A hierarchical analysis of the scaling of force and power production by dragonfly flight motors
208 Mueller Laboratory, Department of Biology, Pennsylvania State University, University Park, PA 16802, USA
* Author for correspondence (e-mail: rjs360{at}psu.edu)
Accepted 25 November 2003
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Summary |
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Key words: force, work, power, scaling, allometry, dynamic force output, dragonfly, flight motor, lever arm, basalar muscle, work loop, load lifting
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Introduction |
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Data from swimming fish (Webb,
1978), running and hopping animals (Full et al., 1991;
Blob and Biewener, 2001
;
Biewener et al., 1988
;
Ritter et al., 2001
) and
flying animals (Marden, 1987
)
indicate that maximum force output of intact musculoskeletal systems scales
very nearly as mass1.0. More recently, Marden and Allen
(2002
) have shown that maximum
force output by all types of rotary motors (musculoskeletal systems and
man-made machines such as piston engines, jets, and electric motors that use
rotary or oscillatory motion to accomplish more than simple translational
motion of a load) scales as motor mass1.0, and can be described by
a single scaling equation in which the motor mass-specific force is
57±14 N kg1 (mean ± S.D.).
The question of why different types of motors show such similarity
in both the magnitude and scaling of maximal force output is a complex
question that is beyond the scope of this study. Here, we focus on
how motor mass1.0 scaling of force output by a biological
motor is achieved by examining the morphology and mechanics of dragonfly
musculoskeletal systems in a hierachical fashion, from the static force output
of single muscles to the dynamic force output of the intact animal. Maximum
force output by dragonfly flight muscle mass has previously be shown to scale
isometrically with flight muscle mass
(Marden, 1987). Our objective
is to identify how an intact musculoskeletal system changes the scaling of
force output from muscle mass0.67 to muscle mass1.0.
The dragonfly flight motor consists mainly of synchronous muscles that act
directly on the wings. Up- and downstroke muscles insert on opposite sides of
an internal pivot or fulcrum (Simmons,
1977). As such, the wings combined with fulcra, and either up- or
downstroke muscles act as first and third order levers, respectively. We have
chosen a downstroke muscle and its lever system as the focus of our analyses.
All references to force output will refer to maximal force output, unless
stated otherwise.
At least two potential mechanisms could account for the difference in
scaling between isometric force output by individual muscles and dynamic force
output by intact musculoskeletal systems. First, although maximum isometric
force output of muscles scales as mass0.67, dynamic force output by
the muscle might scale with a higher exponent due to differences in mechanics
of isometric versus dynamic oscillatory contraction. Small muscles
tend to operate at higher contraction frequencies
(Medler, 2002). At higher
frequencies, transitions from an inactive (relaxation) to an active
(contraction) state and vice versa will constitute a relatively
greater portion of the total contraction cycle of the muscle, assuming that
calcium release and uptake by the sarcoplasmic reticulum occurs at a rate that
scales independently of mass. This would allow relatively less time for
complete cross-bridge activation and relaxation and therefore a lower
proportion of attached crossbridges during each cycle (i.e.
Rome and Lindstedt, 1998
;
Rome et al., 1999
). Dynamic
force output by small muscles could therefore be relatively low compared to
force output by larger muscles, and an increase in the scaling exponent that
relates force output to muscle mass is expected.
Secondly, the scaling of force output of a musculoskeletal system is a
function of the scaling of the dynamic force output and the geometry of lever
arms present in the musculoskeletal system. Biewener
(1989) showed that allometry of
lever arms can compensate for unequal scaling of skeletal cross-sectional area
and body mass in terrestrial vertebrates. Changes in limb posture (and
associated changes in mechanical advantage) toward a more upright position of
the supporting leg bones prevent large animals from operating at very low
safety factors for stresses acting on the skeleton.
Departures from geometric similarity in mechanisms that affect force output need not be restricted to terrestrial vertebrate locomotion. Geometrical changes within musculoskeletal systems could compensate for the loss of force with increasing size (i.e. the mass0.67 scaling of maximum force output) in order to achieve mass1.0 scaling of dynamic force output by the intact musculoskeletal system.
We examine the possibility that dragonflies depart from geometrical
similarity in the scaling of lever arm lengths within their thorax. Insect
wing lengths have previously (Greenewalt,
1962,
1975
) been shown to scale with
mass0.33, i.e. conform to geometric similarity
(length
mass0.33), but the scaling of internal lever arms used
in insect thoracic musculoskeletal systems is unknown.
