1. Simulations for pattern formation and regeneration in hydra From The Algorithmic Beauty of Sea Shells © Hans Meinhardt and Springer Company

2. Essential elements of hydra patterning can be modeled by assuming three pattern-forming systems Three activator-inhibitor systems are assumed; one for the head (green), one for the tentacles (dark red) and one for the foot (pink). These are coupled via the graded competence of the cells to perform the head-forming reaction (blue; source density in our original model, head activation gradient in the literature). The head signal increases this competence, the foot reduces it. As the simulations shows, the model accounts for the activation of these signaling systems at correct positions, e.g., in a fragment of a body column during regeneration. After growth, newly separated fragments will regenerate again with the original polarity. In other words, the intrinsic polarity is part of the dynamic system and will persist.

3. A non-trivial aspect: maintenance of polarity during growth This simulation shows a highly non-trivial feature of the model: the adaptation to changing sizes. After the correct initiation, substantial growth is possible without the formation of additional maxima. Due to the graded competence, the apical dominance extends over a large region. Without this feature… (As mentioned, the signals for hypostome (green), tentacle (red) and foot formation (pink) are activator-inhibitor systems. These systems are liked via source density (blue). The hypostome system elevates the source and appears at the highest source density, the foot system does the opposite. Tentacles emerge at the highest source density not occupied by the hypostome signal).

4. ….without a graded competence, multiple maxima would form

5. Maintenance of polarity during regeneration In addition to the enhanced apical dominance, the graded source density (graded competence, blue) has another important consequence: regeneration occurs always with the correct polarity. Any fragment has an intrinsic asymmetry that orients the emerging pattern. As a sideline and as shown in the next frame, this is not a general feature of developing systems….

6. Sea urchin regeneration occurs - in contrast to hydra - with polarity reversal in one fragment Left: after separation of a sea urchin embryo, the two fragments regenerate. In one embryo the polarity is reversed (Hörstadius 1939; the original orientation is documented by vital staining; stippled). Model: after separation, the remaining inhibitor gradient in the non-activated fragment imposes an asymmetry. Since there is no intrinsic asymmetry, the side most distant to the originally activated region will win the competition (blue arrow). In sea urchins the predicted activator-inhibitor interaction is realized by Nodal/Lefty2. Nodal has a non-linear feedback on its own production. The inhibitor lefty2/antivin is under the same control as Nodal but blocks the Nodal-self-enhancement. In agreement with the model, it is the fragment without Nodal expression (V) that reverses polarity.

7. Hobmayer et al. Nature 407, 186-189 (2000) Organization by the canonical Wnt pathway Hyß-cat / Hy-Tcf Wnt The predicted model accounts for de-novo pattern formation. The formation of complete animals from disaggregated cells is a most convincing example (Gierer et al., 1972). The dynamics observed by Hobmayer et al. (2000) for the appearance of ß-catenin and TCF in aggregates is in full agreement with the expectation of the model: first a more overall activation appears that sharpens in the course of time. The head emerges subsequently at these signalling centers. Most remarkable, the Wnt maximum appears with a different dynamics. It appears somewhat delayed at the smoother ß-catenin maxima as sharp peaks, suggesting that Wnt and ß-catenin are coupled but are not part of the same positive feedback loop. Curr.Op.Genet.Dev. 14, 446-454 (2004).

8. Model for the formation of structures next to each other: tentacle formation as example Tentacles appear at the correct position if the head signal generates on long range the precondition to form the tentacle signal. In contrast, on short range, the head signal inhibits tentacle formation. Thus, tentacle formation can appear only next to the head. In the simulation above, the head signal (green) maintains the source density (blue; competence, head formation gradient). The tentacles appear at the highest blue level that is not occupied by the head signal (green) Dev. Biol. (1993) 157, 321]

9. Dynamics of tentacle regeneration in hydra: in near-head fragments the tentacle signal appears first at the very tip Bode et al (1988). Development 102, 223-235 For near-head fragments, Bode et al. 1988 showed with antibody staining that the tentacle signal appears first at the very tip and becomes subsequently shifted to its final position. An intermediate ring decays into the separate patches. In the model, the source density (blue) has a long time constant. After head removal, the source density is still high enough to trigger tentacle formation directly. This occurs at the oral end of the fragment. Only after the trigger of the head signal, the tentacle signal becomes displaced to the final position….

