s-1 for the frogs at temperatures
measured in the present study (mean temperature = 26°C). It was assumed
that the muscle generated 30 N cm-2 peak isometric stress. The
tendon was modeled as a linear spring. Tendon stiffness was chosen such that a
force equivalent to Po would give a tendon extension
equivalent to 20% of muscle fiber length. Measurements of muscle shortening
in situ using sonomicrometry indicated that plantaris muscle
fascicles shortened by 10-20% in tetanus when the muscletendon unit was
held isometric.
Dimensions of the modeled muscle were chosen to produce a single muscle that represented the average dimensions of the hindlimb musculature. Dimensions from dissection of a 210 g frog were used. A total muscle cross-sectional area of 4 cm2 was used as the estimate for two frog hindlimbs, based on the approximate average of the cross-sectional areas of the ankle, knee and hip muscles, which were 1.9 cm2, 2.1 cm2 and 2.2 cm2, respectively. The muscle cross-sectional area was used to calculate the peak isometric force from the peak isometric stress value given above. The length of the model muscle was based on the sum of the lengths of the hindlimb muscle fascicles, which averaged 36 mm, 26 mm and 19 mm for the hip, knee and ankle, respectively. The total muscle mass was 36 g for two legs.
The load meant to simulate the body of an accelerating frog was 210 g, the mass of the frog used for muscle dimension measurements. A gravitational force equivalent to 0.5 times gravity acted against the load. This value was chosen because it represents the component of the gravitational force that acts, on average, against the direction of movement for a jump with a trajectory of 30° to the horizontal. In an actual frog jump, the animal may work more or less against gravity depending on variation in instantaneous trajectory during the takeoff phase. Because the load moved along a circular trajectory, gravitational forces were modeled with a constant force vector that rotated along with the lever on which the load was mounted.
Effective mechanical advantage (EMA)
The leverage with which a limb muscle produces force against the ground can
be described by its EMA; the ratio of the distance from the muscle line of
action to the joint center of rotation, or muscle moment arm r, and
the orthogonal distance from the joint center of rotation to the ground
reaction force vector, or out-moment arm R
(Biewener, 1989). The effective
mechanical advantage (EMA) that muscles operate with, on average, over the
course of the jump can be estimated from the ratio of total muscle shortening
to the distance the body moves. For the frog used to determine muscle
dimensions, the total hindlimb length was approximately 200 mm. It was assumed
that muscles involved in jumping contracted over a strain of 30% during a
jump. This gives a total muscle shortening of 0.3x80 mm=24 mm, and a
mean EMA of 24/200=0.12.
All model simulations were performed at the same mean EMA; i.e. the load
always moved the same total distance for the complete contraction. However, to
model the effect of a variable EMA throughout the jump, the musclelever
system operated through a controllable gearbox. This gearbox controlled the
ratio of output (body) velocity to input (muscle) velocity, or the reciprocal
of EMA, and could therefore simulate changes in EMA that might occur with
either a change in R or r. For fixed EMA contractions, the
gearing was held constant throughout the contraction. For variable EMA
contractions, an equation was used to vary the gear (1/EMA) as a function of
the length of the muscletendon unit (Lmt) or
velocity of the body (Vb). Gearing was controlled by an
equation for two variable EMA conditions. First, the gear was controlled as a
function of velocity of the load such that the muscle shortening velocity was
maintained at 0.3V/Vmax:
![]() | (6) |
The second equation was chosen to allow a continuously decreasing gearing
throughout the jump:
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Model configurations
We compared the performance of the model in five different configurations.
In all of the configurations, the load and muscle contractile properties were
the same. The model configurations differed in the presence/absence of a
series elastic component and the EMA trajectory:
To simulate the contraction in each configuration, we `stimulated' the muscle to fully activate while the load was at rest. The muscle contracted, producing force according to the combined effects of forcevelocity, lengthtension and activation properties and the interaction with the inertial and gravitational forces on the load. The simulation was allowed to run until muscle fiber strain reached 30%. Muscle strain, velocity, force and power were recorded during the contraction. We also recorded the velocity and force on the load during the contraction. The total work done on the mass and the kinetic energy work were calculated from the energy values at the end of the simulated contraction.