Our hierarchical approach started by measuring the maximum force output
(Fstat) of the basalar muscle while it was held at
constant length (i.e. an isometric contraction). We then determined mean force
output generated by muscles during oscillatory contraction regimes (i.e.
workloops; Josephson, 1985)
that approximated in vivo working conditions during maximally loaded
flight. This force output will be referred to as Fdyn.
Through its lever system, Fdyn produced by the muscle
is utilized to satisfy inertial and aerodynamic force requirements, the latter
of which can be further divided into induced, parasite and profile force
requirements. The mean inertial and aerodynamic forces act at different
distances along the wing. To make our analysis tractable, we chose a single
external lever arm length (d2), the distance along the
wing at which the mean induced aerodynamic force acts (the radius of the
second moment of wing area; Ellington
1984a).
We defined the internal lever arm length as the distance between the muscle apodeme and the forewing fulcrum (d1). Then, assuming that moments are balanced about the wing fulcrum (Fig. 1), we calculated the induced force produced by this muscle-lever system (Find) during maximum performance conditions. Finally, we compared the scaling exponent of Find to that of the induced force output by the intact dragonfly thoracic musculature during maximum load lifting (Flift).
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In addition to analyzing the scaling of maximum force output by the musculoskeletal system, we present scaling relationships for both muscle mass-specific work and power output during maximum load lifting conditions.
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Materials and methods |
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Muscle
We studied the performance of the basalar muscle of the mesothorax
(terminology according to Marden et al.,
2001), which functions as the main depressor of the leading edge
of the forewing. It inserts directly on the humeral plate of the wing base by
means of a tendinous connection to an apodeme
(Snodgrass, 1935
). The basalar
muscle is not the only muscle that contributes to the depression of the
forewing. The first subalar depressor assists the basalar in depression, and
three smaller muscles (first basalar, second subalar depressor and third
subalar depressor) are partly responsible for depression and supination of the
wing (Simmons, 1977
).
Load lifting experiments
In order to obtain Flift, we incrementally increased a
dragonfly's body mass and examined its capability to lift this additional mass
(Marden, 1987). Lead weights
were glued to the abdomen, after which the dragonfly was placed for
approximately 1 min in an incubator set to 36°C to warm the flight muscles
to an appropriately high temperature. Dragonflies were then placed on a white
floor and stimulated to attempt take-off. They cooled quickly when removed
from the incubator and were probably flying at a thoracic temperature of
approximately 3334°C. Added loads were increased until the specimen
was just able to take off from the ground. Take-off attempts were recorded
using a high-speed video camera (Redlake Imaging, San Diego, CA, USA) at 500
frames s1 with the camera situated at an angle that allowed
an approximate head-on view of the body axis. Flift was
calculated as maximum load lifted (body mass + added mass). Mean mass-specific
Flift was obtained by dividing Flift
by total thoracic muscle mass. Video records of flight attempts were analysed
using iMovie© and NIH image© software. Wingbeat frequency and
amplitude were calculated over three consecutive wingbeat cycles and were used
with estimates of the internal lever arm length to calculate the basalar
muscle length changes and muscle contraction frequencies during maximum
load-lifting performance. These values were then used as input values for the
workloop experiments.
Aeshna u. umbrosa specimens were not tested for their load lifting capability. Consequently, no dynamic force measurements (workloops) were performed on these specimens and no values for Find were obtained. Morphological data and Fstat measurements were obtained from Aeshna u. umbrosa specimens, as these data did not depend on input values obtained from load lifting experiments.
Mechanical isolation of the basalar muscle
Dragonflies were decapitated and their legs and wings clipped. The abdomen
was left intact so that it continued to rythmically ventilate the thorax.
Using an epoxy resin, thoraces with the abdomen still attached were glued into
a temperature-controlled aluminum test chamber. The thorax was set into a
position in which the basalar muscle fibres were running vertically. In order
to keep the surrounding air moist and prevent dessication, a wetted tissue was
placed within the chamber. The cuticle surrounding the basalar muscle apodeme
was cut free, thus mechanically isolating the basalar muscle from the rest of
the thorax. A fine suture was tied around the apodeme and glued to a modified
insect pin suspended from the lever arm of a lever system (Cambridge
Technology 300B, Cambridge, MA, USA). Before the onset of
Fstat measurements and workloop experiments, a
micromanipulator controlling the position of the lever arm was used to
carefully stretch the basalar muscle to its original position, which was
determined by comparing it to the position of the neighbouring muscles and
wingbase.
