Bayesian Analysis

Reciprocal Graphical Models for Integrative Gene Regulatory Network Analysis

Yang Ni, Yuan Ji, and Peter Müller

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Constructing gene regulatory networks is a fundamental task in systems biology. We introduce a Gaussian reciprocal graphical model for inference about gene regulatory relationships by integrating messenger ribonucleic acid (mRNA) gene expression and deoxyribonucleic acid (DNA) level information including copy number and methylation. Data integration allows for inference on the directionality of certain regulatory relationships, which would be otherwise indistinguishable due to Markov equivalence. Efficient inference is developed based on simultaneous equation models. Bayesian model selection techniques are adopted to estimate the graph structure. We illustrate our approach by simulations and application in colon adenocarcinoma pathway analysis.

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Bayesian Anal. (2017), 16 pages.

First available in Project Euclid: 1 December 2017

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simultaneous equation models Markov equivalence directed cycles feedback loop multimodal genomic data

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Ni, Yang; Ji, Yuan; Müller, Peter. Reciprocal Graphical Models for Integrative Gene Regulatory Network Analysis. Bayesian Anal., advance publication, 1 December 2017. doi:10.1214/17-BA1087.

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  • Alon, U. (2007). “Network motifs: theory and experimental approaches.”Nature Reviews Genetics, 8(6): 450–461.
  • Bhadra, A. and Mallick, B. K. (2013). “Joint high-dimensional Bayesian variable and covariance selection with an application to eQTL analysis.”Biometrics, 69(2): 447–457.
  • Cai, T. T., Li, H., Liu, W., and Xie, J. (2012). “Covariate-adjusted precision matrix estimation with an application in genetical genomics.”Biometrika, 100(1): 139–156.
  • Cai, X., Bazerque, J. A., and Giannakis, G. B. (2013). “Inference of gene regulatory networks with sparse structural equation models exploiting genetic perturbations.”PLoS Computational Biology, 9(5): e1003068.
  • Chen, M., Ren, Z., Zhao, H., and Zhou, H. (2016). “Asymptotically normal and efficient estimation of covariate-adjusted Gaussian graphical model.”Journal of the American Statistical Association, 111(513): 394–406.
  • Colussi, D., Brandi, G., Bazzoli, F., and Ricciardiello, L. (2013). “Molecular pathways involved in colorectal cancer: implications for disease behavior and prevention.”International Journal of Molecular Sciences, 14(8): 16365–16385.
  • Dhillon, A. S., Hagan, S., Rath, O., and Kolch, W. (2007). “MAP kinase signalling pathways in cancer.”Oncogene, 26(22): 3279–3290.
  • Dobra, A., Lenkoski, A., and Rodriguez, A. (2012). “Bayesian inference for general Gaussian graphical models with application to multivariate lattice data.”Journal of the American Statistical Association.
  • Fang, J. Y. and Richardson, B. C. (2005). “The MAPK signalling pathways and colorectal cancer.”The Lancet Oncology, 6(5): 322–327.
  • Frydenberg, M. (1990). “The chain graph Markov property.”Scandinavian Journal of Statistics, 333–353.
  • Green, P. J. and Thomas, A. (2013). “Sampling decomposable graphs using a Markov chain on junction trees.”Biometrika, 100(1): 91–110.
  • Holt, K. H., Kasson, B. G., and Pessin, J. E. (1996). “Insulin stimulation of a MEK-dependent but ERK-independent SOS protein kinase.”Molecular and Cellular Biology, 16(2): 577–583.
  • Johnson, V. E. and Rossell, D. (2010). “On the use of non-local prior densities in Bayesian hypothesis tests.”Journal of the Royal Statistical Society: Series B (Statistical Methodology), 72(2): 143–170.
  • Koster, J. T. (1996). “Markov properties of nonrecursive causal models.”The Annals of Statistics, 24(5): 2148–2177.
  • Kundu, S. and Kang, J. (2016). “Semiparametric Bayes conditional graphical models for imaging genetics applications.”Stat, 5(1): 322–337.
  • Mendoza, M. C., Er, E. E., and Blenis, J. (2011). “The Ras-ERK and PI3K-mTOR pathways: cross-talk and compensation.”Trends in Biochemical Sciences, 36(6): 320–328.
