Controllability of probabilistic Boolean control networks

Abstract

This paper deals with the controllability of probabilistic Boolean control networks. First, a survey on the semi-tensor product approach to probabilistic Boolean networks is given. Second, the controllability of probabilistic Boolean control networks via two kinds inputs is studied. Finally, examples are given to show the efficiency of the obtained results.

Introduction

An important research issue in systems biology is to understand the nature of cellular function and to study the behavior of genes in a holistic rather than in an individual manner. This leads to the study of the interactions between different genes. Many mathematical models have been proposed to model the gene regulatory networks. These models include Bayesian networks Marínez-Rodríguez, May, and Vargas (2008), differential equations Wang, Lam, Wei, Fraser, and Liu (2008), Boolean networks (BNs) (see, e.g., Kauffman (1969)), and their generalization to probabilistic Boolean networks (PBNs) (see, e.g., Shmulevich, Dougherty, Kim, and Zhang (2002)).

The Boolean network was first introduced by Kauffman (1969) to describe genetic circuits. In Boolean networks, the state of a gene can be described by a Boolean variable: active (1) or inactive (0), hence its product is present or absent. Moreover, interactions between the states of each gene can be determined by Boolean functions, which calculate the state of a gene from the activation of other genes. The study of Boolean networks has received great attention, such as the study of the topological structure of Boolean networks, including the fixed points, cycles, attractors and transient time, for example see Cheng, 2009, Drossel et al., 2005, Farrow et al., 2004, Heidel et al., 2003, Samuelsson and Troein, 2003. Systematic analysis of Boolean networks is also a hot topic, (see e.g. Cheng et al., 2010, Cheng et al., 2010, Cheng and Qi, 2009, Cheng et al., 2011b).

In Boolean networks, the target gene is predicted by several genes, called its input genes, through a Boolean function. Once the input genes and the Boolean functions are given, the Boolean network is defined deterministically. However, the stochastic nature of the genetic regulation and the micro array data used to infer the network structure may have errors due to experimental noise in the complex measurement processes. Thus, a stochastic model is more practical and favorable to such situations, resulting in the development of PBNs.

Shmulevich et al. (2002) proposed the PBNs model, which shares the appealing properties of Boolean networks and also copes with the presence of uncertainty. In PBNs, the stochasticity is introduced into the model by allowing several possible regulatory functions for each gene and allowing random perturbation. The state transits into a number of states with certain probabilities according to the realizations of all possible BNs. Thus, the dynamics of a PBN can be studied in the context of Markov chain. The study of PBNs attracts much attention. A natural and important problem is to study the steady-state probability distribution for PBNs, (see e.g. Brun et al., 2005, Ching et al., 2007, Shmulevich et al., 2003). Stability analysis of a PBN is another interesting topic. The stability and stabilization problem has been studied in Qi, Cheng, and Hu (2010).

While the mechanism of a genetic network can be studied and understood by using PBNs, an ultimate goal of systems biologists is to design therapeutic intervention strategies using information from the network dynamics. To achieve relatively more permanent effect of intervention, optimal control theory finds its application (see e.g. Ching et al., 2008, Faryabi et al., 2007, Pal et al., 2006). As we know, systematic analysis of biological systems is also an important issue in systems biology. Controllability is a structural property of system, and it is one of the fundamental concepts in systematic science and control theory. There have been many results that studied the controllability of dynamic systems (see e.g. Bloch et al., 2010, Shen et al., 2010). When referring to the controllability problem of PBNs, there have been few results in the literature to the best of our knowledge (see e.g. Chen, Akutsu, Tamura, & Ching, 2010). It should be pointed out that, until now, some of the existing literature has dealt with the steady-state probability distribution problem for PBNs, but the important controllability problem for PBNs has been overlooked despite its practical significance. In fact, the controllability and some related problems of PBNs are far from being solved. To study the controllability of PBNs is meaningful and challenging. Motivated by the above analysis, in our paper, we study the controllability of probabilistic Boolean control networks.

In this paper, we consider the controllability of the probabilistic Boolean control network. This paper is organized as follows. In Section 2, we provide a brief review for the semi-tensor product of matrices as well as the vector form of Boolean variables and random Boolean variables. Also, notations and basic properties are introduced. In Section 3, we use the semi-tensor product of matrices to convert the probabilistic Boolean control network into a discrete time system. Then, the controllability of the probabilistic Boolean control network via two types of controls is discussed. One kind of control is the open-loop, another kind is the closed-loop. Some conditions for the controllability of the probabilistic Boolean network are obtained. Examples are also presented to illustrate the efficiency of the obtained results in Section 4. Finally, a brief summary is given in Section 5.