Boolean Networks Research Articles

Boolean network modeling and its integration with experimental read-outs : An interdisciplinary presentation using a leukemia model.

The limited availability of suitable animal models and cell lines often impedes experimental cancer research. Wet-laboratory experiments are also time-consuming and cost-intensive. In this review, we present an in silico modeling strategy, namely, Boolean network (BN) models, and demonstrate how it could be applied to streamline experimental design and to focus the effort of experimental read-outs. Boolean network models allow for the dynamic analysis of large molecular signaling pathways and their crosstalks. After establishing and validating aspecific tumor model, mechanistic insights into the tumor cell behavior can be gained by studying the trajectories of different tumor phenotypes. Also, tumor driver and drug target screenings can be performed. These automatic screenings can help to identify new intervention targets and putative biomarkers for tumor evolution, hence guiding new wet-laboratory experiments. The goal of this round-up is to demonstrate how to establish, validate, and use BN modeling and its crosstalks in classic wet-laboratory research using achronic lymphocytic leukemia (CLL) BN model.

Multi-state sequential Boolean networks: An analysis of its combinatorial dynamics

Multi-state Boolean networks model the behavior of interrelated (non-binary) Boolean entities over time. These networks can be used in several contexts, and extend the usual (simpler) binary-state Boolean networks. Leveraging the established knowledge of binary-state Boolean networks, we explore the dynamics of sequential multi-state Boolean networks, where their entities or nodes have states which belong to a (general) Boolean algebra with more than two elements and whose updates are determined in an specific order. This study contributes to the advancement of multi-state Boolean networks, enriching their theoretical knowledge and practical utility.

Local set stability and target control of probabilistic Boolean networks

This paper analyzes the local set stability of probabilistic Boolean networks (PBNs), including local finite-time set stability with probability one (FSSPO) and local asymptotic set stability with probability one (ASSPO). Firstly, a necessary and sufficient condition is proposed for the local FSSPO of PBNs. Then, a reachable set with probability one is constructed, based on which, the largest domain of attraction for the local FSSPO of PBNs is determined. Secondly, by constructing two parameterized sets, the transition probability matrix of PBNs is properly partitioned, and the largest domain of attraction for the ASSPO is depicted. Finally, for PBNs which are locally stable, the target control is introduced to achieve the global FSSPO and global ASSPO of PBNs, respectively.

New method for stability of probabilistic Boolean networks with inequality constraints

This paper studies the stability of probabilistic Boolean networks (PBNs) with state inequality constraints by establishing a new algebraic representation. On one hand, combining with the semi-tensor product (STP) of matrices and the law of total expectation, the algebraic representation of PBNs is established. On the other hand, based on the STP of matrices, an algorithm is proposed to determine the solution set of the state inequality constraints. Then, based on the algebraic representation and the inequality constrained solution set, a new form of transition probability matrix is constructed for PBNs. Finally, by the proposed transition probability matrix, new criteria are proposed for the finite-time stability with positive probability of PBNs. Several simulation examples are used to illustrate the effectiveness of the obtained results.

Dynamics of ecosystem services along ecological network seascapes

Abstract Human societies depend on services provided by ecosystems, from local needs such as clean water and pest control to global services like the ozone layer and the ocean biological pump. Ecosystem services are linked to the states of the ecosystem, which are, in turn, governed by a web of ecological interactions. These interactions, along with the services they support, are under threat from environmental changes driven by human activities. Therefore, safeguarding these vital services requires an understanding of how the structure and dynamics of ecological interactions are affected by environmental change. A critical step towards this goal is the development of a theoretical framework that can elucidate how ecosystem services are sustained or impaired by interactions within ecosystems in fluctuating environments. Recent years have seen progress in characterizing the organization and dynamics of ecological networks. However, linking temporally varying network structure in fluctuating environments, the seascapes of ecological networks, and their impact on services remains a challenge. We propose an approach based on merging ecological network analysis with Boolean functions and modeling of fluctuating environments to address how services are affected by environmental change. We review aspects of Boolean Network models and illustrate the approach using biologically inspired Boolean rules that involve predator-prey cycles, trophic cascades, and mutualisms formed by plants and their frugivores. This approach aims to contribute to the study of how the organization of ecological interactions affects the persistence of ecosystem services. Specifically, we discuss how this approach can provide new insights into how environmental change affects the relationship between ecological networks and ecosystem services. The combination of information on the natural history of species interactions and ecosystem services, Boolean networks, and models for fluctuating environments may contribute to conservation strategies for preserving biodiversity and ecosystem services in the face of ongoing environmental change.

