## Company Of Biologists

Analysis of the bite force and mechanical design of the feeding mechanism of the durophagous horn shark Heterodontus francisci
Daniel R. Huber, Thomas G. Eason, Robert E. Hueter, Philip J. Motta

## SUMMARY

Three-dimensional static equilibrium analysis of the forces generated by the jaw musculature of the horn shark Heterodontus francisci was used to theoretically estimate the maximum force distributions and loadings on its jaws and suspensorium during biting. Theoretical maximum bite force was then compared with bite forces measured (1) voluntarily in situ, (2) in restrained animals and (3) during electrical stimulation of the jaw adductor musculature of anesthetized sharks. Maximum theoretical bite force ranged from 128 N at the anteriormost cuspidate teeth to 338 N at the posteriormost molariform teeth. The hyomandibula, which connects the posterior margin of the jaws to the base of the chondrocranium, is loaded in tension during biting. Conversely, the ethmoidal articulation between the palatal region of the upper jaw and the chondrocranium is loaded in compression, even during upper jaw protrusion, because H. francisci's upper jaw does not disarticulate from the chondrocranium during prey capture. Maximum in situ bite force averaged 95 N for free-swimming H. francisci, with a maximum of 133 N. Time to maximum force averaged 322 ms and was significantly longer than time away from maximum force (212 ms). Bite force measurements from restrained individuals (187 N) were significantly greater than those from free-swimming individuals (95 N) but were equivalent to those from both theoretical (128 N) and electrically stimulated measurements (132 N). The mean mass-specific bite of H. francisci was greater than that of many other vertebrates and second highest of the cartilaginous fishes that have been studied. Measuring bite force on restrained sharks appears to be the best indicator of maximum bite force. The large bite forces and robust molariform dentition of H. francisci correspond to its consumption of hard prey.

## Introduction

The elasmobranch fishes (sharks, skates and rays) possess highly diverse feeding mechanisms composed of few kinetic elements, making them an ideal group in which to investigate feeding biomechanics and patterns of diversity in cranial morphology, feeding behavior and ecology. Elasmobranchs inhabit nearly all marine environments and have evolved ram, suction, biting and filter feeding mechanisms to exploit prey ranging from plankton to marine mammals (Motta, 2004). Among the diverse feeding mechanisms found in extant elasmobranch taxa are those adapted for durophagy, the consumption of hard prey. While hard' prey of some sort is found in the diets of elasmobranchs from approximately 13 families, it does not comprise a substantial portion of the diet in many of these groups. Genuine durophagy has convergently evolved in the bullhead (Heterodontidae), hammerhead (Sphyrnidae), zebra (Stegostomatidae) and hound sharks (Triakidae), as well as the eagle rays (Myliobatidae) (Compagno, 1984a,b, 2001; Summers et al., 2004).

The heterodontid sharks are the only family of elasmobranchs in which every species is ecologically and functionally specialized for durophagy (Compagno, 1984a, 1999; Taylor, 1972). The suite of morphological characters associated with durophagy in the heterodontid sharks includes robust jaws capable of resisting dorsoventral flexion under high loading, molariform teeth and hypertrophied jaw adductor muscles (Nobiling, 1977; Reif, 1976; Summers et al., 2004). To date, the concept of durophagy in the heterodontid sharks has mostly been examined qualitatively (but see Summers et al., 2004). Neither the bite forces they are capable of producing nor the subsequent loadings on the various articulations within their feeding mechanisms have been quantified in any manner. Bite force is particularly informative in regard to linking morphological, ecological and behavioral variables associated with prey capture because biting capacity is dictated by cranial morphology and is known to affect resource partitioning (Verwaijen et al., 2002; Wiersma, 2001), dietary diversity (Clifton and Motta, 1998; Wainwright, 1988) and ontogenetic changes in feeding ecology (Hernandez and Motta, 1997).

Like most modern elasmobranchs, the heterodontid sharks possess a hyostylic jaw suspension in which the mandibular arch indirectly articulates with the chondrocranium via the hyomandibular cartilages, and the palatal region of the upper jaw is suspended from the ethmoid region of the chondrocranium via ligamentous connections (Fig. 1A). However, a number of variants on this arrangement exist, primarily in the superorder Squalea (Gregory, 1904; Shirai, 1996; Wilga, 2002). The hexanchiform sharks possess an orbitostylic jaw suspension in which the upper jaw articulates with the ethmoidal, orbital and postorbital regions of the chondrocranium, and the hyomandibula contributes little support to the jaws (Fig. 1B). Conversely, the only suspensorial element in the batoids is the hyomandibula (euhyostyly; Fig. 1C; Gregory, 1904; Maisey, 1980; Wilga, 2002). These highly divergent morphologies constitute independent mechanical systems, perhaps with comparably divergent cranial loading regimes occurring during feeding. Determining these loading regimes will help to establish the link, if any, between elasmobranch jaw suspension and the functional diversity of their feeding mechanisms.

Fig. 1.

Left lateral views of representative elasmobranch jaw suspensions. (A) Heterodontus, Heterodontiformes (hyostyly); (B) Heptranchias, Hexanchiformes (amphistyly); (C) Rhinobatos, Batoidea (euhyostyly). Articulation points are marked with arrows. C, ceratohyal; E, ethmoidal; H, hyomandibula; L, lower jaw; O, orbital; P, postorbital; U, upper jaw. Reproduced from Wilga (2002) with permission from Blackwell Publishing.

The purpose of this study was therefore to determine the biomechanical basis of durophagy in the heterodontid sharks, as represented by the horn shark Heterodontus francisci (Girard 1855), a primarily shallow-water, nocturnal forager of molluscs, echinoderms and benthic crustaceans (Segura-Zarzosa et al., 1997; Strong, 1989). Heterodontus francisci uses suction to capture prey, which is grasped by the anterior cuspidate teeth and then crushed by the posterior molariform teeth, effectively combining both suction and biting feeding mechanisms (Edmonds et al., 2001; Summers et al., 2004). Through in situ bite performance measurements and theoretical modeling of the forces generated by the cranial musculature of H. francisci, the specific goals of this study were to: (1) theoretically determine the forces generated by each of the cranial muscles active during the gape cycle; (2) determine the distribution of forces throughout the jaws and suspensorium and discuss the implications of these loadings for jaw suspension; (3) compare theoretical bite force from anatomical measures with those obtained during voluntary unrestrained feeding, restrained biting and electrical stimulation of the jaw adductors; (4) relate its bite performance to feeding ecology and (5) compare the bite force of H. francisci with those of other vertebrates.

## Materials and methods

### Experimental animals

Five horn sharks Heterodontus francisci Girard [63-74 cm total length (TL)] were housed at the University of South Florida in Tampa, FL, USA in accordance with the guidelines of the Institutional Animal Care and Use Committee (IACUC #1882). Individuals were maintained at 20°C in a 1500 liter semicircular tank on a diet of thread herring Opisthonema oglinum and squid Loligo spp. The planar face of the tank held a window for viewing. Five additional H. francisci (55-68 cm TL), obtained as fisheries bycatch off the coast of Los Angeles, CA, USA, were frozen until used for morphological analyses.

### Morphological analysis

A theoretical model of the feeding mechanism of H. francisci was designed by investigating the forces produced by the nine cranial muscles involved in the abduction (coracomandibularis, coracohyoideus, coracoarcualis and coracobranchiales), adduction (adductor mandibulae complex consisting of the quadratomandibularis-preorbitalis complex, quadratomandibularis-γ and preorbitalis-α) and retraction (levator palatoquadrati and levator hyomandibularis) of the jaws and hyobranchial region (Fig. 2). The quadratomandibularis-preorbitalis complex consists of six individual heads of the adductor mandibulae complex (Nobiling, 1977). Difficulty in mechanically separating these heads led to their analysis as a group. Using the tip of the snout as the center of a three-dimensional coordinate system, the three-dimensional position of the origin and insertion of each muscle was determined by measuring the distance of these points from the respective X, Y and Z planes intersecting the tip of the snout (Fig. 3A). Each muscle was then excised (unilaterally where applicable), bisected through its center of mass perpendicular to the principal fiber direction, and digital images of the cross-sections were taken (JVC DVL9800 camera). Cross-sectional areas were measured from these images using Sigma Scan Pro 4.01 (SYSTAT Software Inc., Point Richmond, CA, USA). Center of mass was estimated by suspending the muscle from a pin and tracing a vertical line down the muscle. After repeating this from another point, the intersection of the two line-tracings indicated the center of mass of the muscle.

Fig. 2.

Right lateral (A) and ventral (B) views of the cranial and branchial musculature of a 63 cm male H. francisci. CC, coracoarcualis; CH, coracohyoideus; CHD, dorsal hyoid constrictor; CHV, ventral hyoid constrictor; CM, coracomandibularis; CO, coracoid bar; HM, hyomandibulo-mandibularis; IMD, intermandibularis; LH, levator hyomandibularis; LJ, lower jaw; LP, levator palatoquadrati; QM-PO complex, quadratomandibularis-preorbitalis complex; QM-γ, quadratomandibularis-γ; PO-α, preorbitalis-α; UJ, upper jaw; VSBC, ventral superficial branchial constrictor. The IMD has been partially removed to reveal the ventral musculature. The coracobranchiales (not shown) are located deep to the CC.

The three-dimensional coordinates of the center of rotation of the dual (lateral and medial; Nobiling, 1977) quadratomandibular jaw articulation (hereafter referred to as jaw joint'), the ethmoidal articulation and the lateral and medial articulations of the hyomandibula with the jaws and chondrocranium, respectively, were determined with respect to the right side of the head of each individual. Points corresponding to 0, 25, 50, 75 and 100% of the distance along the functional tooth row on the lower jaw from the posteriormost molariform tooth were also determined; 100% is the anteriormost cuspidate tooth. The in-lever for jaw abduction from the center of rotation of the jaw joint to the point of insertion of the coracomandibularis was determined from the three-dimensional coordinates. In-levers for jaw adduction from the center of rotation of the jaw joint to the points of insertion on the lower jaw of the quadratomandibularis-preorbitalis complex, quadratomandibularis-γ and preorbitalis-α were determined in the same manner. A weighted average of these in-levers was determined based on the forces produced by their respective muscles. The abductive and weighted adductive in-levers were divided by the out-lever distance from the center of rotation of the jaw joint to the tip of the anteriormost tooth of the lower jaw to determine mechanical advantage ratios for jaw opening and closing (Fig. 3B). Due to the quadratomandibularis-preorbitalis complex's broad surface attachment on the lateral face of both the upper and lower jaws, an exact insertion point for this muscle could not be identified. Its center of mass and principal muscle fiber direction relative to the lower jaw were used to approximate its mechanical line of action. The distance from the jaw joint to the intersection of this line of action with the lower jaw served as the in-lever for this muscle. Anatomical nomenclature is based on Daniel (1915), Motta and Wilga (1995, 1999) and Nobiling (1977).

Fig. 3.

(A) Coordinate system for three-dimensional vector analysis of the forces generated by the cranial musculature of H. francisci. Directionality is defined with respect to the head of H. francisci using the right-hand rule'. (B) Schematic diagram of the jaws of H. francisci indicating variables for mechanical lever-ratio analysis. A-B, resolved in-lever for jaw adduction; A-C, out-lever; B-D, resolved adductive muscle force vector; P0, maximum tetanic tension. CT-scan image used with permission of A. Summers.

Fig. 4.

Forces involved in the static equilibrium calculations of the lower and upper jaws of H. francisci. FB, bite reaction force; FE, reaction force at the ethmoidal articulation; FH, reaction force at the hyomandibular articulation; FJR, jaw joint reaction force; FPO-α, force generated by the preorbitalis-α; FQM-PO, force generated by the quadratomandibularis-preorbitalis complex; FQM-γ, force generated by the quadratomandibularis-γ; FR, resultant adductive force; α, angle of incidence of FE relative to the articular surface of the upper jaw at the ethmoidal articulation. Arrow size does not indicate force magnitude, and angles of force vectors are approximate. CT-scan image used with permission of A. Summers.

### Theoretical force generation

Anatomical cross-sectional area (Acs) measurements of the nine parallel fibered muscles were multiplied by the specific tension of elasmobranch white muscle (Tsp; 289 kN m-2; Lou et al., 2002) to determine their theoretical maximum tetanic forces (P0): $Math$(1)

Anatomical cross-sectional area was used in this analysis because theoretical estimates of maximum bite force based on the anatomical cross-sectional area of the parallel fibered jaw adducting musculature of the spiny dogfish Squalus acanthias best approximated bite forces measured during tetanic stimulation of the jaw adducting musculature (Huber and Motta, 2004). Force vectors for each muscle were constructed from their maximum tetanic forces and the three-dimensional coordinates of their origins and insertions. The force vectors of muscles excised unilaterally were reflected about the Y-plane to represent the forces generated by the musculature on the other side of the head.

Mathcad 11.1 software (Mathsoft, Inc., Cambridge, MA, USA) was used to generate a three-dimensional model of the static forces acting on the jaws of H. francisci during prey capture. Summation of the three-dimensional moments acting on the lower jaw about the jaw joints (left and right) determined the theoretical maximum bite force for each individual and the mean maximum bite force for all individuals (B; Fig. 4). Maximum bite force was modeled at points 0, 25, 50, 75 and 100% of the distance along the functional tooth row from the posteriormost tooth to determine a bite force gradient along the lower jaw. Additionally, the reaction force acting on the jaw joints during bites occurring at 0 and 100% of the distance along the functional tooth row was determined (FJR; Fig. 4).

Loadings were determined at the ethmoidal and hyomandibular articulations of the upper jaw with the chondrocranium and hyomandibula, respectively (Figs 1A, 4). For bites occurring at 0% (posteriormost molariform tooth) and 100% (anteriormost cuspidate tooth) of the distance along the functional tooth row, the moments acting on the upper jaw about the ethmoidal articulation were summed to determine the forces acting at the hyomandibular articulation (FH; Fig. 4). In these analyses, the hyomandibula was modeled as a two-force member, moveable about its articulations with both the upper jaw and chondrocranium (Hibbeler, 2004). Static equilibrium analysis of the forces acting on the upper jaw was then used to determine the forces acting at the ethmoidal articulation (FE; Fig. 4). Static equilibrium conditions for the forces acting on the lower (FLJ) and upper jaws (FUJ) were: $Math$(2) $Math$(3)

where FB is the bite reaction force from a prey item, FPO-α is the force generated by the preorbitalis-α, FQM-PO is the force generated by the quadratomandibularis-preorbitalis complex, and FQM-γ is the force generated by the quadratomandibularis-γ. Forces generated by the preorbitalis-α and quadratomandibularis-γ are isolated to the lower jaw because they originate on the chondrocranium and insert only upon the lower jaw (Figs 2A, 4). Joint reaction forces maintain the static equilibrium of feeding mechanisms by balancing the moments acting upon the jaws via their associated musculature and contact with prey items. The moment acting on the lower jaw during jaw opening via the coracomandibularis muscle was used to determine the theoretical maximum jaw opening force of H. francisci.

