Cell-local -catenin/Wnt dynamics.

Eq. (1) describes the dynamics of the cell-local canonical Wnt activity , representing intracellular -catenin/Tcf signalling and expression of canonical HyWnt genes. The production term is promoted by the source density ([S]). The first two inhibitory denominators capture repression by HyDkk1/2/4-A and HyDkk1/2/4-C, consistent with experimental evidence for mutual inhibition between Wnt and Dkk expression [26,27]. The third denominator represents local self-limiting saturation, reflecting constraints on gene expression and protein synthesis, while the final term accounts for degradation of . Additional experimental support is provided in [23–25,33,38,39].

Dkk1/2/4-A dynamics.

Eq. (2) describes the dynamics of HyDkk1/2/4-A ([A]), an inhibitor of canonical Wnt signalling. The first term represents weak diffusion of [A] along the tissue surface. The production term includes basal expression repressed by local Wnt activity (), consistent with experimental evidence for mutual inhibition between HyDkk1/2/4-A and canonical Wnt signalling [26,27]. The final term accounts for degradation of [A]. Supporting evidence is given in [26,27,32].

Diffusible Wnt dynamics.

Eq. (3) describes the dynamics of diffusible Wnt activity [W], representing the extracellular pool of secreted Wnt ligands acting over longer distances. The first term accounts for diffusion along the tissue surface. Production of [W] depends on both the source density ([S]) and local Wnt activity (), where [S] captures permissive conditions arising from slower regulatory layers (such as epigenetic states, chromatin accessibility, or other long-term determinants of cellular identity), and provides the transcriptional drive. The final term accounts for degradation of [W]. Experimental support is provided in [23,25,33,38,39]. While the quantitative propagation properties of Wnt in Hydra remain experimentally unconstrained, assuming moderate diffusivity provides a plausible approximation; the resulting patterns are robust to variations in the diffusion strength (Fig G in S1 Appendix, panels C–D).

Dkk1/2/4-C dynamics.

Eq. (4) describes the dynamics of HyDkk1/2/4-C ([C]), a second Wnt antagonist with distinct regulation compared to [A]. The first term represents weak diffusion along the tissue surface. Production of [C] is positively regulated by diffusible Wnt activity ([W]) and repressed by local Wnt activity (), consistent with experimental observations showing upregulation in regions exposed to secreted Wnt and downregulation in the head region where canonical Wnt/-catenin activity is high [27]. The final term accounts for degradation of [C]. For experimental evidence we refer to [27,32].

Source density dynamics.

Eq. (5) describes the dynamics of the source density ([S]), encoding long-term positional information along the oral–aboral axis of Hydra. The first term represents slow diffusion, modelling gradual redistribution through local cell interactions. Production of [S] depends on local Wnt activity (), reflecting the experimentally observed induction of long-lasting tissue competence by sustained -catenin/Tcf signalling [16]. The decay term captures the gradual relaxation of positional information over several days, consistent with regeneration and grafting experiments [24,34,40]. Conceptually, [S] represents a coarse-grained positional memory that determines whether a region is permissive for organiser formation. Accordingly, [S] evolves on a slower timescale than the Wnt–Dkk signalling system. Experimental evidence is provided in [14,24,32,34,39,40]; further details are given in S1 Appendix.

Tentacle system.

The tentacle subsystem is represented by an activator–inhibitor module that receives positive input from the source density ([S]) and negative input from local Wnt activity (). This captures experimental observations that tentacle primordia form preferentially in regions of intermediate head-forming competence, where [S] is high but -catenin/Wnt locally suppresses this system [45]. The resulting field describes the periodic activation of tentacle-specific genes along the upper body column, consistent with observed tentacle spacing and regenerative behaviour.

Foot system.

The foot subsystem is modelled analogously as an activator–inhibitor pair regulated by [S], but independent of the Wnt–Dkk head system. It represents the basal organiser region, stabilising the aboral pole. The foot activator is enhanced in areas of low [S], supporting the formation of a robust basal identity. Both subsystems are mathematically formulated in S1 Appendix.

