Network models of the tetrapod skull in which nodes represent bones and links represent sutures have recently offered new insights into the structural constraints underlying the evolutionary reduction of bone number in the tetrapod skull, known as Williston's Law. Here, we have built null network model-derived generative morphospaces of the tetrapod skull using random, preferential attachment, and geometric proximity growth rules. Our results indicate that geometric proximity is the best null model to explain the disparity of skull structures under two structural constraints: bilateral symmetry and presence of unpaired bones. The analysis of the temporal occupation of this morphospace, concomitant with Williston's Law, indicates that the tetrapod skull has followed an evolutionary path toward more constrained morphological organizations.

Les modèles de réseaux crâniens dans lesquels les nœuds représentent les os et les liaisons les sutures ont récemment permis d’apporter un nouveau regard sur les contraintes structurales qui sous-tendent la réduction évolutive du nombre d’os du crâne des tétrapodes, connue sous le nom de loi de Williston. Ici ont été construits des espaces morphologiques génératifs de crânes de tétrapodes, dérivés d’un modèle de réseau nul utilisant des lois de croissance à liaison préférentielle et proximité géométrique aléatoires. Nos résultats indiquent que la proximité géométrique est le meilleur modèle nul qui permette d’expliquer la disparité des structures crâniennes sous une double contrainte : symétrie bilatérale et présence d’os non appariés. L’analyse de l’occupation temporelle de cet espace morphologique qu’explique la loi de Williston indique que le crâne de tétrapode a suivi un itinéraire évolutif vers des organisations morphologiques davantage contraintes.

The evolution of the tetrapod skull has been extensively studied in comparative morphology. In the early 20th century, a pivotal analysis of changes in the number and complexity of skull bones in the evolution of Permian reptiles formed the basis for what is now known as the Williston's Law: an evolutionary trend in tetrapods toward reduction in the number of skull bones (

Recent studies on the evolution of the skull have focused on the analysis of morphological integration and modularity in different groups, such as: hominids (

In previous works we have shown that the structure of the tetrapod skull can be efficiently analyzed using network theory (

Theoretical morphology appeared in the 1960s beginning with the seminal work of David Raup on the accretionary growth of coiling shells (

The dimensions of a morphospace are timeless; this makes theoretical morphology suitable to frame evolutionary patterns of morphological change (

The articulation of skull bones was first analyzed in a theoretical morphology framework in

We built four null network model-derived generative morphospaces based on different growth assumptions about how bone connections are established during skull formation: at random, by preferential attachment, and using two different geometric proximity assumptions (

We have used these four null models to build generative morphospaces, analyzing their occupation using an empirical sample of real tetrapod skull networks. The results of this approach will be used to address the following questions: (1) how does the number of connections vary in relation to the number of bones; (2) how is this variation distributed across geological time; and, most importantly, (3) which growth rules are more likely to have been involved in producing the disparity of skull structures found in nature?

Skull network models are a morphological abstraction of the skull suture patterns in which each bone is a node and each suture connection is a link of the network. Methods to build skull network models have been extensively discussed in

We have built four generative morphospaces for two morphological traits: number of bones (

We have set the space of possible networks by imposing the following restrictions (

We built four generative morphospaces: random, preferential, proximal, and symmetric proximal, using different null network models (

The Random morphospace is based on the classic model of

The preferential morphospace is based on the model proposed by

The Proximal morphospace is based on the model proposed by

In addition, we have modified the model of Gabriel and Sokal to build a Symmetric Proximal morphospace, which introduces two additional constraints based on real skull anatomy: (1) the symmetric positioning of bones along a left-right axis (bilateral symmetry) and (2) the presence of unpaired bones positioned in the midline of this axis. We built the Symmetric Proximal morphospace for 0 to 7 unpaired nodes, while the remaining nodes were paired.

An empirical sample of 53 skull networks has been used to explore their occupation within each generative morphospace (

Empirical skull networks have been mapped onto each generative morphospace in order to analyze their occupation. Additionally, a temporal analysis of the morphospace occupation has been carried out for the generative morphospace that shows the best fit to the empirical sample. We have used seven time intervals: Devonian, Carboniferous, Permian, Triassic, Jurassic, Cretaceous, and Cenozoic. Temporal occupation for each empirical skull network was taken at the genus level using origin and extinction occurrence from the Paleobiological Database (available at

Generative morphospaces cover the theoretical morphospace distinctively; in addition, each type of morphospace behaves differently when varying their parameter values (

The occupation of the Random morphospace varies in each restricted region, according to the probability value (

Morphospaces generated with spatial constraints are more uniformly occupied. The proximal morphospace includes 42 out of 53 skull networks (79%) inside its boundaries (

To analyze the temporal occupation of the empirical sample of skull networks within the theoretical morphospace, we have used the extended symmetric proximal morphospace (

We have built four null network model-derived generative morphospaces using three growth rules: randomness, preferential attachment, geometric proximity, and symmetric geometric proximity. By mapping an empirical sample of skull networks onto these morphospaces, we have assessed their plausibility as developmental processes involved in the formation and evolution of the tetrapod skull. Our results indicate that geometric proximity is the best model to explain the disparity of skull structures found in tetrapods. This can only happen when bones are positioned in such a way that bilateral symmetry is kept and only a few of them are unpaired, which is the case of the symmetric proximal morphospace. Further analysis of the temporal occupation of this network morphospace reveals that early skulls, for all major groups, originated in the wider area of the morphospace, in which the variability is potentially greater. Subsequently, skull networks have evolved toward the narrower area of the morphospace, in which the potential skull variability is lower. This fits Williston's Law because the wider area represents skulls with higher number of bones and connections, whereas the narrower area represents skulls with fewer bones and connections (but showing higher density or complexity).