### In situ bite performance measurements

Bite performance measurements were performed using a modified single-point load cell (Amcells Corp., Carlsbad, CA, USA) with custom-designed stainless steel lever arms, which was calibrated using a series of known weights. Free-swimming H. francisci were trained to voluntarily bite the transducer by wrapping the device with squid and presenting it to them after several days of food deprivation. A P-3500 strain indicator (Vishay Measurements Group, Raleigh, NC, USA) was used for transducer excitation and signal conditioning. Data were acquired with a 6020E data acquisition board and LabVIEW 6.0 software (National Instruments Corp., Austin, TX, USA). Fifteen measurements of bite force were taken from each animal. Only events in which the transducer was bitten between the tips of the jaws were kept for analysis. The five largest bite force measurements for each individual were analyzed for the following performance variables, as well as used in the multivariate statistical analyses described below: maximum force (N), duration of force production (ms), time to maximum force (ms), rising slope of force-time curve (N s-1), duration at maximum force (ms), time from maximum force to end of force production [hereafter referred to as time away from maximum force' (ms)], falling slope of force-time curve (N s-1) and impulse (I), which is the integrated area under the force-time curve (kg m s-1) from the initiation of force generation to its cessation: $Math$(4)

where F is force and t is time. The impulse of a force is the extent to which that force changes the momentum of another body, in this case the force transducer, and therefore has the units of momentum (kg m s-1). For each individual, the single largest bite force and its associated performance measurements were used to create a profile of maximum bite performance for H. francisci, to compare the dynamics of the ascending and descending portions of the bite performance waveforms and to compare the maximum bite forces obtained from the theoretical, in situ, restrained and stimulated methods of determining bite force (see below).

In situ bite performance measurements were simultaneously filmed with a Redlake PCI-1000 digital video system (Redlake MASD, San Diego, CA, USA) at 250 frames s-1 to verify that bites on the transducer occurred between the tips of the jaws (hereafter referred to as transducer bites'). The modified single-point load cell used in this study averages the signals generated by four strain gages in a full Wheatstone bridge such that the transducer is insensitive to the position on the lever arms at which the bite is applied. Therefore, the point at which a shark bit the lever arms of the transducer did not need to be determined from the digital video sequences for appropriate calibration. To identify any behavioral artifacts associated with biting a stainless steel transducer, H. francisci were also filmed while consuming pieces of O. oglinum cut to the same size as the biting surface of the force transducer (hereafter referred to as fish bites'). The following kinematic variables were quantified from transducer and fish bites using Motionscope 2.01 (Redlake MASD, San Diego, CA, USA) and SigmaScan Pro 4.01 software: distance, duration, velocity and acceleration of lower jaw depression, lower jaw elevation, upper jaw protrusion, and head depression; maximum gape; time to maximum gape; time to onset of lower jaw elevation; time to onset of head depression; cranial elevation angle. All kinematic variables were quantified using discrete cranial landmarks as reference points (Edmonds et al., 2001).

### Restrained and stimulated bite performance measurements

At least one week after the in situ bite performance measurements, four of the H. francisci were individually removed from the experimental tank and restrained on a table. Once they had opened their jaws an adequate distance, the transducer was placed between the anterior teeth, which elicited an aggressive bite. Following a recovery period of approximately 10-15 min, the shark was again removed from the tank and anaesthetized with MS-222 (0.133 g l-1). The quadratomandibularis-preorbitalis complex, quadratomandibularis-γ and preorbitalis-α were implanted with stainless steel 23-gauge hypodermic needles connected to a SD9 stimulator (Grass Telefactor, West Warwick, RI, USA), and tetanic fusion of these muscles was accomplished via stimulation (10 V, 100 Hz, 0.02 ms delay, 3 ms pulse width) while the bite force transducer was placed between the tips of the anterior teeth. Three measurements were taken from each individual in both of these experimental protocols. Individuals were ventilated with aerated seawater between measurements during muscle stimulation experiments. Maximum bite force, time to maximum force, and time away from maximum force were quantified from all restrained and stimulated bites.

### Statistical analysis

All bite performance and kinematic variables were log10 transformed and linearly regressed against body mass to remove the effects of size. Studentized residuals were saved from each regression for subsequent analysis (Quinn and Keough, 2002). Principal components analyses (PCA) based on correlation matrices were then used to (1) identify covariation in bite performance variables and reduce these variables to a series of non-correlated principal components, which were subsequently analyzed to assess the extent of individual variability in these parameters, (2) identify covariation in performance and kinematic variables from in situ bite performance trials and (3) identify covariation in kinematic variables from fish and transducer bites and reduce these variables to a series of non-correlated principal components, which were subsequently analyzed to determine whether there were any behavioral artifacts associated with biting the steel transducer. Variables were considered to load strongly on a given principal component (PC) if their factor scores were greater than 0.6. Non-rotated axes described the greatest amount of variability in each PCA. For analyses 1 and 3, multivariate analysis of variance (MANOVA) was used to compare the factor scores for the PCs with eigenvalues greater than 1.0. To determine whether fish and transducer bites differed kinematically, a two-way, mixed-model MANOVA was performed on the PCs from PCA 3, with individual' as the random effect and prey type' as the fixed effect, which was tested over the interaction mean square. Kinematic data from four individuals were included in this analysis because a complete data set was lacking for one individual. To determine the extent of individual variability within the bite performance variables, a one-way MANOVA was performed on the PCs from PCA 1.

To determine whether the kinematic variables associated with biting the transducer were predictive of biting performance in H. francisci, stepwise (forward) multiple regressions were performed with kinematic variables measured from transducer bites as the multiple independent factors, and the eight bite performance variables as the individual dependent factors. Data from four individuals were included in this analysis because a complete kinematic data set was lacking for one individual. One-way ANOVA on Studentized residuals was used to identify significant differences among the theoretical, in situ, restrained and electrically stimulated methods of determining maximum bite force. A Student's t-test was used to identify differences between time to maximum force and time away from maximum force and between the rising and falling slopes of the force-time curves for in situ biting trials. One-way ANOVA was used to compare time to maximum force and time away from maximum force within and among in situ, restrained and electrically stimulated bite forces. Finally, bite forces at the anterior jaw (fish, reptiles and birds) or canine teeth (mammals) and body masses for various vertebrates were compiled from the available literature and grouped according to major taxonomic level. These bite forces, along with those of the horn sharks investigated in this study, were linearly regressed against body mass. Studentized residuals from this regression were then coded according to taxonomic level and compared with a one-way ANOVA. All significant differences were investigated post-hoc with Tukey's pairwise comparisons test. Linear regressions were performed in SigmaStat 2.03 (SYSTAT Software, Inc.) in order to obtain Studentized residuals. All other statistical analyses were performed in SYSTAT 10 (SYSTAT Software, Inc.) with a P-value of 0.05.

## Results

### Biomechanical modeling

The quadratomandibularis-preorbitalis complex, which is the primary jaw adductor, generated the greatest force of all muscles investigated (242 N; Table 1). Of the muscles active during jaw and hyobranchial abduction, the coracobranchiales generated the greatest force (107 N; Table 1). The levator hyomandibularis generated more force during the retractive phase (33 N) than the levator palatoquadrati (20 N; Table 1). After resolving the force generated by the adductor musculature into its principal components, the majority of force was directed dorsally (294 N) and anteriorly (128 N). The Z-axis components of this force (19 N per side) were directed laterally on either side of the head and negate each other during jaw adduction (Table 2; Fig. 3A). Thus, the resultant adductive force along the Z-axis was 0 N. The large anterodorsally directed component of this adductive bite force (FR; Fig. 4) drives the lower jaw towards the upper jaw, which is itself driven into the ethmoid region of the chondrocranium (FE; Fig. 4).

View this table:
Table 1.

Theoretical maximum forces generated by the cranial musculature active during the gape cycle in H. francisci

View this table:
Table 2.

Resultant bilateral muscle and jaw forces occurring during prey capture in H. francisci, broken into their principal components

Summation of the moments acting on the lower jaw determined that the maximum theoretical bite force of H. francisci ranged from 128 N at the anterior teeth to 338 N at the posteriormost molariform teeth (Fig. 5; Table 2). The bite force at the posteriormost molariform teeth exceeded the resultant force generated by the adductive musculature (Table 2) because the mechanical advantage at this point along the jaw was 1.06. The resultant jaw closing mechanical advantage at the anterior teeth was 0.51, resulting in a dramatically lower bite force at this point.

The jaw joint reaction forces (FJR; Fig. 4) occurring when prey is captured at the anterior teeth and crushed at the posterior teeth by H. francisci were 106 N and 73 N per side, respectively (Table 3). This force was oriented posteroventrally relative to the articular surface of the lower jaw joint for anterior biting, and consequently oriented anterodorsally relative to the articular surface of the upper jaw joint. The local/internal loadings on the joint between the upper and lower jaws indicate that the jaw joint is globally in compression (Hibbeler, 2004) when prey is bitten at the tips of the jaws. When prey is crushed between the posterior molariform teeth, the orientation of the vertical component of the joint reaction force relative to the lower jaw (25 N) was opposite that for the lower jaw joint during anterior biting (-80 N), indicating tensile loading of the jaw joint during posterior prey capture (Table 3).

View this table:
Table 3.

Unilateral mechanical loadings at articulation points in H. francisci's feeding mechanism, broken into their principal components

The ethmoidal articulation of H. francisci received a loading of 59 N per side during biting, regardless of whether biting occurred at the anterior or posterior margin of the jaws (FE; Fig. 4). The angle of incidence of this force relative to the articular surface of the upper jaw at the ethmoidal articulation was 80° (α; Fig. 4). For both anterior and posterior biting, the majority of loading was directed ventrally into the upper jaw, indicating compression between the ethmoid region of the chondrocranium and the palatal region of the upper jaw (Table 3).

The magnitude of loading at the hyomandibular articulation (36 N) was independent of bite point as well (FH; Fig. 4). The lower jaw was loaded posterodorsally and medially at its articulation with the hyomandibula during both anterior and posterior biting (Table 3). The reaction forces acting on the distal ends of the hyomandibula are equal to and opposite the forces acting at the jaws' articulation with the hyomandibula. Therefore, during biting, the hyomandibula was loaded anteroventrally and laterally. These local/internal loadings between the jaws and hyomandibula indicate that the hyomandibula is globally in tension. Modeling the hyomandibula as a two-force member assumed that the line of action of the force acting on the hyomandibula passed through its articulation with the jaws and chondrocranium. The hyomandibula is therefore loaded in pure tension, and the angle of incidence of the hyomandibular force cannot be determined.

Fig. 5.

Theoretical maximum bite force (N) of five male H. francisci (N=5, TL=55-68 cm) from three-dimensional vector analysis of the jaw adducting musculature measured at 0, 25, 50, 75 and 100% of the length of the functional tooth row of the lower jaw from posterior to anterior. CT-scan image used with permission of A. Summers.

The only muscle involved in abduction of the lower jaw is the coracomandibularis, which was capable of generating 31 N of force (Table 1). This muscle inserts on the caudal aspect of the lower jaw symphysis at 37° below the longitudinal axis of this jaw and has a mechanical advantage of 0.89. Despite this high mechanical advantage, indicative of force amplification in a class III lever system, its acute insertion angle caused the muscular force generating motion about the lower jaw (force component perpendicular to the lower jaw) to be 19 N (Table 2). After accounting for mechanical advantage, the resultant abductive force at the tip of the lower jaw was 16 N (Table 2). The abductive force lacks a component along the Z-axis because the coracomandibularis runs parallel to the longitudinal axis of the body. The other muscles involved in the expansive phase of the gape cycle generated considerably greater forces than the coracomandibularis (Table 1).

Fig. 6.

Bite force waveforms from bite performance trials of three male H. francisci (TL=66-70 cm), illustrating in situ voluntary bites with single (black) and double (light gray) force peaks and a bite from a restrained individual (dark gray).

### Performance measurements

#### In situ measurements

H. francisci approached and bit the force transducer in an attempt to remove the attached food. In most cases, biting continued until the food was removed from the transducer. PCA 1 reduced the performance variables for each individual to three PCs (89.7% of variance explained), each of which indicated considerable overlap among individuals. MANOVA subsequently demonstrated no differences among individuals for bite performance variables using size-corrected data (Wilk's Lambda=0.51, F12,47=1.157, P=0.340). The mean maximum in situ bite force measured at the anterior teeth was 95 N, with an absolute maximum force of 133 N (66 cm male H. francisci). Heterodontus francisci took approximately 322 ms to reach maximum bite force, which was held for 41 ms, and released after an additional 212 ms (Table 4). The mean duration of force application was 535 ms. Time to maximum bite force was longer than the time away from maximum bite force (P=0.049). The mean rising slope of the force-time curve was 300 N s-1 and was lower than the mean falling slope of 457 N s-1 (P=0.048). The mean impulse generated from the beginning of force application until its cessation was 25 kg m s-1 but measured as high as 44 kg m s-1. The majority of bite force waveforms consisted of single peaks associated with single bites. However, in 32% of the bites, multiple peaks occurred, indicating a repetitive crushing behavior during force application (Fig. 6).

View this table:
Table 4.

In situ bite performance data for H. francisci biting at the tips of its jaws

PCA 2 of performance and kinematic variables yielded six PCs with eigenvalues greater than 1.0, which collectively explained 86.7% of the variance. All of the variables that loaded heavily on the first PC (30.5% of variance explained) were kinematic measurements (Table 5). These variables primarily demonstrated covariance in the timings and excursions of lower jaw depression and elevation. Performance measures were the only variables to load heavily on the second PC (19.5% of variance explained), indicating covariance between rates and durations of force application (Table 5). Maximum bite force did not load heavily until the fifth PC (7.8% of variance explained), and impulse did not load heavily on any of the PCs.

View this table:
Table 5.

Principal component loadings of performance and kinematic variables from bite performance trials of H. francisci

Stepwise multiple regressions yielded similar results to PCA 2 on kinematic and performance data. Only three of the bite performance variables were significantly related to individual kinematic variables. Force duration was significantly, though poorly, related to lower jaw elevation velocity (r2=0.226, F1,18=5.268, P=0.034). Similarly, time to maximum force (r2=0.389, F1,18=11.471, P=0.003) and the rising slope of the force-time curve (r2=0.410, F1,18=12.523, P=0.002) were significantly related to lower jaw elevation distance. Inclusion of additional kinematic variables did not improve the predictive ability of these regression models. The two variables indicative of the magnitude of bite force generated (maximum force, impulse) could not accurately be predicted by any combination of kinematic variables. Although kinematic variables were not predictive of bite performance variables, PCA 1 used to assess individual variability (see above) identified notable covariance in performance measures. Maximum in situ bite force exhibited a strong linear relationship with impulse (r2=0.758) and moderate linear relationships with force duration (r2=0.450) (Fig. 7) and time to maximum force (r2=0.489).

