Supplementary Text

1.1 Mathematical formalism We use simplified phenotypic models of transcriptional networks similar to [1]. Activity of tran-scriptional kernels is a function of both activator and repressor concentrations. Transcriptional regulations are modeled by Hill functions. We assume that an " OR " function is implemented for activators, while repressors act multiplicatively. For instance if kernel m is activated by two activators A 1 and A 2 and repressed by one repressor R, corresponding transcriptional activity is Hill thresholds, and ρ m the maximum transcriptional activity. Full differential equation for corresponding protein P with transcriptional activity t m is dP dt = t m − δ p P + ∆ P ∂ 2 P ∂x 2 (2) where δ p is degradation constant associated to P and ∆ P diffusion constant. For the results described here diffusion does not matter and can safely be taken to 0. Typically, gap genes positioning can be well captured with one kernel t m per gap gene. For pair-rule genes (eve, ftz) we need to have several independent stripe kernels. These kernels were modelled as independent 'genes', and then combined with a max function to represent the corresponding pair-rule gene. In particular, this allows us to study evolution of individual kernels and thus infer kernel homologies. Considering eve as an example, each stripe kernel is described by Equations 1-2 with the appropriate activators and inhibitors and zero diffusion constant. The output of each module is then fed through an additional Michaelis-Menten expression to normalize the various kernels to a common scale and the result combined with the same " OR " function as before.