MI model reproduces experimentally observed patterns of Dkk expression.

We start with numerical analysis of the wild-type pattern that resembles experimental observations (Fig 2A–2C and Ref. [26,27]). Such a pattern can be established de novo if the initial conditions provide a sufficiently strong canonical Wnt signal localised at the head end, which is consistent with head-cutting experiments with a localised signal stemming from the injury. With respect to the resulting stable patterns, the simulations predict the lack of both HyDkk1/2/4-related molecules in the hypostome, showing a sharp expression border beneath the tentacles, with HyDkk1/2/4-C expression fading out in the aboral direction (Fig 2A and 2C), and HyDkk1/2/4-A strongly expressed in the entire body column (Fig 2B and 2C). The simulated activity field is restricted to the head region, fading within and below the tentacle zone (Fig 2A–2C). Also, the temporal evolution of canonical Wnt activity matches both qualitatively and quantitatively between experimental observations and our simulations (Fig H in S1 Appendix, panel F, vs. Ref. [23]).

The difference in the two HyDkk patterns results from the qualitative differences in the corresponding production terms in the model equations. While HyDkk1/2/4-A is assumed to be constantly produced in the absence of repressing signals, HyDkk1/2/4-C production is modelled downstream of the HyWnt signalling and thus fades out in the aboral direction. To explore the role of the constant production term, we additionally simulated the system with the HyDkk1/2/4-A production depending on SD instead of considering a constant expression. It led to a graded HyDkk1/2/4-A expression along the body axis fading out in aboral direction (Fig G in S1 Appendix, panels I–J). In this modified system, the expression of appeared to be distinctly increased in the budding zone (compared to the original model), indicating that body-wide expression of HyDkk1/2/4-A might be involved in controlling/suppressing bud formation.

MI model reproduces self-organised axis formation in Hydra aggregates.

To assess whether the MI model captures genuine de novo axis formation, we simulated Hydra aggregate experiments. In these experiments, dissociated cells reassemble into initially symmetric tissue spheres without predefined positional information and subsequently undergo spontaneous symmetry breaking to form one or multiple body axes. Accordingly, we initialise the model with a symmetric domain and a random distribution of all biochemical components, including the SD. Small stochastic variations represent intrinsic heterogeneities that provide local competence for Wnt activation. Importantly, these perturbations do not determine organiser position, but act as transient activation events from which stable axes emerge through the intrinsic dynamics of the Wnt–Dkk network. The simulations reproduce de novo pattern formation consistent with experimental observations (Fig 2D–2F). Small aggregates develop a single axis (Fig 2D), whereas larger aggregates give rise to multiple heads (Fig 2E), in agreement with experimental data (Fig 2F). Thus, the model captures the experimentally observed scaling behaviour of axis formation, with the number of axes increasing with aggregate size.

MI model requires a strong localised signal for head regeneration in cutting experiments.

One of the key challenges for Hydra models of head regeneration is the previously demonstrated role of strong localised signalling in cutting experiments. These experiments show that removing the head without creating a wound does not lead to regeneration; only the activation of strong signalling and a localised increase in Wnt3 expression as part of the wound response enables regeneration [16]. This finding is particularly significant as it directly challenges the classic de novo pattern formation mechanism described by activator-inhibitor models. In former theoretical work, we postulated the necessity of bistability in a model to account for such phenomena [48]. The MI model meets these conditions, describing the absence of pattern formation under certain initial conditions, specifically, when an area with high Wnt3 expression and source density levels is removed, Fig 2G – right hand side. Conversely, the introduction of a strong signal, such as that triggered by the wound response, facilitates pattern formation. A systematic analysis of this mechanism is provided in the next section.

MI model reproduces key features of head transplantation and inhibition experiments.

