Delimiting the boundaries of sesamoid identities under the network theory framework

Sample and data acquisition

We examined six adult specimens of L. latinasus (one dissected and five cleared and stained). We complemented gross anatomy dissections with several previous anatomic descriptions (Gaupp & Ecker, 1896; Dunlap, 1960; Nussbaum, 1982; Burton, 1998; Ponssa, Goldberg & Abdala, 2010; Abdala, Vera & Ponssa, 2017). Due to the bilateral symmetry of the body, we built an adjacency matrix considering the right half of the body as a proxy of the whole configuration. An adjacency matrix defines the connectivity pattern of the anatomical network by indicating pairs of connected elements. Further details are available in Text S1 and Table S1.

Network construction

Networks are appropriate mathematical models for tackling the study of biological systems because they are intrinsically collections of entities (nodes) connected through relationships (links). The identification of nodes and links is a critical issue that depends on what we want to know about the modeled system. We modeled the skeletal network of L. latinasus based on the main adjacency matrix. The network model was constructed to inquire on the topological nature of sesamoids within the skeletal system and the anatomical relation among them. Therefore, in our model the nodes represent canonical skeletal pieces and sesamoids (ossified, cartilaginous and fibrocartilaginous). The dorsal fascia was also modeled as a node due to its great extension and its role as insertion or origin point for muscles. Since joints, muscles, tendo-muscular units, tendons, and aponeurosis are functionally responsible for the flow of mechanical information among skeletal elements they were considered links in our study. Because all anuran sesamoids are postcranial, the consideration of the whole skull is not informative for our purposes, thus cranial bones were collapsed into a single node. We considered the network as undirected and weighted, considering weights as multiple links.

Network modularity

A morphological module is a semi-independent set of densely interconnected skeletal pieces that are only sparsely connected to the rest of the skeleton (Rasskin-Gutman & Esteve-Altava, 2014; Dos Santos et al., 2017). The number and composition of modules were identified using Order Statistics Local Optimization Method (OSLOM) (Lancichinetti et al., 2011). This community detection algorithm allows the detection of the statistical significance (bs) of modules with respect to random fluctuations. OSLOM takes into account the possibility of homeless nodes (i.e., nodes that are not assigned to any module), as well as the detection of overlapping modules, and the presence of hierarchical organization (Lancichinetti et al., 2011). For our analysis we followed the default options of the OSLOM algorithm, considering bs threshold as 0.1 and allowing the detection of singletons (homeless nodes). Because of the stochastic nature of the OSLOM modularity analysis, results may vary depending on the run (Lancichinetti et al., 2011; Esteve-Altava, 2017). We have reported the most stable modules among multiple runs.

Sesamoid characterization

Network characterization

The anatomical network of Leptodactylus latinasus comprises 102 nodes connected by 328 physical connections (Figs. 1 and 2). Regarding centrality parameters, long bones (especially the tibiafibula), the ilium, and the dorsal fascia stand out by showing the highest centrality values (see Table S3). The module detection algorithm applied (OSLOM) revealed five significant and partial overlapping modules plus two singletons in the first hierarchical level (Figs. 1 and 2, Table 1): the pectoral-forelimb module (M1, bs = 0.069), the axial-scapular module (M2, bs = 0.002), the axial-pelvic module (M3, bs = 0.081), the hindlimb module (M4, bs = 0.063), and the IV–V toes module (M5, bs = 0.013). The fourth vertebra and the urostyle (nodes IDs 5 adn 11) are simultaneously members of the two axial modules (M2 and M3). The femur and the ischium (nodes IDs 14 and 77) are simultaneously members of overlaps the axial-pelvic and the hindlimb module, and the fibulare (node ID 75) overlaps between the hindlimb (node ID 92) and the IV–V toes module. The fascia dorsalis and the third phalanx of digit V of the foot (node ID 65) were not assigned to any module in the first hierarchical level (singletons). The configuration of the second hierarchical level presented two modules with a broad overlap of the axial nodes. The third phalanx of digit V of the foot was also not included in any module in the second hierarchical level.

