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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 finite-horizon 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 finite-horizon control, i.e., control over a small number of stages. This paper considers the design of optimal infinite-horizon control for context-sensitive 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 average-cost-per-stage problems. These formulations are used to generate stationary policies for a PBN constructed from melanoma gene-expression 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, infinite-horizon control, optimal control of probabilistic Boolean networks. I.
Inference of a Probabilistic Boolean Network from a Single Observed Temporal Sequence
, 2007
"... The inference of gene regulatory networks is a key issue for genomic signal processing. This paper addresses the inference of probabilistic Boolean networks (PBNs) from observed temporal sequences of network states. Since a PBN is composed of a finite number of Boolean networks, a basic observation ..."
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Cited by 19 (6 self)
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The inference of gene regulatory networks is a key issue for genomic signal processing. This paper addresses the inference of probabilistic Boolean networks (PBNs) from observed temporal sequences of network states. Since a PBN is composed of a finite number of Boolean networks, a basic observation is that the characteristics of a single Boolean network without perturbation may be determined by its pairwise transitions. Because the network function is fixed and there are no perturbations, a given state will always be followed by a unique state at the succeeding time point. Thus, a transition counting matrix compiled over a data sequence will be sparse and contain only one entry per line. If the network also has perturbations, with small perturbation probability, then the transition counting matrix would have some insignificant nonzero entries replacing some (or all) of the zeros. If a data sequence is sufficiently long to adequately populate the matrix, then determination of the functions and inputs underlying the model is straightforward. The difficulty comes when the transition counting matrix consists of data derived from more than one Boolean network. We address the PBN inference procedure in several steps: (1) separate the data sequence into “pure ” subsequences corresponding to constituent Boolean networks; (2) given a subsequence, infer a Boolean network; and (3) infer the probabilities of perturbation, the probability of there being a switch between constituent Boolean networks, and the selection probabilities governing
Robust intervention in probabilistic Boolean networks
- IEEE Trans
, 2008
"... Abstract—Probabilistic Boolean networks (PBNs) have been recently introduced as a paradigm for modeling genetic regulatory networks. One of the objectives of PBN modeling is to use the network for the design and analysis of intervention strategies aimed at moving the network out of undesirable state ..."
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Cited by 17 (8 self)
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Abstract—Probabilistic Boolean networks (PBNs) have been recently introduced as a paradigm for modeling genetic regulatory networks. One of the objectives of PBN modeling is to use the network for the design and analysis of intervention strategies aimed at moving the network out of undesirable states, such as those associated with disease, and into desirable ones. To date, a number of intervention strategies have been proposed in the context of PBNs. However, all these techniques assume perfect knowledge of the transition probability matrix of the PBN. Such an assumption cannot be satisfied in practice since the presence of noise and the availability of limited number of samples will prevent the transition probabilities from being accurately deter-mined. Moreover, even if the exact transition probabilities could be estimated from the data, mismatch between the PBN model and the actual genetic regulatory network will invariably be present. Thus, it is important to study the effect of modeling errors on the final outcome of an intervention strategy and one of the goals of this paper is to do precisely that when the uncertainties are in the entries of the transition probability matrix. In addition, the paper develops a robust intervention strategy that is obtained by minimizing the worst-case cost over the uncertainty set. Index Terms—Control of biological networks, estimation errors, robust dynamic programming, robust minimax control, perturba-tion bounds. I.
Effect of Function Perturbation on the Steady-state Distribution of Genetic regulatory Networks
- Optimal Structural Intervention’, IEEE Transactions and Signal Processing
, 2008
"... Abstract—The dynamics of a rule-based gene regulatory net-work 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 gen-erally, probabilistic Boolean networks (PBNs), the ..."
