Pattern Propagation Driven by Surface Curvature

Abstract

Pattern dynamics on curved surfaces are found everywhere in nature. The geometry of surfaces have been shown to influence dynamics and play a functional role, yet a comprehensive understanding is still elusive. Here, we report for the first time that a static Turing pattern on a flat surface can propagate on a curved surface, as opposed to previous studies, where the pattern is presupposed to be static irrespective of the surface geometry. To understand such significant changes on curved surfaces, we investigate reaction-diffusion systems on axisymmetric curved surfaces. Numerical and theoretical analyses reveal that both the symmetries of the surface and pattern participate in the initiation of pattern propagation. This study provides a novel and generic mechanism of pattern propagation that is caused by surface curvature, as well as insights into the general role of surface geometry.Pattern formation and dynamics on curved surfaces are ubiquitous, particularly in biological systems [1-3]. Recent studies reveal the functional roles of the topology and geometry of surfaces in pattern formation [4-7]. For example, defect dynamics on closed curved surfaces are constrained by the Poincaré-Hopf theorem and have been investigated in liquid crystals [8], flocking [9], and active nematics [10]. Such defects in cortical actin fibers serve as organization centers in the morphogenesis of hydra regeneration [4]. Molecules such as Bin/Amphiphysin/Rvs domain proteins sense the curvature of a cellular membrane and regulate the cellular shape [11]. Cellular migration is guided by the curvature of a substrate, which is a response known as "curvotaxis" [5]. It was theoretically shown that surface curvature can induce splitting [6] and rectification [12] of excitable waves. Rectification by curved surfaces has been reported in the collective motion of self-propelled particles [13].However, a comprehensive and general understanding of the effect of surface geometry on pattern dynamics and its functional role remains elusive.Among pattern formation, Turing patterns are a prominent example arising from reactiondiffusion systems [14][15][16][17]. Turing patterns on curved surfaces [18][19][20], such as spheres [14,[21][22][23][24][25], hemispheres [26,27], toruses [25,28], ellipsoids [28,29], and deformed (but axisymmetric) cylinders and spheres [30] have been investigated previously. These studies revealed how the Turing instability condition changes from the flat plane case and how the position of the pattern is modulated by the inhomogeneity of the surface curvature [19], which is referred to as "pinning" by Frank et al. [30] It is noteworthy that in these studies, the Turing pat-