Boolean Networks Research Articles

Delay Synchronization of Drive-Response Boolean Networks and Boolean Control Networks

This article investigates the delay synchronization of drive Boolean network (DBN) and response BN (RBN) [response Boolean control network (RBCN)]. Synchronization and time delay are two common phenomena in real life. In aircraft formation, the latter's trajectory is consistent with the former's, but there exists a time gap when they arrive at the same position, which illustrates that time delay can happen in the process of synchronization. According to the delay step depending on initial states or not, two types of delay synchronization are proposed: 1) weak and 2) strong delay synchronization. Compared with synchronization, delay synchronization is only the theoretical generalization and has different application fields. Using the results of limit sets, several criteria for delay synchronization of the DBN and RBN are derived. This article provides an important result that delay synchronization of the DBN and RBCN can be equivalent to set stabilization of the augmented BCN, then the necessary and sufficient conditions are obtained. Moreover, two algorithms are established to design all time-optimal state feedback controllers. Finally, two examples are shown to demonstrate the effectiveness of the results.

Maximum-Likelihood State Estimators in Probabilistic Boolean Control Networks.

This study addresses state estimation problems for probabilistic Boolean control networks (PBCNs). Compared with deterministic Boolean networks, PBCNs have the stochastic switching in logical update functions in the state equation. Consequently, statistical analysis is required to estimate unavailable states, which induces an optimization problem called maximum-likelihood estimation. This article mainly focuses on two scenarios: 1) state estimation from partially measured state and 2) state estimation from output data, meaning observer design. The resulting optimization problems are solved using efficient algorithms based on dynamic programming. Concurrently, Dijkstra-type algorithms, which solve equivalent shortest path problems, are also proposed using best-first search. Furthermore, both the proposed algorithms derive novel observer design methods for PBCNs. The proposed algorithms are evaluated with practical estimation problems aiming to the sensor reduction and applied to gene regulatory networks of apoptosis and Lac operon.

Controllability of Boolean control networks with multiple time delays in both states and controls

Controllability of Boolean control networks with multiple time delays in both states and controls

Observability of Probabilistic Boolean Networks via Matrix Approach

In this article, the observability of probabilistic Boolean networks (PBNs) is investigated. First, the distinguishable state pairs are obtained by constructing a matrix sequence, based on which a novel criterion for examining the observability of PBNs is given. Then, an algorithm is proposed to verify the observability of PBNs. Finally, some numerical examples are presented to illustrate the proposed method.

Multiscale model of the different modes of cancer cell invasion.

Mathematical models of biological processes altered in cancer are built using the knowledge of complex networks of signaling pathways, detailing the molecular regulations inside different cell types, such as tumor cells, immune and other stromal cells. If these models mainly focus on intracellular information, they often omit a description of the spatial organization among cells and their interactions, and with the tumoral microenvironment. We present here a model of tumor cell invasion simulated with PhysiBoSS, a multiscale framework, which combines agent-based modeling and continuous time Markov processes applied on Boolean network models. With this model, we aim to study the different modes of cell migration and to predict means to block it by considering not only spatial information obtained from the agent-based simulation but also intracellular regulation obtained from the Boolean model. https://github.com/sysbio-curie/Invasion_model_PhysiBoSS.

Robust flipping stabilization of Boolean networks: A Q-learning approach

Robust flipping stabilization of Boolean networks: A Q-learning approach

Reinforcement Learning Data-Acquiring for Causal Inference of Regulatory Networks.

