Dev Genes Evol (2011) 221:255–279 which has no ideal robust solution. However, the mere fact that such exceptions exist brings into question the blind reliance on robustness for selecting appropriate parameter sets. It seems that assumptions about robustness have to be analysed in each particular case. Second, biochemical studies and additional modelling approaches revealed different interaction networks, which could also produce sharp BMP signalling peaks. An interesting assumption was that Sog acts as a competitive inhibitor of BMP binding to its receptor and that receptor- bound BMP is degraded (Mizutani et al. 2005). Thus, Sog protects BMP from receptor-mediated degradation and thereby enhances its range of action. Another paper showed that the main signalling molecule is a heterodimer of two BMPs and that it is the heterodimer which is bound by the inhibitor, while mathematical modelling had suggested alternative interactions (Shimmi et al. 2005). While the former two papers demonstrated that the interactions of even a small number of secreted proteins and their receptors can lead to several alternative models which can account for observed phenotypes, another publication opened up completely new aspects of the system. Eldar et al. (2002) could still state: ‘No apparent transcriptional feedback, which might account for the robustness of dorsal patterning, has been identified so far’. However, 3 years later, precisely such a feedback loop was identified (Wang and Ferguson 2005). A transcription factor was shown both to be a target of, and to promote, dorsal BMP signalling. Apparently, this feedback loop promotes future BMP binding to its receptor as a function of previous signalling strength. This finding led to a completely new series of modelling approaches (Umulis et al. 2006). In addition, new precise measurements of BMP signalling in whole embryos provided the basis for organism-scale modelling using realistic geometries (Umulis et al. 2010). Unfortu- nately, until now the molecular basis of the positive feedback has remained elusive. The most recent publication introduces eight different models for possible positive feedback mechanisms encompassing a variety of cell biological and biochemical details as well as newly observed interactions with extracellular matrix molecules. The most important outcome of the paper is that positive feedback mechanisms are indeed producing the best fit to the data and that, among these mechanisms, a positive feedback involving a surface-bound BMP binding protein is slightly superior to other mechanisms. However, the modelling approaches veered away from elegant analytical results to numerical solutions of high dimensional systems of more than ten coupled partial differential equations and 17 parameters. Despite whole embryo 3D representations of BMP signalling in wild-type and mutant embryos at different developmental stages, and even comparisons of different Drosophila species, the discriminatory power of 269 modelling results appears to be weak with regard to the mechanistic alternatives. Even the most ardent aficionado of pattern formation theory might come to the conclusion that we have moved further away from understanding the system. The complexity of the embryo appears to evade full mathematical treatment. (2) Segmentation in Drosophila is probably the para- digm for a complex hierarchical patterning process which generates fine-grained spatial information starting from broadly distributed morphogen gradients (Akam 1987). Within 2 h of embryonic development corresponding to the blastoderm stage, maternal gradients activate a cascade of transcription factors with increasingly refined expression domains. First, the gap genes are activated in broad domains and in turn control the pair-rule genes which represent the first level of periodic gene expression in the embryo, albeit with double-segment periodicity. Segmental periodicity is only reached at the next tier, the expression of the segment polarity genes, which receive their regulatory input largely from the pair-rule genes. At the end of the blastoderm stage, the patterning process has reached its highest possible resolution: single-cell wide stripes of segmental polarity gene expression defining the position of segment and compartment boundaries. The gap and pair- rule genes code for transcription factors which can diffuse between nuclei since patterning occurs before the cell membranes are formed. Thus, the system is basically an interaction network of factors which mutually regulate their expression. Besides nuclear divisions, no morphogenetic events take place, allowing a description of the process with one spatial coordinate representing the AP axis (or rather the position of the nuclei along the AP axis). Inspired by models for neural networks (Hopfield 1984), a data-driven modelling approach for gene regulatory networks was devised called the connectionist gene circuit method and was specifically applied to the Drosophila segmentation cascade (Mjolsness et al. 1991; Reinitz and Sharp 1995). Usually, theoretical modelling is based on proposed interactions of the relevant components inferred from genetic and molecular experiments. The gene circuit method uses the opposite strategy and therefore was also characterised as a ‘reverse-engineering’ approach (Perkins et al. 2006). Only minimal a priori assumptions are made about potential regulatory interactions. A system of general model equations is formulated which describes the concen- tration change of each transcription factor in a particular nucleus as a function of gene expression, diffusion and decay. The expression of a particular transcription factor depends on regulatory inputs from other transcription factors. To capture these inputs, an interconnectivity matrix T is introduced, in which the regulatory effect of gene a on gene b is represented by the matrix element Tab. Depending on whether this element is positive or negative, gene a