JOURNAL OF MORPHOLOGY 00:00–00 (2014)

Does Osteoderm Growth Follow Energy Minimization Principles?  n Sensale,1,2* Washington Jones,1 and R. Ernesto Blanco3 Sebastia 1

N ucleo de Biomec anica, Espacio Interdiscipinario, Universidad de la Rep ublica, Montevideo 11200, Uruguay Instituto de Fısica, Facultad de Ingenierıa, Universidad de la Rep ublica, Montevideo 11300, Uruguay 3 Instituto de Fısica, Facultad de Ciencias, Universidad de la Rep ublica, Montevideo 11400, Uruguay 2

ABSTRACT Although the growth and development of tissues and organs of extinct species cannot be directly observed, their fossils can record and preserve evidence of these mechanisms. It is generally accepted that bone architecture is the result of genetically based biomechanical constraints, but what about osteoderms? In this article, the influence of physical constraints on cranial osteoderms growth is assessed. Comparisons among lepidosaurs, synapsids, and archosaurs are performed; according to these analyses, lepidosaur osteoderms growth is predicted to be less energy demanding than that of synapsids and archosaurs. Obtained results also show that, from an energetic viewpoint, ankylosaurid osteoderms growth resembles more that of mammals than the one of reptilians, adding evidence to debate whether dinosaurs were hot or cold blooded. C 2014 Wiley PeriodiJ. Morphol. 000:000–000, 2014. V cals, Inc. KEY WORDS: osteoderms; ankylosaurs; biomechanics; periodic patterns

INTRODUCTION Periodic Patterning Modeling The growth of feathers, scales, hairs, and teeth, among other epidermal tissues, has long fascinated biologists. The presence of patterns in biological tissues demonstrates how extremely organized developmental processes are to understand how these patterns arise, the integration of multiple disciplines (like physics and developmental biology) seems mandatory. Periodic patterning has emerged early in evolution to allow functional redundancy and phenotypic variation (Chuong and Richardson, 2009). In birds and mammals, reaction-diffusion mechanisms (RDM) generate the periodicity of structures such as hairs and feathers (Chuong et al., 2013), but their relation with other tissues that present periodic, repetitive patterns—such as the head scales of crocodiles—is not that direct (Milinkovitch et al., 2013). Three major mechanisms could generate polygonal patterns in biological tissues: the cracking of a material layer into adjacent polygons (as in crocodile head scales), RDM patternings of genetically determined developmental units (as in feathers and hairs), and energy minimizaC 2014 WILEY PERIODICALS, INC. V

tion of contact surfaces among genetically determined elements (as in epithelia; Gibson et al., 2006; Farhadifar et al., 2007; Lecuit and Lenne, 2007). It is important to note that there is always interplay between physically and genetically controlled parameters (Padian et al., 2001; Castanet, 2006). For example, in the cracking of a material layer, composition and thickness of the skin affect the mechanism (Milinkovitch et al., 2013); in RDM models treat developmental steps as if they were isolated from the rest of the organism, leaving aside factors such as spatial and temporal location (Koch and Meinhardt, 1994); and, under closer inspection, in energy minimization models the systems are not perfectly regular and are often somewhat disordered, which reflects the limit of a tight genetic control (Lecuit and Lenne, 2007). Also, these are all theoretical models and, as such, they tend to omit many features, such as the possible influence of electric or thermic properties, in order to focus on the influence of certain parameters. Polygonal Pattern Formation Mechanisms Due to their surfactants, biological membranes behave like elastic membranes (Lautrup, 2004) and cell-surface mechanics are comparable to the physics of foam formation (Taylor, 1976). Although this idea has been around for some time (Thompson, 1917), only recently 2D foam models have been applied to explain the development of Additional Supporting Information may be found in the online version of this article. Contract grant sponsor: Programa de Desarrollo de Ciencias B asicas (PEDECIBA) and Agencia Nacional de Investigaci on e Innovaci on (ANII).. *Correspondence to: S. Sensale; Instituto de Fısica, Facultad de Ingenierıa, Montevideo 11300, Uruguay. E-mail: [email protected] Received 27 August 2013; Revised 16 December 2013; Published online 00 Month 2014 in Wiley Online Library (wileyonlinelibrary.com). DOI 10.1002/jmor.20273

