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First published online October 5, 2007
Journal of Experimental Biology 210, 3677-3688 (2007)
Published by The Company of Biologists 2007
doi: 10.1242/jeb.006486
Linkage mechanics and power amplification of the mantis shrimp's strike
1 Department of Integrative Biology, University of California, Berkeley, CA
94720-3140, USA
2 Ecology and Evolutionary Biology, University of California–Irvine,
Irvine, CA 92697-2525, USA
* Author for correspondence (e-mail: patek{at}berkeley.edu)
Accepted 6 August 2007
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Summary |
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 TOP Summary IntroductionMaterials and methodsResultsDiscussionReferences |
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Key words: power amplification, predation, movement, feeding, speed, acceleration, Crustacea, kinematic transmission, four-bar linkage model
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Introduction |
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TOPSummaryIntroductionMaterials and methodsResultsDiscussionReferences |
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;
Alexander and Bennet-Clark,
1977; Gronenberg,
1996a). Power-amplified animal movements are truly diverse,
ranging from the elastic tendons and springy legs of kangaroo and locust
jumps, to snapping shrimp appendages and spring-loaded vertebrate ballistic
tongues (Alexander, 1990;
Alexander and Bennet-Clark,
1977; de Groot and van
Leeuwen, 2004; Deban et al.,
1997; Heitler,
1974; Lappin et al.,
2006; Nishikawa,
1999; Ritzmann,
1973; Versluis et al.,
2000).
In all of these systems, relatively slow muscle contractions precede rapid
movement. As muscles contract to provide the necessary work for the movement,
elastic potential energy is typically stored in structural elements (e.g.
kangaroo tendons) while latches and/or antagonistic muscle contractions
prevent movement until the animal is ready to jump or strike
(Alexander, 1983;
Gronenberg, 1996a). A good
analogy for these biological principles is found in the crossbow: slow muscle
contractions of a human arm gradually load (`load phase') and store elastic
potential energy in the crossbow and ultimately a latch releases the string,
which in turn drives forward the arrow (`release phase'). In the load phase,
muscle contractions load elastic elements and thereby store potential energy.
In the release phase, fast movement is actuated through the rapid release of
stored potential energy. It is important to note that in the release phase,
muscle activity plays a minimal role or no role at all in actuating the fast
movement; the release of elastic potential energy occurs at far shorter
timescales than muscle contractions. With this mechanism, the arrow
accelerates and flies through the air at far greater speeds than would have
been possible by simply throwing the arrow.
The mysteries of the crossbow – where is energy stored, how release
is triggered, and the mechanics behind the loading or unloading of the bow
– are the same principal questions we ask of a biological energy storage
system. The speed and power of the killing strike of the second thoracic
appendages (the `raptorial appendages') of mantis shrimp (Stomatopoda) are
clear evidence of a power amplification system at work
(Fig. 1)
(Burrows, 1969;
Patek and Caldwell, 2005;
Patek et al., 2004). The
entire strike occurs over several milliseconds and can reach peak speeds of
10–24 m s–1
(Burrows, 1969;
Burrows and Hoyle, 1972;
Patek et al., 2004). Peacock
mantis shrimp Odontodactylus scyllarus can directly deliver impact
forces of over 1000 N (thousands of times its body weight) with an equal or
greater force secondarily caused by cavitation bubble collapse
(Patek and Caldwell,
2005).
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;
Burrows and Hoyle, 1972;
McNeill et al., 1972). When
the extensor muscles have fully contracted and the animal is ready to strike,
the flexor muscles turn off, releasing the sclerites, and the appendage
rapidly rotates outward toward its target
(Burrows, 1969;
Burrows and Hoyle, 1972;
McNeill et al., 1972). It has
been proposed that energy is stored in a saddle-shaped portion of the merus
exoskeleton (Patek et al.,
2004) and in the connective tissue of merus, specifically the
extensor muscles and apodemes (Burrows,
1969).
The morphological complexity and evolutionary diversity of the mechanical
system that drives the raptorial strike raises the possibility that there is a
leverage system, such as a four-bar linkage, underlying the rapid rotation of
the dactyl. While the storage and release of cuticular elastic energy during
the release phase is often observed in arthropods, e.g. locust jumping legs
(Heitler, 1974), linkage
mechanisms in arthropod power amplification mechanisms are not well studied.