Fig. 7.

Maximum in situ bite force (N) from five male H. francisci (TL=63-74 cm) plotted against (A) impulse (kg m s-1) and (B) force duration (ms) on logarithmic axes.

PCA 3 reduced the set of kinematic variables measured from fish and transducer bites to a series of four PCs (73.3% of variance explained). MANOVA indicated no significant differences between the prey capture kinematics of H. francisci while biting fish or the transducer on any of the PCs for all individuals (Wilk's Lambda=1.0, F4,29=0.0, P=1.0). However, a single individual was found to differ from two other individuals on the first PC (F3,32=4.646, P=0.008). Variables that loaded heavily on the first PC were durations and distances of lower jaw depression and elevation, times to maximum gape, onset of lower jaw elevation, completion of lower jaw elevation and maximum gape distance. The acceleration of lower jaw elevation loaded heavily, but negatively, on the first PC.

#### Methodological comparison

In situ measurement of maximum bite force was a reasonably good indicator of the maximum bite force of H. francisci. Using size-corrected data, a single difference was found among the four methods of determining maximum bite force (F3,14=4.358, P=0.023). Restrained bite force (159-206 N) was significantly greater than in situ bite force (60-133 N) (P=0.013). In situ bite force was, however, equivalent to theoretical (107-163 N) and electrically stimulated (62-189 N) bite forces. Restrained, electrically stimulated and theoretical bite forces were equivalent (Table 6). During restrained bites, time to maximum force (522 ms) was greater than time away from maximum force (339 ms) (t4=2.848, P=0.046). Time to maximum force (285 ms) was shorter than time away from maximum force (556 ms) for electrically stimulated bites (t8=-5.476, P<0.001). Significant differences were detected between the in situ, restrained and electrically stimulated methods for time to maximum force (F2,10=4.996, P=0.031) and time away from maximum force (F2,10=58.290, P<0.001). Time to maximum force was greater for restrained bites than for electrically stimulated bites (P=0.030), both of which were equivalent to the time to maximum force of in situ bites. Time away from maximum force was greater for electrically stimulated bites than restrained bites (P=0.001), which was greater than that of in situ bites (P=0.005).

View this table:
Table 6.

Results of one-way ANOVA on different methods of determining bite force at the tips of the jaws in H. francisci

### Bite forces among vertebrates

Bite forces and body masses were compiled for 113 species of vertebrates (including H. francisci) from the available literature (Binder and Van Valkenburgh, 2000; Cleuren et al., 1995; Clifton and Motta, 1998; Erickson et al., 2004; Hernandez and Motta, 1997; Herrel et al., 1999, 2001, 2002; Huber et al., 2004; Huber and Motta, 2004; Korff and Wainwright, 2004; Ringqvist, 1972; Robins, 1977; Thomason et al., 1990; Thompson et al., 2003; van der Meij and Bout, 2004; Weggelaar et al., 2004; Wroe et al., 2005; D. R. Huber, M. N. Dean, and A. P. Summers, unpublished) (Appendix I). Collectively, bite force scaled to body mass with a coefficient of 0.60, which is below the isometric scaling coefficient of 0.67 (Fig. 8). When the mammalian bite forces from Wroe et al. (2005) were excluded from this analysis, bite force scaled with a coefficient of 0.66, approximating isometry. This discrepancy is probably due to Wroe et al. (2005) having used the dry-skull method of estimating muscle Acs, which can underestimate Acs by 1.3-1.5× (Thomason et al., 1991).

View this table:
Appendix I

Fishes collectively had the highest mass-specific bite force of the four vertebrate groups, followed by reptiles, mammals and birds, respectively (F3,130=6.357, P<0.001). Mass-specific bite force of the fishes was greater than those of the birds (P=0.002) and mammals (P=0.013), while reptilian mass-specific bite force was greater than that of the birds (P=0.009). The striped burrfish, Chilomycterus schoepfi, had the highest mass-specific bite force, followed by the Canary Island lizard, Gallottia galloti, and the American alligator, Alligator mississippiensis (Erickson et al., 2004; Herrel et al., 1999; Korff and Wainwright, 2004). The hogfish, Lachnolaimus maximus, had the second highest mass-specific bite force, but for biting with the pharyngeal jaws not the oral jaws (Clifton and Motta, 1998). The three lowest mass-specific bite forces were those of the red-bellied short-necked turtle, Emydura subglobosa, the mata mata turtle, Chelus fimbriatus, and the twist-necked turtle Platemys platycephala (Herrel et al., 2002) (Fig. 8). Of the cartilaginous fishes in this analysis, the mean mass-specific bite force of H. francisci was greater than those of S. acanthias and the blacktip shark, Carcharhinus limbatus, but less than that of the white-spotted ratfish, Hydrolagus colliei.

Fig. 8.

(A) Bite forces (N) of various vertebrates plotted against mass (g). (B) Residuals from regression analysis of log10 bite force versus log10 mass plotted against log10 mass (g). Broken lines indicate ± 1 standard deviation about the residual mean.

## Discussion

### Functional morphology

The jaw adducting cranial musculature (QM-PO complex, QM-γ, PO-α on Fig. 2) of H. francisci generates more force during prey capture than either the jaw and hyobranchial abducting or retracting musculature. The mechanical advantage of H. francisci's jaw adducting mechanisms ranges from 0.51 at the tip of the jaws to 1.06 at the posterior margin of the functional tooth row. In class III lever systems such as shark jaws, a mechanical advantage greater than 1.0 indicates that the point at which force is being applied to a prey item is closer to the jaw joint than the point at which muscular force is being applied to the jaw, resulting in an amplification of the muscular force. Subsequently, the theoretical maximum bite force at the posterior margin of the functional tooth row exceeds the resultant force generated by H. francisci's adductor musculature. This amplification of muscular force is advantageous for the processing of hard prey such as the molluscs, echinoderms and benthic crustaceans consumed by H. francisci (Segura-Zarzosa et al., 1997; Strong, 1989).

The jaw closing mechanical advantage at the anterior teeth of H. francisci is greater than that of the only other elasmobranch for which values have been published, S. acanthias (0.28; Huber and Motta, 2004), which utilizes a combination of ram and suction feeding to consume soft-bodied prey (Wilga and Motta, 1998). Its jaw closing mechanical advantage is greater than those at the anterior teeth of nearly every actinopterygian fish investigated (∼150), which include prey from plankton to hard-shelled species (Durie and Turingan, 2001; Turingan et al., 1995; Wainwright et al., 2004; Westneat, 2004). The durophagous species among these taxa do, however, have the highest jaw adducting mechanical advantages. The durophagous parrot fishes (Scaridae) are the only actinopterygian fishes with jaw adducting mechanical advantages comparable with that of H. francisci (Wainwright et al., 2004; Westneat, 2004). Thus, there is extensive evolutionary convergence on high leverage jaw adducting mechanisms in fishes that consume hard prey.

The jaws of H. francisci are elliptical in transverse-section, with their major axis oriented vertically, in-line with the compressive stresses associated with feeding. Calcium reinforcement in the jaw cortex increases posteriorly as the dentition becomes more molariform and is greatest at the jaw joints (Summers et al., 2004). Calcification and elliptical geometry increase the second moment of area of the jaws with respect to the compressive loading of prey capture and processing, which augments the jaws' ability to resist dorsoventral flexion (Summers et al., 2004). The resolved force vector for jaw adduction also occurs approximately in the region of the most robust molariform teeth of H. francisci, where it can generate upwards of 338 N of bite force. Therefore, maximum bite force is produced where both the dentition and jaw cartilages are best able to resist compressive stresses.

Despite the high mechanical advantage (0.89) of the coracomandibularis muscle in the lower jaw depression mechanism of H. francisci, its acute insertion angle relative to the lower jaw causes most of its force to be directed posteriorly, into the jaw joints (Table 2.) This high mechanical advantage is due to the insertion of the coracomandibularis on the posterior margin of the mandibular symphysis, which is synapomorphic for elasmobranchs (Wilga et al., 2000). Although this mechanism is suited for force production, velocity production is desirable for inertial suction feeding. Heterodontus francisci nonetheless effectively uses suction to initially capture and reorient prey (Edmonds et al., 2001), which may be due in part to its powerful hyoid and branchial abductors (Table 1). These muscles rapidly expand the floor of the buccopharyngeal cavity, which is critical to suction feeding in elasmobranchs (Motta et al., 2002; Svanback et al., 2002). As in the nurse shark, Ginglymostoma cirratum (Motta et al., 2002), the large labial cartilages of H. francisci considerably occlude its lateral gape, theoretically augmenting suction ability (Muller and Osse, 1984; Van Leeuwen and Muller, 1984).

Three-dimensional resolution of the forces generated during jaw adduction may reveal the mechanical basis of upper jaw protrusion in H. francisci. The force driving the upper jaw into the ethmoidal articulation has both dorsal and anterior components, causing the upper jaw to slide through the anteroventrally sloping palatal fossa of the chondrocranium and protrude (Fig. 4; Table 2). This proposed mechanism is based on the resolved force vector for all muscles involved in jaw adduction. Differential activity of the heads of this complex may facilitate modulation of protrusion. The quadratomandibularis-γ is the likely candidate for control over protrusion because its acute insertion angle relative to the lower jaw and anterior insertion point give it high leverage over anterior motion (Fig. 2).

Activity of the quadratomandibularis-preorbitalis complex alone, which has a broad insertion on the lateral face of both the upper and lower jaws, may contribute to protrusion of the upper jaw as well. After the lower jaw has been depressed, contraction of this muscle complex may simultaneously raise the lower jaw and pull the upper jaw away from the skull. This mechanism has been proposed for upper jaw protrusion in S. acanthias, G. cirratum and the lemon shark, Negaprion brevirostris (Moss, 1977; Motta et al., 1997; Wilga and Motta, 1998). Protrusion by H. francisci, which may be used to chisel away at attached benthic prey, occurs after the lower jaw has been depressed (Edmonds et al., 2001), corroborating the role of the quadratomandibularis-preorbitalis complex in this behavior.

Extensive calcification near the jaw joints of H. francisci (Summers et al., 2004) would apparently indicate high joint reaction forces during prey capture. Joint reaction forces can exceed bite forces at the tip of the jaw depending on the mechanical advantage of the given feeding mechanism and the force produced by the associated musculature, which has been identified in numerous reptiles (3-4× greater; Cleuren et al., 1995; Herrel et al., 1998). Although joint reaction force was greater than anterior bite force in H. francisci, the ratio of these values (1.65) is substantially lower than those found for reptiles. The ratio of joint reaction force to posterior bite force in H. francisci was 0.43.

Low ratios of joint reaction force to bite force in H. francisci are due to its high mechanical advantage jaw adducting mechanism. Humans, which share this characteristic, have correspondingly low ratios of joint reaction force to bite force (Koolstra et al., 1988). Although some damping will occur in the connective tissue associated with the jaw joint, loading occurring at the joint will be transmitted to adjacent skeletal elements. Therefore, low ratios of joint reaction force to bite force may be adaptive in H. francisci, and elasmobranchs in general, because the posterior region of their jaws is suspended from the cranium by mobile hyomandibulae, not a stable jaw articulation as in other vertebrates. Minimizing loading at this articulation may stabilize the feeding mechanism during prey capture and processing.

In heterodontiform sharks, the cranial stresses associated with prey capture can be isolated to the ethmoidal and hyomandibular articulations. Unlike carcharhinid sharks (Motta and Wilga, 1995), the upper jaw of H. francisci does not disarticulate from the chondrocranium during feeding, even during upper jaw protrusion (Maisey, 1980). Therefore, in carcharhinid sharks, the hyomandibulae may receive all of the suspensorial loading occurring during prey capture. Optimal loading at the ethmoidal articulation would entail forces directed perpendicularly into the articular surface of the upper jaw because cartilage is strongest in axial compression (Carter and Wong, 2003). The estimated forces at this articulation deviated from optimal orientation by only 10° during anterior and posterior biting. The ethmoidal articulation of H. francisci appears well designed for withstanding this nearly axial compressive loading because the upper jaw calcifies at this articulation early in ontogeny (Summers et al., 2004) and the ethmoid region of the chondrocranium is one of the thickest parts of this structure (Daniel, 1915). Additionally, maintenance of contact between the upper jaw and chondrocranium in H. francisci will distribute stresses from the repetitive loading associated with processing hard prey.

Although it is well known that the hyomandibulae support the posterior margin of the jaws, the nature of the loading they receive has been a matter of speculation. This mechanical analysis indicates that the hyomandibulae of H. francisci are tensile elements, as suggested by Moss (1972) and Frazzetta (1994). Consequently, the hyomandibulae may regulate anterior movement of the jaws during feeding, such as would occur during jaw protrusion. In this regulatory role, activity of the levator hyomandibularis could hypothetically modulate resistance to anterior motion of the jaws. Electromyographic analysis of the feeding musculature of H. francisci would be required to verify this hypothesis.

The ligamentous attachments between the hyoid arch and the posterior end of the jaws stabilize this articulation against the tensile stresses caused by biting. The internal hyomandibular palatoquadrate and hyoideo-mandibulare ligaments resist dorsoventral translation between the hyomandibula and jaws, while the hyomandibuloceratohyal ligament prevents lateral translation between these elements. The two slips of the median ligament (Daniel, 1915) stabilize against dorsoventral and lateral translation, respectively. Although this analysis makes the assumption that the hyomandibulae are loaded as two-force members in axial tension, they probably experience a more diverse loading pattern in nature, necessitating this multidirectional support.

Increased hyomandibular loading may have played a role in the transition from amphistylic to hyostylic jaw suspensions in modern elasmobranchs. As the number and size of the articulations between the jaws and chondrocranium was reduced, the hyomandibulae took on a greater role in suspending the jaws (Carroll, 1988; Maisey, 1980; Schaeffer, 1967). Concomitant with these changes in articulation, the hyomandibulae became shorter and more mobile (Cappetta, 1987; Maisey, 1985; Moy-Thomas and Miles, 1971; Schaeffer, 1967), oriented more orthogonally to the chondrocranium (Stahl, 1988; Wilga, 2002) and had more extensive ligamentous attachments with the jaws and chondrocranium (Gadow, 1888). Collectively, these changes may have been associated with a shifting of the force of jaw adduction to a more posterior region of the jaws. This may have resulted in greater hyomandibular loading as well as a freeing-up' of the anterior margin of the jaws such that upper jaw kinesis was increased, facilitating jaw protrusion and prey gouging (Maisey, 1980; Moss, 1977; Wilga, 2002).

