1. Parallel lines with tongues: stabilization by an oscillating system From “The Algorithmic Beauty of Sea Shells” © Hans Meinhardt and Springer Company

2. Shells of Clithon oualaniensis are decorated with fine parallel lines. These lines may be interrupted by white patches, called “tongues”. These tongues can appear at more or less random positions, between preceding tongues or in rows at particular positions. The parallel lines are records of near synchronous oscillations. After an interruption, the pigment deposition may reoccur with a strong overshoot, causing the dark crescents at the lower border of the tongues.

3. Tongues may occur along rows at particular positions…

4. The parallel lines may merge to a homogeneous background pigmentation. Note the somewhat shadow-like darker pigmentation after a tongue (arrow), This suggests that some sort of recovery of the pigmentation system occurs in phases with no pigment production Patterns on other shells indicate that circular tongues indeed result from an initial retraction and subsequent expansion of synchronous oscillations

5. The more-or-less synchronous oscillations that give rise to the parallel lines may be restricted to particular bands. Regions in which oscillations occur may periodically enlarge, resulting in a pattern of parallel lines with a triangle-shaped outline. If the oscillating regions enlarge, the regions with parallel lines are usually bordered with oblique lines. These oblique lines may extend into the regions that are free of oscillations.

6. On this shell a grey background pigmentation is visible between the parallel lines. This background pigmentation disappears in the tongues. This suggest, that there is a reaction in a steady state, which is a precondition for the oscillations. In turn, the oscillations maintains the precondition pattern in the steady state. If a pulse comes to late, the precondition pattern breaks down, causing a stop of the oscillation and thus the initiation of a tongue.

7. Model: a precondition pattern (green) is required in order that pigment deposition can take place in a synchronously oscillating fashion (dark parallel lines). The oscillating system, in turn, acts to maintain the precondition system (green) in an active state. Due to the accumulation of an inhibitory metabolic product, pulses may appear delayed - too late to refresh the precondition pattern. Non-pigmented regions, the tongues, emerge that are filled by traveling waves from the regions where the precondition and oscillation system survived.

8. The restriction of the parallel lines to bands is a characteristic feature of many Clithon shells. These bands are frequently connected by oblique lines. Model: in regions with a higher production rate of the poison ( b e ), the activation of the precondition pattern is too short to allow several oscillations. Isolated oblique lines (traveling waves) result that always have their origin in one of the parallel lines. Note that these lines do not branch. The parallel lines are not bordered by oblique lines as long as the precondition pattern does not spread (arrows). There are other shells, in which two successive oscillations are possible….

9. On the left shell the faint modulation of the background pigmentation indicate that there is a global oscillation independent of the proper pigmentation reaction. Such a global oscillation was already inferred from the analysis of triangle formation on other shells (Chapter 8). The proper oscillation that leads to pigment deposition occurs in synchrony with the global oscillation. The shell at right, however, shows that this is not a general feature.

10. Traveling tongues in Neritina virginea: Model: after initiation of a tongue, the precondition system (green) may retract further on one or both sides, enlarging the oscillation-free area (red arrows). The reinforcing action of the pigmentation system has a counteracting influence. It allows a wave-like spread of the precondition system (dark oblique lines), reducing the size of the tongue. If the tongue enlarges on one side but shrinks on the other, the position of the tongue shifts. Eventually, a trigger of a traveling wave on the retracting side (blue arrows) causes the closure. (be)(be)

11. The movie and the XT-plot show identical simulations; the closing side of a tongue is bordered by a dark oblique line (strong non-oscillating peaks in the movie). This oblique line is absent at the side at which the precondition pattern retracts (white regions bordered by a green area in the movie). (green = precondition reaction for the oscillation, it needs refreshing by the pigment reaction shown in black) (be)(be)

12. Oscillations that maintain a pigmentation system in a steady state has been discussed already in Chapter 7. What is the difference between these and the Clithon patterns? In Cymbiola verspertilio (left), the oscillating reaction is involved even in the formation of the traveling waves. Therefore, the oblique lines have a fine structure and can branch. In Clithon (center and right), in the oblique lines have not such a fine structure and branching does not occur, suggesting that traveling waves require the support from the non-oscillating steady state system (in the simulation the green and black oblique line run in parallel). The period in which the precondition system is active is too short to allow synchronous oscillations.

13. Equations for the tongue-simulations with An activator –substrate model is used for the pigment system An activator – inhibitor system is used for the precondition pattern The metabolic product, the poison, is produced if the precondition system is active s e = 0.12 s e = 0.08 s e = 0.05 p(x ) can be a stable pattern in space that leads to a higher probability for tongue formation

14. Conclusion: Clithon patterning can be explained by an oscillating systems that keeps a precondition system in an steady-state. Like in a higher organism, the beating heart is required to keep the organism alive, i.e., in a steady state. In turn, being alive, however, is the precondition that the heart can beat. On a mollusk, many of such oscillators along the growing edge are coupled. If a breakdown of the precondition system occurs in one region, the oscillation will stop. The breakdown is a collective process that leads to the large non-pigmented regions, the tongues. After such a breakdown, however, the activity can spread from an adjacent region in which the oscillation survived. This restores the more or less synchronous oscillation.