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against bulk pressure (P) when the volume (V) of the system is expanded by dV dW52PdV This resistance against expansion shows that the surface has an internal surface tension, which equals the surface energy density when under mechanical equilibrium.Considering a lapse of time where surface tension can be considered as constant, if two similar surfaces are in contact, it may be useful for many purposes to treat them as one by just doubling the surface tension constant. Therefore, problems of energy minimization can be reduced to problems of area minimization (Taylor, 1976). Fig. 1. Cross section of an elementary 3D model of a materialvacuum interface. Note how a molecule in the interface has one missing bond in comparison to a molecule in the interior.

biological tissues such as the retina (Hayashi and Carthew, 2004) and epithelial junctional networks (Farhadifar et al., 2007). Under this framework, surface tension properties underlie the control of tissue organization. The mathematics of elastic solids is a well-developed subject, and as such it can provide useful ideas about real-life processes like the morphogenesis of biological tissues. In this section, surface tension will be defined as well as its role in energy minimization. Consider a simple regular “solid” surrounded by vacuum whose molecules are placed in a cubic grid as in Figure 1, with grid length equal to the molecular separation. Each molecule in the interior has six bonds to its neighbors, while those in the surface have only five; this leads to a free surface energy which, divided by the molecular area scale, defines what is called “surface energy density.” Because of the missing negative binding energy of the surface of molecules, this quantity is always positive when considering the density associated with a liquid or solid interface against vacuum or gases, guaranteeing that such interfaces seek toward the minimal area consistent with the other forces that may be at play. This is not necessarily true on interfaces between solids and/or liquids, where its sign depends on the relation between the strength of the cohesive forces holding molecules together and the strength of the adhesive forces between the opposing molecules of the interfacing molecules. Increasing the area of an interface with energy density a, a tiny amount dA requires an amount of work equal to the surface energy contained in the extra piece of interface

Clusters and energy minimizing (foam) models. If a 2D-structure consisting of many surfaces back to back everywhere (Fig. 2A) is considered, minimizing the energy in the surfaces’ junctions will involve minimizing their lengths. In order to achieve this minimization, both the junctions’ perimeter and the area on each vertex where three or more edges meet must be minimal. Therefore, two problems coexist: one involving a minimal perimeter with free topology, and one involving the minimization of the vertices’ area. The areas surrounding each vertex have been proven to be minimal when three minimal surfaces meet smoothly at angles of 120 along a curve (Taylor, 1976; Graner, 2002), condition that will be referred to as “triple junctures.” This condition can also be seen as the result of the three instantaneous equal length-tension force vectors acting at a node. Basic mathematical properties prove that, under this hypothesis, the average surface will have six (when considering the structure as free) or fewer sides (when considering the structure as bordered, because of the defects; Graner et al., 2001; Milinkovitch et al., 2013). It is also worth noting that this result does not state that the mean surface will be a regular hexagon, but that it will be a polygon with six sides. This is also consistent with Von Neumann–Mullins equation (Von Neumann, 1952; Mullins, 1956)

dW 5adA: Note that, except for the sign, this is quite analogous to the mechanical work (dW) performed Journal of Morphology

Fig. 2. Scheme of surfaces back to back everywhere (A) lateral view; (B) dorsal view.

