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The pigment patterns on tropical shells are of great beauty and diversity. Their mixture of regularity and irregularity is fascinating. A particular pattern seems to follow particular rules but these rules allow variations. No two shells are identical. The motionless patterns appear to be static, and, indeed, they consist of calci?ed material. However, as will be shown in this book, the underlying mechanism that generates this beauty is eminently dynamic. It has much in common with other dynamic systems that generate patterns, such as a wind-sand system that forms large dunes, or rain and erosion that form complex rami?ed river systems. On other shells the underlying mechanism has much in common with waves such as those commonly observed in the spread of an epidemic. A mollusk can only enlarge its shell at the shell margin. In most cases, only at this margin are new elements of the pigmentation pattern added. Therefore, the shell pattern preserves the record of a process that took place over time in a narrow zone at the growing edge. A certain point on the shell represents a certain moment in its history. Like a time machine one can go into the past or the future just by turning the shell back and forth. Having this complete historical record opens the possibility of decoding the generic principles behind this beauty.



Chapter 1. Shell patterns - a natural picture book to study dynamic systems and biological pattern formation

Everyday we are confronted with systems that have an inherent tendency to change. The weather, the stock market, or the economic situation are examples. Dramatic changes can be initiated by relatively small perturbations. In the stock market, for instance, even a rumour may be sufficient to trigger sales, lowering quotations and causing panic reactions in other shareholders.
Hans Meinhardt

Chapter 2. Pattern formation by local self-enhancement and long range inhibition

Like other biological processes pattern formation is based on the interaction of molecules. In order to find a mathematical description for a particular process the concentrations of the substances involved must be described as a function of space and time. This is possible by using equations that describe the changes in concentration over a short time interval as a function of other substances. Adding these concentration changes to given initial concentrations provides us with the concentration at a somewhat later time.
Hans Meinhardt

Chapter 3. Oscillations and traveling waves

A very important class of shell patterning is caused by pigment productions that occur only during a short time interval, followed by an inactive period without pigment production. Stripes parallel to the growing edge and oblique lines belong to this class of patterns. Such oscillations can occur if the antagonist reacts too slowly. Then, a change in activator concentration is not immediately backregulated since the antagonist reacts too slowly and the activation will proceed in a burst-like manner.
Hans Meinhardt

Chapter 4. Superposition of stable and periodic patterns

A widely distributed subgroup of shell patterns result from the superposition of a stable and a periodic pattern. The upper shell in Figure 4.1 shows two sets of parallel relief-like lines. One set is oriented parallel to the growing edge and results from a thickening of the shell at periodic time intervals. The other set is oriented parallel to the direction of growth and results from a permanently enhanced deposition of shell material at regularly spaced positions. In this example, the two patterns do not interfere with each other, a situation that is more the exception than the rule, but it shows that the assumption of two superimposed systems is reasonable.
Hans Meinhardt

Chapter 5. Crossings, meshwork of oblique lines and staggered dots: the combined action of two antagonists

Many shells display simple periodic patterns that cannot be accounted for with the elementary mechanisms described so far. Patterns of staggered dots and meshworks belong in this class (Figure 5.1). These patterns are characterized by a periodicity along the time coordinate as well as along the space coordinate. This suggests that two antagonists are involved: a nondiffusible one that is responsible for the periodicity in time, and a second highly diffusible one that causes the pattern through space. The interactions described in this chapter are possible extensions of the activator-substrate and the activator-inhibitor model (see boxes). An important property of such mechanisms is that traveling waves can emerge without pacemaker regions and that colliding waves can penetrate each other without annihilation. In other words, crossings of oblique lines can occur.
Hans Meinhardt

Chapter 6. Branch initiation by global control

Branch formation is a dominating pattern element of Oliva porphyria (Figure 6.1). Considering that the shell patterns resemble space-time plots, it is obvious that the formation of a branch is based on the sudden initiation of a backward-running wave. In the view of the normal behavior of waves in excitable media, this is a very unusual event. A new wave must spread into a region that should be refractory due to the primary wave.
Hans Meinhardt

