A Reduction Method for Boolean Network Models Proven to Conserve Attractors

Abstract

Abstract. Boolean models, wherein each component is characterized with a binary (ON or OFF) variable, have been widely employed for dynamic modeling of biological regulatory networks. However, the exponential dependencse of the size of the state space of these models on the number of nodes in the network can be a daunting prospect for attractor analysis of large-scale systems. We have previously proposed a network reduction technique for Boolean models and demonstrated its applicability on two biological systems, namely, the abscisic acid signal transduction network as well as the T-LGL leukemia survival signaling network. In this paper, we provide a rigorous mathematical proof that this method not only conserves the fixed points of a Boolean network, but also conserves the complex attractors of general asynchronous Boolean models wherein at each time step a randomly selected node is updated. This method thus allows one to infer the long-term dynamic properties of a large-scale system from those of the corresponding reduced model. Key words. Boolean models, Network reduction, Asynchronous methods, Attractors, Biological regulatory networks AMS subject classifications. 92C42, 37G351. Introduction. The ever-accelerating pace of experimental data generation has laid the foundation for developing network models of biological systems wherein the components of a system are represented by nodes and the interactions among them by edges. Analyzing these network models and studying their dynamics can unravel unknown facets of the underlying biological systems. Among different dynamic modeling approaches, discrete models, in which each component is assumed to have a finite number of qualitative states, have been increasingly employed in modeling biological regulatory networks [10,18,19,20,23]. The simplest discrete dynamic models are the so-called Boolean models that assume only two states (ON or OFF) for each component [8,21].Since Boolean models are parameter free, they serve as a suitable starting point for modeling biological systems for which a detailed kinetic characterization of the interactions is not available. In particular, attractor analysis of these models is of immense biological importance as it can provide valuable insights into the long-term behaviors, i.e. observed phenotypes, of these systems in response to environmental stimuli and internal perturbations [1,5,6,9,15,16]. For example, it allows one to predict the long-term activity levels of components or to determine key components influencing different cellular traits. However, the exponential dependence of the size of the state space of Boolean models on the number of nodes in the network makes the identification of all attractors of even relatively small systems computationally intractable. In particular, it has been proven that determination of the existence of fixed points in Boolean networks is a strong NPcomplete problem [24]. There have been several efforts to reduce the state space of Boolean models by simplifying the underlying networks. In [2...

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