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Results |
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Our measurements of muscle shortening revealed a characteristic bimodal pattern of shortening velocity in bullfrog plantaris during jumping (Fig. 3). In jumps greater than 60 cm (Fig. 3AC), active muscle fascicles shortened rapidly early in the jump and late in the jump, with a region of reduced velocity during mid-jump. This pattern was not observed in some shorter jumps, where muscle velocity showed a slow decline or was approximately constant during takeoff (Fig. 3DF). In some cases, the two peaks in muscle shortening velocity corresponded to two separate EMG bursts. More commonly, the EMG activity was approximately constant or gradually increased during the takeoff phase (Fig. 3). The total muscle strain for all jumps greater than 60 cm was 26.4±0.3% (mean ± S.E.M.; N=5 animals).
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Comparison of muscle shortening with body movement during powerful jumps
reveals that much of the shortening of the plantaris occurred before
significant movement of the body (Figs
4,
5). Generally, the pattern of
muscle shortening did not resemble the pattern of body movement
(Fig. 4).
Fig. 4A,C shows that muscle
fascicle velocity reached a maximum when body velocity had just begun to
increase. During much of the period when body velocity increased, muscle
fascicle shortening velocity decreased. The increase in muscle shortening
velocity late in the takeoff phase is more consistent with the increase in
velocity of the body observed during this period. Figs
4B,D,
5 illustrate the lack of
correlation between muscle shortening distance and the movement of the body.
For the jumps shown, half of the shortening of the muscle fascicles occurred
before significant movement of the body (Figs
4B,D,
5). The lack of correlation
between muscle fascicle movement and body movement suggests that muscle
shortening early in the jump occurs against stretching elastic elements.
Muscle fascicle shortening early in the jump in the plantaris is not due to
early extension of the ankle joint (Fig.
5). Kinematics of jumping bullfrogs show that ankle extension
begins at the same time or later than extension at other joints
(Calow and Alexander,
1973).
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Model results
To determine whether the patterns of body movement and muscle shortening in
jumping frogs were consistent with the mechanical behavior of muscles and
series elastic elements, we modeled jumping as a single contraction of a fully
active muscle that contracted by 30% of its length as it accelerated a load
(Fig. 1). We compared the
velocity, power and force on the modeled load with the same quantities for
jumping frogs. These results are shown in
Fig. 6 for four of the five
model configurations.
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The model results show that the patterns of body movement observed in jumping bullfrogs are most consistent with muscle-powered accelerations in which the muscle operates in series with an elastic element and operates through a variable, increasing EMA throughout the acceleration (Fig. 6D). Like the frog jump, this modeled acceleration showed highest forces and powers late in the jump, and peak isotonic powers of approximately 1.5x peak isotonic power. The compliant, increasing EMA configuration also showed a simultaneous increase in both force and velocity during more than half of the jump (Fig. 6D, top two panels) and resulted in the highest final velocity of the load. The agreement between model body movement and frog movement was also good when the model muscle included a series elastic element and a constant EMA (Fig. 6C). When the model muscle operated without a series elastic element (Fig. 6A,B), the timing of force, velocity and power applied to the body was not consistent with the pattern observed during frog jumping. Both non-compliant model configurations produced a force peak early (Fig. 6A,B, middle panel), with force declining throughout most of the contraction. The decline in output force was due to forcevelocity effects in the model with constant EMA (Fig. 6A). When the model operated with a continuously decreasing EMA to maintain a constant muscle shortening velocity, the force applied to the body decreased throughout the contraction as a result of the decreasing leverage (Fig. 6B). In both non-compliant configurations, power outputs were near peak isotonic during much of the contraction but they did not exceed it.
