Analysis of Boolean functions based on interaction graphs and their influence in system biology

Biological regulatory network can be modeled through a set of Boolean functions. These set of functions enable graph representation of the network structure, and hence, the dynamics of the network can be seen easily. In this article, the regulations of such network have been explored in terms of interaction graph. With the help of Boolean function decomposition, this work presents an approach for construction of interaction graphs. This decomposition technique is also used to reduce the network state space of the cell cycle network of fission yeast for finding the singleton attractors. Some special classes of Boolean functions with respect to the interaction graphs have been discussed. A unique recursive procedure is devised which uses the Cartesian product of sets starting from the set of one-variable Boolean function. Interaction graphs generated with these Boolean functions have only positive/negative edges, and the corresponding state spaces have periodic attractors with length one/two.

[1]  Stefan Bornholdt,et al.  Less Is More in Modeling Large Genetic Networks , 2005, Science.

[2]  Gérard Y. Vichniac,et al.  Boolean derivatives on cellular automata , 1991 .

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

[4]  Assieh Saadatpour,et al.  A Reduction Method for Boolean Network Models Proven to Conserve Attractors , 2013, SIAM J. Appl. Dyn. Syst..

[5]  Mithun Chakraborty,et al.  Characterization Of any Non-linear Boolean function Using A Set of Linear Operators , 2008, ArXiv.

[6]  Yuzhen Wang,et al.  Robust stability and stabilisation of Boolean networks with disturbance inputs , 2017, Int. J. Syst. Sci..

[7]  Adrien Richard,et al.  Static Analysis of Boolean Networks Based on Interaction Graphs: A Survey , 2012, SASB.

[8]  Martine Labbé,et al.  Identification of all steady states in large networks by logical analysis , 2003, Bulletin of mathematical biology.

[9]  Simon X. Yang,et al.  On delayed uncertain genetic regulatory networks: robust stability analysis , 2011, Int. J. Comput. Math..

[10]  Miha Mraz,et al.  Computational modelling of genome-scale metabolic networks and its application to CHO cell cultures , 2017, Comput. Biol. Medicine.

[11]  J. Millar,et al.  Fkh2p and Sep1p regulate mitotic gene transcription in fission yeast , 2004, Journal of Cell Science.

[12]  Andrew J. Bulpitt,et al.  A Primer on Learning in Bayesian Networks for Computational Biology , 2007, PLoS Comput. Biol..

[13]  Daizhan Cheng,et al.  A Linear Representation of Dynamics of Boolean Networks , 2010, IEEE Transactions on Automatic Control.

[14]  R. Balkrishnan,et al.  Markov Chain Modelling Analysis of HIV/AIDS Progression: A Race-based Forecast in the United States , 2014, Indian journal of pharmaceutical sciences.

[15]  J. Tyson,et al.  Mathematical model of the cell division cycle of fission yeast. , 2001, Chaos.

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

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

[18]  Pabitra Pal Choudhury,et al.  Classification of Boolean Functions where Affine Functions are Uniformly Distributed , 2013, ArXiv.

[19]  Guozhen Xiao,et al.  On cross-correlation properties of Boolean functions , 2009, 2009 Fourth International Conference on Communications and Networking in China.

[20]  A. Jarrah,et al.  Inferring Biologically Relevant Models: Nested Canalyzing Functions , 2010, 1011.6064.

[21]  Jinde Cao,et al.  Stability and bifurcation of genetic regulatory networks with small RNAs and multiple delays , 2014, Int. J. Comput. Math..

[22]  Yupu Hu,et al.  New constructions of balanced Boolean functions with high nonlinearity and optimal algebraic degree , 2012, Int. J. Comput. Math..

[23]  Denis Thieffry,et al.  Dynamical roles of biological regulatory circuits , 2007, Briefings Bioinform..

[24]  Steffen Klamt,et al.  Computing paths and cycles in biological interaction graphs , 2009, BMC Bioinformatics.

[25]  Pabitra Pal Choudhury,et al.  On Analysis and Generation of some Biologically Important Boolean Functions , 2014, ArXiv.

[26]  Charles M. Grinstead,et al.  Introduction to probability , 1986, Statistics for the Behavioural Sciences.

[27]  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.

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

[29]  Aniruddha Datta,et al.  Generating Boolean networks with a prescribed attractor structure , 2005, Bioinform..

[30]  Debashis Sahoo,et al.  Combined Analysis of Murine and Human Microarrays and ChIP Analysis Reveals Genes Associated with the Ability of MYC To Maintain Tumorigenesis , 2008, PLoS genetics.

[31]  David A. Rosenblueth,et al.  Inference of Boolean Networks from Gene Interaction Graphs Using a SAT Solver , 2014, AlCoB.