Although the jaw suspension mechanism of H. francisci is classified as hyostylic (Gregory, 1904; Wilga, 2002), this analysis indicates that it is functionally amphistylic. Heterodontus francisci exhibits considerable upper jaw kinesis (reduces maximum gape by 39%), similar to other hyostylic carcharhiniform sharks (Edmonds et al., 2001; Wilga et al., 2001). Despite this functional similarity, the upper jaw does not disarticulate from the chondrocranium during protrusion in H. francisci. Furthermore, in contrast to the hypotheses regarding hyomandibular evolution (see above), H. francisci has considerable loading at both the hyomandibular (tensile) and ethmoidal (compressive) articulations. The term hyostyly' should therefore be reserved for taxa in which the upper jaw disarticulates from the chondrocranium during protrusion such that hyomandibulae are the primary means of support and the ethmopalatine ligaments are loaded in tension. Therefore, contemporary definitions of jaw suspension should incorporate functional interpretations of loadings at the various articulations between the jaws and cranium, as well as the relationship between suspension type and upper jaw protrusion.

### Methodological comparison

Although no differences were found between theoretical, restrained and electrically stimulated bite force measurements using size-corrected data, the absolute maximum bite force for each individual occurred during restrained bite force measurements. No differences were found between theoretical and electrically stimulated bite force measurements of S. acanthias either (Huber and Motta, 2004). Therefore, restrained measurements appear to be the best method of obtaining maximum bite force measurements from live elasmobranchs. Small sample size might, however, account for the lack of statistical significance found between different methods of determining bite force. Nonetheless, the results of both this analysis and that of bite force production in S. acanthias (Huber and Motta, 2004) indicate that theoretical estimates of bite force in sharks are accurate in predicting maximum bite forces. This is fortunate given the logistical problems associated with obtaining bite force measurements from live elasmobranchs. Given the appropriate resources, however, maximum bite force can be obtained through in situ methods, as indicated by the equivalence of theoretical, electrically stimulated and in situ bite forces in this study. In situ measurements enable the quantification of biting dynamics as well, which is informative regarding feeding performance and ecology (see below).

Static estimates of force production based on muscle architecture may underestimate actual force production because active stretching of the jaw adductors during the expansive phase of the gape cycle can increase force production (Askew and Marsh, 1997; Josephson, 1999). Furthermore, by modeling the primary jaw adductor as the quadratomandibularis-preorbitalis complex instead of delineating the individual heads of this complex, variations in muscle architecture of these heads such as pinnate insertion points may have been overlooked. If this were the case, a theoretical model of force production based on morphological cross-sectional area alone could underestimate maximum force production.

The ratios of time to and away from maximum force for in situ (1.52) and restrained (1.54) bites suggest that the application of bite force by H. francisci takes longer than its release. However, the opposite relationship for these variables occurred during electrically stimulated bites. The ratio of time to and away from maximum force during electrically stimulated biting (0.51) approximates the ratio of time for twitch tension development to relaxation (0.42) for pectoral fin muscle of the cuckoo ray, Raja naevus (Johnston, 1980). This suggests that force generation during voluntary or stimulated biting is a function of the rate at which the adductor muscles reach tetanic fusion. Gradual summation of motor unit recruitment during voluntary biting results in a prolonged time to maximum force, whereas manual, high-frequency electrical stimulation of the adductor muscles causes more rapid tetani and subsequently shorter times to maximum force. Time away from maximum force was longer for restrained measurements than for in situ measurements, perhaps indicating motivational differences between these two presentation methods.

### Feeding performance

Several bite performance variables demonstrated patterns consistent with the durophagous diet of H. francisci. The time to maximum bite force application by H. francisci was longer than time away from maximum force, the rising slope of the force-time curve was lower than the falling slope, and maximum bite force was positively related to the time to maximum force. These performance characteristics indicate that the application of bite force is a slower, more deliberate action than its release by H. francisci. Linear relationships of maximum bite force with impulse and force duration further indicate that higher bite forces are associated with slower, more deliberate closing of the jaws by H. francisci (Fig. 7).

The impulse generated upon impact between two bodies is a measure of momentum transfer and can be interpreted as the effort' that each body exerts on the other (Nauwelaerts and Aerts, 2003). Because momentum is conserved during impact, larger impulses generated during biting transfer greater quantities of kinetic energy from the jaws to the prey. Optimizing impulse by maximizing bite force output per unit time will increase the amount of energy contributing to the rupture/fracture of a prey item. Heterodontus francisci capitalizes upon this when consuming hard prey with composite exoskeletons. Sustained loading after a high-velocity initial impact is effective at fracturing composite structures such as sea urchin exoskeletons (calcite ossicles linked by collagenous ligaments) because composites harden to a saturation point upon initial compression, after which crack nucleation occurs, followed by structural failure (Christoforou et al., 1989; Ellers et al., 1998; Provan and Zhai, 1985; Strong, 1989).

The prevalence of multiple force peaks within a compressive waveform of a single bite (32% of in situ bites) also indicates H. francisci's behavioral specialization for exploiting hard prey (Fig. 6). This behavior maximizes the damage inflicted upon prey items during a given bite by ramping up the applied force multiple times, especially when there are multiple bites during a feeding event. The rate at which the strength of a composite structure degrades is a power function of both the strain rate and number of strain cycles (Hwang and Han, 1989). Multiple force peaks within a given bite indicate that H. francisci may have evolved motor patterns specialized for durophagy as well. High-frequency bursts of electrical activity associated with rhythmic compression of prey items occur in the jaw adductor musculature of the lungfish Lepidosiren paradoxa (Bemis and Lauder, 1986). Prolonged jaw adductor activity occurs in the queen triggerfish, Balistes vetula (Turingan and Wainwright, 1993), and the bonnethead shark, Sphyrna tiburo, which also uses repeated compressions of the jaws to process prey (Wilga and Motta, 2000). All of these fish include hard prey in their diets (Berra, 2001; Turingan and Wainwright, 1993; Wilga and Motta, 2000). These behavioral attributes demonstrate that the way in which force is applied to prey items, and not just the magnitude of force, is likely to be a determinant of feeding success.

Although covariation in several performance measures appears related to the consumption of hard prey by H. francisci, covariation was lacking between kinematic and performance variables from the in situ bite performance trials. Both principal components and multiple regression analyses demonstrated the inability of kinematic measures to predict bite performance measures with any accuracy (Table 5). These findings beg the question, how are two series of sequential behaviors so unrelated? One would assume that at least the kinematics of lower jaw elevation (e.g. velocity, acceleration) would be predictive of biting performance (e.g. maximum force, impulse). This lack of covariation is probably due to the instantaneous position of the jaw adducting muscles of H. francisci on the force-velocity curve relating muscle tension to contraction velocity (Aidley, 1998). Based on this principle, when the adductor musculature is elevating the lower jaw, it is contracting with high velocity and low force. However, once contact is made with the bite force transducer, movement of the lower jaw is impeded and the jaw adductors shift to the low-velocity, high-force region of the force-velocity curve. In addition to high force, maximum muscle power is generated at low velocity as well (Askew and Marsh, 1997). Because of the dramatic differences in muscle function at either end of the force-velocity curve, jaw kinematics and biting performance may vary conversely and possibly be modulated independently. A predictive relationship between cranial kinesis and performance kinetics is more likely to be found for behaviors such as suction feeding in which kinesis and performance occur simultaneously (cranial expansion and suction generation; Sanford and Wainwright, 2002; Svanback et al., 2002), not sequentially, as is the case in biting performance (jaw adduction and bite force application).

An additional behavior that may augment the biting performance of H. francisci is the use of upper jaw protrusion to dislodge and chisel away at hard prey, as was suggested by Edmonds et al. (2001). While the restrained bite force measurements of H. francisci indicate that they can consume prey capable of resisting over 200 N using this behavior, in situ bite force measurements suggest they would consume smaller, less durable prey. An analysis of the forces necessary to crush various sizes of hard prey items found in H. francisci's diet is needed to delineate the prey it is theoretically capable of consuming (potential niche) from that which it actually consumes (realized niche). Functioning at maximum capacity would typically be an unnecessary expenditure of energy, especially when feeding occurs in a niche such as durophagy that is relatively inaccessible to sympatric taxa.

### Feeding ecology

The cranial architecture and prey capture behavior of H. francisci enable it to exploit hard prey, which is a relatively untapped ecological niche for aquatic vertebrates. In fishes, durophagy has been associated with high bite forces and low dietary diversity (Clifton and Motta, 1998; Wainwright, 1988). Species capable of consuming hard prey are morphologically segregated by relative differences in bite force and ecologically segregated by the hardness of the prey they can consume (Aguirre et al., 2003; Kiltie, 1982). Therefore, durophagy appears to result in niche specialization and competition reduction. This is the case in H. francisci because hard prey (molluscs, echinoderms, benthic crustaceans) comprises approximately 95% of its diet (Segura-Zarzosa et al., 1997; Strong, 1989). However, Summers et al. (2004) suggested that H. francisci goes through an ontogenetic shift to durophagy due to biomechanical changes in its jaw cartilages. It remains to be seen if H. francisci undergoes a reduction in dietary diversity and increased niche specialization over ontogeny, with associated changes in feeding behavior and performance. A more detailed dietary analysis of neonate and juvenile H. francisci would be needed to determine whether these changes occur.

Biomechanical modeling and performance testing provide a morphological and behavioral basis from which to interpret differences in organismal ecology. These analyses determined that H. francisci is capable of generating bite forces an order of magnitude higher than comparably sized S. acanthias (Huber and Motta, 2004) and that H. francisci applies bite force in a way suited for processing hard prey. Differences in the feeding performance of H. francisci and S. acanthias directly coincide with the different feeding niches they occupy (durophagy and piscivory, respectively; Alonso et al., 2002; Segura-Zarzosa et al., 1997). Therefore, these analyses are of utility for understanding the diversity of elasmobranch feeding mechanisms at numerous organismal levels (morphology, behavior, ecology), as well as the selective pressures involved in the evolution of these mechanisms.

Heterodontus francisci has the second highest mass-specific bite force of the cartilaginous fishes in which bite force has been measured or estimated (Huber et al., 2004; Huber and Motta, 2004; Weggelaar et al., 2004; D. R. Huber, M. N. Dean and A. P. Summers, unpublished). Relative to body mass, the hardest biting cartilaginous fish studied thus far is H. colliei, which is also durophagous (Ebert, 2003; Johnson, 1967). Neither H. francisci nor H. colliei were comparable in biting ability to the durophagous teleost fishes C. schoepfi, L. maximus and the sheepshead, Archosargus probatocephalus. The mass-specific bite forces of these teleost fishes, which possess a battery of anatomical specializations associated with durophagy (Clifton and Motta, 1998; Hernandez and Motta, 1997; Korff and Wainwright, 2004), were considerably higher than those of the durophagous cartilaginous fishes (Appendix I). Comparative materials testing of the hard prey items in the diets of these cartilaginous and teleost fishes would be required to determine the ecological relevance of these differences in bite force. Nonetheless, the bite forces of these fishes collectively indicate that high biting performance, in addition to anatomical specialization, are associated with the consumption of hard prey.

### Conclusions

The heterodontiform sharks, as represented by the horn shark H. francisci, possess a unique combination of morphological and behavioral characteristics that enable them to consume hard prey. Although H. francisci bites harder than the average vertebrate of comparable size, on a mass-specific basis it is not the most powerful biter in the animal kingdom (Fig. 8). Reptiles, mammals, other fishes and even some birds are capable of performing as well as or better than H. francisci when body mass is accounted for. These data suggest that factors other than bite force magnitude play a significant role in prey capture and processing ability. For H. francisci, these factors are molariform teeth, robust jaws, a high leverage jaw-adducting mechanism, and long duration, cyclically applied bite forces. The durophagous feeding behavior of H. francisci is reflected in its extensive ethmoidal articulation bracing the anterior portion of the upper jaw against the chondrocranium during prey capture and processing. Although in situ bite force measurements provided valuable information regarding its feeding behavior and ecology, theoretical estimates and restrained bite force measurements were the most effective means of estimating maximum bite force, depending on the availability of deceased specimens and live individuals. Because only a few investigations of biting performance in cartilaginous fishes have been made (Evans and Gilbert, 1971; Huber et al., 2004; Huber and Motta, 2004; Snodgrass and Gilbert, 1967; Weggelaar et al., 2004), little is known about the role that bite force plays in the ecological and evolutionary success of sharks. Combining theoretical and performance analyses provides the basis for an in-depth understanding of the link between morphology, behavior and ecology in sharks, and the role that biomechanics plays in the form and function of shark feeding mechanisms.

List of symbols and abbreviations
α
angle of incidence of the ethmoidal articulation force
ACS
cross-sectional area
C
ceratohyal
CC
coracoarcualis
CH
coracohyoideus
CHD
dorsal hyoid constrictor
CHV
ventral hyoid constrictor
CM
coracomandibularis
CO
coracoid bar
E
ethmoidal articulation
F
force
FB
bite reaction force
FE
ethmoidal articulation force
FH
hyomandibular articulation force
FJR
jaw joint reaction force,
FLJ
static equilibrium of forces acting on the lower jaw
FPO-α
preorbitalis-α force
FQM-PO
FQM-γ
FR
resultant jaw adducting force
FUJ
static equilibrium of forces acting on the lower jaw
H
hyomandibula
HM
hyomandibulo-mandibularis
I
impulse
IMD
intermandibularis
LH
levator hyomandibularis
LJ
lower jaw
LP
O
orbital ariculation
P
postorbital articulation
P0
theoretical maximum tetanic tension
PO-α
preorbitalis-α
QM-PO
QM-γ
Tsp
specific tension
UJ
upper jaw
VSBC
ventral superficial branchial constrictor

## ACKNOWLEDGEMENTS

We would like to thank J. Morris, J. Tyminski and D. Conklin for their assistance in animal care and maintenance, and the Florida Aquarium, Texas A&M University, Los Angeles County Museum of Natural History, Philadelphia Academy of Natural Sciences, C. Lowe and D. Pondella for providing experimental animals and morphological specimens. Special thanks go to C. Weggelaar, M. Dean, D. Lowry, R. Martinez, S. Valenti, Mandica Illustration and Hall Machine Inc. for their generous contributions of time, labor, wisdom and artistry. This manuscript was improved through the advice of S. Pierce. Funding for this project was provided by the University of South Florida Foundation Presidential Fellowship, University of South Florida-Mote Marine Laboratory Graduate Research Fellowship in Elasmobranch Biology and PADI A.W.A.R.E. Foundation grants awarded to D.R.H. and by a University of South Florida Research and Creative Grant Scholarship awarded to P.J.M. and D.R.H.

