In order to understand the functioning of organisms on the molecular level, we need to know which genes are expressed, when and where in the organism, and to which extent. The regulation of gene expression is achieved through genetic regulatory systems structured by networks of interactions between DNA, RNA, proteins, and small molecules. As most genetic regulatory networks of interest involve many components connected through interlocking positive and negative feedback loops, an intuitive understanding of their dynamics is hard to obtain. As a consequence, formal methods and computer tools for the modeling and simulation of genetic regulatory networks will be indispensable. This paper reviews formalisms that have been employed in mathematical biology and bioinformatics to describe genetic regulatory systems, in particular directed graphs, Bayesian networks, Boolean networks and their generalizations, ordinary and partial differential equations, qualitative differential equations, stochastic equations, and rule-based formalisms. In addition, the paper discusses how these formalisms have been used in the simulation of the behavior of actual regulatory systems. Key words: genetic regulatory networks, mathematical modeling, simulation, computational biology.

c biological data sets. The ability to adequately solve the inverse problem may enable in-depth analysis of complex dynamic systems in biology and other fields. Binary models of genetic networks Virtually all molecular and cellular signaling processes involve several inputs and outputs, forming a complex feedback network. The information for the construction and maintenance of this signaling system is stored in the genome. The DNA sequence codes for the structure and molecular dynamics of RNA and proteins, in turn determining biochemical recognition or signaling processes. The regulatory molecules that control the expression of genes are themselves the products of other genes. Effectively, genes turn each other on and off within a proximal genetic network of transcriptional regulators (Somogyi and Sniegoski, 1996). Furthermore, complex webs involving various intra- and extracellular signaling systems on the one hand depend on the expression of the genes that encode them, and on the

... for inferring genetic network architectures from state transition tables which correspond to time series of gene expression patterns, using the Boolean network model. Their results of computational experiments suggested that a small number of state transition (INPUT/OUTPUT) pairs are sufficient in order to infer the original Boolean network correctly. This paper gives a mathematical proof for their observation. Precisely, this paper devises a much simpler algorithm for the same problem and proves that, if the indegree of each node (i.e., the number of input nodes to each node) is bounded by a constant, only O(log n) state transition pairs (from 2n pairs) are necessary and sufficient to identify the original Boolean network of n nodes correctly with high probability. We made computational experiments in order to expose the constant factor involved in O(log n) notation. The computational results show that the Boolean network of size 100,000 can be identified by our algorithm from about 100 INPUT/OUTPUT pairs if the maximum indegree is bounded by 2. It is also a merit of our algorithm that the algorithm is conceptually so simple that it is extensible for more realistic network models.

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Recently, there has been much interest in reverse engineering genetic networks from time series data. In this paper, we show that most of the proposed discrete time models — including the boolean network model [Kau93, SS96], the linear model of D’haeseleer et al. [DWFS99], and the nonlinear model of Weaver et al. [WWS99] — are all special cases of a general class of models called Dynamic Bayesian Networks (DBNs). The advantages of DBNs include the ability to model stochasticity, to incorporate prior knowledge, and to handle hidden variables and missing data in a principled way. This paper provides a review of techniques for learning DBNs. Keywords: Genetic networks, boolean networks, Bayesian networks, neural networks, reverse engineering, machine learning. 1

In order to cope with the large amounts of data that have become available in genomics, mathematical tools for the analysis of networks of interactions between genes, proteins, and other molecules are indispensable. We present a method for the qualitative simulation of genetic regulatory networks, based on a class of piecewise-linear (PL) differential equations that has been well-studied in mathematical biology. The simulation method is well-adapted to state-of-the-art measurement techniques in genomics, which often provide qualitative and coarsegrained descriptions of genetic regulatory networks. Given a qualitative model of a genetic regulatory network, consisting of a system of PL differential equations and inequality constraints on the parameter values, the method produces a graph of qualitative states and transitions between qualitative states, summarizing the qualitative dynamics of the system. The qualitative simulation method has been implemented in Java in the computer tool Genetic Network Analyzer.

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Many natural processes consist of networks of interacting elements which a ect each other's state over time, the dynamics depending on the pattern of connections and the updating rules for each element. Genomic regulatory networks are arguably networks of this sort. An attempt to understand genomic networks would bene t from the context of a general theory of discrete dynamical networks which is currently emerging. A key notion here is global dynamics, whereby state-space is organized into basins of attraction, objects that have only recently become accessible by computer simulation of idealized models 12;13, in particular \random Boolean networks". Cell types have been explained as attractors in genomic networks 5, where the network architecture is biased to achieve a balance between stability and adaptability in response to perturbation 3. Based on computer simulations using the software Discrete Dynamics Lab (DDLab) 15, these ideas are described, as well as order-chaos measures on typical trajectories that further characterize network dynamics. 1

We provide the first classification of different types of RandomBoolean Networks (RBNs). We study the differences of RBNs depending on the degree of synchronicity and determinism of their updating scheme. For doing so, we first define three new types of RBNs. We note some similarities and differences between different types of RBNs with the aid of a public software laboratory we developed. Particularly, we find that the point attractors are independent of the updating scheme, and that RBNs are more different depending on their determinism or non-determinism rather than depending on their synchronicity or asynchronicity. We also show a way of mapping non-synchronous deterministic RBNs into synchronous RBNs. Our results are important for justifying the use of specific types of RBNs for modelling natural phenomena.

this article, I review the theoretical and computational approaches used to: (i) identify genes differentially expressed (across cell types, developmental stages, pathological conditions, etc.); (ii) identify genes expressed in a coordinated manner across a set of conditions; and (iii) delineate clusters of genes sharing coherent expression features, eventually defining global biological pathways