Isometric tetanus experiments
The stimulation frequency used to produce Fstat was 285
Hz, which yielded maximum static tension for all species in preliminary
experiments. An S48 stimulator (Grass Instruments, Quincy, MA, USA) produced
trains of 0.25 ms pulses to the basalar muscle through two fine-gauge
electrodes inserted on each side of the mesothorax. The intensity of the
stimuli was set to a level that produced maximal twitch tension; this level
was generally about 1.0 V. The maximum force produced during a period of 0.5 s
of complete tetanus was recorded. Isometric tetanus experiments were performed
approximately 10 min after workloop experiments (see below); we chose this
order of experiments because our muscle preparations frequently performed
poorly after being subjected to isometric tetanus.
Workloop experiments
The basalar muscle was driven through a series of five sinusoidal length
cycles (Fig. 2). For each
species, the amount of imposed length change during these cycles was
calculated using species-specific lever arm length measurements and the
wingbeat amplitudes obtained from the load-lifting experiments. Similarly, the
muscle contraction frequency used for workloop analyses was the mean wingbeat
frequency used by that species during maximum load lifting. The phase
relationship between electrical stimulation and muscle strain was set at a
value previously determined to be the in vivo phase relationship for
basalar activation in one species of dragonfly (activation at 44% of maximum
length during the lengthening phase;
Marden et al., 2001). Net work
produced by the basalar muscle was measured from the workloop area during the
fourth length cycle. Mean force produced during a length cycle
(Fdyn) was calculated as net work produced divided by the
muscle strain during the loop.
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Stimulation magnitude and duration settings were identical to those described above for the isometric force measurements. Thoracic temperature was monitored using a fine-gauge thermocouple that was inserted into the metathorax and connected to a TC-1000 thermometer (Sable Systems, Las Vegas, NV, USA). Thoracic temperature during workloop experiments was regulated between 3234°C.
Anatomical measurements
Total thorax and basalar muscle wet mass of dragonflies were measured to
the nearest 0.1 mg using an analytical balance. The non-dimensional radius of
the second moment of forewing area was calculated according to Ellington
(1984a) and multiplied by
forewing length in order to obtain d2. To measure
d1, the internal lever arm length, a section of the dorsal
thorax containing a forewing fulcrum and basalar apodeme was cut out of the
thorax, after which all muscle and soft tissue except for the apodeme of the
basalar muscle was removed (Fig.
3). Micrographs were taken using a DC200 digital camera (Leica,
Cambridge, UK) attached to a Leica MZ 125 microscope and analysed using NIH
image© software.
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Calculation of motor force output
Find was calculated according to the following equation
for balanced moments over the wing fulcrum, using measured values of
Fdyn, d1 and d2
for each individual:
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Scaling and statistical analyses
The scaling analyses performed on the variables d1,
d2, Fdyn and Find
were done with respect to the wet mass of the basalar muscle. For
Flift, total thoracic muscle mass was used. Data were
log-transformed and each of the variables measured were fitted using a
least-squares linear regression model. The scaling exponent for
Find was obtained by first calculating values for
Find according to
Equation 1, after which
calculated Find values were regressed with respect to
muscle mass. Scaling exponents were tested against hypothesized exponents
using t-tests (Draper and Smith,
1981; Zar, 1984
).
Statistical analyses were conducted using JMP software (SAS Institute Inc.,
Cary, NC, USA).
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Results |
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Isometric force output
Isometric force output (Fstat) was proportional to
muscle mass0.670 (Fig.
4B). A two-tailed t-test confirmed that the scaling
exponent was not statistically different (=0.05, P=0.948,
N=10) from 0.667, indicating that dragonfly basalar muscles behave
similarly to most other muscles, i.e. maximum isometric force production is
proportional to cross-sectional area.
Lever arms
Lever arm d2 for this group of dragonflies was
proportional to muscle mass0.31
(Fig. 5A). A two-tailed
t-test could not distinguish the d2 scaling
exponent from 0.33, the expected scaling of length with mass for similarly
shaped bodies (=0.05, P=0.173, N=51). In contrast,
lever arm d1 did not scale according to the expectations
of geometrical similarity, as d1 was proportional to
muscle mass0.540 (Fig.
5B). This scaling exponent was significantly higher than the
expected value of 0.33 (two-tailed t-test;
=0.05,
P<<0.001, N=52).
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Dynamic muscle force output
Fig. 6 shows examples of
typical workloops for each species used in this study. Mean dynamic muscle
force output during workloops (Fdyn) was proportional to
muscle mass0.83 (Fig.
