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 Mathematica software package GeneticNetworks. This package accepts as input the interaction graphs of the transcriptional activators and repressors of a biological process 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 concentration dependent 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 Drosophila early development and we calibrate the genetic transcriptional network responsible for the patterning of the gap gene proteins Hunchback and Knirps, along the antero-posterior axis of the Drosophila embryo. In this approach, the zygotically produced proteins Hunchback and Knirps do not diffuse along the antero-posterior axis of the embryo of Drosophila, developing a spatial pattern due to concentration dependent thresholds. This shows that patterning at the gap genes stage can be explained by the concentration gradients along the embryo of the transcriptional regulators.
Highlights
A genetic regulatory network is an ensemble of interactions in a biological process involving proteins, genes and mRNAs
In the Methods subsection, we briefly review the mass action law of chemical kinetics and we introduce the collision graphs associated with the mass action law
Under the hypothesis of homogeneity of the solution where reactions occur, the mass action law asserts that the time evolution of the concentrations of the chemical substances is described by the system of ordinary differential equations, dAj dt
Summary
A genetic regulatory network is an ensemble of interactions in a biological process involving proteins, genes and mRNAs. A genetic regulatory networks is described by a graph where vertices represent genes, proteins, enzymes or other chemical substances. In this example, we have V ~fbcd,hb,BCD,HB,KNI,TLLg, Ea~f(bcd,BCD),(hb,HB), (BCD,HB),(BCD,KNI)g and Er~f(HB,KNI),(KNI,HB),(TLL, KNI )g
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