Further, we simulate transplantation experiments (Fig 2G) motivated by the seminal experiments presented in Ref. [34,49,50]. These experiments showed that, under certain conditions, tissue fragments transplanted from one polyp to another can induce a secondary body axis. Our simulations reproduce key features of this behaviour. In particular, head self-inhibition suppresses secondary axis formation when two initiating signals are too close, whereas in the absence of the oral head, grafts can induce a secondary axis at distant positions. Quantitatively, the simulations predict secondary head formation only when the graft is placed more than approximately 50% of the body length away from the oral pole, in agreement with experimental observations [39]. Historical transplantation and fusion-type experiments in Hydra further suggested that inhibitory effects of an existing head can act over extended tissue regions and become established over time [51,52]. Motivated by these observations, we additionally simulated serially joined Hydra tubes in the one-dimensional model (Fig J in S1 Appendix). In the absence of a terminal head, multiple secondary peaks emerge approximately simultaneously, whereas a pre-existing head at one end suppresses the nearby response such that the more distant secondary peak rises more strongly than the one closer to the existing organiser. In addition, simulations motivated by the tandem ring graft experiments of Ando and Sawada show that repeated pieces of identical axial origin can generate qualitatively different numbers of head-related peaks depending on their original body-axis position (Fig K in S1 Appendix), consistent with the idea that patterning in such extended constructs is governed by the interaction of local activation and long-range inhibition [53]. Thus, the model captures not only distance-dependent suppression of secondary axis formation, but also the temporally developing influence of an existing organiser and the axial-origin dependence of pattern formation in extended tissue constructs.

Further in silico experiments.

In the next step, we verify the role of different model components by simulating their manipulated levels and comparing the resulting patterns to experimental data. First, we examine how system behaviour depends on the presence of the two HyDkk1/2/4-related molecules. Specifically, we simulate the undisturbed polyp system until the head pattern is established, as shown in Figs 2A, 2B and 3A (left-hand side). We then perturb the system by virtually removing the expression of one of the two HyDkks. In line with experiment, we observe a distinct expansion of in the head region following the removal of HyDkk1/2/4-A (Fig 3A (middle) and Ref. [26]) and body-wide ectopic activation of the field after removing HyDkk1/2/4-C (Fig 3A, right-hand side as well as Ref. [27]). The only difference between the simulations and experiments is that production appears relatively diffuse and homogeneous after virtual reduction of HyDkk1/2/4-C expression (Fig 3A, third snapshot), whereas experiments report a more granular pattern of HyWnt3 expression [27]. This discrepancy likely reflects that the experimental patterns correspond to HyWnt3 mRNA expression, whereas the model variable represents an effective signalling activity integrating multiple molecular components. As a result, the simulated patterns appear spatially smoother than the expression pattern of a single gene.

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Fig 3. Simulations and experiments of manipulated -catenin/HyWnt and HyDkk-levels.

(A) Simulation snapshots of the undisturbed polyp vs. virtual removal of HyDkk1/2/4-A vs. HyDkk1/2/4-C expression showing the activity field (corresponding to -catenin/Tcf/bound HyWnt3 domains) and tentacles. (B–C) Simulations of HyDkk1/2/4-A and HyDkk1/2/4-C expression after ALP treatment. (D) Simulation snapshots of and tentacle patterns of the undisturbed system vs. different combinations of ALP-treatment and HyDkk1/2/4-A knockdown. (E) experimental picture of double axis formation after ALP + siDkk1/2/4-A treatment, arrows indicate axes, asterisk represents foot. Colour-scaling is similar in all simulation snapshots in this figure (cf., colour legend).

