SUMMARY
The geometry of an animal’s skeleton governs the transmission of force to its appendages. Joints and rigid elements that create a relatively large output displacement per unit input displacement have been considered to be geared for speed, but the relationship between skeletal geometry and speed is largely untested. The present study explored this subject with experiments and mathematical modeling to evaluate how morphological differences in the raptorial appendage of a mantis shrimp (Gonodactylus smithii) affect the speed of its predatory strike. Based on morphological measurements and material testing, we computationally simulated the transmission of the stored elastic energy that powers a strike and the drag that resists this motion. After verifying the model’s predictions against measurements of strike impulse, we conducted a series of simulations that varied the linkage geometry, but were provided with a fixed amount of stored elastic energy. We found that a skeletal geometry that creates a large output displacement achieves a slower maximum speed of rotation than a lowdisplacement system. This is because a large displacement by the appendage causes a relatively large proportion of its elastic energy to be lost to the generation of drag. Therefore, the efficiency of transmission from elastic to kinetic energy mediates the relationship between the geometry and the speed of a skeleton. We propose that transmission efficiency plays a similar role in form–function relationships for skeletal systems in a diversity of animals.
INTRODUCTION
There is a longstanding practice of inferring the mechanical performance of an animal from the geometry of its skeleton. A skeletal geometry that produces a relatively large output displacement per unit input displacement is traditionally considered to be well suited to highspeed motion (Gregory, 1912; Smith and Savage, 1956; Hildebrand, 1974). However, it is largely untested whether this gearing for speed actually produces faster motion under biologically realistic forces. Therefore, we investigated the relationship between skeletal morphology and speed in an exceptionally fast biomechanical system: the predatory strike by the raptorial appendage of the mantis shrimp Gonodactylus smithii (Crustacea, Stomatopoda; Fig. 1A).
The gearing provided by a skeleton is most clear in a simple lever. Force acting on a lever is generated by muscular contraction or the release of energy by an elastic structure. The force created by such an actuator is amplified by the mechanical advantage of the lever, which is equal to the ratio of inlever to outlever lengths. The mechanical advantage is inversely proportional to the displacement advantage, which is the ratio of output to input displacement. This inverse relationship provides the argument that a lever possesses a tradeoff between force and speed that is inherent to its geometry (Gregory, 1912), which we will henceforth refer to as the gearingforspeed hypothesis. This hypothesis predicts that species possessing a relatively high displacement advantage will move faster than related species with a low displacement advantage.
In the past century, the gearingforspeed hypothesis has been used to interpret the potential for fast motion in species with a variety of skeletal lever systems. A classic example is the leg bones of mammals, where the limbs joints of a horse have a higher displacement advantage than a slowmoving armadillo (Smith and Savage, 1956). This practice of using displacement advantage as an index of speed carries through to the contemporary literature. For example, recent studies have inferred the bite speed of extinct dinosaurs from displacement advantage (Barrett and Reyfeld, 2006; Sakamoto, 2010). Such interpretations have been extended beyond simple lever systems. For example, the ability of some muscular hydrostats, such as squid tentacles and lizard tongues, to perform a rapid predatory strike has been attributed to a high displacement advantage (Kier and Smith, 1985). The same argument has been offered for fourbar linkage systems, which are composed of four rigid skeletal elements (i.e. links) that are connected together by four joints (Paul, 1979). The displacement advantage of a linkage system is defined by its kinematic transmission, T_{k}, which is the ratio of output rotation to input rotation (Anker, 1974; Muller, 1987). As for a linear measure of displacement advantage, a high T_{k} has been interpreted as facilitating rapid motion (Muller, 1987; Westneat, 1995). In support of this view, fish species that feed on evasive prey have higher values of T_{k} than related species that feed on slowmoving prey (Westneat, 1995; Wainwright et al., 2004).
Models that consider the dynamic forces of lever systems offer a more complex view of the relationship between the geometry and the speed of a skeleton. The gearingforspeed hypothesis holds true in cases where an actuator contracts at a fixed rate, irrespective of the load it encounters. Under such conditions, the linear output speed of a system will be proportional to its displacement advantage. However, both elastic actuators and muscles generally shorten more slowly when resisted by a greater load, and such resistance may depend on the lever’s geometry. For example, the leg muscles of the frog Xenopus laevis shorten more slowly when they actuate a lever system with a higher displacement advantage, due to the force–velocity relationship that is intrinsic to muscle (Richards, 2011). Therefore, the speed gains of a high displacement advantage may be offset by a slower rate of shortening by the actuator. To account for how a force transmission system may affect both actuators and loads, forward dynamic models can predict the speed of motion from a consideration of mechanics (Zajac, 1993; Daniel, 1995). The present study has adopted the use of such modeling to examine how speed varies across changes in leverage when the input energy remains fixed.
The predatory strike of a mantis shrimp is powered by elastic energy stored within its exoskeleton. This energy is generated by a lateral extensor muscle that adducts a Vshaped element (the meralV; Fig. 1B) at the distal end of the merus segment of the second thoracic appendages, known as the ‘raptorial appendages’. The strike begins when the meralV is released by a latch mechanism at the start of a strike (Burrows, 1969; Patek et al., 2004; Patek et al., 2007; Zack et al., 2009). During the strike, the merus functions as a spring through the abduction of the meralV, a mechanism that we will henceforth refer to as the ‘meral spring’. Mantis shrimp transmit the force for a predatory strike with a fourbar linkage (Fig. 1A–C) (Patek et al., 2007). The meral spring engages the fourbar linkage (Fig. 1C) (Patek et al., 2007), which is rigidly coupled to distal segments that consequently swing away from the merus to strike a prey. These segments (the carpus, propodus and dactyl; Fig. 1B) rotate as a uniform rigid mass, the striking body, to impact the prey. The mass of the striking body and the fluid forces it generates serve to resist the action of the meral spring to power the strike.
