The effects of a coherent feed-forward loop on Turing pattern formation in the Gierer-Meinhardt model

Turing’s reaction–diffusion framework provides a mathematical basis for diffusion-driven pattern formation in interacting systems. An early biological application of the Turing principles is the Gierer and Meinhardt (GM) model, which was developed to describe hydra head regeneration involving two species: an activator and inhibitor. However, biological networks are far beyond simplicity because they contain numerous interconnected components. In this work, we incorporate a coherent feed-forward loop (FFL) into the GM model by assuming that the activation of the inhibitor is mediated through this motif to better capture the complexity of biological systems. Our results indicate that the FFL reshapes the underlying Turing region in two distinct ways. This not only reduces the Turing instability region but also induces spatial patterns, even when the GM model is stable. This reveals that the network topology, rather than the parameters alone, can reshape the onset and structure of the diffusion-driven instability. Our findings suggest that considering the network topology is crucial for unraveling the behavior of real systems.