Biological shapes, transformations and the ‘laws of growth’. (A) Geometric shape changes during hyacinth leaf growth, which follows a very specific set of spatial rules, e.g. following particular ratios of radial and tangential growth velocities (Fig. 359 from Thompson, 1917a). (B) Geometric morphometric space describing shape alterations in hyena skulls during development from juvenile to adult (Tanner et al., 2010). (C) The ‘laws of growth' interpreted as a broad set of developmental mechanisms translating genetic information and physical forces of the environment into individual biological shapes and contributing to their diversity. Images in C created by Keith Chan; copyright Alexmit (www.fotosearch.com); animal images are illustrations from Seton (1898).

Geometric relations among the beaks of Darwin's finches. (Left) Lateral profiles of beaks of Darwin's finches. (Centre) Group structure under scaling transformations focusing on the upper beak profile: untransformed shapes and shapes collapsed onto a common shape via scaling transformations. (Right) Collapse of all group shapes onto a common shape via a composition of shear and scaling transformations suggesting a two-tier morphological variation (adapted from Campàs et al., 2010).

Evolution of beak shape diversity in songbirds and developmental mechanisms for beak morphogenesis. (A) Exceptionally high diversity of beak shapes in Tholospiza (Darwin's finches on Galápagos and endemic Caribbean relatives, marked with an asterisk) and Hawaiian honeycreepers as compared with the much more limited morphological diversity in Galápagos mockingbirds (and a single Caribbean endemic Bahama mockingbird Mimus gundlachii, also marked with an asterisk). Tholospiza and honeycreepers show much higher degrees of sympatric distribution (related species co-inhabiting the same geographic areas), while Galápagos mockingbirds (and other Galápagos land birds) display allopatric speciation with just four morphologically similar endemic species for the entire Galápagos archipelago. (B) Highly modular multigenic beak developmental programs are found in Darwin's finches and close Tholospiza relatives but not in the more basal species (modified after Mallarino et al., 2012). Darwin's finches and the closely related Caribbean species Loxigilla noctis deploy multiple regulatory genes to control both prenasal cartilage and premaxillary bone parts of the developing beak, whereas the more basal species use a much more limited set of genes to regulate the shape of premaxillary bone (genes expected to be deployed based on Galápagos species are not expressed). The red arrow in the Tholospiza phylogeny points at the phylogenetic position within this group beyond which higher morphological diversification is observed and needs to be explained (phylogeny from Burns et al., 2002; modified after Mallarino et al., 2012). (C) Within Darwin's finches, species with deep beaks have strong expression of Bmp4, TGFβIIr, β-catenin and Dkk3, whereas expression of CaM is upregulated in species with elongated beaks, such as Geospiza scandens and Geospiza conirostris. BMP4 and CaM act independently to alter the growth of the prenasal cartilage and TGFβIIr, β-catenin, and Dkk3 regulate the premaxillary bone. (D) Darwin's finches and the closely related Caribbean L. noctis share beak developmental programs, which are highly modular in terms of both tissue and molecular composition, while the more basal species Loxigilla portoricensis and Loxigilla violacea have simpler beak developmental patterns and represent a simpler starting condition (also found in outgroup species such as zebra finches). Illustrations of the head of each bird species were derived from del Hoyo et al. (2017) with permission from Elisa Badia. Illustrations of Hawaiian honeycreepers are by H. Douglas Pratt (Pratt, 2005).

Principles of beak development in songbirds. (A) Snapshots of growing beak for embryonic day (E) 5-9, showing developing beak outlines (black), the size of the growth zone (red), its centroid (blue) and the relevant length scale for shaping the upper beak profile (yellow). All measures of the growth zone are derived from midsagittal cross-sections of zebra finch embryo beaks, stained to show cell nuclei (blue) and dividing cells (green). Areas with a high density of dividing cells are defined as the growth zone (red outline). (B) The final conical shape of the beak is given by an ‘envelope' of the growth zone observed during developmental time, which decays at a constant rate until it shrinks to size zero generating a tip (adapted and modified from Fritz et al., 2014). All beak shapes are well fitted by equations, as shown, that describe beak shape and the profile [p_u(x) and p_l(x)] generated by a fixed growth law and growth zone decay, where φ is a vector parameterizing the shape and all other free parameters in the problem. The midsagittal sections of songbird beaks are all calculated to be conic sections.

High level of morphological disparity in avian skull shapes is associated with increased modularity. (A) Microcomputed tomography (µCT) scan of a bird skull showing geometric morphometric landmarks (above, red dots on skull scan) and defined modules (beneath, coloured lines). (B) The phylogeny of bird species used for mapping shape data by squared-change parsimony. Representative, more basal, species were used as outgroups. (C) Morphospace occupied by skull shapes. The polygons show parts of the morphospace occupied by different groups revealing that Hawaiian honeycreepers (polygon shaded blue) have some of the most diversified and divergent skulls of all species studied. (D) Darwin's finches show strong covariation between the shape of the whole skull and several skull modules (red thick lines), suggesting strong integration within the skull. By contrast, Hawaiian honeycreepers show strong covariation only between the shape of the whole skull and that of the upper beak (blue thick line), indicating much greater flexibility of evolutionary change. Such a unique pattern of morphological integration in their skulls is likely to reflect an altered and much higher state of developmental modularity within the skull, which contributed to the outstanding level of disparity of skull morphology observed in Hawaiian honeycreepers. Cranial modules in D are colour-coded in the same way as in A. All images adapted and modified from Tokita et al. (2017).