The present study considered the interacting dynamics of the meral spring, the striking body and fourbar linkage to determine how the rotation speed of a strike depends on the morphology of the raptorial appendage. This was achieved by developing a mathematical model of these dynamics and performing a series of experimental measurements to characterize the system. We tested the model by comparing its prediction of strike impulse with measurements. Once verified, the model provided the basis for exploring how morphological differences affect strike speed.
MATERIALS AND METHODS
Our mathematical model required a variety of experimental techniques to comprehensively characterize the mechanics of a raptorial appendage. To calculate the torque generated by the meral spring during a strike (τ_{spring}; Fig. 1D), we measured its stiffness by materials testing. Measurements of the linkage lengths and the angle of the meral spring were used to calculate the gearing of the fourbar linkage over the course of a strike (Fig. 1C). According to our model, the torque applied to the striking body (τ_{applied}) is inversely proportional to the T_{k} of the fourbar linkage (T_{k}=dθ/dγ, where θ and γ are the output and input angles, respectively; Fig. 1C) and is resisted by the moment of inertia of the striking body and the drag generated by the surrounding water (Fig. 1D). We modeled the drag force from morphometrics of the striking body and measured the moment of inertia using density and shape information from microcomputed tomography (CT). Once verified with strike impulse measurements, a series of simulations were conducted to evaluate how differences in the meral spring, fourbar linkage and striking body affect the rotation speed of a strike.
Mathematical model
Linkage geometry
The geometry of the fourbar linkage system was measured from the coordinates of each joint [as in Patek et al. (Patek et al., 2007)]. These coordinates were defined within a twodimensional Cartesian coordinate system with the unit vectors x and y and an origin, a, that was fixed to the merus and denoted the pivot point for the meralV (Fig. 1B,C). The joint d was also fixed to the merus and defined the direction of y. Joints b and c moved relative to the merus, but maintained fixed distances from adjacent points over the course of a strike because of the rigidity of skeletal elements in each link. The angle between links 1 and 2 served as the input angle (θ) to the linkage system, which increased in value over the course of a strike as the meralV rotated away from the merus. All angles were recorded in radians for our calculations, but were presented in degrees to conform with the convention in the literature. The input angle between the meralV and the carpus determined the coordinates of joint b, as follows: where l_{2} is the length of link 2 (Fig. 1C). The distance between b and d (h_{b,d}) was useful for defining the rest of the linkage system and was calculated as follows: where l_{1} is the length of link 1. The geometry of a fourbar linkage requires that this distance occurs within bounds defined by the following inequalities: h_{b,d}<l_{3}+l_{4}, h_{b,d}>l_{4}–l_{3}, h_{b,d}<l_{1}+l_{2}, and h_{b,d}>l_{1}–l_{2}, where l_{3} and l_{4} are the lengths of links 3 and 4, respectively. The distance h_{b,d} factors into the calculation of the angle between links 1 and 4 (Ψ) and the position of the second carpus joint (point c) (Fig. 1C):
Using these geometric definitions, we calculated the T_{k} of the linkage systems computationally. The output angle, γ, was defined as the angle of link 3, about joint c and relative to the xaxis (Fig. 1C). We calculated values for γ from coordinates b (Eqn 1) and c (Eqn 4) for 1000 values of θ over the entire range allowed by the abovementioned inequalities for h_{b,d}. Cubic splines were fit to the numerical values of these angles in order to find the continuous functions γ(θ) and θ(γ), which were used in our simulations (described below). T_{k} was calculated as the first derivative of the output angle function [T_{k}(θ)=γ′(θ)]. We used the minimum value of T_{k} (T_{k,min}) to provide a single value to characterize a particular geometry. An alternative to our numerical approach would have been to calculate γ(θ) analytically, which requires a series of large equations that do not offer a more elegant expression than our computational method. All analyses and modeling for the present study were performed with scripted programs within MATLAB (R2009a, MathWorks, Natick, MA, USA), unless mentioned otherwise.
Strike dynamics
The torque created by the meral spring functions to abduct the meralV. We modeled this torque with the following equation: where k_{torsion} is the stiffness of the torsion spring (N m deg^{–1}) and θ_{rest} is the resting angle for the spring, which we measured from photographs of the appendage at rest (described below). The meralV is a flexible strip of exoskeleton that extends when it is released during a strike and maintains a fixed position when at rest (Patek et al., 2007). The torque created by the meral spring is applied to the striking body to a degree that depends on the instantaneous T_{k} of the fourbar linkage system. Using the abovementioned spline functions for T_{k}(θ), we calculated the torque applied to the striking body through the linkage system using the following relationship (Fig. 1D): This torque was considered to act at joint b on the striking body, which we modeled as a pin joint with a fixed position in the global frame of reference. This assumes that translation of the proximal base of the striking body (at b) has a negligible influence on strike dynamics. We evaluated this assumption with the preliminary results of a more complicated model that allowed translation. Considering the particular geometry of G. smithii fourbar linkage, we found that the kinetic energy created by translation was more than two orders of magnitude below the kinetic energy generated by rotation. By neglecting this small component of motion, the mathematics of our model were greatly simplified.
We modeled the torque created by drag on the striking body with a bladeelement approach that integrates the force generated by elements along the length of the striking body (Patek et al., 2004). This torque was calculated with the following equation (Hoerner, 1965): where ρ_{water} is water density (1024 kg m^{–3}), dγ/dt is the speed of output angle rotation, H is the total length of the striking body and D is the drag torque index. We define the drag torque index as a dimensionless parameter that indicates the capacity for the shape of the striking body to generate torque from drag. We calculated this parameter by approximating the shape of the striking body as an elliptical cylinder, which we varied by modifying its dimensions along its length. The drag torque index was calculated with the following integral: where each element has a position along the length of the dactyl, h_{dac} (up to the total length of the striking body, l_{SB}), a width, T (perpendicular to flow), a distance from b, r, and a drag coefficient, C_{d}. The drag coefficient for an elliptical cylinder at subcritical Reynolds numbers is approximated by the following equation (Hoerner, 1965): where C is the chord width (in the direction of flow) of the elliptical section.
