Estimating cranial musculoskeletal constraints in theropod dinosaurs

Many inferences on the biology, behaviour and ecology of extinct vertebrates are based on the reconstruction of the musculature and rely considerably on its accuracy. Although the advent of digital reconstruction techniques has facilitated the creation and testing of musculoskeletal hypotheses in recent years, muscle strain capabilities have rarely been considered. Here, a digital modelling approach using the freely available visualization and animation software Blender is applied to estimate cranial muscle length changes and optimal and maximal possible gape in different theropod dinosaurs. Models of living archosaur taxa (Alligator mississippiensis, Buteo buteo) were used in an extant phylogenetically bracketed framework to validate the method. Results of this study demonstrate that Tyrannosaurus rex, Allosaurus fragilis and Erlikosaurus andrewsi show distinct differences in the recruitment of the jaw adductor musculature and resulting gape, confirming previous dietary and ecological assumptions. While the carnivorous taxa T. rex and Allo. fragilis were capable of a wide gape and sustained muscle force, the herbivorous therizinosaurian E. andrewsi was constrained to small gape angles.


Introduction
The anatomy, size and arrangement of the jaw muscles are important functional factors underpinning an animal's capability for vocalization [1], social signalling [2][3][4] andmost importantly-food acquisition (i.e. capture, manipulation, processing) [5][6][7]. These factors determine jaw closing and opening speeds, jaw closing force (=bite force), jaw gape, and horizontal and vertical jaw movements, thereby constraining food selection and prey or food size [8,9], feeding behaviour [4,[10][11][12] and the occupation of ecological niches [13,14]. In extinct animals, these parameters are often difficult to determine. Soft-tissue structures are rarely preserved and information on myological structures has to be inferred from preserved hard tissues  either form a phylogenetic bracket or a functional analogue [15,16]. As such soft-tissue reconstructions lie at the base of functional inferences, accurate reconstructions are paramount for wider ecological or behavioural deductions in extinct vertebrates.
Recent advances in digital imaging techniques have enabled detailed studies of the cranial musculature in living vertebrates [17][18][19], providing a steadily increasing catalogue of extant anatomical data. Similarly, novel approaches for three-dimensional, digital muscle reconstructions have provided detailed models and musculoskeletal hypotheses for extinct taxa [20,21]. In contrast with traditional descriptions and two-dimensional reconstructions, the digital nature of this data permits not only the extraction and quantification of muscle properties (length, physiological cross-section area, volume), but also the functional analysis of cranial systems [22][23][24]. Among the growing number of these studies, though, the maximal possible strain (or excursion) of cranial muscles is rarely considered and the few exceptions [25] do not take three-dimensional morphology into account. Similarly, studies on (fossil) taxa implementing muscle information have focused on locomotory function and the measurement of moment arms [26][27][28] rather than muscle excursion. However, muscles can only stretch a certain amount before they tear, thus limiting the maximum gape angle and jaw opening. Furthermore, muscular performance is closely related to the extension of muscle fibres [29][30][31]. Detailed information on these factors can, therefore, provide a better understanding on the feeding behaviour of extinct organisms and complement existing analyses.
Cranial function of theropod dinosaurs has been extensively studied in the past based on direct inferences through tooth marks and feeding traces [32,33], theoretical considerations derived from functional measurements [34,35], and most recently by using computational biomechanical analyses [36][37][38]. Yet, debate remains regarding the feeding behaviour of different theropods [32,39]. Furthermore, different feeding styles have been suggested ranging from puncture-and-pull feeders (e.g. Tyrannosaurus rex) to strike-and-tear feeders (e.g. Allosaurus fragilis) to secondarily herbivorous, specialized forms (e.g. Erlikosaurus andrewsi) [37,40,41]. Here, the musculoskeletal constraints imposed on the crania of selected theropod dinosaurs are evaluated supported by an extant phylogenetically bracketed approach. By applying a novel approach to estimate maximum muscle strain using the three-dimensional modelling and animation software BLENDER, digital models of T. rex, Allo. fragilis and E. andrewsi are studied to test the hypothesis that musculoskeletal constraints reflect different feeding behaviours and dietary adaptations.

