Growth and patterning in vertebrate limb developmentA timescale perspective on skeletal specification

1The vertebrate limb provides a powerful system to study how growth and molecular signaling interact to shape complex skeletal patterns. However, 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-Processing-Propagation (GPP) framework, a reaction-diffusion system within a growing domain, in which the relative contribution of growth, processing (reaction) and propagation (diffusion) is modulated through two non-dimensional parameters, informed by experimental morphogen maps. This formulation normalizes the reaction-diffusion equation relative to growth, enabling investigation of how different spatiotemporal regimes of growth, processing and propagation interact to produce whole limb patterning. The GPP framework captures the progressive formation of limb segments and digit patterns by varying the relative contributions of reaction, diffusion, and growth to the pattern. Our models 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 (hands) the reaction and diffusion contributions are much greater than the contribution of growth to formation of the digits. A single framework predicts whole-limb skeletal patterns in both mice and axolotls, despite their morphological differences, highlighting its potential to explore conserved and divergent features of limb development from an evolutionary perspective through a unified mechanism across species. O_TEXTBOXSignificanceUnderstanding limb skeletal patterning is a central question in developmental biology. Numerical models provide a means to explore this process. We introduce the Growth-Processing-Propagation (GPP) framework, which integrates tissue expansion (growth), gene regulation and cellular specification (processing), and signaling spread (propagation). By normalizing patterning dynamics relative to growth, the framework reveals how the relative contributions of growth, processing, and propagation vary across the limb, influencing the timing and positioning of skeletal elements. The model successfully predicts whole-limb skeletal patterning in both mice and axolotls, supporting the idea of a shared underlying mechanism. This cross-species framework offers a versatile computational tool for studying fundamental processes in morphogenesis, with potential applications in developmental biology, and evolutionary studies. C_TEXTBOX