Controlling large Boolean networks with single-step perturbations

Abstract Motivation The control of Boolean networks has traditionally focussed on strategies where the perturbations are applied to the nodes of the network for an extended period of time. In this work, we study if and how a Boolean network can be controlled by perturbing a minimal set of nodes for a single-step and letting the system evolve afterwards according to its original dynamics. More precisely, given a Boolean network (BN), we compute a minimal subset Cmin of the nodes such that BN can be driven from any initial state in an attractor to another ‘desired’ attractor by perturbing some or all of the nodes of Cmin for a single-step. Such kind of control is attractive for biological systems because they are less time consuming than the traditional strategies for control while also being financially more viable. However, due to the phenomenon of state-space explosion, computing such a minimal subset is computationally inefficient and an approach that deals with the entire network in one-go, does not scale well for large networks. Results We develop a ‘divide-and-conquer’ approach by decomposing the network into smaller partitions, computing the minimal control on the projection of the attractors to these partitions and then composing the results to obtain Cmin for the whole network. We implement our method and test it on various real-life biological networks to demonstrate its applicability and efficiency. Supplementary information Supplementary data are available at Bioinformatics online.

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Finding the fixed points of a Boolean network from a positive feedback vertex set

Bioinformatics ◽

10.1093/bioinformatics/btaa922 ◽

2020 ◽

Author(s):

Julio Aracena ◽

Luis Cabrera-Crot ◽

Lilian Salinas

Keyword(s):

Fixed Points ◽

Positive Feedback ◽

Boolean Network ◽

Dynamical Behavior ◽

Boolean Networks ◽

Supplementary Information ◽

Feedback Vertex Set ◽

Executable File ◽

Vertex Set ◽

Regulatory Functions

Abstract Motivation In the modeling of biological systems by Boolean networks, a key problem is finding the set of fixed points of a given network. Some constructed algorithms consider certain structural properties of the regulatory graph like those proposed by Akutsu et al. and Zhang et al., which consider a feedback vertex set of the graph. However, these methods do not take into account the type of action (activation and inhibition) between its components. Results In this article, we propose a new algorithm for finding the set of fixed points of a Boolean network, based on a positive feedback vertex set P of its regulatory graph and which works, by applying a sequential update schedule, in time O(2|P|·n2+k), where n is the number of components and the regulatory functions of the network can be evaluated in time O(nk), k≥0. The theoretical foundation of this algorithm is due a nice characterization, that we give, of the dynamical behavior of the Boolean networks without positive cycles and with a fixed point. Availability and implementation An executable file of FixedPoint algorithm made in Java and some examples of input files are available at: www.inf.udec.cl/˜lilian/FPCollector/. Supplementary information Supplementary material is available at Bioinformatics online.

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Atlas: automatic modeling of regulation of bacterial gene expression and metabolism using rule-based languages

Bioinformatics ◽

10.1093/bioinformatics/btaa1040 ◽

2020 ◽

Author(s):

Rodrigo Santibáñez ◽

Daniel Garrido ◽

Alberto J M Martin

Keyword(s):

Gene Expression ◽

Biological Networks ◽

Regulatory Networks ◽

Dynamic Models ◽

Stochastic Dynamics ◽

Divide And Conquer ◽

Supplementary Information ◽

Bacterial Gene ◽

Rule Based ◽

Gene Regulatory

Abstract Motivation Cells are complex systems composed of hundreds of genes whose products interact to produce elaborated behaviors. To control such behaviors, cells rely on transcription factors to regulate gene expression, and gene regulatory networks (GRNs) are employed to describe and understand such behavior. However, GRNs are static models, and dynamic models are difficult to obtain due to their size, complexity, stochastic dynamics and interactions with other cell processes. Results We developed Atlas, a Python software that converts genome graphs and gene regulatory, interaction and metabolic networks into dynamic models. The software employs these biological networks to write rule-based models for the PySB framework. The underlying method is a divide-and-conquer strategy to obtain sub-models and combine them later into an ensemble model. To exemplify the utility of Atlas, we used networks of varying size and complexity of Escherichia coli and evaluated in silico modifications, such as gene knockouts and the insertion of promoters and terminators. Moreover, the methodology could be applied to the dynamic modeling of natural and synthetic networks of any bacteria. Availability and implementation Code, models and tutorials are available online (https://github.com/networkbiolab/atlas). Supplementary information Supplementary data are available at Bioinformatics online.