View Abstract

## SUMMARY

Three-dimensional static equilibrium analysis of the forces generated by the jaw musculature of the horn shark Heterodontus francisci was used to theoretically estimate the maximum force distributions and loadings on its jaws and suspensorium during biting. Theoretical maximum bite force was then compared with bite forces measured (1) voluntarily in situ, (2) in restrained animals and (3) during electrical stimulation of the jaw adductor musculature of anesthetized sharks. Maximum theoretical bite force ranged from 128 N at the anteriormost cuspidate teeth to 338 N at the posteriormost molariform teeth. The hyomandibula, which connects the posterior margin of the jaws to the base of the chondrocranium, is loaded in tension during biting. Conversely, the ethmoidal articulation between the palatal region of the upper jaw and the chondrocranium is loaded in compression, even during upper jaw protrusion, because H. francisci's upper jaw does not disarticulate from the chondrocranium during prey capture. Maximum in situ bite force averaged 95 N for free-swimming H. francisci, with a maximum of 133 N. Time to maximum force averaged 322 ms and was significantly longer than time away from maximum force (212 ms). Bite force measurements from restrained individuals (187 N) were significantly greater than those from free-swimming individuals (95 N) but were equivalent to those from both theoretical (128 N) and electrically stimulated measurements (132 N). The mean mass-specific bite of H. francisci was greater than that of many other vertebrates and second highest of the cartilaginous fishes that have been studied. Measuring bite force on restrained sharks appears to be the best indicator of maximum bite force. The large bite forces and robust molariform dentition of H. francisci correspond to its consumption of hard prey.

KEY WORDS
• bite force
• elasmobranch
• feeding biomechanics
• performance
• durophagy
• jaw suspension
• Heterodontus francisci

## Introduction

The elasmobranch fishes (sharks, skates and rays) possess highly diverse feeding mechanisms composed of few kinetic elements, making them an ideal group in which to investigate feeding biomechanics and patterns of diversity in cranial morphology, feeding behavior and ecology. Elasmobranchs inhabit nearly all marine environments and have evolved ram, suction, biting and filter feeding mechanisms to exploit prey ranging from plankton to marine mammals (Motta, 2004). Among the diverse feeding mechanisms found in extant elasmobranch taxa are those adapted for durophagy, the consumption of hard prey. While hard' prey of some sort is found in the diets of elasmobranchs from approximately 13 families, it does not comprise a substantial portion of the diet in many of these groups. Genuine durophagy has convergently evolved in the bullhead (Heterodontidae), hammerhead (Sphyrnidae), zebra (Stegostomatidae) and hound sharks (Triakidae), as well as the eagle rays (Myliobatidae) (Compagno, 1984a,b, 2001; Summers et al., 2004).

The heterodontid sharks are the only family of elasmobranchs in which every species is ecologically and functionally specialized for durophagy (Compagno, 1984a, 1999; Taylor, 1972). The suite of morphological characters associated with durophagy in the heterodontid sharks includes robust jaws capable of resisting dorsoventral flexion under high loading, molariform teeth and hypertrophied jaw adductor muscles (Nobiling, 1977; Reif, 1976; Summers et al., 2004). To date, the concept of durophagy in the heterodontid sharks has mostly been examined qualitatively (but see Summers et al., 2004). Neither the bite forces they are capable of producing nor the subsequent loadings on the various articulations within their feeding mechanisms have been quantified in any manner. Bite force is particularly informative in regard to linking morphological, ecological and behavioral variables associated with prey capture because biting capacity is dictated by cranial morphology and is known to affect resource partitioning (Verwaijen et al., 2002; Wiersma, 2001), dietary diversity (Clifton and Motta, 1998; Wainwright, 1988) and ontogenetic changes in feeding ecology (Hernandez and Motta, 1997).

Like most modern elasmobranchs, the heterodontid sharks possess a hyostylic jaw suspension in which the mandibular arch indirectly articulates with the chondrocranium via the hyomandibular cartilages, and the palatal region of the upper jaw is suspended from the ethmoid region of the chondrocranium via ligamentous connections (Fig. 1A). However, a number of variants on this arrangement exist, primarily in the superorder Squalea (Gregory, 1904; Shirai, 1996; Wilga, 2002). The hexanchiform sharks possess an orbitostylic jaw suspension in which the upper jaw articulates with the ethmoidal, orbital and postorbital regions of the chondrocranium, and the hyomandibula contributes little support to the jaws (Fig. 1B). Conversely, the only suspensorial element in the batoids is the hyomandibula (euhyostyly; Fig. 1C; Gregory, 1904; Maisey, 1980; Wilga, 2002). These highly divergent morphologies constitute independent mechanical systems, perhaps with comparably divergent cranial loading regimes occurring during feeding. Determining these loading regimes will help to establish the link, if any, between elasmobranch jaw suspension and the functional diversity of their feeding mechanisms.

Fig. 1.

Left lateral views of representative elasmobranch jaw suspensions. (A) Heterodontus, Heterodontiformes (hyostyly); (B) Heptranchias, Hexanchiformes (amphistyly); (C) Rhinobatos, Batoidea (euhyostyly). Articulation points are marked with arrows. C, ceratohyal; E, ethmoidal; H, hyomandibula; L, lower jaw; O, orbital; P, postorbital; U, upper jaw. Reproduced from Wilga (2002) with permission from Blackwell Publishing.

The purpose of this study was therefore to determine the biomechanical basis of durophagy in the heterodontid sharks, as represented by the horn shark Heterodontus francisci (Girard 1855), a primarily shallow-water, nocturnal forager of molluscs, echinoderms and benthic crustaceans (Segura-Zarzosa et al., 1997; Strong, 1989). Heterodontus francisci uses suction to capture prey, which is grasped by the anterior cuspidate teeth and then crushed by the posterior molariform teeth, effectively combining both suction and biting feeding mechanisms (Edmonds et al., 2001; Summers et al., 2004). Through in situ bite performance measurements and theoretical modeling of the forces generated by the cranial musculature of H. francisci, the specific goals of this study were to: (1) theoretically determine the forces generated by each of the cranial muscles active during the gape cycle; (2) determine the distribution of forces throughout the jaws and suspensorium and discuss the implications of these loadings for jaw suspension; (3) compare theoretical bite force from anatomical measures with those obtained during voluntary unrestrained feeding, restrained biting and electrical stimulation of the jaw adductors; (4) relate its bite performance to feeding ecology and (5) compare the bite force of H. francisci with those of other vertebrates.

## Materials and methods

### Experimental animals

Five horn sharks Heterodontus francisci Girard [63-74 cm total length (TL)] were housed at the University of South Florida in Tampa, FL, USA in accordance with the guidelines of the Institutional Animal Care and Use Committee (IACUC #1882). Individuals were maintained at 20°C in a 1500 liter semicircular tank on a diet of thread herring Opisthonema oglinum and squid Loligo spp. The planar face of the tank held a window for viewing. Five additional H. francisci (55-68 cm TL), obtained as fisheries bycatch off the coast of Los Angeles, CA, USA, were frozen until used for morphological analyses.

### Morphological analysis

A theoretical model of the feeding mechanism of H. francisci was designed by investigating the forces produced by the nine cranial muscles involved in the abduction (coracomandibularis, coracohyoideus, coracoarcualis and coracobranchiales), adduction (adductor mandibulae complex consisting of the quadratomandibularis-preorbitalis complex, quadratomandibularis-γ and preorbitalis-α) and retraction (levator palatoquadrati and levator hyomandibularis) of the jaws and hyobranchial region (Fig. 2). The quadratomandibularis-preorbitalis complex consists of six individual heads of the adductor mandibulae complex (Nobiling, 1977). Difficulty in mechanically separating these heads led to their analysis as a group. Using the tip of the snout as the center of a three-dimensional coordinate system, the three-dimensional position of the origin and insertion of each muscle was determined by measuring the distance of these points from the respective X, Y and Z planes intersecting the tip of the snout (Fig. 3A). Each muscle was then excised (unilaterally where applicable), bisected through its center of mass perpendicular to the principal fiber direction, and digital images of the cross-sections were taken (JVC DVL9800 camera). Cross-sectional areas were measured from these images using Sigma Scan Pro 4.01 (SYSTAT Software Inc., Point Richmond, CA, USA). Center of mass was estimated by suspending the muscle from a pin and tracing a vertical line down the muscle. After repeating this from another point, the intersection of the two line-tracings indicated the center of mass of the muscle.

Fig. 2.

Right lateral (A) and ventral (B) views of the cranial and branchial musculature of a 63 cm male H. francisci. CC, coracoarcualis; CH, coracohyoideus; CHD, dorsal hyoid constrictor; CHV, ventral hyoid constrictor; CM, coracomandibularis; CO, coracoid bar; HM, hyomandibulo-mandibularis; IMD, intermandibularis; LH, levator hyomandibularis; LJ, lower jaw; LP, levator palatoquadrati; QM-PO complex, quadratomandibularis-preorbitalis complex; QM-γ, quadratomandibularis-γ; PO-α, preorbitalis-α; UJ, upper jaw; VSBC, ventral superficial branchial constrictor. The IMD has been partially removed to reveal the ventral musculature. The coracobranchiales (not shown) are located deep to the CC.

The three-dimensional coordinates of the center of rotation of the dual (lateral and medial; Nobiling, 1977) quadratomandibular jaw articulation (hereafter referred to as jaw joint'), the ethmoidal articulation and the lateral and medial articulations of the hyomandibula with the jaws and chondrocranium, respectively, were determined with respect to the right side of the head of each individual. Points corresponding to 0, 25, 50, 75 and 100% of the distance along the functional tooth row on the lower jaw from the posteriormost molariform tooth were also determined; 100% is the anteriormost cuspidate tooth. The in-lever for jaw abduction from the center of rotation of the jaw joint to the point of insertion of the coracomandibularis was determined from the three-dimensional coordinates. In-levers for jaw adduction from the center of rotation of the jaw joint to the points of insertion on the lower jaw of the quadratomandibularis-preorbitalis complex, quadratomandibularis-γ and preorbitalis-α were determined in the same manner. A weighted average of these in-levers was determined based on the forces produced by their respective muscles. The abductive and weighted adductive in-levers were divided by the out-lever distance from the center of rotation of the jaw joint to the tip of the anteriormost tooth of the lower jaw to determine mechanical advantage ratios for jaw opening and closing (Fig. 3B). Due to the quadratomandibularis-preorbitalis complex's broad surface attachment on the lateral face of both the upper and lower jaws, an exact insertion point for this muscle could not be identified. Its center of mass and principal muscle fiber direction relative to the lower jaw were used to approximate its mechanical line of action. The distance from the jaw joint to the intersection of this line of action with the lower jaw served as the in-lever for this muscle. Anatomical nomenclature is based on Daniel (1915), Motta and Wilga (1995, 1999) and Nobiling (1977).

Fig. 3.

(A) Coordinate system for three-dimensional vector analysis of the forces generated by the cranial musculature of H. francisci. Directionality is defined with respect to the head of H. francisci using the right-hand rule'. (B) Schematic diagram of the jaws of H. francisci indicating variables for mechanical lever-ratio analysis. A-B, resolved in-lever for jaw adduction; A-C, out-lever; B-D, resolved adductive muscle force vector; P0, maximum tetanic tension. CT-scan image used with permission of A. Summers.

Fig. 4.

Forces involved in the static equilibrium calculations of the lower and upper jaws of H. francisci. FB, bite reaction force; FE, reaction force at the ethmoidal articulation; FH, reaction force at the hyomandibular articulation; FJR, jaw joint reaction force; FPO-α, force generated by the preorbitalis-α; FQM-PO, force generated by the quadratomandibularis-preorbitalis complex; FQM-γ, force generated by the quadratomandibularis-γ; FR, resultant adductive force; α, angle of incidence of FE relative to the articular surface of the upper jaw at the ethmoidal articulation. Arrow size does not indicate force magnitude, and angles of force vectors are approximate. CT-scan image used with permission of A. Summers.

### Theoretical force generation

Anatomical cross-sectional area (Acs) measurements of the nine parallel fibered muscles were multiplied by the specific tension of elasmobranch white muscle (Tsp; 289 kN m-2; Lou et al., 2002) to determine their theoretical maximum tetanic forces (P0): \batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} $\ P_{0}=A_{\mathrm{cs}}{\times}T_{\mathrm{sp}}.$ \end{document}(1)

Anatomical cross-sectional area was used in this analysis because theoretical estimates of maximum bite force based on the anatomical cross-sectional area of the parallel fibered jaw adducting musculature of the spiny dogfish Squalus acanthias best approximated bite forces measured during tetanic stimulation of the jaw adducting musculature (Huber and Motta, 2004). Force vectors for each muscle were constructed from their maximum tetanic forces and the three-dimensional coordinates of their origins and insertions. The force vectors of muscles excised unilaterally were reflected about the Y-plane to represent the forces generated by the musculature on the other side of the head.

Mathcad 11.1 software (Mathsoft, Inc., Cambridge, MA, USA) was used to generate a three-dimensional model of the static forces acting on the jaws of H. francisci during prey capture. Summation of the three-dimensional moments acting on the lower jaw about the jaw joints (left and right) determined the theoretical maximum bite force for each individual and the mean maximum bite force for all individuals (B; Fig. 4). Maximum bite force was modeled at points 0, 25, 50, 75 and 100% of the distance along the functional tooth row from the posteriormost tooth to determine a bite force gradient along the lower jaw. Additionally, the reaction force acting on the jaw joints during bites occurring at 0 and 100% of the distance along the functional tooth row was determined (FJR; Fig. 4).

Loadings were determined at the ethmoidal and hyomandibular articulations of the upper jaw with the chondrocranium and hyomandibula, respectively (Figs 1A, 4). For bites occurring at 0% (posteriormost molariform tooth) and 100% (anteriormost cuspidate tooth) of the distance along the functional tooth row, the moments acting on the upper jaw about the ethmoidal articulation were summed to determine the forces acting at the hyomandibular articulation (FH; Fig. 4). In these analyses, the hyomandibula was modeled as a two-force member, moveable about its articulations with both the upper jaw and chondrocranium (Hibbeler, 2004). Static equilibrium analysis of the forces acting on the upper jaw was then used to determine the forces acting at the ethmoidal articulation (FE; Fig. 4). Static equilibrium conditions for the forces acting on the lower (FLJ) and upper jaws (FUJ) were: \batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} $\ {\Sigma}\mathbf{F}_{\mathrm{LJ}}=\mathbf{F}_{\mathrm{JR}}+\mathbf{F}_{\mathrm{QM}\mathrm{-}\mathrm{PO}}+\mathbf{F}_{\mathrm{QM}\mathrm{-}{\gamma}}+\mathbf{F}_{\mathrm{PO}\mathrm{-}{\alpha}}+\mathbf{F}_{\mathrm{B}}=0,$ \end{document}(2) \batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} $\ {\Sigma}\mathbf{F}_{\mathrm{UJ}}=\mathbf{F}_{\mathrm{JR}}+\mathbf{F}_{\mathrm{H}}+\mathbf{F}_{\mathrm{QM}\mathrm{-}\mathrm{PO}}+\mathbf{F}_{\mathrm{E}}+\mathbf{F}_{\mathrm{B}}=0,$ \end{document}(3)

where FB is the bite reaction force from a prey item, FPO-α is the force generated by the preorbitalis-α, FQM-PO is the force generated by the quadratomandibularis-preorbitalis complex, and FQM-γ is the force generated by the quadratomandibularis-γ. Forces generated by the preorbitalis-α and quadratomandibularis-γ are isolated to the lower jaw because they originate on the chondrocranium and insert only upon the lower jaw (Figs 2A, 4). Joint reaction forces maintain the static equilibrium of feeding mechanisms by balancing the moments acting upon the jaws via their associated musculature and contact with prey items. The moment acting on the lower jaw during jaw opening via the coracomandibularis muscle was used to determine the theoretical maximum jaw opening force of H. francisci.

