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.