These jointed leverage systems amplify rotational motion and are typically
characterized in terms of kinematic transmission (KT; angular output of the
linkage mechanism divided by angular input)
(Barel et al., 1977;
Westneat, 1994)
(Fig. 2). Thus, KT provides a
heuristic measure of speed- versus force-modification of the linkage
system, such that a high KT system delivers a large angular output (e.g.
angular velocity) for a small angular input and can therefore be considered
`angular velocity-modified' (in the same sense that the mechanical advantage
provided by a long output lever relative to input lever is speed-modified).
Linkage models of fish jaws have proved to be powerful tools for examining the
evolution and performance in force- versus speed-modified feeding
mechanisms within and across species
(Alfaro et al., 2004;
Collar et al., 2005;
Hulsey and Wainwright, 2002;
Muller, 1996;
Westneat, 1991;
Westneat, 1995;
Westneat et al., 1993).
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; Kunze,
1981), muscle anatomy and activity patterns during the strike
(Burrows, 1969;
Burrows and Hoyle, 1972;
McNeill et al., 1972), and a
proposed linkage system and energy storage mechanism
(Patek et al., 2004). Here, we
build on these previous studies by examining the raptorial morphology and
mechanics of peacock mantis shrimp Odontodactylus scyllarus from
several new perspectives, including the use of CT scan technology to
characterize cuticular mineralization patterns and functional morphology of
the latches as well as high-speed video analysis to measure changing
conformations of the appendage segments and strike kinematics. In addition, we
quantitatively test the previously proposed linkage mechanism
(Patek et al., 2004) and
assess whether the proposed elastic energy storage mechanism could function
given the mineralization patterns of the merus. The goals of the present study were to examine the anatomy of the raptorial appendage and the kinematics of the release phase of the strike mechanism from these functional perspectives. (1) Energy storage: what is the distribution of mineralization in the merus and how does this mineralization pattern contribute to the elasticity and stabilization of the appendage? (2) Latching mechanism and pre-strike stabilization: what are the shapes and orientations of the sclerites and how might they control the preparation for and release of the strike? (3) Kinematic transmission: does a four-bar exoskeletal linkage system mediate the storage and release of potential energy in this system?
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Materials and methods |
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TOPSummaryIntroductionMaterials and methodsResultsDiscussionReferences |
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Mineralization patterns were visualized using 3-D reconstructions of CT scans (Amira software, v. 3.1.1, Mercury Computer Systems, Berlin, Germany). A freshly frozen individual was micro-CT scanned at the University of Texas with a slice thickness of 0.0585 mm. Slice images were reconstructed at a resolution of 1024x1024 pixels over a 50 mm field of view. Voxels were 0.0488x0.0488x0.0585 and bit depth was either 8 or 16, depending on the need for visualizing soft tissue. The elements of the raptorial appendage were segmented out of the CT scans and separate images of each were created. Isosurface renderings were utilized to show the outer surfaces of each structure and the articulations between these structures. These surface renderings were also useful in identifying areas of reduced mineralization. In addition, volume renderings of each structure were created to further visualize areas of greater and lesser mineralization. In volume rendered images, brighter structures are more highly mineralized. From the surface renderings a VRML file was used as input to a rapid prototype 3-dimensional printer (Z-Corp 310, Burlington, MA, USA) to produce large-scale models of each structure. These models were helpful in deciphering the articulations between the different structures of the raptorial appendage.
Transmission: kinematics and linkage mechanics
We analyzed high-speed images of raptorial strikes and noted the changing
configurations of the merus in order to characterize the dynamics of the
flexible elements and linkages of the raptorial appendage. Animals regularly
struck objects coated with shrimp paste and most animals were willing to
strike objects under bright video lights after a period of training. A
high-speed imaging system (5000 frames s–1, 35 µs shutter
speed; 640x480 pixel resolution; HG100K Redlake Systems, San Diego, CA,
USA) recorded stomatopods striking a snail shell coated in shrimp paste and
wired to a stick. The snail shell was presented to animals within confined
burrows and aligned parallel to the glass wall of the aquarium, thereby
allowing us to film strikes with the animal positioned laterally. Sequences in
which strikes were directed out of the camera's plane of view were excluded
from the dataset.