Gene regulatory networks (GRNs) consist of multiple interacting genes whose activities govern various cellular processes. The limitations in genomics data and the complexity of the interactions between components often pose huge uncertainties in the models of these biological systems. Meanwhile, inferring/estimating the interactions between components of the GRNs using data acquired from the normal condition of these biological systems is a challenging or, in some cases, an impossible task. Perturbation is a well-known genomics approach that aims to excite targeted components to gather useful data from these systems. This paper models GRNs using the Boolean network with perturbation, where the network uncertainty appears in terms of unknown interactions between genes. Unlike the existing heuristics and greedy data-acquiring methods, this paper provides an optimal Bayesian formulation of the data-acquiring process in the reinforcement learning context, where the actions are perturbations, and the reward measures step-wise improvement in the inference accuracy. We develop a semi-gradient reinforcement learning method with function approximation for learning near-optimal data-acquiring policy. The obtained policy yields near-exact Bayesian optimality with respect to the entire uncertainty in the regulatory network model, and allows learning the policy offline through planning. We demonstrate the performance of the proposed framework using the well-known p53-Mdm2 negative feedback loop gene regulatory network.

Multi-sensor fusion particle filtering for Boolean networks with multi-step randomly-delayed measurements

This paper is concerned with developing particle filters for Boolean networks with multi-step randomly-delayed measurements. Based on the semi-tensor product of matrices and dummy selection matrices, a generalized measurement model including multi-step randomly-delayed measurements is introduced. Besides, multi-sensor fusion particle filtering is proposed and two recursive algorithms are designed to minimize the mean-square estimation errors by fusing multi-sensor measurements. Simulation results show the effectiveness of the proposed multi-sensor fusion particle filters. It can be seen from the simulation results that the estimation performance of multi-sensor fusion has been improved compared to that of one sensor.

Reducing Boolean networks with backward equivalence

BackgroundBoolean Networks (BNs) are a popular dynamical model in biology where the state of each component is represented by a variable taking binary values that express, for instance, activation/deactivation or high/low concentrations. Unfortunately, these models suffer from the state space explosion, i.e., there are exponentially many states in the number of BN variables, which hampers their analysis.ResultsWe present Boolean Backward Equivalence (BBE), a novel reduction technique for BNs which collapses system variables that, if initialized with same value, maintain matching values in all states. A large-scale validation on 86 models from two online model repositories reveals that BBE is effective, since it is able to reduce more than 90% of the models. Furthermore, on such models we also show that BBE brings notable analysis speed-ups, both in terms of state space generation and steady-state analysis. In several cases, BBE allowed the analysis of models that were originally intractable due to the complexity. On two selected case studies, we show how one can tune the reduction power of BBE using model-specific information to preserve all dynamics of interest, and selectively exclude behavior that does not have biological relevance.ConclusionsBBE complements existing reduction methods, preserving properties that other reduction methods fail to reproduce, and vice versa. BBE drops all and only the dynamics, including attractors, originating from states where BBE-equivalent variables have been initialized with different activation values The remaining part of the dynamics is preserved exactly, including the length of the preserved attractors, and their reachability from given initial conditions, without adding any spurious behaviours. Given that BBE is a model-to-model reduction technique, it can be combined with further reduction methods for BNs.

Uncoding the interdependency of tumor microenvironment and macrophage polarization: insights from a continuous network approach.

The balance between pro- and anti-inflammatory immune system responses is crucial to preventing complex diseases like cancer. Macrophages are essential immune cells that contribute to this balance constrained by the local signaling profile of the tumor microenvironment. To understand how pro- and anti-inflammatory unbalance emerges in cancer, we developed a theoretical analysis of macrophage differentiation that is derived from activated monocytes circulating in the blood. Once recruited to the site of inflammation, monocytes can be polarized based on the specific interleukins and chemokines in the microenvironment. To quantify this process, we used a previous regulatory network reconstructed by our group and transformed Boolean Network attractors of macrophage polarization to an ODE scheme, it enables us to quantify the activation of their genes in a continuous fashion. The transformation was developed using the interaction rules with a fuzzy logic approach. By implementing this approach, we analyzed different aspects that cannot be visualized in the Boolean setting. For example, this approach allows us to explore the dynamic behavior at different concentrations of cytokines and transcription factors in the microenvironment. One important aspect to assess is the evaluation of the transitions between phenotypes, some of them characterized by an abrupt or a gradual transition depending on specific concentrations of exogenous cytokines in the tumor microenvironment. For instance, IL-10 can induce a hybrid state that transits between an M2c and an M2b macrophage. Interferon- can induce a hybrid between M1 and M1a macrophage. We further demonstrated the plasticity of macrophages based on a combination of cytokines and the existence of hybrid phenotypes or partial polarization. This mathematical model allows us to unravel the patterns of macrophage differentiation based on the competition of expression of transcriptional factors. Finally, we survey how macrophages may respond to a continuously changing immunological response in a tumor microenvironment.