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Generalized Boolean Networks

Handbook of Research on Computational Methodologies in Gene Regulatory Networks ◽

10.4018/978-1-60566-685-3.ch018 ◽

2010 ◽

pp. 429-449 ◽

Cited By ~ 1

Author(s):

Christian Darabos ◽

Mario Giacobini ◽

Marco Tomassini

Keyword(s):

Biological Networks ◽

Regulatory Networks ◽

Dynamical Behavior ◽

Real Life ◽

Network Models ◽

Cell Types ◽

Boolean Networks ◽

Genetic Regulatory Networks ◽

Point Of View ◽

Scale Free

Random Boolean Networks (RBN) have been introduced by Kauffman more than thirty years ago as a highly simplified model of genetic regulatory networks. This extremely simple and abstract model has been studied in detail and has been shown capable of extremely interesting dynamical behavior. First of all, as some parameters are varied such as the network’s connectivity, or the probability of expressing a gene, the RBN can go through a phase transition, going from an ordered regime to a chaotic one. Kauffman’s suggestion is that cell types correspond to attractors in the RBN phase space, and only those attractors that are short and stable under perturbations will be of biological interest. Thus, according to Kauffman, RBN lying at the edge between the ordered phase and the chaotic phase can be seen as abstract models of genetic regulatory networks. The original view of Kauffman, namely that these models may be useful for understanding real-life cell regulatory networks, is still valid, provided that the model is updated to take into account present knowledge about the topology of real gene regulatory networks, and the timing of events, without loosing its attractive simplicity. According to present data, many biological networks, including genetic regulatory networks, seem, in fact, to be of the scale-free type. From the point of view of the timing of events, standard RBN update their state synchronously. This assumption is open to discussion when dealing with biologically plausible networks. In particular, for genetic regulatory networks, this is certainly not the case: genes seem to be expressed in different parts of the network at different times, according to a strict sequence, which depends on the particular network under study. The expression of a gene depends on several transcription factors, the synthesis of which appear to be neither fully synchronous nor instantaneous. Therefore, we have recently proposed a new, more biologically plausible model. It assumes a scale-free topology of the networks and we define a suitable semi-synchronous dynamics that better captures the presence of an activation sequence of genes linked to the topological properties of the network. By simulating statistical ensembles of networks, we discuss the attractors of the dynamics, showing that they are compatible with theoretical biological network models. Moreover, the model demonstrates interesting scaling abilities as the size of the networks is increased.

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On the Number of Driver Nodes for Controlling a Boolean Network to Attractors

10.1101/395442 ◽

2018 ◽

Author(s):

Wenpin Hou ◽

Peiying Ruan ◽

Wai-Ki Ching ◽

Tatsuya Akutsu

Keyword(s):

Biological Networks ◽

Boolean Network ◽

Network Models ◽

Boolean Networks ◽

Reasonable Assumption ◽

Control Problems ◽

Expected Number ◽

Linear Complex ◽

Simulation Results ◽

Control Complex

AbstractIt is known that many driver nodes are required to control complex biological networks. Previous studies imply that O(N) driver nodes are required in both linear complex network and Boolean network models with N nodes if an arbitrary state is specified as the target. In this paper, we mathematically prove under a reasonable assumption that the expected number of driver nodes is only O(log2N + log2M) for controlling Boolean networks if the targets are restricted to attractors, where M is the number of attractors. Since it is expected that M is not very large in many practical networks, this is a significant improvement. This result is based on discovery of novel relationships between control problems on Boolean networks and the coupon collector’s problem, a well-known concept in combinatorics. We also provide lower bounds of the number of driver nodes as well as simulation results using artificial and realistic network data, which support our theoretical findings.