CRANIAL OSTEODERM GROWTH

dS5

3

Ap ðn26Þ 6

which states that, for foams and grain growth, surfaces with more than six sides (n) tend to grow, therefore being forced by their neighbors to shrink, while surfaces with less than six sides tend to disappear, therefore being less stressed by their neighbors. Although many models of minimal perimeter can be applied depending on the geometry (Bleicher, 1987; Morgan, 1994; Hales, 2000; Graner et al., 2001; Cox and Graner, 2003; Cox et al., 2003; Cox, 2006; Cox and Shuttleworth, 2006; Cox and Flikkema, 2010; Cox et al., 2013a,b), most of them work under theoretical hypotheses that do not seem applicable to real-life systems. Nevertheless, it has been demonstrated that the minimal perimeter configuration of a system of N deformable areas with triple junctures is satisfied when the average area ratio of perimeter to square root of area is along the order of 23=2  31=4  3:7224 (Graner et al., 2001; Cox et al., 2013b). Note that, although this number is not sensitive to the number of sides of an area, this is the same ratio that would be obtained if all the areas were regular hexagons. It is also worth noting that, as the number of areas increases, this average ratio converges to 3.7224; nevertheless, this value is appropriate enough even for a few areas (Cox et al., 2003). Summarizing, a system of deformable surfaces back to back everywhere that follows minimal energy principles presents triple junctions with angles of 120 and its average surface not only has around six sides, but its energy is that of a regular hexagon. Cracking models. Unlike energy minimizing (foam) models, cracking models can generate various different angle distributions (Fig. 3).When fractures propagate simultaneously, junctions tend to form at 120 , but not all fractures propagate this way (Jagla and Rojo, 2002). Due to the fact that cracks propagate perpendicularly to the direction of the maximum strength component (Cotterell and Rice, 1980), secondary cracks can turn when they approach an older one and tend to join in at 90 . Also, if a crack starts on the side of an older crack, it will initially tend to propagate to a new angle (Shorlin et al., 2000). Therefore, it is expected for cracking models not to follow the same energy minimization principles that arise for developmental units (Milinkovitch et al., 2013). RDM. Turing (1952) proposed that a homogeneous surface could develop periodic patterns through molecular RDM. These mechanisms have since explained the growth of mammalian hair, avian feathers and feet scales, and the spatial organization of scales on the largest portion of reptilian bodies (Chang et al., 2009), but not every periodic process is the result of a RDM. A revision

Fig. 3. Crack patterns on a dried mud. Note how chaotic and irregular these patterns are.

of the literature shows that RDM focus on the spatial organization of appendages but do not focus on growing tissues. Therefore, although RDM could control processes where tissues grow, many other physical parameters could contribute to their development as well. In this article, the contribution of energy minimization principles on the growth of periodic patterns is studied, either via physical parameters or via higher principles that could define RDM. To show this influence, a particular case will be analyzed: the growth of osteoderms in the cranium of some tetrapods. Osteoderms and the Integumentary Skeleton of Tetrapods Osteoderms are dermal ossifications that support epidermal scales (Fig. 4). According to Moss, the histological structure of this dermal skeletal tissue is not homogeneous, ranging from dense calcified tendon to true bone, although most osteoderms are basically made of bone tissue comprising osteocyte Iacunae and a collagenous meshwork (Moss, 1969; Vickaryous and Sire, 2009). Osteoderms’ gross morphology may also be highly variable in size, shape, articulation, and Journal of Morphology

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Fig. 4. Dorsal and lateral view of a glyptodont’s osteoderm (Glyptodon sp. MNHN w/n).

geometry both between and within taxa (Grant, 1944; Hoffstetter, 1962). However, examination of cranial osteoderms of some tetrapods identifies a common pattern of development even among distantly related taxa. Osteoderms are usually documented as adult stage taxonomic characters. It has been proposed that they may have arisen independently at least five times in amniotes (Hill, 2005), exemplifying what has been termed “deep homology”: a latent but plesiomorphic ability to form structure and organs (Main et al., 2005; Hill, 2006; Vickaryous and Hall, 2008). The main processes behind osteoderms skeletogenesis are intramembranous ossification and bone metaplasia (Vickaryous and Sire, 2009). Metaplastic ossification is the direct mineralization of the existing connective tissue without the involvement of a primordium or osteoid (Haines and Mohuiddin, 1968). This development is consistent with the osteoderms development of some lepidosaurs such as Gila monsters and beaded lizards (Moss, 1969). Intramembranous ossification is

Fig. 5. Schematic drawing of the cranial osteoderm cluster of an Heloderma horridum in dorsal view.

Journal of Morphology

Fig. 6. Schematic drawing of the cranial osteoderm cluster of an Euphractus sexcinctus in dorsal view.

the direct bone formation within a condensation of osteoblasts depositing bone matrix without a cartilaginous precursor (Vickaryous and Sire, 2009). The initial mode of osteoderm formation in synapsids, such as armadillos, is consistent with intramembranous ossification; nevertheless, metaplastic ossification plays an important role in synapsid osteoderm development as well (Wolf et al., 2011). Cranial osteoderms on tetrapods: State of the art. Osteoderms are taxonomically widespread; nevertheless, their specific phylogenetic distribution is highly irregular. Simple observation can lead to the undeniable idea that one of their

Fig. 7. Schematic drawing of the cranial osteoderm cluster of an Ankylosaurus magniventris in dorsal view.

