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

TL;DR

This paper introduces a computational framework for modeling genetic regulatory networks using mass action laws and operon models, demonstrating how threshold effects in gene regulation emerge and applying it to Drosophila embryonic development.

Contribution

The authors develop a symbolic modeling approach with Mathematica that automatically generates differential equations from interaction graphs, revealing emergent biological thresholds.

Findings

Threshold effects arise from gene catalysis and conservation laws.

Validated models for Drosophila gap gene patterning.

Patterning depends on transcriptional regulator relations and initial conditions.

Abstract

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.