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In addition, we simulate the activation of canonical HyWnt signalling via ALP treatment (Fig 3B and 3C, as well as Fig H in S1 Appendix, panels A–D). Experimentally, both HyDkks are suppressed by this treatment, along with the development of ectopic tentacles along the body column [26,27]. HyDkk1/2/4-C expression appears to be more sensitive to -catenin/HyWnt3 levels, as the reduction in HyDkk1/2/4-A expression occurs later than the reduction in HyDkk1/2/4-C expression [26,27]. These observations align with our simulation results during ectopic tentacle development (Fig 3B, 3C). However, we note that in later stages of the simulations, Dickkopf patterns re-establish (1D concentration profiles in Fig H in S1 Appendix, panels A–D), suggesting that the model can reproduce the data only as transient patterns. This may suggest that ALP-driven effects are initially present but reversible. Finally, we simulate recent HyDkk1/2/4-A knockdown experiments [32] with and without ALP treatment (Fig 3D). Similar to the experimental results (Fig 3E), the combination of ALP treatment and HyDkk1/2/4-A knockdown leads to the development of a secondary ectopic axis, marked by an additional region with high activity of the canonical HyWnt signalling field in our simulation results, but again, this is observed as a transient pattern.

One-dimensional model reduction.

In this section (with more details given in S1 Appendix, section ‘One-dimensional model reduction’), we focus on a one-dimensional version of the model given by equation (1)–(5) to systematically analyse its dynamics depending on the parameters and initial conditions. Our goal is to provide a deeper understanding of the simulation results discussed earlier. We begin by examining the model’s ability to generate patterns. This involves analysing the mathematical structure of the model, the ability of symmetry breaking and the stability of the emerging patterns.

The reduced model is obtained by approximating the complex domain of Hydra by an interval [0,1] and rescaling time and model variables in equations (1)–(5). The technical details of the model reduction, analytical approach, and numerical implementation are provided in S1 Appendix. To compare the reduced system with the original model, we apply a computational fitting procedure using data derived from the pseudo-3D simulation (Fig 2C). These data are obtained by averaging along the axis perpendicular to the body axis, followed by min–max normalisation. Parameter estimates are identified by minimising the least-squares residual between the data profiles and the output of the reduced model. This procedure demonstrates that the reduced model reproduces the same dynamics as the pseudo-3D model, with appropriate parameter adjustments. Further numerical details are included in Table A in S1 Appendix.

MI model reveals bistable behaviour in uniform steady-state structures.

Analysis of the structure of the spatially uniform steady states demonstrates a bifurcation from a semi-trivial state , where , resulting in the existence of additional spatially homogeneous solutions and a change of stability, see Fig 4. We identify two complementary bifurcation parameters, and , which describe the effective production rates of [A] and , respectively. More precisely, linear stability analysis of the semi-trivial steady state yields an instability condition, , which marks the bifurcation point and highlights the complementary relationship between the two bifurcation parameters. The corresponding bifurcation diagrams are shown in Fig 4A and 4B. The remaining parameter values are fixed according to the parameter fit to the three-dimensional model discussed above (see Table A in S1 Appendix).

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Fig 4. Bifurcation diagram illustrating the qualitative changes in pattern formation in the one-dimensional version of model (1)–(5), induced by variations in the rescaled reaction parameters (effective production rate of [A]) and (effective production rate of ).

The vertical axis represents the first component of the homogeneous steady states. Arrows indicate whether initial conditions converge to the pattern formation regime or decay towards the trivial steady state. Coloured bullets mark the boundaries of the bistable region: and in panel (A); and in panel (B). All other model parameters are set to the values listed in Table A in S1 Appendix.

