How Dickkopf molecules and Wnt/β-catenin interplay to self-organise the Hydra body axis

The antagonising interplay between canonical Wnt signalling and Dickkopf (Dkk) molecules has been identified in various processes involved in tissue organisation, such as stem cell differentiation and body-axis formation. Disruption of the interplay between these molecules is related to several diseases in humans. However, the detailed molecular mechanisms of the β -catenin/Wnt-Dkk interplay leading to robust formation of the body axis remain elusive. Although the β -catenin/Wnt signalling system has been shown in the pre-bilaterian model organism Hydra to interact with two ancestral Dkks (HyDkk1/2/4-A and -C) to self-organise and regenerate the body axis, the observed Dkk expression patterns do not match any current pattern-formation theory, such as the famous activator-inhibitor model. To explore the function of Dkk in Hydra patterning process, we propose a new mathematical model which accounts for the two Dkks in interplay with HyWnt3/ β -catenin. Using a systematic numerical study, we demonstrate that the chosen set of interactions is sufficient to explain de novo body-axis gradient formation in Hydra . The presented mutual inhibition model goes beyond the classical activator-inhibitor model and shows that a molecular mechanism based on mutual inhibition may replace the local activation/long-range inhibition loop. The new model is validated using a range of perturbation experiments. It resolves several contradictions between previous models and experimental data, and provides an explanation for the interplay between injury response and pattern formation. ### Competing Interest Statement The authors have declared no competing interest.