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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 84 (17 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 46 (17 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.
Control of Stationary Behavior in Probabilistic Boolean Networks by Means of Structural Intervention
 Biological Systems
, 2002
"... 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 steadystate or longrun behavior of PBNs ..."
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Cited by 30 (11 self)
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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 steadystate or longrun behavior of PBNs may reflect the phenotype or functional state of the cell. Approaches to alter the steadystate 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 steadystate probabil ities of certain states or sets of states with minimal modifications to the underlying rulebased 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.
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 26 (9 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.
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 19 (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.
Optimal infinitehorizon control for probabilistic Boolean networks
 IEEE Transactions on Signal Processing
"... Abstractâ€”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 19 (9 self)
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Abstractâ€”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 Termsâ€”Altering steady state, genetic network intervention, infinitehorizon control, optimal control of probabilistic Boolean networks. I.
Can Markov Chain Models Mimic Biological Regulation?
, 2002
"... this paper is relatively small, it suggests that models incorporating rulebased 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 ..."
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Cited by 18 (8 self)
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this paper is relatively small, it suggests that models incorporating rulebased 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 switchlike 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
A Bayesian connectivitybased approach to constructing probabilistic gene regulatory networks
, 2004
"... 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, ..."
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Cited by 16 (7 self)
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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.
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 16 (8 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.
Optimal FiniteHorizon Control for Probabilistic Boolean Networks with Hard Constraints
 The International Symposium on Optimization and Systems Biology (OSB 2007), Lecture Notes in Operations Research
, 2007
"... Abstract In this paper, we study optimal control policies for Probabilistic Boolean Networks (PBNs) with hard constraints. Boolean Networks (BNs) and PBNs are useful and effective tools for modelling genetic regulatory networks. A PBN is essentially a collection of BNs driven by a Markov chain proce ..."
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Cited by 7 (5 self)
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Abstract In this paper, we study optimal control policies for Probabilistic Boolean Networks (PBNs) with hard constraints. Boolean Networks (BNs) and PBNs are useful and effective tools for modelling genetic regulatory networks. A PBN is essentially a collection of BNs driven by a Markov chain process. It is wellknown that the control/intervention of a genetic regulatory network is useful for avoiding undesirable states associated with diseases like cancer. Therefore both optimal finitehorizon control and infinitehorizon control policies have been proposed to achieve the purpose. Actually the optimal control problem can be formulated as a probabilistic dynamic programming problem. In many studies, the optimal control problems did not consider the case of hard constraints, i.e., to include a maximum upper bound for the number of controls that can be applied to the PBN. The main objective of this paper is to introduce a new formulation for the optimal finitehorizon control problem with hard constraints. Experimental results are given to demonstrate the efficiency of our proposed formulation.