- Research Article
- Open Access
Inference of a Probabilistic Boolean Network from a Single Observed Temporal Sequence
EURASIP Journal on Bioinformatics and Systems Biology volume 2007, Article number: 32454 (2007)
The inference of gene regulatory networks is a key issue for genomic signal processing. This paper addresses the inference of probabilistic Boolean networks (PBNs) from observed temporal sequences of network states. Since a PBN is composed of a finite number of Boolean networks, a basic observation is that the characteristics of a single Boolean network without perturbation may be determined by its pairwise transitions. Because the network function is fixed and there are no perturbations, a given state will always be followed by a unique state at the succeeding time point. Thus, a transition counting matrix compiled over a data sequence will be sparse and contain only one entry per line. If the network also has perturbations, with small perturbation probability, then the transition counting matrix would have some insignificant nonzero entries replacing some (or all) of the zeros. If a data sequence is sufficiently long to adequately populate the matrix, then determination of the functions and inputs underlying the model is straightforward. The difficulty comes when the transition counting matrix consists of data derived from more than one Boolean network. We address the PBN inference procedure in several steps: (1) separate the data sequence into "pure" subsequences corresponding to constituent Boolean networks; (2) given a subsequence, infer a Boolean network; and (3) infer the probabilities of perturbation, the probability of there being a switch between constituent Boolean networks, and the selection probabilities governing which network is to be selected given a switch. Capturing the full dynamic behavior of probabilistic Boolean networks, be they binary or multivalued, will require the use of temporal data, and a great deal of it. This should not be surprising given the complexity of the model and the number of parameters, both transitional and static, that must be estimated. In addition to providing an inference algorithm, this paper demonstrates that the data requirement is much smaller if one does not wish to infer the switching, perturbation, and selection probabilities, and that constituent-network connectivity can be discovered with decent accuracy for relatively small time-course sequences.
Dougherty ER, Datta A, Sima C: Research issues in genomic signal processing. IEEE Signal Processing Magazine 2005, 22(6):46-68.
Akutsu T, Miyano S, Kuhara S: Identification of genetic networks from a small number of gene expression patterns under the Boolean network model. Proceedings of the 4th Pacific Symposium on Biocomputing (PSB '99), Mauna Lani, Hawaii, USA, January 1999 17-28.
Lähdesmäki H, Shmulevich I, Yli-Harja O: On learning gene regulatory networks under the Boolean network model. Machine Learning 2003, 52(1-2):147-167.
Liang S, Fuhrman S, Somogyi R: REVEAL, a general reverse engineering algorithm for inference of genetic network architectures. Proceedings of the 3rd Pacific Symposium on Biocomputing (PSB '98), Maui, Hawaii, USA, January 1998 18-29.
Pal R, Ivanov I, Datta A, Bittner ML, Dougherty ER: Generating Boolean networks with a prescribed attractor structure. Bioinformatics 2005, 21(21):4021-4025. 10.1093/bioinformatics/bti664
Zhou X, Wang X, Dougherty ER: Construction of genomic networks using mutual-information clustering and reversible-jump Markov-chain-Monte-Carlo predictor design. Signal Processing 2003, 83(4):745-761. 10.1016/S0165-1684(02)00469-3
Hashimoto RF, Kim S, Shmulevich I, Zhang W, Bittner ML, Dougherty ER: Growing genetic regulatory networks from seed genes. Bioinformatics 2004, 20(8):1241-1247. 10.1093/bioinformatics/bth074
Zhou X, Wang X, Pal R, Ivanov I, Bittner ML, Dougherty ER: A Bayesian connectivity-based approach to constructing probabilistic gene regulatory networks. Bioinformatics 2004, 20(17):2918-2927. 10.1093/bioinformatics/bth318
Dougherty ER, Xiao Y: Design of probabilistic Boolean networks under the requirement of contextual data consistency. IEEE Transactions on Signal Processing 2006, 54(9):3603-3613.
Pe'er D, Regev A, Elidan G, Friedman N: Inferring subnetworks from perturbed expression profiles. Bioinformatics 2001, 17(supplement 1):S215-S224.
Husmeier D: Sensitivity and specificity of inferring genetic regulatory interactions from microarray experiments with dynamic Bayesian networks. Bioinformatics 2003, 19(17):2271-2282. 10.1093/bioinformatics/btg313
Peña JM, Björkegren J, Tegnér J: Growing Bayesian network models of gene networks from seed genes. Bioinformatics 2005, 21(supplement 2):ii224-ii229.
