Abstract
Boolean networks have been widely used to model biological processes lacking detailed kinetic information. Despite their simplicity, Boolean network dynamics can still capture some important features of biological systems such as stable cell phenotypes represented by steady states. For small models, steady states can be determined through exhaustive enumeration of all state transitions. As the number of nodes increases, however, the state space grows exponentially thus making it difficult to find steady states. Over the last several decades, many studies have addressed how to handle such a state space explosion. Recently, increasing attention has been paid to a satisfiability solving algorithm due to its potential scalability to handle large networks. Meanwhile, there still lies a problem in the case of large models with high maximum node connectivity where the satisfiability solving algorithm is known to be computationally intractable. To address the problem, this paper presents a new partitioning-based method that breaks down a given network into smaller subnetworks. Steady states of each subnetworks are identified by independently applying the satisfiability solving algorithm. Then, they are combined to construct the steady states of the overall network. To efficiently apply the satisfiability solving algorithm to each subnetwork, it is crucial to find the best partition of the network. In this paper, we propose a method that divides each subnetwork to be smallest in size and lowest in maximum node connectivity. This minimizes the total cost of finding all steady states in entire subnetworks. The proposed algorithm is compared with others for steady states identification through a number of simulations on both published small models and randomly generated large models with differing maximum node connectivities. The simulation results show that our method can scale up to several hundreds of nodes even for Boolean networks with high maximum node connectivity. The algorithm is implemented and available at http://cps.kaist.ac.kr/∼ckhong/tools/download/PAD.tar.gz.
Highlights
Modeling of biological systems as a network of interacting components has received increasing attention in various areas, such as computational and systems biology since it allows to analyze biological phenomena systematically at various scales including molecular and cellular levels [1]
6 and 7, we show the runtime of BNS, ST, and PAD, respectively
The results indicate that the proposed tool performs better than BNS by 2.91 times, and ST by 3.27 times in average
Summary
Modeling of biological systems as a network of interacting components has received increasing attention in various areas, such as computational and systems biology since it allows to analyze biological phenomena systematically at various scales including molecular and cellular levels [1]. The BN is a discrete model of biological system that comprises of a number of nodes and corresponding update rules. Each node represents a gene and takes on a value of 1 or 0, meaning that the gene is expressed or unexpressed, respectively. Each update rule represents interactions between genes. The state of a gene at a given time step is determined by its update rule and the state of its input genes at the previous time step. In synchronous BNs, the states of all nodes are updated simultaneously at each time step, and it directly induces global state transitions. An important characteristic of BNs is that any sequence of consecutive global state transitions eventually converges to either a single state (i.e., steady state) or a cycle of states (i.e., cyclic attractors)
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