Boolean Networks Research Articles

Biologically meaningful regulatory logic enhances the convergence rate in Boolean networks and bushiness of their state transition graph.

Boolean models of gene regulatory networks (GRNs) have gained widespread traction as they can easily recapitulate cellular phenotypes via their attractor states. Their overall dynamics are embodied in a state transition graph (STG). Indeed, two Boolean networks (BNs) with the same network structure and attractors can have drastically different STGs depending on the type of Boolean functions (BFs) employed. Our objective here is to systematically delineate the effects of different classes of BFs on the structural features of the STG of reconstructed Boolean GRNs while keeping network structure and biological attractors fixed, and explore the characteristics of BFs that drive those features. Using $10$ reconstructed Boolean GRNs, we generate ensembles that differ in BFs and compute from their STGs the dynamics' rate of contraction or 'bushiness' and rate of 'convergence', quantified with measures inspired from cellular automata (CA) that are based on the garden-of-Eden (GoE) states. We find that biologically meaningful BFs lead to higher STG 'bushiness' and 'convergence' than random ones. Obtaining such 'global' measures gets computationally expensive with larger network sizes, stressing the need for feasible proxies. So we adapt Wuensche's $Z$-parameter in CA to BFs in BNs and provide four natural variants, which, along with the average sensitivity of BFs computed at the network level, comprise our descriptors of local dynamics and we find some of them to be good proxies for bushiness. Finally, we provide an excellent proxy for the 'convergence' based on computing transient lengths originating at random states rather than GoE states.

Parameter Learning of Probabilistic Boolean Control Networks with Input-Output Data

Article Parameter Learning of Probabilistic Boolean Control Networks with Input-Output Data Hongwei Chen 1,*, Qi Chen 1, Bo Shen 1, and Yang Liu 2 1 College of Information Science and Technology, Donghua University, Shanghai 201620, China 2 School of Mathematical Sciences, Zhejiang Normal University, Jinhua 321004, China * Correspondence: hongwei@dhu.edu.cn Received: 23 September 2023 Accepted: 27 November 2023 Published: 26 March 2024 Abstract: This paper investigates the parameter learning problem for the probabilistic Boolean control networks (PBCNs) with input-output data. Firstly, an algebraic expression of the PBCNs is obtained by taking advantage of the semi-tensor product technique, and then, the parameter learning problem is transformed into an optimal problem to reveal the parameter matrices of a linear system in a computationally efficient way. Secondly, two recursive semi-tensor product based algorithms are designed to calculate the forward and backward probabilities. Thirdly, the expectation maximization algorithm is proposed as an elaborate technique to address the parameter learning problem. In addition, a useful index is introduced to describe the performance of the proposed parameter learning algorithm. Finally, two numerical examples are employed to demonstrate the reliability of the proposed parameter learning approach.

Set stabilization of logical control networks: A minimum node control approach

In network systems, control using minimum nodes or pinning control can be effectively used for stabilization problems to cut down the cost of control. In this paper, we investigate the set stabilization problem of logical control networks. In particular, we study the set stabilization problem of probabilistic Boolean networks (PBNs) and probabilistic Boolean control networks (PBCNs) via controlling minimal nodes. Firstly, an algorithm is given to search for the minimum index set of pinning nodes. Then, based on the analysis of its high computational complexity, we present optimized algorithms with lower computational complexity to ascertain the network control using minimum node sets. Moreover, some sufficient and necessary conditions are proposed to ensure the feasibility and effectiveness of the proposed algorithms. Furthermore, a theorem is presented for PBCNs to devise all state-feedback controllers corresponding to the set of pinning nodes. Finally, two models of gene regulatory networks are considered to show the efficacy of obtained results.

