Preferential positioning and positional transitions of localized Turing patterns on curved surfaces.

Abstract

Turing patterns, arising from a reaction-diffusion system, play a crucial role in comprehending diverse patterns in nature. These patterns often manifest new features on curved surfaces that are absent on flat surfaces. With this motivation, the present work systematically analyzes the phenomena of preferential positioning and positional transitions of localized Turing patterns or a single spot on curved surfaces, emphasizing the interplay between the surface shape and the parameters in the reaction-diffusion system. In particular, we examine analytically and numerically single-spot formation in the Turing system evolving with Schnakenberg kinetics on nonaxisymmetric ellipsoids to illustrate the remarkable effect of surface shapes and reaction-diffusion parameters in tuning the positions and their transitions on curved surfaces. The work attempts to provide a generic understanding of the interplay between reaction-diffusion models and the system geometry crucial in deciding specific positions and positional transitions of localized Turing patterns on curved surfaces. The present analysis may provide useful insights into the role of cellular surface shape on various symmetry-breaking mechanisms in cellular processes.