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.

Given the immanent gene expression mapping covering whole genomes during development, health and disease, we seek computational methods to maximize functional inference from such large data sets. Is it possible, in principle, to completely infer a complex regulatory network architecture from input/output patterns of its variables? We investigated this possibility using binary models of genetic networks. Trajectories, or state transition tables of Boolean nets, resemble time series of gene expression. By systematically analyzing the mutual information between input states and output states, one is able to infer the sets of input elements controlling each element or gene in the network. This process is unequivocal and exact for complete state transition tables. We implemented this REVerse Engineering ALgorithm (REVEAL) in a C program, and found the problem to be tractable within the conditions tested so far. For n=50 (elements) and k=3 (inputs per element), the analysis of incomplete state transition tables (100 state transition pairs out of a possible 10 15) reliably produced the original rule and wiring sets. While this study is limited to synchronous Boolean

... 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

The inference of genetic interactions from measured expression data is one of the most challenging tasks of modern functional genomics. When successful, the learned network of regulatory interactions yields a wealth of useful information. An inferred genetic network contains information about the pathway to which a gene belongs and which genes it interacts with. Furthermore, it explains the gene's function in terms of how it influences other genes and indicates which genes are pathway initiators and therefore potential drug targets. Obviously, such wealth comes at a price and that of genetic network modeling is that it is an extremely complex task. Therefore, it is necessary to develop sophisticated computational tools that are able to extract relevant information from a limited set of microarray measurements and integrate this with different information sources, to come up with reliable hypotheses of a genetic regulatory network. Thus far, a multitude of modeling approaches has been proposed for discovering genetic networks. However, it is unclear what the advantages and disadvantages of each of the different approaches are and how their results can be compared. In this review, genetic network models are put in a historical perspective that explains why certain models were introduced. Various modeling assumptions and their consequences are also highlighted. In addition, an overview of the principal differences and similarities between the approaches is given by considering the qualitative properties of the chosen models and their learning strategies. In pharmacogenomics and related areas, a lot of research is directed towards discovering, understanding and/or controlling the outcome of some particular biological pathway. Numerous examples exist where the manipulation of a key enzyme in such a pathway did not lead to the desired effect We know that the structure of complex genetic and biochemical networks lies hidden in the sequence information of our DNA but it is far from trivial to predict gene expression from the sequence code alone. The current availability of microarray measurements of thousands of gene expression levels during the course of an experiment or after the knockout of a gene provides a wealth of complementary information that may be exploited to unravel the complex interplay between genes. It now becomes possible to start answering some of the truly challenging questions in systems biology. For example, is it possible to model these genetic interactions as a large network of interacting elements and can these interactions be effectively learned from measured expression data? Since Kauffman Although the behavior and properties of artificial networks match the observations made in real biological systems well, the field of genetic network modeling has yet to reach its full maturity. The automatic discovery of genetic networks from expression data alone is far from trivial because of the combinatorial nature of the problem and the poor information content of 1 For reasons of brevity, the authors consistently refer only to the first author of each reference.

Mathematical models are useful for providing a framework for integrating data and gaining insights into the static and dynamic behavior of complex biological sys-tems such as networks of interacting genes. We review the dynamic behaviors ex-pected from model gene networks incorporating common biochemical motifs, and we compare current methods for modeling genetic networks. A common modeling technique, based on simply modeling genes as ON–OFF switches, is readily imple-mented and allows rapid numerical simulations. However, this method may predict dynamic solutions that do not correspond to those seen when systems are modeled with a more detailed method using ordinary differential equations. Until now, the majority of gene network modeling studies have focused on determining the types of dynamics that can be generated by common biochemical motifs such as feed-back loops or protein oligomerization. For example, these elements can generate multiple stable states for gene product concentrations, state-dependent responses to stimuli, circadian rhythms and other oscillations, and optimal stimulus frequencies for maximal transcription. In the future, as new experimental techniques increase the ease of characterization of genetic networks, qualitative modeling will need to be supplanted by quantitative models for specific systems. c © 2000 Society for Mathematical Biology 1.

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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.

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