The following parameters were measured using high-speed imaging: angular velocity, acceleration and strike duration of the dactyl heel (the bulbous structure at the base of the dactyl segment of the raptorial appendage; Fig. 1) (50–58 strikes; 6 individuals; 7–12 strikes per individual), and rotation of the meral-V (a moveable element in the merus segment of the raptorial appendage; Fig. 1) (24 strikes; 6 individuals; 3–7 strikes per individual) (Matlab v. 6.5 and v. 7.0.4). Meral-V rotation was calculated by measuring the change in angle of the meral-V relative to horizontal across each video frame. The acceleration and speed of the dactyl heel were derived from the arc distance traveled by the heel across video image intervals. Two points were digitized along the propodus/dactyl axis formed by the distal two segments of the raptorial appendage, which remain folded during a smashing strike (Fig. 1). The angular change of this line was calculated across video frames. This angle was multiplied by the distance between the propodus/carpus joint and dactyl/propodus joint, which yielded the arc distance moved by the heel of the dactyl.
Speed and acceleration were calculated as the first and second derivatives
of distance, respectively. A drawback to computing derivatives from kinematic
data is that they only provide average kinematic estimates. Even with
curve-fitting and spline methods (e.g.
Walker, 1998), the transient
and non-sinusoidal movement of the mantis shrimp's strike caused the filtered
and smoothed data to fail to track the displacement of the appendage.
Specifically, the movement of the limb follows a gradual path interrupted by a
sudden impact and reverse in direction. The smoothing and spline algorithms
applied to these data failed to track this transient movement and instead
continued to follow the initial path of the appendage. Nonetheless, the
animals typically struck the target when the appendage was moving the greatest
distance in the arc and the frame rate of 5000 frames s–1
under-sampled the movement. Thus, the distances measured were underestimates
and the resulting velocities and accelerations should also underestimate the
rate of movement. Given the uncertainty of deriving accelerations from these
data, we report acceleration in orders of magnitude. The relative movement
across frames was converted to SI units by measuring the pixel distance of
known structures on the raptorial appendage in each frame and converting
pixels to meters using the calibrated distance. We estimated digitizing
measurement error by digitizing 5 sequences, 10 times each.
Four-bar linkage pivot points were identified based on high-speed videos,
functional morphological observations and manual manipulations of the
specimens. Using a standard four-bar linkage configuration
(Uicker et al., 2003;
Westneat, 1990), we identified
four pivot points defining four `links': a fixed link, input link, follower
link and coupler link (Fig. 2,
also see Results). We measured link lengths using photographs of raptorial
appendages at rest (13 individuals) and in digital video images when all the
pivot points were visible (images from 21 video sequences of strikes performed
by 4 individuals). Using t-tests (JMP v. 5.0.1), we tested whether
specimen measurements were equivalent to video-based measurements. In
addition, we compared the length of Link 4 between these two datasets, given
that Link 4 is formed by the contracted extensor muscle and, therefore, should
be longer in the photographs of relaxed appendages than in the video images of
appendages prepared to strike.
Based on the above morphological and kinematic analyses, we tested the
hypothesis that a four-bar linkage system mechanically couples this system
(Fig. 2). With the known length
of the diagonal bar (Fig. 2),
the law of cosines was used to calculate the angles between any of the links
during a given input bar rotation. The lengths of each of the four links
(Ln) and the input angle of between link1 and
link2 (
input) were entered into the following
equations to calculate the length of the diagonal bar
(Ldb) (Fig.
2):
 | (1) |
output):
 | (2) |
Depending on the relative lengths of the links, a four-bar linkage system may allow a 360° rotation of the input link, but a more common case is that the input link `jams' after some amount of rotation. This range of input angles for which there is movement of the output link is called the `operational' range of the four-bar linkage. We used a mathematical model to determine the operational range of the input linkage and compared it to the input range actually used by the mantis shrimp. The input range used by the mantis shrimp yielded an output of the four-bar model that was effectively approximated as a line (see Results) with a slope equivalent to the predicted KT of the system.
