A software tool to model genetic regulatory networks. Applications to the modeling of threshold phenomena and of spatial patterning in Drosophila

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.

Figure 1. Graph describing the genetic network associated with the production of proteins Bicoid (BCD), Hunchback (HB), Knirps (KNI) and Tailless (TLL) in Drosophila early development.

mRNAs hunchback and bicoid are represented by hb and bcd respectively. Arrows represent activations and are listed in the set of ordered pairs . Lines with perpendicular endings represent repressions and are listed in the set .

Figure 2. Jacob and Monod operon model for the regulation of protein production.

The transcription is regulated by the activators and the repressors binding to the binding sites of the gene.

Figure 3. Double graph associated with the input strings (13) for the GeneticNetworks software package.

Figure 4. Regulation graph describing a self-activating protein.

Figure 5. Dependence of the protein steady state on the total concentration of the gene.

Below the bifurcation or threshold , the equilibrium value of protein concentration is zero, while above it takes the value . The parameters are: , , and .

Figure 6. Genetic regulatory network for the production of protein with one activator and two repressors and .

Figure 7. Steady states of proteins , and for the genetic regulatory network of Figure 6.

The steady state of protein shows a spiky profile, resulting from the inhibitory action of proteins and . In this model, we have considered that the concentrations of and are constant in time and non-homogeneous in space. The activator protein has been considered constant along the spatial region.

Figure 8. Concentration of protein Hunchback (HB) at the end of cleavage cycle 13, and of Bicoid (BCD) and Tailless (TLL) proteins at the cleavage cycle 14, along the antero-posterior axis of the embryo of Drosophila.

The embryo length has been scaled from to , and the units in the vertical axis are proportional to protein concentration. The data has been taken from the FlyEx database. The continuous curves represent the fitted mean distribution of the concentration of proteins calculated from the data of embryos. These curves are the initial conditions for a model obtained with the software package GeneticNetworks for the production of the proteins KNI and zygotically produced HB, during cleavage cycle 14.

Figure 9. The dots with error bars are the mean experimental profiles of the proteins HB and KNI at cleavage cycle 14, taken along the antero-posterior axis of the embryo, for several Drosophila embryos.

The mean distribution of the concentration of proteins has been calculated from the data of embryos, taken from the FlyEx database. The embryo length has been scaled from to , and the units in the vertical axis are proportional to protein concentration. The continuous curves show the predictions of the model of production of proteins HB and KNI. The model equations have been obtained with the software package GeneticNetworks.