Sesamoid patterns

Sesamoids are widely distributed through the network, being present in all modules except in the axial-pectoral (M2). GSs are arranged peripherally in the network topology. The pectoral-forelimb module (M1) includes two ESs and six GSs. In the axial-pelvic module (M3), there are three ESs. The hindlimb module (M4) shows the highest number of sesamoids (nine), and the IV–V toes module contains only GSs (five).

There are significant differences among node skeletal categories for the all centrality indicators measured. GSs exhibit the lowest average values for every centrality indicator, while CSEs exhibit the highest centrality indicator average values except for eigen-centrality. ESs do not differ significantly from CSEs in any of the centrality indicators. GSs are significantly lower both from SEs and from CSEs in degree and closeness. GSs do not differ significantly from ESs regarding betweenness centrality, but they do differ from CSEs in this parameter. GSs do not differ significantly from CSEs in eigen-centrality, but they do differ from ESs in this parameter (Fig. 3; Table 2; see also Table S2 and Fig. S1).

I. General network properties

In the first hierarchical level, and because the skull has been simplified to a single node, the modules are mainly related to the girdles and limbs. Anatomically, the pectoral girdle is divided in a ventral region, composed by the coracoid elements, and in a dorsal region, composed by the scapular elements (Baleeva, 2009). This regionalization could explain why the pectoral girdle is arranged in two modules: the axial-scapular module and the pectoral-forelimb module. In the axial-scapular module, the dorsal elements of the pectoral girdle are more connected to the cranium and to the first vertebrae than to the rest of the pectoral girdle. Indeed, both scapular elements and the cranium are connected by several muscles, such as the M. depressor maxillae which inserts in the lower jaw (Ecker, 1889; Manzano, Moro & Abdala, 2003). The cartilaginous connection between the scapular and coracoid elements provides a considerable strength of connection while admitting some mobility and decreasing compression forces (Baleeva, 2001). The pectoral-forelimb module is formed by the coracoid elements and the forelimb. This region has an important and complex function absorbing the stress of the impact in the landing phase of the jump (Emerson, 1982). Also, both modules are involved in the complex mechanism of anuran acoustic perception (Lombard & Straughan, 1974; Kardong, 2012).

The axial-pelvic module is where most of the overlapping occurs in both hierarchical levels of our modularity analysis. This particular configuration of the anuran pelvic girdle seems to represent a topological transitional region between an anterior and posterior region of the network. The elements of this module are deeply affected by developmental changes during metamorphosis from a swimming tadpole into a tailless jumping adult (Soliz & Ponssa, 2016). These transformations result in a compact and highly integrated vertebral column plus pelvic girdle, epitomized in the network by the membership of the 4th vertebra and the urostyle to both the axial-pectoral and axial-pelvic modules. The femur and the ischium are also simultaneously members of two modules, the axial-pelvic and the hindlimb module. Previous studies have shown that, at the beginning of embryonic formation, the femur develops in close contact with the future acetabulum of the pelvic girdle (Pomikal, Blumer & Streicher, 2011). Furthermore, functional co-dependencies exist in the pelvic-hindlimb boundary, in which sequential movement of the hip and leg joints were identified during a typical frog jump (Astley & Roberts, 2014; Nauwelaerts, Stamhuis & Aerts, 2005).