Computational Complexity of Minimal Trap Spaces in Boolean Networks

Computational Complexity of Minimal Trap Spaces in Boolean Networks

Dynamically equivalent disjunctive networks

The study of the dynamical behavior of Boolean networks with different update schedules has so far focused primarily on the possible dynamics and equivalent networks that can be obtained. However, few studies have been done about which networks can be obtained from another network with a non-parallel schedule.In this article, we define the problem of finding a Boolean network that is dynamically equivalent to another network. For the general case, it is shown that the problem is NP-Hard. However, if the problem is restricted to disjunctive Boolean networks, it can be solved in polynomial time.

Pinning control design for the output regulation of Boolean networks

In previous studies, the output regulation problem (ORP) of Boolean networks (BNs) is usually studied by the augmented system method, however, the state transition matrix dimension of augmented system is growing exponentially as the number of network nodes increases. To reduce the computational complexity, a new method to solve the ORP is adopted in this paper. Firstly, the relationship among the attractors of two BNs and their augmented system is studied, based on which, the ORP is converted into the set stabilisation problem of the original BN, and some necessary and sufficient conditions are given to detect the solvability of the ORP. Secondly, a pinning control for the ORP is considered according to the attractors of the original BNs, and an algorithm to design this kind of controls is provided. Finally, a practical example of signal transduction network is given to illustrate the effectiveness of the method in this paper.

Long-Run Behavior Estimation of Temporal Boolean Networks With Multiple Data Losses.

This brief devotes to investigating the long-run behavior estimation of temporal Boolean networks (TBNs) with multiple data losses, especially the asymptotical stability. The information transmission is modeled by Bernoulli variables, based on which an augmented system is constructed to facilitate the analysis. A theorem guarantees that the asymptotical stability of the original system can be converted to that of the augmented system. Subsequently, one necessary and sufficient condition is obtained for asymptotical stability. Furthermore, an auxiliary system is derived to study the synchronization issue of the ideal TBNs with normal data transmission and TBNs with multiple data losses, as well as an effective criterion for verifying synchronization. Finally, numerical examples are given to illustrate the validity of the theoretical results.

Intrinsic Universality in Automata Networks III: On Symmetry versus Asynchrony

An automata network is a finite assembly of interconnected entities endowed with a set of local maps defined over a common finite alphabet. These relationships are represented through an interaction graph. Together with the local functions, an assignment known as an update scheme directs the evolution of the network by updating specific subsets of entities at discrete time steps. Despite the scrutiny of interaction graphs and update schemes, their profound impact on automata network dynamics remains largely open. This work investigates the intricate interplay between these aspects, with a focus on how update schemes can counterbalance constraints stemming from symmetric local interactions. This paper is the third of a series about intrinsic universality, a notion that assesses both dynamical and computational complexity, encompassing transient behaviors, attractors, and prediction or reachability problems. We consider four update modes—parallel, block-sequential, local clocks, and general periodic— along with several families of symmetric signed conjunctive boolean networks defined by local constraints on signs. Our main result is to show a diagonal complexity leap in this two-dimensional landscape: the stronger the local constraints the higher the level of asynchrony required to obtain intrinsic universality or increase in complexity. We also show how in some cases asynchronism allows to simulate directed interactions from undirected ones with the same local rules.

