Asynchronous Boolean Networks Research Articles

Reduction for asynchronous Boolean networks: elimination of negatively autoregulated components

To simplify the analysis of Boolean networks, a reduction in the number of components is often considered. A popular reduction method consists in eliminating components that are not autoregulated, using variable substitution. In this work, we show how this method can be extended, for asynchronous dynamics of Boolean networks, to the elimination of vertices that have a negative autoregulation, and study the effects on the dynamics and interaction structure. For elimination of non-autoregulated variables, the preservation of attractors is in general guaranteed only for fixed points. Here we give sufficient conditions for the preservation of complex attractors. The removal of so called mediator nodes (i.e. vertices with indegree and outdegree one) is often considered, and frequently does not affect the attractor landscape. We clarify that this is not always the case, and in some situations even subtle changes in the interaction structure can lead to a different asymptotic behaviour. Finally, we use properties of the more general elimination method introduced here to give an alternative proof for a bound on the number of attractors of asynchronous Boolean networks in terms of the cardinality of positive feedback vertex sets of the interaction graph.

A Framework of Pinning Control for Non-periodical Stable Behaviors of Large-scale Asynchronous Boolean Networks

A Framework of Pinning Control for Non-periodical Stable Behaviors of Large-scale Asynchronous Boolean Networks

Controllability of generalized asynchronous Boolean networks with periodical impulsive control

The objective of this paper is to investigate the controllability of generalized asynchronous Boolean networks under periodical impulsive control. In order to attain the objective, the semi-tensor product is employed to transform generalized asynchronous Boolean networks that have periodic impulsive control into linear representations. The conversion provides the basis for deriving the general formula of the network transition matrix’s dependence on the controller. Then three different control modes are used to define the controllability of generalized asynchronous Boolean networks: free Boolean sequence periodic impulsive control, input networked periodic impulsive control and feedback networked periodic impulsive control. For each mode, reachable sets of time s under a specific set of controls and arbitrary control are provided. The likelihood of transitioning from one state to another is also calculable. Finally, instances are given for a variety of control modes to illustrate the theorems’ effectiveness and robustness.

Lipschitz continuity under toric equivalence for asynchronous Boolean networks.

Mathematical models rooted in network representations are becoming increasingly more common for capturing a broad range of phenomena. Boolean networks (BNs) represent a mathematical abstraction suited for establishing general theory applicable to such systems. A key thread in BN research is developing theory that connects the structure of the network and the local rules to phase space properties or so-called structure-to-function theory. While most theory for BNs has been developed for the synchronous case, the focus of this work is on asynchronously updated BNs (ABNs) which are natural to consider from the point of view of applications to real systems where perfect synchrony is uncommon. A central question in this regard is sensitivity of dynamics of ABNs with respect to perturbations to the asynchronous update scheme. Macauley & Mortveit [Nonlinearity 22, 421-436 (2009)] showed that the periodic orbits are structurally invariant under toric equivalence of the update sequences. In this paper and under the same equivalence of the update scheme, the authors (i) extend that result to the entire phase space, (ii) establish a Lipschitz continuity result for sequences of maximal transient paths, and (iii) establish that within a toric equivalence class the maximal transient length may at most take on two distinct values. In addition, the proofs offer insight into the general asynchronous phase space of Boolean networks.

Attractor separation and signed cycles in asynchronous Boolean networks

The structure of the graph defined by the interactions in a Boolean network can determine properties of the asymptotic dynamics. For instance, considering the asynchronous dynamics, the absence of positive cycles guarantees the existence of a unique attractor, and the absence of negative cycles ensures that all attractors are fixed points. In presence of multiple attractors, one might be interested in properties that ensure that attractors are sufficiently “isolated”, that is, they can be found in separate subspaces or even trap spaces, subspaces that are closed with respect to the dynamics. Here we introduce notions of separability for attractors and identify corresponding necessary conditions on the interaction graph. In particular, we show that if the interaction graph has at most one positive cycle, or at most one negative cycle, or if no positive cycle intersects a negative cycle, then the attractors can be separated by subspaces. If the interaction graph has no path from a negative to a positive cycle, then the attractors can be separated by trap spaces. Furthermore, we study networks with interaction graphs admitting two vertices that intersect all cycles, and show that if their attractors cannot be separated by subspaces, then their interaction graph must contain a copy of the complete signed digraph on two vertices, deprived of a negative loop. We thus establish a connection between a dynamical property and a complex network motif. The topic is far from exhausted and we conclude by stating some open questions.

AEON.py: Python library for attractor analysis in asynchronous Boolean networks.

AEON.py is a Python library for the analysis of the long-term behaviour in very large asynchronous Boolean networks. It provides significant computational improvements over the state-of-the-art methods for attractor detection. Furthermore, it admits the analysis of partially specified Boolean networks with uncertain update functions. It also includes techniques for identifying viable source-target control strategies and the assessment of their robustness with respect to parameter perturbations. All relevant results are available in Supplementary Materials. The tool is accessible through https://github.com/sybila/biodivine-aeon-py. Supplementary data are available at Bioinformatics online.

