Turing patterns on non-fluctuating surfaces under mechanical stresses

This paper presents a numerical investigation of Turing patterns (TPs) utilizing the reaction-diffusion equation for the activator $u$ and the inhibitor $v$ on two- and three-dimensional lattices, discarding vertex fluctuations. The absence of vertex fluctuations means the absence of positional movement of $u$ and $v$. Consequently, $u$ and $v$ have values at spatially discrete points, such as the pigment cells in zebrafish and sea shells. Furthermore, the mechanical property is implemented through the Finsler geometry modeling technique. This technique incorporates the internal degree of freedom $\vec{\tau}$, corresponding to the direction of mechanical stress. Additionally, a stress formula based on Gaussian bond potential is shown to be well-defined on the non-fluctuating lattices, and therefore, it enables the calculation of entropy for capturing the stress relaxation phenomenon in a manner analogous to that on fluctuating surfaces. The results of the study indicate that these biological patterns also exhibit responses to external mechanical forces similar to TPs on fluctuating membranes.