Steady-State Design of Large-Dimensional Boolean Networks

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.

Multi-Sensor Fusion Boolean Bayesian Filtering for Stochastic Boolean Networks

Stochastic Boolean networks (SBNs) take process noise into account, so it is better to fit the actual situation and has a wider application background than Boolean networks (BNs). However, the presence of noise influences us to estimate the real state of the system. To minimize the inaccuracies caused by the presence of noise, an optimal state estimation problem is studied in this article. The multi-sensor fusion Boolean Bayesian filtering is proposed and a recursive algorithm is provided to calculate the prior and posterior belief of system state by fusing multi-sensor measurements based on the algebraic form of the SBN and Bayesian law. Then, the optimal state estimator is obtained, which minimizes the mean-square estimation error. Finally, a simulation example is carried out to demonstrate the performance of the proposed methodology. It has been shown through the simulation experiment that it increases the confidence level of the state estimation and improves the estimation performance using multi-sensor fusion compared with using single sensor.

Minimal Pinning Control for Oscillatority of Boolean Networks

In this article, minimal pinning control for oscillatority (i.e., instability) of Boolean networks (BNs) under algebraic state space representations method is studied. First, two criteria for oscillatority of BNs are obtained from the aspects of state transition matrix (STM) and network structure (NS) of BNs, respectively. A distributed pinning control (DPC) from these two aspects is proposed: one is called STM-based DPC and the other one is called NS-based DPC, both of which are only dependent on local in-neighbors. As for STM-based DPC, one arbitrary node can be chosen to be controlled, based on certain solvability of several equations, meanwhile a hybrid pinning control (HPC) combining DPC and conventional pinning control (CPC) is also proposed. In addition, as for NS-based DPC, pinning control nodes (PCNs) can be found using the information of NS, which efficiently reduces the high computational complexity. The proposed STM-based DPC and NS-based DPC in this article are shown to be simple and concise, which provide a new direction to dramatically reduce control costs and computational complexity. Finally, gene networks are simulated to discuss the effectiveness of theoretical results.

Stabilizing Large-Scale Probabilistic Boolean Networks by Pinning Control

This article aims to stabilize probabilistic Boolean networks (PBNs) via a novel pinning control strategy. In a PBN, the state evolution of each gene switches among a collection of candidate Boolean functions with preassigned probability distributions, which govern the activation frequency of each Boolean function. Due to the existence of stochasticity, the mode-independent pinning controller might be disabled. Thus, both mode-independent and mode-dependent pinning controller are required here. Moreover, a criterion is derived to determine whether mode-independent controllers are applicable while the pinned nodes are given. It is worth pointing out that this pinning control is based on the n×n network structure rather than 2ⁿ ×2ⁿ state transition matrix. Therefore, compared with the existing results, this pinning control strategy is more practicable and has the ability to handle large-scale networks, especially sparsely connected networks. To demonstrate the effectiveness of the designed control scheme, a PBN that describes the mammalian cell-cycle encountering a mutated phenotype is discussed as a simulation.

Exploring attractor bifurcations in Boolean networks

Background

Boolean networks (BNs) provide an effective modelling formalism for various complex biochemical phenomena. Their long term behaviour is represented by attractors–subsets of the state space towards which the BN eventually converges. These are then typically linked to different biological phenotypes. Depending on various logical parameters, the structure and quality of attractors can undergo a significant change, known as a bifurcation. We present a methodology for analysing bifurcations in asynchronous parametrised Boolean networks.

Results

In this paper, we propose a computational framework employing advanced symbolic graph algorithms that enable the analysis of large networks with hundreds of Boolean variables. To visualise the results of this analysis, we developed a novel interactive presentation technique based on decision trees, allowing us to quickly uncover parameters crucial to the changes in the attractor landscape. As a whole, the methodology is implemented in our tool AEON. We evaluate the method’s applicability on a complex human cell signalling network describing the activity of type-1 interferons and related molecules interacting with SARS-COV-2 virion. In particular, the analysis focuses on explaining the potential suppressive role of the recently proposed drug molecule GRL0617 on replication of the virus.

Conclusions

The proposed method creates a working analogy to the concept of bifurcation analysis widely used in kinetic modelling to reveal the impact of parameters on the system’s stability. The important feature of our tool is its unique capability to work fast with large-scale networks with a relatively large extent of unknown information. The results obtained in the case study are in agreement with the recent biological findings.

Graph-Based Function Perturbation Analysis for Observability of Multivalued Logical Networkss

Observability is a fundamental concept for the synthesis of both linear systems and nonlinear systems. This article devotes to discussing the robustness of observability for multivalued logical networks (MVLNs) subject to function perturbation and establishing a graph-based framework. First, based on the transition graph of undistinguishable pairs of states, a new graph-based criterion is presented for the observability of MVLNs. Second, a candidate set consisting of all suspicious undistinguishable pairs of states is defined, based on the cardinality of which and the graph-based condition, a series of effective criteria are proposed for the robustness of observability subject to function perturbations. Finally, the obtained results are applied to the robust observability analysis of the p53-MDM2 negative feedback regulatory loop.