The analysis of genetic regulatory networks responsible for cell differentiation and development in prokaryotes and eukaryotes will much benefit from the recent upscaling to the genomic level of experimental methods in molecular biology. Hardly imaginable only 20 years ago, the sequencing of complete genomes has become a routine job, highly automated and realized in a quasi-industrial environment. The miniaturization of techniques for the hybridization of labelled nucleic acids in solution to DNA molecules attached to a surface has given rise to DNA chips, tools for measuring the level of gene expression in a massively parallel way. The development of proteomic methods based on two-dimensional electrophoresis, mass spectrometry, and the double-hybrid system allows the identification of proteins and their interactions at the genomic scale.

In addition to high-throughput experimental methods, mathematical and bioinformatical approaches are indispensable for the analysis of genetic regulatory networks. Given the large number of components of most networks of biological interest, connected by positive and negative feedback loops inside and between cells, an intuitive comprehension of the spatiotemporal evolution of a developmental system is often difficult, if not impossible to obtain. Mathematical modeling supported by computer tools can contribute to the analysis of a regulatory network by allowing the biologist to focus on a restricted number of plausible hypotheses. The formulation of a mathematical model requires an explicit and non-ambiguous description of the hypotheses being made on the regulatory mechanisms under study. Furthermore, simulation using the model yields predictions on the behavior of the cell or embryo that can be verified experimentally.

A variety of methods for the modeling and simulation of genetic regulatory networks have been proposed in the literature. The use of formal methods to study regulatory networks is currently subject to two major constraints. First of all, the biochemical reaction mechanisms underlying the interactions are usually not or incompletely known. This prevents the formulation of detailed kinetic models, such as those developed for the genetic switch controlling phage lambda growth or the feedback mechanisms regulating tryptophan synthesis in E. coli. A second constraint arises from the general absence of quantitative information on most kinetic parameters and molecular concentrations. As a consequence, traditional methods for numerical analysis are difficult to apply.

Few of the modeling and simulation methods that have been developed so far are capable of handling these constraints. In this chapter, we review two related methods that form an exception to the rule. On the one hand, we present the qualitative simulation of genetic regulatory networks described by piecewise-linear (PL) differential equations. On the other hand, we provide an overview of the analysis of genetic regulatory networks by means of asynchronous, multivalued logic. Both methods are based on coarse-grained models that, while abstracting from the precise molecular mechanisms involved, capture essential aspects of gene regulation. Moreover, these methods allow a qualitative analysis of the dynamics of the genetic regulatory systems to be carried out. Although the methods are based on different formalisms, differential and logical equations, they share important biological intuitions, in particular the description of gene activation in terms of on/off-switches.

Both methods are supported by computer tools that allow the user to enter a model of a genetic regulatory network, simulate or analyze its qualitative behavior, and interpret the results in biological terms. We will illustrate the method and the tools by their application to two model systems for development: the choice between vegetative growth and sporulation in B. subtilis and the genetic control of the segmentation in the early Drosophila embryo. These examples show that, in order to understand the functioning of an organism in terms of the interactions in regulatory networks, it is not always necessary to model the process down to individual biochemical reactions. In fact, when a global understanding of the evolution of spatiotemporal patterns of gene expression is sought, we suggest that it might be more profitable to employ coarse-grained and qualitative models of the type discussed in this chapter.