Turing pattern prediction in three-dimensional domains: the role of initial conditions and growth

Reaction-diffusion systems have been widely used to model pattern formation in biological systems. However, the emergence of Turing patterns in three-dimensional (3D) domains remains relatively unexplored. A few studies on this topic have shown that extending pattern formation from 2D to 3D is not straightforward. Linear stability analysis, which is commonly used to associate admissible wave modes with predicted patterns in 1D and 2D, has yet to be applied in 3D. We have used this approach, together with finite element modeling of a Turing system with Schnakenberg kinetics, to investigate the effects of initial conditions and growing domains on the competition between admissible modes in 3D Turing pattern emergence. Our results reveal that non-random initial conditions on the activator play a stronger role than those on the substrate. We also observe a path dependency of the evolving pattern within a growing domain. Our findings shed new light on the mechanisms ensuring reliable pattern formation in 3D domains and have important implications for the development of more robust models of morphogen patterning in developmental processes. ### Competing Interest Statement The authors have declared no competing interest.