7A). A one-tailed t-test indicated that this scaling
exponent was significantly higher than 0.667 (=0.05, P=0.029,
N=33), but significantly lower than 1.0 (one-tailed t-test;
=0.05, P=0.027, N=33). This result shows that during
realistic dynamic contraction, the scaling of muscle force output was
different from that during static isometric conditions
(Fstat) and from the scaling of intact flight motor force
output (Flift).
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Muscle-lever system force output
The calculated force output (Find) produced by the
muscle-lever system during maximum performance scaled as muscle
mass1.04 (Fig. 7B).
This scaling exponent was not statistically different from 1.0 (two-tailed
t-test; =0.05, P=0.670, N=33) or from the
scaling exponent (1.035) that we found for Flift
production by intact dragonflies (two-tailed t-test;
=0.05,
P=0.982, N=33). Mean muscle mass-specific
Find was 138.3±38.2 N kg1 (mean
± S.D.).
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Discussion |
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Like other animal muscles, dragonfly basalar muscles held at constant length produce maximum forces that are proportional to muscle mass0.67 (Fig. 8). However, the average force produced by the basalar muscle during realistic dynamic working conditions (i.e. during workloops) is proportional to muscle mass0.83 (Fig. 8), a significant increase from the scaling of muscle force during isometric tetanus. The difference in scaling between isometric and dynamic force production is by itself insufficient to explain the mass1.0 scaling of maximum force production by an intact dragonfly flight musculoskeletal system, since the scaling exponent 0.83 was significantly lower than 1.0.
|
Our data show that there is a departure from geometrical similarity in one
of the lever arms within the musculoskeletal system, as d1
scaled as muscle mass0.54. This departure was specific to the small
internal lever arm because d2, the larger external lever
arm, scaled as muscle mass0.31, which was not significantly
different from mass0.33. The departure from geometric similarity
for d1 indicates allometry for a skeletal element rather
than adjustments in posture or alignment of motor parts as has been reported
for terrestrial mammals (Biewener,
1989). It is the combination of the allometry of
d1 and the scaling of Fdyn that causes
the scaling of Find to be very close to muscle
mass1.0 (Fig.
8).
It should be noted that the particular combination of scaling parameters
for dragonfly flight musculoskeletal systems is probably idiosyncratic to this
system and that other combinations of scaling relationships could also result
in mass1.0 scaling of force output. If dynamic force output in
another system should scale with an exponent other than 0.83, then we expect a
compensating difference in the scaling of at least one of the lever arms, in
order to maintain mass1.0 scaling of total system force output. It
remains to be seen how general the value of 0.83 is for
Fdyn in other taxa, but if differences between the scaling
of isometric and dynamic force output are common, then scaling models that
assume a value of 0.67 (e.g. Wakeling et
al., 1999; Hutchinson and Garcia, 2000) should be used with
caution.
Our analyses treated all data points as independent; however, the
phylogenetic relatedness within and between species makes this assumption
worth examining (Pagel and Harvey,
1988; Felsenstein,
1985
). We repeated the regression analyses using mean values
calculated for each species. No substantial change in the scaling exponents
was detected [for example, the scaling exponent for Find
became 1.058 for species means (N=8) instead of 1.036 for all
individuals], indicating that the use of individual datapoints in the
regression analyses did not bias our results. Similarly, when the data set was
collapsed to mean values for genera (N=7) or family (N=2)
there was no substantial change in the estimated scaling exponent of
Find (Table
2 shows a full list of scaling exponents for
d1, d2, Fdyn and
Find generated by these different analyses).
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Our focus in this study was to examine the scaling of force output;
however, it is also interesting to compare the magnitude of the forces we
measured and calculated with measures from previous studies. Although the
scaling exponent for maximum load lifting force output agrees with the scaling
slopes found by a previous survey of load lifting by flying animals
(Marden, 1987) and a broader
survey of net force output by nearly all types of motors used for animal and
mechanized transportation (Fig.
8; Marden and Allen,
2002
), our measure of mean mass-specific Flift
is low compared to the mean mass-specific maximum force output (57±14 N
kg1) found in those studies. In our calculations of
Flift, we used the maximum weight carried by dragonflies,
whereas previous studies used the midway point between the maximum weight
carried and the next incremental load that could not be carried
(Marden, 1987
). This
difference is at least partly responsible for the somewhat lower mean
mass-specific Flift data presented in this study.
