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

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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FINDING CYCLES IN SYNCHRONOUS BOOLEAN NETWORKS WITH APPLICATIONS TO BIOCHEMICAL SYSTEMS

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127403006765 ◽

2003 ◽

Vol 13 (03) ◽

pp. 535-552 ◽

Cited By ~ 133

Author(s):

JACK HEIDEL ◽

JOHN MALONEY ◽

CHRISTOPHER FARROW ◽

J. A. ROGERS

Keyword(s):

Signal Transduction ◽

Fixed Points ◽

Boolean Network ◽

Boolean Networks ◽

Large Network ◽

Logical Function ◽

Scalar Form ◽

Biochemical Systems ◽

Sigmoidal Kinetics ◽

Periodic Cycles

This paper is an analytical study of Boolean networks. The motivation is our desire to understand the large, complicated and interconnected pathways which comprise intracellular biochemical signal transduction networks. The simplest possible conceptual model that mimics signal transduction with sigmoidal kinetics is the n-node Boolean network each of whose elements or nodes has the value 0 (off) or 1 (on) at any given time T = 0, 1, 2, …. A Boolean network has 2nstates all of which are either on periodic cycles (including fixed points) or transients leading to cycles. Thus one understands a Boolean network by determining the number and length of its cycles. The problem one must circumvent is the large number of states (2n) since the networks we are interested in have 100 or more elements. Thus we concentrate on developing size n methods rather than the impossible task of enumerating all 2nstates. This is done as follows: the dynamics of the network can be described by n polynomial equations which describe the logical function which determines the interaction at each node. Iterating the equations one step at a time finds all fixed points, period two cycles, period three cycles, etc. This is a general method that can be used to determine the fixed points and moderately large periodic cycles of any size network, but it is not useful in finding the largest cycles in a large network. However, we also show that the network equations can often be reduced to scalar form, which makes the cycle structure much more transparent. The scalar equations method is a true "size n" method and several examples (including nontrivial biochemical systems) are examined.

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Controlling large Boolean networks with single-step perturbations

Bioinformatics ◽

10.1093/bioinformatics/btz371 ◽

2019 ◽

Vol 35 (14) ◽

pp. i558-i567 ◽

Cited By ~ 2

Author(s):

Alexis Baudin ◽

Soumya Paul ◽

Cui Su ◽

Jun Pang

Keyword(s):

Biological Networks ◽

Boolean Network ◽

Real Life ◽

Extended Period ◽

Boolean Networks ◽

Single Step ◽

Divide And Conquer ◽

Supplementary Information ◽

Initial State ◽

Minimal Subset

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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Design of Fixed Points in Boolean Networks Using Feedback Vertex Sets and Model Reduction

Complexity ◽

10.1155/2019/9261793 ◽

2019 ◽

Vol 2019 ◽

pp. 1-9 ◽

Cited By ~ 1

Author(s):

Koichi Kobayashi

Keyword(s):

Fixed Points ◽

Model Reduction ◽

Design Method ◽

Cell Types ◽

Boolean Networks ◽

Graph Structure ◽

Integer Linear Programming Problem ◽

Vertex Set ◽

External Stimuli ◽

Vertex Sets

Fixed points in Boolean networks (BNs) represent cell types or states of cells and are important to decide characteristics of cells. As the control problem on fixed points, it is important to consider the problem of changing fixed points by using external stimuli (i.e., control inputs). In this paper, we propose two methods for designing fixed points. First, a design method using model reduction is proposed. Using the reduced model, the problem of placing fixed points can be rewritten as an integer linear programming problem. Next, we consider the design problem using only the graph structure of a given BN and derive some results. In both methods, a feedback vertex set of a directed graph plays an important role. Finally, a biological example is presented.

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Parameterized complexity of fair feedback vertex set problem

Theoretical Computer Science ◽

10.1016/j.tcs.2021.03.008 ◽

2021 ◽

Vol 867 ◽

pp. 1-12

Author(s):

Lawqueen Kanesh ◽

Soumen Maity ◽

Komal Muluk ◽

Saket Saurabh

Keyword(s):

Parameterized Complexity ◽

Feedback Vertex Set ◽

Vertex Set ◽

Feedback Vertex Set Problem

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Parameterized Algorithms for Generalizations of Directed Feedback Vertex Set

Lecture Notes in Computer Science - Algorithms and Complexity ◽

10.1007/978-3-030-17402-6_21 ◽

2019 ◽

pp. 249-261

Author(s):

Alexander Göke ◽

Dániel Marx ◽

Matthias Mnich

Keyword(s):

Parameterized Algorithms ◽

Feedback Vertex Set ◽

Vertex Set ◽

Directed Feedback Vertex Set

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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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On Structural Parameterizations of the Edge Disjoint Paths Problem

Algorithmica ◽

10.1007/s00453-020-00795-3 ◽

2021 ◽

Author(s):

Robert Ganian ◽

Sebastian Ordyniak ◽

M. S. Ramanujan

Keyword(s):

Polynomial Time ◽

Disjoint Paths ◽

Structural Parameter ◽

Input Graph ◽

Feedback Vertex Set ◽

Fpt Algorithms ◽

Edge Disjoint ◽

Fpt Algorithm ◽

Vertex Set ◽

Terminal Pair

AbstractIn this paper we revisit the classical edge disjoint paths (EDP) problem, where one is given an undirected graph G and a set of terminal pairs P and asks whether G contains a set of pairwise edge-disjoint paths connecting every terminal pair in P. Our focus lies on structural parameterizations for the problem that allow for efficient (polynomial-time or FPT) algorithms. As our first result, we answer an open question stated in Fleszar et al. (Proceedings of the ESA, 2016), by showing that the problem can be solved in polynomial time if the input graph has a feedback vertex set of size one. We also show that EDP parameterized by the treewidth and the maximum degree of the input graph is fixed-parameter tractable. Having developed two novel algorithms for EDP using structural restrictions on the input graph, we then turn our attention towards the augmented graph, i.e., the graph obtained from the input graph after adding one edge between every terminal pair. In constrast to the input graph, where EDP is known to remain -hard even for treewidth two, a result by Zhou et al. (Algorithmica 26(1):3--30, 2000) shows that EDP can be solved in non-uniform polynomial time if the augmented graph has constant treewidth; we note that the possible improvement of this result to an FPT-algorithm has remained open since then. We show that this is highly unlikely by establishing the [1]-hardness of the problem parameterized by the treewidth (and even feedback vertex set) of the augmented graph. Finally, we develop an FPT-algorithm for EDP by exploiting a novel structural parameter of the augmented graph.

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