The motion of the striking body is also resisted by its mass. The total mass and the proximity of its center to the axis of rotation determine the degree of the resistance provided by the striking body, as indicated by its moment of inertia, I_{SB}. The striking body also accelerates a mass of water as it moves that provides additional resistance that is proportional to its moment of inertia, I_{water}. As detailed below, I_{SB} was measured from CT scans of the striking body. I_{water} was calculated from the same measurements of T used to calculate drag (Eqns 8 and 9), using the following equation (Sarpkaya, 1986):
We formulated an equation for the dynamics of the striking body as the basis for our simulations. This governing equation was formulated as the sum of torques acting at b, set equal to the product of the moment of inertia and the rotational acceleration: This is a secondorder, ordinary differential equation, where torque generated by drag (τ_{drag}) depends on the square of dγ/dt (Eqn 7) and τ_{applied} depends on θ, which varies with γ (Eqns 5 and 6).
We simulated individual strikes by finding solutions to the governing equation numerically. This was achieved with an explicit fourthorder Runge–Kutta method [the ‘ode45’ function in MATLAB, based on Dormand and Prince (Dorman and Prince, 1980)], with a zero initial velocity and an initial position (γ_{0}) that was calculated [using spline functions for γ(θ), described above] from the initial position of the meralV (θ_{0}). For each simulation we calculated the angular momentum, L, as follows: We used this calculation to test the accuracy of our simulations against strike impulse measurements (described below). This was achieved by estimating the linear momentum, P, of the striking body at the time of impact as P=L/h_{out}, where h_{out} is the distance between b and of the center of mass of the striking body. This distance was calculated with the following equation (McHenry, 2011): Strike energetics For each of our simulations, we calculated strike energetics to examine how the morphology of the raptorial appendage affects the efficiency of a strike. The elastic energy stored within the meral spring was calculated with the following equation: We calculated the kinetic energy of the striking body with the following equation: The power output of the striking body was calculated as the first time derivative of kinetic energy. The energy lost from the appendage due to drag was calculated with the following equation: where γ_{0} and γ_{t} are the output angles at the start and at time t, respectively.
We considered strike energetics for versions of our model that both included and excluded drag. In the absence of drag, energy conservation was assumed and the ultimate rotation speed generated by the striking body was predicted by setting the elastic energy at the start of the strike equal to the kinetic energy at the end of the strike (i.e. when the meral spring has reached its resting position). This equality permits the following prediction of the maximum rotation speed for a simulation without drag (McHenry, 2011): where θ_{0} is the initial position of the meral spring. From this equation, the maximum angular momentum generated by the striking body, L_{max}, may be calculated as the product of I_{SB} and Eqn 17: Energy conservation was not assumed in simulations that considered the drag generated on the striking body. In this version of the model, drag acts to reduce the efficiency of a strike. We calculated this effect of drag with the transmission efficiency, η, defined by the following equation:
Measurements
We conducted a series of measurements to provide parameter values for our model and to test its accuracy. For these measurements, individuals of Gonodactylus smithii Pocock 1893 were collected on Lizard Island, Queensland, Australia (Great Barrier Reef Marine Parks Permit #G07/23055.1) and transported to the University of California, Berkeley, for experimentation. The animals were maintained in individual plastic cups (2 l) with artificial seawater (salinity: 32–36 ppt; 22°C) and were fed fresh grass shrimp twice per week until strike impulse experiments were performed (described below). Materials tests were then conducted to establish the stiffness of the meral spring (described below). Body length and body mass were measured for each individual. A total of 10 animals (five males and five females), ranging from 58 to 64 mm in length, were used.
Fourbar linkage geometry
The geometry of the linkage system was determined from coordinates that were visually selected from highresolution photographs of the raptorial appendage (12 megapixel DSLR camera, Nikon D300, AF MicroNIKKOR 60 mm f/2.8D or 105 mm f/2.8D macro lenses, Nikon Inc., Melville, NY, USA; and EM140 DG macroflash, Sigma Corp., Ronkonkoma, NY, USA) (Fig. 2A). The lateral sides of the left and right appendages were each photographed three times and we used the mean values of coordinates recorded from all of these measurements. The four coordinates for the fourbar linkage (a–d in Fig. 1B) were recorded with the software TpsDig2 (Rohlf, 2005) for the morphology in its loaded configuration (i.e. the coordinates of the meralV were rotated proximally after digitizing to represent the contracted configuration of the fourbar linkage).
Stiffness measurements
We measured the stiffness of the meral spring that drives the motion of the raptorial appendage. Measurements were performed as in previous studies (Zack et al., 2009; Claverie et al., 2011) by pulling the carpus and the meralV towards the merus – the same movement as performed by living mantis shrimp prior to a strike (Patek et al., 2007). Forcedisplacement measurements on the spring were acquired using a materials testing machine (Blue Hill v. 2.12 Instron model 5544, Instron Corp., Canton, MA, USA; load cell model 2530416, 0.5 kN maximum, 0.125 N resolution or 0.25% of load). As previously reported (Zack et al., 2009), the force created by the meral spring may be approximated as a linear function of displacement with a slope equal to the linear spring stiffness, k_{linear}. We transformed these measurements of linear stiffness into torsion spring stiffness (k_{torsion}) for our model, using the following equation: where h_{load} is the distance between the base of the meralV (a; Fig. 1B) and the point of force application on the carpus and ν is the angle between the direction of the load and the meralV (link 2; Fig. 1C). This relationship assumed that the deflections are small relative to h_{load}. We measured ν and h_{load} before loading began from video stills of the materials testing setup using ImageJ (National Institutes of Health, Bethesda, MD, USA).
