Delimiting the boundaries of sesamoid identities under the network theory framework

Sesamoid identity has long been the focus of debate, and how they are linked to other elements of the skeleton has often been considered relevant to their definition. A driving hypothesis of our work was that sesamoids’ nature relies deeply on their connections, and thus we propose an explicit network framework to investigate this subject in Leptodactylus latinasus (Anura: Leptodactylidae). Through the dissection of L. latinasus’ skeleton, we modeled its anatomical network where skeletal elements were considered nodes while joints, muscles, tendons, and aponeurosis were considered links. The skeletal elements were categorized into canonical skeletal pieces, embedded sesamoids, and glide sesamoids. We inquired about the general network characterization and we have explored further into sesamoid connectivity behavior. We found that the network is structured in a modular hierarchical organization, with five modules on the first level and two modules on the second one. The modules reflect a functional, rather than a topological proximity clustering of the skeleton. The 25 sesamoid pieces are members of four of the first-level modules. Node parameters (centrality indicators) showed that: (i) sesamoids are, in general terms, peripheral elements of the skeleton, loosely connected to the canonical bone structures; (ii) embedded sesamoids are not significantly distinguishable from canonical skeletal elements; and (iii) glide sesamoids exhibit the lowest centrality values and strongly differ from both canonical skeletal elements and embedded sesamoids. The loose connectivity pattern of sesamoids, especially glides, could be related to their evolvability, which in turn seems to be reflected in their morphological variation and facultative expression. Based on the connectivity differences among skeletal categories found in our study, an open question remains: can embedded and glide sesamoids be defined under the same criteria? This study presents a new approach to the study of sesamoid identity and to the knowledge of their morphological evolution.

