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 by which β -catenin/Wnt–Dkk interactions give rise to robust spatial patterning 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 within the activator–inhibitor framework, a model that directly integrates experimentally identified molecular components has been lacking. Here, we introduce a mathematical model incorporating both Dkk molecules and their experimentally established interactions with Wnt signalling. Numerical simulations and analytical results show that the Wnt–Dkk network alone is sufficient to drive de novo body-axis formation across a broad parameter range. The model provides a biologically grounded realisation of the general local activation–long-range inhibition (LALI) principle, in which effective local activation emerges from mutual inhibition rather than molecular self-activation. In contrast to previous Hydra models, it explicitly links experimentally characterised Wnt–Dkk interactions to pattern formation, accounts for the experimentally observed role of injury-induced activation, and exhibits robust behaviour under perturbations.