Estimating maximum bite performance in Tyrannosaurus rex using multi-body dynamics

K. T. Bates, P. L. Falkingham

Abstract

Bite mechanics and feeding behaviour in Tyrannosaurus rex are controversial. Some contend that a modest bite mechanically limited T. rex to scavenging, while others argue that high bite forces facilitated a predatory mode of life. We use dynamic musculoskeletal models to simulate maximal biting in T. rex. Models predict that adult T. rex generated sustained bite forces of 35 000–57 000 N at a single posterior tooth, by far the highest bite forces estimated for any terrestrial animal. Scaling analyses suggest that adult T. rex had a strong bite for its body size, and that bite performance increased allometrically during ontogeny. Positive allometry in bite performance during growth may have facilitated an ontogenetic change in feeding behaviour in T. rex, associated with an expansion of prey range in adults to include the largest contemporaneous animals.

1. Introduction

Attaining quantitative measures of mechanical performance for extinct animals is notoriously difficult and involves numerous assumptions, particularly when a close living analogue is lacking. This has frustrated attempts to definitely assess the feeding strategies of large carnivorous dinosaurs, particularly one of the largest and best-known taxa, Tyrannosaurus rex [14]. Some studies contend that a modest bite limited T. rex to scavenging [3], while others have argued that high ‘bone-crushing bite’ forces facilitated a ‘puncture–pull’ mode of predation [1,2,4]. Resolving this issue would not only enhance understanding of the palaeobiology of T. rex, but would also provide a clearer understanding of trophic ecology in dinosaurian ecosystems.

Addressing this issue directly requires a model to be constructed that makes explicit assumptions about musculoskeletal geometry, mass and inertia, together with muscle properties in T. rex (figure 1). These factors have a determinate effect on mechanical performance [58] and such a model would provide reasonable estimates of the quantitative data required to test hypotheses regarding bite performance, facilitating meaningful comparisons between T. rex and other taxa, both living and extinct. Here, we use the computational engineering technique multi-body dynamic analysis (MDA) to investigate bite performance in T. rex. Specifically, we test the hypotheses that T. rex had an extremely (i) high absolute and (ii) relative bite force, and (iii) that bite performance increased in relative terms during ontogeny.

Figure 1.

MDA model of adult T. rex. (a) Three-dimensional digitized skull with soft tissues reconstructed for mass and muscle properties (red, adductor mandibulae externus group; blue, adductor mandibulae posterior group; purple, pterygoideus group). (b) MDA model with joint centres (green circle), muscles (red cylinders), and ‘contact’ springs (blue spheres and cylinder) on the teeth in the initial simulation starting pose. (c) MDA model during sustained biting.

2. Material and methods

A Z&F 5600i laser scanner was used to digitize skulls of an adult Alligator mississippiensis (skull length 0.53 m), Allosaurus (SMA0005), and near-complete juvenile (BMRP2002.4.1) and adult (BHI3033) T. rex. A juvenile Alligator (skull length 0.188 m) and adult human model were derived from computer tomography data. Volumetric reconstructions of the upper and lower jaw segments were produced in Maya to provide inertial properties ([9], figure 1a). Upper and lower jaw segments were joined by a single hinge joint, orientated perpendicular to the cranial–caudal axis of the skull (figure 1). We reconstructed the main jaw-closing musculature in Alligator and the dinosaurs using simplified functional groups ([2]; figure 1). The temporalis, masseter and pterygoideus muscles were reconstructed in the human model.