Digital models: extant taxa
To obtain information about muscle behaviour in extant archosaurs and to provide an extant phylogenetically bracketed framework for the fossil taxa, digital models of a common buzzard (Buteo buteo) [18] and an alligator (Alligator mississippiensis) [42] were used. These taxa were selected as accurate data on muscle architecture and morphology had been gained in previous studies employing contrast-enhanced computed tomography (CT) scanning [17,18].   and was digitized using a photogrammetry approach [43]   combined into a movie file (see the electronic supplementary material). The python script was written to run for the duration of a complete cycle, but alternatively a stop command can be activated to interrupt the cycle when a specific muscle strain factor is reached. Given the length-tension relationship of muscles, the resting length of cranial muscles has to lie at a small gape angle over 0 • in order to generate the necessary force during biting [ 28.0°F igure 4. Gape angles at optimal and maximum tension limit for Alligator mississippiensis with muscle resting lengths at a gape angle of (a) 3.0 • , (b) 6.0 • and (c) 9.0 • . Bar diagrams show strain factors of individual muscles at optimal and maximum tension limit. Muscle abbreviations as in table 1.

Digital models: theropod dinosaurs
with resting length set at an angle of 3.0 • , 6.0 • and 9.0 • . Results then informed the settings for the theropod taxa, for which the analyses were run with resting lengths at 3.0 • and 6.0 • . A resting length set at 9.0 • turned out to produce strain factors substantially lower than in the extant taxa and were therefore not considered (see Results and Discussion sections for details).

Extant taxa
Strain analyses were performed for Alli. mississippiensis and B. buteo with the muscle resting length set at a gape angle of 3.0 • , 6.0 • and 9.0 • . The plotted ratios between stretched and relaxed muscle lengths (=strain factor) against gape angle show a consistent pattern for each taxon regardless of resting length ( figure 3). However, owing to the changed resting length, the strain values at 60.0 • (maximum gape angle for each analysis) are highest with the resting length at a low gape (3.0 • ) and, respectively, lowest at higher gape (9.0 • ). For Alli. mississippiensis strain factors range from 103 to 193% (resting length at 3.0 • ) to 102 to 175% (resting length at 9.0 • ; figure 3a,c,e). Values are somewhat higher for B. buteo with strain between 101 and 204% (resting length at 3 • ) and, respectively, between 100 and 188% (resting length at 9 • ; figure 3b,d,f ).
Owing to their internal structure consisting of overlapping filament cross-bridges, muscles have a strain range in which maximal tetanic contraction can be achieved (optimal tension limit, 100-130% of resting length) and a maximum tension limit (170% of resting length) above which contraction is no longer possible [30,31]. When these limits are applied to the strain results for the extant taxa in this study, the optimal tension limit would be reached at gape angles of 20.

Theropod dinosaurs
Strain ratios for the theropod taxa were recorded with resting lengths set at a gape angle of 3.0 • and 6.0 • . As found for the extant taxa, the strain patterns are unchanged for the different resting lengths ( figure 6), but strain value changes according to resting length. At a gape angle of 80.0 • strain factors are recorded between 117 and 170% (resting length at 3.0 • ) and between 116 and 164% (resting length at 6.0 • ) for Allo. fragilis ( figure 6a,b). For T. rex similar values are found ranging from 122 to 172% (resting length at 3.0 • ) and from 121 to 170% (resting length at 6.0 • ; figure 6c,d). Strain factors for E. andrewsi are considerably higher at 122 to 216% (resting length at 3.0 • ) and 121 to 205% (resting length at 6.0 • ; figure 6e,f ). Accordingly, the optimal tension limit is reached at a jaw gape of 28