Minimal observability of switching Boolean networks

In this paper, the minimal observability of switching Boolean networks (SBNs) is investigated. Firstly, applying the semi‐tensor product (STP) method of matrices, a parallel extension system is constructed, based on which a necessary and sufficient condition to detect the observability of the SBNs is given. Secondly, when an SBN is unobservable, the specific steps to obtain the required measurements to make the system observable are given using the set reachable method; however, the measurements given in this part are not necessarily the fewest. Then, a criterion for determining the minimum number of measurements is further proposed through a constructed indicator matrix. Lastly, the effectiveness of the new results is verified by an example.

Lightweight High‐Throughput TRNG Based on Single‐Node Boolean Chaotic Structure

ABSTRACTWith the rise of the Internet and electronic devices, the security of network information is gaining attention, and the true random number generator (TRNG) is playing an increasingly crucial role in information security. TRNG, based on Boolean chaotic entropy source, has drawn significant interest due to its uncomplicated circuit design and minimal hardware resource usage. However, most existing structures consist of two‐input or three‐input logic devices, forming a complex multinode, geometrically symmetric Boolean chaotic network using multiple logic devices. This network configuration results in increased complexity and reduced throughput. This study introduces an entropy source based on Boolean chaos utilizing single‐node and four‐input XOR gates, which can be easily placed and routed on Xilinx Artix‐7 FPGA. It requires only 29 LUTs and 5 DFFs without any postprocessing, achieving a throughput of up to 700 Mb/s. The output of TRNG has successfully passed various tests including the autocorrelation test, NIST SP800‐22, NIST SP800‐90B, AIS‐31, and TESTU01 tests with favorable results. Furthermore, by applying a three‐stage XOR chain postprocessing on Xilinx Spartan‐6 FPGA and Xilinx Virtex‐6 FPGA, it has passed the NIST SP800‐22 and NIST SP800‐90B tests at 300 Mb/s. The structure was also tested using Xilinx Virtex‐6 FPGA under different temperature and voltage conditions, passing the NIST SP800‐90B IID test.

Set Restabilization of Perturbed Boolean Control Networks

This paper develops a parameter tuning method for solving the set restabilization problem of perturbed Boolean control networks (BCNs). First, the absorbable attractor, which we previously proposed, is recalled. Based on the relationship between attractors, a necessary and sufficient restabilizability criterion is derived. This criterion is used to check whether a perturbed BCN can be stabilized to the original target set by modifying the least number of parameters to the old controller. Furthermore, a constructive method for fine-tuning the old controller is provided if the criterion condition derived above is satisfied. Compared with the existing relevant results, ours have clear advantages, since they can address the set restabilization problem of BCNs subject to multi-column function perturbations, which has not been solved yet. Finally, two examples are employed to show the effectiveness and advantages of our results.

Combining Zhegalkin Polynomials and SAT Solving for Context-specific Boolean Modeling of Biological Systems.

Large amounts of knowledge regarding biological processes are readily available in the literature and aggregated in diverse databases. Boolean networks are powerful tools to render that knowledge into models that can mimic and simulate biological phenomena at multiple scales. Yet, when a model is required to understand or predict the behavior of a biological system in given conditions, existing information often does not completely match this context. Networks built from only prior knowledge can overlook mechanisms, lack specificity, and just partially recapitulate experimental observations. To address this limitation, context-specific data needs to be integrated. However, the brute-force identification of qualitative rules matching these data becomes infeasible as the number of candidates explodes for increasingly complex systems. Here, we used Zhegalkin polynomials to transform this identification into a binary value assignment for exponentially fewer variables, which we addressed with a state-of-the-art SAT solver. We evaluated our implemented method alongside two widely recognized tools, CellNetOptimizer and Caspo-ts, on both artificial toy models and large-scale models based on experimental data from the HPN-DREAM challenge. Our approach demonstrated benchmark-leading capabilities on networks of significant size and intricate complexity. It thus appears promising for the in silico modeling of ever more comprehensive biological systems.