Our loaded dragonflies flew forward at velocities between 1 and 1.8 m
s1. This implies that whole motor force output is higher
than the maximum weight carried, by an amount equal to the force necessary to
overcome parasite drag and profile drag at such speed. We used published
values for mean parasite and profile drag forces on Sympetrum
sanguineum dragonflies that were gliding at 2 m s1
(Wakeling and Ellington,
1997a) to estimate this additional force. The adjusted force
output was not substantially different from the value for mass-specific
Flift (40.2 N kg1 instead of 40 N
kg1), indicating that average parasite and profile force
requirements at these low speeds are negligible in comparison with induced
force requirements. However, wing kinematics are different during maximum load
lifting compared to those during free gliding flight, and mean parasite and
profile drag values could therefore be different.
Mean mass-specific Find was calculated to be 138.3 N
kg1, which is higher than the mean value for
Flift for loaded dragonflies and the mass-specific force
output by motors in general (Marden and
Allen, 2002). At least part of this difference is due to the fact
our calculated value for Find assumes that all of the
force output is used to create induced lift. However, this simplification
ignores the fact that force output by the flight motor must also meet
inertial, parasite and profile force requirements.
Inertial force requirements especially are known to be substantial during
dragonfly flight. Work needed to accelerate wings and virtual masses during
hovering flight can be between 1.4 and 5.9 times the work done against
aerodynamic forces (Ellington,
1984b). During forward flight, however, inertial force
requirements have been shown to be lower than aerodynamic requirements
(Wakeling and Ellington,
1997b
). Although we did not measure inertial requirements
quantitatively in this study, they are particularly interesting with regard to
the results of our scaling analysis. If we assume that average parasite and
profile force requirements are negligible (see above), then the total moment
required from the muscle and lever arms can be described as:
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This study demonstrates a mechanism by which dragonflies achieve the
`universal' mass1.0 scaling of maximum force output, but it cannot
explain why this is true for most types of biological and engineered
motors. Marden and Allen
(2002) have discussed the idea
that mass1.0 scaling of force output and the universal upper limit
of mass-specific force output by rotational motors represents a failure mode
above which there may be a drop-off in durability. Perhaps complex stress
regimes (Marden and Allen,
2002
) require a motor to have a constant ratio of mass to net
force output in order to be durable and successful. This remains a highly
speculative idea, but no other hypotheses have been put forward to explain the
universal mass1.0 scaling of force output by rotational motors.
Scaling of muscle mass-specific work and power
Ellington (1991) used
previously published data (Marden,
1987
) to estimate that mass-specific muscle power output available
during maximally loaded flight scales as mass0.13 for flying
animals spanning 19 mg to 920 g body mass. By assuming that wingbeat frequency
scales as mass0.33, muscle mass-specific work was estimated
to be proportional to mass0.46. For Anisoptera within that sample,
mass-specific muscle power was estimated to scale as mass0.27,
which implies a mass scaling exponent of approximately 0.60. In contrast to
these results, Tobalske and Dial
(2000
) proposed that maximum
mass-specific work by muscles could be invariant with size (i.e. scaling as
mass0) for Phasianidae, as they showed that pectoralis
mass-specific take-off power in this group scaled approximately as
mass0.33, i.e. maximum mass-specific power available was
proportional to wingbeat frequency. However, take-off power analysed by the
latter study represented an unknown fraction of the power available from the
muscles and the scaling of total mechanical power output by Phasianid muscles
during maximum performance could be different from mass0.33.
Askew et al. (2001
) reported a
lower scaling exponent for mass-specific muscle power (i.e.
mass0.14) available from Phasianid flight muscles, but the
allometric scaling of wing beat frequency (i.e. mass0.247
scaling instead of the predicted mass0.33 scaling) found for
this group indicated that mass-specific muscle work would indeed be largely
independent of body mass. However, the inclusion of previously published data
for hummingbirds, Harris' hawk and bees changed the scaling relationship of
mass-specific work to mass0.336. So, while mass-specific force
output by different sized flight motors shows remarkable consistency in its
scaling with mass (e.g. Marden,
1987
; this study), it seems that the scaling of other important
indicators of flight performance, muscle mass-specific work and power, shows
more variation amongst different taxonomic groups.
Mass-specific power available from dragonfly basalar muscles during maximum
performance (calculated using muscle work during strain regimes and
contraction frequencies that matched maximally loaded flight) increased
significantly with increasing body mass and was proportional to muscle
mass0.24 (Fig. 9A).
Mass-specific work during maximum performance was proportional to muscle
mass0.43 (Fig. 9B),
while wingbeat frequency scaled as mass0.20. Both of these
values are in rough agreement with estimates by Ellington
(1991) for Anisoptera, and
provide some of the first directly measured data concerning the scaling of
mass-specific work and power available from insect flight muscles during
maximum performance.
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List of symbols
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Acknowledgments |
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References |
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