Dimension Reduction of Large Sparse AND-NOT Network Models

In this manuscript we propose and implement a dimension reduction algorithm of AND-NOT networks for the purpose of steady state computation. Our method of network reduction consists in using "steady state approximations" that do not change the number of steady states. The algorithm is designed to work at the wiring diagram level without the need to evaluate or simplify Boolean functions. Also, our implementation of the algorithm takes advantage of the sparsity typical of discrete models of biological systems.The main features of our reduction algorithm are that it works at the wiring diagram level and it preserves the number of steady states. Furthermore, the steady states of the original network can be recovered from the steady states of the reduced network; thus, all steady states are found. Also, heuristic analysis and simulations show that it runs in polynomial time. We used our results to study AND-NOT network models of gene networks and showed that our algorithm greatly simplifies steady state analysis. Furthermore, our algorithm can handle sparse AND-NOT networks with up to 1,000,000 nodes.

[1]  M. Huynen,et al.  The frequency distribution of gene family sizes in complete genomes. , 1998, Molecular biology and evolution.

[2]  Qianchuan Zhao,et al.  A remark on "Scalar equations for synchronous Boolean networks with biological Applications" by C. Farrow, J. Heidel, J. Maloney, and J. Rogers , 2005, IEEE Transactions on Neural Networks.

[3]  Abdul Salam Jarrah,et al.  The Dynamics of Conjunctive and Disjunctive Boolean Network Models , 2010, Bulletin of mathematical biology.

[4]  Q. Ouyang,et al.  The yeast cell-cycle network is robustly designed. , 2003, Proceedings of the National Academy of Sciences of the United States of America.

[5]  Michael K. Ng,et al.  Algorithms for Finding Small Attractors in Boolean Networks , 2007, EURASIP J. Bioinform. Syst. Biol..

[6]  M. Aldana.  Boolean dynamics of networks with scale-free topology , 2003 .

[7]  Alan Veliz-Cuba,et al.  An Algebraic Approach to Reverse Engineering Finite Dynamical Systems Arising from Biology , 2012, SIAM J. Appl. Dyn. Syst..

[8]  Robert E. Tarjan,et al.  Depth-First Search and Linear Graph Algorithms , 1972, SIAM J. Comput..

[9]  H. Othmer,et al.  The topology of the regulatory interactions predicts the expression pattern of the segment polarity genes in Drosophila melanogaster. , 2003, Journal of theoretical biology.

[10]  R. Albert.  Scale-free networks in cell biology , 2005, Journal of Cell Science.

[11]  Minping Qian,et al.  Stochastic model of yeast cell-cycle network , 2006, q-bio/0605011.

[12]  W. Just,et al.  The number and probability of canalizing functions , 2003, math-ph/0312033.

[13]  Wen Li,et al.  A matrix perturbation method for computing the steady-state probability distributions of probabilistic Boolean networks with gene perturbations , 2011, J. Comput. Appl. Math..

[14]  Seda Arat,et al.  Modeling stochasticity and variability in gene regulatory networks , 2012, EURASIP J. Bioinform. Syst. Biol..

[15]  Carsten Peterson,et al.  Random Boolean network models and the yeast transcriptional network , 2003, Proceedings of the National Academy of Sciences of the United States of America.

[16]  Alan Veliz-Cuba,et al.  ADAM: Analysis of Discrete Models of Biological Systems Using Computer Algebra , 2010, BMC Bioinformatics.

[17]  Michael K. Ng,et al.  On computation of the steady-state probability distribution of probabilistic Boolean networks with gene perturbation , 2012, J. Comput. Appl. Math..

[18]  Holger Fröhlich,et al.  Modeling ERBB receptor-regulated G1/S transition to find novel targets for de novo trastuzumab resistance , 2009, BMC Systems Biology.

[19]  Reinhard C. Laubenbacher,et al.  Structure and dynamics of acyclic networks , 2014, Discret. Event Dyn. Syst..

[20]  Alan Veliz-Cuba.  Reduction of Boolean network models. , 2011, Journal of theoretical biology.

[21]  R. Laubenbacher,et al.  The number of multistate nested canalyzing functions , 2011, 1108.0206.

[22]  Tatsuya Akutsu,et al.  Detecting a Singleton Attractor in a Boolean Network Utilizing SAT Algorithms , 2009, IEICE Trans. Fundam. Electron. Commun. Comput. Sci..

[23]  Ioannis Xenarios,et al.  A method for the generation of standardized qualitative dynamical systems of regulatory networks , 2005, Theoretical Biology and Medical Modelling.

[24]  Abdul Salam Jarrah,et al.  The effect of negative feedback loops on the dynamics of boolean networks. , 2007, Biophysical journal.

[25]  Reinhard Laubenbacher,et al.  AND-NOT logic framework for steady state analysis of Boolean network models , 2012, 1211.5633.

[26]  Abdul Salam Jarrah,et al.  Nested Canalyzing, Unate Cascade, and Polynomial Functions. , 2006, Physica D. Nonlinear phenomena.

[27]  Réka Albert,et al.  An effective network reduction approach to find the dynamical repertoire of discrete dynamic networks. , 2013, Chaos.

[28]  B. Stigler,et al.  Boolean Models Can Explain Bistability in the lac Operon , 2008, J. Comput. Biol..

[29]  Steffen Klamt,et al.  A methodology for the structural and functional analysis of signaling and regulatory networks , 2006, BMC Bioinformatics.

[30]  Matthew R Bennett,et al.  Library of synthetic transcriptional AND gates built with split T7 RNA polymerase mutants , 2013, Proceedings of the National Academy of Sciences.

[31]  Aurélien Naldi,et al.  A Reduction of Logical Regulatory Graphs Preserving Essential Dynamical Properties , 2009, CMSB.

[32]  Alan Veliz-Cuba,et al.  Steady state analysis of Boolean molecular network models via model reduction and computational algebra , 2014, BMC Bioinformatics.

[33]  R. Laubenbacher,et al.  On the computation of fixed points in Boolean networks , 2012 .

[34]  I. Albert,et al.  Attractor analysis of asynchronous Boolean models of signal transduction networks. , 2010, Journal of theoretical biology.

[35]  S. Kauffman,et al.  Genetic networks with canalyzing Boolean rules are always stable. , 2004, Proceedings of the National Academy of Sciences of the United States of America.

[36]  Abdul Salam Jarrah,et al.  Polynomial algebra of discrete models in systems biology , 2010, Bioinform..