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.

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this paper is the interactions occurring within specific complexes. These were obtained from the MIPS complexes catalog (Fellenberg et al. 2000), which represents a carefully annotated, comprehensive data set of protein complexes culled from the scientific literature. In addition, we looked at other types of protein-protein interactions from large "aggregated" data sets collecting many heterogeneous pair-wise interactions. We collected these from the MIPS catalogs of physical and genetic interactions (Fellenberg et al. 2000), databases of interacting proteins (DIP and BIND) (Bader and Hogue 2000; Xenarios 2000), and a comprehensive collection of yeast two-hybrid experiments (Cagney et al. 2000; lto et al. 2000; Schwikowski et al. 2000; Uetz et al. 2000; Uetz and Hughes 2000; lto et al. 2001). These interactions are subdivided into groups based on their method of discovery. They include physical interactions (e.g., collected through coimmunoprecipitation and copurification), genetic interactions (e.g., determined through genetic means such as synthetic lethality or suppression experiments), and yeast twohybrid pairs

Motivation: We describe a new approach to the analysis of gene expression data coming from DNA array experiments, using an unsupervised neural network. DNA array technologies allow monitoring thousands of genes rapidly and efficiently. One of the interests of these studies is the search for correlated gene expression patterns, and this is usually achieved by clustering them. The Self-Organising Tree Algorithm, (SOTA) (Dopazo,J. and Carazo,J.M. (1997) J. Mol. Evol., 44, 226--233), is a neural network that grows adopting the topology of a binary tree. The result of the algorithm is a hierarchical cluster obtained with the accuracy and robustness of a neural network. Results: SOTA clustering confers several advantages over classical hierarchical clustering methods. SOTA is a divisive method: the clustering process is performed from top to bottom, i.e. the highest hierarchical levels are resolved before going to the details of the lowest levels. The growing can be stopped at the desired hierarchical level. Moreover, a criterion to stop the growing of the tree, based on the approximate distribution of probability obtained by randomisation of the original data set, is provided. By means of this criterion, a statistical support for the definition of clusters is proposed. In addition, obtaining average gene expression patterns is a built-in feature of the algorithm. Different neurons defining the different hierarchical levels represent the averages of the gene expression patterns contained in the clusters. Since SOTA runtimes are approximately linear with the number of items to be classified, it is especially suitable for dealing with huge amounts of data. The method proposed is very general and applies to any data providing that they can be coded as a series of numbers and t...

Motivation: Clustering is a useful exploratory technique for the analysis of gene expression data. Many different heuristic clustering algorithms have been proposed in this context. Clustering algorithms based on probability models offer a principled alternative to heuristic algorithms. In particular, model-based clustering assumes that the data is generated by a finite mixture of underlying probability distributions such as multivariate normal distributions. The issues of selecting a 'good' clustering method and determining the 'correct' number of clusters are reduced to model selection problems in the probability framework. Gaussian mixture models have been shown to be a powerful tool for clustering in many applications.

Motivation: Increasingly, biological processes are being studied through time series of RNA expression data collected for large numbers of genes. Because common processes may unfold at varying rates in different experiments or individuals, methods are needed that will allow corresponding expression states in different time series to be mapped to one another. Results: We present implementations of time warping algorithms applicable to RNA and protein expression data and demonstrate their application to published yeast RNA expression time series. Programs executing two warping algorithms are described, a simple warping algorithm and an interpolative algorithm, along with programs that generate graphics that visually present alignment information. We show time warping to be superior to simple clustering at mapping corresponding time states. We document the impact of statistical measurement noise and sample size on the quality of time alignments, and present issues related to statistical assessment of alignment quality through alignment scores. We also discuss directions for algorithm improvement including development of multiple time series alignments and possible applications to causality searches and non-temporal processes (`concentration warping'). Availability: Academic implementations of alignment programs genewarp and genewarpi and the graphics generation programs grphwarp and grphwarpi are available as Win32 system DOS box executables on our web site along with documentation on their use. The publicly available data on which they were demonstrated may be found at http://genome-www.stanford.edu/cellcycle/. Postscript files generated by grphwarp and grphwarpi may be directly printed or viewed using GhostView software available at http://www.cs.wisc.edu/#ghost/. Con...

Many clustering algorithms have been proposed to analyze gene expression data, but little guidance is available to help choose among them. We provide a systematic and quantitative framework to assess the results of clustering algorithms. A typical gene expression data set contains measurements of the expression levels of a fixed set of genes under various experimental conditions. Clustering algorithms attempt to partition the genes into groups exhibiting similar patterns of variation in expression level, hopefully revealing biologically meaningful patterns of activity or control. Our methodology is to apply a clustering algorithm to the data from all but one experimental condition. The remaining condition is used to assess the predictive power of the resulting clusters---meaningful clusters should exhibit less variation in the remaining condition than clusters formed by coincidence. We have successfully applied the methodology to compare three clustering algorithms on three p...

Microarray experiments have been extensively used for simultaneously measuring DNA expression levels of thousands of genes in genome research. A key step in the analysis of gene expression data is the clustering of genes into groups that show similar expression values over a range of conditions. Since only a small subset of the genes participate in any cellular process of interest, by focusing on subsets of genes and conditions, we can lower the noise induced by other genes and conditions — a co-cluster characterizes such a subset of interest. Cheng and Church [3] introduced an effective measure of co-cluster quality based on mean squared residue. In this paper, we use two similar squared residue measures and propose two fast k-means like co-clustering algorithms corresponding to the two residue measures. Our algorithms discover k row clusters and l column clusters simultaneously while monotonically decreasing the respective squared residues. Our co-clustering algorithms inherit the simplicity, efficiency and wide applicability of the k-means algorithm. Minimizing the residues may also be formulated as trace optimization problems that allow us to obtain a spectral relaxation that we use for a principled initialization for our iterative algorithms. We further enhance our algorithms by an incremental local search strategy that helps avoid empty clusters and escape poor local minima. We illustrate co-clustering results on a yeast cell cycle dataset and a human B-cell lymphoma dataset. Our experiments show that our co-clustering algorithms are efficient and are able to discover coherent co-clusters. Keywords: Gene-expression, co-clustering, biclustering, residue, spectral relaxation

Abstract—DNA microarray technology has now made it possible to simultaneously monitor the expression levels of thousands of genes during important biological processes and across collections of related samples. Elucidating the patterns hidden in gene expression data offers a tremendous opportunity for an enhanced understanding of functional genomics. However, the large number of genes and the complexity of biological networks greatly increases the challenges of comprehending and interpreting the resulting mass of data, which often consists of millions of measurements. A first step toward addressing this challenge is the use of clustering techniques, which is essential in the data mining process to reveal natural structures and identify interesting patterns in the underlying data. Cluster analysis seeks to partition a given data set into groups based on specified features so that the data points within a group are more similar to each other than the points in different groups. A very rich literature on cluster analysis has developed over the past three decades. Many conventional clustering algorithms have been adapted or directly applied to gene expression data, and also new algorithms have recently been proposed specifically aiming at gene expression data. These clustering algorithms have been proven useful for identifying biologically relevant groups of genes and samples. In this paper, we first briefly introduce the concepts of microarray technology and discuss the basic elements of clustering on gene expression data. In particular, we divide cluster analysis for gene expression data into three categories. Then, we present specific challenges pertinent to each clustering category and introduce several representative approaches. We also discuss the problem of cluster validation in three aspects and review various methods to assess the quality and reliability of clustering results. Finally, we conclude this paper and suggest the promising trends in this field. Index Terms—Microarray technology, gene expression data, clustering.