We statistically evaluated the fit between the predicted four-bar model
behavior and the measured kinematics of the raptorial appendage. Given that
the carpus, propodus and dactyl are tightly coupled once the dactyl begins to
sweep toward its target, we assumed that these three segments share the same
pivot point and rotate an equivalent number of degrees during the sweeping
phase of the strike. This allowed us to measure the rotation of the propodus
as the output angle equivalent to the rotation of Link 3 (carpus); the
propodus is larger, visible in a greater proportion of video sequences and can
be more accurately digitized than the carpus. We tested whether the slope of
the relationship between the input (Link 2) and output angles (Link 3=propodus
rotation) measured from the high-speed videos was significantly different than
the slope predicted by the four-bar model [modified t-test, see p.
32, Grafen and Hails (Grafen and Hails,
2002)]; incorporating individual effects and treating video
sequences nested within individuals as random effects using Residual Maximum
Likelihood method in JMP statistical software (v. 5.0.1).
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Results |
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TOPSummaryIntroductionMaterials and methodsResultsDiscussionReferences |
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). In O.
scyllarus, sclerite 1 is a small, rod-shaped structure positioned
medially along the ventral edge of the merus and embedded in the thick
arthrodial membrane that connects the merus and carpus (Figs
3,
4). The medial flexor apodeme
attaches just below the sclerite's tip and forms a thin, pinnate layer over
the medial surface of the merus.
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Merus mineralization and articulations
The joint articulations between the merus and carpus range from robust to
nearly absent (Figs 3,
5). The lateral meral-carpal
articulation is visible externally and is only loosely articulated
(Fig. 5A). Opposing this
somewhat unconstrained connection on the lateral side are two medial
meral-carpal articulations (Fig.
5B,C). The internal medial articulation forms a smooth,
channel-like surface with a stop at the end, which both stabilizes the carpus
to move only along the dorso-ventral axis and also prevents the carpus from
adducting beyond a particular angle.
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The lateral and medial sides of the merus are distinctly asymmetric in terms of mineralization and flexibility. The medial side is stiff and robust whereas the lateral side is thin and flexible (Figs 1, 3, 5). Similarly, the dorsal surface is thin and flexible while the ventral surface consists of large, stiff buttresses running along the distal-proximal axis. There are also unmineralized regions on the medial and dorsal surfaces where cuticle is replaced by arthrodial membrane.
There are two independently mobile components of the merus' cuticle – the meral-V and the saddle (Figs 5, 6). The meral-V is a thick, triangular-shaped structure that connects to one of the large ventral buttresses forming the underside of the merus. It has a flexion point along the ventral-lateral margin of the merus, such that the meral-V bends at this junction and rotates proximally (Figs 1, 5, 6). The meral-V is flanked by arthrodial membrane and the ventral buttress also has a region of low-mineralization adjacent to it (Fig. 1), allowing this structure room to flex.
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Transmission: kinematics
At the onset of a strike, the propodus and dactyl slide distally along the
merus and then transition to a sweeping movement with a large rotational
velocity (Figs 7,
8). The duration of the
sweeping movement averaged 1.8±0.4 ms (± s.d.) and the sliding
movement averaged 0.9±0.5 ms (6 individuals, 6–12 strikes per
individual). The dactyl heel reached peak speeds of 14.7–23.5 m
s–1 (mean peak speed: 13.7±3.3 m s–1;
mean median speed: 3.4±1.7 m s–1), peak angular speeds
of 670–990 rad s–1 (mean peak angular speed:
608.9± 147.0 rad s–1; mean median angular speed:
155.7±79.6 rad s–1), and mean peak accelerations on
the order of 104 m s–2. Digitizing measurement
error at maximal speeds was on average ±4%.
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The rotation of the dactyl heel and meral-V were variable and correlated with each other. In all sequences, the dactyl heel struck the prey item while the meral-V was still rotating; thus, during these smashing strikes, the propodus and dactyl did not transition to a ballistic, unpowered phase prior to impact. The dactyl heel struck the snail across a range of excursion angles, such that in some strikes the dactyl/propodus rotated outward only 7° whereas in other strikes the dactyl struck the snail with a maximum extension of 42° (mean: 25±9°). Similarly, the net meral-V rotation ranged widely depending on the excursion of the dactyl when it struck the snail. The net meral-V rotation was on average 9° (range: 3–17°; ±5° s.d.; 5 individuals, 4–6 strikes per individual) (Fig. 7). Values were not significantly different across individuals for propodus rotation (one-way ANOVA; F=0.6943; P=0.60) nor for meral-V rotation (one-way ANOVA; F=1.1424; P=0.37).
