Pattern formation on a growing oblate spheroid. an application to adult sea urchin development

Abstract

<p style='text-indent:20px;'>In this study, the formation of the adult sea urchin shape is rationalized within the Turing's theory paradigm. The emergence of protrusions from the expanding underlying surface is described through a reaction-diffusion model with Gray-Scott kinetics on a growing oblate spheroid. The case of slow exponential isotropic growth is considered. The model is first studied in terms of the spatially homogenous equilibria and of the bifurcations involved. Turing diffusion-driven instability is shown to occur and the impact of the slow exponential growth on the resulting Turing regions adequately discussed. Numerical investigations validate the theoretical results showing that the combination between an inhibitor and an activator can result in a distribution of spot concentrations that underlies the development of ambulacral tentacles in the sea urchin's adult stage. Our findings pave the way for a model-driven experimentation that could improve the current biological understanding of the gene control networks involved in patterning.</p>

Virtual element method for elliptic bulk‐surface PDEs in three space dimensions

“…More generally, given a number $$$ d\in \mathbb{N} $$$ of space dimensions, a system of BSPDEs comprises of $$$ m\in \mathbb{N} $$$ PDEs posed in the bulk $$$ \Omega \subset {\mathbb{R}}^d $$$, coupled with $$$ n\in \mathbb{N} $$$ PDEs posed on the surface $$$ \Gamma := \mathrm{\partial \Omega } $$$ through (i) either linear or non‐linear coupling conditions [51], (ii) linear or nonlinear coupled kinetics [37] and possibly (iii) cross‐diffusion [38]. The quickly growing interest toward stationary or time‐dependent BSPDEs arises from the numerous applications of such PDE problems in different areas, such as cellular biological systems [27, 34, 50, 55], fluid dynamics [18, 22, 48], plant biology [56], biological patterning [36, 47], and electrochemistry [46] among many other applications.…”

“…More generally, given a number 𝑑 ∈ N of space dimensions, a system of BSPDEs comprises of m ∈ N PDEs posed in the bulk Ω ⊂ R 𝑑 , coupled with n ∈ N PDEs posed on the surface Γ ∶= 𝜕Ω through (i) either linear or non-linear coupling conditions [51], (ii) linear or nonlinear coupled kinetics [37] and possibly (iii) cross-diffusion [38]. The quickly growing interest toward stationary or time-dependent BSPDEs arises from the numerous applications of such PDE problems in different areas, such as cellular biological systems [27,34,50,55], fluid dynamics [18,22,48], plant biology [56], biological patterning [36,47], and electrochemistry [46] among many other applications. Among the various state-of-the art numerical methods for the spatial discretization of BSPDEs existing in the literature we mention bulk-surface finite elements (BSFEM) [33,45,51,52], trace finite elements [44], cut finite elements [22], discontinuous Galerkin methods [26], kernel collocation method [25], and closest point method [49].…”