In addition to the pattern of movement of the model load, we also compared the pattern of shortening in the model muscle to that observed for the plantaris muscle in jumping frogs. The different model configurations resulted in strikingly different patterns of muscle shortening and force (Fig. 7).Without a series elastic component, muscle velocity increased in parallel with body velocity throughout the contraction (Fig. 7A), and muscle force declined during most of the contraction due to forcevelocity effects (Fig. 7A, middle panel). When EMA was adjusted to hold shortening velocity constant, muscle force and power output were nearly constant throughout the contraction, except for the decline due to lengthtension effects (Fig. 7B). When a series elastic component was present, muscle shortening velocity was high early in the contraction as the muscle shortened against the stretch of the series elastic component, then declined as force began to reach a maximum and the spring no longer stretched (Fig. 7C,D). During the last part of the contraction, an increase in both tendon and muscle velocity contributed to the increase in velocity of the load. Under these conditions, muscle velocity bore little resemblance to load velocity during much of the acceleration. The bimodal pattern of muscle shortening velocity observed for the models that included a series elastic element was very similar to that observed for the plantaris in jumping bullfrogs (Figs 3, 4). The muscle maintained a relatively high power output during the entire contraction (Fig. 7D, bottom panel), as energy was loaded into the spring during the first half of the contraction and was applied directly to the load during the second half.
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The model that most closely reproduced the dynamics of a frog jump also performed the greatest work to increase the velocity of the body. Jump distance is proportional to the velocity of the body at takeoff (as well as takeoff angle), and these results suggest that operating muscles with tendons in series and a decreasing EMA can increase jump distance. Table 1 shows the work performed during jumps for each model configuration. The variable EMA and elastic configuration produced approximately 13% more work than the configuration in which the muscle acts with a constant EMA and no spring.
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Discussion |
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A simple acceleration model reproduces the dynamics of a frog
jump
When our modeled muscle included a series elastic element, there was a
striking agreement between the modeled acceleration and the mechanical
behavior of jumping bullfrogs; without a series elastic element many of the
features of the jump could not be reproduced. The only inputs to the model
were estimates of the dimensions and physiological properties of the muscles
involved (peak tetanic force, maximum shortening velocity, etc.), and the
remaining behavior of the model resulted from the interaction of these
properties with the inertial and gravitational forces on the accelerated load.
When the model included a series elastic element and a variable EMA, the model
output matched a typical bullfrog jump in jump duration, magnitude and timing
of peak force on the body, magnitude and timing of peak power output and
pattern of change in body velocity (Fig.
6D). Under these conditions, there was also a remarkable agreement
between the pattern of shortening of the modeled muscle and the pattern of
shortening measured in the plantaris muscle (Figs
3,
4,
7D). By contrast, without a
series elastic element, the model produced peak forces and powers early in the
jump, rather than late as observed in frogs, and magnitudes of peak power
output were lower than for frogs (Fig.
6A,B). The measured pattern of muscle shortening velocity in the
plantaris was not reproduced in models without a series elastic element
(Fig. 7A,B).
The results from our model indicate how elastic elements can allow movements that would otherwise be incompatible with the mechanical behavior of fully active muscle contractile elements. In the case of the dynamics of the body in jumping frogs, the simultaneous increase in body force and velocity cannot be powered directly by muscle contractile elements because the forcevelocity relation dictates that force must decline as velocity increases. A pattern of decreasing mechanical advantage cannot solve this problem; as EMA decreases to allow a constant muscle velocity, the force applied to the body per unit muscle force must decrease (Fig. 6B). The observation that the appropriate pattern of force and power output could only be reproduced by a model with a series elastic component and a constantly increasing EMA illustrates the importance of the interaction of muscle properties and the forces acting on the load.
How should muscles shorten when accelerating inertial loads?
Our model predicts a distinct pattern of muscle fiber shortening during
contractions involving an elastic element in series with an inertial load.
Initially, shortening velocity is predicted to be high, followed by declining
velocity as the spring becomes fully stretched. Later in the movement,
shortening velocity increases again as the load is accelerated. During this
period of increasing shortening velocity late in the jump, muscle forces drop
due to forcevelocity and lengthtension effects and energy is
released from the tendon.