### In situ bite performance measurements

Bite performance measurements were performed using a modified single-point load cell (Amcells Corp., Carlsbad, CA, USA) with custom-designed stainless steel lever arms, which was calibrated using a series of known weights. Free-swimming H. francisci were trained to voluntarily bite the transducer by wrapping the device with squid and presenting it to them after several days of food deprivation. A P-3500 strain indicator (Vishay Measurements Group, Raleigh, NC, USA) was used for transducer excitation and signal conditioning. Data were acquired with a 6020E data acquisition board and LabVIEW 6.0 software (National Instruments Corp., Austin, TX, USA). Fifteen measurements of bite force were taken from each animal. Only events in which the transducer was bitten between the tips of the jaws were kept for analysis. The five largest bite force measurements for each individual were analyzed for the following performance variables, as well as used in the multivariate statistical analyses described below: maximum force (N), duration of force production (ms), time to maximum force (ms), rising slope of force-time curve (N s-1), duration at maximum force (ms), time from maximum force to end of force production [hereafter referred to as time away from maximum force' (ms)], falling slope of force-time curve (N s-1) and impulse (I), which is the integrated area under the force-time curve (kg m s-1) from the initiation of force generation to its cessation: \batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} $\ I={{\int}}F{\,}\mathrm{d}t,$ \end{document}(4)

where F is force and t is time. The impulse of a force is the extent to which that force changes the momentum of another body, in this case the force transducer, and therefore has the units of momentum (kg m s-1). For each individual, the single largest bite force and its associated performance measurements were used to create a profile of maximum bite performance for H. francisci, to compare the dynamics of the ascending and descending portions of the bite performance waveforms and to compare the maximum bite forces obtained from the theoretical, in situ, restrained and stimulated methods of determining bite force (see below).

In situ bite performance measurements were simultaneously filmed with a Redlake PCI-1000 digital video system (Redlake MASD, San Diego, CA, USA) at 250 frames s-1 to verify that bites on the transducer occurred between the tips of the jaws (hereafter referred to as transducer bites'). The modified single-point load cell used in this study averages the signals generated by four strain gages in a full Wheatstone bridge such that the transducer is insensitive to the position on the lever arms at which the bite is applied. Therefore, the point at which a shark bit the lever arms of the transducer did not need to be determined from the digital video sequences for appropriate calibration. To identify any behavioral artifacts associated with biting a stainless steel transducer, H. francisci were also filmed while consuming pieces of O. oglinum cut to the same size as the biting surface of the force transducer (hereafter referred to as fish bites'). The following kinematic variables were quantified from transducer and fish bites using Motionscope 2.01 (Redlake MASD, San Diego, CA, USA) and SigmaScan Pro 4.01 software: distance, duration, velocity and acceleration of lower jaw depression, lower jaw elevation, upper jaw protrusion, and head depression; maximum gape; time to maximum gape; time to onset of lower jaw elevation; time to onset of head depression; cranial elevation angle. All kinematic variables were quantified using discrete cranial landmarks as reference points (Edmonds et al., 2001).

### Restrained and stimulated bite performance measurements

At least one week after the in situ bite performance measurements, four of the H. francisci were individually removed from the experimental tank and restrained on a table. Once they had opened their jaws an adequate distance, the transducer was placed between the anterior teeth, which elicited an aggressive bite. Following a recovery period of approximately 10-15 min, the shark was again removed from the tank and anaesthetized with MS-222 (0.133 g l-1). The quadratomandibularis-preorbitalis complex, quadratomandibularis-γ and preorbitalis-α were implanted with stainless steel 23-gauge hypodermic needles connected to a SD9 stimulator (Grass Telefactor, West Warwick, RI, USA), and tetanic fusion of these muscles was accomplished via stimulation (10 V, 100 Hz, 0.02 ms delay, 3 ms pulse width) while the bite force transducer was placed between the tips of the anterior teeth. Three measurements were taken from each individual in both of these experimental protocols. Individuals were ventilated with aerated seawater between measurements during muscle stimulation experiments. Maximum bite force, time to maximum force, and time away from maximum force were quantified from all restrained and stimulated bites.

### Statistical analysis

All bite performance and kinematic variables were log10 transformed and linearly regressed against body mass to remove the effects of size. Studentized residuals were saved from each regression for subsequent analysis (Quinn and Keough, 2002). Principal components analyses (PCA) based on correlation matrices were then used to (1) identify covariation in bite performance variables and reduce these variables to a series of non-correlated principal components, which were subsequently analyzed to assess the extent of individual variability in these parameters, (2) identify covariation in performance and kinematic variables from in situ bite performance trials and (3) identify covariation in kinematic variables from fish and transducer bites and reduce these variables to a series of non-correlated principal components, which were subsequently analyzed to determine whether there were any behavioral artifacts associated with biting the steel transducer. Variables were considered to load strongly on a given principal component (PC) if their factor scores were greater than 0.6. Non-rotated axes described the greatest amount of variability in each PCA. For analyses 1 and 3, multivariate analysis of variance (MANOVA) was used to compare the factor scores for the PCs with eigenvalues greater than 1.0. To determine whether fish and transducer bites differed kinematically, a two-way, mixed-model MANOVA was performed on the PCs from PCA 3, with individual' as the random effect and prey type' as the fixed effect, which was tested over the interaction mean square. Kinematic data from four individuals were included in this analysis because a complete data set was lacking for one individual. To determine the extent of individual variability within the bite performance variables, a one-way MANOVA was performed on the PCs from PCA 1.

To determine whether the kinematic variables associated with biting the transducer were predictive of biting performance in H. francisci, stepwise (forward) multiple regressions were performed with kinematic variables measured from transducer bites as the multiple independent factors, and the eight bite performance variables as the individual dependent factors. Data from four individuals were included in this analysis because a complete kinematic data set was lacking for one individual. One-way ANOVA on Studentized residuals was used to identify significant differences among the theoretical, in situ, restrained and electrically stimulated methods of determining maximum bite force. A Student's t-test was used to identify differences between time to maximum force and time away from maximum force and between the rising and falling slopes of the force-time curves for in situ biting trials. One-way ANOVA was used to compare time to maximum force and time away from maximum force within and among in situ, restrained and electrically stimulated bite forces. Finally, bite forces at the anterior jaw (fish, reptiles and birds) or canine teeth (mammals) and body masses for various vertebrates were compiled from the available literature and grouped according to major taxonomic level. These bite forces, along with those of the horn sharks investigated in this study, were linearly regressed against body mass. Studentized residuals from this regression were then coded according to taxonomic level and compared with a one-way ANOVA. All significant differences were investigated post-hoc with Tukey's pairwise comparisons test. Linear regressions were performed in SigmaStat 2.03 (SYSTAT Software, Inc.) in order to obtain Studentized residuals. All other statistical analyses were performed in SYSTAT 10 (SYSTAT Software, Inc.) with a P-value of 0.05.

## Results

### Biomechanical modeling

The quadratomandibularis-preorbitalis complex, which is the primary jaw adductor, generated the greatest force of all muscles investigated (242 N; Table 1). Of the muscles active during jaw and hyobranchial abduction, the coracobranchiales generated the greatest force (107 N; Table 1). The levator hyomandibularis generated more force during the retractive phase (33 N) than the levator palatoquadrati (20 N; Table 1). After resolving the force generated by the adductor musculature into its principal components, the majority of force was directed dorsally (294 N) and anteriorly (128 N). The Z-axis components of this force (19 N per side) were directed laterally on either side of the head and negate each other during jaw adduction (Table 2; Fig. 3A). Thus, the resultant adductive force along the Z-axis was 0 N. The large anterodorsally directed component of this adductive bite force (FR; Fig. 4) drives the lower jaw towards the upper jaw, which is itself driven into the ethmoid region of the chondrocranium (FE; Fig. 4).

View this table:
Table 1.

Theoretical maximum forces generated by the cranial musculature active during the gape cycle in H. francisci

View this table:
Table 2.

Resultant bilateral muscle and jaw forces occurring during prey capture in H. francisci, broken into their principal components

Summation of the moments acting on the lower jaw determined that the maximum theoretical bite force of H. francisci ranged from 128 N at the anterior teeth to 338 N at the posteriormost molariform teeth (Fig. 5; Table 2). The bite force at the posteriormost molariform teeth exceeded the resultant force generated by the adductive musculature (Table 2) because the mechanical advantage at this point along the jaw was 1.06. The resultant jaw closing mechanical advantage at the anterior teeth was 0.51, resulting in a dramatically lower bite force at this point.

The jaw joint reaction forces (FJR; Fig. 4) occurring when prey is captured at the anterior teeth and crushed at the posterior teeth by H. francisci were 106 N and 73 N per side, respectively (Table 3). This force was oriented posteroventrally relative to the articular surface of the lower jaw joint for anterior biting, and consequently oriented anterodorsally relative to the articular surface of the upper jaw joint. The local/internal loadings on the joint between the upper and lower jaws indicate that the jaw joint is globally in compression (Hibbeler, 2004) when prey is bitten at the tips of the jaws. When prey is crushed between the posterior molariform teeth, the orientation of the vertical component of the joint reaction force relative to the lower jaw (25 N) was opposite that for the lower jaw joint during anterior biting (-80 N), indicating tensile loading of the jaw joint during posterior prey capture (Table 3).

View this table:
Table 3.

Unilateral mechanical loadings at articulation points in H. francisci's feeding mechanism, broken into their principal components

The ethmoidal articulation of H. francisci received a loading of 59 N per side during biting, regardless of whether biting occurred at the anterior or posterior margin of the jaws (FE; Fig. 4). The angle of incidence of this force relative to the articular surface of the upper jaw at the ethmoidal articulation was 80° (α; Fig. 4). For both anterior and posterior biting, the majority of loading was directed ventrally into the upper jaw, indicating compression between the ethmoid region of the chondrocranium and the palatal region of the upper jaw (Table 3).

The magnitude of loading at the hyomandibular articulation (36 N) was independent of bite point as well (FH; Fig. 4). The lower jaw was loaded posterodorsally and medially at its articulation with the hyomandibula during both anterior and posterior biting (Table 3). The reaction forces acting on the distal ends of the hyomandibula are equal to and opposite the forces acting at the jaws' articulation with the hyomandibula. Therefore, during biting, the hyomandibula was loaded anteroventrally and laterally. These local/internal loadings between the jaws and hyomandibula indicate that the hyomandibula is globally in tension. Modeling the hyomandibula as a two-force member assumed that the line of action of the force acting on the hyomandibula passed through its articulation with the jaws and chondrocranium. The hyomandibula is therefore loaded in pure tension, and the angle of incidence of the hyomandibular force cannot be determined.

Fig. 5.

Theoretical maximum bite force (N) of five male H. francisci (N=5, TL=55-68 cm) from three-dimensional vector analysis of the jaw adducting musculature measured at 0, 25, 50, 75 and 100% of the length of the functional tooth row of the lower jaw from posterior to anterior. CT-scan image used with permission of A. Summers.

The only muscle involved in abduction of the lower jaw is the coracomandibularis, which was capable of generating 31 N of force (Table 1). This muscle inserts on the caudal aspect of the lower jaw symphysis at 37° below the longitudinal axis of this jaw and has a mechanical advantage of 0.89. Despite this high mechanical advantage, indicative of force amplification in a class III lever system, its acute insertion angle caused the muscular force generating motion about the lower jaw (force component perpendicular to the lower jaw) to be 19 N (Table 2). After accounting for mechanical advantage, the resultant abductive force at the tip of the lower jaw was 16 N (Table 2). The abductive force lacks a component along the Z-axis because the coracomandibularis runs parallel to the longitudinal axis of the body. The other muscles involved in the expansive phase of the gape cycle generated considerably greater forces than the coracomandibularis (Table 1).

Fig. 6.

Bite force waveforms from bite performance trials of three male H. francisci (TL=66-70 cm), illustrating in situ voluntary bites with single (black) and double (light gray) force peaks and a bite from a restrained individual (dark gray).

### Performance measurements

#### In situ measurements

H. francisci approached and bit the force transducer in an attempt to remove the attached food. In most cases, biting continued until the food was removed from the transducer. PCA 1 reduced the performance variables for each individual to three PCs (89.7% of variance explained), each of which indicated considerable overlap among individuals. MANOVA subsequently demonstrated no differences among individuals for bite performance variables using size-corrected data (Wilk's Lambda=0.51, F12,47=1.157, P=0.340). The mean maximum in situ bite force measured at the anterior teeth was 95 N, with an absolute maximum force of 133 N (66 cm male H. francisci). Heterodontus francisci took approximately 322 ms to reach maximum bite force, which was held for 41 ms, and released after an additional 212 ms (Table 4). The mean duration of force application was 535 ms. Time to maximum bite force was longer than the time away from maximum bite force (P=0.049). The mean rising slope of the force-time curve was 300 N s-1 and was lower than the mean falling slope of 457 N s-1 (P=0.048). The mean impulse generated from the beginning of force application until its cessation was 25 kg m s-1 but measured as high as 44 kg m s-1. The majority of bite force waveforms consisted of single peaks associated with single bites. However, in 32% of the bites, multiple peaks occurred, indicating a repetitive crushing behavior during force application (Fig. 6).

View this table:
Table 4.

In situ bite performance data for H. francisci biting at the tips of its jaws

PCA 2 of performance and kinematic variables yielded six PCs with eigenvalues greater than 1.0, which collectively explained 86.7% of the variance. All of the variables that loaded heavily on the first PC (30.5% of variance explained) were kinematic measurements (Table 5). These variables primarily demonstrated covariance in the timings and excursions of lower jaw depression and elevation. Performance measures were the only variables to load heavily on the second PC (19.5% of variance explained), indicating covariance between rates and durations of force application (Table 5). Maximum bite force did not load heavily until the fifth PC (7.8% of variance explained), and impulse did not load heavily on any of the PCs.

View this table:
Table 5.

