Elementary pigment patterns on shells
of tropical mollusks
From “The Algorithmic Beauty of Sea Shells”
© Hans Meinhardt and Springer Company
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.
Pairs of
stripes
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
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.
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
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)
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.
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
If the antagonistic reaction is too slow: oscillations occur;
stripes parallel to the growing edge emerge
Time
Time
Position
In the activator-depletion mechanism, oscillations occur if the
substrate production is insufficient to maintain a steady state
Time
Time
Position
Oblique lines are time records of traveling waves: one cell infects
the next; colliding waves extinguish each other
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.
Pattern regulation after a global perturbation: trigger of
new waves that might collide with surviving waves
Collision and annihilation
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.
= spontaneous trigger,
extinction of two colliding waves
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
Another example of traveling waves
Although the shell at left shows
parallel lines, the relation to the
faint growth lines in the background 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)
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
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 pigmentation system does not return rapidly to the ground state
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.
The real challenge is to understand the complex patterns…