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TABLE 1. Institutional abbreviations AMNH

American Museum of Natural History, New York, USA

CMN ETVP

Canadian Museum of Nature Ottawa, Ontario, CA East Tennessee State University, Vertebrate Paleontology Laboratory, Department of Geosciences, Johnson City, Tennessee, USA La Ferme aux Crocodiles, Pierrelatte, FR Museo Argentino de Ciencias Naturales “Bernardino Rivadavia.” Buenos Aires, AR Museum of Comparative Zoology, Herpetology Department, Harvard University, Cambridge, Massachusetts, USA Museo Municipal de Colonia Dr. Buatista Rebuffo, Colonia del Sacramento, Colonia, UY Museo Nacional de Historia Natural, Montevideo, Montevideo, UY Museo Nacional de Historia Natural, Asunci on, PY Texas Memorial Museum, Texas Natural Science Center, University of Texas at Austin, Austin, Texas, USA Texas Natural Science Center, University of Texas, Austin, Texas, USA Royal Tyrrell Museum of Palaeontology Drumheller, Alberta, CA University of Alberta Laboratory for Vertebrate Paleontology Edmonton, Alberta, CA University of California Museum of Paleontology, Berkeley, California, USA

FC MACN MCZ MMC MNHN MNHNP TMM TMM TMP UALVP UCMP

main functions is that of providing protection against mechanical, environmental, and physiological stress (Moss, 1969; Daly and Buffenstein, 1998; Chuong et al., 2002; Menon, 2004). While osteoderms can be distributed all along the entire body, cephalic osteoderms are quite interesting due to the fact that, in most groups, they form a solid, bony pavement that obscures sutural contacts and cranial openings, obstructing possible movements (for exceptions see Bever et al., 2005). Three main taxa present cranial osteoderms of this kind: synapsids, archosaurs, and lepidosaurs (Vickaryous and Sire, 2009). Lepidosauria. Among lepidosaurs, osteoderms are most commonly localized on the dorsal surface of the body and head (Gadow, 1901; Camp, 1923; Read, 1986; Estes et al., 1988), and are well represented in scleroglossan lizards such as anguids, helodermatids, and a few gekkonids. 1. Gekkonidae (Gekkota) Dermal ossifications are rare among gekkonids, and their development has yet to be investigated, although it has been suggested that it could be due to metaplastic ossification (LevratCalviac, 1986; Levrat-Calviac and Zylberberg, 1986; Levrat-Calviac et al., 1986). Cranial osteoderms bear a one-to-one correspondence with epidermal scales, while those on the bodies do not (Parker, 1942; Levrat-Calviac and Zylberberg, 1986; Bauer and Russell, 1989). In Tarentola and Geckonia, osteoderms are regular and polygonal in shape (Parker, 1942). 2. Helodermatidae (Anguimorpha) Gila monsters and beaded lizards, both in the genus Heloderma (Wiegmann, 1829), are the only extant taxa of the genus Heloderma, fossils of which date from the Miocene (Stevens, 1977). In helodermatid lizards, osteoderms are one of the last skeletal elements to develop. Prior to their formation, the stratum superficiale has already conformed to the osteoderms pattern across the body, but this does not apply to cranial osteoderms (Vickaryous and Sire, 2009).

