In many branches of sciences, a precise mathematical description is a key to obtain a consistent understanding of the underlying principles. The complexity of the development of a higher organism seems to preclude such an approach in this discipline. However, recently it was shown that basic types of the molecular interaction allowing pattern formation can be described by sets of coupled partial equations. They allow computer simulations that mimic the observation rather precisely. Meanwhile these theories found direct support by observation on the molecular-genetic level.

The generation of the complex structure of a higher organism within each life cycle is certainly one of the most fascinating aspects of biology. Development starts, as the rule, with a single cell, the fertilized egg. At an early stage, many embryos can be fragmented and each fragment forms a complete organism. This indicates that a communication exists between different parts of the developing embryo. The removal of some parts is detected and the missing structures become replaced. This regulation makes development to a surprisingly robust process. However, from the reaction of the organism upon such an experimental perturbation one cannot directly deduce the molecular basis on which development is based.

Recently it was possible to clone genes and to isolate the substances involved. However, even if a particular gene is found to be expressed at a high level at a particular position and a mutation shows that this gene expression is crucial for a particular step, we have no information on how this local maximum is generated.
As in other branches of science, theories provide the bridges between observations and an understanding of the underlying mechanisms. The development of a higher organism may appear far too complex to allow such an approach in this discipline. However, this process can be separated into a number of elementary steps. A key process is the generation of local concentration maxima that act as signaling centers. The concentration of the signaling molecules declines with increasing distance from organizing regions, which allows a position-dependent activation of genes.

Such organizing regions where first found in classical transplantation experiments. Examples are the gastric opening of the freshwater polyp Hydra [1] and the organizer in amphibians [2]. Upon transplantation, such organizing regions are able to instruct the surrounding cells. Meanwhile several genes are known that are expressed at high levels in such signaling centers.

A Theory of Biological Pattern Formation

There are good reasons to assume that every step in development is accomplished by molecules. Since many molecules interact, the description of development will involve many coupled differential equations that describe the production, spread, removal and, most importantly, the mutual regulation of molecules. Thus, the aim was to find hypothetical molecular interactions that reproduce the capability to generate structures in space, i.e., to accomplish pattern formation. Long before the molecular approach became feasible together with Alfred Gierer I have shown which type of molecular interaction allows the generation of such signaling centers and a concentration-dependent gene activation [3, 4]. These theories found meanwhile direct support by experimental observations [5, 6]. In particular, the theory accounts for the basic observations in the perturbation experiments mentioned above.

Local Self-enhancement and Long-range Antagonistic Effects as the Driving Force of Pattern Formation

Local concentration maxima can be generated if, and only if, strong (nonlinear) and local-acting positive feedback loops exist that are antagonized by a reaction that acts on a longer range [3-6]. In such an interaction the homogeneous distribution is instable. Small deviations from a homogeneous distribution further grow due to the local positive feedback while the long-ranging antagonistic effect keeps the emergent maxima localized and suppresses the onset of a similar patterning processes at larger distance.

Pattern formation starting from almost homogeneous initial conditions is also very common in non-animated systems; sand dunes, rivers, clouds, and lightning are examples. It is easy to see that these patterning processes are based on the same principle. For a sand dune a stone may provide a windshield that accelerates the local deposition of more sand. Erosion proceeds faster at an initial injury since more water collects there. Since the total amount of water or sand is limited, a local accumulation must be accompanied by an overall decrease elsewhere. The same type of interaction - self-re-enforcement and inhibition of a larger domain - is also the basis in many social interactions.