To investigate bite performance, we used the MDA package GaitSym. Models required physiological cross-sectional area (PCSA), fibre length (FL), maximum contraction velocity (Vmax) and force per unit area (FPUA) to be specified for each muscle. FPUA was initially set to 300 000 N m2 [10,11] and Vmax to 8 s−1 to represent mixed slow–fast twitch fibre types [10,11]. For our human and juvenile Alligator models, all subsequent input data were available in the literature [6,12]. Quantitative muscle data were unavailable for adult Alligator and parameters were estimated in the same manner as for the dinosaurs. FLs of jaw-closing muscles in our human model ranged between 10 and 40 per cent of maximum muscle length across the range of joint motion investigated. We therefore set FLs in our adult Alligator and dinosaur models to 25 per cent maximum length over the same range of joint motion, testing 10–40% maximum lengths in our sensitivity analysis (table 1). Muscle PCSAs were calculated by dividing a volume for each muscle by its FL. Initial muscle volumes were derived by reconstructing three-dimensional volumes within the skull models in Maya (figure 1). Repeating this approach with our human and juvenile Alligator models resulted in volumes with 5–12% error in most muscles, but a 22 per cent overestimation in the human medial pterygoid muscle relative to measured values [6,12]. We therefore tested muscle volumes ±20% initial estimates for the dinosaurs and adult Alligator (table 1). Pennation effects were tested in two ways: first we produced models in which PCSAs were increased by 6.04 per cent, equivalent to that expected from 20° pennation angle (3° greater than the maximum angle reported in Alligator [12]). Second, we produced models with both increased PCSAs and shorter FLs to mimic the expected contractile effects of pennation angle [10,11]. We varied Vmax between 4 and 12 s−1, consistent with the range found in the literature ([10,11]; table 1). For biting simulations, the jaws in each model were opened to 45° and all jaw-closing muscles activated maximally for 1 s. Forces were measured in stiff springs (stiffness 2 000 000 N m2) positioned on the teeth.

View this table:
Table 1.

Outline of the sensitivity analysis carried out on the adult Alligator and dinosaur MDA models.

To assess our scaling hypotheses, we isometrically scaled (i.e. length ∝ mass1/3, areas ∝ mass1/3) all our MDA models to the mass of the adult T. rex model and repeated our analysis. This allows direct comparison of size-normalized bite performance, and consideration of soft tissue uncertainties when testing scaling hypotheses.

3. Results

Initial simulations produced the bite forces shown in figure 2 and summarized in table 2. Muscle FL and Vmax had little effect on sustained bite force (±5% initial models) across a large range of input values tested. However, both parameters did have a considerable impact on bite velocities (figure 2b). PCSA had a significant effect on absolute bite force; ±20% PCSA yielding forces 18–25% higher and lower than initial models (figure 2b). Increasing PCSA to mimic the effects of a 20° pennation angle increased bite force by approximately 6 per cent in any given model (figure 2b). Increasing PCSA and decreasing FL produced increasingly forceful but lower velocity bites, respectively (figure 2b). Overall this yielded a range of maximum bite forces at posterior tooth positions of 2565–4012 N and 35 640–57 158 N for the juvenile and adult T. rex models, respectively (figure 2).

View this table:
Table 2.

Summary of initial, minimum and maximum bite force results.

Figure 2.

(a) Bite force versus time for mid-jaw positions in the MDA models, showing an initial force peak representing the bite impact, before levelling off to record the sustained bite force (time normalized across models so that initiation of biting coincides). (b) Range in bite performance predictions for adult T. rex about the initial model for each parameter in the sensitivity analysis. Blue diamonds, initial model force; red squares, initial model velocity. (c) Mechanical advantage of jaw-closing muscles in the models. (d) Results of isometrically scaling all models to the mass of adult T. rex. White bars, force; grey bars, velocity. (e) Summed muscle force in the isometrically scaled models. (f) Comparison of biting forces predicted in this study with some of the largest values in the literature for terrestrial vertebrates ([48,12,13], grey bars are values from previous studies; white bars are from this analysis). (a,c) Lines: black, human; green, juvenile Alligator; red, adult Alligator; yellow, Allosaurus; blue, adult T. rex; purple, juvenile T. rex.

Bite force remained lower than adult T. rex in all taxa when MDA models were isometrically scaled to the same mass (figure 2d,e). However, uncertainties in muscle PCSA means error bars show considerable overlap with Allosaurus and slight overlap with adult Alligator (figure 2d,e). Large error bars on bite velocity predictions mean all taxa show considerable overlap (figure 2d).

4. Discussion

(a) Validation

Predicted bite force in our human and Alligator models closely match previous values (figure 2a,f). Our human bite forces (700–1020 N) overlap experimental values of 730–749 N [5,6] and lie within estimates in earlier modelling studies of 678–1080 N [7,8], with variability across studies attributed to significant variation in muscle size between individuals and the level of muscle activation associated with different biting activities [58]. Erickson et al. [13] reported peak forces of 500–765 N and 9000–11 125 N in alligators of equivalent size to our models. Peak forces in the initial biting impact overlap these values (figure 2a,f; juvenile 601–801 N, adult 9800–13 135 N). Sustained bite force in the juvenile model (202–266 N) overlaps the higher end values of Porro et al. ([12]; 120–215 N), and shows even closer agreement (148–209 N) if we use the lower value for FPUA (250 000 N m−2) preferred by these authors.

