Complex patterning: if two or more time-dependent systems interfere with each other

From “The Algorithmic Beauty of Sea Shells” © Hans Meinhardt and Springer-Company

Conus marmoreusThe pattern on Conus marmoreus can be explained by assuming a slowly spreading pigmentation system (black triangles). Whenever the pigmentation reaction was active for a certain period, a second patterning mechanism is triggered that extinguishes the pigmentation reaction (red in the simulation). The pigment reaction remains active at the flanking regions, from which the next spread starts

Conus marmoreus roeder

If the extinguishing reaction is stronger, pigment production terminates more frequently. This occurs preferentially at the sides that had less time to recover from the last activation. Chains of triangles can result.

Other shells show lines that branch frequently. The fine lines indicate traveling waves. Branching occurs if particular cells remain in a steady state until a backwards wave was triggered…

Model: one system generates traveling waves, a second (green in the upper part and in the movie) shifts the system temporarily into a steady state. The cells remain long enough active such that a backwards-running wave can be triggered.

Other shell patterns suggest that both the enhancement reaction (branching) and the extinguishing reaction is involved. Cases for the latter are marked by arrows

Enhancement and extinguishing on the same shell: dark lines with white spots on a pigmented background. An enhanced pigment production (dark lines) makes a breakdown (white patches) more likely. Note that the white patches are larger then the dark lines, indicating that the breakdown reaction can spread.

A oscillation enhances the pigment reaction

If the pigment reaction is in a steady state, a more or less synchronous oscillation of an enhancing reaction leads to dark parallel lines on a pigmented background. In the remaining regions, the pigment reaction leads to traveling waves (thin lines). The enhancing reaction triggers branch formation.

An oscillating enhancing reaction keeps the pigmentation reaction in the active state

The pattern on Cymbiola vespertilio shows triangle-like elements that are connected by oblique lines. Both display a fine structure of fine parallel lines. This suggest that an enhancing oscillation is at work (green in the simulation) that keeps the pigment reaction active. If an oscillation comes too late, pigment production terminates. For the simulation, the same interaction is used as before for Conus textile, although both patterns look overtly very different.

….if the maintenance effect is stronger….

If the maintenance effect is stronger, larger pigmented regions result. The fine structure of parallel lines is clearly visible in this specimen. The enhancing effect is stronger in two bands, as it is frequent on many shell patterns.

Here, the oblique lines are dominating….

Larger pigmented regions result if the maintenance reaction has a stronger effect

The enhancing reaction is locally stable

In Conus bednalli a stable enhancing reaction leads to two or three lines parallel to the direction of growth. These zones of stable activations are the starting points of traveling waves that annihilate each other upon collision. Branching and attempts to form branches along the oblique lines are clearly visible, indicating that the traveling waves result from three-component systems

Conclusion

Complex patterns result from time-dependent modifying reactions that either enhance or extinguish the pigmentation reaction. Evidence for both types of modification can be found on the same shell.

The analysis of these patterns is difficult since the modifying reaction is, as the rule, not visible but has to be inferred from the unusual behavior of the pigment reaction.

Very different-appearing patterns can be simulated with the same interaction using different parameters. Thus, overtly different patterns are not an indicator of how closely related the underlying patterning mechanisms are.

The equations(for most of the simulations either the enhancing or the extinguishing reaction is used)

2 2

2 21 a a e aa

a a as b b r a c e a D

t s a x

2 2

2 2(1 )

1b b a b ba

b a bb s c s b b r b D

t s a x

2

22 )(

x

cDcr

dsd

bcar

t

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2

22 )(

x

dDbdrbcar

t

ddddcc

2

2

2

2

1 x

eDerb

es

efr

t

eeee

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2

2

2

2

1)(

x

fDfrb

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efrxpcab

t

fffe

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The pigment-forming reaction (an activator-depleted substrate system):

The enhancing reaction leads to a higher production of the substrate production b for the pigment reaction (term sb ).

The reaction is an activator-inhibitor system that depends on the a-system

The extinguishing reaction leads to a reduced half life of activator a of the pigment reaction (term ce ). It is

an activator-substrate system; the production of the substance f depends on a . The production rate may be different along the growing edge [term p(x)]: