Abstract

The synchronous Boolean network (SBN) is a simple and powerful model for describing, analyzing, and simulating cellular biological networks. This paper seeks a complete understanding of the dynamics of such a model by employing a matrix method that relies on relating the network transition matrix to its function matrix via a self-inverse state matrix. A recursive ordering of the underlying basis vector leads to a simple recursive expression of this state matrix. Hence, the transition matrix is computed via multiplication of binary matrices over the simplest finite (Galois) field, namely the binary field GF(2), i.e., conventional matrix multiplication involving modulo-2 addition, or XOR addition. We demonstrate the conceptual simplicity and practical utility of our approach via an illustrative example, in which the transition matrix is readily obtained, and subsequently utilized (via its powers, characteristic equation, minimal equation, 1-eigenvectors, and 0-eigenvectors) to correctly predict both the transient behavior and the cyclic behavior of the network. Our matrix approach for computing the transition matrix is superior to the approach of scalar equations, which demands cumbersome manipulations and might fail to predict the exact network behavior. Our approach produces result that exactly replicate those obtained by methods employing the semi-tensor product (STP) of matrices, but achieves that without sophisticated ambiguity or unwarranted redundancy.

Highlights

  • 2.3 Cyclic and Transient behavior We review some of the mathematics utilized in the study of the cyclic and transient behaviors of an synchronous Boolean network (SBN)

  • We utilized well-known results in the seminal work of Cull (1971), together with the improvement added by Rushdi and Al- Otaibi (2007), to present a method for constructing the transition matrix [T] of an SBN

  • Our work leads to a better understanding of the relation between the matrix and scalar approaches for studying SBNs

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Summary

Introduction

Many biological systems, such as cellular networks and genetic regulatory networks, are too complex to allow exact or nearly exact analysis. These systems are usually studied in terms of synchronous Boolean networks (SBNs) as approximating models. An SBN is a set of n nodes, each of which is either in state 1 (On) or state 0 (Off) at any given time t. Each node is updated at time (t+1) by inputs from any fixed subset of the set of nodes according to any desired logical rule. All possible trajectories of the network are either cycles (loops or attractors) of any length from size 1 (a fixed point) to a maximum length of 2n, or transient states leading eventually to a cycle (attractor)

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