### Power cascade growth is not accounted for by existing models of tooth growth

There are currently two general models that each strive to describe or explain various aspects of tooth development. Enamel knots produce inhibitory signals that prevent new enamel knots forming close to an existing knot [17]. The ‘patterning cascade’ model describes how this inhibition, along with the folding of the epithelial-mesenchyme interface, creates limitations on the size and position of successive cusps during development [33]. First described in seal postcanine teeth, the patterning cascade model has since been extended to primate molars [34, 35]. The second model, the ‘inhibitory cascade’, describes the relative size of sequentially produced teeth, such as molars, as a linear change in size along a tooth row [2, 36]. Neither of these models addresses the shape of cusps.

The power cascade model proposed here is a third general model of tooth development complementary to the two existing models, indicating how the shapes of unicuspid teeth and individual cusps are generated. After determination of cusp shape by the power cascade model, we postulate that cusp spacing is dictated by inhibition of enamel knots according to the patterning cascade [33], and number of cusps is controlled by the number of enamel knots that can fit in the total area of the tooth. The sizes of sequential teeth are then directed by the inhibitory cascade [2, 36]. Therefore, cusp shape, cusp number, and tooth size can be simulated according to this trio of models to generate the main features of an entire tooth row.

The power function has been used to represent or measure a limited set of teeth in previous studies, including the tips of shapes designed for mechanical penetration testing [37], using an average *Slope* of 0.5. Felid canine profiles measured using power functions [38] showed that they generally had a *Slope* of ~ 0.55. Both of these studies are consistent with the current findings in many mammal canines, but they did not generalise this pattern to all teeth or cusps.

Detailed developmental computer simulations of tooth morphogenesis have used a 3D reaction-diffusion-like model that calculates bending stresses to form cusps and teeth [39, 40]. This model produces cusp positions that can have morphological variation similar to biological teeth [39, 41]. Here we tested whether the cusp shapes produced by that model conform to the power cascade model. Varying five parameters of the model that simulates the development of ringed seal postcanine teeth [41] shows that most of the cusp shapes produced do not closely resemble the expected power cascade, with *R*^{2} between 0.59 and 0.97 (Additional file 1: Figure S6). Therefore, the power cascade model describes cusp shape (or cross-sectional profile) substantially better than complex in silico models, although this may be a result of the limited number of cells in the simulations.

Given the power of this new model to define the limits of tooth shape in animals, we expanded our focus to compare it with existing models of growth in other morphological systems. Wren’s [19] model of shells growing as a cone bending to form a logarithmic spiral has since been used to model shells and teeth [6, 7, 20]. Starting with a cone, a logarithmic spiral is generated when one side grows faster than the other, causing the cone to bend to one side (Fig. 1b; Additional file 1: Figure S7d). A mechanism to generate a logarithmic spiral is the unequal growth rates of the two sides A and B. Logarithmic spirals have a formula in polar coordinates *S = a e*^{b θ}, where *θ* is the angle of rotation around the origin, *S* is the resulting radius of the logarithmic spiral, and *a* and *b* are parameters affecting the size and rate of expansion of the spiral, respectively (Additional file 1: Figure S7a). The radius of the shell opening expands linearly with the angle of rotation (*Radius* = *c θ*, where *c* is a parameter affecting the rate of growth of the shell opening), which creates a cone spiralling around the central axis (Additional file 1: Figure S7b). This model was used to generate shell shapes of many types by modifying relative rates of growth [7, 42, 43].

