Global stabilization of Boolean networks with applications to biomolecular network control

Boolean networks (BNs) are vital modeling tools in systems biology for biomolecular regulatory networks. After a transient phase, BNs converge to attractors that represent distinct cell types or conditions. Therefore, methods to control the long-term behavior of BNs have important implications for biological and genetic applications. In this paper, we propose a method to enforce convergence of a BN to a desired attractor from any initial state through a simple intervention: fixing a specific subset of network variables at definite values. We refer to this method as the global stabilization of a BN to a target attractor. Utilizing the algebraic state space representation of BNs, we introduce novel matrix tools to formulate this intervention method, as well as develop a foundation for analyzing the stabilizability of BNs. We derive necessary and sufficient conditions for the global stabilizability of BNs and utilize these criteria to identify a minimal subset of network variables-termed the global stabilizing kernel-whose regulation ensures that the BN converges to the desired attractor. Finally, we apply our proposed method to determine the stabilizing kernels of several biomolecular regulatory network models and demonstrate how they can be steered to their target attractors, showcasing the applicability of our approach. We also apply our method to identify the stabilizing kernels of 480 randomly generated BNs. Our experiments suggest that, on average, only a relatively small portion (approximately 25%) of the network nodes need to be manipulated for the networks to converge to their primary attractors.

Identification of Boolean control networks with time delay

This paper investigates the identification of time-delay Boolean networks (TBNs) and time-delay Boolean control networks (TBCNs) via Cheng product. According to all admissible (input-)output sequences, definition on identifiability of the (TBCN) TBN is given. Two algorithms are designed to select suitable delay parameters of the TBN and TBCN, respectively. Based on these, the original systems are divided into several subsystems. Then by virtue of observability, the criteria for identifiability of the TBN and TBCN are obtained. Moreover, the corresponding constructing processes are presented to establish the internal structures of the TBN and TBCN. Finally, two illustrative examples are given to show the feasibility of the proposed methods.

Observability of Boolean Control Networks Using Parallel Extension and Set Reachability

This brief reviews various definitions of observability for Boolean control networks (BCNs) and proposes a new one: output-feedback observability. This new definition applies to all BCNs whose initial states can be identified from the history of output measurements. A technique called parallel extension is then proposed to facilitate observability analysis. Furthermore, a technique called state transition graph reconstruction is proposed for analyzing the set reachability of BCNs, based on which new criteria for observability, single-input sequence observability, and arbitrary-input observability, are obtained. Using the proposed techniques, this brief proves that the problem of output-feedback observability can be recast as that of stabilizing a logic dynamical system with output feedback. Then, a necessary and sufficient condition for static output feedback observability is proposed. The relationships between the different definitions of observability are discussed, and the main results are illustrated with examples.

Stability and Guaranteed Cost Analysis of Time-Triggered Boolean Networks

This paper investigates stability and guaranteed cost of time-triggered Boolean networks (BNs) based on the semitensor product of matrices. The time triggering is generated by mode-dependent average dwell-time switching signals in the BNs. With the help of the copositive Lyapunov function, a sufficient condition is derived to ensure that the considered network is globally stable under a designed average dwell-time switching signal. Subsequently, an infinite time cost function is further discussed and its bound is presented according to the obtained stability result. Numerical examples are finally given to show the feasibility of the theoretical results.

Policy Iteration Algorithm for Optimal Control of Stochastic Logical Dynamical Systems

This brief investigates the infinite horizon optimal control problem for stochastic multivalued logical dynamical systems with discounted cost. Applying the equivalent descriptions of stochastic logical dynamics in term of Markov decision process, the discounted infinite horizon optimal control problem is presented in an algebraic form. Then, employing the method of semitensor product of matrices and the increasing-dimension technique, a succinct algebraic form of the policy iteration algorithm is derived to solve the optimal control problem. To show the effectiveness of the proposed policy iteration algorithm, an optimization problem of p53-Mdm2 gene network is investigated.

