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On Generating Optimal Sparse Probabilistic Boolean Networks with Maximum Entropy from a Positive Stationary Distribution

Published online by Cambridge University Press:  28 May 2015

Hao Jiang ,
Xi Chen ,
Yushan Qiu  and
Wai-Ki Ching
Hao Jiang*
Affiliation:
Advanced Modeling and Applied Computing Laboratory, Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong
Xi Chen*
Affiliation:
Advanced Modeling and Applied Computing Laboratory, Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong
Yushan Qiu*
Affiliation:
Advanced Modeling and Applied Computing Laboratory, Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong
Wai-Ki Ching*
Affiliation:
Advanced Modeling and Applied Computing Laboratory, Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong
*
Corresponding author. Email: [email protected]
Corresponding author. Email: [email protected]
Corresponding author. Email: [email protected]
Corresponding author. Email: [email protected]
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Abstract.

To understand a genetic regulatory network, two popular mathematical models, Boolean Networks (BNs) and its extension Probabilistic Boolean Networks (PBNs) have been proposed. Here we address the problem of constructing a sparse Probabilistic Boolean Network (PBN) from a prescribed positive stationary distribution. A sparse matrix is more preferable, as it is easier to study and identify the major components and extract the crucial information hidden in a biological network. The captured network construction problem is both ill-posed and computationally challenging. We present a novel method to construct a sparse transition probability matrix from a given stationary distribution. A series of sparse transition probability matrices can be determined once the stationary distribution is given. By controlling the number of nonzero entries in each column of the transition probability matrix, a desirable sparse transition probability matrix in the sense of maximum entropy can be uniquely constructed as a linear combination of the selected sparse transition probability matrices (a set of sparse irreducible matrices). Numerical examples are given to demonstrate both the efficiency and effectiveness of the proposed method.

Keywords

65C2092B05Boolean Networks (BNs)EntropyProbabilistic Boolean Networks (PBNs)genetic regulatory networkssparsitystationary probability distributiontransition probability matrix
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 2 , Issue 4 , November 2012 , pp. 353 - 372
Copyright
Copyright © Global-Science Press 2012

