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.

Motivation: We have used state-space models to reverse engineer transcriptional networks from highly replicated gene expression profiling time series data obtained from a wellestablished model of T-cell activation. State space models are a class of dynamic Bayesian networks that assume that the observed measurements depend on some hidden state variables that evolve according to Markovian dynamics.These hidden variables can capture effects that cannot be measured in a gene expression profiling experiment, e.g. genes that have not been included in the microarray, levels of regulatory proteins, the effects of messenger RNA and protein degradation, etc. Results: Bootstrap confidence intervals are developed for parameters representing ‘gene–gene ’ interactions over time. Our models represent the dynamics of T-cell activation and provide a methodology for the development of rational and experimentally testable hypotheses. Availability: Supplementary data and Matlab computer source code will be made available on the web at the URL given below.

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Introduction Inference of gene networks from gene expression measurements is a major challenge in Systems Biology. If gene networks can be infered correctly, it can lead to a better understanding of cellular processes, and, therefore, have applications to drug discovery, disease studies, and other areas. Bayesian networks are a widely used approach to model gene networks 3,4,7,9,11,13 ,14,17 . In Bayesian networks, the behaviour of the gene network is modeled as a joint probability distribution for all genes. This allows a very general modeling of gene interactions. The joint probability distribution can be decomposed as a product of conditional probabilities P (X g |X 1 ,...,X n ), representing the regulation of a gene g by some genes g 1 ,...,g n . This decomposition can be represented as a directed acyclic graph. The Bayesian network model has been shown to allow finding biologically plausible gene networks 4,9 . However, the di#culty of learning Bayesian networks lies in

Abstract—In this paper, the expectation maximization (EM) algorithm is applied for modeling the gene regulatory network from gene time-series data. The gene regulatory network is viewed as a stochastic dynamic model, which consists of the noisy gene measurement from microarray and the gene regulation first-order autoregressive (AR) stochastic dynamic process. By using the EM algorithm, both the model parameters and the actual values of the gene expression levels can be identified simultaneously. Moreover, the algorithm can deal with the sparse parameter identification and the noisy data in an efficient way. It is also shown that the EM algorithm can handle the microarrary gene expression data with large number of variables but a small number of observations. The gene expression stochastic dynamic models for four real-world gene expression data sets are constructed to demonstrate the advantages of the introduced algorithm. Several indices are proposed to evaluate the models of inferred gene regulatory networks, and the relevant biological properties are discussed. Index Terms—Clustering, DNA microarray technology, expectation maximization (EM) algorithm, gene expression, modeling, time-series data. I.

The accurate estimation of gene networks from gene expression measurements is a major challenge in the field of Bioinformatics. Since the problem of estimating gene networks is NP-hard and exhibits a search space of super-exponential size, researchers are using heuristic algorithms for this task. However, little can be said about the accuracy of heuristic estimations. In order to overcome this problem, we present a general approach to reduce the search space to a biologically meaningful subspace and to find optimal solutions within the subspace in linear time. We show the e#ectiveness of this approach in application to yeast and Bacillus subtilis data.

Currently, the need arises for tools capable of unraveling the functionality of genes based on the analysis of microarray measurements. Modeling genetic interactions by means of genetic network models provides a methodology to infer functional relationships between genes. Although a wide variety of different models have been introduced so far, it remains, in general, unclear what the strengths and weaknesses of each of these approaches are and where these models overlap and differ. This paper compares different genetic modeling approaches that attempt to extract the gene regulation matrix from expression data. A taxonomy of continuous genetic network models is proposed and the following important characteristics are suggested and employed to compare the models: (1) inferential power; (2) predictive power; (3) robustness; (4) consistency; (5) stability and (6) computational cost. Where possible, synthetic time series data are employed to investigate some of these properties. The comparison shows that although genetic network modeling might provide valuable information regarding genetic interactions, current models show disappointing results on simple artificial problems. For now, the simplest models are favored because they generalize better, but more complex models will probably prevail once their bias is more thoroughly understood and their variance is better controlled.

We have developed a bioinformatics tool named PAINT that automates the promoter analysis of a given set of genes for the presence of transcription factor binding sites. Based on coincidence of regulatory sites, this tool produces an interaction matrix that represents a candidate transcriptional regulatory network. This tool currently consists of (1) a database of promoter sequences of known or predicted genes in the Ensembl annotated mouse genome database, (2) various modules that can retrieve and process the promoter sequences for binding sites of known transcription factors, and (3) modules for visualization and analysis of the resulting set of candidate network connections. This information provides a substantially pruned list of genes and transcription factors that can be examined in detail in further experimental studies on gene regulation. Also, the candidate network can be incorporated into network identification methods in the form of constraints on feasible structures in order to render the algorithms tractable for large-scale systems. The tool can also produce output in various formats suitable for use in external visualization and analysis software. In this manuscript, PAINT is demonstrated in two case studies involving analysis of differentially regulated genes chosen from two microarray data sets. The first set is from a neuroblastoma N1E-115 cell differentiation experiment, and the second set is from neuroblastoma N1E-115 cells at different time intervals following exposure to neuropeptide angiotensin II. PAINT is available for use as an agent in BioSPICE simulation and analysis framework (www.biospice.org), and can also be accessed via a WWW interface at www.dbi.tju.edu/dbi/tools/paint/.

Recent advances in gene-expression profiling technologies provide large amounts of gene expression data. This raises the possibility for a functional understanding of genome dynamics by means of mathematical modelling. As gene expression involves intrinsic noise, stochastic models are essential for better descriptions of gene regulatory networks. However, stochastic modelling for large scale gene expression data sets is still in the very early developmental stage. In this paper we present some stochastic models by introducing stochastic processes into neural network models that can describe intermediate regulation for large scale gene networks. Poisson random variables are used to represent chance events in the processes of synthesis and degradation. For expression data with normalized concentrations, exponential or normal random variables are used to realize fluctuations. Using a network with three genes, we show how to use stochastic simulations for studying robustness and stability properties of gene expression patterns under the influence of noise, and how to use stochastic models to predict statistical distributions of expression levels in a population of cells. The discussion suggests that stochastic neural network models can give better descriptions of gene regulatory networks and provide criteria for measuring the reasonableness of mathematical models.

The most fundamental problem in genetic network modeling is generally known as the dimensionality problem. Typical gene expression matrices contain measurements of thousands of genes taken over fewer than twenty time-steps. A large dynamic network cannot be learned from data with such a limited number of time-steps without the use of additional constraints, preferably derived from biological knowledge. In this paper, we present an approach that can find rough estimates of the underlying genetic network based on limited time-course gene expression data by employing the fact that gene expression measurements are relatively noisy and genetic networks are thought to be robust. The method expands the data-set by adding noisy duplicates, thereby simultaneously tackling the dimensionality problem and making the solutions more robust against (the already large) noise in the data. This simple concept is similar to adding a Tikhonov regularization term in the optimization process. In the case of linear models, the addition of noisy duplicates is equivalent to ridge regression, i.e. the sum of the squared weights is minimized as well as the prediction error. In the limiting case, it becomes even equivalent to the application of the Moore-Penrose Pseudo-Inverse to the original data. The strength of the proposed concept of adding noisy duplicates lies in the fact that it can be employed to all modelling approaches, including non-linear models. 1