Particle-Level simulations using diffusiophoresis and cellular automata to create dynamic Turing patterns

Turing patterns, emerging from short-range activation and long-range inhibition in reaction-diffusion systems, are commonly observed in biological systems. Recent research has shown that incorporating diffusiophoretic transport can enhance Turing theory, enabling a degree of control that allows for robust modeling of biological patterns. However, such chemical-mechanical mechanisms usually yield static structures, failing to capture the dynamic behaviors typical of biological systems. Cellular automata, on the other hand, are dynamic discrete models capable of reproducing complex evolutionary and growth-like behaviors given an initial state. Traditionally, these two classes of models have been treated separately. Here, we introduce a novel framework where the cells migrate through diffusiophoresis, in response to the biomolecular signals, and then evolve via cellular automata. Our large-scale Eulerian-Lagrangian simulations reveal that this unified framework generates intriguing evolving structures, whose characteristics are controllable by both the reaction-diffusion model and cellular automata principles. Additionally, we quantify how cell concentration and size limitations impact pattern formation and its subsequent evolution.