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Cited by 14 (8 self)
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Abstract—The dynamics of a rule-based gene regulatory net-work 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 gen-erally, probabilistic Boolean networks (PBNs), there has been a sig-nificant amount of investigation into the effect of perturbing gene states, in particular, the design of intervention strategies based on finite- or infinite-horizon control polices. This paper considers the less investigated issue of function perturbations. A single function perturbation affects network dynamics and alters the long-run dis-tribution, whereas any individual gene perturbation has only tran-sient effects and does not change the long-run distribution. We de-rive analytic results for changes in the steady-state distributions of PBNs resulting from modifications to the underlying regulatory rules and apply the derived results to find optimal structural in-terventions to avoid undesirable states. The results are applied to a WNT5A network and a mammalian cell cycle related network, re-spectively, to achieve more favorable steady-state distributions and reduce the risk of getting into aberrant phenotypes. Index Terms—Boolean networks (BNs), genetic regulatory net-works, Markov chains, metastasis, optimal structural intervention, probabilistic Boolean networks (PBNs), steady-state distribution. I.
Bayesian robustness in the control of gene regulatory networks
- Signal Processing, IEEE Transactions on 2009
"... Abstract—The errors originating in the data extraction process, gene selection and network inference prevent the transition probabilities of a gene regulatory network from being accurately estimated. Thus, it is important to study the effect of modeling errors on the final outcome of an intervention ..."
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Cited by 13 (7 self)
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Abstract—The errors originating in the data extraction process, gene selection and network inference prevent the transition probabilities of a gene regulatory network from being accurately estimated. Thus, it is important to study the effect of modeling errors on the final outcome of an intervention strategy and to design robust intervention strategies. Two major approaches applied to the design of robust policies in general are the minimax (worst case) approach and the Bayesian approach. The minimax control approach is typically conservative because it gives too much importance to the scenarios which hardly occur in practice. Consequently, in this paper, we formulate the Bayesian approach for the control of gene regulatory networks. We characterize the errors emanating from the data extraction and inference processes and compare the performance of the minimax and Bayesian designs based on these uncertainties. Index Terms—Bayesian robustness, gene regulatory networks, intervention, parameter estimation, robust control. I.
Intervention in Context-Sensitive Probabilistic Boolean Networks Revisited
, 2009
"... An approximate representation for the state space of a context-sensitive probabilistic Boolean network has previously been proposed and utilized to devise therapeutic intervention strategies. Whereas the full state of a context-sensitive probabilistic Boolean network is specified by an ordered pair ..."
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Cited by 11 (3 self)
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An approximate representation for the state space of a context-sensitive probabilistic Boolean network has previously been proposed and utilized to devise therapeutic intervention strategies. Whereas the full state of a context-sensitive probabilistic Boolean network is specified by an ordered pair composed of a network context and a gene-activity profile, this approximate representation collapses the state space onto the gene-activity profiles alone. This reduction yields an approximate transition probability matrix, absent of context, for the Markov chain associated with the context-sensitive probabilistic Boolean network. As with many approximation methods, a price must be paid for using a reduced model representation, namely, some loss of optimality relative to using the full state space. This paper examines the effects on intervention performance caused by the reduction with respect to various values of the model parameters. This task is performed using a new derivation for the transition probability matrix of the context-sensitive probabilistic Boolean network. This expression of transition probability distributions is in concert with the original definition of context-sensitive probabilistic Boolean network. The performance of optimal and approximate therapeutic strategies is compared for both synthetic networks and a real case study. It is observed that the approximate representation describes the dynamics of the context-sensitive probabilistic Boolean network through the instantaneously random probabilistic Boolean network with similar parameters.
Recent development and biomedical applications of probabilistic Boolean networks
, 2013
"... Probabilistic Boolean network (PBN) modelling is a semi-quantitative approach widely used for the study of the topology and dynamic aspects of biological systems. The combined use of rule-based representation and probability makes PBN appealing for large-scale modelling of biological networks where ..."
