Quotients of Probabilistic Boolean Networks

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.

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 context-sensitive 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 context-sensitive PBNs. This paper treats intervention via external control variables in context-sensitive PBNs by extending the results for instantaneously random PBNs in several directions. First, and most importantly, whereas an instantaneously random PBN yields a Markov chain whose state space is composed of gene vectors, each state of the Markov chain corresponding to a context-sensitive PBN is composed of a pair, the current gene vector occupied by the network and the current constituent Boolean network. Second, the analysis is applied to PBNs with perturbation, meaning that random gene perturbation is permitted at each instant with some probability. Third, the (mathematical) influence of genes within the network is used to choose the particular gene with which to intervene. Lastly, PBNs are designed from data using a recently proposed inference procedure that takes steady-state considerations into account. The results are applied to a context-sensitive PBN 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, since the available data suggest that disruption of this influence could reduce the chance of a melanoma metastasizing.

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.

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

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.

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.

This article aims to stabilize probabilistic Boolean networks (PBNs) via a novel pinning control strategy. In a PBN, the state evolution of each gene switches among a collection of candidate Boolean functions with preassigned probability distributions, which govern the activation frequency of each Boolean function. Due to the existence of stochasticity, the mode-independent pinning controller might be disabled. Thus, both mode-independent and mode-dependent pinning controller are required here. Moreover, a criterion is derived to determine whether mode-independent controllers are applicable while the pinned nodes are given. It is worth pointing out that this pinning control is based on the n×n network structure rather than 2n ×2n state transition matrix. Therefore, compared with the existing results, this pinning control strategy is more practicable and has the ability to handle large-scale networks, especially sparsely connected networks. To demonstrate the effectiveness of the designed control scheme, a PBN that describes the mammalian cell-cycle encountering a mutated phenotype is discussed as a simulation.

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 the coupled dynamics of gene expression patterns is an important task in systems biology. It is most accurately performed via systems of coupled differential equations, derived by analyzing involved biochemical reactions of the cell cycle (bottom-up approach). Probabilistic Boolean networks (PBN) represent stochastic extensions of Boolean models [5, 7] that allow inference by reverse engineering from a given data set (top-down approach). In a PBN, a list of Boolean functions is associated with each node in the network, and each time the state of a gene is updated, only one of these functions is randomly chosen to compute the new state of the gene [7]. Recently [1], we presented a constructive approach for reverse engineering gene expression dynamics casted within the algebraic framework developed in [6] that can be seen as a generalization of PBN. We showed that, in a probabilistic framework, reverse engineering under this model is closely related to problems arising in coding theory. In particular, we applied list-decoding of Reed-Muller codes to address randomness, measurement errors, and small sample size issues. In this contribution we show how the concept of local decoding can be used to reduce the decoding complexity and aid experimental design.

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

The ability to direct a probabilistic Boolean network (PBN) to the desired state is important to applications such as targeted therapeutics in cancer biology. Reinforcement learning (RL) has been proposed as a framework that solves a discrete-time optimal control problem cast as a Markov decision process. We focus on an integrative framework powered by a model-free deep RL method that can address different flavors of the control problem (e.g., with <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">or</i> without control inputs; having attractor states <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">or</i> a subset of the state space as the target domain). The method is agnostic to the distribution of probabilities for the next state; hence, it does not use the probability transition matrix. The time complexity is only <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">linear</i> on the time steps or interactions between the agent (deep RL) and the environment (PBN), during training. Indeed, we explore the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">scalability</i> of the deep RL approach to (set) stabilization of large-scale PBNs and demonstrate successful control on large networks, including a metastatic melanoma PBN with <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">200 nodes</i> .

Recent research has suggested that disrupted interactions between certain brain regions may contribute to the symptoms of neurological diseases such as Parkinson's disease (PD). It is therefore important to develop models for inferring brain functional connectivity from non-invasive imaging data, such as functional magnetic resonance imaging (fMRI). In this paper, we propose applying probabilistic Boolean networks (PBNs) for modeling brain connectivity due to its solid stochastic properties, computational simplicity, robustness to uncertainty, and ability to deal with small-size data, typical for fMRI data sets. Applying the proposed PBN framework to real fMRI data recorded from PD and control subjects, the PBN method detected statistically significant differing interactions between task-related regions of interest (ROIs) across groups. Comparing the PBN results in PD subjects before and after they had taken L-dopa medication, the principal treatment for PD, suggests that a key mechanism of action of this medication is relative normalization of disrupted brain connectivity.

Probabilistic Boolean networks model biological processes with the network dynamics. This paper studies the network sensitivity with respect to perturbations to networks, including regulatory rules and the involved parameters, in the long run. We define the network sensitivity based on the steady-state distributions of probabilistic Boolean networks as their underlying model is a finite Markov chain. The steady-state distribution reflects the long-run behavior of the network and the change of steady-state distribution caused by possible perturbations is the key measure for intervention.

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