[32]  Zalmiyah Zakaria,et al.  A review on the computational approaches for gene regulatory network construction , 2014, Comput. Biol. Medicine.

[33]  Manuel A. Martins,et al.  Boolean dynamics revisited through feedback interconnections , 2018, Natural Computing.

[34]  J. Demongeot,et al.  Positive and negative feedback: striking a balance between necessary antagonists. , 2002, Journal of theoretical biology.

[35]  R. Thomas,et al.  Boolean formalization of genetic control circuits. , 1973, Journal of theoretical biology.

[36]  Akutsu,et al.  A System for Identifying Genetic Networks from Gene Expression Patterns Produced by Gene Disruptions and Overexpressions. , 1998, Genome informatics. Workshop on Genome Informatics.

[37]  John N. Tsitsiklis,et al.  Introduction to Probability , 2002 .

[38]  Tamer Kahveci,et al.  Scalable Steady State Analysis of Boolean Biological Regulatory Networks , 2009, PloS one.

[39]  Aniruddha Datta,et al.  Stationary and structural control in gene regulatory networks: basic concepts , 2010, Int. J. Syst. Sci..

[40]  D. Slepian.  On The Number of Symmetry Types of Boolean Functions of n Variables , 1953, Canadian Journal of Mathematics.

[41]  M. Aldana,et al.  From Genes to Flower Patterns and Evolution: Dynamic Models of Gene Regulatory Networks , 2006, Journal of Plant Growth Regulation.

[42]  W. Ching,et al.  Generating probabilistic Boolean networks from a prescribed transition probability matrix. , 2009, IET systems biology.

[43]  Daizhan Cheng,et al.  Model Construction of Boolean Network via Observed Data , 2011, IEEE Transactions on Neural Networks.

[44]  Madalena Chaves,et al.  Analysis Tools for Interconnected Boolean Networks With Biological Applications , 2018, Front. Physiol..

[45]  Enrico Giampieri,et al.  A Cross-Sectional Analysis of Body Composition Among Healthy Elderly From the European NU-AGE Study: Sex and Country Specific Features , 2018, Front. Physiol..

[46]  Pauli Sipari.  Structured system models Part 2. Directed graphs and boolean matrices , 1991 .

[47]  Bo Li,et al.  Graphical reduction of probabilistic boolean networks , 2017, 2017 36th Chinese Control Conference (CCC).

[48]  Benjamin D. Greenberg,et al.  Partitioning the Heritability of Tourette Syndrome and Obsessive Compulsive Disorder Reveals Differences in Genetic Architecture , 2013, PLoS genetics.

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

[50]  Adrien Richard,et al.  Positive circuits and maximal number of fixed points in discrete dynamical systems , 2008, Discret. Appl. Math..

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

[52]  René Thomas,et al.  Biological Feedback , 2019 .

[53]  Pabitra Pal Choudhury,et al.  Partitioning 1-variable Boolean functions for various classification of n-variable Boolean functions , 2015, Int. J. Comput. Math..

[54]  Edward R. Dougherty,et al.  Probabilistic Boolean Networks - The Modeling and Control of Gene Regulatory Networks , 2010 .

[55]  John Maloney,et al.  Finding Cycles in Synchronous Boolean Networks with Applications to Biochemical Systems , 2003, Int. J. Bifurc. Chaos.

[56]  Sui Huang.  Cell State Dynamics and Tumorigenesis in Boolean Regulatory Networks , 2006 .

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

[58]  Solomon W. Golomb,et al.  On the classification of Boolean functions , 1959, IRE Trans. Inf. Theory.

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

[60]  Maximilien Gadouleau,et al.  Reduction and Fixed Points of Boolean Networks and Linear Network Coding Solvability , 2014, IEEE Transactions on Information Theory.

[61]  Bianca Zadrozny,et al.  A Bayesian network decision model for supporting the diagnosis of dementia, Alzheimer's disease and mild cognitive impairment , 2014, Comput. Biol. Medicine.

[62]  Robert Gentleman,et al.  Graphs in molecular biology , 2007, BMC Bioinformatics.

[63]  Xiangzhen Zan,et al.  Signalling pathway impact analysis based on the strength of interaction between genes , 2016, IET systems biology.

[64]  S. Bornholdt,et al.  Boolean Network Model Predicts Cell Cycle Sequence of Fission Yeast , 2007, PloS one.

[65]  Denis Thieffry,et al.  Graphic requirements for multistability and attractive cycles in a Boolean dynamical framework , 2008, Adv. Appl. Math..

[66]  A. Mochizuki.  An analytical study of the number of steady states in gene regulatory networks. , 2005, Journal of theoretical biology.

[67]  Junior Barrera,et al.  A pattern-oriented specification of gene network inference processes , 2013, Comput. Biol. Medicine.