Dev Genes Evol (2011) 221:255–279 discrepancy between low diffusivity and large length scale. (2) Bicoid movement might be more complex, including both space- and time-dependent changes in the diffusion constant as well as convective streaming. Diffusivity of Bicoid has so far been measured only at the surface of the syncytial blastoderm in embryos. It could be larger in earlier stages or in the inner part of the embryo as opposed to the cortex. Recent measurements of Bicoid diffusion using different biophysical methods have questioned previous data (Abu-Arish et al. 2010). In addition, a certain amount of cytoplasmic streaming occurs in the embryo which has been used to explain the length scale of the gradient (Hecht et al. 2009). (3) Bicoid is a target of post- translational modifications which may occur in a spatially controlled way. For example, the phosphorylation of Bicoid by the receptor tyrosine kinase Torso occurs only in a restricted anterior zone (Janody et al. 2000; Ronchi et al. 1993). Recently, a ubiquitinylation of Bicoid was demon- strated, which targets the protein for degradation (Liu and Ma 2011). Thus, many different protein forms of Bicoid might exist in the embryo, each with particular diffusion and decay rates or even with particular spatial distributions. Taken together, organismic complexity impinges on multiple levels even for the simple process of forming a protein gradient. The most difficult problem for theory lies in the fact that no a priori physicochemical argument exists that allows us to decide which aspects are relevant and which can be neglected. Thus, for some time, bicoid mRNA distribution was not considered to be important for gradient formation since the mRNA is clearly more locally restricted than the protein. Given the suggested low diffusivity of Bicoid protein, the potential contribution of mRNA spreading becomes more important. Conversely, nucleocytoplasmic shuttling of Bicoid appeared to be of pivotal importance for understanding gradient stability and scaling, leading to very impressive theoretical approaches (Gregor et al. 2007). However, recent experimental evi- dence undermines the importance of this process (Grimm and Wieschaus 2010). Evolutionary processes can affect and fine-tune each level of a developmental process. From the physical point of view, no simple process exists in an organism since not a single protein or RNA molecule can be treated separately from the organismic context. This is impressively shown by the example of Bicoid proteins from large eggs that form normal gradients within smaller eggs (Gregor et al. 2008). Since evolution can operate at each level (e.g. changing the localisation or stability of an mRNA, adding a modification to a protein, changing the size or the physical property of a compartment), multiple ways exist in which the organism can overcome physicochemical constraints. Bicoid shows us even more impressively than the other examples, how the organism evades the attempts of the theoretician. 273 Conclusions: Kant revisited The history of pattern formation theory presented here reveals the necessity of mathematical modelling already for simple chemical and even more so for biological systems. The oscillatory behaviour Lotka found by solving the differential equations is not evident if one just looks at the reaction scheme. It becomes plausible by using ‘graphical’ interpretations such as predator–prey relationships. Howev- er, understanding the important difference between conser- vative and dissipative systems, reflected in the difference between structurally unstable and structurally stable limit cycle oscillations, requires relatively abstract mathematical arguments. Nevertheless, this point is absolutely crucial; all interesting chemical and biological pattern-forming systems have the property of structural stability. The demonstration by Turing at the beginning of his paper that differences in diffusion rates can lead to unequal distributions of components starting from homogeneity is rather counter- intuitive. Is not diffusion a process that diminishes concentration differences? Finally, the rich pattern-forming capabilities of systems combining non-linear autocatalysis, inhibition and differential diffusion rates (Gierer–Meinhardt model) could have never been demonstrated without computer simulations. The need for mathematical model- ling is even more apparent in modern data-driven approaches. Already data acquisition requires sophisticated algorithms. A large part of the efforts of the Reinitz group was devoted to the question of how to reliably retrieve expression data from microscopic images (Janssens et al. 2005; Myasnikova et al. 2009; Surkova et al. 2008). Non- linear fitting or screening parameter spaces require elabo- rate programming. But even in rather simple cases such as Bicoid gradient formation, a model is needed to decide whether the measured diffusion constant is appropriate for a postulated mechanism or whether a substantial deviation from the expectation indicates the existence of unknown processes. One can easily predict that with our increased ability to acquire highly accurate measurements of molec- ular properties in living systems, the need for model building will increase. Pattern formation theory seem thus to be a prime example for Kant’s statement that ‘…there is present only so much real science, as there is mathematics’ (Kant 1900ff Vol 4, 470). However, in the face of organismic complexity, the work of the theoretician can appear to be a Sisyphean ordeal. In the examples we have described, successful and internally consistent modelling approaches were called into question by re-interpretations of underlying mecha- nisms or by new data sets, which contradicted previous assumptions. In the case of the gene circuit approach, a huge modelling effort was undertaken over more than a decade to produce one of the most impressive results in