Scalable Enumeration of Trap Spaces in Boolean Networks via Answer Set Programming

Boolean Networks (BNs) are widely used as a modeling formalism in several domains, notably systems biology and computer science. A fundamental problem in BN analysis is the enumeration of trap spaces, which are hypercubes in the state space that cannot be escaped once entered. Several methods have been proposed for enumerating trap spaces, however they often suffer from scalability and efficiency issues, particularly for large and complex models. To our knowledge, the most efficient and recent methods for the trap space enumeration all rely on Answer Set Programming (ASP), which has been widely applied to the analysis of BNs. Motivated by these considerations, our work proposes a new method for enumerating trap spaces in BNs using ASP. We evaluate the method on a mix of 250+ real-world and 400+ randomly generated BNs, showing that it enables analysis of models beyond the capabilities of existing tools (namely pyboolnet, mpbn, trappist, and trapmvn).

Optimal control for reachability of Markov jump switching Boolean control networks subject to output trackability

This article aims at designing the finite horizon optimal control sequence (FHOCS) for ensuring the reachability of Markov jump switching Boolean control networks (MJSBCNs) subject to output trackability. The key step in solving such a problem is to construct a Luenberger-like observer (LLO). Our approach includes a few steps as follows. We firstly propose the formulation of the optimal control problem (OCP) of maximising the probability of finite-time reachability subject to output trackability. Then, by converting the dynamics of MJSBCNs into an equivalent expectation system, an LLO is established for MJSBCNs. In this setup, a Luenberger-like observation matrix is constructed, which is an effect tool to solve the finite-time reachability problem. Finally, an LLO method is developed to find the FHOCS by maximising the T -step Luenberger-like observation matrix. In a biological example, we show the effectiveness of our developed methods.

Development of a Probabilistic Boolean network (PBN) to model intraoperative blood pressure management

BackgroundBlood pressure is a vital sign for organ perfusion that anesthesiologists measure and modulate during surgery. However, current decision-making processes rely heavily on clinicians’ experience, which can lead to variability in treatment across surgeries. With the advent of machine learning, we can now create models to predict the outcomes of interventions and guide perioperative decision-making. The first step in this process involves translating the clinical decision-making process into a framework understood by an algorithm. Probabilistic Boolean networks (PBNs) provide an information-rich approach to this problem. A PBN trends toward a steady state, and its decisions are easily understood via its Boolean predictor functions. We hypothesize that a PBN can be developed that corrects hemodynamic instability in patients by selecting clinical interventions to maintain blood pressure within a given range. MethodsData on patients over the age of 65 undergoing surgery with general anesthesia from 2018 to 2020 were drawn from the UF Health PRECEDE data set with IRB approval (IRB201700747). Parameters examined included heart rate, blood pressure, and frequency of medications given 15 min after anesthetic induction and 15 min before awakening. The medication frequency data were truncated into a 66/33 split for the training and validation set used in the PBN. The model was coded using Python 3 and evaluated by comparing the frequency of medications chosen by the program to the values in the testing set via linear regression analysis. ResultsThe network developed successfully models a hemodynamically unstable patient and corrects the imbalance by administering medications. This is evidenced by the model achieving a stable, steady state matrix in all iterations. However, the model's ability to emulate clinical drug selection was variable. It was successful with its use of vasodilator selection but struggled with the appropriate selection of vasopressors. ConclusionsThe PBN has demonstrated the ability to choose appropriate interventions based on a patient's current vitals. Additional work must be done to have the network emulate the frequency at which drugs are selected from in clinical practice. In its current state, the model provides an understanding of how a PBN behaves in the context of correcting hemodynamic instability and can aid in developing more robust models in the future.

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.

