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

Abstract

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.

Machine learning-assisted discovery of growth decision elements by relating bacterial population dynamics to environmental diversity

“…The sum of S τ , S r , and S K , which was defined as the global sensitivity ( S g ) of the component across the three growth phases, showed a similar long-tailed distribution shape. The four distributions were all likely to follow the power law 44, 45 , which agreed well with the ML-predicted conclusion that only a few components determined the growth. This finding strongly suggested that the fate decision components were present among the 41 components, regardless of the complex interactions among these components.…”

Ecological signal in the size and shape of marine amniote teeth

“…With that said, it is also important to note that factors beyond functional constraints could influence the patterns of morphospace occupation observed in our results. Teeth can develop into an amazing array of shapes, but conical teeth are still limited in form by developmental constraints that prevent certain extreme areas of morphospace being occupied [58]. Similarly, other factors such as phylogenetic inertia could drive patterns observed in the results.…”

Whole-limb scaling of muscle mass and force-generating capacity in amniotes

“…It implies the tendency towards some adaptive optimum in organismal ‘design’ or, more likely, the presence of one or more constraints that prevent significant and systematic deviation from a common pattern. These constraints may be functional in nature, such as trade-offs that could occur between conflicting requirements of individual muscles in the execution of disparate tasks (see Introduction), or may have a developmental basis ( e.g ., Evans et al, 2021 ). Given the multidimensional and nonlinear aspects of muscle architecture and function, and terrestrial locomotor biomechanics in general, it would be naïve to suggest that a bivariate statistical model such as those derived here can sufficiently represent the mechanical phenomena involved ( Taylor & Thomas, 2014 ).…”