Probabilistic Boolean networks: a rule-based uncertainty model for gene regulatory networks

MOTIVATION Our goal is to construct a model for genetic regulatory networks such that the model class: (i) incorporates rule-based dependencies between genes; (ii) allows the systematic study of global network dynamics; (iii) is able to cope with uncertainty, both in the data and the model selection; and (iv) permits the quantification of the relative influence and sensitivity of genes in their interactions with other genes. RESULTS We introduce Probabilistic Boolean Networks (PBN) that share the appealing rule-based properties of Boolean networks, but are robust in the face of uncertainty. We show how the dynamics of these networks can be studied in the probabilistic context of Markov chains, with standard Boolean networks being special cases. Then, we discuss the relationship between PBNs and Bayesian networks--a family of graphical models that explicitly represent probabilistic relationships between variables. We show how probabilistic dependencies between a gene and its parent genes, constituting the basic building blocks of Bayesian networks, can be obtained from PBNs. Finally, we present methods for quantifying the influence of genes on other genes, within the context of PBNs. Examples illustrating the above concepts are presented throughout the paper.

[1]  Sui Huang.  Gene expression profiling, genetic networks, and cellular states: an integrating concept for tumorigenesis and drug discovery , 1999, Journal of Molecular Medicine.

[2]  S. Kauffman.  Metabolic stability and epigenesis in randomly constructed genetic nets. , 1969, Journal of theoretical biology.

[3]  D. Thieffry,et al.  Dynamical behaviour of biological regulatory networks—I. Biological role of feedback loops and practical use of the concept of the loop-characteristic state , 1995 .

[4]  A. Gartel,et al.  Transcriptional regulation of the p21((WAF1/CIP1)) gene. , 1999, Experimental cell research.

[5]  T. Mestl,et al.  A mathematical framework for describing and analysing gene regulatory networks. , 1995, Journal of theoretical biology.

[6]  D. A. Baxter,et al.  Mathematical Modeling of Gene Networks , 2000, Neuron.

[7]  E. Dougherty,et al.  Multivariate measurement of gene expression relationships. , 2000, Genomics.

[8]  Erhan Çinlar,et al.  Introduction to stochastic processes , 1974 .

[9]  A Wuensche,et al.  Genomic regulation modeled as a network with basins of attraction. , 1998, Pacific Symposium on Biocomputing. Pacific Symposium on Biocomputing.

[10]  Roland Somogyi,et al.  Modeling the complexity of genetic networks: Understanding multigenic and pleiotropic regulation , 1996, Complex..

[11]  Patrik D'haeseleer,et al.  Genetic network inference: from co-expression clustering to reverse engineering , 2000, Bioinform..

[12]  Z. Szallasi,et al.  Modeling the normal and neoplastic cell cycle with "realistic Boolean genetic networks": their application for understanding carcinogenesis and assessing therapeutic strategies. , 1998, Pacific Symposium on Biocomputing. Pacific Symposium on Biocomputing.

[13]  K Sivakumar,et al.  General nonlinear framework for the analysis of gene interaction via multivariate expression arrays. , 2000, Journal of biomedical optics.

[14]  E. Davidson,et al.  Genomic cis-regulatory logic: experimental and computational analysis of a sea urchin gene. , 1998, Science.

[15]  Eugene Charniak,et al.  Bayesian Networks without Tears , 1991, AI Mag..

[16]  Michal Linial,et al.  Using Bayesian Networks to Analyze Expression Data , 2000, J. Comput. Biol..

[17]  Gary D. Stormo,et al.  Modeling Regulatory Networks with Weight Matrices , 1998, Pacific Symposium on Biocomputing.

[18]  G. Moran.  ON THE PERIOD-TWO-PROPERTY OF THE MAJORITY OPERATOR IN INFINITE GRAPHS , 1995 .

[19]  Judea Pearl,et al.  Probabilistic reasoning in intelligent systems , 1988 .

[20]  Satoru Miyano,et al.  Identification of gene regulatory networks by strategic gene disruptions and gene overexpressions , 1998, SODA '98.

[21]  Farren J. Isaacs,et al.  Computational studies of gene regulatory networks: in numero molecular biology , 2001, Nature Reviews Genetics.

[22]  Satoru Miyano,et al.  Inferring qualitative relations in genetic networks and metabolic pathways , 2000, Bioinform..

[23]  Stuart A. Kauffman,et al.  The origins of order , 1993 .

[24]  A. Arkin,et al.  Stochastic kinetic analysis of developmental pathway bifurcation in phage lambda-infected Escherichia coli cells. , 1998, Genetics.

[25]  Marcel J. T. Reinders,et al.  Linear Modeling of Genetic Networks from Experimental Data , 2000, ISMB.

[26]  Edward R. Dougherty,et al.  Coefficient of determination in nonlinear signal processing , 2000, Signal Process..

[27]  Edward R. Dougherty,et al.  Parallel computing methods for analyzing gene expression relationships , 2001, SPIE BiOS.

[28]  Satoru Miyano,et al.  Identification of Genetic Networks from a Small Number of Gene Expression Patterns Under the Boolean Network Model , 1998, Pacific Symposium on Biocomputing.

[29]  Tommi S. Jaakkola,et al.  Using Graphical Models and Genomic Expression Data to Statistically Validate Models of Genetic Regulatory Networks , 2000, Pacific Symposium on Biocomputing.

[30]  M. Montenarh,et al.  Regulation of CAK kinase activity by p53 , 1998, Oncogene.

[31]  S Fuhrman,et al.  Reveal, a general reverse engineering algorithm for inference of genetic network architectures. , 1998, Pacific Symposium on Biocomputing. Pacific Symposium on Biocomputing.

[32]  Michael L. Bittner,et al.  Nonlinear filters in genomic control , 1999, NSIP.

[33]  I. Mian,et al.  Integrating naive Bayes models and external knowledge to examine copper and iron homeostasis in S. cerevisiae. , 2000, Physiological genomics.

[34]  Finn Verner Jensen,et al.  Introduction to Bayesian Networks , 2008, Innovations in Bayesian Networks.

[35]  Michael Krauthammer,et al.  A knowledge model for analysis and simulation of regulatory networks , 2000, Bioinform..

[36]  David Maxwell Chickering,et al.  Learning Bayesian Networks is NP-Complete , 2016, AISTATS.

[37]  J. Astola,et al.  INFERENCE OF GENETIC REGULATORY NETWORKS UNDER THE BEST-FIT EXTENSION PARADIGM , 2001 .

[38]  L. Glass,et al.  The logical analysis of continuous, non-linear biochemical control networks. , 1973, Journal of theoretical biology.

[39]  Nathan Linial,et al.  The influence of variables on Boolean functions , 1988, [Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science.

[40]  Patrik D'haeseleer,et al.  Linear Modeling of mRNA Expression Levels During CNS Development and Injury , 1998, Pacific Symposium on Biocomputing.