Sea-shell patterns are Nature’s way of depicting what a great artist (and mathematician) she is. How do sea-shell patterns look the way they do? We don’t completely know yet, but Biology and Maths may be bringing us closer to the answer. Pattern generation on shells is a very special case of how complex organisms develop structure or how dynamic systems evolve. These learnings can be used in understanding other systems as well. Initial notes from The Algorithmic Beauty of Sea-shells by Hans Meinhardt :

Let’s dive in!

Some biology basics

The functional significance of pigment patterns on cells is not clear. One hypothesis suggests that molluscs dispose of waste products into their shells. If this is the case, there is no strong selective pressure on a particular shell pattern. The shell of a mollusc is made up of calcified material. When the organism grows, the shell size increases by addition of new material at the margin. The incorporation of pigments during this growth process results in patterns. Once made, they usually remain unchanged. (A shell growing: http://www.youtube.com/watch?v=aSzp_aV998I)

What can you tell at first glance?

Pattern similarities have been found in non-related species indicating the existence of an underlying common mechanism and that diversity is generated by minor modifications and deviation.

Looking at some elementary patterns:

-Lines parallel to the direction of growth indicated a spatially-periodic pattern of pigment production that is stable in time. This is what we see in morphogenesis- how leaves, hair or feathers are produced at regularly spaced intervals. Here it is groups of cells in the mantle gland that do so.

-Lines perpendicular to the direction of growth indicate pigment deposition that is oscillatory in time. This could imply that the cell producing the pigment goes inactive at regular intervals resulting in stripes parallel to the axis.

-Oblique lines are a result of travelling waves of pigment production. These waves arise when pigment producing cells trigger their neighbouring cells so that after a certain delay, the neighbours start producing the pigment

Since the shell grows in several rounds around the axis, the progressing shell comes in direct contact with the shell material from the previous round. In some cases patterns were observed to continue from the older region to the newer region without major discontinuity indicating that the older region can initiate a travelling wave in the newer region (they can in some way ‘taste’ the older pigments). Perturbations from the normal pattern could be due to environmental factors as well such as lack of food, physical injury or even ion concentration. Spontaneous reaction to other triggers influence the pattern

Most shells show highly complex patterns. This complexity can arise from superimposing the influence of multiple pattern-forming reactions. Different reactions can also affect each other.





Studying sea-shells patterns

The following is a brief snapshot of a one of the many schemes used to describe certain kinds of patterns of sea-shells

Activator scheme pattern formation



The activator-inhibitor scheme:

Pattern formation is based on the interaction of molecules. To mathematically model the process, concentrations of substances involved are described as function of space and time.

Pattern formation starting from homogenous conditions requires local self-enhancement coupled with long range antagonistic effects. Patterns result when small deviations from a homogenous distribution create strong positive feedback. The long range antagonistic effects restrict the self enhancement resulting in localizations.

The activator is a short ranged substance that promotes its own production as well as that of its inhibitor which is a rapidly diffusing, long ranged antagonist. The diffusion of the inhibitor is much higher than that of the activator- this is a crucial condition for pattern formation.

Stable patterns require rapid antagonistic reaction: For patterns that are stable in time, the inhibitor must react very quickly to activator concentration, otherwise oscillations will occur- which will form another pattern. Rapid adaptation of the inhibitor regulates deviation from steady state concentration. For this to happen, the rate of removal of inhibitor must be greater than that of activator.

Differential equations describing the possible interaction between an autocatalytic activator a(x,t) and it’s inhibitor b(x,t) as functions of space and time:

Periodic patterns in space: As mentioned earlier, patterns at regular intervals are characteristic of morphogenetic situations. Inhibitor maximas are assumed to be centred around activator maximas but with a more shallow distribution. Irregularities can be explained by the fact that some maximas may be arranged too close and don’t survive the competition with long range inhibitors.

Width of stripes and saturation: A high activator diffusion rate leads to broader stripes where as a higher range of the inhibitor creates a larger region over which the activator cells can be suppressed.

Autocatalysis is limited by inhibitor production (as above) and saturation. At higher activator concentrations, reactions may slow down as available enzyme molecules may be limited in number- resulting in saturation i.e. a maxima in activator concentration.

Activator Depleted Substrate scheme:

An antagonistic effect can also result from the substrate b(x,t) depletion during activator a(x,t) production. For stable patterns, sufficient substrate must be present to maintain steady activator production. Diffusion of the substrate must also be faster than that of the activator. For example a higher activator concentration in a region (forming a maxima) will result in a depression of substrate concentration- and so a second maxima can develop only at a distance form this region. Activator production comes to rest if sufficient substrate is not available

If this has peaked your interest, go ahead and read the book The Algorithmic Beauty of Sea-shells by Hans Meinhard !

Some pictures, graphs and formulae taken from The Algorithmic Beauty of Sea-shells by Hans Meinhard