Typically, osteoderms covering the cranial bones are thicker, more polygonal in shape, and larger than those from the nuchal region or the rest of the body, which usually are smaller and have a more circular outline. Sometimes, some cranial osteoderms abut others and co-ossify as a result of their development (Mead et al., 2012). Osteoderms develop as domed regions of thick collagen within the dermis, first appearing over the ossified head skeleton, then spreading caudally in a process consistent with metaplastic ossification (Moss, 1969). As the animals mature, the collagen domes increase in size before finally becoming mineralized. During this process, the incipient osteoderms (centers of mineralization within individual collagen domes) expand radially, beginning to encroach on one another, constraining their growth and, thereby, giving rise to a polygonal morphology (see Figure 5). 3. Anguidae (Anguimorpha) Glyptosaurinae are large lizards occurring trough the Eocene in the Holarctic region, and are among the squamates that possess the thickest and most broadly developed osteoderm shields (Buffrenil et al., 2010). Although the pattern and mode of anguid osteoderms skeletogenesis is incompletely known, glyptosaurine anguids present morphologically similar elements to helodermatid osteoderms. The complex structure of glyptosaurine osteoderms is poorly compatible with a growth pattern based exclusively on metaplastic ossification (Buffrenil et al., 2011); nevertheless, it has been suggested that the early stages of glyptosaurine osteoderm development may have been the result of a metaplastic process (Buffrenil et al., 2011). Evidence suggests that intramembranous ossification may also contribute on glyptosaurine osteoderm development, at least in latter stages (Zylberberg and Castanet, 1985). Synapsida. Osteoderms are rare among synapsids. Among xenarthans, osteoderms are present Journal of Morphology

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in several species of ground sloths and in all Cingulata, a clade consisting of pampatheres, glyptodonts and armadillos (Gaudin andWible, 2006; Hill, 2006). Most osteoderms conform to either a rectangular or polygonal morphology; while rectangular osteoderms imbricate, polygonal osteoderms form juxtaposed pavements (see Figure 6) and lack the lap articulations of the rectangular form. Osteoderm development in synapsids has been studied for the nine-banded armadillo Dasypus novemcinctus (Vickaryous and Hall, 2006), although evidence suggests that a similar developmental pathway may have led to the elaboration of the carapace in extinct cingulates (Hill, 2006). Dasypus novemcinctus. Osteoderms on Dasypus novemcinctus grow asynchronously across the body; across the head shield, they first appear in the area lying over the frontals and parietals, and they are covered by a usually congruent single epidermal scale (Vickaryous and Sire, 2009). Prior to their formation, the conformation of the integument surface is well established and readily distinguishable (Vickaryous and Hall, 2006). Initial osteoderm development begins deep within the stratum superficiale as a discrete aggregation of osteoblasts oriented parallel to the epidermis; cells of this primordium secrete osteoid and, after continued centrifugal growth, mineralization takes place (Vickaryous and Sire, 2009); in other words, after initial stages of intramembranous growth, metaplastic ossification plays an important role in osteoderm development (Wolf et al., 2011). Polygonal elements ossify centrifugally whereas in rectangular ones the caudal region mineralizes prior to the cranial region. Archosauria. Diversity among archosaurs is vast, and so is the range of morphologies among their osteoderms. Most studies hypothesize that cranial osteoderms in ankylosaurs (see Figure 7) may be comparable to those of extant squamates (Vickaryous and Russell, 2003; Scheyer and Sander, 2004), as all amniote lineages appear to maintain the capacity to develop osteoderms by similar processes (Moss, 1972); nevertheless, evidence of extensive mineralized inclusions from the dermis suggests they could also be comparable to those of crocodylians and Cingulata (Vickaryous and Russell, 2003; Vickaryous and Hall, 2008), while there is also evidence pointing toward more complex mechanisms such as neoplasia (Ricqle`s et al., 2001). As the mode of development of tissues and organs cannot be directly observed in fossils, osteoderm growth in ankylosaurs is far from understood. Osteoderm Development and Foam Models As cephalic osteoderms develop within the dermis, external to the dermatocranium, they are not Journal of Morphology

Fig. 8. Chart including the APA of each studied specimen, ordered according to Table 2. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

confined to the topographic limitations of individual cranial elements, originating in positions that may overlap several elements or sutural boundaries (Vickaryous et al., 2001). Therefore, these clusters are not confined to any particular geometry, being most appropriately modeled as a nonconstrained cluster consisting of N deformable areas. In this article, we assume that, once osteoderms encroach on one another, the process is driven by the principle of energy minimization: the same basic mechanical principle that rules processes