The shortening pattern of the muscle fibers in the plantaris, an ankle
extensor with a long in-series tendon, agrees with the model predictions. The
presence of an elastic tendon uncouples fiber shortening in the plantaris from
movement of the body. According to our model, muscle contractile elements
perform more work when coupled with an elastic component because they can
operate on average at relatively slower velocities and higher forces when
shortening. The pattern of muscle shortening velocity measured in the
plantaris and predicted by the model to produce high work output is unusual
and would have been difficult to predict a priori from physiological
principles. Not all of the hindlimb muscles in frogs shorten like the
plantaris. Previous work has reasonably predicted that jumping animals should
operate their muscles at velocities that maximize power, because acceleration
requires that force must be produced at the same time that the body undergoes
a rapid movement (Hill, 1950b;
Lutz and Rome, 1994
). Lutz and
Rome (1994
) found support for
this prediction in the shortening pattern of the semimembranosus in leopard
frogs (Rana pipiens). Their measurements indicated that the leopard
frog semimembranosus operated at a constant shortening velocity of
approximately 30% Vmax. Olson and Marsh
(1998
) also found shortening
velocities in the semimembranosus and gluteus medius muscles of bullfrogs that
were more uniform than those measured for the plantaris. Thus, proximal
muscles with limited capacity for elastic energy storage may not undergo the
pattern of shortening observed in the present study. However, results from our
model illustrate that a muscle operating at a constant shortening velocity
cannot power movements with the force and velocity trajectory observed in
jumping bullfrogs. These results also indicate that the maximum accelerations
were not necessarily those with the highest power output
(Table 1). We predict that the
pattern of shortening observed in the plantaris of jumping bullfrogs is common
in frog hindlimb muscles and, more generally, will be found in muscles with
substantial in-series tendons specialized to accelerate inertial loads.
Elastic structures improve jumping performance
Our model indicates that bullfrog muscles can power longer jumps by
operating in series with elastic elements that store and release the work done
by contracting muscle fibers. This is consistent with Alexander's
(1995) model demonstrating that
compliance in series with a muscle can improve jump height in vertical jumpers
over a wide size range. Bobbert et al.
(1986
) demonstrated that in
jumping humans energy loaded into the tendons of the triceps surae early in
the jump is released rapidly late in the jump to develop high power output at
the ankle. Using a model of the human squat jump, Bobbert
(2001
) demonstrated that jump
performance improved with increasing series elasticity in the triceps surae.
Bobbert's results indicated that the series elasticity improved the work
output of the hindlimb because it improved the coordination of velocities
between segments and maximized the energy applied to the center of mass rather
than the limb segments (Bobbert,
2001
). Our single-lever model does not address inter-segment
coordination but rather indicates that series elastic components can improve
muscle work output even in a single muscle accelerating a load. Recently,
ultrasound measurements on the gastrocnemius muscle in humans have been used
to demonstrate that most of the muscle contractile element shortening occurs
against the stretch of elastic elements early in a squat jump
(Kurokawa et al., 2001
). This
observation is generally consistent with the results from the present model,
although the gastrocnemius shortening velocity in jumping humans does not show
a period of high velocity late in the jump
(Kurokawa et al., 2001
).
Insect jumpers use a catch mechanism to power jumping in a catapult-like
manner, pre-loading elastic energy before any movement and releasing it
explosively to power jumping (Bennet-Clark,
1975; Alexander,
1995
). Our results suggest that frogs also use a catapult-like
mechanism, pre-loading elastic energy in the early part of the jump. Yet there
are differences between the catapult mechanism proposed for jumping frogs and
that of insects. Bullfrogs appear to perform significant muscle work during
the entire jump, whereas in insect jumpers it is thought that the majority of
muscle work is performed before body movement during the pre-loading stage
(Alexander, 1995
;
Bennet-Clark, 1975
). Our model
results also suggest that jumping frogs can pre-load elastic energy in tendons
even without a functioning physical catch, although we cannot rule out the
possibility that a physical catch mechanism might further enhance elastic
energy storage. Simply redistributing the muscle work during shortening by
elastic storage (Marsh and John-Alder,
1994
) can improve performance.