(b) Bite performance in Tyrannosaurus rex

We are aware of two previous quantitative estimates of bite force in T. rex. Indentation experiments, which estimated the force required to replicate a fossilized bite mark, produced values of 6400–13 400 N for a single posterior tooth [4]. Meers [1] used allometric scaling arguments to suggest that T. rex could have generated on average 7600–9800 N at a single tooth. Even with more conservative estimates for muscle mass our model suggests that these earlier analyses significantly underestimated bite performance in adult T. rex. Even the lowest forces predicted at the most anterior teeth in our model are nearly double that predicted by indentation simulations [4], while forces at posterior teeth are three times higher. Adult T. rex magnitudes are 8–10 times greater than those predicted for Alligator, Allosaurus and juvenile T. rex, and consequently are significantly higher than values measured or predicted for any terrestrial animal (figure 2f).

(c) Scaling bite performance

Uncertainty in muscle properties means we are unable to reject the hypothesis that T. rex had a relatively high bite performance for its body size. Considerable overlap in error bars with Allosaurus, owing largely to the latter's high mechanical advantage (figure 2c), perhaps suggests that large-bodied theropods in general were capable of generating proportionally high bite forces. Our initial adult T. rex reconstruction has a higher proportion of muscle mass (6.89% skull mass, versus 1.2–5.64% in other models), consistent with anatomical studies that have postulated relatively large jaw-closing muscles, based on lateral expansion of the temporal box and parietal crests, deepening of the posterior mandible and overall massive skull construction [1,2]. Our prediction of relatively higher muscle mass in our adult T. rex model appears insensitive to the scaling metric used; for example, PSCA remains considerably higher in adult T. rex if linear skull dimensions, rather than mass, are used to derive scaling exponents (electronic supplementary material, table S14). Slight overlap in bite force error bars with the adult Alligator model (figure 2d) offers weak support for a proportionally higher bite performance relative to living carnivorous archosaurs. Muscle data for adult Alligator would refine this prediction.

Our models provide strong support for the hypothesis that bite performance experienced a large absolute and relative increase during tyrannosaur ontogeny. Juvenile T. rex possessed a lower, longer snouted skull than adult T. rex. Deepening of the rostrum and expansion of the posterior skull with maturity potentially provided greater volume for jaw-closing musculature and increased moment arms (figure 2). Relative shortening of anterior elements brought the tooth row closer to the jaw joint, further increasing mechanical advantage (figure 2).

The age of BMR P2002.4.1 has been estimated at 11 years indicating that this animal died at the beginning of the exponential growth phase [9]. Tyrannosaur body proportions changed appreciably during ontogeny, and biomechanical analyses have provided strong evidence that young individuals were more athletic than adults [9]. With bite forces comparable to Alligator and lions (figure 2g), combined with relatively long forelimbs and cursorial hindlimbs, juvenile T. rex appears well equipped to pursue and dispatch small to medium-sized prey. Forelimbs became significantly reduced (approx. 3% body mass in BMR P2002.4.1, approx. 0.1–0.4% in adults [9]), and our results strongly suggest increasing importance of the skull in food procurement as compensation.

In living carnivores, such as Alligator, strong positive allometry in bite performance is typically associated with an ontogenetic shift in feeding ecology [13]. Enhancement of an already highly competent biting apparatus during the exponential growth phase may be indicative of a change in feeding behaviour as T. rex reached adulthood. Living carnivores preying on large animals have relatively high bite forces, while carnivores preying on small prey have more moderate bite forces for their size, suggesting that bite force represents an important adaptation to differing feeding ecologies, at least throughout carnivoran evolution [14]. Attaining adult body sizes, combined with specialized craniofacial anatomy underpinning high mechanical performance, may have allowed adult T. rex to function as a ‘large prey specialist’, alleviating direct competition from smaller, more agile carnivores, including juvenile T. rex.

  • Received January 19, 2012.
  • Accepted February 7, 2012.

References

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