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Cited by 8 (3 self)
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Probabilistic Boolean network (PBN) modelling is a semi-quantitative approach widely used for the study of the topology and dynamic aspects of biological systems. The combined use of rule-based representation and probability makes PBN appealing for large-scale modelling of biological networks where degrees of uncertainty need to be considered. A considerable expansion of our knowledge in the field of theoretical research on PBN can be observed over the past few years, with a focus on network inference, network intervention and control. With respect to areas of applications, PBN is mainly used for the study of gene regulatory networks though with an increasing emergence in signal transduction, metabolic, and also physiological networks. At the same time, a number of computational tools, facilitating the modelling and analysis of PBNs, are continuously developed. A concise yet comprehensive review of the state-of-the-art on PBN modelling is offered in this article, including a comparative discussion on PBN versus similar models with respect to concepts and biomedical applications. Due to their many advantages, we consider PBN to stand as a suitable modelling framework for the description and analysis of complex biological systems, ranging from molecular to physiological levels.
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 well-known that the control/intervention of a genetic regulatory network is useful for avoiding undesirable states associated with diseases like cancer. Therefore both optimal finite-horizon control and infinite-horizon 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 finite-horizon control problem with hard constraints. Experimental results are given to demonstrate the efficiency of our proposed formulation.
Design of probabilistic Boolean networks under the requirement of contextual data consistency
- IEEE Trans. Signal Process
, 2006
"... Abstract—A key issue of genomic signal processing is the design of gene regulatory networks. A probabilistic Boolean network (PBN) is composed of a family of Boolean networks. It stochastically switches between its constituent networks (contexts). For network design, connectivity and transition rule ..."
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Cited by 6 (5 self)
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Abstract—A key issue of genomic signal processing is the design of gene regulatory networks. A probabilistic Boolean network (PBN) is composed of a family of Boolean networks. It stochastically switches between its constituent networks (contexts). For network design, connectivity and transition rules must be inferred from data via some optimization criterion. Except rarely, the optimal rule for a gene will not be a perfect predictor because there will be inconsistencies in the data. It would be natural to model these inconsistencies to reflect changes in PBN contexts. If we assume inconsistencies result from the data arising from a random function, then design involves finding the realizations of a random function and the probability mass on those realizations so that the resulting random function best fits the data relative to the expectation of its output and does so using a minimal number of realizations. We propose PBN design satisfying the biological assumption that data are consistent within a context, for which the distribution of the network agrees with the empirical distribution of the data, and such that this is accomplished with a minimal number of contexts. The design also satisfies the biological constraint that, because the network spends the great majority of time in its attractors, all data states should be attractor states in the model. Index Terms—Data consistency, gene regulatory network, graphical model, network inference. I.
Characterizing the effect of coarse-scale pbn modeling on dynamics and intervention performance of genetic regulatory networks represented by stochastic master equation models
- IEEE Transactions on Signal Processing
, 2010
"... Abstract—Fine-scale models based on stochastic master equa-tions can provide the most detailed description of the dynamics of gene expression and imbed, in principle, all the information about the biochemical reactions involved in gene interactions. However, there is limited time-series experimental ..."
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Cited by 6 (2 self)
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Abstract—Fine-scale models based on stochastic master equa-tions can provide the most detailed description of the dynamics of gene expression and imbed, in principle, all the information about the biochemical reactions involved in gene interactions. However, there is limited time-series experimental data available for infer-ence of such fine-scale models. Furthermore, the computational complexity involved in the design of optimal intervention strate-gies to favorably effect system dynamics for such detailed models is enormous. Thus, there is a need to design mappings from fine-scale models to coarse-scale models while maintaining sufficient structure for the problem at hand and to study the effect of inter-vention policies designed using coarse-scale models when applied to fine-scale models. In this paper, we propose a mapping from a fine-scale model represented by a stochastic master equation to a coarse-scale model represented by a probabilistic Boolean network that maintains the collapsed steady state probability distribution of the detailed model. We also derive bounds on the performance of the intervention strategy designed using the coarse-scale model when applied to the fine-scale model. Index Terms—Markov chain models, probabilistic Boolean net-works, relationships between genetic regulatory network models, stochastic master equations. I.