Principal component loadings of performance and kinematic variables from bite performance trials of H. francisci

Stepwise multiple regressions yielded similar results to PCA 2 on kinematic and performance data. Only three of the bite performance variables were significantly related to individual kinematic variables. Force duration was significantly, though poorly, related to lower jaw elevation velocity (r2=0.226, F1,18=5.268, P=0.034). Similarly, time to maximum force (r2=0.389, F1,18=11.471, P=0.003) and the rising slope of the force-time curve (r2=0.410, F1,18=12.523, P=0.002) were significantly related to lower jaw elevation distance. Inclusion of additional kinematic variables did not improve the predictive ability of these regression models. The two variables indicative of the magnitude of bite force generated (maximum force, impulse) could not accurately be predicted by any combination of kinematic variables. Although kinematic variables were not predictive of bite performance variables, PCA 1 used to assess individual variability (see above) identified notable covariance in performance measures. Maximum in situ bite force exhibited a strong linear relationship with impulse (r2=0.758) and moderate linear relationships with force duration (r2=0.450) (Fig. 7) and time to maximum force (r2=0.489).

Fig. 7.

Maximum in situ bite force (N) from five male H. francisci (TL=63-74 cm) plotted against (A) impulse (kg m s-1) and (B) force duration (ms) on logarithmic axes.

PCA 3 reduced the set of kinematic variables measured from fish and transducer bites to a series of four PCs (73.3% of variance explained). MANOVA indicated no significant differences between the prey capture kinematics of H. francisci while biting fish or the transducer on any of the PCs for all individuals (Wilk's Lambda=1.0, F4,29=0.0, P=1.0). However, a single individual was found to differ from two other individuals on the first PC (F3,32=4.646, P=0.008). Variables that loaded heavily on the first PC were durations and distances of lower jaw depression and elevation, times to maximum gape, onset of lower jaw elevation, completion of lower jaw elevation and maximum gape distance. The acceleration of lower jaw elevation loaded heavily, but negatively, on the first PC.

#### Methodological comparison

In situ measurement of maximum bite force was a reasonably good indicator of the maximum bite force of H. francisci. Using size-corrected data, a single difference was found among the four methods of determining maximum bite force (F3,14=4.358, P=0.023). Restrained bite force (159-206 N) was significantly greater than in situ bite force (60-133 N) (P=0.013). In situ bite force was, however, equivalent to theoretical (107-163 N) and electrically stimulated (62-189 N) bite forces. Restrained, electrically stimulated and theoretical bite forces were equivalent (Table 6). During restrained bites, time to maximum force (522 ms) was greater than time away from maximum force (339 ms) (t4=2.848, P=0.046). Time to maximum force (285 ms) was shorter than time away from maximum force (556 ms) for electrically stimulated bites (t8=-5.476, P<0.001). Significant differences were detected between the in situ, restrained and electrically stimulated methods for time to maximum force (F2,10=4.996, P=0.031) and time away from maximum force (F2,10=58.290, P<0.001). Time to maximum force was greater for restrained bites than for electrically stimulated bites (P=0.030), both of which were equivalent to the time to maximum force of in situ bites. Time away from maximum force was greater for electrically stimulated bites than restrained bites (P=0.001), which was greater than that of in situ bites (P=0.005).

View this table:
Table 6.

Results of one-way ANOVA on different methods of determining bite force at the tips of the jaws in H. francisci

### Bite forces among vertebrates

Bite forces and body masses were compiled for 113 species of vertebrates (including H. francisci) from the available literature (Binder and Van Valkenburgh, 2000; Cleuren et al., 1995; Clifton and Motta, 1998; Erickson et al., 2004; Hernandez and Motta, 1997; Herrel et al., 1999, 2001, 2002; Huber et al., 2004; Huber and Motta, 2004; Korff and Wainwright, 2004; Ringqvist, 1972; Robins, 1977; Thomason et al., 1990; Thompson et al., 2003; van der Meij and Bout, 2004; Weggelaar et al., 2004; Wroe et al., 2005; D. R. Huber, M. N. Dean, and A. P. Summers, unpublished) (Appendix I). Collectively, bite force scaled to body mass with a coefficient of 0.60, which is below the isometric scaling coefficient of 0.67 (Fig. 8). When the mammalian bite forces from Wroe et al. (2005) were excluded from this analysis, bite force scaled with a coefficient of 0.66, approximating isometry. This discrepancy is probably due to Wroe et al. (2005) having used the dry-skull method of estimating muscle Acs, which can underestimate Acs by 1.3-1.5× (Thomason et al., 1991).

View this table:
Appendix I

Fishes collectively had the highest mass-specific bite force of the four vertebrate groups, followed by reptiles, mammals and birds, respectively (F3,130=6.357, P<0.001). Mass-specific bite force of the fishes was greater than those of the birds (P=0.002) and mammals (P=0.013), while reptilian mass-specific bite force was greater than that of the birds (P=0.009). The striped burrfish, Chilomycterus schoepfi, had the highest mass-specific bite force, followed by the Canary Island lizard, Gallottia galloti, and the American alligator, Alligator mississippiensis (Erickson et al., 2004; Herrel et al., 1999; Korff and Wainwright, 2004). The hogfish, Lachnolaimus maximus, had the second highest mass-specific bite force, but for biting with the pharyngeal jaws not the oral jaws (Clifton and Motta, 1998). The three lowest mass-specific bite forces were those of the red-bellied short-necked turtle, Emydura subglobosa, the mata mata turtle, Chelus fimbriatus, and the twist-necked turtle Platemys platycephala (Herrel et al., 2002) (Fig. 8). Of the cartilaginous fishes in this analysis, the mean mass-specific bite force of H. francisci was greater than those of S. acanthias and the blacktip shark, Carcharhinus limbatus, but less than that of the white-spotted ratfish, Hydrolagus colliei.

Fig. 8.

(A) Bite forces (N) of various vertebrates plotted against mass (g). (B) Residuals from regression analysis of log10 bite force versus log10 mass plotted against log10 mass (g). Broken lines indicate ± 1 standard deviation about the residual mean.

## Discussion

### Functional morphology

The jaw adducting cranial musculature (QM-PO complex, QM-γ, PO-α on Fig. 2) of H. francisci generates more force during prey capture than either the jaw and hyobranchial abducting or retracting musculature. The mechanical advantage of H. francisci's jaw adducting mechanisms ranges from 0.51 at the tip of the jaws to 1.06 at the posterior margin of the functional tooth row. In class III lever systems such as shark jaws, a mechanical advantage greater than 1.0 indicates that the point at which force is being applied to a prey item is closer to the jaw joint than the point at which muscular force is being applied to the jaw, resulting in an amplification of the muscular force. Subsequently, the theoretical maximum bite force at the posterior margin of the functional tooth row exceeds the resultant force generated by H. francisci's adductor musculature. This amplification of muscular force is advantageous for the processing of hard prey such as the molluscs, echinoderms and benthic crustaceans consumed by H. francisci (Segura-Zarzosa et al., 1997; Strong, 1989).

The jaw closing mechanical advantage at the anterior teeth of H. francisci is greater than that of the only other elasmobranch for which values have been published, S. acanthias (0.28; Huber and Motta, 2004), which utilizes a combination of ram and suction feeding to consume soft-bodied prey (Wilga and Motta, 1998). Its jaw closing mechanical advantage is greater than those at the anterior teeth of nearly every actinopterygian fish investigated (∼150), which include prey from plankton to hard-shelled species (Durie and Turingan, 2001; Turingan et al., 1995; Wainwright et al., 2004; Westneat, 2004). The durophagous species among these taxa do, however, have the highest jaw adducting mechanical advantages. The durophagous parrot fishes (Scaridae) are the only actinopterygian fishes with jaw adducting mechanical advantages comparable with that of H. francisci (Wainwright et al., 2004; Westneat, 2004). Thus, there is extensive evolutionary convergence on high leverage jaw adducting mechanisms in fishes that consume hard prey.

The jaws of H. francisci are elliptical in transverse-section, with their major axis oriented vertically, in-line with the compressive stresses associated with feeding. Calcium reinforcement in the jaw cortex increases posteriorly as the dentition becomes more molariform and is greatest at the jaw joints (Summers et al., 2004). Calcification and elliptical geometry increase the second moment of area of the jaws with respect to the compressive loading of prey capture and processing, which augments the jaws' ability to resist dorsoventral flexion (Summers et al., 2004). The resolved force vector for jaw adduction also occurs approximately in the region of the most robust molariform teeth of H. francisci, where it can generate upwards of 338 N of bite force. Therefore, maximum bite force is produced where both the dentition and jaw cartilages are best able to resist compressive stresses.

Despite the high mechanical advantage (0.89) of the coracomandibularis muscle in the lower jaw depression mechanism of H. francisci, its acute insertion angle relative to the lower jaw causes most of its force to be directed posteriorly, into the jaw joints (Table 2.) This high mechanical advantage is due to the insertion of the coracomandibularis on the posterior margin of the mandibular symphysis, which is synapomorphic for elasmobranchs (Wilga et al., 2000). Although this mechanism is suited for force production, velocity production is desirable for inertial suction feeding. Heterodontus francisci nonetheless effectively uses suction to initially capture and reorient prey (Edmonds et al., 2001), which may be due in part to its powerful hyoid and branchial abductors (Table 1). These muscles rapidly expand the floor of the buccopharyngeal cavity, which is critical to suction feeding in elasmobranchs (Motta et al., 2002; Svanback et al., 2002). As in the nurse shark, Ginglymostoma cirratum (Motta et al., 2002), the large labial cartilages of H. francisci considerably occlude its lateral gape, theoretically augmenting suction ability (Muller and Osse, 1984; Van Leeuwen and Muller, 1984).

Three-dimensional resolution of the forces generated during jaw adduction may reveal the mechanical basis of upper jaw protrusion in H. francisci. The force driving the upper jaw into the ethmoidal articulation has both dorsal and anterior components, causing the upper jaw to slide through the anteroventrally sloping palatal fossa of the chondrocranium and protrude (Fig. 4; Table 2). This proposed mechanism is based on the resolved force vector for all muscles involved in jaw adduction. Differential activity of the heads of this complex may facilitate modulation of protrusion. The quadratomandibularis-γ is the likely candidate for control over protrusion because its acute insertion angle relative to the lower jaw and anterior insertion point give it high leverage over anterior motion (Fig. 2).

Activity of the quadratomandibularis-preorbitalis complex alone, which has a broad insertion on the lateral face of both the upper and lower jaws, may contribute to protrusion of the upper jaw as well. After the lower jaw has been depressed, contraction of this muscle complex may simultaneously raise the lower jaw and pull the upper jaw away from the skull. This mechanism has been proposed for upper jaw protrusion in S. acanthias, G. cirratum and the lemon shark, Negaprion brevirostris (Moss, 1977; Motta et al., 1997; Wilga and Motta, 1998). Protrusion by H. francisci, which may be used to chisel away at attached benthic prey, occurs after the lower jaw has been depressed (Edmonds et al., 2001), corroborating the role of the quadratomandibularis-preorbitalis complex in this behavior.

Extensive calcification near the jaw joints of H. francisci (Summers et al., 2004) would apparently indicate high joint reaction forces during prey capture. Joint reaction forces can exceed bite forces at the tip of the jaw depending on the mechanical advantage of the given feeding mechanism and the force produced by the associated musculature, which has been identified in numerous reptiles (3-4× greater; Cleuren et al., 1995; Herrel et al., 1998). Although joint reaction force was greater than anterior bite force in H. francisci, the ratio of these values (1.65) is substantially lower than those found for reptiles. The ratio of joint reaction force to posterior bite force in H. francisci was 0.43.

Low ratios of joint reaction force to bite force in H. francisci are due to its high mechanical advantage jaw adducting mechanism. Humans, which share this characteristic, have correspondingly low ratios of joint reaction force to bite force (Koolstra et al., 1988). Although some damping will occur in the connective tissue associated with the jaw joint, loading occurring at the joint will be transmitted to adjacent skeletal elements. Therefore, low ratios of joint reaction force to bite force may be adaptive in H. francisci, and elasmobranchs in general, because the posterior region of their jaws is suspended from the cranium by mobile hyomandibulae, not a stable jaw articulation as in other vertebrates. Minimizing loading at this articulation may stabilize the feeding mechanism during prey capture and processing.

In heterodontiform sharks, the cranial stresses associated with prey capture can be isolated to the ethmoidal and hyomandibular articulations. Unlike carcharhinid sharks (Motta and Wilga, 1995), the upper jaw of H. francisci does not disarticulate from the chondrocranium during feeding, even during upper jaw protrusion (Maisey, 1980). Therefore, in carcharhinid sharks, the hyomandibulae may receive all of the suspensorial loading occurring during prey capture. Optimal loading at the ethmoidal articulation would entail forces directed perpendicularly into the articular surface of the upper jaw because cartilage is strongest in axial compression (Carter and Wong, 2003). The estimated forces at this articulation deviated from optimal orientation by only 10° during anterior and posterior biting. The ethmoidal articulation of H. francisci appears well designed for withstanding this nearly axial compressive loading because the upper jaw calcifies at this articulation early in ontogeny (Summers et al., 2004) and the ethmoid region of the chondrocranium is one of the thickest parts of this structure (Daniel, 1915). Additionally, maintenance of contact between the upper jaw and chondrocranium in H. francisci will distribute stresses from the repetitive loading associated with processing hard prey.

Although it is well known that the hyomandibulae support the posterior margin of the jaws, the nature of the loading they receive has been a matter of speculation. This mechanical analysis indicates that the hyomandibulae of H. francisci are tensile elements, as suggested by Moss (1972) and Frazzetta (1994). Consequently, the hyomandibulae may regulate anterior movement of the jaws during feeding, such as would occur during jaw protrusion. In this regulatory role, activity of the levator hyomandibularis could hypothetically modulate resistance to anterior motion of the jaws. Electromyographic analysis of the feeding musculature of H. francisci would be required to verify this hypothesis.

The ligamentous attachments between the hyoid arch and the posterior end of the jaws stabilize this articulation against the tensile stresses caused by biting. The internal hyomandibular palatoquadrate and hyoideo-mandibulare ligaments resist dorsoventral translation between the hyomandibula and jaws, while the hyomandibuloceratohyal ligament prevents lateral translation between these elements. The two slips of the median ligament (Daniel, 1915) stabilize against dorsoventral and lateral translation, respectively. Although this analysis makes the assumption that the hyomandibulae are loaded as two-force members in axial tension, they probably experience a more diverse loading pattern in nature, necessitating this multidirectional support.

Increased hyomandibular loading may have played a role in the transition from amphistylic to hyostylic jaw suspensions in modern elasmobranchs. As the number and size of the articulations between the jaws and chondrocranium was reduced, the hyomandibulae took on a greater role in suspending the jaws (Carroll, 1988; Maisey, 1980; Schaeffer, 1967). Concomitant with these changes in articulation, the hyomandibulae became shorter and more mobile (Cappetta, 1987; Maisey, 1985; Moy-Thomas and Miles, 1971; Schaeffer, 1967), oriented more orthogonally to the chondrocranium (Stahl, 1988; Wilga, 2002) and had more extensive ligamentous attachments with the jaws and chondrocranium (Gadow, 1888). Collectively, these changes may have been associated with a shifting of the force of jaw adduction to a more posterior region of the jaws. This may have resulted in greater hyomandibular loading as well as a freeing-up' of the anterior margin of the jaws such that upper jaw kinesis was increased, facilitating jaw protrusion and prey gouging (Maisey, 1980; Moss, 1977; Wilga, 2002).

