TY - JOUR

T1 - Stability structures of conjunctive Boolean networks

AU - Gao, Zuguang

AU - Chen, Xudong

AU - Başar, Tamer

N1 - Funding Information:
This research was supported in part by the Office of Naval Research (ONR) MURI grant N-00014-16-1-2710. The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Sophie Tarbouriech under the direction of Editor Richard Middleton.
Publisher Copyright:
© 2017 Elsevier Ltd

PY - 2018/3

Y1 - 2018/3

N2 - A Boolean network is a finite dynamical system, whose variables take values from a binary set. The value update rule for each variable is a Boolean function, depending on a selected subset of variables. Boolean networks have been widely used in modeling gene regulatory networks. We focus in this paper on a special class of Boolean networks, termed as conjunctive Boolean networks. A Boolean network is conjunctive if the associated value update rule is comprised of only AND operations. It is known that any trajectory of a finite dynamical system will enter a periodic orbit. We characterize in this paper all periodic orbits of a conjunctive Boolean network whose underlying graph is strongly connected. In particular, we establish a bijection between the set of periodic orbits and the set of binary necklaces of a certain length. We further investigate the stability of a periodic orbit. Specifically, we perturb a state in the periodic orbit by changing the value of a single entry of the state. The trajectory, with the perturbed state being the initial condition, will enter another (possibly the same) periodic orbit in finite time steps. We then provide a complete characterization of all such transitions from one periodic orbit to another. In particular, we construct a digraph, with the vertices being the periodic orbits, and the (directed) edges representing the transitions among the orbits. We call such a digraph the stability structure of the conjunctive Boolean network.

AB - A Boolean network is a finite dynamical system, whose variables take values from a binary set. The value update rule for each variable is a Boolean function, depending on a selected subset of variables. Boolean networks have been widely used in modeling gene regulatory networks. We focus in this paper on a special class of Boolean networks, termed as conjunctive Boolean networks. A Boolean network is conjunctive if the associated value update rule is comprised of only AND operations. It is known that any trajectory of a finite dynamical system will enter a periodic orbit. We characterize in this paper all periodic orbits of a conjunctive Boolean network whose underlying graph is strongly connected. In particular, we establish a bijection between the set of periodic orbits and the set of binary necklaces of a certain length. We further investigate the stability of a periodic orbit. Specifically, we perturb a state in the periodic orbit by changing the value of a single entry of the state. The trajectory, with the perturbed state being the initial condition, will enter another (possibly the same) periodic orbit in finite time steps. We then provide a complete characterization of all such transitions from one periodic orbit to another. In particular, we construct a digraph, with the vertices being the periodic orbits, and the (directed) edges representing the transitions among the orbits. We call such a digraph the stability structure of the conjunctive Boolean network.

KW - Discrete time dynamics

KW - Networked control systems

KW - Stability analysis

KW - Systems biology

UR - http://www.scopus.com/inward/record.url?scp=85038014725&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85038014725&partnerID=8YFLogxK

U2 - 10.1016/j.automatica.2017.11.017

DO - 10.1016/j.automatica.2017.11.017

M3 - Article

AN - SCOPUS:85038014725

SN - 0005-1098

VL - 89

SP - 8

EP - 20

JO - Automatica

JF - Automatica

ER -