Emergent thresholds in genetic regulatory networks: Protein patterning in Drosophila morphogenesis

We present a general methodology in order to build mathematical models of genetic regulatory networks. This approach is based on the mass action law and on the Jacob and Monod operon model. The mathematical models are built symbolically by the \emph{Mathematica} software package \emph{GeneticNetworks}. This package accepts as input the interaction graphs of the transcriptional activators and repressors and, as output, gives the mathematical model in the form of a system of ordinary differential equations. All the relevant biological parameters are chosen automatically by the software. Within this framework, we show that threshold effects in biology emerge from the catalytic properties of genes and its associated conservation laws. We apply this methodology to the segment patterning in \emph{Drosophila} early development and we calibrate and validate the genetic transcriptional network responsible for the patterning of the gap proteins Hunchback and Knirps, along the antero-posterior axis of the \emph{Drosophila} embryo. This shows that patterning at the gap genes stage is a consequence of the relations between the transcriptional regulators and their initial conditions along the embryo.