Striking body morphology
We used density and shape information from CT scans to measure the moment of inertia (I_{SB}) of the striking body. CT scans of the carpus, dactyl and propodus were acquired at Harvard University Center for Nanoscale Systems (Model HMXST225, XTek, Nikon Metrology NV, Leuven, Belgium). Images were analyzed and produced using 3D reconstruction software (VGStudio Max v. 2.0.5, Volume Graphics GmbH, Heidelberg, Germany). The pixel intensity of voxels within a CT scan indicates the radiodensity of a sample, which is proportional to the mass density of biological tissues, ρ_{tissue} (kg m^{–3}) [i.e. ρ_{tissue}=C_{pix}p, where p is pixel intensity (dimensionless) and C_{pix} is a constant (kg m^{–3})] (Seeram, 2000). Therefore, I_{SB} may be calculated from a CT scan using the following equation: where v is the volume of voxel i, n is the total number of voxels, and r_{vox} is the distance between the voxel and the axis of rotation (at b; Fig. 1C). We identified the medial and lateral joint articulations between the carpus and merus as the point of rotation for the dactyl plus the propodus. CT scans can provide a measure of the total mass of a specimen with the following relationship: We used Eqns 21 and 22 to formulate a nondimensional moment of inertia, I_{SB}^{*}, for the striking body (, where m_{SB} is the mass of the striking body) that varies with the shape, but not the size of the structure. We defined this quantity as follows: where H is the total length of the striking body. This measure of the moment of inertia has the advantage of not requiring the calibration of the CT scans for tissue density (i.e. measuring C_{pix}). Furthermore, I_{SB}^{*} allows for the moment of inertia to be defined in our modeling for a striking body of any size if the mass and dactyl length are known. We measured I_{SB}^{*} from three scans of striking bodies of different size (m_{SB}=31.6, 211.0 and 402.2 mg and H=0.57, 1.10 and 1.27 cm, respectively).
Measurements of the shape of the striking body were used to calculate the drag torque index (Eqn 8). This was achieved by thresholding images of the striking body from headon and side views to produce a binary image with the striking body appearing black on a white field. This procedure and masking of the proximal section of the appendage was completed in Photoshop (CS5, Adobe, San Jose, CA, USA). We then analyzed the contours of these images to calculate the chord width (C) and width (T) of the striking body along its length.
Scaling analysis
Morphological parameters of the raptorial appendage have the potential to vary with the size and sex of a mantis shrimp (Claverie et al., 2011). In order to consider the scaledependency of the dynamics of the raptorial appendage, we measured the scaling of parameters in our model (carapace length, total length, l_{1}, l_{2}, l_{3}, l_{4}, H, θ_{0}, θ_{rest}, k_{torsion}, D and T_{k,min}) with respect to body mass (m_{b}) for individuals of each sex. Each parameter was considered to have the potential for exponential scaling, as given in the following relationship: where y is a morphological parameter, B is the scaling factor and A is a scaling constant. We performed a reducedmajor axis regression on logtransformed values of body mass and each parameter in order to find values for A and B. The degree of certainty in these values was determined by calculating values of lower and upper 95% confidence intervals (CIs). Scaling was considered isometric if the 95% CIs included the scaling factor predicted by isometry, B_{iso} (Rayner, 1985; Sokal and Rohlf, 1995). For example, linear dimensions were predicted to scale to the 1/3 power of body mass (i.e. B_{iso}=1/3), but angles and dimensionless parameters were predicted to be invariant with mass (i.e. B_{iso}=0).
Strike impulse measurements
Impact force was measured from a threeaxis force sensor (stainless steel, 16.5 mm diameter load surface, 170 N μm^{–1} stiffness for the x and yaxes and 740 N μm^{–1} for the zaxis, force range between 2 N and 2 kN, Model 9017B, linked to a charge amplifier model 5073B; Kistler Instruments, Winterthur, Switzerland) that was struck by each individual at least 10 times. Data were collected at 500 kHz using a data acquisition board (NI USB6251, National Instruments, Austin, TX, USA). We recorded the natural frequency of the submerged transducer to be 23.4 kHz, which is more than 100 times the frequency calculated from the period of impact (mean±s.d.=0.15±0.13 kHz). We calculated the impulse of a strike by numerically integrating the peak force generated by the initial impact. The mean values and 95% CIs of measurements of strike impulse were compared with model predictions of linear momentum and thus used to validate the model.
RESULTS
Measurements
We examined the scaling of the fourbar linkage system for both males and females of G. smithii. With only one exception, morphological parameters for both sexes (Table 1) were found to scale isometrically with respect to body mass (Table 2). The only exception was T_{k,min}, which approximately scaled with body length (B≈1/3; Table 2), rather than the isometric prediction (B_{iso}=0). In addition, we found that all linkage lengths in G. smithii were sexually dimorphic, with males possessing slightly larger linkage lengths than females of comparable size (Fig. 2).
