Mathematical and computational modeling of genetic regulatory networks promises to uncover the fundamental principles governing biological systems in an integrarive and holistic manner. It also paves the way toward the development of systematic approaches for effective therapeutic intervention in disease. The central theme in this paper is the Boolean formalism as a building block for modeling complex, large-scale, and dynamical networks of genetic interactions. We discuss the goals of modeling genetic networks as well as the data requirements. The Boolean formalism is justified from several points of view. We then introduce Boolean networks and discuss their relationships to nonlinear digital filters. The role of Boolean networks in understanding cell differentiation and cellular functional states is discussed. The inference of Boolean networks from real gene expression data is considered from the viewpoints of computational learning theory and nonlinear signal processing, touching on computational complexity of learning and robustness. Then, a discussion of the need to handle uncertainty in a probabilistic framework is presented, leading to an introduction of probabilistic Boolean networks and their relationships to Markov chains. Methods for quantifying the influence of genes on other genes are presented. The general question of the potential effect of individual genes on the global dynamical network behavior is considered using stochastic perturbation analysis. This discussion then leads into the problem of target identification for therapeutic intervention via the development of several computational tools based on first-passage times in Markov chains. Examples from biology are presented throughout the paper. 1

Probabilistic Boolean Networks, which form a subclass of Markovian Genetic Regulatory Networks, have been recently introduced as a rule-based paradigm for modeling gene regulatory networks. In an earlier paper, we introduced external control into Markovian Genetic Regulatory networks. More precisely, given a Markovian genetic regulatory network whose state transition probabilities depend on an external (control) variable, a Dynamic Programming-based procedure was developed by which one could choose the sequence of control actions that minimized a given performance index over a finite number of steps. The control algorithm of that paper, however, could be implemented only when one had perfect knowledge of the states of the Markov Chain.This paper presents a control strategy that can be implemented in the imperfect information case, and makes use of the available measurements which are assumed to be probabilistically related to the states of the underlying Markov Chain.

Probabilistic Boolean Networks (PBNs) were recently introduced as mod- els of gene regulatory networks. The dynamical behavior of PBNs, which are probabilistic generalizations of Boolean networks, can be studied using Markov chain theory. In particular, the steady-state or long-run behavior of PBNs may reflect the phenotype or functional state of the cell. Approaches to alter the steady-state behavior in a specific prescribed manner, in cases of aberrant cellular states, such as tumorigenesis, would be highly beneficial. This paper develops a methodology for altering the steady-state probabil- ities of certain states or sets of states with minimal modifications to the underlying rule-based structure. This approach is framed as an optimization problem that we propose to solve using genetic algorithms, which are well suited for capturing the underlying structure of PBNs and are able to locate the optimal solution in a highly efficient manner. Several computer simulation experiments support the proposed methodology.

this paper is relatively small, it suggests that models incorporating rule-based transitions among states have a capacity to mimic biology. The ability of such models to enhance our understanding of biological regulation should be further tested by systematically examining the characteristics of the rules and interconnections that lead to stabilization and switch-like transitions, and by building larger networks that incorporate more extensive prior knowledge of regulatory relationships and more extensive experimental observations of the di#erent stable states the network can occupy. Acknowledgments The authors wish to thank Dr. Shmulevich for his insightful suggestions on the areas of probabilistic Boolean network and Markov chain simulation. Appendix A. Proof Related to Eq. (4) Proof. The following is to prove the sum of Eq. (4) over all possible states, i.e., the sum of transition probability from a state to all possible states, is 1

Motivation: Intervention in a gene regulatory network is used to help it avoid undesirable states, such as those associated with a disease. Several types of intervention have been studied in the framework of a probabilistic Boolean network (PBN), which is essentially a finite collection of Boolean networks in which at any discrete time point the gene state vector transitions according to the rules of one of the constituent networks. For an instantaneously random PBN, the governing Boolean network is randomly chosen at each time point. For a context-sensitive PBN, the governing Boolean network remains fixed for an interval of time until a binary random variable determines a switch. The theory of automatic control has been previously applied to find optimal strategies for manipulating external (control) variables that affect the transition probabilities of an instantaneously random PBN to desirably affect its dynamic evolution over a finite time horizon. This paper extends the methods of external control to context-sensitive PBNs.

Probabilistic Boolean Networks (PBNs) comprise a graphical model based on uncertain rule-based dependencies between nodes and have been proposed as a model for genetic regulatory networks. As with any algebraic strucicf theckxx--zkfjx#[xk of important mappings between PBNs isckT--#G for both theory andapplic-kfjj This paper treats the ckxjH[[kfjj of mappings to alter PBNstruc-#V while at the same time maintaining cintaining with the original probability strucilit It ctkx[[jH projecHkfj onto sub-networks, adjuncwork of new nodes, resolution reducuti mappings formed by merging nodes, and morphological mappings on the graph structure of the PBN. It places PBNs in the framework of many-sorted algebras and in that context defines homomorphisms between PBNs.

Motivation: We have hypothesized that the construction of transcriptional regulatory networks using a method that optimizes connectivity would lead to regulation consistent with biological expectations. A key expectation is that the hypothetical networks should produce a few, very strong attractors, highly similar to the original observations, mimicking biological state stability and determinism. Another central expectation is that, since it is expected that the biological control is distributed and mutually reinforcing, interpretation of the observations should lead to a very small number of connection schemes.

Probabilistic Boolean networks (PBNs) have recently been introduced as a promising class of models of genetic regulatory networks. The dynamic behaviour of PBNs can be analysed in the context of Markov chains. A key goal is the determination of the steady-state (long-run) behaviour of a PBN by analysing the corresponding Markov chain. This allows one to compute the long-term influence of a gene on another gene or determine the long-term joint probabilistic behaviour of a few selected genes. Because matrix-based methods quickly become prohibitive for large sizes of networks, we propose the use of Monte Carlo methods. However, the rate of convergence to the stationary distribution becomes a central issue. We discuss several approaches for determining the number of iterations necessary to achieve convergence of the Markov chain corresponding to a PBN. Using a recently introduced method based on the theory of two-state Markov chains, we illustrate the approach on a sub-network designed from human glioma gene expression data and determine the joint steadystate probabilities for several groups of genes. Copyright # 2003 John Wiley & Sons, Ltd.

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This paper considers key issues in the emerging field of genomic signal processing and its relationship to functional genomics. It focuses on some of the biological mechanisms driving the development of genomic signal processing, in addition to their manifestation in gene-expression-based classification and genetic network modeling. Certain problems are inherent. For instance, small-sample error estimation, variable selection, and model complexity are important issues for both phenotype classification and expression prediction used in network inference. A long-term goal is to develop intervention strategies to drive network behavior, which is briefly discussed. It is hoped that this nontechnical paper demonstrates that the field of signal processing has the potential to impact and help drive genomics research