On optimal control policy for Probabilistic Boolean Network: a state reduction approach

BACKGROUND: Probabilistic Boolean Network (PBN) is a popular model for studying genetic regulatory networks. An important and practical problem is to find the optimal control policy for a PBN so as to avoid the network from entering into undesirable states. A number of research works have been done by using dynamic programming-based (DP) method. However, due to the high computational complexity of PBNs, DP method is computationally inefficient for a large size network. Therefore it is natural to seek for approximation methods. RESULTS: Inspired by the state reduction strategies, we consider using dynamic programming in conjunction with state reduction approach to reduce the computational cost of the DP method. Numerical examples are given to demonstrate both the effectiveness and the efficiency of our proposed method. CONCLUSIONS: Finding the optimal control policy for PBNs is meaningful. The proposed problem has been shown to be ∑ p 2 - hard . By taking state reduction approach into consideration, the proposed method can speed up the computational time in applying dynamic programming-based algorithm. In particular, the proposed method is effective for larger size networks.published_or_final_versio

Algorithms and Complexity Analyses for Control of Singleton Attractors in Boolean Networks

A Boolean network (BN) is a mathematical model of genetic networks. We propose several algorithms for control of singleton attractors in BN. We theoretically estimate the average-case time complexities of the proposed algorithms, and confirm them by computer experiments. The results suggest the importance of gene ordering. Especially, setting internal nodes ahead yields shorter computational time than setting external nodes ahead in various types of algorithms. We also present a heuristic algorithm which does not look for the optimal solution but for the solution whose computational time is shorter than that of the exact algorithms

Effect of coarse-scale modeling on control outcome of genetic regulatory networks

Abstract-Fine-scale models represented by stochastic master equations can provide a very accurate description of the real genetic regulatory system but inadequate time series data and technological limitations on cell specific measurements in cancer related experiments prevent the accurate inference of the parameters of such a finescale model. Furthermore, the computational complexity involved in the design of optimal intervention strategies to favorably effect system dynamics for such detailed models is enormous. Thus, it is imperative to study the effect of intervention policies designed using coarse-scale models when applied to the fine-scale models. In this paper, we map a fine-scale model represented by a Stochastic Master Equation to a coarse-scale model represented by a Probabilistic Boolean Network and derive bounds on the performance of the intervention strategy designed using the coarse scale model when applied to the fine-scale model

Finding and Analyzing the Minimum Set of Driver Nodes in Control of Boolean Networks

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Genomic Regulatory Networks, Reduction Mappings and Control

All high-level living organisms are made of small cell units, containing DNA, RNA, genes, proteins etc. Genes are important components of the cells and it is necessary to understand the inter-gene relations, in order to comprehend, predict and ultimately intervene in the cells’ dynamics. Genetic regulatory networks (GRN) represent the gene interactions that dictate the cell behavior. Translational genomics aims to mathematically model GRNs and one of the main goals is to alter the networks’ behavior away from undesirable phenotypes such as cancer. The mathematical framework that has been often used for modeling GRNs is the probabilistic Boolean network (PBN), which is a collection of constituent Boolean networks with perturbation, BNp. This dissertation uses BNps, to model gene regulatory networks with an intent of designing stationary control policies (CP) for the networks to shift their dynamics toward more desirable states. Markov Chains (MC) are used to represent the PBNs and stochastic control has been employed to find stationary control policies to affect steady-state distribution of the MC. However, as the number of genes increases, it becomes computationally burdensome, or even infeasible, to derive optimal or greedy intervention policies. This dissertation considers the problem of modeling and intervening in large GRNs. To overcome the computational challenges associated with large networks, two approaches are proposed: first, a reduction mapping that deletes genes from the network; and second, a greedy control policy that can be directly designed on large networks. Simulation results show that these methods achieve the goal of controlling large networks by shifting the steady-state distribution of the networks toward more desirable states. Furthermore, a new inference method is used to derive a large 17-gene Boolean network from microarray experiments on gastrointestinal cancer samples. The new algorithm has similarities to a previously developed well-known inference method, which uses seed genes to grow subnetworks, out of a large network; however, it has major differences with that algorithm. Most importantly, the objective of the new algorithm is to infer a network from a seed gene with an intention to derive the Gene Activity Profile toward more desirable phenotypes. The newly introduced reduction mappings approach is used to delete genes from the 17-gene GRN and when the network is small enough, an intervention policy is designed for the reduced network and induced back to the original network. In another experiment, the greedy control policy approach is used to directly design an intervention policy on the large 17-gene network to beneficially change the long-run behavior of the network. Finally, a novel algorithm is developed for selecting only non-isomorphic BNs, while generating synthetic networks, using a method that generates synthetic BNs, with a prescribed set of attractors. The goal of the new method described in this dissertation is to discard isomorphic networks

A control model for Markovian genetic regulatory networks

In this paper, we study a control model for gene intervention in a genetic regulatory network. At each time step, a finite number of controls are allowed to drive to some target states (i.e, some specific genes are on, and some specific genes are off) of a genetic network. We are interested in determining a minimum amount of control cost on a genetic network over a certain period of time such that the probabilities of obtaining such target states are as large as possible. This problem can be formulated as a stochastic dynamic programming model. However, when the number of genes is n, the number of possible states is exponentially increasing with n, and the computational cost of solving such stochastic dynamic programming model would be very huge. The main objective of this paper is to approximate the above control problem and formulate as a minimization problem with integer variables and continuous variables using dynamics of states probability distribution of genes. Our experimental results show that our proposed formulation is efficient and quite effective for solving control gene intervention in a genetic network. © Springer-Verlag Berlin Heidelberg 2006.link_to_subscribed_fulltex