Role of curvature and domain shape on Turing patterns

Nampoothiri, Sankaran; Medhi, Amal

doi:10.48550/arxiv.1705.02119

Cited by 4 publications

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“…Owing to the importance of understanding RD on complex geometries, some of the previous studies have analyzed the effect of geometry on RD systems. [13][14][15][16][17][18][19][20][21][22][23][24] For example, the parr-marks formation on fish skins are studied in the work 15 where the shape of the skin is modeled as a growing elliptic cylinder which indicates the importance of surface shape in understanding the patterns observed in nature. Some of the recent work about the study on nucleation of RD waves on curved surfaces, 16 spiral waves on curved surfaces 25 and the effect of geometry on Min-protein dynamics 26,27 again suggest the importance of surface shape.…”

Section: Introductionmentioning

confidence: 99%

Preferential localization of a single spot in reaction–diffusion systems on non-spherical surfaces

Nampoothiri

2023

Soft Matter

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Section: Introductionmentioning

confidence: 99%

Preferential localization of a single spot in reaction–diffusion systems on non-spherical surfaces

Nampoothiri

2023

Soft Matter

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“…Owing to the importance of understanding RD on complex geometries, some of the previous studies have analyzed the effect of geometry in RD systems [11][12][13][14][15][16][17][18][19][20]. For example, the parr-marks formation on fish skins are studied in the work [12] where shape of the skin is modeled as growing elliptic cylinder which indicates the importance of surface shape in understanding the patterns observed in nature.…”

Section: Introductionmentioning

confidence: 99%

Effect of geometry on the positioning of a single spot in reaction-diffusion systems

Nampoothiri

2019

Preprint

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We consider the formation of a single spot (localized solution) in reaction-diffusion (RD) equation on a curved manifold. Specifically, we study the direction (alignment) of the normal to interface between maxima and minima of concentration in the steady-state on a prolate and on an oblate ellipsoid. We further analyse the effect of shape asymmetry on l = 1 eigenmode of the sphere by assuming a small deformation from the spherical geometry. Our analysis shows that the eigenfunction corresponding to highest eigenvalue align along the symmetry axis for a prolate ellipsoid, and perpendicular to the symmetry axis for an oblate ellipsoid. Finally, we compare the direction of variation of the most unstable mode (eigenfunction with highest growth rate) in the system obtained by assuming a small deformation from the sphere and the alignment of interface normal obtain from the numerical simulations.

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“…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-…”

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Pattern Propagation Driven by Surface Curvature

Nishide,

Ishihara

2022

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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-

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Role of curvature and domain shape on Turing patterns

Cited by 4 publications

References 0 publications

Preferential localization of a single spot in reaction–diffusion systems on non-spherical surfaces

Preferential localization of a single spot in reaction–diffusion systems on non-spherical surfaces

Effect of geometry on the positioning of a single spot in reaction-diffusion systems

Pattern Propagation Driven by Surface Curvature

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