Stability structures of conjunctive Boolean networks

A Boolean network is a finite dynamical system, whose variables take values from a binary set. The value update rule for each variable is a Boolean function, depending on a selected subset of variables. Boolean networks have been widely used in modeling gene regulatory networks. We focus in this paper on a special class of Boolean networks, termed as conjunctive Boolean networks. A Boolean network is conjunctive if the associated value update rule is comprised of only AND operations. It is known that any trajectory of a finite dynamical system will enter a periodic orbit. We characterize in this paper all periodic orbits of a conjunctive Boolean network whose underlying graph is strongly connected. In particular, we establish a bijection between the set of periodic orbits and the set of binary necklaces of a certain length. We further investigate the stability of a periodic orbit. Specifically, we perturb a state in the periodic orbit by changing the value of a single entry of the state. The trajectory, with the perturbed state being the initial condition, will enter another (possibly the same) periodic orbit in finite time steps. We then provide a complete characterization of all such transitions from one periodic orbit to another. In particular, we construct a digraph, with the vertices being the periodic orbits, and the (directed) edges representing the transitions among the orbits. We call such a digraph the stability structure of the conjunctive Boolean network.

[1]  Seyed Rasoul Etesami,et al.  Complexity of equilibrium in competitive diffusion games on social networks , 2016, Autom..

[2]  Bernd Sturmfels,et al.  Monomial Dynamical Systems over Finite Fields , 2006, Complex Syst..

[3]  Tamer Basar,et al.  Optimal control of LTI systems over unreliable communication links , 2006, Autom..

[4]  Joel Franklin,et al.  A Finite-Difference Sieve to Count Paths and Cycles by Length , 1996, Inf. Process. Lett..

[5]  Yuichi Nakamura,et al.  Approximation of dynamical systems by continuous time recurrent neural networks , 1993, Neural Networks.

[6]  Eric Goles Ch.,et al.  On the preservation of limit cycles in Boolean networks under different updating schemes , 2013, ECAL.

[7]  Damien Regnault,et al.  Boolean networks synchronism sensitivity and XOR circulant networks convergence time , 2012, AUTOMATA & JAC.

[8]  Tamer Basar,et al.  Asymptotic behavior of a reduced conjunctive Boolean network , 2017, 2017 IEEE 56th Annual Conference on Decision and Control (CDC).

[9]  Denis Thieffry,et al.  A description of dynamical graphs associated to elementary regulatory circuits , 2003, ECCB.

[10]  Tamer Basar,et al.  Controllability of Formations Over Directed Time-Varying Graphs , 2017, IEEE Transactions on Control of Network Systems.

[11]  Tamer Basar,et al.  Orbit-controlling sets for conjunctive Boolean networks , 2017, 2017 American Control Conference (ACC).

[12]  Frank Ruskey,et al.  An efficient algorithm for generating necklaces with fixed density , 1999, SODA '99.

[13]  Tamer Başar,et al.  Controllability of Conjunctive Boolean Networks With Application to Gene Regulation , 2017, IEEE Transactions on Control of Network Systems.

[14]  Donald B. Johnson,et al.  Finding All the Elementary Circuits of a Directed Graph , 1975, SIAM J. Comput..

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

[16]  E. Gilbert,et al.  Symmetry types of periodic sequences , 1961 .

[17]  Herbert Weinblatt,et al.  A New Search Algorithm for Finding the Simple Cycles of a Finite Directed Graph , 1972, JACM.

[18]  Elizabeth-sharon Fung,et al.  A gene regulatory network armature for T-lymphocyte specification , 2008 .

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

[20]  Eric T. Bax.  Algorithms to Count Paths and Cycles , 1994, Inf. Process. Lett..

[21]  L. Raeymaekers,et al.  Dynamics of Boolean networks controlled by biologically meaningful functions. , 2002, Journal of theoretical biology.

[22]  Narsingh Deo,et al.  On Algorithms for Enumerating All Circuits of a Graph , 1976, SIAM J. Comput..

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

[24]  Alan Veliz-Cuba,et al.  Dimension Reduction of Large Sparse AND-NOT Network Models , 2015, SASB.

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

[26]  K Varadarajan,et al.  Aperiodic rings, necklace rings, and Witt vectors , 1990 .

[27]  Shuhong Gao,et al.  Monomial Dynamical Systems in # P-complete , 2012 .

[28]  Tamer Basar,et al.  Consensus with linear objective maps , 2015, 2015 54th IEEE Conference on Decision and Control (CDC).

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

[30]  Bahman Gharesifard,et al.  Stability properties of infected networks with low curing rates , 2014, 2014 American Control Conference.

[31]  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.

[32]  Xudong Chen,et al.  State-controlling Sets for Conjunctive Boolean Networks , 2017 .

[33]  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.

[34]  Mathilde Noual,et al.  Updating Automata Networks , 2012 .

[35]  Nils M. Kriege,et al.  A General Purpose Algorithm for Counting Simple Cycles and Simple Paths of Any Length , 2016, Algorithmica.

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

[37]  Tamer Basar,et al.  Periodic behavior of a diffusion model over directed graphs , 2016, 2016 IEEE 55th Conference on Decision and Control (CDC).

[38]  S. Kauffman.  Homeostasis and Differentiation in Random Genetic Control Networks , 1969, Nature.

[39]  Noga Alon,et al.  Finding and counting given length cycles , 1997, Algorithmica.

[40]  Dana Ron,et al.  Finding cycles and trees in sublinear time , 2010, Random Struct. Algorithms.

[41]  Kévin Perrot,et al.  On the Flora of Asynchronous Locally Non-monotonic Boolean Automata Networks , 2016, Electron. Notes Theor. Comput. Sci..

[42]  R. Laubenbacher,et al.  Boolean Monomial Dynamical Systems , 2004, math/0403166.

[43]  Damien Regnault,et al.  About non-monotony in Boolean automata networks , 2011, Theor. Comput. Sci..

[44]  Reinhard Laubenbacher,et al.  The Dynamics of Semilattice Networks , 2010 .

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

[46]  Adrien Richard,et al.  On the Convergence of Boolean Automata Networks without Negative Cycles , 2013, Automata.

[47]  Denis Thieffry,et al.  Genetic control of flower morphogenesis in Arabidopsis thaliana: a logical analysis , 1999, Bioinform..

[48]  Eric Goles Ch.,et al.  Disjunctive networks and update schedules , 2012, Adv. Appl. Math..

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