Discussion
In  performance. Since then, their cross-bridge model of sliding actin and myosin filaments during muscle contraction has been widely accepted and associated length changes of muscle fibres have been studied in great detail [52,53]. It is generally assumed that highest tetanic tension can be achieved in a range up to 30% length changes and a maximal tetanic tension limit occurs at 70% length changes [30,31]. However, these relationships have rarely been tested in vivo for complete muscle lengths. Based on theoretical calculations a maximum strain of 140% was suggested before irreversible damage occurs in mammalian jaw adductor muscles [25]. Experimentally derived measurements, however, indicate maximal possible strain factors up to or slightly above 170% are possible, but with the majority of values well below this limit in mammalian taxa [48,54].
Published measurements on maximum gape angles in extant vertebrates, and particular in birds and crocodilians, are rare. Experimental results show that mandibular angles lie between 25 • and 35 • but can reach angles of around 40 • in birds [55][56][57][58]. When a maximum muscle strain factor of 170% is applied, the data presented here for B. buteo would result in a similar possible gape angle of 43.0 • for a muscle resting length set at 3.0 • . For resting lengths at larger gape angles, the results from this study are moderately higher with up to 50.5 • . However, despite the fact that actual gape can be increased in birds because of the kinetic coupling of the upper and lower beaks, the postorbital ligament considerably restricts theoretically possible gape angles [59].
In comparison, data for crocodilian jaw kinematics reveal a wide range of gape angles in Caiman crocodilus, ranging from 25 • to 35 • when biting and crushing prey, to a maximum of 45 • to 50 • when transporting prey [60]. Very similar maximum gape angles (43.5 • and 49.5 • ) were found here for   Figure 8. Gape angles at optimal and maximum tension limit for (a) Allosaurus fragilis, (b) Tyrannosaurus rex and (c) Erlikosaurus andrewsi with muscle resting length at a gape angle of 6.0 • . Bar diagrams show strain factors of individual muscles at optimal and maximum tension limit. Muscle abbreviations as in table 1.
Alli. mississippiensis for muscle resting lengths at gapes of 3.0 • and 6.0 • , respectively. Further experimental data for Alli. mississippiensis obtained through biomechanical modelling and experimentally derived length-tension curves [49], recorded strain factors between 110 and 151% at a gape angle of 30 • . The strain analyses presented herein recorded comparable strain values between 104 and 147% (resting length at 3.0 • ) and between 103 and 140% (resting length at 6.0 • ). This suggests that the resting lengths at gape angles between 3.0 • and 6.0 • approach realistic values.
In living vertebrates muscles are often not simple and straight point to point connections as modelled here, but would curve around other muscles and bony structures or could attach to connective tissue. This could potentially increase or decrease the muscle length affecting muscle strain factors. Similarly, the microstructure of muscle bodies, such as fibre length, can partially influence the strain capability of muscles. However, despite these simplifications, there is a good correspondence between gape angles obtained for the extant taxa in this study and the (admittedly limited) number of published data. Furthermore, the results show a consistent strain-gape pattern regardless of resting length within the individual taxa. Therefore, the comparative approach of this method further ascertains meaningful results.
Different feeding styles and dietary adaptations have been suggested for the three theropod taxa in this study. While it has been assumed that T. rex relied on its powerful bite and robust, conical teeth in a puncture-and-pull fashion to crush bone and soft-tissues, the comparably weak muscle-driven bite in Allo. fragilis was used in combination with the neck musculature in a strike-and-tear mode to attack prey   [40,41]. In order to be able to hunt prey in such a manner, Allo. fragilis possessed a jaw joint configuration which allowed wide gapes without the risk of dislocation [61], but would also require a muscle arrangement to permit large gape angles. Results of this study lend support to this assumption. Among the studied taxa, Allo. fragilis recorded the highest gape angle (79.0 • -92.0 • ) when reaching the maximum tension limit, and therefore considerably larger than T. rex (63.5 • -80.0 • ). However, both theropods reached the optimal tension limit at gape angles of 28.0 • and 32.5 • , which would allow high muscle efficiency within this range. In T. rex, however, the muscle strain curves for the majority of the different muscles lie on a narrow trajectory (figure 6c,d). This suggests that these muscles had a homogenous muscle performance and provided a sustained bite force, as necessary to crush bone and dismember prey.
The therizinosaurian E. andrewsi stands in stark contrast with other theropods because of its unusual cranial anatomy, which includes an edentulous premaxilla and small, densely packed, leaf-shaped teeth [44]. These anatomical modifications are thought to represent adaptations to an herbivorous diet, and biomechanical models suggest that E. andrewsi recruited the postcranial musculature to compensate for low bite forces to crop foliage and strip leaves of branches [37]. The data presented here shows that E. andrewsi could achieve only a comparably low gape of 43.5 • -49.0 • before reaching the maximum tension limit, which is consistent with the inferred feeding style. In extant mammals, herbivorous taxa generally have a distinctly smaller maximum gape than carnivores [25]. In the carnivorous grasshopper mouse (Onychomuys leucogaster) and the granivorous deer mouse (Peromyscus maniculatus), two sympatric and closely related murid species, dietary specialization is reflected by gape angle [62], and the same pattern of niche separation appears to apply to the theropods in this study. However, while carnivorous and herbivorous taxa show distinct differences in gape angle and muscle strain trajectories, there is no consistent pattern regarding the strain of individual muscle groups (table 3). While parts of the m. pterygoideus ventralis (m. PTv) are generally among the muscles that show the least strain, different muscle groups were found to experience highest strain across the studied taxa.

Conclusion
The analyses of musculoskeletal structures and muscle strain trajectories presented herein not only demonstrate distinct differences in the feeding styles of theropod dinosaurs, but also confirm previous