Turing patterns on non-fluctuating surfaces under mechanical stresses

Abstract

This paper presents a numerical study of Turing patterns (TPs) governed by reaction diffusion equations for the activator $u$ and the inhibitor $v$ on two- and three-dimensional lattices without vertex fluctuations. In this framework, $u$ and $v$ are fixed at discrete spatial locations, as pigment cells on zebrafish skin or shell patterns. Mechanical effects are incorporated through the Finsler geometry modeling formulation, which introduces an internal degree of freedom, $\vec{\ddot{I}}„$, representing the direction of mechanical stress. A tensile-stress formula based on the Gaussian bond potential is shown to be well defined on non-fluctuating lattices, enabling the entropy associated with stress relaxation to be evaluated in a manner analogous to that on fluctuating surfaces. The results indicate that biological TPs respond to external mechanical forces in much the same way as TPs on fluctuating membranes. Simulation codes are provided in the Supplementary Material.

25 pages, 11 figures, with supplementary PDF files and Fortran f90 files

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