Observability of Periodically Switched Boolean Control Networks

Observability of Periodically Switched Boolean Control Networks

Stability of linear Boolean networks

Stability is an important characteristic of network models that has implications for other desirable aspects such as controllability. The stability of a Boolean network (BN) depends on various factors, such as the topology of its wiring diagram and the type of the functions describing its dynamics. Linear Boolean networks can be completely described by their wiring diagram, and therefore the structure of linear networks plays a prominent role in determining their stability. In this paper, we study the stability of linear Boolean networks by computing Derrida curves and quantifying the number of attractors and cycle lengths imposed by their network topologies. Derrida curves are commonly used to measure the stability of Boolean networks and several parameters such as the average in-degree K and the output bias p can indicate if a network is stable, critical, or unstable. For random unbiased Boolean networks there is a critical connectivity value Kc=2 such that if K<Kc networks operate in the ordered regime, and if K>Kc networks operate in the chaotic regime. Here, we show that for linear networks, which are the least canalizing and most unstable, the phase transition from order to chaos already happens at an average in-degree of Kc=1. Consistently, we also show that unstable networks exhibit a large number of attractors with very long limit cycles while stable and critical networks exhibit fewer attractors with shorter limit cycles. Additionally, we present theoretical results to quantify important dynamical properties of linear networks. First, we present a formula for the proportion of attractor states in linear systems. Second, we show that the expected number of fixed points in linear systems is 2, while general Boolean networks possess on average one fixed point. Third, we present a formula to quantify the number of bijective linear Boolean networks and provide a lower bound for the percentage of this type of network.

From Boolean networks to linear dynamical systems: a simplified route

ABSTRACT Boolean Networks can be converted to discrete linear dynamical systems on finite spaces by a semi-tensor-product approach. This approach has been used by many to study the dynamics and control of Boolean systems. However, the process of getting the linear representation using the semi-tensor-product method is complicated even for a simple three-node network and requires the help of a computer program. In this work, we show that we can skip the semi-tensor process and obtain the same linear representation with a straightforward mapping. Moreover, our approach produces a large number of isomorphic representations which provides a flexible framework. Importantly, it could simplify the analytical study of networks with unspecified number of nodes that have some structure.

Finite-Time Disturbance Decoupling of Boolean Control Networks

In this article, the finite-time disturbance decoupling problem (FTDDP) of Boolean networks (BNs) and Boolean control networks (BCNs) is studied. A novel necessary and sufficient condition is first provided to determine whether a given BN is finite-time disturbance decoupled. Then, an algorithm is devised to find the shortest time that guarantees that a given BN is finite-time disturbance decoupled. In addition, using a constructed truth matrix, a necessary and sufficient condition for solving the disturbance decoupling problem of BCNs is proposed, and all possible time optimal state feedback controllers are designed. Finally, two examples are introduced to illustrate the feasibility of the obtained results.

Fixed Point Preserving Model Reduction of Boolean Networks Focusing on Complement and Absorption Laws

A Boolean network (BN) is well known as a discrete model for analysis and control of complex networks such as gene regulatory networks. Since complex networks are large-scale in general, it is important to consider model reduction. In this paper, we consider model reduction that the information on fixed points (singleton attractors) is preserved. In model reduction studied here, the interaction graph obtained from a given BN is utilized. In the existing method, the minimum feedback vertex set (FVS) of the interaction graph is focused on. The dimension of the state is reduced to the number of elements of the minimum FVS. In the proposed method, we focus on complement and absorption laws of Boolean functions in substitution operations of a Boolean function into other one. By simplifying Boolean functions, the dimension of the state may be further reduced. Through a numerical example, we present that by the proposed method, the dimension of the state can be reduced for BNs that the dimension of the state cannot be reduced by the existing method.

