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Issue No.05 - Sept.-Oct. (2012 vol.9)
pp: 1410-1421
Tatsuya Akutsu , Inst. for Chem. Res., Kyoto Univ., Kyoto, Japan
Sven Kosub , Dept. of Comput. & Inf. Sci., Univ. of Konstanz, Konstanz, Germany
Avraham A. Melkman , Dept. of Comput. Sci., Ben-Gurion Univ. of the Negev, Beer-Sheva, Israel
Takeyuki Tamura , Inst. for Chem. Res., Kyoto Univ., Kyoto, Japan
ABSTRACT
In this paper, we study the problem of finding a periodic attractor of a Boolean network (BN), which arises in computational systems biology and is known to be NP-hard. Since a general case is quite hard to solve, we consider special but biologically important subclasses of BNs. For finding an attractor of period 2 of a BN consisting of n OR functions of positive literals, we present a polynomial time algorithm. For finding an attractor of period 2 of a BN consisting of n AND/OR functions of literals, we present an O(1.985n) time algorithm. For finding an attractor of a fixed period of a BN consisting of n nested canalyzing functions and having constant treewidth w, we present an O(n2p(w+1)poly(n)) time algorithm.
INDEX TERMS
polynomials, biology computing, Boolean functions, genetic algorithms, time algorithm, periodic attractor, Boolean network, computational systems biology, positive literals, polynomial time algorithm, AND-OR functions, nested canalyzing functions, constant treewidth, Boolean functions, Computational systems biology, Polynomials, treewidth., Boolean network, periodic attractor, SAT, nested canalyzing function
CITATION
Tatsuya Akutsu, Sven Kosub, Avraham A. Melkman, Takeyuki Tamura, "Finding a Periodic Attractor of a Boolean Network", IEEE/ACM Transactions on Computational Biology and Bioinformatics, vol.9, no. 5, pp. 1410-1421, Sept.-Oct. 2012, doi:10.1109/TCBB.2012.87
REFERENCES
 [1] T. Akutsu, S. Kuhara, O. Maruyama, and S. Miyano, "A System for Identifying Genetic Networks from Gene Expression Patterns Produced by Gene Disruptions and Overexpressions," Genome Informatics, vol. 9, pp. 151-160, 1998. [2] T. Akutsu, A.A. Melkman, T. Tamura, and M. Yamamoto, "Determining a Singleton Attractor of a Boolean Network with Nested Canalyzing Functions," J. Computational Biology, vol. 18, pp. 1275-1290, 2011. [3] T. Akutsu, A.A. Melkman, and T. Tamura, "Singleton and 2-Periodic Attractors of Sign-Definite Boolean Networks," Information Processing Letters, vol. 112, pp. 35-38, 2012. [4] J. Aracena, J. Demongeot, and E. Goles, "Positive and Negative Circuits in Discrete Neural Networks," IEEE Trans. Neural Networks, vol. 15, no. 1, pp. 77-83, Jan. 2004. [5] C. Barrett, C.H.B. HuntIII, M.V. Marathe, S.S. Ravi, D.J. Rosenkrantz, R.E. Stearns, and M. Thakur, "Predecessor Existence Problems for Finite Discrete Dynamical Systems," Theoretical Computer Science, vol. 386, pp. 3-37, 2007. [6] J. Böttcher, K.P. Pruessmann, A. Taraz, and A. Würfl, "Bandwidth, Expansion, Treewidth, Separators and Universality for Bounded-Degree Graphs," European J. Combinatorics, vol. 31, pp. 1217-1227, 2010. [7] Q. Cheng, P. Berman, R. Harrison, and A. Zelikovsky, "Efficient Alignments of Metabolic Networks with Bounded Treewidth," Proc. IEEE Int'l Conf. Data Mining Workshops, pp. 687-694, 2010. [8] V. Devloo, P. Hansen, and M. Labbé, "Identification of All Steady States in Large Networks by Logical Analysis," Bull. Math. Biology, vol. 65, pp. 1025-1051, 2003. [9] E. Dubrova and M. Teslenko, "A SAT-Based Algorithm for Finding Attractors in Synchronous Boolean Networks," IEEE/ACM Trans. Computational Biology and