An algorithm for detecting fixed points of boolean network

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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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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