A variable mechanical advantage provides an inertial catch
mechanism
The pattern of continuously increasing EMA during a jump that was most
effective for increasing muscle work output in our model may provide an
inertial catch mechanism for elastic energy storage and recovery. The physical
catch mechanisms employed by insect jumpers provide resistance to allow
muscles to contract to a high force without causing movement at a joint
(Gronenberg, 1996). The delay
in applying force to the body is necessary to stretch the spring. In our
model, the inertia of the body early in the jump allows muscle force to rise
to a high level before significant displacement of the body occurs. During the
period of increasing force, the series elastic element is stretched, and
energy is stored. A poor EMA early in the jump enhances this effect, because
the force transferred to the body is relatively low and thus accelerations are
low. In order to release the energy stored in the elastic elements before the
movement is completed, force must necessarily decline. The more rapidly force
declines the more rapidly the energy will be released. The increasing EMA as
the movement progresses accentuates the rapid increase in muscle fiber
shortening, and consequent decline in force, thus facilitating the release of
the stored energy.
Importantly, the most effective pattern of change in mechanical advantage
in our model is the opposite of the strategy widely accepted as favorable for
accelerations. Other workers have suggested that muscular systems ought to be
arranged to allow for constant shortening velocity during movement (Lutz and
Rome, 1994,
1996
;
Carrier et al., 1994
). To
achieve this end, the EMA must decrease during an accelerative movement; i.e.
the muscle gearing must continuously increase
(Carrier et al., 1994
). Many
motor-driven machines and human-powered vehicles (e.g. bicycles) utilize this
strategy to increase output velocity for a given motor velocity. Our results
suggest that for some activities the unique behavior of a muscle motor in
series with an elastic element may operate best with a counterintuitive use of
variable mechanical advantage, one that decreases the output (body) velocity
for a given input (muscle) velocity.
A variable mechanical advantage during muscle contraction has also been
proposed as a mechanism to maximize muscle efficiency and power during
steady-speed running. Carrier et al.
(1998) examined the pattern of
mechanical advantage change at individual joints in running dogs and found
that the pattern of EMA at the shoulder and knee was consistent with the idea
that EMA decreases to maintain a constant muscle velocity. However, they found
that at the hip, wrist, elbow and ankle joints the EMA increased during the
contraction. Thus, the pattern of EMA at some joints in running dogs resembles
the pattern predicted by our model to maximize work output in a
muscletendon unit acting to accelerate an inertial load. However,
during running, muscles do not operate at maximal power or work outputs
(Farley, 1997
), and the
accelerations of the body powered by a single muscle contraction are
relatively small. It has been suggested that the design of the musculoskeletal
system may favor minimization of muscle work during running to improve energy
economy (Taylor, 1994
;
Roberts et al., 1997
).
Therefore, although the pattern of EMA change at some of the joints in running
dogs may resemble that suggested for jumping frogs, it remains to be seen
whether the pattern of decreasing EMA that maximizes work output in our model
of contraction will apply generally to activities where maximum work is not
the desired mechanical output.
Model constraints
In modeling, a trade-off often exists between complexity and generality.
Our goal was to capture in a simple form the essential features of an
accelerative contraction driven by skeletal muscle. This simple model
accurately characterizes the complex interaction of the properties of a single
muscle, tendon and lever with the inertial and gravitational forces acting on
an accelerated mass. The model does not include the coordinated function of
many muscles operating over several joints. Undoubtedly, some of the dynamics
of any particular type of acceleration, such as a frog jump, result from the
variation in muscle properties, architecture and gearing for individual
muscles at different joints. Anatomically precise models of frog jumping are
providing, and will continue to provide, insight into the importance of the
integrated function of multiple muscles for the dynamics of frog jumping
(Kargo et al., 2002;
Kargo and Rome, 2002
).