Preponderance of generalized chain functions in reconstructed Boolean models of biological networks

Boolean networks (BNs) have been extensively used to model gene regulatory networks (GRNs). The dynamics of BNs depend on the network architecture and regulatory logic rules (Boolean functions (BFs)) associated with nodes. Nested canalyzing functions (NCFs) have been shown to be enriched among the BFs in the large-scale studies of reconstructed Boolean models. The central question we address here is whether that enrichment is due to certain sub-types of NCFs. We build on one sub-type of NCFs, the chain functions (or chain-0 functions) proposed by Gat-Viks and Shamir. First, we propose two other sub-types of NCFs, namely, the class of chain-1 functions and generalized chain functions, the union of the chain-0 and chain-1 types. Next, we find that the fraction of NCFs that are chain-0 (also holds for chain-1) functions decreases exponentially with the number of inputs. We provide analytical treatment for this and other observations on BFs. Then, by analyzing three different datasets of reconstructed Boolean models we find that generalized chain functions are significantly enriched within the NCFs. Lastly we illustrate that upon imposing the constraints of generalized chain functions on three different GRNs we are able to obtain biologically viable Boolean models.

Entropy Model of Rosin Autonomous Boolean Network Digital True Random Number Generator

A True Random Number Generator (TRNG) is an important component in cryptographic algorithms and protocols. The Rosin Autonomous Boolean Network (ABN) digital TRNG has been widely studied due to its nice properties, such as low energy consumption, high speed, strong platform portability, and strong randomness. However, there is still a lack of suitable entropy models to deduce the requirement of design parameters to ensure true randomness. The current model to evaluate the entropy of oscillator-based TRNGs is not applicable for Rosin ABN TRNGs due to low-frequency noise. This work presents a new, suitable stochastic model to evaluate the entropy of Rosin ABN TRNGs. Theoretical analysis and simulation experiments verify the correctness and the effectiveness of the model, and, finally, the appropriate sampling parameters for Rosin ABN TRNGs are given for sufficient entropy per random bit to ensure true randomness.

Metabolic symbiosis between oxygenated and hypoxic tumour cells: An agent-based modelling study.

Deregulated metabolism is one of the hallmarks of cancer. It is well-known that tumour cells tend to metabolize glucose via glycolysis even when oxygen is available and mitochondrial respiration is functional. However, the lower energy efficiency of aerobic glycolysis with respect to mitochondrial respiration makes this behaviour, namely the Warburg effect, counter-intuitive, although it has now been recognized as source of anabolic precursors. On the other hand, there is evidence that oxygenated tumour cells could be fuelled by exogenous lactate produced from glycolysis. We employed a multi-scale approach that integrates multi-agent modelling, diffusion-reaction, stoichiometric equations, and Boolean networks to study metabolic cooperation between hypoxic and oxygenated cells exposed to varying oxygen, nutrient, and inhibitor concentrations. The results show that the cooperation reduces the depletion of environmental glucose, resulting in an overall advantage of using aerobic glycolysis. In addition, the oxygen level was found to be decreased by symbiosis, promoting a further shift towards anaerobic glycolysis. However, the oxygenated and hypoxic populations may gradually reach quasi-equilibrium. A sensitivity analysis using Latin hypercube sampling and partial rank correlation shows that the symbiotic dynamics depends on properties of the specific cell such as the minimum glucose level needed for glycolysis. Our results suggest that strategies that block glucose transporters may be more effective to reduce tumour growth than those blocking lactate intake transporters.

The connectivity degree controls the difficulty in reservoir design of random boolean networks.

Reservoir Computing (RC) is a paradigm in artificial intelligence where a recurrent neural network (RNN) is used to process temporal data, leveraging the inherent dynamical properties of the reservoir to perform complex computations. In the realm of RC, the excitatory-inhibitory balance b has been shown to be pivotal for driving the dynamics and performance of Echo State Networks (ESN) and, more recently, Random Boolean Network (RBN). However, the relationship between b and other parameters of the network is still poorly understood. This article explores how the interplay of the balance b, the connectivity degree K (i.e., the number of synapses per neuron) and the size of the network (i.e., the number of neurons N) influences the dynamics and performance (memory and prediction) of an RBN reservoir. Our findings reveal that K and b are strongly tied in optimal reservoirs. Reservoirs with high K have two optimal balances, one for globally inhibitory networks (b < 0), and the other one for excitatory networks (b > 0). Both show asymmetric performances about a zero balance. In contrast, for moderate K, the optimal value being K = 4, best reservoirs are obtained when excitation and inhibition almost, but not exactly, balance each other. For almost all K, the influence of the size is such that increasing N leads to better performance, even with very large values of N. Our investigation provides clear directions to generate optimal reservoirs or reservoirs with constraints on size or connectivity.

