Properties of Boolean networks and methods for their tests

Transcriptional regulation networks are often modeled as Boolean networks. We discuss certain properties of Boolean functions (BFs), which are considered as important in such networks, namely, membership to the classes of unate or canalizing functions. Of further interest is the average sensitivity (AS) of functions. In this article, we discuss several algorithms to test the properties of interest. To test canalizing properties of functions, we apply spectral techniques, which can also be used to characterize the AS of functions as well as the influences of variables in unate BFs. Further, we provide and review upper and lower bounds on the AS of unate BFs based on the spectral representation. Finally, we apply these methods to a transcriptional regulation network of Escherichia coli, which controls central parts of the E. coli metabolism. We find that all functions are unate. Also the analysis of the AS of the network reveals an exceptional robustness against transient fluctuations of the binary variables.a

[1]  J. Hammann,et al.  Ageing Process and Response Function in Spin Glasses: An Analysis of the Thermoremanent Magnetization Decay in Ag:Mn (2.6%) , 1986 .

[2]  James F. Lynch.  DYNAMICS OF RANDOM BOOLEAN NETWORKS , 2007 .

[3]  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.

[4]  Olli Yli-Harja,et al.  ANALYZING DYNAMICS OF BOOLEAN NETWORKS WITH CANALYZING FUNCTIONS USING SPECTRAL METHODS , 2005 .

[5]  Sean W. Smith,et al.  Improved learning of AC0 functions , 1991, COLT '91.

[6]  Hans A. Kestler,et al.  Inferring Boolean network structure via correlation , 2011, Bioinform..

[7]  Adam M. Feist,et al.  A genome-scale metabolic reconstruction for Escherichia coli K-12 MG1655 that accounts for 1260 ORFs and thermodynamic information , 2007, Molecular systems biology.

[8]  S. Kauffman,et al.  Activities and sensitivities in boolean network models. , 2004, Physical review letters.

[9]  Martin Bossert,et al.  Harmonic analysis of Boolean networks: determinative power and perturbations , 2011, EURASIP J. Bioinform. Syst. Biol..

[10]  Ryan O'Donnell,et al.  Some topics in analysis of boolean functions , 2008, STOC.

[11]  Nader H. Bshouty,et al.  On the Fourier spectrum of monotone functions , 1996, JACM.

[12]  Martin Bossert,et al.  Detecting controlling nodes of boolean regulatory networks , 2011, EURASIP J. Bioinform. Syst. Biol..

[13]  John L. Shanks,et al.  Computation of the Fast Walsh-Fourier Transform , 1969, IEEE Transactions on Computers.

[14]  Karen O. Egiazarian,et al.  Spectral methods for testing membership in certain post classes and the class of forcing functions , 2004, IEEE Signal Processing Letters.

[15]  S. Kauffman,et al.  An analysis of the class of gene regulatory functions implied by a biochemical model. , 2006, Bio Systems.

[16]  Andrew Wuensche,et al.  A model of transcriptional regulatory networks based on biases in the observed regulation rules , 2002, Complex..

[17]  Martin Bossert,et al.  Analysis of random Boolean networks using the average sensitivity , 2007 .

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

[19]  Markus J. Herrgård,et al.  Integrating high-throughput and computational data elucidates bacterial networks , 2004, Nature.

[20]  B. Derrida,et al.  Random networks of automata: a simple annealed approximation , 1986 .

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

[22]  Steffen Schober.  Analysis and identification of boolean networks using harmonic analysis , 2011 .

[23]  Ehud Friedgut,et al.  Boolean Functions With Low Average Sensitivity Depend On Few Coordinates , 1998, Comb..

[24]  Sanjay Jain,et al.  The regulatory network of E. coli metabolism as a Boolean dynamical system exhibits both homeostasis and flexibility of response , 2007 .

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

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

[27]  I. Benjamini,et al.  Noise sensitivity of Boolean functions and applications to percolation , 1998 .

[28]  Pao-Ta Yu,et al.  Convergence behavior and root signal sets of stack filters , 1992 .