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From Boolean to Probabilistic Boolean Networks as Models of Genetic Regulatory Networks
 Proc. IEEE
, 2002
"... 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 di ..."
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Cited by 124 (23 self)
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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, largescale, 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 firstpassage times in Markov chains. Examples from biology are presented throughout the paper. 1
External control in Markovian genetic regulatory networks: the imperfect information case
 Machine Learning
, 2004
"... Probabilistic Boolean Networks, which form a subclass of Markovian Genetic Regulatory Networks, have been recently introduced as a rulebased paradigm for modeling gene regulatory networks. In an earlier paper, we introduced external control into Markovian Genetic Regulatory networks. More precisely ..."
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Cited by 79 (27 self)
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Probabilistic Boolean Networks, which form a subclass of Markovian Genetic Regulatory Networks, have been recently introduced as a rulebased 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 Programmingbased 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.
Intervention in contextsensitive probabilistic Boolean networks
, 2005
"... 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 n ..."
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Cited by 45 (15 self)
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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 contextsensitive 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 contextsensitive PBNs.
Optimal infinitehorizon control for probabilistic Boolean networks
 IEEE Transactions on Signal Processing
"... Abstract—External control of a genetic regulatory network is used for the purpose of avoiding undesirable states, such as those associated with disease. Heretofore, intervention has focused on finitehorizon control, i.e., control over a small number of stages. This paper considers the design of opt ..."
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Cited by 35 (14 self)
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Abstract—External control of a genetic regulatory network is used for the purpose of avoiding undesirable states, such as those associated with disease. Heretofore, intervention has focused on finitehorizon control, i.e., control over a small number of stages. This paper considers the design of optimal infinitehorizon control for contextsensitive probabilistic Boolean networks (PBNs). It can also be applied to instantaneously random PBNs. The stationary policy obtained is independent of time and dependent on the current state. This paper concentrates on discounted problems with bounded cost per stage and on averagecostperstage problems. These formulations are used to generate stationary policies for a PBN constructed from melanoma geneexpression data. The results show that the stationary policies obtained by the two different formulations are capable of shifting the probability mass of the stationary distribution from undesirable states to desirable ones. Index Terms—Altering steady state, genetic network intervention, infinitehorizon control, optimal control of probabilistic Boolean networks. I.
Steadystate analysis of genetic regulatory networks modelled by probabilistic Boolean networks
, 2003
"... 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 steadystate (longrun) behaviour of a PBN by analy ..."
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Cited by 25 (1 self)
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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 steadystate (longrun) behaviour of a PBN by analysing the corresponding Markov chain. This allows one to compute the longterm influence of a gene on another gene or determine the longterm joint probabilistic behaviour of a few selected genes. Because matrixbased 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 twostate Markov chains, we illustrate the approach on a subnetwork designed from human glioma gene expression data and determine the joint steadystate probabilities for several groups of genes. Copyright # 2003 John Wiley & Sons, Ltd.
Mappings between Probabilistic Boolean Networks
, 2003
"... Probabilistic Boolean Networks (PBNs) comprise a graphical model based on uncertain rulebased dependencies between nodes and have been proposed as a model for genetic regulatory networks. As with any algebraic strucicf theckxxzkfjx#[xk of important mappings between PBNs isckT#G for both theory ..."
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Cited by 22 (12 self)
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Probabilistic Boolean Networks (PBNs) comprise a graphical model based on uncertain rulebased dependencies between nodes and have been proposed as a model for genetic regulatory networks. As with any algebraic strucicf theckxxzkfjx#[xk of important mappings between PBNs isckT#G for both theory andapplickfjj 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 subnetworks, 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 manysorted algebras and in that context defines homomorphisms between PBNs.
Effect of Function Perturbation on the Steadystate Distribution of Genetic regulatory Networks
 Optimal Structural Intervention’, IEEE Transactions and Signal Processing
, 2008
"... Abstract—The dynamics of a rulebased gene regulatory network are determined by the regulatory functions in conjunction with whatever probability distributions are involved in network transitions. In the case of Boolean networks (BNs) and, more generally, probabilistic Boolean networks (PBNs), the ..."
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Cited by 14 (8 self)
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Abstract—The dynamics of a rulebased gene regulatory network are determined by the regulatory functions in conjunction with whatever probability distributions are involved in network transitions. In the case of Boolean networks (BNs) and, more generally, probabilistic Boolean networks (PBNs), there has been a significant amount of investigation into the effect of perturbing gene states, in particular, the design of intervention strategies based on finite or infinitehorizon control polices. This paper considers the less investigated issue of function perturbations. A single function perturbation affects network dynamics and alters the longrun distribution, whereas any individual gene perturbation has only transient effects and does not change the longrun distribution. We derive analytic results for changes in the steadystate distributions of PBNs resulting from modifications to the underlying regulatory rules and apply the derived results to find optimal structural interventions to avoid undesirable states. The results are applied to a WNT5A network and a mammalian cell cycle related network, respectively, to achieve more favorable steadystate distributions and reduce the risk of getting into aberrant phenotypes. Index Terms—Boolean networks (BNs), genetic regulatory networks, Markov chains, metastasis, optimal structural intervention, probabilistic Boolean networks (PBNs), steadystate distribution. I.