Mathematical modeling in autoimmune diseases: from theory to clinical application.

The research & development (R&D) of novel therapeutic agents for the treatment of autoimmune diseases is challenged by highly complex pathogenesis and multiple etiologies of these conditions. The number of targeted therapies available on the market is limited, whereas the prevalence of autoimmune conditions in the global population continues to rise. Mathematical modeling of biological systems is an essential tool which may be applied in support of decision-making across R&D drug programs to improve the probability of success in the development of novel medicines. Over the past decades, multiple models of autoimmune diseases have been developed. Models differ in the spectra of quantitative data used in their development and mathematical methods, as well as in the level of "mechanistic granularity" chosen to describe the underlying biology. Yet, all models strive towards the same goal: to quantitatively describe various aspects of the immune response. The aim of this review was to conduct a systematic review and analysis of mathematical models of autoimmune diseases focused on the mechanistic description of the immune system, to consolidate existing quantitative knowledge on autoimmune processes, and to outline potential directions of interest for future model-based analyses. Following a systematic literature review, 38 models describing the onset, progression, and/or the effect of treatment in 13 systemic and organ-specific autoimmune conditions were identified, most models developed for inflammatory bowel disease, multiple sclerosis, and lupus (5 models each). ≥70% of the models were developed as nonlinear systems of ordinary differential equations, others - as partial differential equations, integro-differential equations, Boolean networks, or probabilistic models. Despite covering a relatively wide range of diseases, most models described the same components of the immune system, such as T-cell response, cytokine influence, or the involvement of macrophages in autoimmune processes. All models were thoroughly analyzed with an emphasis on assumptions, limitations, and their potential applications in the development of novel medicines.

Identification of edge removal fault in Boolean networks and disjunctive Boolean networks

It is meaningful to identify various faults occurring in dynamic systems as early as possible. This paper explores the identification of edge removal fault in Boolean networks (BNs). At first, with the help of semi-tensor product of matrices, the algebraic formulation is presented for the faulty BNs subject to an edge removal fault. After that, a fault-triggered matrix is constructed to uncover the relationship of algebraic formulations between non-faulty BN and faulty BN, based on which, a kind of fault identification equation is established to identify the two faulty nodes in the single edge removal fault. Furthermore, a fault identification matrix is constructed to identify the faulty nodes in the single and multiple edge removal faults of a disjunctive BN, whose dimension is much smaller than the fault-triggered matrix. Finally, the obtained results are demonstrated by the Drosophila Spz processing model.

An optimal Bayesian intervention policy in response to unknown dynamic cell stimuli

Interventions in gene regulatory networks (GRNs) aim to restore normal functions of cells experiencing abnormal behavior, such as uncontrolled cell proliferation. The dynamic, uncertain, and complex nature of cellular processes poses significant challenges in determining the best interventions. Most existing intervention methods assume that cells are unresponsive to therapies, resulting in stationary and deterministic intervention solutions. However, cells in unhealthy conditions can dynamically respond to therapies through internal stimuli, leading to the recurrence of undesirable conditions. This paper proposes a Bayesian intervention policy that adaptively responds to cell dynamic responses according to the latest available information. The GRNs are modeled using a Boolean network with perturbation (BNp), and the fight between the cell and intervention is modeled as a two-player zero-sum game. Assuming an incomplete knowledge of cell stimuli, a recursive approach is developed to keep track of the posterior distribution of cell responses. The proposed Bayesian intervention policy takes action according to the posterior distribution and a set of Nash equilibrium policies associated with all possible cell responses. Analytical results demonstrate the superiority of the proposed intervention policy against several existing intervention techniques. Meanwhile, the performance of the proposed policy is investigated through comprehensive numerical experiments using the p53-MDM2 negative feedback loop regulatory network and melanoma network. The results demonstrate the empirical convergence of the proposed policy to the optimal Nash equilibrium policy.

