Mutual Inhibition Model of pattern formation: The role of Wnt-Dickkopf interactions in driving Hydra body axis formation

The antagonistic interplay between canonical Wnt signalling and Dickkopf (Dkk) proteins is fundamental to tissue organisation, including stem cell differentiation and body-axis formation. Disruptions in this interaction are linked to various human diseases, yet the mechanisms enabling robust body-axis formation through β -catenin/Wnt–Dkk interactions remain unclear. A key model system for Wnt-driven pattern formation is the pre-bilaterian organism Hydra , where two ancestral Dkk proteins interact with Wnt signalling to self-organise the body axis. While Hydra patterning has been extensively studied using the activator–inhibitor framework, a model integrating experimentally identified molecules has been lacking. Here, we introduce a mathematical model that incorporates both Dkks and their experimentally observed interactions with Wnt signalling. Numerical and analytical studies show that this network alone is sufficient to drive de novo body-axis formation across a broad parameter range. Our mutual inhibition model provides a biologically grounded realization of the general local-activation/long-range-inhibition (LALI) principle of de novo pattern formation, offering a mechanistic explanation for the observed Dkk and Wnt expression patterns under various conditions. Unlike previous models, it is directly grounded in experimental data, links injury response to pattern formation, and remains robust against perturbations. Author Summary How organisms form and regenerate complex body structures is a fundamental question in biology. In the freshwater animal Hydra , which can regenerate its entire body from a small tissue fragment, a molecular signalling system involving Wnt proteins and their inhibitors, the Dickkopf (Dkk) family, plays a central role in organising the body axis. While these molecules are known to interact, their exact roles and how they collectively shape large-scale patterns have remained unclear—especially since their activity does not fully align with established pattern formation models. In this study, we present a new mathematical model that captures the observed interactions between Wnt and two Dkk molecules in Hydra . We show that a mechanism based on mutual inhibition—rather than the traditional interplay between activator and inhibitor molecules—can explain the emergence of a stable body axis and the results of various perturbation experiments. Our work offers new insights into the design principles of biological pattern formation and emphasizes the importance of exploring alternative mechanisms beyond classical theories.