We examined how T_{k} changes over the range of motion for linkage systems of different geometry. The geometry was altered in these comparisons by varying the length of individual links about the measured values (Table 1). The linkage system was modeled as a massless component of the system, such that these changes in linkage lengths influenced only T_{k} and not our calculations of the moment of inertia or hydrodynamic forces. Low and highT_{k} systems were created by respectively increasing and decreasing l_{3} beyond measured values by 0.25 mm (Fig. 3A,B). For the same input displacement (Δθ), the highT_{k} system exhibited a much larger output displacement (Δγ) than the lowT_{k} system (Fig. 3A,B). The relationship between input and output displacements is nonlinear, as revealed by calculating the output angle predicted over the range of possible input values (Fig. 3C). The T_{k} calculated from this relationship (Fig. 3D) exhibited a Ushaped curve, where T_{k} may theoretically be increased by either increasing or decreasing the input angle beyond the value for minimum T_{k}. For a highT_{k} system, T_{k} exhibited a pronounced increase within the range of motion measured for G. smithii. In contrast, a lowT_{k} system did not achieve substantial change in T_{k} over the same range (Fig. 3D). We found that a reduction in l_{3} caused change in T_{k,min} with a strong nonlinear dependency not exhibited in any of the other linkage lengths (±0.5 mm; Fig. 3E). Because of the strong sensitivity of T_{k} to differences in link 3, we manipulated l_{3} in our subsequent simulations to evaluate the effects of T_{k}. The implications of these changes in geometry to the range of motion were also determined (Fig. 3F).
The capacity of a raptorial appendage to generate drag was indicated by the drag torque index, D. According to our bladeelement model of drag, the striking body encounters resistive torque in proportion to D (Eqn 7). To calculate this index for an appendage, we used photographs of the striking body (Fig. 4A,B) to measure the chord width and thickness along the dactyl length from silhouettes of the side (Fig. 4C) and headon (Fig. 4D) views. These measurements (Fig. 4E) provided the basis for calculations of the drag coefficient, C_{d} (Eqn 9), at each position along the length of the striking body (Fig. 4F). The drag torque index (Eqn 8) was calculated as the integral of the product C_{d}Tr^{3} along the length of the striking body (Fig. 4G). This was achieved for all individuals modeled (Table 1) to yield a mean drag torque index of 0.072 (95% CI=0.069, 0.075, N=10), with no significant differences due to body size or sex (Table 2).
We used CT scans to measure the nondimensional moment of inertia for the striking body (Fig. 5A,B). Our three specimens spanned more than an order of magnitude in mass, yet yielded I^{*} with similar values (0.26, 0.31 and 0.28; Fig. 5C) that produced a mean value with relatively small 95% confidence intervals (I^{*}=0.28, 95% CI=0.22, 0.35). This value is not significantly different than a uniform slender rod of uniform density (I^{*}=1/3) (Meriam and Kraige, 1997). Therefore, the mass distribution of the striking body is similar to a homogenous structure, despite the substantial variation in the distribution of highdensity calcified regions that we observed within the striking body (Fig. 5B).
Materials testing experiments determined the stiffness of the meral spring. The mean stiffness for males (k_{torsion}=0.014 N m deg^{–1}, 95% CI=0.006 and 0.020 N m deg^{–1}, N=5) was not significantly different from that of females (k_{torsion}=0.015 N m deg^{–1}, 95% CI=0.010 and 0.019 N m deg^{–1}, N=5; Table 1). Within each sex we found relatively minor variation among individuals that showed no significant effect of body size (Table 2).
Force measurements were used to determine the strike impulse generated by a raptorial appendage (Fig. 6A). The strike impulse was calculated as the integrated area within the initial force peak created upon impact (Fig. 6B); the mean impulse was 3.1 g m s^{–1} (95% CI=1.3, 4.9 g m s^{–1}, N=10). These measurements provided a basis for testing the accuracy of our mathematical model. From these results, we calculated the momentum of the striking body (Eqn 12) generated by changes in the output angle (Fig. 6D) and interpreted the maximum of these values as the predicted strike impulse. The mean value of this prediction (2.2 g m s^{–1}, N=10) and its 95% confidence intervals (1.7, 2.7 g m s^{–1}) were within the 95% confidence intervals of the measurements and therefore not significantly different (Fig. 6E). By this criterion, the model provides an accurate approximation of the dynamics of the strike by a raptorial appendage.
Mathematical model
Our simulations provided the opportunity to examine the energetics over the time course of a strike. The elastic energy stored within the meral spring at the start of a strike was converted into kinetic energy and lost to drag as the striking body moved through the water (Fig. 7A). In contrast, all energy stored within the meral spring was converted to kinetic energy in simulations that did not include drag (Fig. 7B). Drag generates inefficiency in the conversion from elastic to kinetic energy by two mechanisms. First, the creation of E_{drag} (energy due to drag) necessitated a direct loss of energy from the appendage to the surrounding water. Second, as more energy was lost to drag, the timing of peak kinetic energy occurred earlier in the strike. As a consequence, the meral spring did not reach its resting length at the moment of maximum angular velocity, and therefore fails to release all of its stored energy at the time of peak kinetic energy (Fig. 7A).
The effects of elastic energy storage on strike performance were investigated by running a series of simulations with variable amounts of elastic energy within the meral spring prior to the strike. We conducted a set of 10 simulations that each began at the parameter values for a particular individual (Table 1). Each simulation in a set differed by only the spring stiffness, which we increased at even intervals up to 0.013 N m deg^{–1} above the measured value. For example, we ran a series for individual 4 using its measured parameter values (k_{torsion}=0.030 N m deg^{–1}, purple curve in Fig. 8) and then repeated simulations with incrementally increased spring stiffness (up to k_{torsion}=0.043 N m deg^{–1}). In simulations without drag, the calculated speed and momentum (Fig. 8A,B) were predictable from our analytical model of strike energetics. This model shows that both rotation speed and angular momentum varied with the square root of spring stiffness (Eqns 17 and 18). Although drag decreased the speed and momentum of simulated strikes, spring stiffness had a similar effect on these variables (Fig. 8Aii,Bii) as in the simulations without drag (Fig. 8Ai,Bi). In simulations both with and without drag, the time to reach maximum rotation speed decreased monotonically with stiffness (Fig. 8C) because greater spring stiffness created a larger torque to rotate the striking body (Eqns 5 and 6).
