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He L: An efficient algorithm for computing attractors of synchronous and asynchronous Boolean networks
 PLoS ONE
"... Biological networks, such as genetic regulatory networks, often contain positive and negative feedback loops that settle down to dynamically stable patterns. Identifying these patterns, the socalled attractors, can provide important insights for biologists to understand the molecular mechanisms und ..."
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Biological networks, such as genetic regulatory networks, often contain positive and negative feedback loops that settle down to dynamically stable patterns. Identifying these patterns, the socalled attractors, can provide important insights for biologists to understand the molecular mechanisms underlying many coordinated cellular processes such as cellular division, differentiation, and homeostasis. Both synchronous and asynchronous Boolean networks have been used to simulate genetic regulatory networks and identify their attractors. The common methods of computing attractors are that start with a randomly selected initial state and finish with exhaustive search of the state space of a network. However, the time complexity of these methods grows exponentially with respect to the number and length of attractors. Here, we build two algorithms to achieve the computation of attractors in synchronous and asynchronous Boolean networks. For the synchronous scenario, combing with iterative methods and reduced order binary decision diagrams (ROBDD), we propose an improved algorithm to compute attractors. For another algorithm, the attractors of synchronous Boolean networks are utilized in asynchronous Boolean translation functions to derive attractors of asynchronous scenario. The proposed algorithms are implemented in a procedure called geneFAtt. Compared to existing tools such as genYsis, geneFAtt is
2013): Dynamics of Random Boolean Networks under Fully Asynchronous Stochastic Update Based on Linear Representation. PLoS ONE 8(6): e66491. doi:10.1371/journal.pone
"... A novel algebraic approach is proposed to study dynamics of asynchronous random Boolean networks where a random number of nodes can be updated at each time step (ARBNs). In this article, the logical equations of ARBNs are converted into the discretetime linear representation and dynamical behaviors ..."
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A novel algebraic approach is proposed to study dynamics of asynchronous random Boolean networks where a random number of nodes can be updated at each time step (ARBNs). In this article, the logical equations of ARBNs are converted into the discretetime linear representation and dynamical behaviors of systems are investigated. We provide a general formula of network transition matrices of ARBNs as well as a necessary and sufficient algebraic criterion to determine whether a group of given states compose an attractor of lengths in ARBNs. Consequently, algorithms are achieved to find all of the attractors and basins in ARBNs. Examples are showed to demonstrate the feasibility of the proposed scheme.
State Space Analysis of Boolean Networks
"... AbstractThis paper provides a comprehensive framework for the state space approach to Boolean networks. First, it surveys the authors' recent work on the topic: Using semitensor product of matrices and the matrix expression of logic, the logical dynamic equations of Boolean (control) networks ..."
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AbstractThis paper provides a comprehensive framework for the state space approach to Boolean networks. First, it surveys the authors' recent work on the topic: Using semitensor product of matrices and the matrix expression of logic, the logical dynamic equations of Boolean (control) networks can be converted into standard discretetime dynamics. To use the state space approach, the state space and its subspaces of a Boolean network have been carefully defined. The basis of a subspace has been constructed. Particularly, the regular subspace, friendly subspace, and invariant subspace are precisely defined, and the verifying algorithms are presented. As an application, the indistinct rolling gear structure of a Boolean network is revealed. Index TermsBoolean (control) network, state space, subspace, basis, indistinct rolling gear structure.
An Algorithm to Find Cycles of Biochemical Systems
"... Genetic regulatory systems, selforganized systems and other living systems can be modeled as synchronous Boolean networks with stable states which are also called cycles. This paper devises two algorithms based on BDD to compute all the cycles in synchronous Boolean networks and enumerate all state ..."
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Genetic regulatory systems, selforganized systems and other living systems can be modeled as synchronous Boolean networks with stable states which are also called cycles. This paper devises two algorithms based on BDD to compute all the cycles in synchronous Boolean networks and enumerate all states in those cycles. Empirical experiments with biochemical systems demonstrate the feasibility and efficiency of our algorithms. It also shows that the two algorithms are conceptually so simple and efficient that they can be extensible to other realistic biochemical systems.
On the Linear Analysis of Synchronous Switching Networks
"... Abstract. An exposition of an earlier seminal paper on the linear analysis of synchronous switching networks is presented. This analysis, based on the use of the finite or Galois field GF(2), resembles the linear analysis of continuous systems and has important applications in genetics and biochemis ..."
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Abstract. An exposition of an earlier seminal paper on the linear analysis of synchronous switching networks is presented. This analysis, based on the use of the finite or Galois field GF(2), resembles the linear analysis of continuous systems and has important applications in genetics and biochemistry. A synchronous switching network is represented by a function matrix or by a transition matrix, which are related by a similarity transformation in terms of a state matrix. Our use of a novel recursive ordering for the keys or indices of these matrices reveals several new and interesting features and properties. The state matrix is observed to depend not on the particular network but merely on its number of nodes, and is further given a novel interpretation via the modern concept of subsumption of a logical product by another. This reveals a recursive structure of the state matrix and leads to a proof that it is involutory (selfinverse). The autonomous behavior of synchronous switching networks is studied via the characteristic equations and eigenvectors of the aforementioned matrices. In general, the classical ideas are enriched with modern concepts and terminology, supported with correct proofs, and clarified with detailed tutorial examples. 1.