1. Elementary pigment patterns on shells of tropical mollusks From “The Algorithmic Beauty of Sea Shells” © Hans Meinhardt and Springer Company

2. Stripes parallel to the direction of growth are time records of periodic patterns that are stable in time A mollusk can enlarge its shell only by accretion of new material at the growing edge. As the rule, new pigment is only deposited in this growth zone. Therefore, the two-dimensional pigment patterns are time records of one-dimensional patterning processes.

3. Pairs of stripes can be explained by an early switching off of diffusion. The pigment reaction may only survive at the margins of former stripes. The switching off in the center of the stripes results from to the temporary accumulating inhibitor. Alternatively, one system could trigger on long range a second system but exclude this locally. The pattern on the inner side of the shell remains invisible as long as the mollusk is alive, indicating that the pattern has no signalling function Pairs of stripes

4. Behavior upon growth: insertion of new maxima New activation centers are inserted if, due to growth, the distance between existing maxima becomes too large. At a distance from existing maxima, the inhibitor drops to such a low level that the autocatalysis triggers.

5. Behavior upon growth: split of existing maxima If the activator autocatalysis saturates at low activator concentration, the peak becomes broader. With growth, the region into which the inhibitor can escape becomes larger; the maximum reaches the saturation concentration. The center of a broad peak may become disfavored since the cells at the flank can better dump the inhibitor into the non-activated surroundings. The activation ceases in the center, causing a split of the peak. Just a single split

6. Splitting is prevalent in the activator-depleted substrate interaction In the activator-substrate interaction the limitation of the activator autocatalysis is inherent. The activator maximum cannot increase further if most of the substrate is consumed. Therefore, splits of maxima during growth are dominating in this interaction. Peaks shift to regions in which more substrate is available. (For this reason, the activator-substrate model is inconvenient to form organizing regions. Organizing regions should remain local and neither move nor split)

7. Behavior upon growth: early fixation If the coupling by diffusion is switched off, a once obtained pattern can be maintained; activated and non-activated cells keep their state. A condition is that a spontaneous trigger is suppressed by a moderate baseline inhibitor production.

8. Stripes of different thickness On this shells, some stripes are much thinner than others. In the model, this can occur if the autocatalysis saturates. This interpretation is supported by the close proximity of some stripes (arrow). If autocatalysis saturates, the lateral inhibition is no longer fully effective; activated cells have to tolerate other activated cells nearby

9. If the antagonistic reaction is too slow: oscillations occur; stripes parallel to the growing edge emerge Position Time

10. In the activator-depletion mechanism, oscillations occur if the substrate production is insufficient to maintain a steady state Position Time

11. Oblique lines are time records of traveling waves: one cell infects the next; colliding waves extinguish each other

12. Traveling waves emerge if the individual cells oscillate or are excitable, the activator diffuses slowly and the antagonist remains local Like with a forest fire or in the spread of an epidemic, if the self-enhancing reaction can infect adjacent regions, the activation will spread. The overall pattern depends strongly on whether a spontaneous trigger is possible. If two waves collide, the two become extinct since a wave cannot enter the region that became refractory by the counter wave. In this example, the activation (brown) shifts into regions of higher substrate concentration (green). Occasionally the substrate concentration reaches a level such that a spontaneous trigger occurs.

13. Pattern regulation after a global perturbation: trigger of new waves that might collide with surviving waves = spontaneous trigger, extinction of two colliding waves The horizontal line at which many oblique lines terminated indicate that the mollusk was exposed to a global perturbation. Many traveling waves terminated simultaneously. Cells that were non-activated for a longer period could spontaneously trigger a new activation. Collision and annihilation

14. Entrainment of the new waves by the old pattern: Some lines continue from an older round of shell building to the new one. In this shell, between two connected lines (arrowheads), one additional line is inserted. This illustrates that wave initiation is based on a free-running oscillator, which is sensitive to an external synchronizing signal

15. Another example of traveling waves Although the shell at left shows parallel lines, the relation to the faint growth lines in the back- ground pattern indicates that the dark lines are oblique to the growth lines and are thus the result of traveling waves. Note that the background pattern disappears shortly before the pigmentation was triggered, causing the thin white lines above the thick black lines. This is a frequent phenomenon (arrow below)

16. Conclusion: elementary patterns can be explained by assuming that a standard pattern-forming system is at work: an autocatalytic reaction is coupled with an antagonistic reaction Lines perpendicular to the edge: patterning in space, stable in time; occurs if the antagonistic reaction is fast and spreads rapidly Lines parallel to the edge: oscillations, i.e., patterning in time; occurs if the antagonistic reaction has a longer half-life than the activator Oblique lines: traveling waves: occurs if the self-enhancing reaction spreads slowly and the antagonistic reaction is local and slow

17. Even if a shell displays an elementary pattern: details show that the actual mechanism can be more complex Periodic modulations (red arrows) show that a periodic pattern is superimposed on the stable pattern. The periodic pattern is even dominating on the shoulder of the shell (green arrow). Abrupt phase changes in the oscillation (arrow) suggest a re-synchronization with an additional and invisible oscillation. Oblique lines with a fuzzy lower border: after passing of a wave, small patches of cells can remain activated for variable periods (red arrow). The spacing seems to be regular. Two approaching traveling waves may become slower or do not collide at all. A higher background pigmentation indicates that the activation of the pigmen- tation system does not return rapidly to the ground state

18. Conclusions: Most of the pigmentation patterns on tropical sea shells preserve a historical record of their generation. This unique feature invites to decode the dynamics of the underlying mechanism. Obviously there was no strong selective pressure to produce particular patterns; nature was able to play. The resulting variability is of great help since it is to be expected that the patterns, even if they look overtly very different, are based on closely related interactions.

19. The real challenge is to understand the complex patterns…