Development of whole-limb skeletal patterning through the coordination of growth and self-organization models.

The vertebrate limb provides an interesting system to study how tissue growth and molecular signaling interact to shape complex skeletal patterns. How these processes are coordinated across space and time is not fully understood. This study introduces a computational tool to examine how growth interacts with positional cues and self-organizing patterning mechanisms to shape skeletal structures in both mice and axolotl limbs. We developed the Growth-Reaction-Diffusion (GRD) framework, a reaction-diffusion system within a growing domain, where reaction represents the regulation of patterning cues and diffusion captures their spatial propagation. The relative contribution of growth, reaction and diffusion is modulated through two non-dimensional parameters, whose spatial variation is informed by positional cues derived from experimental morphogen maps. This formulation normalizes the reaction-diffusion equation relative to growth, enabling investigation of how different spatiotemporal regimes of growth interact with reaction and diffusion to produce whole limb patterning. The GRD framework captures the progressive formation of all limb segments: the humerus, radius/ulna, and the digits patterns. Our simulations indicate that in the proximal region (humerus, radius/ulna) the contributions of growth, reaction and diffusion are equally important to patterning, but in the distal elements (digits) the reaction and diffusion contributions are much greater than the contribution of growth to the formation of the digits. Through a single framework, we simulate the whole-limb skeletal patterns in both mice and axolotls, despite their morphological differences. These results highlight the model's potential to explore conserved and divergent features of limb development from an evolutionary perspective through a unified mechanism across species.