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FINDING AND ANALYZING THE MINIMUM SET OF DRIVER NODES IN CONTROL OF BOOLEAN NETWORKS

Advances in Complex Systems ◽

10.1142/s0219525916500065 ◽

2016 ◽

Vol 19 (03) ◽

pp. 1650006 ◽

Cited By ~ 6

Author(s):

WENPIN HOU ◽

TAKEYUKI TAMURA ◽

WAI-KI CHING ◽

TATSUYA AKUTSU

Keyword(s):

Linear Programming ◽

Integer Linear Programming ◽

Boolean Network ◽

Boolean Networks ◽

Np Hard ◽

Initial State ◽

Random Boolean Networks ◽

Minimum Number ◽

Average Size ◽

Theoretical Analyses

We study the minimum number of driver nodes control of which leads a Boolean network (BN) from an initial state to a target state in a specified number of time steps. We show that the problem is NP-hard and present an integer linear programming-based method that solves the problem exactly. We mathematically analyze the average size of the minimum set of driver nodes for random Boolean networks with bounded in-degree and with a small number of time steps. The results of computational experiments using randomly generated BNs show good agreements with theoretical analyses. A further examination in realistic BNs demonstrates the efficiency and generality of our theoretical analyses.

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Algorithmic and Stochastic Representations of Gene Regulatory Networks and Protein-Protein Interactions

Current Topics in Medicinal Chemistry ◽

10.2174/1568026619666190311125256 ◽

2019 ◽

Vol 19 (6) ◽

pp. 413-425 ◽

Cited By ~ 3

Author(s):

Athanasios Alexiou ◽

Stylianos Chatzichronis ◽

Asma Perveen ◽

Abdul Hafeez ◽

Ghulam Md. Ashraf

Keyword(s):

Protein Interactions ◽

Biological Networks ◽

Regulatory Networks ◽

Review Paper ◽

Boolean Networks ◽

Diagnostic Tools ◽

Complex Nature ◽

Cellular Interactions ◽

Protein Protein Interactions ◽

Deterministic Models

Background:Latest studies reveal the importance of Protein-Protein interactions on physiologic functions and biological structures. Several stochastic and algorithmic methods have been published until now, for the modeling of the complex nature of the biological systems.Objective:Biological Networks computational modeling is still a challenging task. The formulation of the complex cellular interactions is a research field of great interest. In this review paper, several computational methods for the modeling of GRN and PPI are presented analytically.Methods:Several well-known GRN and PPI models are presented and discussed in this review study such as: Graphs representation, Boolean Networks, Generalized Logical Networks, Bayesian Networks, Relevance Networks, Graphical Gaussian models, Weight Matrices, Reverse Engineering Approach, Evolutionary Algorithms, Forward Modeling Approach, Deterministic models, Static models, Hybrid models, Stochastic models, Petri Nets, BioAmbients calculus and Differential Equations.Results:GRN and PPI methods have been already applied in various clinical processes with potential positive results, establishing promising diagnostic tools.Conclusion:In literature many stochastic algorithms are focused in the simulation, analysis and visualization of the various biological networks and their dynamics interactions, which are referred and described in depth in this review paper.

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Controllability of Boolean networks with intermittent disturbances

Modern Physics Letters B ◽

10.1142/s0217984920503091 ◽

2020 ◽

Vol 34 (28) ◽

pp. 2050309

Author(s):

Tao You ◽

Hailun Zhang ◽

Mingyu Yang ◽

Xiao Wang ◽

Yangming Guo

Keyword(s):

Gene Expression ◽

Boolean Network ◽

Boolean Networks ◽

Genetic Mutations ◽

Intermittent Control ◽

Mathematical Tool ◽

Pulse Control ◽

Expression Of Genes ◽

Complicated Process ◽

Living Being

In biological systems, gene expression is an important subject. In order to clarify the specific process of gene expression, mathematical tools are needed to simulate the process. The Boolean network (BN) is a good mathematical tool. In this paper, we study a Boolean network with intermittent perturbations. This is of great significance for studying genetic mutations in bioengineering. The expression of genes in the internal system of a living being is a very complicated process, and it is clear that the process is trans-ageal for humans. Through the intermittent control and pulse control of the BN, we can obtain the trajectory of gene expression better and faster, which will provide a very important theoretical basis for our next research.

