268 their strength—they could be applied to many cases, and their weakness—they could not capture any specific molecular process. With detailed molecular data, in partic- ular from Drosophila, becoming available in the mid- 1990s, a new generation of theoretical approaches became possible. Modelling was no longer restricted to general mechanism, such as local activation and lateral inhibition, but could incorporate genetic interaction data or even real measurements of concentrations of respective components. Now finally, one could hope, the real molecular processes could be simulated in the computer. This should lead to realistic models providing a deep understanding of multi- component systems. The step from the linear stability analysis to which Turing was largely restricted to the simulation of non-linear systems by Gierer and Meinhardt was linked to increased computer power. Likewise, the new simulations again were only possible by another large leap in simulation capacity. In particular, now it was possible to perform systematic explorations of parameter spaces or to apply complex algorithms of non-linear fitting (simulated annealing). The idea of robustness started to play an important role. By systematic variation of the parameters of given model equations, one could identify those conditions which provided the most robust mechanism. In the following, three examples from recent work on Drosophila pattern formation will be discussed which reveal the power and limitations of the new approaches. Each of them demonstrates the need for mathematical modelling. They will be presented in a sequence of increasing closeness to quantitative experimental data: the first example deals with models for the formation of a BMP morphogen gradient in the embryo, the second with the gene circuit approach to explain the segmentation process, and the third with models for the formation of the Bicoid morphogen gradient. (1) In Drosophila, the transcription factor Dorsal is responsible for establishing the entire dorsoventral axis (Moussian and Roth 2005). Dorsal forms a nuclear concentration gradient with peak levels along the ventral midline of the early embryo. Dorsal activates and represses target genes in a concentration-dependent manner and thereby specifies different gene expression domains along the DV axis. Interestingly, the Dorsal gradient is confined to the ventral half of the embryonic circumference, but nevertheless is also required for patterning the dorsal half. In particular, the Dorsal gradient indirectly establishes high levels of BMP signalling in a narrow, precisely defined stripe along the Dorsal midline (O’Connor et al. 2006). How is such a precisely controlled long-range influence possible? Mutant analysis showed that, paradoxically, a BMP inhibitor expressed ventrally under the control of Dorsal is required for establishing BMP peak levels at the Dev Genes Evol (2011) 221:255–279 dorsal midline. This observation remained a conundrum until the suggestion was made that BMP–inhibitor com- plexes forming in ventral regions diffuse dorsally and thereby transport BMP to the dorsal side (Ashe and Levine 1999; Ferguson and Anderson 1991, 1992). Here, the inhibitor is cleaved and BMP is released. Since the protease cleaving the inhibitor is not confined to the dorsal midline, it was still hard to understand how this mechanism accounts for spatial precision. A general mathematical treatment of the reaction-diffusion system encompassing BMP, the inhibitor and the protease revealed that rather simple interactions are sufficient to reproduce the experimentally observed patterning capability (Eldar et al. 2002; Meinhardt and Roth 2002). The authors started with general model equations for the three-component reaction-diffusion sys- tem and calculated the steady-state BMP activation profile for thousands of randomly chosen parameter sets. The parameters included rate constants for production, decay and diffusion of the individual proteins and complexes. Parameter sets, which produced steady-state solutions mirroring the experimentally observed BMP activation pattern, were tested for robustness using the following rationale: Genetic experiments show that reducing the gene dosage of some network components by a factor of two does not significantly change the final BMP activation pattern. Thus, realistic parameter sets should also lead to solutions which are robust, given twofold changes in network components. Parameter sets fulfilling this condi- tion were identified and suggested that free BMP does not significantly diffuse and the free inhibitor is not a good target for proteolysis. Thus, BMP should be predominantly transported in a complex with the inhibitor, and this complex should be the main target of proteolysis. This assumption led to simplified equations which could be treated analytically, providing precise mathematical argu- ments for the proposed mechanism. The entire approach seemed to be highly convincing (Meinhardt and Roth 2002). The derivation of an analytical expression capturing the mechanism of robustness could be almost regarded as a proof of the correctness of the proposed interactions. However, several complications arose that revealed the intricateness and unpredictability of biological systems. First, one component of the network, the BMP ligand Dpp was known for a long time to be dosage-sensitive. Indeed, dpp is the only developmental gene in Drosophila which is haplolethal, meaning that a twofold reduction of its dose leads to lethality (Irish and Gelbart 1987). Irrespective of the particular implications for the proposed mechanism, this observation raises the general question of how valid robustness is as a universal criterion for identifying appropriate modelling parameters. In the case of dpp, one could argue that its employment in different patterning contexts leads to an evolutionary optimisation problem