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TABLE 2. Results of average perimeter to square root of the osteoderms area (APA), SD of the ratios, and percentage of the interval APA 6 SD that falls inside the interval (3.6972, 3.7672) of the studied specimens Clade Lepidosaur

Synapsida

Archosaur

Crocodylia

Sampled taxa

Collection number

APA

SD

%

Heloderma horridum Heloderma horridum Heloderma horridum Heloderma horridum Heloderma suspectum Heloderma supectum Heloderma texana Tarentola annularis Glyptosaurus sylvestris Euphractus sexcinctus Euphractus sexcinctus Euphractus sexcinctus Euphractus sexcinctus Euphractus sexcinctus Dasypus novemcinctus Dasypus novemcinctus Dasypus novemcinctus Dasypus novemcinctus Dasypus hybridus Dasypus hybridus Dasypus hybridus Cabassous tatouay Cabassous tatouay Cabassous tatouay Chaetophractus vellerosus Dasypus (fetus) Glyptodon Panochtus tuberculatus Panochtus Panochtus Panochtus frenzelianus Peltephilus ferox Prozaedius exilis Ankylosaurus magniventris Euoplocephalus tutus Euoplocephalus tutus Euoplocephalus tutus Euoplocephalus tutus Euoplocephalus tutus Anodontosaurus lambei Crocodylus niloticus

UCMP 668DRL ETVP 7081 MCZ R-27899 MCZ R-5009 ETVP 7096 MCZ R-4315 TMM 40635-230 Bauer and Russell (1989) UCMP 126000 MNHN 682 MNHN 1105 MNHN 1242 MNHN 2434 MNHN 4697 MNHN 959 MNHN 1313 MNHN 2775 MNHN 4698 MNHNP TK1403 MNHN 1386 MNHN 2761 MNHN 1284 MNHN 2961 MNHN 4700 MNHN 2676 MNHN W/N Lydekker (1894), Plate V. A Scott (1903), Plate LXXXVIII MMC W/N Lydekker (1894), Plate V. A Scott (1903), Plate LXXXVIII MACN 7784 Scott (1903), Plate VI AMNH 5214 AMNH 5405 AMNH 5337 TMP 1991.27.1 CMN 0210 UALVP 31 CMN 8530 FC W/N, Cover of Science 339

3.7537 3.7237 3.7240 3.7415 3.7263 3.7242 3.7247 3.7492 3.7567 3.7989 3.8145 3.7956 3.7869 3.8516 3.7901 3.7968 3.8179 3.8153 3.8148 3.9185 3.9033 3.7728 3.7755 3.8365 3.8294 3.7977 3.8487 3.9446 3.7841 3.8297 3.8294 3.8681 3.9276 3.8040 3.9162 3.8973 3.8834 3.8039 3.8294 3.8953 3.9105

0.0527 0.0554 0.0550 0.0715 0.0476 0.0645 0.0671 0.0380 0.0758 0.1048 0.0828 0.1003 0.0659 0.1018 0.0964 0.1262 0.0882 0.1057 0.1119 0.1968 0.1598 0.1055 0.0548 0.1049 0.0773 0.0890 0.1362 0.1890 0.0773 0.0994 0.1481 0.1223 0.0988 0.1200 0.2029 0.1517 0.1336 0.1112 0.1367 0.2556 0.1287

62.8 63.2 63.6 48.9 73.5 54.3 52.2 73.7 46.2 33.4 21.5 34.9 35.1 08.6 36.3 27.7 21.2 27.2 03.1 11.6 07.4 33.2 42.4 16.9 09.7 32.9 20.1 03.1 39.1 18.6 23.6 08.8 0.0 29.2 13.3 07.1 06.5 31.5 25.6 13.7 0.0

with such different origins as the coarsening of two-dimensional fluid foams and movement of grain boundaries of a recrystallized metal (Gottstein et al., 2004). This assumption will be tested by comparing the results that derive from the energy minimizing model with the osteoderm patterns observed on real-life specimens.

result of a process which is not expected to optimize its energy (Milinkovitch et al., 2013), the scales of a specimen will also be analyzed in order to compare our results with those obtained from a physical cracking process. Measurements of the perimeter and areas were taken with image processing software ImageJ 1.46 (Abr amoff et al., 2004). Although osteoderms are irregular in shape and this software works by approximating curves through lines, accurate measurements are possible at sufficient magnifications.

