Global regulations Many shell patterns indicate that regulatory events occurred simultaneously along the whole growing edge, indicating the involvement of a global control. Global termination of pigment production is especially frequent. The pattern of Oliva porphyria is an intricate example, in which this global control less obvious. From “The Algorithmic Beauty of Sea Shells” © Hans Meinhardt and Springer-Company

Branch formation on Oliva porphyria is based on the trigger of backwards-running waves Branches are the dominating pattern elements on the shell of Oliva porphyria. This pattern indicates the occasional initiation of backwards-running waves. This is unusual since after passing of a wave, the region should be refractory for another wave. Most remarkably, many branches are formed simultaneously at distant positions (green dotted lines)…

Branch formation by a temporary transition into a steady state Backwards-running waves are triggered whenever the hormone concentration (green) becomes too low Model: each activated cell produce a hormone (green) that stabilizes the inhibitor (red). When the number of waves become too low, cells remain activated until a backwards-running wave is triggered. Many branches form simultaneously (red dotted line )

Simulation of Oliva porphyria

Details are well described A large non-pigmented region can lead to stronger pigmentations in other regions After the trigger of backwards-running waves, often an attempt can be seen to form a further wave running into the original direction. A hook-like pattern results… …sometimes the second wave survives, causing two parallel lines close to each other. The faint pigmentation resulting from the temporary steady state is clearly visible (arrow)

The system is more complex: the spontaneous initiation of new pairs of waves indicate the involvement of a second and long-lasting inhibitor d: the long-lasting second inhibitor New pairs of two diverging waves are triggered whenever the long-lasting inhibitor drops to a low level.

A very different pattern can be described by the same interaction Parameters are changed such that wave termination can occur without collision whenever a certain wave density is obtained (changed is the Michaelis - Menten constant in the action of the inhibitor; (sb in the equation at the preceeding page).

Conclusion A common scheme in the patterning of shells is a global control of pigmentation. A globally distributed hormone may be the controlling agent. Whenever the pigment density becomes too high or too low, hormone-controlled changes of parameters may lead to global pigment termination or to transitions from a pulse-like to a steady state activation. In this way, an average level of pigment deposition can be maintained. For Oliva porphyria the best fit is obtained if a hormone-controlled change of the half-life of the inhibitor is assumed.