Similar to our investigation into the effects of spring stiffness, we ran a series of simulations that varied the mass of the striking body (Fig. 9). This manipulation influenced the governing equation (Eqn 11) by changing the moment of inertia (, where and H were assumed to be constant). In the absence of drag, rotation speed is inversely proportional to the square root of mass (Eqn 17), and momentum (Eqn 18) is directly proportional to the square root of mass (Fig. 9Ai,Bi). Similar effects of mass were apparent in simulations that included drag (Fig. 9Aii,Bii).
We examined two simulations that demonstrated the effects of a difference in linkage geometry. Both received an equal amount of stored elastic energy, but they differed in T_{k} because of a difference in the length of link 3 (Fig. 1C). Because of this difference, the highT_{k} system swept the striking body with a greater displacement of output angle than a lowT_{k} system (Fig. 10A). However, the lowT_{k} system achieved a higher rotation speed (Fig. 10B) and power output (Fig. 10C) over its shorter displacement. This result was a consequence of drag generation. Drag energy (Fig. 10D) was calculated as the integral of the torque created by drag over the output angle displacement (Eqn 17). The lowT_{k} system spanned a more narrow range of the output angle that generated relatively low drag energy (red area in Fig. 11). Therefore, the lowT_{k} system generated a higher transmission efficiency (Eqn 20) than the highT_{k} system (blue area in Fig. 11).
The effects of T_{k} on strike performance were further examined with a series of simulations. Similar to the abovedescribed comparison, these simulations began with an equivalent amount of stored elastic energy and differed only in the length of link 3. These simulations revealed that the time to maximum speed increased (Fig. 12A), and the maximum speed of the input angle (Fig. 12B) and power output (Fig. 12C) decreased monotonically with greater T_{k,min}. In simulations without drag, differences in T_{k} had no effect on the maximum rotation speed (Fig. 12D). However, when drag was included, the maximum rotation speed decreased monotonically with increasing T_{k,min}. This trend was attributable to the effect of T_{k} on transmission efficiency. Linkage systems with greater T_{k} achieved a greater output angle excursion, causing more energy to be lost to drag (Fig. 11), which caused a reduction in efficiency and, in the end, a slower strike (Fig. 12D).
DISCUSSION
The present study demonstrates that the predatory strike of the mantis shrimp can be modeled from measurements of its morphology and structural properties (Fig. 6). An analysis of this model offers insight into the roles of elastic energy storage, the striking body and the force transmission system in a predatory strike. These results illustrate how the speed of a force transmission system is mediated by transmission efficiency, which is a general principle that may be applied to the biomechanics of a broad diversity of animals.
Elastic energy storage
The rotation speed of a strike is largely determined by the amount of stored elastic energy imparted to the raptorial appendage. The meral spring may theoretically store a greater amount of energy by elevating the stiffness of the spring or increasing its degree of compression prior to the strike (Eqn 14). As a consequence, the maximum speed of output rotation is proportional to the prestrike compression and the square root of stiffness (Eqn 17) in the absence of drag (Fig. 8Ai), with a similar relationship in the presence of drag (Fig. 8Aii). Therefore, mantis shrimp may have evolved to enhance their strike speed by increasing the elastic storage capacity of the meralV and the work output of the extensor muscles that primarily deflect this structure (Claverie et al., 2011).
The architecture of the extensor muscles has likely influenced the evolution of the meral spring. The fibers of these muscles have a pinnate architecture (Burrows, 1969; Burrows and Hoyle, 1972; McNeill et al., 1972), which is a shape that is ubiquitous among crustaceans and is thought to offer an advantage over parallelfibered muscle because it creates a relatively modest bulging in diameter. This allows a muscle to occupy a relatively large proportion of the crosssectional area within the rigid exoskeleton (Alexander, 1968). Pinnate muscles generate relatively high force, but must do so over a low strain. Therefore, the extensor muscle likely constrains the range that the meral spring may be deflected in preparation for a strike.
The striking body
A strike is resisted by the inertia and drag torque generated by the striking body. This resistance is largely generated in the distal region, where the striking body’s mass and projected area are large (Fig. 5C, Fig. 4D). Despite these effects of shape, the striking body may be approximated by a simple geometry, as in previous models (e.g. Patek et al., 2004). For example, our analysis of CT scans revealed that the dimensionless moment of inertia for the striking body is similar to a cylinder of uniform density (I^{*}≈1/3) (Meriam and Kraige, 1997) and this result holds true over more than an order of magnitude in mass (Fig. 5C). Similarly, the torque drag product (Fig. 4) does not vary substantially among individuals (Table 1) and could be approximated by modeling the striking body as a uniform cylinder. Therefore, differences in the shape of the striking body among individuals of G. smithii do not have a large effect on strike dynamics. Given the large morphological diversity among mantis shrimp species (Ahyong, 2001), interspecific differences in the shape of the striking body may have a more substantial influence on the predatory strike than what we have found with a single species.
Our model of drag neglects some fluid forces that could arguably affect strike dynamics. This includes skin friction generated at the distal end of the striking body due to large shear stress in that region. Our bladeelement approach also ignores any effect of spanwise flow along the length of the striking body. Although such factors may contribute higherorder effects, the matching of our results with measured kinematics (Fig. 6) suggest that our model offers an accurate firstorder approximation of the major forces generated during a strike. Furthermore, additional fluid forces would serve to enhance the loss of energy to drag and thereby produce an even stronger negative relationship between T_{k,min} and maximum rotation speed (Fig. 12).