Transmission: linkage mechanics
The structural asymmetries of the merus, as described above, generate two
distinct functional regions of the merus. Specifically, the robust
mineralization and paired meral-carpal articulations on the medial side yield
stability and resistance to flexion. By contrast, on the lateral side of the
merus, considerable flexion occurs via the rotating meral-V, which
allows transmission of forces distally to the carpus and proximally to the
meral bridge and saddle. It is on this lateral side of the merus that we
identified the four-bar linkage system which actuates the spring-loaded
raptorial strike during the release phase (Figs
2,
8).
The links comprising the four-bar linkage model are designated as follows
(Fig. 2): Link 1, fixed link:
proximal merus exoskeleton; Link 2, input link: meral-V; Link 3, coupler link:
carpus; and Link 4, follower link: contracted extensor muscle. Previous work
showed that the lateral extensor muscle (Link 4,
Fig. 2) remains contracted
throughout the release phase (Burrows,
1969); contracted muscles are commonly used in biological linkage
systems as fixed-length links (e.g.
Muller, 1987;
Westneat, 1990;
Westneat, 1994). Pivot A is a
fixed pivot point located at the meral-V articulation and located between
Links 1 and 2. Pivot B is not fixed in space and is formed by the lateral
meral-carpal articulation between Links 2 and 3. Pivot C also is not fixed in
space and is located at the lateral extensor apodeme attachment on the carpus
between Links 3 and 4. Pivot D is a fixed pivot formed by the lateral extensor
muscle attachment immediately proximal to the saddle between Links 4 and 1
(there are no muscle attachments to the saddle itself).
Relative link lengths, as measured from photographs of resting appendages and video images of loaded appendages, were statistically indistinguishable with the exception of Link 4 (Table 1). As described above, Link 4 is formed by the contracted extensor muscle and thus is relaxed in the photographs and contracted in the video images of loaded appendages. Link 4 was an average of 14% shorter in an appendage prepared to strike as compared to a resting appendage. The mean starting angle between Links 1 and 2 was 64±5° (± s.d.; mean median 64°; range 40–77°; 24 video sequences, 6 individuals, 1–6 videos per individual).
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Based on the average link lengths and starting angles measured in the high-speed video sequences, we developed an average four-bar model to generate predictions with which to compare the high-speed video data (Figs 9, 10, 11). This four-bar model is operational when input angles range from 63–99° and the input angles used by the mantis shrimp most often range from 64–73°. Within this limited range of input angles, the model output can be approximated as a line with a slope of 3.56 (least-squares linear regression, R2=0.9970, P<0.0001) (Fig. 10). In other words, the model predicts a greater than threefold amplification of an input rotation (Fig. 11). This slope is equivalent to the predicted KT (kinematic transmission) of the system and was used to test whether the measured KT was correlated with predicted KT.
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A four-bar linkage might work over the full 360° rotation of the input link, but that requires a certain set of linkage lengths. A more likely outcome is that that the model is `operational' over a small range of input angles and `jams' at either end of this range. We found that inter-individual variation in link lengths did not change the shape of the model curve substantially; however, the operational range of input angle values shifted substantially (Fig. 9). The input range over which the four-bar is operational if the relaxed extensor lengths are used as Link 4 was from 64° to 82° (mean ± s.d., 74±5°). When that link is contracted, however, as it is before the strike, the operational range was from 40° to 75° (mean 63±9°). The input angles for the contracted Link 4 measured from the video images were not significantly different than those predicted by the model (one-way ANOVA; F=0.5214, P=0.5), whereas when the relaxed Link 4 was used, the input angles were significantly different from the model predictions (one-way ANOVA; F=17.22, P=0.0002) and from the digitized input angles (one-way ANOVA; F=22.04, P<0.0001). When the extensor muscle was relaxed, most of the low end of the operational range exceeded the meral-V input angles measured in the high-speed video sequences and predicted by the model. This means that, for most of the observed input angles, the linkage mechanism would not function when the extensor muscle is relaxed.
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Discussion |
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TOPSummaryIntroductionMaterials and methodsResultsDiscussionReferences |
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)].
