A universal power law for modelling the growth and form of teeth, claws, horns, thorns, beaks, and shells.

BACKGROUND: A major goal of evolutionary developmental biology is to discover general models and mechanisms that create the phenotypes of organisms. However, universal models of such fundamental growth and form are rare, presumably due to the limited number of physical laws and biological processes that influence growth. One such model is the logarithmic spiral, which has been purported to explain the growth of biological structures such as teeth, claws, horns, and beaks. However, the logarithmic spiral only describes the path of the structure through space, and cannot generate these shapes. RESULTS: Here we show a new universal model based on a power law between the radius of the structure and its length, which generates a shape called a 'power cone'. We describe the underlying 'power cascade' model that explains the extreme diversity of tooth shapes in vertebrates, including humans, mammoths, sabre-toothed cats, tyrannosaurs and giant megalodon sharks. This model can be used to predict the age of mammals with ever-growing teeth, including elephants and rodents. We view this as the third general model of tooth development, along with the patterning cascade model for cusp number and spacing, and the inhibitory cascade model that predicts relative tooth size. Beyond the dentition, this new model also describes the growth of claws, horns, antlers and beaks of vertebrates, as well as the fangs and shells of invertebrates, and thorns and prickles of plants. CONCLUSIONS: The power cone is generated when the radial power growth rate is unequal to the length power growth rate. The power cascade model operates independently of the logarithmic spiral and is present throughout diverse biological systems. The power cascade provides a mechanistic basis for the generation of these pointed structures across the tree of life.

Global elongation and high shape flexibility as an evolutionary hypothesis of accommodating mammalian brains into skulls.

Little is known about how the large brains of mammals are accommodated into the dazzling diversity of their skulls. It has been suggested that brain shape is influenced by relative brain size, that it evolves or develops according to extrinsic or intrinsic mechanical constraints, and that its shape can provide insights into its proportions and function. Here, we characterize the shape variation among 84 marsupial cranial endocasts of 57 species including fossils, using three-dimensional geometric morphometrics and virtual dissections. Statistical shape analysis revealed four main patterns: over half of endocast shape variation ranges from elongate and straight to globular and inclined; little allometric variation with respect to centroid size, and none for relative volume; no association between locomotion and endocast shape; limited association between endocast shape and previously published histological cortex volumes. Fossil species tend to have smaller cerebral hemispheres. We find divergent endocast shapes in closely related species and within species, and diverse morphologies superimposed over the main variation. An evolutionarily and individually malleable brain with a fundamental tendency to arrange into a spectrum of elongate-to-globular shapes-possibly mostly independent of brain function-may explain the accommodation of brains within the enormous diversity of mammalian skull form.