Time-optimal open-loop set stabilization of Boolean control networks

We show that for stabilization of Boolean control networks (BCNs) with unobservable initial states, open-loop control and close-loop control are not equivalent. An example is given to illustrate the nonequivalence. Enlightened by the nonequivalence, we explore open-loop set stabilization of BCNs with unobservable initial states. More specifically, this issue is to investigate that for a given BCN, whether there exists a unified free control sequence that is effective for all initial states of the system to stabilize the system states to a given set. The criteria for open-loop set stabilization is derived and for any open-loop set stabilizable BCN, every time-optimal open-loop set stabilizer is proposed. Besides, we obtain the least upper bounds of two integers, which are respectively related to the global stabilization and partial stabilization of BCNs in the results of two literature articles. Using the methods in the two literature articles, the least upper bounds of the two integers cannot be obtained.

A sensitivity analysis of cellular automata and heterogeneous topology networks: partially-local cellular automata and homogeneous homogeneous random boolean networks

ABSTRACT Elementary Cellular Automata (ECA) are well-studied computational universes capable of impressive computational variety, but harnessing their potential has been challenging. When combined with Reservoir Computing (RC), harnessing this computation becomes feasible, and furthermore enables energy efficient AI. This study compares ECA reservoirs to topological heterogeneous and more biological plausible counterparts of Partially-Local CA (PLCA) and Homogeneous Homogeneous Random Boolean Networks (HHRBN). Using the 5-bit memory benchmark, Temporal Derrida plots and collapse rate, finding are that more disordered topology does not equate to more disordered computation and moreover the evidence suggest this heterogeneity shrinks the critical range.

Pinning Asymptotic Observability of Distributed Boolean Networks.

Asymptotic observability of distributed Boolean networks (DBNs) is studied in this article. Via a parallel extension method, asymptotic observability of the original system is converted to reachability at a fixed point of the extended system. Based on the structure matrix of the extended system, a necessary and sufficient condition is presented for asymptotic observability. Further, for unobservable systems, mode-dependent pinning control is first introduced and applied to achieve asymptotic observability, including the selections of pinning nodes, the design of output feedback controls, and the adding approaches. Then, a set of matrices is defined for the construction of the desired structure matrix. Based on it, a necessary condition is given to guarantee the solvability of the corresponding output feedback controls and the adding approaches. Finally, a numerical example is presented to show the effectiveness of the obtained results.

A Polynomial-Time Criterion for Stability in Distribution of Probabilistic Boolean Networks

A Polynomial-Time Criterion for Stability in Distribution of Probabilistic Boolean Networks

Bayesian Lookahead Perturbation Policy for Inference of Regulatory Networks.

The complexity, scale, and uncertainty in regulatory networks (e.g., gene regulatory networks and microbial networks) regularly pose a huge uncertainty in their models. These uncertainties often cannot be entirely reduced using limited and costly data acquired from the normal condition of systems. Meanwhile, regulatory networks often suffer from the non-identifiability issue, which refers to scenarios where the true underlying network model cannot be clearly distinguished from other possible models. Perturbation or excitation is a well-known process in systems biology for acquiring targeted data to reveal the complex underlying mechanisms of regulatory networks and overcome the non-identifiability issue. We consider a general class of Boolean network models for capturing the activation and inactivation of components and their complex interactions. Assuming partial available knowledge about the interactions between components of the networks, this paper formulates the inference process through the maximum aposteriori (MAP) criterion. We develop a Bayesian lookahead policy that systematically perturbs regulatory networks to maximize the performance of MAP inference under the perturbed data. This is achieved by optimally formulating the perturbation process in a reinforcement learning context and deriving a scalable deep reinforcement learning perturbation policy to compute near-optimal Bayesian policy. The proposed method learns the perturbation policy through planning without the need for any real data. The high performance of the proposed approach is demonstrated by comprehensive numerical experiments using the well-known mammalian cell cycle and gut microbial community networks.

A Framework of Pinning Control for Nonperiodical Stable Behaviors of Large-Scale Asynchronous Boolean Networks

A Framework of Pinning Control for Nonperiodical Stable Behaviors of Large-Scale Asynchronous Boolean Networks