Through the basic physical principles of power amplification, the raptorial
appendage generates remarkable forces and speeds
(Burrows, 1969;
Patek and Caldwell, 2005;
Patek et al., 2004). Our
results show that key modifications of the raptorial appendage's merus segment
permit localized flexion, elastic energy storage and transmission of stored
potential energy via linkages. Specifically, the meral-V both acts as
an elastic energy storage device and one of the links in the linkage
mechanism. In addition, the asymmetries in mineralization and joint
architecture allow flexion of the lateral side of the merus that is offset by
the constrained joint system and stiff exoskeleton of the medial side. A
linkage mechanism is formed by the flexible components of the medial side,
while a pair of sclerites controls the release of the strike by sliding
against a simple brace formed by the merus exoskeleton. Thus, through the
actuation of the strike by a pair of sclerites and the stabilization provided
by the medial articulations and reinforcement of the merus, these animals can
strike with remarkable precision to spear elusive fish or hammer hard-shelled
molluscs (Caldwell and Dingle,
1976; Patek and Caldwell,
2005).
Stabilization and control: mineralization, articulations and sclerites
Analysis of the mineralization patterns in the merus and functional
morphology of the joints provide new insights into the stabilization and
articulations of the appendage. Arthropod appendages are usually perceived as
a uniform series of hinged cylinders, yet CT scans (Figs
1,
3,
4) and kinematic analyses (Figs
7,
8,
9) revealed complex joints,
distinct asymmetries in mineralization, and flexion on the lateral and medial
sides of the merus (Fig. 6).
For example, the lateral meral-carpal articulation couples the rotation of the
meral-V to the carpus; in contrast, the medial meral-carpal articulations form
a stabilizing channel, which restricts the carpus to dorsal/ventral movements
(Fig. 5). Perhaps most
surprising is the presence of a flexion point within the merus segment at the
base of the rotatable meral-V; no equivalent flexion point exists on the
opposite side of the merus. Instead, a thickened bar of exoskeleton on the
medial side of the merus opposes the lateral flexion of the meral-V
(Fig. 1). Thus, the dynamic
linkages present on the lateral side are mirrored by a stiff medial wall of
the merus, which lacks any flexion points.
The CT scans also permitted visualization of the highly mineralized
sclerites. These images (Figs
3,
4) depicted the sclerites'
orientation in an undisturbed specimen and suggested a somewhat different
orientation and mechanism of action than previously proposed
(Burrows, 1969). Rather than
using a catch to lock the raptorial appendage during the loading phase,
sclerite 2 slides smoothly over a bracing surface formed by an infolding of
the merus (Figs 3,
4). Sclerite 1 folds above
sclerite 2 and does not have a comparable bracing surface. The use of a smooth
brace, rather than binary latch, explains why previous electromyographic
analyses showed that both the extensor and flexor muscles remain contracted
when the appendage is in a loaded state and why mantis shrimp typically hold
the cocked position for only a brief time period
(Burrows, 1969). This
arrangement also permits mantis shrimp to disengage the system without firing;
the extensor and flexor muscles can simply slowly relax to release the stored
energy over a longer time period. At present, it is not clear whether the two
sclerites have distinct functions or whether sclerite 1 simply serves to
increase the mechanical advantage of the larger sclerite 2
(Burrows, 1969) relative to
the considerable force generated by the opposing extensor muscle
contraction.
These latches may be similar in origin to other latch systems in arthropods
(reviewed in Gronenberg,
1996a). For example, trap-jaw ants generate extreme speeds and
accelerations during their mandible strikes and have evolved latch systems
multiple times using various modifications of joints and mouthparts
(Gronenberg, 1995a;
Gronenberg, 1995b;
Gronenberg, 1996b;
Gronenberg et al., 1998;
Patek et al., 2006). The flea
also uses modifications of the exoskeleton to lock a compressed block of
resilin in place prior to a jump
(Rothschild et al., 1975;
Rothschild and Schlein, 1975).
Similarly, the mantis shrimp's sclerites appear to be mineralized
modifications of the flexor apodemes.