The Raup [7] shell equation describes shell growth using a cone, which is the shape where *Slope* = 1 in our *Log Distance-Radius* plots (Fig. 6; Additional file 1: Figure S2). If this model accurately describes shell growth, all shells should fall on the right-hand edge of the morphospace in Fig. 6. The shells of molluscs (scaphopod *Dentalium* sp. and gastropod *Bembicium auratum*) and cephalopods (nautilus *Nautilus pompilius* and ram’s head squid *Spirula spirula*) each apparently form logarithmic spirals, but follow the power cascade with *Slope* between 0.37 and 0.88 (Fig. 6). This shows that power cones can bend to form logarithmic spirals in an analogous manner to that first proposed by Wren [19] for cones (a specific power cone; Fig. 1d). It also establishes that not all shell shapes can be generated by the existing model of development [7]. In order to accommodate such shapes, the Raup [7] model must have the *Slope* parameter added, such that *Radius* = *c θ*^{Slope}. In the first description of the shell growth model, Raup [44] assumes that ‘the rate of expansion of the generating curve is approximately constant’, i.e. *Slope* = 1, and so this parameter was not included in his model. In contrast, Thompson [6] suggested that the growth may not be constant in some shells but in fact vary ‘in accordance with some simple law’, and Ackerly [45] showed that for some shells there is an allometric component to the change in radius. Our power cascade model accounts for this important feature of growth.

The long axis of each tooth grows as a logarithmic spiral [6, 46], which can be seen in an extreme form in the curved upper tusks of the babirusa *Babyrousa celebensis*. However, we find that the *Slope* of these tusks (0.25) is considerably less than 1, and therefore, they are not conical (Fig. 5): their high *Aspect Ratio* can make them appear more conical. This means that teeth cannot be modelled by the Raup [7] shell equation. The radius of the circle must change logarithmically with the angle of rotation to form a power cone, rather than a straight-sided cone with *Slope* = 1.

### A general model of growth for horns, claws, spines, beaks, and thorns

Thompson [6] expected that pointed and spiral structures such as horns and claws would follow the same growth pattern as shells, which has been used to model some horn-like structures [47]. If horns grow according to the shell model and are spiralled cones, then their *Slope* parameter will be 1. From measurements of bony horn cores from vertebrates including mammals, non-avian dinosaurs (referred to here as dinosaurs) and reptiles, we have found that *Log Distance-Radius* plots are linear and the *Slope* is typically between 0.4 and 0.8 (Fig. 6; Additional file 1: Figure S8), demonstrating that they do follow the power cascade but are not growing according to the original conical shell model.

Other structures throughout vertebrates also show power cascade growth, including mammal, bird and dinosaur claw and hoof bones (unguals), the bony beaks of birds and dinosaurs, and spines of fish (Fig. 6). Outside vertebrates, the power cascade model is also followed in arthropod fangs and cephalopod beaks. Beyond animals, it is found in thorns and prickles in plants (Fig. 6).

The rose prickle (generally called a thorn) represents an interesting exception. While the concave shape of a mature prickle does not follow the power cascade prediction, a young prickle does (Additional file 1: Figure S9). It appears that the prickle is initially generated following the power cascade growth with *Slope* = 0.6, but then as the stem to which it is attached grows, the base of the prickle is stretched along the long axis of the branch. The result is the typical concave shape of a rose prickle, where only the top half follows the power cascade, not the basal half that has been stretched (Additional file 1: Figure S9). In general, it appears that deviations from the power cascade are more likely in pointed structures controlled by multiple growth processes.

The power cascade model can be added to the logarithmic spiral model to generate a ‘power spiral’ that can simulate realistic shapes of pointed, curved structures (Additional file 1: Figure S7c). Figure 7 shows some comparisons between real teeth and power spiral models, using both circular cross-sections that would be generated in surfaces of revolution and other cross-sectional shapes (elliptical, lenticular, truncated circle) implemented in a Mathematica notebook (v. 12.0, Wolfram Research Inc., Champaign, IL) available in the Supplementary Information (see also Additional file 1: Figure S10).

The majority of the structures that are closely emulated by the power cascade grow from tip to base, including teeth, horns, thorns, and prickles. These shapes are presumably formed as each addition of material increases the radius by a constant proportion for a proportional increase in length. For example, bovid horns grow from tip to base, increasing in radius down the horn, and they generally follow the power cascade model. In contrast, cervid antlers grow from base to tip, with the growing antler branching, and the antler points being the last structures to form. Despite this directional difference in growth—and the antler starting from a wider base and narrowing towards the tip—antler points also follow the power cascade (Fig. 6). This shows that the proportional growth pattern can act both when increasing the radius of the structure as it cascades downwards from the tip to the base, and also when decreasing the radius to cascade upwards from base to tip. It appears that only the direction of radial growth differs between these two scenarios.

