JOURNAL OF MORPHOLOGY 00:00–00 (2014) Does Osteoderm Growth Follow Energy Minimization Principles? Sebastian Sensale,1,2* Washington Jones,1 and R. Ernesto Blanco3 1Nucleo de Biomecanica, Espacio Interdiscipinario, Universidad de la Republica, Montevideo 11200, Uruguay 2Instituto de Fısica, Facultad de Ingenierıa, Universidad de la Republica, Montevideo 11300, Uruguay 3Instituto de Fısica, Facultad de Ciencias, Universidad de la Republica, Montevideo 11400, Uruguay ABSTRACT Although the growth and development of tion of contact surfaces among genetically deter- tissues and organs of extinct species cannot be directly mined elements (as in epithelia; Gibson et al., observed, their fossils can record and preserve evidence 2006; Farhadifar et al., 2007; Lecuit and Lenne, of these mechanisms. It is generally accepted that bone 2007). architecture is the result of genetically based biome- It is important to note that there is always chanical constraints, but what about osteoderms? In this article, the influence of physical constraints on cra- interplay between physically and genetically con- nial osteoderms growth is assessed. Comparisons trolled parameters (Padian et al., 2001; Castanet, among lepidosaurs, synapsids, and archosaurs are per- 2006). For example, in the cracking of a material formed; according to these analyses, lepidosaur osteo- layer, composition and thickness of the skin affect derms growth is predicted to be less energy demanding the mechanism (Milinkovitch et al., 2013); in RDM than that of synapsids and archosaurs. Obtained models treat developmental steps as if they were results also show that, from an energetic viewpoint, isolated from the rest of the organism, leaving ankylosaurid osteoderms growth resembles more that aside factors such as spatial and temporal location of mammals than the one of reptilians, adding evidence (Koch and Meinhardt, 1994); and, under closer to debate whether dinosaurs were hot or cold blooded. inspection, in energy minimization models the sys- J. Morphol. 000:000–000, 2014. VC 2014 Wiley Periodi- cals, Inc. tems are not perfectly regular and are often some- what disordered, which reflects the limit of a tight KEY WORDS: osteoderms; ankylosaurs; biomechanics; genetic control (Lecuit and Lenne, 2007). Also, periodic patterns 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 INTRODUCTION to focus on the influence of certain parameters. Periodic Patterning Modeling Polygonal Pattern Formation Mechanisms The growth of feathers, scales, hairs, and teeth, among other epidermal tissues, has long fasci- Due to their surfactants, biological membranes nated biologists. The presence of patterns in bio- behave like elastic membranes (Lautrup, 2004) logical tissues demonstrates how extremely and cell-surface mechanics are comparable to the organized developmental processes are to under- physics of foam formation (Taylor, 1976). Although stand how these patterns arise, the integration of this idea has been around for some time (Thomp- multiple disciplines (like physics and developmen- son, 1917), only recently 2D foam models have tal biology) seems mandatory. been applied to explain the development of Periodic patterning has emerged early in evolu- tion to allow functional redundancy and pheno- typic variation (Chuong and Richardson, 2009). In Additional Supporting Information may be found in the online ver- birds and mammals, reaction-diffusion mecha- sion of this article. nisms (RDM) generate the periodicity of structures Contract grant sponsor: Programa de Desarrollo de Ciencias such as hairs and feathers (Chuong et al., 2013), Basicas (PEDECIBA) and Agencia Nacional de Investigacion e but their relation with other tissues that present Innovacion (ANII).. periodic, repetitive patterns—such as the head scales of crocodiles—is not that direct (Milinko- *Correspondence to: S. Sensale; Instituto de Fısica, Facultad de vitch et al., 2013). Three major mechanisms could Ingenierıa, Montevideo 11300, Uruguay. E-mail: ssensale@fing.edu.uy generate polygonal patterns in biological tissues: the cracking of a material layer into adjacent poly- Received 27 August 2013; Revised 16 December 2013; gons (as in crocodile head scales), RDM pattern- Published online 00 Month 2014 in ings of genetically determined developmental units Wiley Online Library (wileyonlinelibrary.com). (as in feathers and hairs), and energy minimiza- DOI 10.1002/jmor.20273 VC 2014 WILEY PERIODICALS, INC. 2 S. SENSALE ET AL. 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). Clusters and energy minimizing (foam) Fig. 1. Cross section of an elementary 3D model of a material- models. If a 2D-structure consisting of many vacuum interface. Note how a molecule in the interface has one missing bond in comparison to a molecule in the interior. surfaces back to back everywhere (Fig. 2A) is con- sidered, minimizing the energy in the surfaces’ biological tissues such as the retina (Hayashi and junctions will involve minimizing their lengths. In Carthew, 2004) and epithelial junctional networks order to achieve this minimization, both the junc- (Farhadifar et al., 2007). Under this framework, tions’ perimeter and the area on each vertex surface tension properties underlie the control of where three or more edges meet must be minimal. tissue organization. The mathematics of elastic Therefore, two problems coexist: one involving a solids is a well-developed subject, and as such it minimal perimeter with free topology, and one can provide useful ideas about real-life processes involving the minimization of the vertices’ area. like the morphogenesis of biological tissues. In The areas surrounding each vertex have been proven to be minimal when three minimal surfa- this section, surface tension will be defined as well as its role in energy minimization. ces meet smoothly at angles of 120 along a curve Consider a simple regular “solid” surrounded by (Taylor, 1976; Graner, 2002), condition that will be vacuum whose molecules are placed in a cubic grid referred to as “triple junctures.” This condition can as in Figure 1, with grid length equal to the molecu- also be seen as the result of the three instantane- lar separation. Each molecule in the interior has six ous equal length-tension force vectors acting at a bonds to its neighbors, while those in the surface node. Basic mathematical properties prove that, have only five; this leads to a free surface energy under this hypothesis, the average surface will which, divided by the molecular area scale, defines have six (when considering the structure as free) what is called “surface energy density.” Because of or fewer sides (when considering the structure as the missing negative binding energy of the surface bordered, because of the defects; Graner et al., of molecules, this quantity is always positive when 2001; Milinkovitch et al., 2013). It is also worth considering the density associated with a liquid or noting that this result does not state that the solid interface against vacuum or gases, guarantee- mean surface will be a regular hexagon, but that ing that such interfaces seek toward the minimal it will be a polygon with six sides. This is also con- area consistent with the other forces that may be at sistent with Von Neumann–Mullins equation (Von play. This is not necessarily true on interfaces Neumann, 1952; Mullins, 1956) 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 oppos- ing 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 dW 5adA: Note that, except for the sign, this is quite anal- Fig. 2. Scheme of surfaces back to back everywhere (A) lateral ogous to the mechanical work (dW) performed view; (B)dorsalview. Journal of Morphology CRANIAL OSTEODERM GROWTH 3 Ap dS5 ðÞn26 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; Gra- ner 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. Neverthe- less, it has been demonstrated that the minimal perimeter configuration of a system of N deforma- ble 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 Ã 31=4 3:7224 (Graner et al., 2001; Cox et al., 2013b). Note that, although this number is not sensitive to the num- ber 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 num- ber 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 Fig. 3. Crack patterns on a dried mud. Note how chaotic and back to back everywhere that follows minimal irregular these patterns are.