10. Dynamics of tentacle regeneration in hydra: in near-head fragments the tentacle signal appears first at the very tip Bode et al (1988). Development 102, 223-235 This works also in a two-dimensional simulation.

11. Tentacle regeneration in hydra: in more aboral fragments and in buds tentacle signal appears directly at the correct position The prediction* was that the sequence of events is other way round in fragments obtained from a more distal position. There, the competence is too low to allow a direct trigger of the tentacle signal. First the head (hypostome-) signal triggers that leads to an elevation of the competence (blue) After reaching a certain level, the tentacles are triggered. Since the head signal is already there, the tentacle signal appear directly at the correct position. (*) Dev. Biol. (1993) 157, 321; Evidence for this prediction has been found first by Technau and Holstein (1995): Head formation in hydra is different at apical and basal levels. Development 121,1273-1282. Further evidence came from Smith, Gee and Bode (2000). Hyalx, an aristaless-related gene... Development 127, 4743-4752:

12. Hyalx in tentacle formation Hyalx, an aristaless-related gene...Smith, Gee and Bode (2000). Development 127, 4743-4752 The same dynamics was convincingly demonstrated by Smith et al.(2000) with the Hyalx gene that is expressed at the tentacle base. In near-head fragments (upper row) its appears first at the tip that become subsequently shifted to the correct position. In contrast, during budding, the tentacle signal appears as a ring at the final position. Later it breaks up into individual small rings that forms the base of the tentacles. The model accounts for these different modes …

13. Hyalx in tentacle formation Hyalx, an aristaless-related gene...Smith, Gee and Bode (2000). Development 127, 4743-4752 In this simulation, the regenerating apical end is approximated by a sheet; the center corresponds to the tip of the hypostome. The source density (blue) is high enough to trigger a single tentacle signal (red) directly. This occurs, therefore in the center. Only after the subsequent trigger of the hypostome signal (green, top), the tentacle signal is displaced to the final position and splits up. Dev. Biol. (1993) 157, 321

14. Hyalx in tentacle formation In more aboral fragments and during budding, the source density (blue, bottom) is too low to trigger the tentacle signal directly. After elevation due to the newly formed head signal, the tentacle signal appears in a ring surround the head signal. The ring disintegrates into discrete spots. Hyalx, an aristaless-related gene...Smith, Gee and Bode (2000). Development 127, 4743-4752 H.M., Dev. Biol. (1993) 157, 321

15. From: Hobmayer et al.. (2000), WNT signalling act in axis formation in the diploblastic metazoan Hydra. Nature 407, 186-189 (2000) The signal for the first bud has initially a belt-shaped distribution that sharpens subsequently Due to the absence of a strong circum- ferential asymmetry, the first ß-cat / TCF signal that leads to bud formation has initially nearly a ring-shaped extension that sharpens subsequently to a lateral spot. This is reproduced by the model. Wnt Tcf ß-catenin

16. The localization of the tentacles depends on the graded source density (graded competence) Treatment with the drug Alsterpaullone stabilizes ß-catenin. This leads to tentacle formation everywhere. At early stages supernumerary tentacles form at some distance from the head, later they are found allover (Top left: control, top right: after treatment). In the simulation, first normal tentacles are formed during a regeneration. Later, the drug Alsterpaullone is added, assumed to elevate the source density everywhere (blue). Supernumerary tentacles form first at a distance from the existing tentacles. In this simulation also shifts of the tentacle signals play a role. Broun et al., (2005). Development 132,2907-2916.