Controllability and stabilization of periodic switched Boolean control networks with application to asynchronous updating

This paper investigates the periodic switching point controllability and stabilization of periodic switched Boolean control networks (PSBCNs), and applies the obtained results to the stabilization of deterministic asynchronous Boolean control networks (DABCNs). Firstly, using the algebraic state space representation of PSBCNs, a kind of periodic switching point controllability matrix is constructed, based on which, a necessary and sufficient condition is presented for the periodic switching point reachability and controllability of PSBCNs. Secondly, using the reachable set of PSBCNs, a constructive procedure is proposed to design time-variant state feedback controllers for the periodic switching point stabilization of PSBCNs. Finally, by converting the dynamics of DABCNs into the form of PSBCNs, the time-variant state feedback stabilization problem of DABCNs is solved.

Random Networks with Quantum Boolean Functions

We propose quantum Boolean networks, which can be classified as deterministic reversible asynchronous Boolean networks. This model is based on the previously developed concept of quantum Boolean functions. A quantum Boolean network is a Boolean network where the functions associated with the nodes are quantum Boolean functions. We study some properties of this novel model and, using a quantum simulator, we study how the dynamics change in function of connectivity of the network and the set of operators we allow. For some configurations, this model resembles the behavior of reversible Boolean networks, while for other configurations a more complex dynamic can emerge. For example, cycles larger than 2N were observed. Additionally, using a scheme akin to one used previously with random Boolean networks, we computed the average entropy and complexity of the networks. As opposed to classic random Boolean networks, where “complex” dynamics are restricted mainly to a connectivity close to a phase transition, quantum Boolean networks can exhibit stable, complex, and unstable dynamics independently of their connectivity.

Target Control of Asynchronous Boolean Networks.

We study the target control of asynchronous Boolean networks, to identify interventions that can drive the dynamics of a given Boolean network from any initial state to the desired target attractor. Based on the application time, the control can be realised with three types of perturbations, including instantaneous, temporary and permanent perturbations. We develop efficient methods to compute the target control for a given target attractor with these three types of perturbations. We compare our methods with the stable motif-based control method on a variety of real-life biological networks to evaluate their performance. We show that our methods scale well for large Boolean networks and they are able to identify a rich set of solutions with a small number of perturbations.

An Efficient Approach Towards the Source-Target Control of Boolean Networks.

We study the problem of computing a minimal subset of nodes of a given asynchronous Boolean network that need to be perturbed in a single-step to drive its dynamics from an initial state to a target steady state (or attractor), which we call the source-target control of Boolean networks. Due to the phenomenon of state-space explosion, a simple global approach that performs computations on the entire network may not scale well for large networks. We believe that efficient algorithms for such networks must exploit the structure of the networks together with their dynamics. Taking this view, we derive a decomposition-based solution to the minimal source-target control problem which can be significantly faster than the existing approaches on large networks. We then show that the solution can be further optimized if we take into account appropriate information about the source state. We apply our solutions to both real-life biological networks and randomly generated networks, demonstrating the efficiency and efficacy of our approach.

CABEAN: a software for the control of asynchronous Boolean networks.

Direct cell reprogramming, also called transdifferentiation, has great potential for tissue engineering and regenerative medicine. Boolean networks, a popular modelling framework for gene regulatory networks, make it possible to identify intervention targets for direct cell reprogramming with computational methods. In this work, we present our software, CABEAN, for the control of asynchronous Boolean networks. CABEAN identifies efficacious nodes, whose perturbations can drive the dynamics of a network from a source attractor (the initial cell type) to a target attractor (the desired cell type). CABEAN provides several control methods integrating practical constraints. Thus, it has the ability to provide a rich set of control sets, such that biologists can select suitable ones for validation based on specific experimental settings. The executable binary and the user guide of the software are publicly available at https://satoss.uni.lu/software/CABEAN/.

On Attractor Detection and Optimal Control of Deterministic Generalized Asynchronous Random Boolean Networks.

Deterministic asynchronous Boolean networks play a crucial role in modeling and analysis of gene regulatory networks. In this paper, we focus on a typical type of deterministic asynchronous Boolean networks called deterministic generalized asynchronous random Boolean networks (DGARBNs). We first formulate the extended state transition graph, which captures the whole dynamics of a DGARBN and paves potential ways to analyze this DGARBN. We then propose two SMT-based methods for attractor detection and optimal control of DGARBNs. These methods are implemented in a JAVA tool called DABoolNet. Two experiments are designed to highlight the scalability of the proposed methods. We also formally state and prove several relations between DGARBNs and other models including deterministic asynchronous models, block-sequential Boolean networks, generalized asynchronous random Boolean networks, and mixed-context random Boolean networks. Several case studies are presented to show the applications of our methods.