Attractor controllability of Boolean networks by flipping a subset of their nodes

The controllability analysis of Boolean networks (BNs), as models of biomolecular regulatory networks, has drawn the attention of researchers in recent years. In this paper, we aim at governing the steady-state behavior of BNs using an intervention method which can easily be applied to most real system, which can be modeled as BNs, particularly to biomolecular regulatory networks. To this end, we introduce the concept of attractor controllability of a BN by flipping a subset of its nodes, as the possibility of making a BN converge from any of its attractors to any other one, by one-time flipping members of a subset of BN nodes. Our approach is based on the algebraic state-space representation of BNs using semi-tensor product of matrices. After introducing some new matrix tools, we use them to derive necessary and sufficient conditions for the attractor controllability of BNs. A forward search algorithm is then suggested to identify the minimal perturbation set for attractor controllability of a BN. Next, a lower bound is derived for the cardinality of this set. Two new indices are also proposed for quantifying the attractor controllability of a BN and the influence of each network variable on the attractor controllability of the network and the relationship between them is revealed. Finally, we confirm the efficiency of the proposed approach by applying it to the BN models of some real biomolecular networks.

Observability of Boolean multiplex control networks

Boolean multiplex (multilevel) networks (BMNs) are currently receiving considerable attention as theoretical arguments for modeling of biological systems and system level analysis. Studying control-related problems in BMNs may not only provide new views into the intrinsic control in complex biological systems, but also enable us to develop a method for manipulating biological systems using exogenous inputs. In this article, the observability of the Boolean multiplex control networks (BMCNs) are studied. First, the dynamical model and structure of BMCNs with control inputs and outputs are constructed. By using of Semi-Tensor Product (STP) approach, the logical dynamics of BMCNs is converted into an equivalent algebraic representation. Then, the observability of the BMCNs with two different kinds of control inputs is investigated by giving necessary and sufficient conditions. Finally, examples are given to illustrate the efficiency of the obtained theoretical results.

Controllability and observability of Boolean networks arising from biology

Boolean networks are currently receiving considerable attention as a computational scheme for system level analysis and modeling of biological systems. Studying control-related problems in Boolean networks may reveal new insights into the intrinsic control in complex biological systems and enable us to develop strategies for manipulating biological systems using exogenous inputs. This paper considers controllability and observability of Boolean biological networks. We propose a new approach, which draws from the rich theory of symbolic computation, to solve the problems. Consequently, simple necessary and sufficient conditions for reachability, controllability, and observability are obtained, and algorithmic tests for controllability and observability which are based on the Gröbner basis method are presented. As practical applications, we apply the proposed approach to several different biological systems, namely, the mammalian cell-cycle network, the T-cell activation network, the large granular lymphocyte survival signaling network, and the Drosophila segment polarity network, gaining novel insights into the control and/or monitoring of the specific biological systems.

Synchronization in an array of output-coupled Boolean networks with time delay

This brief presents an analytical study of synchronization in an array of coupled deterministic Boolean networks (BNs) with time delay. Two kinds of models are considered. In one model, the outputs contain time delay, while in another one, the outputs do not. One restriction in this brief is that the state delay and output delay are restricted to be equal. By referring to the algebraic representations of logical dynamics and using the techniques of semitensor product of matrices, some necessary and sufficient conditions are derived for the synchronization of delay-coupled BNs. Examples including a practical epigenetic example are given for illustration.

Synchronization of Boolean Networks with Different Update Schemes

In this paper, the synchronizations of Boolean networks with different update schemes (synchronized Boolean networks and asynchronous Boolean networks) are investigated. All nodes in Boolean network are represented in terms of semi-tensor product. First, we give the concept of inner synchronization and observe that all nodes in a Boolean network are synchronized with each other. Second, we investigate the outer synchronization between a driving Boolean network and a corresponding response Boolean network. We provide not only the concept of traditional complete synchronization, but also the anti-synchronization and get the anti-synchronization in simulation. Third, we extend the outer synchronization to asynchronous Boolean network and get the complete synchronization between an asynchronous Boolean network and a response Boolean network. Consequently, theorems for synchronization of Boolean networks and asynchronous Boolean networks are derived. Examples are provided to show the correctness of our theorems.