MATERIALS AND METHODS In order to verify that the morphology of cranial osteoderms can be modeled as a surface tension phenomenon, the quotient between perimeter and the square root of the area of the osteoderms will be used as an indicator. A review of pictures found on national exhibitions, published articles (Lydeker, 1894; Scott, 1903; Bauer and Russell, 1989; Carpenter, 2004; Hill, 2005; Mead et al., 2012; Arbour and Currie, 2013; Milinkovitch et al., 2013) and online (DigiMorph Staff, 2010; Museum of Comparative Zoology; O’Leary and Kaufman, 2012) was considered (see Table 1 for a full list of the institutions where the studied specimens can be found). As the scales of Nile crocodiles are the

RESULTS AND DISCUSSION Given that regular pentagons, hexagons, and heptagons have average perimeter to square root of the area (APA) ratios of 3.8119, 3.7224, and 3.672, respectively, we assume that a cluster of osteoderms has minimal energy when the APA falls closer to that expected for a regular hexagon than a regular pentagon or heptagon; that is, the APA indicator falls inside the interval (3.6972; 3.7672), Journal of Morphology

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Fig. 9. Dasypodid fetus (MNHN w/n). Its APA value (3.7977) is of the order of the values obtained for adult dasypodids. Although this image illustrates the pattern of scalation, it also gives an idea of how the underlying osteoderms look like at fetal stage, as osteoderms are covered by a usually congruent single epidermal scale.

interval determined by the midpoints between the APA ratio for pentagons to hexagons, and for hexagons and heptagons. These values are shown in Figure 8. The values of the APA indicator and the standard deviations (SDs) of each specimen are presented on Table 2; so is the percentage of the interval APA 6 SD that falls inside the interval (3.6972; 3.7672), a confidence interval defined to show how close the osteoderms APA are to those expected for a regular hexagon in comparison to a regular pentagon or a heptagon. The data show that, while osteoderm growth in general does not seem to be driven by energy minimizing principles, lepidosaurian osteoderms are significantly more optimal than any other studied osteoderms, within safe margins. We suggest that, as cold-blooded animals are able to produce significantly less energy from aerobic and anaerobic energy sources than mammals (Seymour, 2013), they could compensate this condition through optimizing their energy usage even more than warmblooded animals. Osteoderms on Heloderma horridum are mainly developed through metaplastic ossification. This Journal of Morphology

mechanism has also been suggested as the main mechanism behind osteoderm formation on other helodermatid lizards and geckos (Vickaryous and Sire, 2009). On Dasypus novemcinctus, after initial stages of intramembranous ossification, metaplastic ossification plays an important role (Hill, 2006; Vickaryous and Hall, 2006). Literature suggests these same mechanisms behind osteoderms development on Dasypus novemcinctus for other synapsids (Vickaryous and Sire, 2009). Physical cracking is not driven by energy minimizing principles but by optimal release of stress (Cotterell and Rice, 1980), and this is seen both in the literature (Milinkovitch et. al, 2013) and in the study of the APA indicator on the crocodilian scales. According to our results, osteoderms clusters whose development has been suggested to begin with intramembranous ossification demand more energy to develop than those where metaplastic ossification takes place from the start. Therefore, from a strictly energetic viewpoint, osteoderm growth on ankylosaurs resembles more that of synapsids than that of lepidosaurs. However, osteoderms on Dasypus novemcinctus develop in utero, while the development of osteoderms on ankylosaurs is delayed in comparison to the rest of the skeleton, being osteoderms frequently absent from juvenile specimens (Marya~ nska, 1977; Jacobs et al., 1994; Vickaryous et al., 2001; Vickaryous and Hall, 2008). Due to this asynchronicity between bone growth and dermal armor development, recent studies suggest that remodeling cycles may take place in the development of both osteoderms and bone structures (Stein et al., 2013), mobilizing minerals in the same sense that calcium is mobilized from the skeleton in egg-laying alligators (Schweitzer et al., 2007). We suggest that this mobilization of minerals could imply an initial development of cranial ankylosaur osteoderms consistent with intramembranous ossification, which, as in synapsids, may have a great influence on osteoderm configurations. This hypothesis is supported by the fact that ornithischian dinosaurs’ bone tissues (which include ankylosaurs) resemble the histology in mammals (Enlow and Brown, 1957), which, among other factors (Ricqle`s et al., 1991), supports the hypothesis that large dinosaurs grew more like extant birds and mammals than like extant reptiles, even large ones (Ricqle`s, 1980; Reid, 1997; Varricchio, 1997; Horner et. al, 2001; Seymour, 2013). Another interesting though more debatable suggestion is that the main configuration of the cranial osteoderms that are initially developed through intramembranous ossification is mostly determined by RDM, as prior to their formation the conformation of the future osteoderms is well established, readily distinguishable and already seems to show a higher APA than that determined