Lähdesmäki H, Hautaniemi S, Shmulevich I, Yli-Harja O: Relationships between probabilistic Boolean networks and dynamic Bayesian networks as models of gene regulatory networks. Signal Processing 2006, 86(4):814-834. 10.1016/j.sigpro.2005.06.008
Kauffman SA: Metabolic stability and epigenesis in randomly constructed genetic nets. Journal of Theoretical Biology 1969, 22(3):437-467. 10.1016/0022-5193(69)90015-0
Kauffman SA: Homeostasis and differentiation in random genetic control networks. Nature 1969, 224(5215):177-178. 10.1038/224177a0
Kauffman SA: The Origins of Order: Self-Organization and Selection in Evolution. Oxford University Press, New York, NY, USA; 1993.
Huang S: Gene expression profiling, genetic networks, and cellular states: an integrating concept for tumorigenesis and drug discovery. Journal of Molecular Medicine 1999, 77(6):469-480. 10.1007/s001099900023
Shmulevich I, Dougherty ER, Kim S, Zhang W: Probabilistic Boolean networks: a rule-based uncertainty model for gene regulatory networks. Bioinformatics 2002, 18(2):261-274. 10.1093/bioinformatics/18.2.261
Kim S, Li H, Dougherty ER, et al.: Can Markov chain models mimic biological regulation? Journal of Biological Systems 2002, 10(4):337-357. 10.1142/S0218339002000676
Shmulevich I, Dougherty ER, Zhang W: From Boolean to probabilistic Boolean networks as models of genetic regulatory networks. Proceedings of the IEEE 2002, 90(11):1778-1792. 10.1109/JPROC.2002.804686
Shmulevich I, Dougherty ER: Modeling genetic regulatory networks with probabilistic Boolean networks. In Genomic Signal Processing and Statistics, EURASIP Book Series on Signal Processing and Communication, Hindawi, New York, NY, USA Edited by: Dougherty ER, Shmulevich I, Chen J, Wang ZJ. 2005, 241-279.
Datta A, Choudhary A, Bittner ML, Dougherty ER: External control in Markovian genetic regulatory networks. Machine Learning 2003, 52(1-2):169-191.
Pal R, Datta A, Bittner ML, Dougherty ER: Intervention in context-sensitive probabilistic Boolean networks. Bioinformatics 2005, 21(7):1211-1218. 10.1093/bioinformatics/bti131
Pal R, Datta A, Dougherty ER: Optimal infinite-horizon control for probabilistic Boolean networks. IEEE Transactions on Signal Processing 2006, 54(6, part 2):2375-2387.
Datta A, Pal R, Dougherty ER: Intervention in probabilistic gene regulatory networks. Current Bioinformatics 2006, 1(2):167-184. 10.2174/157489306777011978
Albert R, Othmer HG: The topology of the regulatory interactions predicts the expression pattern of the segment polarity genes in Drosophila melanogaster . Journal of Theoretical Biology 2003, 223(1):1-18. 10.1016/S0022-5193(03)00035-3
Langholz G, Kandel A, Mott JL: Foundations of Digital Logic Design. World Scientific, River Edge, NJ, USA; 1998.
Ivanov I, Dougherty ER: Modeling genetic regulatory networks: continuous or discrete? Journal of Biological Systems 2006, 14(2):219-229. 10.1142/S0218339006001763
Ivanov I, Dougherty ER: Reduction mappings between probabilistic Boolean networks. EURASIP Journal on Applied Signal Processing 2004, 2004(1):125-131. 10.1155/S1110865704309182
Ching W-K, Ng MK, Fung ES, Akutsu T: On construction of stochastic genetic networks based on gene expression sequences. International Journal of Neural Systems 2005, 15(4):297-310. 10.1142/S0129065705000256
Ng MK, Zhang S-Q, Ching W-K, Akutsu T: A control model for Markovian genetic regulatory networks. Transactions on Computational Systems Biology V, Lecture Notes in Computer Science 2006, 36-48.
About this article
Cite this article
Marshall, S., Yu, L., Xiao, Y. et al. Inference of a Probabilistic Boolean Network from a Single Observed Temporal Sequence. J Bioinform Sys Biology 2007, 32454 (2007). https://doi.org/10.1155/2007/32454
- Data Sequence
- Selection Probability
- Network State
- Gene Regulatory Network
- Unique State