Weak Stabilization of Boolean Networks Under State-Flipped Control.

In this brief, stabilization of Boolean networks (BNs) by flipping a subset of nodes is considered, here we call such action state-flipped control. The state-flipped control implies that the logical variables of certain nodes are flipped from 1 to 0 or 0 to 1 as time flows. Under state-flipped control on certain nodes, a state-flipped-transition matrix is defined to describe the impact on the state transition space. Weak stabilization is first defined and then some criteria are presented to judge the same. An algorithm is proposed to find a stabilizing kernel such that BNs can achieve weak stabilization to the desired state with in-degree more than 0. By defining a reachable set, another approach is proposed to verify weak stabilization, and an algorithm is given to obtain a flip sequence steering an initial state to a given target state. Subsequently, the issue of finding flip sequences to steer BNs from weak stabilization to global stabilization is addressed. In addition, a model-free reinforcement algorithm, namely the Q -learning ( [Formula: see text]) algorithm, is developed to find flip sequences to achieve global stabilization. Finally, several numerical examples are given to illustrate the obtained theoretical results.

Recent advances on disturbance decoupling of Boolean control networks

Recent advances on disturbance decoupling of Boolean control networks

Asynchronous Control of Stochastic Switched Boolean Control Networks With Piecewise-Homogeneous Dwell Time.

In this article, the l₁-induced performance of the stochastic switched Boolean control network (BCN) is investigated. The switched signal is considered to follow a time-varying probability distribution, the switching of which is considered to have a random dwell time. The asynchronous state feedback control (SFC) is studied to achieve the control objective. This kind of control can avoid the failure of the control due to the inconsistency between the system mode and the control mode, so the results obtained are more general. Using the semitensor product of matrices, the algebraic form of the considered BCN is represented. Under this framework, sufficient conditions are obtained to ensure that the closed-loop system is stochastic stabilized with a prescribed l₁-induced performance level ɣ. Parameters can be solved by inequalities. In addition, when the dwell time converges to infinity, the probability distribution of the switched signal becomes fixed. Necessary and sufficient conditions are presented to ensure the stabilization of the closed system under asynchronous SFC as well as the design of the asynchronous SFC. Then, sufficient condition is obtained for the prescribed l₁-induced performance level. Examples are presented to show the effectiveness of the obtained results.

Leveraging developmental landscapes for model selection in Boolean gene regulatory networks.

Boolean models are a well-established framework to model developmental gene regulatory networks (DGRNs) for acquisition of cellular identities. During the reconstruction of Boolean DGRNs, even if the network structure is given, there is generally a large number of combinations of Boolean functions that will reproduce the different cell fates (biological attractors). Here we leverage the developmental landscape to enable model selection on such ensembles using the relative stability of the attractors. First we show that previously proposed measures of relative stability are strongly correlated and we stress the usefulness of the one that captures best the cell state transitions via the mean first passage time (MFPT) as it also allows the construction of a cellular lineage tree. A property of great computational importance is the insensitivity of the different stability measures to changes in noise intensities. That allows us to use stochastic approaches to estimate the MFPT and thereby scale up the computations to large networks. Given this methodology, we revisit different Boolean models of Arabidopsis thaliana root development, showing that a most recent one does not respect the biologically expected hierarchy of cell states based on relative stabilities. We therefore developed an iterative greedy algorithm that searches for models which satisfy the expected hierarchy of cell states and found that its application to the root development model yields many models that meet this expectation. Our methodology thus provides new tools that can enable reconstruction of more realistic and accurate Boolean models of DGRNs.

Editorial for the special issue “Control‐theoretic approaches for systems in the life sciences”

Editorial for the special issue “Control<scp>‐theoretic</scp> approaches for systems in the life sciences”