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References

[1]Arnone, M. and Davidson, M., The Hardwiring of Development: Organization and Function of Genomic Regulatory Systems, Development, 124 (1997), pp. 1851–1864.CrossRefGoogle ScholarPubMed
[2]Chen, X., Ching, W., Chen, X.S., Cong, Y. and Tsing, N., Construction of Probabilistic Boolean Networks from a Prescribed Transition ProbabilityMatrix: A Maximum Entropy Rate Approach, East Asian Journal of Applied Mathematics, 1 (2011), pp. 132–154.Google Scholar
[3]Chen, X., Jiang, H. and Ching, W., On Construction of Sparse Probabilistic Boolean Networks, East Asian Journal of Applied Mathematics, 2 (2012), pp. 1–18.Google Scholar
[4]Candes, E. and Tao, T., Near-optimal Signal Recovery from Random Projections: Universal Encoding Strategies, IEEE Transactions on Information Theory, 52 (2006), pp. 5406–5425.CrossRefGoogle Scholar
[5]Ching, W. and Ng, M., Markov Chains: Models, Algorithms and Applications, International Series on operations Research and Management Science, Springer, New York, (2006).Google Scholar
[6]Ching, W., Zhang, S., Ng, M. and Akutsu, T., An Approximation Method for Solving the Steady-state Probability Distribution of Probabilistic Boolean Networks, Bioinformatics, 23 (2007), pp.1511–1518.Google Scholar
[7]Ching, W., Chen, X. and Tsing, N., Generating Probabilistic Boolean Networks from a Prescribed Transition Probability Matrix, IET on Systems Biology, 6 (2009), pp. 453–464.CrossRefGoogle Scholar
[8]Ching, W., Zhang, S., Jiao, Y., Akutsu, T., Tsing, N. and Wong, A., Optimal Control Policy for Probabilistic Boolean Networks with Hard Constraints, IET on Systems Biology, 3 (2009), pp. 90–99.Google Scholar
[9]Ching, W., Fung, E., Ng, M. and Akutsu, T., On Construction of Stochastic Genetic Networks Based on Gene Expression Sequences, International Journal of Neural Systems, 15 (2005), pp. 297–310.Google Scholar
[10]Cong, Y., Ching, W., Tsing, N. and Leung, H., On Finite-Horizon Control of Genetic Regulatory Networks with Multiple Hard-Constraints, BMC Systems Biology, 4(Suppl 2) (2010), S14, doi:10.1186/1752-0509-4-S2-S14.Google Scholar
[11]Cover, T. and Thomas, J., Elements of Information Theory, John Wiley and Sons, Inc., (1991).Google Scholar
[12]De Jong, H., Modeling and Simulation of Genetic Regulatory Systems: a Literature Review, Journal of Computational Biology, 9 (2002), pp. 69–103.CrossRefGoogle ScholarPubMed
[13]Friedman, N., Linial, M., Nachman, I. and Pe'er, D., Using Bayesian Networks to Analyze Expression Data, Journal of Computational Biology, 7 (2000), pp. 601–620.Google Scholar
[14]Gibson, M. and Mjolsness, E.Modeling the Activity of Single Genes, in Computational Modeling of Genetic and Biochemical Networks, Bower, J. and Bolouri, H. (Eds.), Cambridge MA: MIT Press, (2001), Chapter 1.Google Scholar
[15]Huang, S., Gene Expression Profiling, Genetic Networks, and Cellular States: An Integrating Concept for Tumorigenesis and Drug Discovery, Journal of Molecular Medicine, 77 (1999), pp. 469–480.CrossRefGoogle ScholarPubMed
[16]Huang, S. and Ingber, D., Shape-dependent Control of Cell Growth, Differentiation, and Apoptosis: Switching Between Attractors in Cell Regulatory Networks, Experimental Cell Research, 261 (2000), pp. 91–103.CrossRefGoogle ScholarPubMed
[17]Jaynes, E.Information Theory and Statistical Mechanics, The Physical Review, 106 (1957), pp. 620–630.CrossRefGoogle Scholar
[18]Kauffman, S., Metabolic Stability and Epigenesis in Randomly Constructed Gene Nets, Journal of Theoretical Biology, 22 (1969), pp. 437–467.Google Scholar
[19]Kauffman, S., Homeostasis and Differentiation in Random Genetic Control Networks, Nature, 224 (1969), pp. 177–178.CrossRefGoogle ScholarPubMed
[20]Kauffman, S., The Origins of Order: Self-organization and Selection in Evolution, New York: Oxford univ. Press, (1993).Google Scholar
[21]Li, W., Cui, L. and Ng, M., On Computation of the Steady-State Probability Distribution of Probabilistic Boolean Networks with Gene Perturbation, Journal of Computational and Applied Mathematics, 236 (2012), pp. 4067–4081.Google Scholar
[22]Linden, R. and Bhaya, A., Evolving Fuzzy Rules to Model Gene Expression, Biosystems, 88 (2007), pp. 76–91.Google Scholar
[23]Song, M., Ouyang, Z. and Liu, Z., Discrete Dynamical System Modeling for Gene Regulatory Networks of 5-hxymethylfurfural Tolerance for Ethanologenic Yeast, IET on Systems Biology, 3 (2009), pp. 203–218.Google Scholar
[24]Martin, S., Zhang, A., Martino, A. and Faulon, J., Boolean Dynamics of Genetic Regulatory Networks inferred from Microarraytime Series Data, Bioinformatics, 23 (2007), pp. 886–874.CrossRefGoogle ScholarPubMed
[25]Pal, R., Ivanov, I., Datta, A., Bittner, M. and Dougherty, E., Generating Boolean Networks with a Prescribed Attractor Structure, Bioinformatics, 21 (2005), pp. 4021–4025.CrossRefGoogle ScholarPubMed
[26]Ribeiro, A., Zhu, R, and Kauffman, S., A Genetic Modeling Strategy for Gene Regulatory Networks with Stochastic Dynamics, Journal of Computational Biology, 13 (2006), pp. 1630–1639.CrossRefGoogle ScholarPubMed
[27]Shannon, C., A Mathematical Theory of Communication, Bell System Technical Journal, 27 (1948), pp. 379–423.Google Scholar
[28]Shin, Y. and Bleris, L., Linear Control Theory for Gene Network Modeling, PLoS ONE 5(9): e12785. doi:10.1371/journal.pone.0012785.CrossRefGoogle Scholar
[29]Shmulevich, I., Dougherty, E., Kim, S. and Zhang, W., Probabilistic Boolean Networks: A Rule-based Uncertainty Model for Gene Regulatory Networks, Bioinformatics, 18 (2002), pp. 261–274.CrossRefGoogle ScholarPubMed
[30]Shmulevich, I., Dougherty, E., Kim, S. and Zhang, W., Control of Stationary Behavior in Probabilistic Boolean Networks by Means of Structural Intervention, Journal of Biological Systems, 10 (2002), pp. 431–445.Google Scholar
[31]Shmulevich, I., Dougherty, E., Kim, S. and Zhang, W.From Boolean to Probabilistic Boolean Networks as Models of Genetic Regulatory Networks, Proceedings of the IEEE, 90 (2002), pp. 1778–1792.Google Scholar
[32]Shmulevich, I. and Dougherty, E., Probabilistic Boolean Networks: The Modeling and Control of Gene Regulatory Networks, Philadelphia: Society for industrial and Applied Mathematics, (2010).Google Scholar
[33]Smolen, P., Baxter, D. and Byrne, J., Mathematical Modeling of Gene Networks, Neuron, 26 (2000), pp. 567–580.CrossRefGoogle ScholarPubMed
[34]Xu, Z., Zhang, H., Wang, Y. and Chang, X., L1/2 Regularization, Science in China Series F -Information Sciences, 40 (2010), pp. 1–11.Google Scholar
[35]Xu, W., Ching, W., Zhang, S., Li, W. and Chen, X., A Matrix Perturbation Method for Computing the Steady-state Probability Distributions of Probabilistic Boolean Networks with Gene Perturbations, Journal of Computational and Applied Mathematics, 235 (2011), pp. 2242–2251.Google Scholar
[36]Zhang, S., Ching, W., Ng, M. and Akutsu, T., Simulation Study in Probabilistic Boolean Network Models for Genetic Regulatory Networks, Journal of Data Mining and Bioinformatics, 1 (2007), pp. 217–240.Google Scholar
[37]Zhang, S., Ching, W., Chen, X. and Tsing, N., Generating Probabilistic Boolean Networks from a Prescribed Stationary Distribution, Information Sciences, 180 (2010), pp. 2560–2570.Google Scholar
[38]Zhang, S., Ching, W., Tsing, N., Leung, H. and Guo, D., A New Multiple Regression Approach for the Construction of Genetic Regulatory Networks, Journal of Artificial Intelligence in Medicine, 48 (2010), pp. 153–160.Google Scholar