  • Mitra, R., Müller, P., Liang, S., Yue, L., and Ji, Y. (2013). “A Bayesian graphical model for ChIP-seq data on histone modifications.”Journal of the American Statistical Association, 108(501): 69–80.
  • Müller, P., Parmigiani, G., and Rice, K. (2006). “FDR and Bayesian multiple comparisons rules.”
  • Newton, M. A., Noueiry, A., Sarkar, D., and Ahlquist, P. (2004). “Detecting differential gene expression with a semiparametric hierarchical mixture method.”Biostatistics, 5(2): 155–176.
  • Ni, Y., Ji, Y., and Müller, P. (2017). “Supplementary Material for “Reciprocal Graphical Models for Integrative Gene Regulatory Network Analysis”.”Bayesian Analysis.
  • Ni, Y., Stingo, F., and Baladandayuthapani, V. (2018). “Bayesian graphical regression.”Journal of the American Statistical Association, just accepted.
  • Ni, Y., Stingo, F. C., and Baladandayuthapani, V. (2015). “Bayesian nonlinear model selection for gene regulatory networks.”Biometrics, 71(3): 585–595.
  • Oates, C. J., Smith, J. Q., and Mukherjee, S. (2016). “Estimating causal structure using conditional DAG models.”Journal of Machine Learning Research, 17(54): 1–23.
  • Plotnikov, A., Zehorai, E., Procaccia, S., and Seger, R. (2011). “The MAPK cascades: signaling components, nuclear roles and mechanisms of nuclear translocation.”Biochimica et Biophysica Acta (BBA)-Molecular Cell Research, 1813(9): 1619–1633.
  • Rothman, A. J., Levina, E., and Zhu, J. (2010). “Sparse multivariate regression with covariance estimation.”Journal of Computational and Graphical Statistics, 19(4): 947–962.
  • Rudelson, M. (2008). “Invertibility of random matrices: norm of the inverse.”Annals of Mathematics, 575–600.
  • Rudelson, M. and Vershynin, R. (2008). “The Littlewood–Offord problem and invertibility of random matrices.”Advances in Mathematics, 218(2): 600–633.
  • Shin, S.-Y., Rath, O., Zebisch, A., Choo, S.-M., Kolch, W., and Cho, K.-H. (2010). “Functional roles of multiple feedback loops in ERK and Wnt signaling pathways that regulate epithelial-mesenchymal transition.”Cancer Research, 70(17): 6715.
  • Spirtes, P. (1995). “Directed cyclic graphical representations of feedback models.” InProceedings of the Eleventh Conference on Uncertainty in Artificial Intelligence, 491–498. Morgan Kaufmann Publishers Inc.
  • Stingo, F. C., Chen, Y. A., Vannucci, M., Barrier, M., and Mirkes, P. E. (2010). “A Bayesian graphical modeling approach to microRNA regulatory network inference.”The Annals of Applied Statistics, 4(4): 2024.
  • TCGA (2012). “Comprehensive molecular characterization of human colon and rectal cancer.”Nature, 487(7407): 330–337.
  • Telesca, D., Müller, P., Kornblau, S. M., Suchard, M. A., and Ji, Y. (2012a). “Modeling protein expression and protein signaling pathways.”Journal of the American Statistical Association, 107(500): 1372–1384.
  • Telesca, D., Müller, P., Parmigiani, G., and Freedman, R. S. (2012b). “Modeling dependent gene expression.”The Annals of Applied Statistics, 6(2): 542–560.
  • Wang, H. and West, M. (2009). “Bayesian analysis of matrix normal graphical models.”Biometrika, 96(4): 821–834.
  • Wang, W., Baladandayuthapani, V., Holmes, C. C., and Do, K.-A. (2013). “Integrative network-based Bayesian analysis of diverse genomics data.”BMC Bioinformatics, 14(Suppl 13): S8.
  • Yajima, M., Telesca, D., Ji, Y., and Müller, P. (2015). “Detecting differential patterns of interaction in molecular pathways.”Biostatistics, 16(2): 240–251.
  • Zenonos, K. and Kyprianou, K. (2013). “RAS signaling pathways, mutations and their role in colorectal cancer.”World Journal of Gastrointestinal Oncology, 5(5): 97–101.
  • Zhang, D., Wells, M. T., Turnbull, B. W., Sparrow, D., and Cassano, P. A. (2005). “Hierarchical graphical models: An application to pulmonary function and cholesterol levels in the normative aging study.”Journal of the American Statistical Association, 100(471): 719–727.
  • Zhang, L. and Kim, S. (2014). “Learning gene networks under SNP perturbations using eQTL datasets.”PLoS Computational Biology, 10(2): e1003420.
  • Zhu, Y., Qiu, P., and Ji, Y. (2014). “TCGA-assembler: open-source software for retrieving and processing TCGA data.”Nature Methods, 11(6): 599–600.

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