The Algorithmic Beauty of Sea Shells (The Virtual Laboratory)

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The book is accompanied by computer programs for performing the simulations on a PC. Most simulations shown can be reproduced. Seeing these patterns emerge on the screen provides a much more intuitive feel for the dynamics of the system.

Since minor fluctuation can play a decisive role, even the repetition of the same simulation can produce a somewhat different pattern. This corresponds to the fact that the patterns on any two shells are never identical. The programs allow you to change parameters such as the half-life of a substance or its spread by diffusion. The consequences of these changes can be seen immediately as an alteration of the pattern. Therefore, new model interactions can be easily inserted.

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What has been added in the new editions As mentioned already in the first edition, the models describing shell patterning are only special applications of mechanisms developed to account for biological pattern formation in general. To illustrate these close connections, in the second edition a new chapter was added in which models are discussed that describe, for instance, how an embryo can obtain its primary axes, how legs and wings obtain their own coordinate system at the correct positions, and how gene activation can proceed under the influence of graded signals in a position-dependent manner.

We started our modeling with the freshwater polyp hydra. Meanwhile it turned out that these primitive organisms can be regarded, as far as the body plan is concerned, as living fossils that provide information of how the formation of body axes of higher organisms evolved. As explained now in Chapter 12, pattern formation in such simple organisms provides a key to understanding how of the body axes are established in higher organisms.

Models originally developed to describe some sea shell patterns turned out to provide a key to understanding other developmental systems, it turned out.

Overtly the arrangement of leaves on a growing shoot, the patterning of bird feathers, the localization of the plane of cell division in a bacterium, blood coagulation, and the chemotactic orientation of cells seem to have little in common with patterns on a Preface ix sea shell.

Now, in the fourth edition, a new chapter shows that these systems have a common logical basis. They depend on signals that become quenched shortly after they are generated. This leads to highly dynamic pattern-forming systems that never reach a stable state; avoiding to enter into a state in which the system is irreversibly trapped. Such flexibility then accounts for biological systems that require a permanent adaptation to changing situations. To provide easier access to the highly dynamic behavior of the interactions on which shell patterning is based, animated simulations were added to the accompanying CD-ROM in the third editions.

These can be inspected like conventional websites with any browser in any system. For most of these reactions only a single click is required to see the equations and parameters. The animated simulations are given as separate files that can be integrated, e.

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Also included are animated simulations of models that describe general steps in the development of higher organisms, for instance, regeneration and transplantation experiments with hydra. In the fourth edition PowerPoint presentations were added for the major topics. They also contain many animated simulations. The best way to become familiar with these reactions is to run these programs enclosed.

The corresponding compilers are freely available on the Web and working versions are supplied. Moreover, it is now straightforward to save simulation results in a form that allows printing or integration into other applications. Acknowledgements This book could not have been completed without encouragement from many quarters, foremost from my wife Edeltraud Putz-Meinhardt. I would like to express my thanks to those who contributed to the book.

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The basic ideas grew out of a theory I developed with Alfred Gierer. His concept of local autocatalysis and longranging inhibition has formed the basis of most of my work on biological pattern formation. In his diploma thesis Martin Klingler described many interactions capable of reproducing shell patterns in fine detail. Discussions with Andre Koch and Kai Kumpf have been stimulating for me. I thank Drs. Christa Hug helped to prepare the manuscript and Karl Heinz Nill made several of the drawings.

I am very grateful to Deborah Fowler and Przemyslaw Prusinkiewicz from the University of Calgary, Canada, for contributing their chapter on shell shapes Chapter Lynn Mercer and Sherryl Sundell provided very careful corrections of the manuscript.

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During the preparation of a German edition, Isolde Tegtmeier found several minor errors in the first edition that are now corrected. Jens Hemmen provided some corrections in the nomenclature of shells. Contents 1 Shell patterns - a natural picture book to study dynamic systems and biological pattern formation. Oblique lines.

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  6. Relief-like patterns follow the same rules. Many open questions and some hints. The hard problem: complex patterns. Earlier attempts to understand shell patterns. The formation of undulating lines and the partial synchronization of cells by activator diffusion. Reducing wave termination with a longer activation period. Interconnecting wavy lines and the formation of arches. Hidden waves. Pattern on the shell of Nautilus pompilius. Stabilizing an otherwise oscillating pattern by diffusion.

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    Combinations of oscillating and nonoscillating patterns. Rows of patches parallel to the direction of growth. The possible role of a central oscillator. Displacement of stable maxima or enforced de-synchronization by a second antagonist. Pattern variability. Global pattern rearrangements. Traces of the additional inhibition: oblique lines initiated or terminated out of phase. Crossings and branching.

    Changing the wave speed before and during collisions. Parallel and oblique rows of staggered dots. A very different pattern generated by the same interaction. Intimate coupling of an enhancing and an extinguishing reaction. Extinguishing that results from a depletion of resources due to an enhancing reaction. Related patterns reveal unsolved problems. Apparently different patterns can be simulated by closely related models Conclusion. Conus textile: tongues and branches on the same shell. Missing elements, missing links. Implementation of the interactions. Numerical instabilities that may cause errors.

    The Algorithmic Beauty of Sea Shells (The Virtual Laboratory)

    Compilers and versions. Parameters used in the program. The evolution of the main body axes. Gene activation under the control of a morphogen gradient. Position-dependent activation of several genes. Out-of-phase oscillations in E. 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. An essential element of dynamic systems is a positive feedback that self-enhances the initial deviation from the mean.

    The avalanche is proverbial. Cities grow since they attract more people, bacteria or viruses can replicate and the progeny start replicating too. In the universe a local accumulation of matter may attract more dust, eventually leading to the birth of a star.