CRANIAL OSTEODERM GROWTH

by metaplastic ossification (see Fig. 9). This seems more debatable due to the fact that the APA indicator theoretically works better in relatively flat surfaces, and the skull of the unborn armadillo is quite curve. Overall, when analyzing relatively flat surfaces, the APA seems to be a good indicator for optimal surface energy on clusters, in consistence with the physical-mathematical theory, and could be applied both to determine whether minimal energy configurations are given, and to study other systems where surface tension affects the concerning dynamics as well. Theoretically, this study is only a starting point for more thorough study and experimental tests, as it could be altered to fit more complex situations where mathematical elements are harder to determine, such as the osteoderms growth on curved carapaces. These are therefore future avenues of study for these biological surface tension phenomena models. ACKNOWLEDGMENTS Authors would like to thank Gustavo Grinspan, Andr es Rinderknecht (MNHN), Gabriel Lambach, and Joaquın Villamil for their help and support during the manuscript elaboration. LITERATURE CITED Abr amoff MD, Magalh~ aes PJ, Ram SJ. 2004. Image Processing with ImageJ. Biophotonics Int 11(7):36–42. Arbour VM, Currie PJ. 2013. Euoplocephalus tutus and the diversity of Ankylosaurid dinosaurs in the Late Cretaceous of Alberta, Canada, and Montana, USA. PLoS ONE 8(5):e62421. Bauer AM, Russell AP. 1989. Supraorbital ossifications in geckos (Reptilia: Gekkonidae). Can J Zool 67:678–684. Bever GS, Bell CJ, Maisano JA. 2005. The ossified braincase and cephalic osteoderms of Shinisaurus crocodylurus (Squamata, Shinisauridae). Palaeontol Electron 8:1–36. Bleicher MN. 1987. Isoperimetric divisions into several cells with natural boundary. In: Intuitive Geometry, Vol. 48. North-Holland, Amsterdam: Colloquium Mathematical Society, J anos Bolyai. pp. 63–84. Buffrenil V de , Sire JY, Rage JC. 2010. The histological structure of glyptosaurine osteoderms (squamata: anguidae), and the problem of osteoderm development in squamates. J Morphol 271:729–737. Buffrenil V de , Dauphin Y, Rage JC, Sire JY. 2011. An enamellike tissue, osteodermine, on the osteoderms of a fossilanguid (Glyptosaurinae) lizard. C R Palevol 10:427–437. Camp CL. 1923.Classification of the lizards. Bull Am Mus Nat Hist 48:289–482. Carpenter K. 2004. Redescription of Ankylosaurusmagniventris Brown 1908 (Ankylosauridae) from the Upper Cretaceous of the Western Interior of North America. Can J Earth Sci 41:961–986. Castanet J. 2006. Time recording in bone microstructuresof endothermic animals; functional relationships. C R Palevol 5: 629–636. Chang C, Wu P, Ruth Baker RE, Maini PK, Alibardi L, Chuong CM. 2009. Reptile scale paradigm: Evo-Devo, pattern formation and regeneration. Int J Dev Biol 53:813–826. Chuong CM, Richardson MK. 2009.Pattern formation today. Int J Dev Biol 53:653–658. Chuong CM, Nickoloff BJ, Elias PM, Goldsmith LA, Macher E, Maderson PA, Sundberg JP, Tagami H, Plonka PM, ThestrupPedersen K, Bernard BA, Schroder KR, Williams ML,

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