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The reliable development of highly complex organisms is an intriguing and fascinating problem. The genetic material is, as a rule, the same in each cell of an organism. How then do cells, under the influence of their common genes, produce spatial patterns' Simple models are discussed that describe the generation of patterns out of an initially nearly homogeneous state. They are based on nonlinear interactions of at least two chemicals and on their diffusion. The concepts of local autocatalysis and of long-range inhibition play a fundamental role. Numerical simulations show that the models account for many basic biological observations such as the regeneration of a pattern after excision of tissue or the production of regular (or nearly regular) arrays of organs during (or after) completion of growth. Very complex patterns can be generated in a reproducible way by hierarchical coupling of several such elementary reactions. Applications to animal coats and to the generation of polygonally shaped patterns are provided. It is further shown how to generate a strictly periodic pattern of units that themselves exhibit a complex and polar fine structure. This is illustrated by two examples: the assembly of photoreceptor cells in the eye of Drosophila and the positioning of leaves and axillary buds in a growing shoot. In both cases, the substructures have to achieve an internal polarity under the influence of some primary pattern-forming system existing in the fly s eye or in the plant. The fact that similar models can describe essential steps in organisms as distantly related as animals and plants suggests that they reveal some universal mechanisms. CONTENTS I. Introduction II. Gradients in Biological Systems III. Simple Models for Pattern Formation A. Activator-inhibitor systems B. Activator-substrate systems C. Biochemical switches D. Other realizations of local autocatalysis and long-range inhibition IV. Some Regulatory Properties of Pattern-Forming Reactions A. Insertion of new maxima during isotropic growth B. Strictly periodic patterns C. Regeneration properties and polarity V. Prom Simple Gradients to Complex Structures A. Animal coat patterns B. Reticulated structures C. The faceted eye of Dv osophila flies 1. The morphogenetic furrow

Proper cell division requires an accurate definition of the division plane. In bacteria, this plane is determined by a polymeric ring of the FtsZ protein. The site of Z ring assembly in turn is controlled by the Min system, which suppresses FtsZ polymerization at noncentral membrane sites. The Min proteins in Escherichia coli undergo a highly dynamic localization cycle, during which they oscillate between the membrane of both cell halves. By using computer simulations we show that Min protein dynamics can be described accurately by using the following assumptions: (i) the MinD ATPase self-assembles on the membrane and recruits both MinC, an inhibitor of Z ring formation, and MinE, a protein required for MinC͞MinD oscillation, (ii) a local accumulation of MinE is generated by a pattern formation reaction that is based on local self-enhancement and a long range antagonistic effect, and (iii) it displaces MinD from the membrane causing its own local destabilization and shift toward higher MinD concentrations. This local destabilization results in a wave of high MinE concentration traveling from the cell center to a pole, where it disappears. MinD reassembles on the membrane of the other cell half and attracts a new accumulation of MinE, causing a wave-like disassembly of MinD again. The result is a pole-to-pole oscillation of MinC͞D. On time average, MinC concentration is highest at the poles, forcing FtsZ assembly to the center. The mechanism is self-organizing and does not require any other hypothetical topological determinant.bacteria ͉ cell division ͉ polar pattern ͉ FtsZ ͉ center finding