Transmission: kinematics and linkage mechanics
The raptorial strikes follow a characteristic series of movements,
beginning with a brief, 0.9 ms `slide phase' when the propodus slides several
millimeters distally along the merus and no movement of external meral
structures is visible (Figs 7,
8). Then, the saddle begins to
lengthen, the meral-V rotates distally and the propodus, dactyl and carpus
transition to a sweeping rotational movement (Figs
7,
8), which lasts an average 1.8
ms and brings the dactyl heel to an average speed of 14 m s–1
(609 rad s–1). The magnitude and timing of the meral-V and
propodus rotations are correlated, such that greater rotations of the propodus
are correlated with larger meral-V rotations. Furthermore, the propodus
rotates at least twice the meral-V rotation over the course of an entire
strike (Fig. 11).
A four-bar linkage mechanism and the mechanical coupling proposed
previously (Patek et al.,
2004) are generally supported by the transmission of a twofold
rotational amplification of the meral-V to the propodus (Figs
9,
10,
11). However, the KT of the
empirical data is lower than predicted by the model, raising the question as
to whether an alternative model should be considered or, instead, that the
four-bar model is appropriate for the system and some additional effect is
absorbing rotational input of the merus. For example, the incomplete fit of
the model may be caused by non-planar orientation of linkages and the presence
of a sliding cam-type joint between the merus and carpus
(Fig. 5A); this joint could
yield shifting force vectors or lever arms during meral-V rotation. Shifting
mechanical advantage of the contracted extensor muscle relative to the
relaxing flexor muscles during latch release
(Burrows, 1969) may influence
the momentum of the dactyl/propodus/carpus unit as it rotates around this
point [e.g. in bush crickets (Burrows and
Morris, 2003)]. In addition, Burrows noted that strike speeds were
influenced by duration, frequency and timing of both flexor and extensor
muscle activity (Burrows,
1969). Thus, this variable control of muscle activity could cause
a change in length of Link 4, resulting in variable meral-V rotation and
saddle-shortening, again influencing the output of the system
(Fig. 9). All of these
potential variations on the model should be addressed in future studies, and,
although infrequently performed, alternative models to this four-bar linkage
mechanism should be evaluated (e.g. Hoese
and Westneat, 1996; Muller,
1996).
In most systems, a high KT is associated with high speeds, whereas a low KT
is found in systems with large forces. Perhaps counter-intuitively, even with
a relatively high KT of approximately 2, the high-speed system of O.
scyllarus can also generate large forces. Such extreme accelerations,
coupled with an impact between two hard, massive surfaces, cause the strikes
to yield high transient forces that can exceed 1000 N
(Patek and Caldwell, 2005).
Linkage systems that yield a high angular output rotation relative to input
rotation are considered `speed-modified'; however, mantis shrimp produce both
high speeds and forces through extreme acceleration. Thus, these high
transient forces are due to rotational amplification rather than a low KT.
One strength of evaluating linkage mechanics in an arthropod system is the ability to use exoskeletal markers during actual strikes, unlike vertebrate linkage systems in which the link lengths and positions have traditionally been limited to inferences from dissection and external soft markers. Specifically, we were able to measure the effects of varying Link 4 (formed by the contracted extensor muscle) as well as the range of input angles actually used by the mantis shrimp (Fig. 9). Link 4 was 14% shorter in contracted, loaded appendages than in relaxed appendages. When entered into the four-bar model, these longer link lengths yielded greater predicted input angles that were significantly different than the observed and predicted input angle range of a contracted Link 4 length (Fig. 9). Furthermore, we were able to measure actual input angles (Fig. 9) in order to evaluate the mechanical space within the model that is actually used by the mantis shrimp. Both of these approaches offered insights into the variability of the link lengths and input angles across and within individuals, suggesting that rotational amplification is robust across a range of parameters while, at the same time, yielding slightly different performance output.
The twofold KT found in mantis shrimp is high relative to four-bar linkages
evaluated across fish which range, for example, from 0.5 to 1.29 in labrid
fish jaws (Alfaro et al., 2004;
Hulsey and Wainwright, 2002).
In addition, some bony fishes may use a spring-loaded four-bar configuration
by storing elastic energy in the linkage system and then relying on small
shifts in relative position of the links to release the system
(Muller, 1987). Surprisingly,
we were unable to find any published arthropod systems in which a four-bar
linkage mechanism has been analyzed.