The hindlimb module is constituted by pieces of the stylopodium, zeugopodium, and autopodium, most of them are long bones with high centrality values. Long bones have been recurrently recovered as central nodes in network analysis of tetrapod limbs (Diogo et al., 2015; Dos Santos et al., 2017). The zeugopodization hypothesis in anurans postulates the elongation of the tibiale and fibulare and a consequent distal shift in the zeugo-autopodial border (Diogo & Ziermann, 2014; Dos Santos et al., 2017). This could explain why the tibiale and fibulare show high centrality values, that in turn, could be associated with their functional importance as an extra site for muscle attachment (Handrigan & Wassersug, 2007). Moreover, the fibulare is also part of the IV–V toes module, which is composed of those elements of the foot aligned with this bone. The elements of IV–V toes module also share a common origin of their flexor tendons, which arise from the flexor digitorum brevis superficialis. Contrary to those of the toes I–III, which originate from the aponeurosis plantaris (Gaupp & Ecker, 1896).

II. Sesamoid Nature, as revealed by node parameters

Sesamoids were often conceptually placed outside or beyond the skeleton (Vickaryous & Olson, 2007). Diogo et al. (2015), indeed modeled many sesamoids as isolated nodes (i.e., without any connection to other pieces) in their strictly skeletal network. On the contrary, our anatomical network does include sesamoids as inner pieces of the skeleton, linked by tendons and muscles to the other skeletal elements. Surprisingly, all of them are unambiguously integrated within four of the modules of the first hierarchical level of the network, being absent only in the axial-pectoral module. Connectivity patterns arrange GSs as peripheral elements of the system, while the ESs are more variably distributed through the network. Both are, in general terms, weakly connected with the canonical pieces of the skeleton as recovered by the centrality indicators. In this sense, sesamoids seem to be lowly burdened structures, sensu Riedl (1978). Connections between elements are established during embryological development (Rasskin-Gutman & Esteve-Altava, 2018). Sesamoids develop independently and relatively late in comparison to other skeletal elements, only later becoming associated with the primary skeleton (Hall, 2005; Vera, Ponssa & Abdala, 2015). Although sesamoids may have an imperfect fossil record, probably due to their loose connectivity pattern (this work), their earliest fossil reports predate the Jurassic period (200+ mya) (Vickaryous & Olson, 2007). By then, most of the skeletal pieces would have been evolving for at least 420 million years (Ravi & Venkatesh, 2008). In turn, sesamoids’ high diversity in size, shape, number, and distribution (Abdala et al., 2019) could be a consequence of being low burdened structures. In fact, high rates of evolution of sesamoid bones were reported by Baum & Smith (2013). Thus, the loose connectivity pattern characterizing sesamoids seems to be a consequence of their delay in ontogeny and phylogeny.

The facultative expression of many sesamoids in the phenotype as a response to continuous mechanical stress (i.e., epigenetic influence) (Abdala & Ponssa, 2012; Abdala et al., 2019) could be, at least in part, a consequence of the low burden of these skeletal pieces. Perturbations on sesamoids development is unlikely to be accompanied by systemic consequences for an organism, as could be the case of perturbations on the development of canonical elements. As lowly constrained pieces, sesamoids may be labile evolutionary elements, with a relatively high capacity of generating heritable phenotypic variation (Kirschner & Gerhart, 1998), in turn the path of the appearance of evolutionary novelties would be facilitated. This rationale is in accordance with the dynamic model stated by Abdala et al. (2019) in which sesamoids are proposed as a source of new skeletal morphologies available to natural selection processes.

II-1. Embedded sesamoids

ESs centralities values were not significantly different from CSEs elements, but they resulted to be significantly more central than those of the GSs (except for betweenness). The fact of being embedded in the connective tissue of the most powerful muscles of the limbs (Jerez, Mangione & Abdala, 2010), which were considered as network links, straightforwardly contributes to the higher centrality values of ESs when compared to GSs. ESs are distributed in three modules related to the limbs and the pelvic girdle (M1, M3, and M4), and absent from axial-scapular and IV–V toes modules (M2 and M5). Most ESs are included within the hindlimb module, coincidentally, this module is subject to the highest mechanical forces during the take-off phase of the jump (Nauwelaerts & Aerts, 2006). The palmar sesamoid (pectoral-forelimb module) showed a notably high betweenness value among embedded sesamoids, and surprisingly similar to top-ranked canonical elements (Table S3). This could be associated with the fact that the palmar sesamoid is embedded in the m. flexor digitorum longus which is the source of the flexor tendons of digits II–V (Ponssa, Goldberg & Abdala, 2010; Diogo & Abdala, 2010).

