Abstract

Pattern formation plays a pivotal role in morphogenesis as it leads to the emergence of structures that later support function. Alan Turing's theory stands out as a prominent explanation for morphogen patterning. From a nearly uniform initial state, and through suitable nonlinear reaction-diffusion interactions between morphogens, stable spatial patterns are established. Turing patterns have been used to investigate the formation of skeletal structures such as digit formation in mice. Recent research suggests that while a Turing-like mechanism may lead to periodic pattern formation, mechanical cues also drive the formation of organizing centres, revealing a coupling between digit specification and elongation. On the other hand, from a mathematical perspective, domain growth is known to confer robustness to Turing pattern formation. We use morphological data from developing axolotl salamander limb buds to explore how domain size, shape and growth rate impact limb skeletal patterning through a data- informed computational model. Limb bud growth is simulated using a finite element model with a moving boundary condition. At each growth stage, corresponding to an experimental measure, an inverse elastic-growth optimization algorithm is applied to determine the new domain mesh and tissue growth distribution. In our Turing model, one species represents a group of molecules that instructs cells to differentiate into cartilage, while the other species represents a group of molecules that signal cells not to differentiate into cartilage. Therefore, the patterns that we are targeting are the ones observed during skeletal limb development. A sensitivity analysis is conducted to understand the impact of various parameters in the reaction-diffusion system, the initial and boundary conditions of the computational model, and the spatial scaling as well as the growth rate on the predicted patterns and bifurcations. Our analysis reveals a limited range of patterns (dots, single stripes, double stripes, and intermediate patterns). We also make use of stability analysis techniques to associate the admissible wave modes of the reaction-diffusion system with the predicted patterns in the numerical simulations. Through these explorations, we seek to better understand the stability of the Turing patterns within the process of skeletal limb development.