Because our goal was to determine maximal muscle performance under conditions of constant muscle strain, our model did not include a muscle deactivation function. Deactivation is time dependent, and other model parameters influenced the time to complete the contraction. Thus, including deactivation would have resulted in variation in total muscle strain between conditions (depending on the velocity of the muscle at the onset of deactivation). Deactivation would have improved the performance of the compliant configurations because it would have allowed the release of all of the stored elastic energy. The model also did not include any regulation of level of muscle recruitment during a jump; it was assumed that the entire muscle mass was fully stimulated at time zero and, after an initial period of activation, was maintained at full activity throughout the jump. The very high power and work outputs observed during some jumps suggest that full muscle recruitment is a reasonable assumption. However, clearly, the level of activation in frog muscles can be modulated to produce jumps of differing distances. During some frog jumps measured, distinct bursts of EMG activity occurred, suggesting that activity may be modulated to fine-tune the jumping movement. Because we were interested in the limit to performance set by muscle contractile properties, we did not attempt to model any modulation in activity during the jump. The functions used to set EMA during our modeled contraction were chosen to represent two extremes. The first function modeled the pattern of EMA that has been proposed for accelerative contractions, where EMA decreases in proportion to the velocity of the body to maintain constant muscle velocity. To facilitate comparison, the other EMA function was effectively the reciprocal of this function, i.e. mechanical advantage increased in direct proportion to the velocity of the load. Experimentation with other functions for the increase in EMA did not result in dramatically different performance of muscle work. In a jumping animal such as the bullfrog, the exact pattern of EMA at any given joint will depend upon how the muscle moment arm changes with joint angle and how the ground reaction force moment arm changes throughout the jump. Although the exact function is unknown, all joints probably experience a decreasing ground reaction force moment arm throughout the jump as leg straightening causes the legs to move towards the midline, shortening the out-lever arm (the distance between the ground reaction force and the joint center of rotation). Thus, a pattern of increasing EMA is not only advantageous for work production during jumping but may also be a necessary consequence of powering jumping with jointed limbs that must transition from fully flexed to nearly straight during a jump.
All simulated accelerations were performed at a single value for mean EMA
realistic for a jumping frog. Model simulations at mean EMAs other than the
one used here indicate that the magnitude of performance benefit from an
elastic mechanism is sensitive to the value of mean EMA used. At higher
effective mechanical advantages, the increase in muscle work output when an
elastic component is included was greater than the 15% enhancement observed in
the present results. At lower EMAs, the increase in work between these
conditions was lower. However, at EMAs lower than those used for the
simulations presented here the force produced against the body was less than
two times body weight throughout the contraction. Such low forces would be
inconsistent with powering rapid jumps
(Marsh, 1994).
Patterns of work and power output in jumping frogs
The observation that frogs jump farther than they should is based upon the
discrepancy between the measured capacity for power production in their
hindlimb musculature and the power produced during the takeoff phase of a jump
(Marsh and John-Alder, 1994;
Peplowski and Marsh, 1997
;
Navas et al., 1999
). A similar
discrepancy in power output has been measured in a small mammalian jumper, the
galago (Galago senegalensis;
Aerts, 1997
). The present
results support the proposal (Marsh and
John-Alder, 1994
) that this discrepancy can be explained in
jumping bullfrogs by the release of elastic energy late in the jump. Peak
power outputs during maximal jumps in bullfrogs were approximately 1.5 times
their estimated peak muscle power, and the peak power output of the
muscletendon unit in our model was also 1.5 times the peak muscle power
(Table 1). The largest
documented discrepancy in power output in jumping frogs was recorded in Cuban
tree frogs (Peplowski and Marsh,
1997
). Calculations of takeoff power suggest that Cuban tree frogs
develop average powers in excess of seven times their capacity for muscle
power production. The power amplification for bullfrogs was much less than
this value. However, results from the same model with muscle and body
parameters scaled to values appropriate for Cuban tree frogs indicate that a
sevenfold power amplification can be obtained with an elastic element and the
inertial catch mechanism proposed here (T. J. Roberts and R. L. Marsh,
unpublished).
Conclusions
Much of the design of the non-muscular components of the musculoskeletal
system has likely been shaped through evolution by the limits to performance
imposed by the rather conservative contractile properties of skeletal muscle.
Thus, muscles, their naturally occurring loads and the linkages between these
two must be approached as integrated systems.
In the present study, this integrated approach yielded the following conclusions about muscle-powered accelerations:
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References |
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