Intervention in Gene Regulatory Networks Via a Stationary Meanfirstpassagetime Control Policy
 IEEE Transactions on Biomedical Engineering
, 2008
"... Abstract—A prime objective of modeling genetic regulatory networks is the identification of potential targets for therapeutic intervention. To date, optimal stochastic intervention has been studied in the context of probabilistic Boolean networks, with the control policy based on the transition p ..."
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Cited by 12 (6 self)
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Abstract—A prime objective of modeling genetic regulatory networks is the identification of potential targets for therapeutic intervention. To date, optimal stochastic intervention has been studied in the context of probabilistic Boolean networks, with the control policy based on the transition probability matrix of the associated Markov chain and dynamic programming used to find optimal control policies. Dynamical programming algorithms are problematic owing to their high computational complexity. Two additional computationally burdensome issues that arise are the potential for controlling the network and identifying the best gene for intervention. This paper proposes an algorithm based on mean firstpassage time that assigns a stationary control policy for each gene candidate. It serves as an approximation to an optimal control policy and, owing to its reduced computational complexity, can be used to predict the best control gene. Once the best control gene is identified, one can derive an optimal policy or simply utilize the approximate policy for this gene when the network size precludes a direct application of dynamic programming algorithms. A salient point is that the proposed algorithm can be modelfree. It can be directly designed from timecourse data without having to infer the transition probability matrix of the network. Index Terms—Dynamic programming, genetic regulatory networks, mean firstpassage time, probabilistic Boolean networks, stochastic optimal control. I.
Intervention in Gene Regulatory Networks via Greedy
 Control Policies Based on LongRun Behavior,” BMC Systems Biology
"... Abstract—A salient purpose for studying gene regulatory networks is to derive intervention strategies to identify potential drug targets and design genebased therapeutic intervention. Optimal and approximate intervention strategies based on the transition probability matrix of the underlying Markov ..."
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Cited by 10 (5 self)
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Abstract—A salient purpose for studying gene regulatory networks is to derive intervention strategies to identify potential drug targets and design genebased therapeutic intervention. Optimal and approximate intervention strategies based on the transition probability matrix of the underlying Markov chain have been studied extensively for probabilistic Boolean networks. While the key goal of control is to reduce the steadystate probability mass of undesirable network states, in practice it is important to limit collateral damage and this constraint should be taken into account when designing intervention strategies with network models. In this paper, we propose two new phenotypically constrained stationary control policies by directly investigating the effects on the network longrun behavior. They are derived to reduce the risk of visiting undesirable states in conjunction with constraints on the shift of undesirable steadystate mass so that only limited collateral damage can be introduced. We have studied the performance of the new constrained control policies together with the previous greedy control policies to randomly generated probabilistic Boolean networks. A preliminary example for intervening in a metastatic melanoma network is also given to show their potential application in designing genetic therapeutics to reduce the risk of entering both aberrant phenotypes and other ambiguous states corresponding to complications or collateral damage. Experiments on both random network ensembles and the melanoma network demonstrate that, in general, the new proposed control policies exhibit the desired performance. As shown by intervening in the melanoma network, these control policies can potentially serve as future practical gene therapeutic intervention strategies. Index Terms—Gene regulatory networks, probabilistic Boolean networks, network intervention, Markov chain, stationary control policy, melanoma. Ç
A tutorial on analysis and simulation of boolean gene regulatory network models
 Curr Genomics
"... Abstract: Driven by the desire to understand genomic functions through the interactions among genes and gene products, the research in gene regulatory networks has become a heated area in genomic signal processing. Among the most studied mathematical models are Boolean networks and probabilistic Boo ..."
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Cited by 8 (0 self)
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Abstract: Driven by the desire to understand genomic functions through the interactions among genes and gene products, the research in gene regulatory networks has become a heated area in genomic signal processing. Among the most studied mathematical models are Boolean networks and probabilistic Boolean networks, which are rulebased dynamic systems. This tutorial provides an introduction to the essential concepts of these two Boolean models, and presents the uptodate analysis and simulation methods developed for them. In the Analysis section, we will show that Boolean models are Markov chains, based on which we present a Markovian steadystate analysis on attractors, and also reveal the relationship between probabilistic Boolean networks and dynamic Bayesian networks (another popular genetic network model), again via Markov analysis; we dedicate the last subsection to structural analysis, which opens a door to other topics such as network control. The Simulation section will start from the basic tasks of creating state transition diagrams and finding attractors, proceed to the simulation of network dynamics and obtaining the steadystate distributions, and finally come to an algorithm of generating artificial Boolean networks with prescribed attractors. The contents are arranged in a roughly logical order, such that the Markov chain analysis lays the basis for the most part of Analysis section, and also prepares the readers to the topics in Simulation section.