Although the jaw suspension mechanism of H. francisci is classified as hyostylic (Gregory, 1904; Wilga, 2002), this analysis indicates that it is functionally amphistylic. Heterodontus francisci exhibits considerable upper jaw kinesis (reduces maximum gape by 39%), similar to other hyostylic carcharhiniform sharks (Edmonds et al., 2001; Wilga et al., 2001). Despite this functional similarity, the upper jaw does not disarticulate from the chondrocranium during protrusion in H. francisci. Furthermore, in contrast to the hypotheses regarding hyomandibular evolution (see above), H. francisci has considerable loading at both the hyomandibular (tensile) and ethmoidal (compressive) articulations. The term hyostyly' should therefore be reserved for taxa in which the upper jaw disarticulates from the chondrocranium during protrusion such that hyomandibulae are the primary means of support and the ethmopalatine ligaments are loaded in tension. Therefore, contemporary definitions of jaw suspension should incorporate functional interpretations of loadings at the various articulations between the jaws and cranium, as well as the relationship between suspension type and upper jaw protrusion.

### Methodological comparison

Although no differences were found between theoretical, restrained and electrically stimulated bite force measurements using size-corrected data, the absolute maximum bite force for each individual occurred during restrained bite force measurements. No differences were found between theoretical and electrically stimulated bite force measurements of S. acanthias either (Huber and Motta, 2004). Therefore, restrained measurements appear to be the best method of obtaining maximum bite force measurements from live elasmobranchs. Small sample size might, however, account for the lack of statistical significance found between different methods of determining bite force. Nonetheless, the results of both this analysis and that of bite force production in S. acanthias (Huber and Motta, 2004) indicate that theoretical estimates of bite force in sharks are accurate in predicting maximum bite forces. This is fortunate given the logistical problems associated with obtaining bite force measurements from live elasmobranchs. Given the appropriate resources, however, maximum bite force can be obtained through in situ methods, as indicated by the equivalence of theoretical, electrically stimulated and in situ bite forces in this study. In situ measurements enable the quantification of biting dynamics as well, which is informative regarding feeding performance and ecology (see below).

Static estimates of force production based on muscle architecture may underestimate actual force production because active stretching of the jaw adductors during the expansive phase of the gape cycle can increase force production (Askew and Marsh, 1997; Josephson, 1999). Furthermore, by modeling the primary jaw adductor as the quadratomandibularis-preorbitalis complex instead of delineating the individual heads of this complex, variations in muscle architecture of these heads such as pinnate insertion points may have been overlooked. If this were the case, a theoretical model of force production based on morphological cross-sectional area alone could underestimate maximum force production.

The ratios of time to and away from maximum force for in situ (1.52) and restrained (1.54) bites suggest that the application of bite force by H. francisci takes longer than its release. However, the opposite relationship for these variables occurred during electrically stimulated bites. The ratio of time to and away from maximum force during electrically stimulated biting (0.51) approximates the ratio of time for twitch tension development to relaxation (0.42) for pectoral fin muscle of the cuckoo ray, Raja naevus (Johnston, 1980). This suggests that force generation during voluntary or stimulated biting is a function of the rate at which the adductor muscles reach tetanic fusion. Gradual summation of motor unit recruitment during voluntary biting results in a prolonged time to maximum force, whereas manual, high-frequency electrical stimulation of the adductor muscles causes more rapid tetani and subsequently shorter times to maximum force. Time away from maximum force was longer for restrained measurements than for in situ measurements, perhaps indicating motivational differences between these two presentation methods.

### Feeding performance

Several bite performance variables demonstrated patterns consistent with the durophagous diet of H. francisci. The time to maximum bite force application by H. francisci was longer than time away from maximum force, the rising slope of the force-time curve was lower than the falling slope, and maximum bite force was positively related to the time to maximum force. These performance characteristics indicate that the application of bite force is a slower, more deliberate action than its release by H. francisci. Linear relationships of maximum bite force with impulse and force duration further indicate that higher bite forces are associated with slower, more deliberate closing of the jaws by H. francisci (Fig. 7).

The impulse generated upon impact between two bodies is a measure of momentum transfer and can be interpreted as the effort' that each body exerts on the other (Nauwelaerts and Aerts, 2003). Because momentum is conserved during impact, larger impulses generated during biting transfer greater quantities of kinetic energy from the jaws to the prey. Optimizing impulse by maximizing bite force output per unit time will increase the amount of energy contributing to the rupture/fracture of a prey item. Heterodontus francisci capitalizes upon this when consuming hard prey with composite exoskeletons. Sustained loading after a high-velocity initial impact is effective at fracturing composite structures such as sea urchin exoskeletons (calcite ossicles linked by collagenous ligaments) because composites harden to a saturation point upon initial compression, after which crack nucleation occurs, followed by structural failure (Christoforou et al., 1989; Ellers et al., 1998; Provan and Zhai, 1985; Strong, 1989).

The prevalence of multiple force peaks within a compressive waveform of a single bite (32% of in situ bites) also indicates H. francisci's behavioral specialization for exploiting hard prey (Fig. 6). This behavior maximizes the damage inflicted upon prey items during a given bite by ramping up the applied force multiple times, especially when there are multiple bites during a feeding event. The rate at which the strength of a composite structure degrades is a power function of both the strain rate and number of strain cycles (Hwang and Han, 1989). Multiple force peaks within a given bite indicate that H. francisci may have evolved motor patterns specialized for durophagy as well. High-frequency bursts of electrical activity associated with rhythmic compression of prey items occur in the jaw adductor musculature of the lungfish Lepidosiren paradoxa (Bemis and Lauder, 1986). Prolonged jaw adductor activity occurs in the queen triggerfish, Balistes vetula (Turingan and Wainwright, 1993), and the bonnethead shark, Sphyrna tiburo, which also uses repeated compressions of the jaws to process prey (Wilga and Motta, 2000). All of these fish include hard prey in their diets (Berra, 2001; Turingan and Wainwright, 1993; Wilga and Motta, 2000). These behavioral attributes demonstrate that the way in which force is applied to prey items, and not just the magnitude of force, is likely to be a determinant of feeding success.

Although covariation in several performance measures appears related to the consumption of hard prey by H. francisci, covariation was lacking between kinematic and performance variables from the in situ bite performance trials. Both principal components and multiple regression analyses demonstrated the inability of kinematic measures to predict bite performance measures with any accuracy (Table 5). These findings beg the question, how are two series of sequential behaviors so unrelated? One would assume that at least the kinematics of lower jaw elevation (e.g. velocity, acceleration) would be predictive of biting performance (e.g. maximum force, impulse). This lack of covariation is probably due to the instantaneous position of the jaw adducting muscles of H. francisci on the force-velocity curve relating muscle tension to contraction velocity (Aidley, 1998). Based on this principle, when the adductor musculature is elevating the lower jaw, it is contracting with high velocity and low force. However, once contact is made with the bite force transducer, movement of the lower jaw is impeded and the jaw adductors shift to the low-velocity, high-force region of the force-velocity curve. In addition to high force, maximum muscle power is generated at low velocity as well (Askew and Marsh, 1997). Because of the dramatic differences in muscle function at either end of the force-velocity curve, jaw kinematics and biting performance may vary conversely and possibly be modulated independently. A predictive relationship between cranial kinesis and performance kinetics is more likely to be found for behaviors such as suction feeding in which kinesis and performance occur simultaneously (cranial expansion and suction generation; Sanford and Wainwright, 2002; Svanback et al., 2002), not sequentially, as is the case in biting performance (jaw adduction and bite force application).

An additional behavior that may augment the biting performance of H. francisci is the use of upper jaw protrusion to dislodge and chisel away at hard prey, as was suggested by Edmonds et al. (2001). While the restrained bite force measurements of H. francisci indicate that they can consume prey capable of resisting over 200 N using this behavior, in situ bite force measurements suggest they would consume smaller, less durable prey. An analysis of the forces necessary to crush various sizes of hard prey items found in H. francisci's diet is needed to delineate the prey it is theoretically capable of consuming (potential niche) from that which it actually consumes (realized niche). Functioning at maximum capacity would typically be an unnecessary expenditure of energy, especially when feeding occurs in a niche such as durophagy that is relatively inaccessible to sympatric taxa.

### Feeding ecology

The cranial architecture and prey capture behavior of H. francisci enable it to exploit hard prey, which is a relatively untapped ecological niche for aquatic vertebrates. In fishes, durophagy has been associated with high bite forces and low dietary diversity (Clifton and Motta, 1998; Wainwright, 1988). Species capable of consuming hard prey are morphologically segregated by relative differences in bite force and ecologically segregated by the hardness of the prey they can consume (Aguirre et al., 2003; Kiltie, 1982). Therefore, durophagy appears to result in niche specialization and competition reduction. This is the case in H. francisci because hard prey (molluscs, echinoderms, benthic crustaceans) comprises approximately 95% of its diet (Segura-Zarzosa et al., 1997; Strong, 1989). However, Summers et al. (2004) suggested that H. francisci goes through an ontogenetic shift to durophagy due to biomechanical changes in its jaw cartilages. It remains to be seen if H. francisci undergoes a reduction in dietary diversity and increased niche specialization over ontogeny, with associated changes in feeding behavior and performance. A more detailed dietary analysis of neonate and juvenile H. francisci would be needed to determine whether these changes occur.

Biomechanical modeling and performance testing provide a morphological and behavioral basis from which to interpret differences in organismal ecology. These analyses determined that H. francisci is capable of generating bite forces an order of magnitude higher than comparably sized S. acanthias (Huber and Motta, 2004) and that H. francisci applies bite force in a way suited for processing hard prey. Differences in the feeding performance of H. francisci and S. acanthias directly coincide with the different feeding niches they occupy (durophagy and piscivory, respectively; Alonso et al., 2002; Segura-Zarzosa et al., 1997). Therefore, these analyses are of utility for understanding the diversity of elasmobranch feeding mechanisms at numerous organismal levels (morphology, behavior, ecology), as well as the selective pressures involved in the evolution of these mechanisms.

Heterodontus francisci has the second highest mass-specific bite force of the cartilaginous fishes in which bite force has been measured or estimated (Huber et al., 2004; Huber and Motta, 2004; Weggelaar et al., 2004; D. R. Huber, M. N. Dean and A. P. Summers, unpublished). Relative to body mass, the hardest biting cartilaginous fish studied thus far is H. colliei, which is also durophagous (Ebert, 2003; Johnson, 1967). Neither H. francisci nor H. colliei were comparable in biting ability to the durophagous teleost fishes C. schoepfi, L. maximus and the sheepshead, Archosargus probatocephalus. The mass-specific bite forces of these teleost fishes, which possess a battery of anatomical specializations associated with durophagy (Clifton and Motta, 1998; Hernandez and Motta, 1997; Korff and Wainwright, 2004), were considerably higher than those of the durophagous cartilaginous fishes (Appendix I). Comparative materials testing of the hard prey items in the diets of these cartilaginous and teleost fishes would be required to determine the ecological relevance of these differences in bite force. Nonetheless, the bite forces of these fishes collectively indicate that high biting performance, in addition to anatomical specialization, are associated with the consumption of hard prey.

### Conclusions

The heterodontiform sharks, as represented by the horn shark H. francisci, possess a unique combination of morphological and behavioral characteristics that enable them to consume hard prey. Although H. francisci bites harder than the average vertebrate of comparable size, on a mass-specific basis it is not the most powerful biter in the animal kingdom (Fig. 8). Reptiles, mammals, other fishes and even some birds are capable of performing as well as or better than H. francisci when body mass is accounted for. These data suggest that factors other than bite force magnitude play a significant role in prey capture and processing ability. For H. francisci, these factors are molariform teeth, robust jaws, a high leverage jaw-adducting mechanism, and long duration, cyclically applied bite forces. The durophagous feeding behavior of H. francisci is reflected in its extensive ethmoidal articulation bracing the anterior portion of the upper jaw against the chondrocranium during prey capture and processing. Although in situ bite force measurements provided valuable information regarding its feeding behavior and ecology, theoretical estimates and restrained bite force measurements were the most effective means of estimating maximum bite force, depending on the availability of deceased specimens and live individuals. Because only a few investigations of biting performance in cartilaginous fishes have been made (Evans and Gilbert, 1971; Huber et al., 2004; Huber and Motta, 2004; Snodgrass and Gilbert, 1967; Weggelaar et al., 2004), little is known about the role that bite force plays in the ecological and evolutionary success of sharks. Combining theoretical and performance analyses provides the basis for an in-depth understanding of the link between morphology, behavior and ecology in sharks, and the role that biomechanics plays in the form and function of shark feeding mechanisms.

List of symbols and abbreviations
α
angle of incidence of the ethmoidal articulation force
ACS
cross-sectional area
C
ceratohyal
CC
coracoarcualis
CH
coracohyoideus
CHD
dorsal hyoid constrictor
CHV
ventral hyoid constrictor
CM
coracomandibularis
CO
coracoid bar
E
ethmoidal articulation
F
force
FB
bite reaction force
FE
ethmoidal articulation force
FH
hyomandibular articulation force
FJR
jaw joint reaction force,
FLJ
static equilibrium of forces acting on the lower jaw
FPO-α
preorbitalis-α force
FQM-PO
FQM-γ
FR
resultant jaw adducting force
FUJ
static equilibrium of forces acting on the lower jaw
H
hyomandibula
HM
hyomandibulo-mandibularis
I
impulse
IMD
intermandibularis
LH
levator hyomandibularis
LJ
lower jaw
LP
O
orbital ariculation
P
postorbital articulation
P0
theoretical maximum tetanic tension
PO-α
preorbitalis-α
QM-PO
QM-γ
Tsp
specific tension
UJ
upper jaw
VSBC
ventral superficial branchial constrictor

## ACKNOWLEDGEMENTS

We would like to thank J. Morris, J. Tyminski and D. Conklin for their assistance in animal care and maintenance, and the Florida Aquarium, Texas A&M University, Los Angeles County Museum of Natural History, Philadelphia Academy of Natural Sciences, C. Lowe and D. Pondella for providing experimental animals and morphological specimens. Special thanks go to C. Weggelaar, M. Dean, D. Lowry, R. Martinez, S. Valenti, Mandica Illustration and Hall Machine Inc. for their generous contributions of time, labor, wisdom and artistry. This manuscript was improved through the advice of S. Pierce. Funding for this project was provided by the University of South Florida Foundation Presidential Fellowship, University of South Florida-Mote Marine Laboratory Graduate Research Fellowship in Elasmobranch Biology and PADI A.W.A.R.E. Foundation grants awarded to D.R.H. and by a University of South Florida Research and Creative Grant Scholarship awarded to P.J.M. and D.R.H.

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