17. Foot formation or how to generate a second organizer to the opposite position of a field In many systems different organizers form at antipodal positions. The head and the foot of hydra is an example. A simple cross-inhibition of the two pattern-forming systems is insufficient since then, for instance, foot-regeneration would be impossible in a small fragment containing the head. The cross-inhibition of the head would suppress the trigger of the foot signal. The problem disappears if no direct inhibition exists but the formation of the foot signal occurs preferentially at a low source density. The lowest source density occurs at the largest distance from the head. Therefore, the foot appears at the maximum distance. The foot, in turn, is assumed to lower the source density, stabilizing in this way its own precondition. Since no direct inhibition of the head onto the foot exists, foot regeneration can also occur close to an existing head….

18. Although forced to be formed at an antipodal position: a foot can regenerate close to a head Note that in this simulation the trigger of a new foot signal close to the existing head has almost no feedback on the head signal. The same would be true if a new head signal regenerates close to an existing foot. A candidate for an autocatalytic loop for foot formation is the positive feedback between Nkx2.5 and pedibin [Thomsen et al., Mech Dev 121,195-204 (2004)] The reappearance of the Cn-nk-2 signal during foot regeneration Grens, Gee, Fisher and Bode (1996) Dev. Biol. 180,473-488.

19. The regeneration of a second head instead of a foot after DAG treatment. Experiments: Müller, W.A. (1989) Diacylglycerol – induced multihead formation in Hydra Development 105, 309-316 Model: H.M., Dev. Biol. (1993) 157, 321 After treatment with Diacylglycerol (DAG) supernumerary tentacles are formed (as in the Alsterpaullone experiment mentioned above and presumably for the same reason). After removal of the foot, a head regenerates instead. Model: with the elevation of the source density, the tendency for foot formation is reduced in favor of head formation.

20. Not yet understood: unexpectedly after washing out DAG, first buds appear in the center; a new foot comes later. Experiments: Müller, W.A. (1989) Diacylglycerol – induced multihead formation in Hydra. Development 105, 309-316 In the model, after termination of DAG treatment, first a foot signal is expected to reappear. However, first buds are formed in the centre. Explanation by Werner Müller (who did the experiments): with the lowering of the head activation gradient, first a threshold for bud formation is reached (dashed line). Only after further lowering, the level required for foot formation is obtained (left). This leaves the question open of how foot regeneration can take place in a region close to the head, i.e., in a region of high head activation gradient

21. Also unexpected: new heads and feet appear close to each other when body columns are grafted together; Head and foot are expected to emerge at maximum distance from each other. However, after grafting several body columns to each other (Tardent, 1954), the terminal organizers appear close to each other. In the model, due to an increase in the source density by the head signal, the foot signal is expected to shift away from the head signaling center… After Tardent (1954). Roux Arch. 146, 593

22. Also unexpected: Tardent grafted body columns together; new heads and feet appear close to each other. After Tardent (1954). Roux Arch. 146, 593

23. Conclusion: Many basic hydra experiments are explicable by the assumption that the formation of the head, the tentacles and the foot are under control of activator- inhibitor mechanisms. These three systems are coupled by the source density (head activation gradient, competence to form the head signal) The model accounts for the generation and regeneration of all signals at the correct place, for the reformation of the signals during re-aggregation, for the maintenance of polarity during regeneration, for the different dynamics of tentacle formation during head regeneration and budding and for the formation of tentacles all over the body after Alsterpaullone treatment. The canonical Wnt pathway with ß- catenin und Wnt plays certainly a central role in the formation of the primary hydra organizer. The molecular basis of the long-range inhibition is not yet clear.

24. The equations used for the hydra-simulations Head activator a Source density c Foot activator f Tentacle activator d H.M., Dev. Biol. (1993) 157, 321