Our results do not support random and preferential growth rules as plausible processes of skull network formation. The analysis of the occupation of these morphospaces show that: (1) different skulls need different values for basic generative parameters, which are linked to their number of bones without any developmental or phylogenetic basis; and (2) their extended regions cover the full range of possible forms, which clearly limits their explanatory power. A common characteristic of both morphospaces is that none of their restricted regions can include completely the empirical sample of skull networks; full sample inclusion occurs only when taking all extended regions. This result entails that if skull structure (as modeled by networks) were produced by random or preferential mechanisms for establishing connections between bones, then the basic generative parameters (

In contrast, geometric proximity rules are well supported by our results. Even though, in these models, the positioning of bones imposes physical constraints for establishing bone connections during skull growth, they are able to generate highly bounded regions of the theoretical morphospace, which fit the empirical data very well. Thus, the proximal morphospace includes most of the real skull networks, except several skulls that have a higher number of connections (and consequently, more density) than expected for their number of bones, namely,

The different occupation of the proximal and the symmetric proximal morphospaces can be better understood if we interpret nodes in growing networks as analogous to ossification centers in skulls. For these null models, this interpretation implies also an idealized mechanism of homogeneous bone growth both in speed and direction. This is so because by connecting nodes using the Gabriel & Sokal model, we are assuming that each node is a center of growth that extends spatially until it contacts another growth front. Since some empirical skulls are not included in the Proximal morphospace, this indicates that this model is unable to predict some connections between bones. These skulls deviate from the growth assumptions of this null model because some of their bones might grow in size, have more irregular shapes, or have different developmental timing (

The generative region in the symmetric proximal morphospace is narrower for lower values of

Generative morphospaces, as hypotheses of developmental constraints, have allowed us to show a directional pattern of morphospace occupation in macroevolutionary time scales, further suggesting that the tetrapod skull has evolved in most lineages under the influence of structural constraints acting on the formation of new patterns of connectivity. These structural constraints are also related with mechanisms that favor the random loss of poorly connected bones and the selective fusion of the most connected ones, incidentally increasing morphological complexity, and providing a mechanistic basis for Williston's Law (

This research project was supported by grant (

Analysis of form using theoretical morphospaces. The empirical morphospace of morphological traits is mapped onto the broader framework provided by the theoretical construction, in which possible and impossible forms can be generated. Below, a hypothetical three-dimensional parameter space (

Analyse de forme utilisant les espaces morphologiques théoriques. L’espace morphologique empirique des caractères morphologiques est disposé dans le cadre le plus large fourni par la construction théorique, dans laquelle les formes possibles ou impossibles peuvent être générées. En bas, espace à paramètre tridimensionnel (

Boundaries constraining the theoretical morphospace of skull networks. Possible skull networks occupy the white region between both boundaries, whereas the grey regions contain only impossible networks. Based on measures of density, number of connections (

Liaisons contraignant l’espace morphologique théorique des réseaux crâniens. Les réseaux crâniens possibles occupent la zone blanche entre deux liaisons, tandis que les régions grises ne comportent que les réseaux impossibles. Sur la base de mesures de densité, du nombre de connexions (

Coverage of the theoretical morphospace by the four generative morphospaces. In grey, the region of impossible forms; in white, the region of possible forms, which each model covers distinctively. Solid dots, adult empirical skull networks; empty dots, human newborns. For each morphospace, the regions generated for different values of the generative parameter delimit restricted morphospace regions (grey line patterns), while the sum of all restricted regions configures an extended morphospace region (black continuous lines). A. The random morphospace; forms can be generated in three restricted regions according to the probability value,

Recouvrement de l’espace morphologique théorique par quatre espaces morphologiques génératifs. En gris, la région de formes impossibles ; en blanc, la région de formes possibles que chaque modèle recouvre de manière distinctive. Points noirs, réseaux crâniens empiriques adultes ; points blancs, nouveau-nés humains. Pour chaque espace, les régions générées pour les différentes valeurs du paramètre génératif délimitent des régions réduites de l’espace morphologique (lignes grises), la somme des régions réduites configurant une région élargie de l’espace morphologique (lignes noires continues). A. Espace morphologique aléatoire ; les formes peuvent être générées dans trois régions réduites, selon la valeur de probabilité,

Temporal occupation of the theoretical morphospace. Black lines delimit the area generated by the symmetric proximal morphospace. Skull networks of major groups originate in the wider area of the morphospace (right; higher

Occupation dans le temps de l’espace morphologique théorique. Les lignes noires délimitent la surface générée par l’espace morphologique proximal symétrique. Les réseaux crâniens des principaux groupes proviennent de la surface la plus large de l’espace morphologique (à droite, valeurs de

Properties of the four generative morphospaces.

Propriétés des quatre espaces morphologiques génératifs.

Empirical skull network sample.

Échantillon empirique de réseau crânien.

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