Synchronization of Coupled Boolean Networks With Different Update Scheme

In this paper, the synchronization of coupled Boolean networks with asynchronous update scheme is investigated. First, the asynchronous Boolean dynamics is transformed into a linear expression with the semi-tensor product technique. Then, the synchronization cycle is proposed to establish the synchronization conditions of asynchronous Boolean networks. Furthermore, an algorithm is given, which can be used to quickly check whether there exists the synchronization cycle in the coefficient matrix of the Boolean function. Finally, an example is provided to verify the efficiency of the theoretical analysis.

An ASP-based Approach for Attractor Enumeration in Synchronous and Asynchronous Boolean Networks

Boolean networks are conventionally used to represent and simulate gene regulatory networks. In the analysis of the dynamic of a Boolean network, the attractors are the objects of a special attention. In this work, we propose a novel approach based on Answer Set Programming (ASP) to express Boolean networks and simulate the dynamics of such networks. Our work focuses on the identification of the attractors, it relies on the exhaustive enumeration of all the attractors of synchronous and asynchronous Boolean networks. We applied and evaluated the proposed approach on real biological networks, and the obtained results indicate that this novel approach is promising.

A Linear Formulation of Asynchronous Boolean Networks

A new linear approach is proposed to model the dynamics of asynchronous Boolean networks (ABNs). The continuous-time discrete-state model definition makes the ABN a suitable formulation to describe gene regulatory networks. The semi-tensor product of matrices is used as the algebraic tool for the linear formulation. The results on the dynamics of the ABN are elaborated with synthetic and biological examples.

Positive and negative cycles in Boolean networks

We review and discuss some results about the influence of positive and negative feedback cycles in asynchronous Boolean networks. These results merge several ideas of Thomas: positive and negative feedback cycles have been largely emphasized by Thomas, through the so called Thomas’ rules, and asynchronous Boolean networks have been introduced by Thomas as a model for the dynamics of gene networks, which is nowadays very popular.

Synchronization of asynchronous probabilistic boolean network

This paper researches the synchronization of asynchronous probabilistic Boolean network and synchronization theorems are provided. Inner synchronization and outer synchronization are considered, respectively. It should be pointed that synchronization theorems and corollaries with probability one and probabilities unequal to one are both taken into consideration. What’s more, the asynchronous multivalued Boolean network is also studied as an extension. Finally, some examples for asynchronous probabilistic Boolean network synchronizations are provided in the numerical simulation.

Taming Asynchrony for Attractor Detection in Large Boolean Networks.

Boolean networks is a well-established formalism for modelling biological systems. A vital challenge for analyzing a Boolean network is to identify all the attractors. This becomes more challenging for large asynchronous Boolean networks, due to the asynchronous scheme. Existing methods are prohibited due to the well-known state-space explosion problem in large Boolean networks. In this paper, we tackle this challenge by proposing a SCC-based decomposition method. We prove the correctness of our proposed method and demonstrate its efficiency with two real-life biological networks.

Therapeutic target discovery using Boolean network attractors: improvements of kali.

In a previous article, an algorithm for identifying therapeutic targets in Boolean networks modelling pathological mechanisms was introduced. In the present article, the improvements made on this algorithm, named kali, are described. These improvements are (i) the possibility to work on asynchronous Boolean networks, (ii) a finer assessment of therapeutic targets and (iii) the possibility to use multivalued logic. kali assumes that the attractors of a dynamical system, such as a Boolean network, are associated with the phenotypes of the modelled biological system. Given a logic-based model of pathological mechanisms, kali searches for therapeutic targets able to reduce the reachability of the attractors associated with pathological phenotypes, thus reducing their likeliness. kali is illustrated on an example network and used on a biological case study. The case study is a published logic-based model of bladder tumorigenesis from which kali returns consistent results. However, like any computational tool, kali can predict but cannot replace human expertise: it is a supporting tool for coping with the complexity of biological systems in the field of drug discovery.

Reduction of Qualitative Models of Biological Networks for Transient Dynamics Analysis.

Qualitative models of dynamics of signalling pathways and gene regulatory networks allow for the capturing of temporal properties of biological networks while requiring few parameters. However, these discrete models typically suffer from the so-called state space explosion problem which makes the formal assessment of their potential behaviors very challenging. In this paper, we describe a method to reduce a qualitative model for enhancing the tractability of analysis of transient reachability properties. The reduction does not change the dimension of the model, but instead limits its degree of freedom, therefore reducing the set of states and transitions to consider. We rely on a transition-centered specification of qualitative models by the mean of automata networks. Our framework encompasses the usual asynchronous Boolean and multi-valued network, as well as 1-bounded Petri nets. Applied to different large-scale biological networks from the litterature, we show that the reduction can lead to a drastic improvement for the scalability of verification methods.