Elastic energy storage
In a system as small as this one, the definitive determination of where
elastic energy is being stored is challenging. Two factors determine storage
capacity – the amount of deformation of an element and its stiffness. As
in the crossbow, either character alone is not sufficient. Energy is not
stored to an appreciable extent in the string; although it is bent at an acute
angle, string has little flexural stiffness. There is also little energy
stored in the stiff catch mechanism; it does not deform substantially. The
energy storage is principally in the limbs of the bow; this can be shown by
determining the mechanical properties of these structural elements and
measuring their deflection when the bow is cocked. For the mantis shrimp
strike, some deformations are too small to fully characterize from the video
and the extent of mineralization offers a proxy for stiffness. Here we will
propose a principal storage mechanism, but the testing of the mechanism awaits
nanoindention studies and finer scale resolution of strain in the various
parts of the merus.
Previous research suggested that elastic energy storage in the mantis
shrimp system was provided by the extensor apodeme
(Burrows, 1969), saddle
(Patek et al., 2004) and
unspecified cuticular elements (Currey et
al., 1982) (Fig.
6). Apodeme elasticity was calculated to be insufficient to power
the extreme kinematics of these strikes
(Patek et al., 2004) and it
was suggested that the saddle could provide the additional needed power. We
propose an additional or alternative energy storage structure: the meral-V.
The poor mineralization of the saddle (Fig.
1) means that although the saddle is flexible, it is unlikely that
a substantial amount of energy can be stored through conformational changes of
this structure. Instead, elastic potential energy is probably stored
via multiple sites of cuticular deformation, most likely concentrated
in the meral-V (Fig. 6).
Ultimately, to resolve this debate, mechanical and material tests must be made
directly on the system as a whole and on each of these structures.
The shape of arthropod cuticle, as well as its composition, influences the
presence and degree of elastic energy storage
(Vincent, 1990;
Vincent and Wegst, 2004;
Wainwright et al., 1976).
While the presence of resilin, the arthropod rubber-like protein
(Weis-Fogh, 1960), has not yet
been determined in this system, the shape of the meral-V suggests an elastic
function. The meral-V and the ventral bar extending from its lateral flexion
point resembles the human-engineered tape spring, i.e. a thin strip with a
bend or fold at which elastic energy is stored
(Seffen and Pellegrino, 1999;
Vehar et al., 2004;
Vincent and Wegst, 2004). The
flexion point at the base of the meral-V is similar to the elastic bend in a
tape spring and the poorly mineralized area adjacent to this bar should permit
flexion (Fig. 1). Furthermore,
when manipulated, the meral-V strongly resists flexion and springs back into
an open position when released. The saddle's function, given the intriguing
hyperbolic–paraboloid shape and considerable flexion during the load
phase, remains to be determined. Hyperbolic–paraboloid shells often are
used in engineered systems to reduce local buckling through the presence of
two opposite and transverse curves. Thus, the saddle may provide a flexible,
yet strong, region of cuticle that allows the necessary space on the medial
side of the merus equivalent to the amount of shortening occurring when the
meral-V closes on the lateral side of the merus. However, while the meral-V is
highly variable across stomatopods, the saddle is highly conserved, and
retains its elegant, hyperbolic–paraboloid form across all mantis shrimp
(R.L.C. and S.N.P., personal observation), thus suggesting an important, and
as yet not fully determined, function.
The integration of elastic energy storage and force transmission through
specialized joint articulations is a hallmark of arthropod power amplification
systems (Bennet-Clark, 1975;
Bennet-Clark, 1976a;
Bennet-Clark, 1976b;
Bennet-Clark and Lucey, 1967;
Blickhan and Barth, 1985;
Sensenig and Shultz, 2003).
Not only is there a rich diversity of power amplification systems across
arthropods, including fleas, locusts and snapping shrimp, but even within the
mantis shrimp there is substantial morphological diversity of the saddle,
meral-V and linkage articulations (S.N.P., personal observation)
(Ahyong, 2001). Integrative
analyses of the kinematics, material properties and conformational changes of
these systems will continue to reveal new insights into the origins and
evolutionary diversification of powerful animal movements.
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Acknowledgments |
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TOPSummaryIntroductionMaterials and methodsResultsDiscussionReferences |
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