It is logical to think that nearby pieces will tend to be more connected than distant pieces. Thus, a correspondence between network modules and euclidean regions of the body is expected (Dos Santos et al., 2017). Sesamoids, in general, have links other than joints connecting them to other skeletal pieces (Vickaryous & Olson, 2007; Table S1). This property allows them to defy the general proximity imposition, in such a way that they are able to share a module with spatially distant pieces. In fact, the patella and the graciella sesamoids, located in the knee joint, are co-opted by a more proximal module (axial-pelvic module) instead of the hindlimb module, as we could expect following a spatial neighborhood criterion. These sesamoids constitute the only elements in the network with such kind of behavior. This pattern could be explained by their remote connection with the pelvic girdle by the cruralis and the gracilis major muscles, respectively, which form a set of muscles required for the extension and flexion of the knee joint (Abdala, Vera & Ponssa, 2017; Table S1). Additionally, Eyal et al. (2015) show that, in mice, the patella arises as part of the femur but from a distinct pool of progenitors. Thus, probably, the patella membership to the axial-pelvic module can be explained by complex cellular and genetic mechanisms during the morphogenesis process.

II-2. Glide sesamoids

Centrality indicators mainly segregated the GSs from the other skeletal categories. Frequently, GSs are implicitly excluded from sesamoid conceptual delimitation, due to definitions typically consider sesamoids as elements surrounded by tendinous or ligamentous structures (e.g., Hall, 2005); “(...) sesamoids are independent ossifications/chondrifications within tendons”). Moreover, developmental evidence has shown that although ESs and GSs share the same progenitor cells, they have different developmental signaling paths (Eyal et al., 2019).

Glide sesamoids are significatively less connected than ESs and CSEs when comparing degree and closeness. A different trend is revealed by the eigen-centrality indicator, which is similar between ESs and CSEs, but distinguishes the two sesamoid categories highlighting the particularities of GSs. Low eigen-centrality indicates that not only GSs, but also that their neighbor nodes have few connections. The unusually low centrality indicators of GSs could be a proxy of a high evolvability of those bones following the burden theory (Rasskin-Gutman & Esteve-Altava, 2018). Indeed, high intraspecific variation in number and morphology has been reported in glides (Ponssa, Goldberg & Abdala, 2010). Therefore, low connectivity could represent an alternative strategy to modularity in order to increase evolvability.

The anatomical distribution, shape, constitution, and the paired condition of GSs of the forelimb in L. latinasus (Ponssa, Goldberg & Abdala, 2010), is similar to those of paraphalangeal elements that characterize many pad-bearing geckos (Squamata). The multiple origins of paraphalanges plus their extremely variable morphology (Wellborn, 1933; Russell & Bauer, 1988; Gamble et al., 2012; Fontanarrosa, Daza & Abdala, 2018) supports the idea of their lability in evolutionary terms. Curiously, lizards that lack paraphalanges also lack GSs related to interphalangeal joints (Fontanarrosa, 2018). Additional network analysis, modeling a species with paraphalanges, would most likely indicate that they are relatively disconnected structures of the main skeleton. Connectivity patterns have long been a criterion for the recognition of homologies (Geoffroy Saint-Hilaire, 1818). Thus, identifying common connections to specific elements, could reveal putative homologous structures through distant related lineages such as paraphalanges and GSs. Furthermore, dissimilar connectivity patterns between ESs and GSs found in this study suggest that they may not be members of the same hierarchical category. Future studies based on complementary sources of evidence, such as development or evolution, are required to test this hypothesis.