271 II-1. Embedded sesamoids. 272 ESs centralities values were not significantly different from CSEs elements, but they resulted to 273 be significantly more central than those of the GSs (except for betweenness). The fact of being 274 embedded in the connective tissue of the most powerful muscles of the limbs (Jerez, Mangione & 275 Abdala, 2010), which were considered as network links, straightforwardly contributes to the 276 higher centrality values of ESs when compared to GSs. ESs are distributed in three modules 277 related to the limbs and the pelvic girdle (M1, M3, and M4), and absent from axial-scapular and 278 IV-V toes modules (M2 and M5). Most ESs are included within the hindlimb module, 279 coincidentally, this module is subject to the highest mechanical forces during the take-off phase 280 of the jump (Nauwelaerts & Aerts, 2006). The palmar sesamoid (pectoral-forelimb module) 281 showed a notably high betweenness value among embedded sesamoids, and surprisingly similar 282 to top-ranked canonical elements (SI : Table S3). This could be associated with the fact that the 283 palmar sesamoid is embedded in the m. flexor digitorum longus which is the source of the flexor 284 tendons of digits II-V (Ponssa, Goldberg & Abdala 2010; Diogo & Abdala, 2010). 285 It is logical to think that nearby pieces will tend to be more connected than distant pieces. Thus, a 286 correspondence between network modules and euclidean regions of the body is expected (Dos 287 . Sesamoids, in general, have links other than joints connecting them to other 288 skeletal pieces (Vickaryous & Olson, 2007; Table S1). This property allows them to defy the 289 general proximity imposition, in such a way that they are able to share a module with spatially 290 distant pieces. In fact, the patella and the graciella sesamoids, located in the knee joint, are co-291 opted by a more proximal module (axial-pelvic module) instead of the hindlimb module, as we 292 could expect following a spatial neighborhood criterion. These sesamoids constitute the only 293 elements in the network with such kind of behavior. This pattern could be explained by their 294 remote connection with the pelvic girdle by the cruralis and the gracilis major muscles, 295 respectively, which form a set of muscles required for the extension and flexion of the knee joint 296 (Abdala, Vera & Ponssa, 2017; Table S1). Additionally, Eyal et al. (2015) show that, in mice, the 297 patella arises as part of the femur but from a distinct pool of progenitors. Thus, probably, the 298 patella membership to the axial-pelvic module can be explained by complex cellular and genetic 299 mechanisms during the morphogenesis process. 300 301 II-2. Glide sesamoids 302 Centrality indicators mainly segregated the GSs from the other skeletal categories. Frequently, 303 GSs are implicitly excluded from sesamoid conceptual delimitation, due to definitions typically 304 consider sesamoids as elements surrounded by tendinous or ligamentous structures (e.g., Hall, 305 2005; "(...) sesamoids are independent ossifications/chondrifications within tendons"). 306 Moreover, developmental evidence has shown that although ESs and GSs share the same 307 progenitor cells, they have different developmental signaling paths (Eyal et al., 2019). 308 Glide sesamoids are significatively less connected than ESs and CSEs when comparing degree 309 and closeness. A different trend is revealed by the eigen-centrality indicator, which is similar 310 between ESs and CSEs, but distinguishes the two sesamoid categories highlighting the 311 particularities of GSs. Low eigen-centrality indicates that not only GSs, but also that their 312 neighbor nodes have few connections. The unusually low centrality indicators of GSs could be a 313 proxy of a high evolvability of those bones following the burden theory (Rasskin-Gutman & 314 Esteve-Altava, 2018). Indeed, high intraspecific variation in number and morphology has been 315 reported in glides (Ponssa, Goldberg & Abdala, 2010). Therefore, low connectivity could 316 represent an alternative strategy to modularity in order to increase evolvability. 317 The anatomical distribution, shape, constitution, and the paired condition of GSs of the forelimb 318 in L. latinasus (Ponssa, Goldberg & Abdala, 2010), is similar to those of paraphalangeal 319 elements that characterize many pad-bearing geckos (Squamata). The multiple origins of 320 paraphalanges plus their extremely variable morphology (Wellborn, 1933 Abdala, 2018) supports the idea of their lability 322 in evolutionary terms. Curiously, lizards that lack paraphalanges also lack GSs related to 323 interphalangeal joints (Fontanarrosa, 2018). Additional network analysis, modeling a species 324 with paraphalanges, would most likely indicate that they are relatively disconnected structures of 325 the main skeleton. Connectivity patterns have long been a criterion for the recognition of 326 homologies (Geoffroy Saint-Hilaire, 1818). Thus, identifying common connections to specific 327 elements, could reveal putative homologous structures through distant related lineages such as 328 paraphalanges and GSs. Furthermore, dissimilar connectivity patterns between ESs and GSs 329 found in this study suggest that they may not be members of the same hierarchical category. 330 Future studies based on complementary sources of evidence, such as development or evolution, 331 are required to test this hypothesis. 332 333 Conclusions 334 Here we presented a new approach to the study of sesamoid identity and hope to contribute to the 335 current research on their morphological evolution. Our findings raise interesting questions to be 336 investigated in other species of tetrapods, as well as by complementary areas of research, such as 337 developmental or evolutionary biology. Multiple sesamoid definitions based on their relations 338 with canonical bones and connective tissue were calling for their explicit framing under network 339 theory. After performing an anatomical network analysis of a model anuran species (L. 340 latinasus), we inquired first on the general topology, and more specifically on sesamoid 341 connectivity patterns. The main conclusions that emerged from this approach are: 342 1.
The skeletal elements were clustered in five modules that reflect a functional 343 organization. Four modules contain at least one sesamoid. 344 2.
Sesamoids, in general terms, are peripheral elements of the network, with few 345 connections to the canonical skeleton. This could explain their considerable variation on size, 346 shape, number, distribution and high evolvability. These results support the hypothesis of 347 sesamoids as morphological innovations generators. 348 3.
Embedded sesamoids have, on average, similar centrality values to the canonical skeletal 349 elements. These sesamoids are surrounded by connective tissue, thus are prone to have more 350 connections than glide sesamoids. While glides are adjacent to tendons, but not fixed to them.

Figure 1
The anatomical network of Leptodactylus latinasus with an inset (i) providing a schematic representation of the network relative to the species body Links are weighted by the number of connections, and nodes are colored according to membership to modules: pectoral-forelimb module (green), axial-scapular module (orange), axial-pelvic module (yellow), hindlimb module (blue), IV-V toes module (purple), and homeless pieces (grey). Mix coloured nodes are simultaneously members of two modules.
Different shapes distinguish among skeletal categories of nodes: non-sesamoids, glide sesamoids, and embedded sesamoids. ID numbers are shown in Table 1.