Since many of the structures examined here (including teeth and claws) are used to penetrate food or other materials, it may be argued that selection to maximise penetration ability or structural strength is the cause of the underlying similarity in shape as described by the power cascade model. However, many structures that are not for penetration (such as shells, rounded teeth or backward-curving horns) still follow the power cascade pattern. Given that structures that conform to the power cone can vary from sharp and long to blunt and short, we argue that the most parsimonious explanation for the model fit is an underlying biophysical or developmental mechanism rather than strong selection for shapes that coincidentally fit a power cascade-like pattern. The power cascade generates a base set of allowed variations (Fig. 5), and selection chooses from among these shapes, as occurs with the selection of relative tooth size in hominins according to the inhibitory cascade [36].

### Mechanism and generality of power cascade

The log-log linear pattern of the power cascade can be compared with allometric plots of the relative sizes of body components during growth [20], such as head size versus body size in humans. A linear allometric relationship is produced when two components grow exponentially at different rates. The power cascade relationship shows that there is an allometric relationship within the same structure due to differential growth rates of *Radius* and *Distance*.

We can demonstrate this growth process by examining power function growth in *Distance* and *Radius* over time (Fig. 8a): *Distance* ∝ *Time*^{rD} and *Radius* ∝ *Time*^{rR}, where *rD* and *rR* are the growth rates for *Distance* and *Radius*, respectively. Power function growth is very common in biology, including for human height [48] and elephant tusks (Fig. 4b). When both axes of the growth over time curves are logged, the plot log(*Distance*) vs log(*Time*) is linear with slope *rD* (similarly for *Radius* and *rR*; Fig. 8b). By solving the log(*Distance*) equation for log(*Time*) and substituting into the log(*Radius*) equation, the relationship between log(*Distance*) and log(*Radius*) through time becomes apparent (Fig. 8c). If *rD* and *rR* are equal, then *Radius* increases linearly with *Distance* (Fig. 8d) and produces a conical shape (with *Log Distance-Radius* power cascade *Slope* of 1). If instead the rates of growth of *Distance* and *Radius* differ (e.g. *rD* = 2*rR*), then the log-log growth over time trajectories will not be parallel (Fig. 8e-f), and the result will be a power cone such as a paraboloid (Fig. 8h). The *Log Distance-Radius* power cascade *Slope* of such a structure will be *rR*/*rD* = 0.5 (Fig. 8g).

Therefore, the power cascade is an expression of allometry as a shape: power cones show unequal power growth within the same structure, or ‘constant differential growth-ratios’ in the terminology of Huxley [20]. The cone is produced through isometric growth between *Distance* and *Radius*, while a power cone results from allometric growth (*rD* ≠ *rR*). The same shapes can also be generated through exponential (as opposed to power) growth of body parts, although this is not commonly found in organisms. Constant differential growth of the two sides of a structure must generate a logarithmic spiral ([20]; Fig. 1b). In the same manner, differential power growth of *Distance* and *Radius* must generate a power cone (Fig. 1c). Both mechanisms could operate at the same time, forming a power cone on a logarithmic spiral, or a power spiral (Fig. 1d).

The power cascade, and likewise the logarithmic spiral, can be seen as ‘dynamical patterning modules’ [49] that generate patterns and structures in metazoans and plants. Despite over three centuries of research [19], the specific molecules driving logarithmic spiral growth are not known (although recent work has begun to reveal some components in gastropod shells [50]). Likewise, the identity of signalling molecules and genes that influence the differential growth of the power cascade very likely must vary widely across animals and plants. Here we show that common growth patterns in animals and plants generate power cones. These shapes may be considered the default family of shapes for pointed structures, meaning they are more likely to independently evolve multiple times and will be a likely source of homoplasy in evolution.