Intervention in context-sensitive probabilistic Boolean networks

Intervention in a gene regulatory network is used to 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 a collection of Boolean networks in which the gene state vector transitions according to the rules of one of the constituent networks and where network choice is governed by a selection distribution. The theory of automatic control has been applied to find optimal strategies for manipulating external control variables that affect the transition probabilities to desirably affect dynamic evolution over a finite time horizon. In this paper we treat a case in which we lack the governing probability structure for Boolean network selection, so we simply have a family of Boolean networks, but where these networks possess a common attractor structure. This corresponds to the situation in which network construction is treated as an ill-posed inverse problem in which there are many Boolean networks created from the data under the constraint that they all possess attractor structures matching the data states, which are assumed to arise from sampling the steady state of the real biological network. Given a family of Boolean networks possessing a common attractor structure composed of singleton attractors, a control algorithm is derived by minimizing a composite finite-horizon cost function that is a weighted average over all the individual networks, the idea being that we desire a control policy that on average suits the networks because these are viewed as equivalent relative to the data. The weighting for each network at any time point is taken to be proportional to the instantaneous estimated probability of that network being the underlying network governing the state transition. The results are applied to a family of Boolean networks derived from gene-expression data collected in a study of metastatic melanoma, the intent being to devise a control strategy that reduces the WNT5A gene's action in affecting biological regulation

A genetic regulatory network (GRN) is a dynamic system to describe interactions among a large number of different substances in a living biological cell. There have been numerous attempts to model the dynamical behaviour of genetic regulatory networks. Boolean models are a class of discrete models for modelling genetic regulatory networks, which are conditioned on the premise that genes interact with each other through Boolean logic. As mentioned previously, Boolean models include Boolean Networks, Instantaneously Random Probabilistic Boolean Networks and Context-Sensitive Boolean Networks. There are a large number of issues surrounding Boolean models, including Boolean Networks and Probabilistic Boolean Networks. The model inference is the procedure of determining whether the modelled networks are consistent with the given data sample, and choosing one model, from among many, that makes more sense for the data. This paper has reviewed the research into modelling gene regulatory networks, especially upon Boolean models.

An integer programming approach to optimal control problems in context-sensitive probabilistic Boolean networks

BackgroundIn a gene regulatory network (GRN), gene expressions are affected by noise, and stochastic fluctuations exist in the interactions among genes. These stochastic interactions are context dependent, thus it becomes important to consider noise in a context-sensitive manner in a network model. As a logical model, context-sensitive probabilistic Boolean networks (CSPBNs) account for molecular and genetic noise in the temporal context of gene functions. In a CSPBN with n genes and k contexts, however, a computational complexity of O(nk222n ) (or O(nk2 n )) is required for an accurate (or approximate) computation of the state transition matrix (STM) of the size (2 n ∙ k) × (2 n ∙ k) (or 2 n × 2 n ). The evaluation of a steady state distribution (SSD) is more challenging. Recently, stochastic Boolean networks (SBNs) have been proposed as an efficient implementation of an instantaneous PBN.ResultsThe notion of stochastic Boolean networks (SBNs) is extended for the general model of PBNs, i.e., CSPBNs. This yields a novel structure of context-sensitive SBNs (CSSBNs) for modeling the stochasticity in a GRN. A CSSBN enables an efficient simulation of a CSPBN with a complexity of O(nLk2 n ) for computing the state transition matrix, where L is a factor related to the required sequence length in CSSBN for achieving a desired accuracy. A time-frame expanded CSSBN can further efficiently simulate the stationary behavior of a CSPBN and allow for a tunable tradeoff between accuracy and efficiency. The CSSBN approach is more efficient than an analytical method and more accurate than an approximate analysis.ConclusionsContext-sensitive stochastic Boolean networks (CSSBNs) are proposed as an efficient approach to modeling the effects of gene perturbation and intervention in gene regulatory networks. A CSSBN analysis provides biologically meaningful insights into the oscillatory dynamics of the p53-Mdm2 network in a context-switching environment. It is shown that random gene perturbation has a greater effect on the final distribution of the steady state of a network compared to context switching activities. The CSSBN approach can further predict the steady state distribution of a glioma network under gene intervention. Ultimately, this will help drug discovery and develop effective drug intervention strategies.

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 rule-based dynamic systems. This tutorial provides an introduction to the essential concepts of these two Boolean models, and presents the up-to-date 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 steady-state 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 steady-state 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.

Boolean networks are well-known as one of the models of biological networks such as gene regulatory networks. In this paper, we propose a solving method of the optimal control problem of context-sensitive probabilistic Boolean networks (CS-PBNs), which are one of the extended models of Boolean networks. In the existing solving methods, it is necessary to compute state transition diagrams with 2n nodes for a given CS-PBN with n states. So the existing methods cannot be applied to large-scale networks. To avoid the computation of state transition diagrams, an integer programming-based approach is proposed. In the proposed method, a CS-PBN is transformed into a linear system with binary variables, and the optimal control problem is reduced to an integer linear programming problem, which can be computed relatively easier than the existing methods using state transition diagrams.

A probabilistic Boolean network (PBN) is a discrete-time system composed of a collection of Boolean networks between which the PBN switches in a stochastic manner. This article focuses on the study of quotients of PBNs. Given a PBN and an equivalence relation on its state set, we consider a probabilistic transition system that is generated by the PBN; the resulting quotient transition system then automatically captures the quotient behavior of this PBN. We therefore describe a method for obtaining a probabilistic Boolean system that generates the transitions of the quotient transition system. Applications of this quotient description are discussed, and it is shown that for PBNs, controller synthesis can be performed easily by first controlling a quotient system and then lifting the control law back to the original network. A biological example is given to show the usefulness of the developed results.