Robust Stabilizing Control of Perturbed Biological Networks via Coordinate Transformation and Algebraic Analysis.

This article investigates robust stabilizing control of biological systems modeled by Boolean networks (BNs). A population of BNs is considered where a majority of BNs have the same BN dynamics, but some BNs are inflicted by mutations damaging particular nodes, leading to perturbed dynamics that prohibit global stabilization to the desired attractor. The proposed control strategy consists of two steps. First, the nominal BN is transformed and curtailed into a sub-BN via a simple coordinate transformation and network reduction associated with the desired attractor. The feedback vertex set (FVS) control is then applied to the reduced BN to determine the control inputs for the nominal BN. Next, the control inputs derived in the first step and mutated nodes are applied to the nominal BN so as to identify residual dynamics of perturbed BNs, and additional control inputs are selected according to the canalization effect of each node. The overall control inputs are applied to the BN population, so that the nominal BN converges to the desired attractor and perturbed BNs to their own attractors that are the closest possible to the desired attractor. The performance of the proposed robust control scheme is validated through numerical experiments on random BNs and a complex biological network.

Design of all feasible output feedback controllers for robust output tracking of Boolean control networks

This paper devotes to the design of all feasible output feedback controllers for the robust output tracking of Boolean control networks with disturbances (DBCNs). First, the output feedback–based robust control invariant set (RCIS) is defined, and a calculation method to find all output feedback–based RCISs of a given set is derived via the truth matrix technique. Second, a class of complete family of robust reachable sets (CFRRSs) is developed; the solvability of robust output tracking problem by output feedback control is verified. Furthermore, a constructive algorithm of all feasible output feedback controllers is presented. Finally, the study of a network about Escherichia coli lactose operon shows the effectiveness of the theoretical results.

Set Stabilization of Large-Scale Stochastic Boolean Networks: A Distributed Control Strategy

Set Stabilization of Large-Scale Stochastic Boolean Networks: A Distributed Control Strategy

On Connections Between Invariant Subspace and Quotient Approaches for Analysis and Control of Boolean Networks

On Connections Between Invariant Subspace and Quotient Approaches for Analysis and Control of Boolean Networks

Stabilization of Probabilistic Boolean Networks via State-Flipped Control and Reinforcement Learning

Stabilization of Probabilistic Boolean Networks via State-Flipped Control and Reinforcement Learning

Computational inference of chemokine-mediated roles for the vagus nerve in modulating intra- and inter-tissue inflammation

Introduction: The vagus nerve innervates multiple organs, but its role in regulating cross-tissue spread of inflammation is as yet unclear. We hypothesized that the vagus nerve may regulate cross-tissue inflammation via modulation of the putatively neurally regulated chemokine IP-10/CXCL10.Methods: Rate-of-change analysis, dynamic network analysis, and dynamic hypergraphs were used to model intra- and inter-tissue trends, respectively, in inflammatory mediators from mice that underwent either vagotomy or sham surgery.Results: This analysis suggested that vagotomy primarily disrupts the cross-tissue attenuation of inflammatory networks involving IP-10 as well as the chemokines MIG/CXCL9 and CCL2/MCP-1 along with the cytokines IFN-γ and IL-6. Computational analysis also suggested that the vagus-dependent rate of expression of IP-10 and MIG/CXCL9 in the spleen impacts the trajectory of chemokine expression in other tissues. Perturbation of this complex system with bacterial lipopolysaccharide (LPS) revealed a vagally regulated role for MIG in the heart. Further, LPS-stimulated expression of IP-10 was inferred to be vagus-independent across all tissues examined while reducing connectivity to IL-6 and MCP-1, a hypothesis supported by Boolean network modeling.Discussion: Together, these studies define novel spatiotemporal dimensions of vagus-regulated acute inflammation.