A semi-tensor product approach for 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.

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

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> .

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.

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.

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.

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.

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

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.

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.

Sampling-rate-dependent probabilistic Boolean networks

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.

BackgroundA salient purpose for studying gene regulatory networks is to derive intervention strategies, the goals being to identify potential drug targets and design gene-based therapeutic intervention. Optimal stochastic control based on the transition probability matrix of the underlying Markov chain has been studied extensively for probabilistic Boolean networks. Optimization is based on minimization of a cost function and a key goal of control is to reduce the steady-state probability mass of undesirable network states. Owing to computational complexity, it is difficult to apply optimal control for large networks.ResultsIn this paper, we propose three new greedy stationary control policies by directly investigating the effects on the network long-run behavior. Similar to the recently proposed mean-first-passage-time (MFPT) control policy, these policies do not depend on minimization of a cost function and avoid the computational burden of dynamic programming. They can be used to design stationary control policies that avoid the need for a user-defined cost function because they are based directly on long-run network behavior; they can be used as an alternative to dynamic programming algorithms when the latter are computationally prohibitive; and they can be used to predict the best control gene with reduced computational complexity, even when one is employing dynamic programming to derive the final control policy. We compare the performance of these three greedy control policies and the MFPT policy using randomly generated probabilistic Boolean networks and give a preliminary example for intervening in a mammalian cell cycle network.ConclusionThe newly proposed control policies have better performance in general than the MFPT policy and, as indicated by the results on the mammalian cell cycle network, they can potentially serve as future gene therapeutic intervention strategies.

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.