Virtual simulation methods
This protocol is extracted from research article:
High-performance suction feeding in an early elasmobranch
Sci Adv, Sep 11, 2019; DOI: 10.1126/sciadv.aax2742

To virtually simulate mouth opening and closing in T. arcuatus, we first arranged the computed tomography segmented skeletal elements (i.e., digital meshes) in an approximately closed mouth pose. On the basis of the articular anatomy of T. arcuatus and the cranial mechanics and kinematics of present-day sharks (63), we concluded that the skull of T. arcuatus likely had at least six primary degrees of freedom (DoFs) of motion (fig. S6). These are translation of the basihyal dorsoventrally and craniocaudally (fig. S6, A and B), hyoid arch abduction (fig. S6C), Meckel’s cartilage depression (fig. S6D), mandibular arch abduction (fig. S6E), and labial cartilage flaring (fig. S6F).

Given these multiple DoFs, many potential conformations of the skull are possible. Thus, we simulated a single sequence of opening and closing of the mouth by charting a single path or trajectory through this multidimensional “skull conformation space.” We created an initial 5D skull conformation space by simulating motion along five of six cranial DoFs (all but labial cartilage flaring): The basihyal was translated ventrally over a range of 35 mm and caudally over a range of 10 mm, the hyoid arch was abducted over a range of 25°, the Meckel’s cartilage was depressed over a range of 20°, and the mandibular arch was abducted over a range of 5° (we assumed bilateral symmetry throughout all virtual simulation). For each of these motions, we simulated nine positions evenly spaced along the total range of motion and combined them in a full pairwise manner, creating a total of 95 (59,049) conformations. We performed all simulations in R (64), using the R package “linkR” to perform the motion transformations (65, 66) and the R package “svgViewR” (67) to create all mesh visualizations and animations (movies S1 to S5).

Not all of these conformations are biologically reasonable or biomechanically feasible. We next filtered out these conformations based on two criteria. The first criterion was that when the Meckel’s cartilage is near fully depressed, the basihyal would not be near fully elevated and that when the basihyal was near fully depressed, the Meckel’s cartilage would not be near fully elevated (fig. S7A). This follows from the assumption that during prey capture mandibular and hyoid arch rotations are coordinated (although not necessarily synchronous). This first criterion reduced the number of conformations from 59,049 to 22,491.

The second filtering criterion assumed the presence of a ligamentous connection between the medial surface of the retroarticular process of the Meckel’s cartilage and the opposing lateral surface at the proximal (caudal) end of the ceratohyal. We removed conformations in which the distance between these two surfaces exceeded 4 mm by taking the distance between two planes, one on each articular surface. Because we simulated hyoid arch abduction (fig. S6C) independently of basihyal translation (fig. S6, A and B) and mandibular arch rotations (fig. S6, D and E), for some conformations, the hyoid arch and mandibular arch meshes penetrated (i.e., intersected) each other (fig. S7B). We also removed these conformations by detecting intersections of the two articular planes. This second criterion reduced the number of conformations from 22,491 to 1042.

We next defined a trajectory curve through the filtered conformation space (1042 total conformations) that represents a possible mouth opening-closing motion sequence (red lines in fig. S7, C to E) validated by reference to in vivo kinematics described in present-day sharks (7, 2224, 63). In particular, we defined the trajectory through conformation space such that peak hyoid depression follows peak Meckel’s cartilage depression (fig. S7C). In addition, we conjectured that mandibular arch abduction (i.e., palatoquadrate mediolateral sliding; fig. S7D) and hyoid arch abduction (fig. S7E) both reached their respective maxima at peak hyoid depression. To constrain the trajectory curve (e.g., make sure the path passes through particular points), we used Bézier splines. We then found the points in conformation space closest to evenly spaced points along the motion trajectory and smoothed the transformations between consecutive conformations to create the final mouth open-close simulation. Labial cartilage flaring was added last and with the assumption that flaring occurs simultaneous with Meckel’s cartilage depression. Thus, the labial cartilage transformation was essentially a copy of Meckel’s cartilage depression, rescaled to a maximum of 35°.

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