Differences in the size of the striking body affect the strike performance of G. smithii. The mass and length of the striking body are major factors in the more than twofold range of variation in the maximum rotation speed, angular momentum and time to maximum speed predicted by the model (Fig. 9). Because the strike is modeled as moving with pure rotation, its inertial resistance is determined solely by the sum of the moment of inertia of the striking body (I_{SB}) and the surrounding water (I_{water}, Eqn 10). We varied I_{SB} in a series of simulations that differed only in the mass of the striking body (), which yielded analytically predictable results without drag (Eqn 17) and similar results with drag (Fig. 9). These results suggest that large individuals encounter disproportionately high inertial resistance that slows the strike, but the greater mass of the striking body generates more momentum during a strike. Specifically, the angular momentum is nearly proportional to the square root of the moment of inertia (Eqn 18). However, because these higher momentum values are achieved at slower rotation speed (Fig. 9A,B), it would be more difficult for a massive striking body to create the cavitation bubbles that contribute a substantial force when striking prey (Patek and Caldwell, 2005). Furthermore, the torque created by drag scales with length to the power of 5 (Eqn 7), which suggests that a larger striking body may be substantially less efficient than a smaller one.
The linkage system
In the absence of drag, differences in the geometry of the fourbar linkage system have no effect on the speed of rotation for a strike when stored elastic energy is constant. This conclusion is based on the results of a series of simulations run with different T_{k,min} (by varying l_{3}) and the same amount of stored elastic energy. In simulations that excluded drag, all of the elastic energy at the start of the strike was converted into the kinetic energy of the striking body (Eqn 17), irrespective of differences in T_{k,min} (Fig. 12D). As a consequence, the maximum speed was independent of differences in linkage geometry, where a constant moment of inertia was assumed. Therefore, the maximum rotation speed of an elastic system that conserves energy is independent of geometry whether force transmission is achieved by a linear lever or a fourbar linkage system.
Our simulation that included drag demonstrated that drag causes a loss of energy that substantially affects strike speed. This loss is greater for a highT_{k} system, which sweeps the striking body through a longer angular displacement than in a lowT_{k} system with the same input displacement (Fig. 10A). The angular displacement defines the range over which the drag torque is integrated to determine the energy lost during a strike (i.e. the area under the curve in Fig. 11; Eqn 16). As a highT_{k} system transfers relatively more energy to drag, less elastic energy is available for the kinetic energy of the striking body, which slows the strike. This effect is reflected in the transmission efficiency (Eqn 19): increasing T_{k,min} creates a monotonic decrease in the transmission efficiency of a strike, which causes a decrease in maximum power and rotation speed (Fig. 12C,D). Therefore, strike speed is affected by T_{k} because differences in linkage geometry affect the generation of drag and, consequently, transmission efficiency.
Efficiency in force transmission systems
Our finding that a highT_{k} system moves more slowly than a lowT_{k} system is surprising in the context of some research on skeletal mechanics. As described in the Introduction, the linear speed of a simple lever system is related to its displacement advantage, because it is proportional to the ratio of output to input displacement (Gregory, 1912; Smith and Savage, 1956; Hildebrand, 1974). If different levers are actuated with the same input velocity, then their output velocity will be proportional to displacement advantage. Similarly, the velocity of output rotation created by a fourbar linkage system is proportional to T_{k}, if the input rotation velocity is fixed across differences in geometry (Muller, 1987; Westneat, 1995). Therefore, this gearingforspeed hypothesis predicts that linkage systems with greater T_{k} and lever systems with greater displacement advantage should move at faster speeds.
The present model differs from a model that supports the gearingforspeed hypothesis because we have not assumed that a fixed input velocity drives raptorial appendages of different geometry. Velocity was instead determined by the dynamic torques created by the meral spring and the moment of inertia and drag on the striking body (Fig. 1D). Therefore, our simulations permitted the input velocity and power output to vary with time within a simulation (Fig. 10) and to vary with morphology across simulations (Fig. 12). In all simulations, we provided the force transmission system with an equal amount of energy, which is particularly appropriate for comparisons of elastically driven systems. Once released, a spring has no inherent constraint on the rate at which it may deliver energy. Therefore, the input velocity for an elastic system was not fixed with respect to time and no a priori value for velocity was assumed to model the motion of the system.
Our model further deviates from many previous considerations of lever mechanics by examining the influence of hydrodynamic forces. Drag functions as an energetic sink that reduces the kinetic energy available to the appendage and thereby slows its motion. This effect is indicated by the transmission efficiency, which declines with the higher integrated drag torque created by a linkage system having a relatively high T_{k} (Fig. 11, Fig. 12D). Therefore, the relationship between the geometry of a force transmission system and its rotation speed is mediated by transmission efficiency.
Transmission efficiency may play a similar role in the feeding apparatus of fishes, which is a classic subject for the study of linkage biomechanics. Like a mantis shrimp, a striking fish predator must accelerate the mass of its skeleton and overcome hydrodynamic forces in order to capture prey. If there is a significant amount of energy lost to the surrounding water during a strike, then speed may be hindered by a strike exhibiting a high displacement. Therefore, fish species possessing a highT_{k} feeding apparatus may be hindered in their ability to move faster, compared with closely related species having lower T_{k}. Seemingly at odds with this prediction is the observation that fish that feed on evasive prey possess a higher T_{k} than relatives that feed on slow prey (Westneat, 1995; Wainwright et al., 2004). However, species that prey on evasive prey may not be protruding their jaws any faster than their relatives. Instead, highT_{k} species may be taking advantage of the superior displacement of the jaw apparatus that is provided by a highT_{k} system. Jaw protrusion has been identified as a major benefit to the performance of suctionfeeding predators (Holzman et al., 2008). Although the previous interpretations of evolutionary variation in fish jaw T_{k} have largely focused on speed as the relevant parameter, it may instead be the benefits conferred by a greater reach that determine the selection for high T_{k} in fish jaw mechanics.