Bioinformatics, vol. 8, no. 5, pp. 1393-1398, Sept./Oct. 2011. [10] B. Drossel, T. Mihaljev, and F. Greil, "Number and Length of Attractors in a Critical Kauffman Model with Connectivity One," Physical Rev. Letters, vol. 94, article 088701, 2005. [11] J. Flum and M. Grohe, Parameterized Complexity Theory. Springer, 2006. [12] E.C. Freuder, "Complexity of $k$ -Tree Structured Constraint Satisfaction Problems," Proc. Eighth Nat'l Conf. Artificial Intelligence (AAAI '90), pp. 4-9, 1990. [13] A. Garg, A. DiCara, I. Xenarios, L. Mendoza, and G. DeMichel, "Synchronous versus Asynchronous Modeling of Gene Regulatory Networks," Bioinformatics, vol. 24, pp. 1917-1925, 2008. [14] E. Goles and L. Salinas, "Sequential Operator for Filtering Cycles in Boolean Networks," Advances Applied Math., vol. 45, pp. 346-358, 2010. [15] S.E. Harris, B.K. Sawhill, A. Wuensche, and S.A. Kauffman, "A Model of Transcriptional Regulatory Networks Based on Biases in the Observed Regulation Rules," Complexity, vol. 7, pp. 23-40, 2002. [16] D.J. Irons, "Improving the Efficiency of Attractor Cycle Identification in Boolean Networks," Physica D, vol. 217, pp. 7-21, 2006. [17] A.S. Jarrah, R. Laubenbacher, and A. Veliz-Cuba, "The Dynamics of Conjunctive and Disjunctive Boolean Network Models," Bull. Math. Biology, vol. 72, pp. 1425-1447, 2010. [18] A.S. Jarrah, B. Raposa, and R. Laubenbacher, "Nested Canalyzing, Unate Cascade, and Polynomial Functions," Physica D, vol. 233, pp. 167-174, 2007. [19] W. Just, "The Steady State System Problem is NP-Hard Even for Monotone Quadratic Boolean Dynamical Systems," rapid post, http://www.ohio.edu/people/justpubl.html , 2006. [20] S.A. Kauffman, The Origins of Order: Self-Organization and Selection in Evolution. Oxford Univ. Press, 1993. [21] S. Kosub, "Dichotomy Results for Fixed-Points Existence Problems for Boolean Dynamical Systems," Math. Computer Science, vol. 1, pp. 487-505, 2008. [22] L. Layne, E. Dimitrova, and M. Macauley, "Nested Canalyzing Depth and Network Stability," Bull. Math. Biology, vol. 74, pp. 422-433, 2012. [23] M. Leone, A. Pagnani, G. Parisi, and O. Zagordi, "Finite Size Corrections to Random Boolean Networks," J. Statistical Mechanics: Theory and Experiment, vol. 2006, article P12012, 2006. [24] A.A. Melkman, T. Tamura, and T. Akutsu, "Determining a Singleton Attractor of an AND/OR Boolean Network in $O(1.587^n)$ Time," Information Processing Letters, vol. 110, pp. 565-569, 2010. [25] B. Samuelsson and C. Troein, "Superpolynomial Growth in the Number of Attractors in Kauffman Networks," Physics Rev. Letters, vol. 90, article 098701, 2003. [26] H. Tamaki, "A Directed Path-Decomposition Approach to Exactly Identifying Attractors of Boolean Networks," Proc. 10th Int'l Symp. Comm. and Information Technologies, pp. 844-849, 2010. [27] T. Tamura and T. Akutsu, "Detecting a Singleton Attractor in a Boolean Network Utilizing SAT Algorithms," IEICE Trans. Fundamentals, vol. E92-A, pp. 493-501, 2009. [28] T. Tamura and T. Akutsu, "Algorithms for Singleton Attractor Detection in Planar and Non-Planar AND/OR Boolean Networks," Math. Computer Science, vol. 2, pp. 401-420, 2009. [29] M. Yamamoto, "An Improved $O^{\ast }(1.234^m)$ -Time Deterministic Algorithm for SAT," Proc. 16th Int'l Symp. Algorithms and Computation, pp. 644-653, 2005. [30] S-Q. Zhang, M. Hayashida, T. Akutsu, W-K. Ching, and M.K. Ng, "Algorithms for Finding Small Attractors in Boolean Networks," EURASIP J. Bioinformatics System Biology, vol. 2007, article 20180, 2007.
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