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TreeMerge: a new method for improving the scalability of species tree estimation methods

Bioinformatics ◽

10.1093/bioinformatics/btz344 ◽

2019 ◽

Vol 35 (14) ◽

pp. i417-i426 ◽

Cited By ~ 7

Author(s):

Erin K Molloy ◽

Tandy Warnow

Keyword(s):

Large Scale ◽

Species Tree ◽

New Method ◽

Divide And Conquer ◽

Supplementary Information ◽

Estimation Methods ◽

Running Time ◽

Tree Estimation ◽

Computationally Intensive ◽

A Minor

Abstract Motivation At RECOMB-CG 2018, we presented NJMerge and showed that it could be used within a divide-and-conquer framework to scale computationally intensive methods for species tree estimation to larger datasets. However, NJMerge has two significant limitations: it can fail to return a tree and, when used within the proposed divide-and-conquer framework, has O(n5) running time for datasets with n species. Results Here we present a new method called ‘TreeMerge’ that improves on NJMerge in two ways: it is guaranteed to return a tree and it has dramatically faster running time within the same divide-and-conquer framework—only O(n2) time. We use a simulation study to evaluate TreeMerge in the context of multi-locus species tree estimation with two leading methods, ASTRAL-III and RAxML. We find that the divide-and-conquer framework using TreeMerge has a minor impact on species tree accuracy, dramatically reduces running time, and enables both ASTRAL-III and RAxML to complete on datasets (that they would otherwise fail on), when given 64 GB of memory and 48 h maximum running time. Thus, TreeMerge is a step toward a larger vision of enabling researchers with limited computational resources to perform large-scale species tree estimation, which we call Phylogenomics for All. Availability and implementation TreeMerge is publicly available on Github (http://github.com/ekmolloy/treemerge). Supplementary information Supplementary data are available at Bioinformatics online.

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Introduction

Modern Applications of Graph Theory ◽

10.1093/oso/9780198856740.003.0001 ◽

2021 ◽

pp. 1-36

Author(s):

Vadim Zverovich

Keyword(s):

Graph Theory ◽

Biological Networks ◽

Real Life ◽

Scale Free ◽

Graph Theoretic ◽

Life Problems ◽

Scale Free Networks ◽

Web Graph ◽

The Web

This chapter gives a brief overview of selected applications of graph theory, many of which gave rise to the development of graph theory itself. A range of such applications extends from puzzles and games to serious scientific and real-life problems, thus illustrating the diversity of applications. The first section is devoted to the six earliest applications of graph theory. The next section introduces so-called scale-free networks, which include the web graph, social and biological networks. The last section describes a number of graph-theoretic algorithms, which can be used to tackle a number of interesting applications and problems of graph theory.

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On the Lyapunov Exponent of Monotone Boolean Networks †

Mathematics ◽

10.3390/math8061035 ◽

2020 ◽

Vol 8 (6) ◽

pp. 1035

Author(s):

Ilya Shmulevich

Keyword(s):

Lyapunov Exponent ◽

Dynamical Systems ◽

Boolean Functions ◽

Boolean Network ◽

Discrete Dynamical Systems ◽

Boolean Networks ◽

Asymptotic Formulas ◽

Small Perturbations ◽

Monotone Boolean Functions ◽

Almost All

Boolean networks are discrete dynamical systems comprised of coupled Boolean functions. An important parameter that characterizes such systems is the Lyapunov exponent, which measures the state stability of the system to small perturbations. We consider networks comprised of monotone Boolean functions and derive asymptotic formulas for the Lyapunov exponent of almost all monotone Boolean networks. The formulas are different depending on whether the number of variables of the constituent Boolean functions, or equivalently, the connectivity of the Boolean network, is even or odd.

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