Chapter 7. The big problem: two or more time-dependent patterns that interfere with each other

Many shells show patterns far more complex than those simulated so far. Figure 7.1 contains a collection of typical complex shell patterns. To show their inherent similarities, they are arranged such that each subsequent pattern contains elements of the preceding pattern as well as new features. Conus marmoreus (Figure 7.1a) shows white drop-like regions on a dark pigmented background. In Conus marchionatus (Figure 7.1b) the white drops are enlarged at the expense of the pigmented regions. The pattern is reminiscent of staggered wine glasses. Conus pennaceus (Figure 7.1c) shows, in addition, dark lines on a pigmented background, occasionally interrupted by small white drops.
Hans Meinhardt

Chapter 8. Triangles

Several mollusks display triangles as their basic pattern element. The triangles may be connected to each other to form oblique lines with a triangular substructure. If both corners of the lower edge give rise to new triangles, the white regions in between also have a triangular shape although with opposite orientation. The triangles may cover different portions of the shell. If they are densely packed, it appears as if white triangles are arranged on a black background. The triangles can also be of very different sizes. On some shells they are a prominent pattern element, on others they appear more as a roughness in the oblique lines but are clearly visible on closer inspection.
Hans Meinhardt

Chapter 9. Parallel lines with tongues

The upper shell in Figure 9.1 is decorated with many fine parallel lines. This pattern suggests the same synchronous oscillations as described earlier (see Figure 3.4). However, at particular positions, the parallel lines are deformed into U- or Vshaped gaps. The pattern on the lower shell is based on the same principle; only the size and regularity of the gaps are different. On the upper shell the gaps are restricted to particular positions.On the lower shell two broad bands are nearly free of parallel lines while smaller gaps appear at more scattered positions. The shells belong to the species Clithon oualaniensis (in older literature also termed Neritina or Theodoxus oualaniensis). These small brackwater snails are frequent on shores around India and Sri Lanka and display an incredible richness of patterns.
Hans Meinhardt

Chapter 10. Shell models in three dimensions

Inspired by the models of pigmentation patterns developed by Dr. Meinhardt, we pursued a further goal — to create a comprehensive model of seashells that would incorporate these patterns into three-dimensional shell shapes. Our motivation was twofold. On the one hand, in the absence of a formal measure of what makes two forms and patterns look alike, it is often necessary to rely on visual inspection when comparing models with nature (Prusinkiewicz, 1994). Realistic presentation adds credibility to such comparisons by removing potentially misleading artifacts.
Hans Meinhardt

Chapter 11. The computer programs

The programs supplied on the CD are modied versions of my own working programs. Originally written in FORTRAN, they have been translated into BASIC. The program sp is a Windows/Unix version, compiled with FreeBasic, a compiler that is freely available on the Web. Two-dimensional simulations can be performed with the program xy. DOS versions are mentioned further below. Do not expect the programs to be as perfect as a commercial product. Consider them as an extension of the book and as a tool to develop some intuition about the complex world of nonlinear interactions.
Hans Meinhardt

Chapter 12. Pattern formation in the development of higher organisms

The development of higher organisms out of a single cell is a most fascinating process. As an example, Figure 12.1 shows stages in the early development of a chick embryo. It was pointed out repeatedly that the mechanisms discussed for shell patterning are more general and of vital importance for pattern formation during embryonic development in higher organisms. In this chapter, some important steps in this development will be outlined and compared with corresponding models. In the subsequent chapter, some special biological phenomena will be discussed for which the lessons learned from the shell patterns were decisive for deriving the corresponding models.
Hans Meinhardt

Chapter 13. Pattern formation in development in which shell-related mechanisms are implicated

Many shell patterns can be explained by the assumption that peaks of signaling molecules are locally quenched shortly after they emerge. As shown in Chapter 5, this can lead either to traveling waves or to maxima, which disappear and reappear at more or less regularly displaced positions. In this chapter it will be shown that this mechanism is not restricted to shell patterning.
Hans Meinhardt


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