Geometric Characterization of Hereditarily Bijective Boolean Networks

The study of relationships between structure and dynamics of asynchronous Boolean networks has recently led to the introduction of hereditarily bijective maps and even or odd self-dual networks. We show here that these two notions can be simply characterized geometrically: through orthogonality between certain affine subspaces. We also use this characterization to provide a construction of the class of hereditarily bijective maps, and to study its stability properties.

[1]  Hidde de Jong,et al.  Modeling and Simulation of Genetic Regulatory Systems: A Literature Review , 2002, J. Comput. Biol..

[2]  El Houssine Snoussi.  Qualitative dynamics of piecewise-linear differential equations: a discrete mapping approach , 1989 .

[3]  Stuart A. Kauffman,et al.  ORIGINS OF ORDER IN EVOLUTION: SELF-ORGANIZATION AND SELECTION , 1992 .

[4]  Claudine Chaouiya,et al.  Logical Modelling of Regulatory Networks, Methods and Applications , 2013, Bulletin of Mathematical Biology.

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

[6]  Adrien Richard,et al.  Fixed point theorems for Boolean networks expressed in terms of forbidden subnetworks , 2013, Theor. Comput. Sci..

[7]  Adrien Richard,et al.  From kernels in directed graphs to fixed points and negative cycles in Boolean networks , 2013, Discret. Appl. Math..

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

[9]  Eric Goles Ch.,et al.  Comparison between parallel and serial dynamics of Boolean networks , 2008, Theor. Comput. Sci..

[10]  R. Thomas,et al.  Multistationarity, the basis of cell differentiation and memory. I. Structural conditions of multistationarity and other nontrivial behavior. , 2001, Chaos.

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

[12]  J. Schwartz,et al.  Theory of Self-Reproducing Automata , 1967 .

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

[14]  René Thomas.  Regulatory networks seen as asynchronous automata: A logical description , 1991 .

[15]  René Thomas.  On the Relation Between the Logical Structure of Systems and Their Ability to Generate Multiple Steady States or Sustained Oscillations , 1981 .

[16]  J. Demongeot,et al.  Robustness in Regulatory Networks: A Multi-Disciplinary Approach , 2008, Acta biotheoretica.

[17]  Mau-Hsiang Shih,et al.  A combinatorial analogue of the Jacobian problem in automata networks , 2005, Adv. Appl. Math..

[18]  Paul Ruet,et al.  Local cycles and dynamical properties of Boolean networks , 2014, Mathematical Structures in Computer Science.