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The structure of the bifurcation diagram and the stability of different branches of spatially uniform solutions are determined based on the model analysis presented in S1 Appendix (Lemma S1–S4 and Corollary S1) and numerical computations, employed whenever a complete analytical understanding was not feasible. To this end, we investigate the model for varying either or , while fixing all other parameters. As theoretically predicted, for sufficiently small values of , we observe instability of the semi-trivial stationary solution, denoted by , and existence of exactly one non-trivial spatially uniform steady state which is stable for the system without diffusion. Numerical computation of the system linearised at this positive steady state indicates Turing instability. This stays in agreement with the model simulations showing globally stable pattern formation. Above the critical value (transcritical bifurcation point), the semi-trivial state becomes stable and gives branching to an unstable homogeneous steady state to eventually collide with for the parameter value corresponding to the saddle-node bifurcation point. This results in a bistable regime for the parameter . This means that for initial concentration values in the basin of attraction of the trivial steady state , there is no pattern formation, and initial data decay to the trivial steady state . On the other hand, if initial data lie in the basin of attraction of the steady state , then we observe the formation of patterns. Here, unstable steady state acts like a separation hyperplane between two regimes, see Fig 5. For no pattern formation occurs and the only existing spatially uniform steady state becomes a stable attractor. Analogous results are obtained with respect to the bifurcation parameter , as shown in the bifurcation diagram Fig 4B.

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Fig 5. Example of bistable behaviour in the one-dimensional reduced model.

Initial data are taken in vicinity of the homogeneous steady state . When small perturbations lie in the basin of attraction of , pattern formation occurs and initial data evolve to the steady-state patterns (first row). Otherwise, when the steady state is perturbed in the direction of the trivial steady state , no pattern formation occurs and initial concentrations decay to the trivial steady state (second row). Parameter values used in the simulations are given in Table A in S1 Appendix, except . Blue colour shows the final concentration profiles, while red colour denotes the initial concentration values.

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Translated into the language of experimental research, this means that pattern formation or regeneration does not occur for arbitrary perturbations of the initial conditions, but critically depends on the nature and strength of those perturbations. This, in turn, explains why the MI model aligns with the experimental observation that regeneration only occurs when the initial conditions provide a sufficiently strong stimulus through wound signalling.

MI model exhibits Turing pattern formation.

Numerical simulations show emergence of stable stationary patterns. To gain analytical insight, we perform a bifurcation analysis near the onset of diffusion-driven instability (see S1 Appendix for details). Since not all expressions can be obtained explicitly, key coefficients are evaluated numerically. We consider two representative parameter regimes–one exhibiting hysteresis (bistability) and one without–and use the dominant diffusion coefficient as the bifurcation parameter. The critical value is computed numerically, and bifurcation theory is then applied to derive asymptotic expressions for the branching stationary solutions and assess their linear stability. In both regimes, the branching solutions are found to be stable, in agreement with direct simulations. While this analysis is not fully rigorous, it provides strong evidence for the existence of Turing patterns in the MI system.

MI model is robust with respect to parameter perturbations.

Next, we numerically investigate the robustness of pattern formation with respect to variations in model parameters. To demonstrate this robustness, we use the parameter values listed in Table A in S1 Appendix. We then explore the hypercube in parameter space for rescaled reaction rates within a large range , parameters are therefore varied by a factor of 10⁴. To estimate the regions within this hypercube where pattern formation occurs, we employ Markov Chain Monte Carlo (MCMC) methods. The criterion for admissibility is the existence of at least one homogeneous steady state exhibiting Turing instability. For the chosen parameter values, we did not identify any cases where this criterion was not met, indicating the robustness of the observed pattern formation. Of course, the range of parameters studied here did not cross the previously discussed points of transcritical bifurcation, beyond which the only stable equilibrium state of the system would be the semi-trivial state. Another critical parameter is the largest diffusion coefficient in the system, which must be sufficiently large to permit Turing instability.

MI model is robust with respect to a range of qualitative changes.

Finally, we assess the robustness of the reduced model (for equations, cf. S1 Appendix) with respect to modifications in the reaction terms. We consider two variations of the model. In the first, the reaction terms are multiplied by their respective denominator expressions. This modification can be interpreted as describing inhibition at the level of degradation or receptor binding, rather than at the level of production. Another system’s perturbation concerns using higher-order terms in Hill functions, i.e., modulating the inhibition process. For the parameter values examined, both variations exhibit the same qualitative pattern formation behaviour as the original one-dimensional model, suggesting its robustness. For further technical details, see S1 Appendix.