Transmission efficiency may be an important factor for lever systems in animals that are not aquatic. For example, a toad (Bufo alvarius) generates a ballistic mouth opening by storing elastic energy within its depressor mandibulae muscle prior to mouth opening. During the strike, the force generated by this muscle may be modeled as a spring in parallel with a dashpot (Lappin et al., 2006). This dashpot serves to dissipate energy in a manner that is analogous to drag on the raptorial appendage of a mantis shrimp. Unlike drag on an appendage, the toad’s energetic loss occurs on the input end of the lever system. Therefore, the transmission efficiency and speed of a toad jaw may be maximized by moving the jaws with a relatively small input displacement. As a consequence, increases in displacement advantage should increase transmission efficiency and result in faster motion.
The significance of transmission efficiency may be different for lever systems that contribute to animal locomotion. Our results suggest that it is beneficial for a predatory strike in a mantis shrimp to occur over a short distance in order to maximize rotation speed. In contrast, a swimming or flying animal aims to impart momentum to the fluid in order to propel the body forward. It may therefore be advantageous in locomotion to move with a low transmission efficiency by sweeping an appendage over a relatively large excursion.
One of the benefits of examining mechanics and drag in a springdriven system is the simplicity of such an actuator. A spring generates force in proportion to its length, so the energy that it imparts to a system consequently does not vary with the rate of length change (Eqn 14). In contrast, muscular force has a nonlinear dependency on both length and its rate of shortening (Aidley, 1978). Given that the contractile properties of a muscle govern the energy that may be imparted to a system, the dynamics of skeletal geometry have the potential to yield a different relationship between speed and geometry than that found in the mantis shrimp raptorial appendage in the present study. Despite this possibility, an experimental increase of displacement advantage was recently shown to decrease shortening velocity in the leg of a frog (Richards, 2011). In this system, the force–velocity relationship intrinsic to muscle causes speed to vary with displacement advantage in a manner that is similar to what we have found for the raptorial appendage when loaded with drag.
The central importance of drag in the rotation speed and T_{k} dynamics of an aquatic mantis shrimp and the organizing principle of transmission efficiency in the evolution of fish jaws strongly suggest that some of the most fundamental assumptions made in biomechanical and functional morphological analyses of speed, force and lever tradeoffs may need to be reexamined in the light of the energetics of movement. Perhaps these discoveries will open new doors into the interpretations of evolutionary variation in mechanical systems in biology.
ACKNOWLEDGEMENTS
We extend special thanks to E. Staaterman and M. deVries for helping to collect the specimens and to R. Caldwell for providing advice for finding the animals. We appreciate the help acquiring the force data provided by E. Campos, C. Chuter, M. deVries, M. Mendoza Blanco, J. Ng and, especially, E. Staaterman. We are also grateful to J. R. A. Taylor for her assistance with acquiring the spring mechanics data. The authors thank A. Summers, M. Westneat, J. Strother and W. Van Trump for their thoughts on lever mechanics. We thank T. F. Kosar for providing technical assistance with the microCT scans, which were performed at the Center for Nanoscale Systems at Harvard University, a member of the National Nanotechnology Infrastructure Network (National Science Foundation award ECS0335765).
FOOTNOTES

FUNDING
Funding was provided by the National Science Foundation [IOS0952344 to M.J.M. and IOS1014573 to S.N.P.], an internship from the Northeast Alliance for Graduate Education and the Professoriate [to M.V.R.] and a Radcliffe Fellowship [to S.N.P.].
LIST OF SYMBOLS
 A
 scaling constant
 a–d
 position vectors for joint positions in fourbar linkage (m)
 B
 scaling factor
 B_{iso}
 scaling factor predicted by isometry
 C
 chord width of an element of the striking body (m)
 C_{d}
 drag coefficient (dimensionless)
 C_{pix}
 density constant (kg m^{–3})
 D
 drag torque index (dimensionless)
 E_{drag}
 energy lost to drag (J)
 E_{elastic}
 stored elastic energy (J)
 E_{kine}
 kinetic energy (J)
 H
 length of the striking body (m)
 h_{b,d}
 distance between b and d (m)
 h_{dac}
 position along the dactyl (m)
 h_{load}
 distance between the point of loading and a (m)
 h_{out}
 distance between striking body center of mass and b (m)
 I_{SB}
 moment of inertia of striking body (kg m^{2})
 moment of inertia of striking body (dimensionless)
 I_{water}
 moment of inertia of the water (m^{4})
 k_{linear}
 linear stiffness of the meral spring (N rad^{–1})
 k_{torsion}
 torsion stiffness of the meral spring (N m deg^{–1})
 l_{1}–l_{4}
 lengths of links in the fourbar linkage (m)
 L_{max}
 maximum angular momentum of the striking body (kg m^{2} deg s^{–1})
 l_{SB}
 total length of striking body (m)
 m_{b}
 body mass (g)
 m_{SB}
 mass of striking body (g)
 p
 pixel intensity (dimensionless)
 P
 linear momentum of the striking body (g m s^{–1})
 r
 distance between an element of the striking body and b (m)
 r_{vox}
 distance between a voxel and b (m)
 t
 time (s)
 T
 width of an element of the striking body (m)
 T_{k}
 kinematic transmission (dimensionless)
 v
 volume of voxel
 xˆ
 unit vector (dimensionless)
 ŷ
 unit vector, orthogonal to xˆ (dimensionless)
 γ
 output angle of the linkage system (deg)
 η
 transmission efficiency (dimensionless)
 θ
 input angle of a lever or linkage system (deg)
 θ0
 initial position of the input angle (deg)
 θrest
 position of the input angle at rest (deg)
 ρtissue
 density of tissue (kg m^{–3})
 ρwater
 density of water (kg m^{–3})
 τapplied
 torque applied to the striking body (N m)
 τdrag
 torque generated by drag (N m)
 τspring
 torque generated by meral spring (N m)
 ν
 angle between the direction of the load and the meralV (deg)
 Ψ
 angle between links 1 and 4 (rad)
 © 2012.