We present a major new release of ASSA-PBN, a software tool for modelling, simulation, and analysis of probabilistic Boolean networks (PBNs). The new version enables the support for context-sensitive PBNs (CPBNs), which can well balance the uncertainty and stability of the modelled biological systems. It contributes mainly in three aspects. Firstly, it designs a high-level language for specifying CPBNs. Secondly, it implements various simulation-based methods for simulating CPBNs and analysing their long-run dynamics. Last but not least, it provides an efficient method to identify all the attractors of a CPBN. Thanks to its divide and conquer strategy, the implemented detection algorithm can deal with large and realistic biological networks under both synchronous and asynchronous updating schemes.

One of the major issues in systems biology is developing control theory for gene regulatory networks (GRNs). Particularly, an important objective is to develop therapeutic intervention strategies to alter regulatory dynamics of GRNs to avoid undesired or diseased states. Several optimal control strategies have been developed to find therapeutic intervention strategies for small (or medium) sized GRNs modeled as probabilistic Boolean control networks (PBCNs). However, due to the humongous nature of GRNs, we require an optimal control strategy that scales to large systems without posing any constraints on the network dynamics. In this article, we formulate the infinite horizon discounted cost optimal control problem by leveraging the Markov decision process (MDP) based PBCN structure to model GRNs. Further, we design a stationary optimal control strategy, thereby avoiding the states associated with a disease. An augmented state space is defined to facilitate the control design method. By exploiting Kullback–Leibler (KL) divergence and the log transformation of the cost function referred to as the desirability function, the optimal control is devised using the path integral (PI) approach. We propose a sampling‐based technique for the approximation of the PI and hence the optimal control of PBCNs. The sampling‐based strategy is amicable to parallel implementation, thereby addressing the optimal control problem of large‐scale PBCNs. Finally, the viability of the results in the article is shown by some illustrative examples.

Computational modeling of genomic regulation has become an important focus of systems biology and genomic signal processing for the past several years. It holds the promise to uncover both the structure and dynamical properties of the complex gene, protein or metabolic networks responsible for the cell functioning in various contexts and regimes. This, in turn, will lead to the development of optimal intervention strategies for prevention and control of disease. At the same time, constructing such computational models faces several challenges. High complexity is one of the major impediments for the practical applications of the models. Thus, reducing the size/complexity of a model becomes a critical issue in problems such as model selection, construction of tractable subnetwork models, and control of its dynamical behavior. We focus on the reduction problem in the context of two specific models of genomic regulation: Boolean networks with perturbation (BNP) and probabilistic Boolean networks (PBN). We also compare and draw a parallel between the reduction problem and two other important problems of computational modeling of genomic networks: the problem of network inference and the problem of designing external control policies for intervention/altering the dynamics of the model.

A probabilistic Boolean network (PBN) is a collection of Boolean networks endowed with a probability structure describing the likelihood with which a constituent network is active at each time step. This paper proposes a definition of bisimulation for PBNs. The notion is inspired by the analogous notions for probabilistic chains and for stochastic linear systems. Necessary and sufficient conditions to check the proposed notion are derived, model reduction of PBNs via bisimulation is addressed, and a discussion on the use of probabilistic bisimulation for optimal control design is given. The present results extend the theory of bisimulation known for deterministic Boolean networks to a stochastic setting.

In this paper, we study control problems of Boolean Networks (BNs) and Probabilistic Boolean Networks (PBNs). For BN CONTROL, by applying external control, we propose to derive the network to the desired state within a few time steps. For PBN CONTROL, we propose to find a control sequence such that the network will terminate in the desired state with a maximum probability. Also, we propose to minimize the maximum cost of the terminal state to which the network will enter. Integer linear programming and dynamic programming in conjunction with hard constraints are then employed to solve the above problems. Numerical experiments are given to demonstrate the effectiveness of our algorithms. We also present a hardness result suggesting that PBN CONTROL is harder than BN CONTROL.

Structural Control of Probabilistic Boolean Networks and Its Application to Design of Real-Time Pricing Systems

The construction and control of genetic regulatory networks using gene expression data is an important research topic in bioinformatics. Probabilistic Boolean Networks (PBNs) have been served as an effective tool for this purpose. However, PBNs are difficult to be used in practice when the number of genes is large because of the huge computational cost. In this paper, we propose a simplified multivariate Markov model for approximating a PBN. The new model can preserve the strength of PBNs and at the same time reduce the complexity of the network and therefore the computational cost. We then present an optimal control model with hard constraints for the purpose of control/intervention of a genetic regulatory network. Numerical experimental examples based on the yeast data are then given to demonstrate the effectiveness of our proposed model and control policy.

Modeling genetic regulatory networks is an important issue in systems biology. Various models and mathematical formalisms have been proposed in the literature to solve the capture problem. The main purpose in this paper is to show that the transition matrix generated under semi-tensor product approach (Here we call it the probability structure matrix for simplicity) and the traditional approach (Transition probability matrix) are similar to each other. And we shall discuss three important problems in Probabilistic Boolean Networks (PBNs): the dynamic of a PBN, the steady-state probability distribution and the inverse problem. Numerical examples are given to show the validity of our theory. We shall give a brief introduction to semi-tensor and its application. After that we shall focus on the main results: to show the similarity of these two matrices. Since the semi-tensor approach gives a new way for interpreting a BN and therefore a PBN, we expect that advanced algorithms can be developed if one can describe the PBN through semi-tensor product approach.