Synchronization of coupled large-scale Boolean networks

This paper investigates the complete synchronization and partial synchronization of two large-scale Boolean networks. First, the aggregation algorithm towards large-scale Boolean network is reviewed. Second, the aggregation algorithm is applied to study the complete synchronization and partial synchronization of large-scale Boolean networks. Finally, an illustrative example is presented to show the efficiency of the proposed results.

Controllability and observability of Boolean control networks with time-variant delays in states

This brief investigates the controllability and observability of Boolean control networks with (not necessarily bounded) time-variant delays in states. After a brief introduction to converting a Boolean control network to an equivalent discrete-time bilinear dynamical system via the semi-tensor product of matrices, the system is split into a finite number of subsystems (constructed forest) with no time delays by using the idea of splitting time that is proposed in this brief. Then, the controllability and observability of the system are investigated by verifying any so-called controllability constructed path and any so-called observability constructed paths in the above forest, respectively, which generalize some recent relevant results. Matrix test criteria for the controllability and observability are given. The corresponding control design algorithms based on the controllability theorems are given. We also show that the computing complexity of our algorithm is much less than that of the existing algorithms.

Algebraic representation of asynchronous multiple-valued networks and its dynamics

In this paper, dynamics of asynchronous multiple-valued networks (AMVNs) are investigated based on linear representation. By semitensor product of matrices, we convert AMVNs into the discrete-time linear representation. A general formula to calculate all of network transition matrices of a specific AMVN is achieved. A necessary and sufficient algebraic criterion to determine whether a given state belongs to loose attractors of length s is proposed. Formulas for the numbers of attractors in AMVNs are provided. Finally, algorithms are presented to detect all of the attractors and basins. Examples are shown to demonstrate the feasibility of the proposed scheme.

Dynamics of random Boolean networks under fully asynchronous stochastic update based on linear representation

A novel algebraic approach is proposed to study dynamics of asynchronous random Boolean networks where a random number of nodes can be updated at each time step (ARBNs). In this article, the logical equations of ARBNs are converted into the discrete-time linear representation and dynamical behaviors of systems are investigated. We provide a general formula of network transition matrices of ARBNs as well as a necessary and sufficient algebraic criterion to determine whether a group of given states compose an attractor of length[Formula: see text] in ARBNs. Consequently, algorithms are achieved to find all of the attractors and basins in ARBNs. Examples are showed to demonstrate the feasibility of the proposed scheme.

Complete synchronization of temporal Boolean networks

This letter studies complete synchronization of two temporal Boolean networks coupled in the drive-response configuration. Necessary and sufficient conditions are provided based on the algebraic representation of Boolean networks. Moreover, the upper bound to check the criterion is given. Finally, an illustrative example shows the efficiency of the proposed results.

Controllability and optimal control of a temporal Boolean network

This paper investigates the controllability and optimal control of a temporal Boolean network, where the time delays are time variant. First, using the theory of semi-tensor product of matrices, the logical systems can be converted into a discrete time variant system. Second, necessary and sufficient conditions for the controllability via two types of controls are provided respectively. Third, optimal control design algorithms are presented. Finally, examples are given to illustrate the proposed results.

Complete synchronization of Boolean networks

We examine complete synchronization of two deterministic Boolean networks (BNs) coupled unidirectionally in the drive-response configuration. A necessary and sufficient criterion is presented in terms of algebraic representations of BNs. As a consequence, we show that complete synchronization can occur only between two conditionally identical BNs when the transition matrix of the drive network is nonsingular. Two examples are worked out to illustrate the obtained results.