Multi-stability in monotone input/output systems

This paper studies the emergence of multi-stability and hysteresis in those systems that arise, under positive feedback, starting from monotone systems with well-defined steady-state responses. Such feedback configurations appear routinely in several fields of application, and especially in biology. Characterizations of global stability behavior are stated in terms of easily checkable graphical conditions.

[1]  U. Bhalla,et al.  Emergent properties of networks of biological signaling pathways. , 1999, Science.

[2]  David Angeli,et al.  A remark on monotone control systems , 2002, Proceedings of the 41st IEEE Conference on Decision and Control, 2002..

[3]  Xinzhi Liu Comparison methods and stability theory , 1994 .

[4]  Robert J. Plemmons,et al.  Nonnegative Matrices in the Mathematical Sciences , 1979, Classics in Applied Mathematics.

[5]  J. Collins,et al.  Construction of a genetic toggle switch in Escherichia coli , 2000, Nature.

[6]  N. Rouche,et al.  Stability Theory by Liapunov's Direct Method , 1977 .

[7]  J. Borwein,et al.  Convex Analysis And Nonlinear Optimization , 2000 .

[8]  M. Hirsch Stability and convergence in strongly monotone dynamical systems. , 1988 .

[9]  Thomas Mestl,et al.  FEEDBACK LOOPS, STABILITY AND MULTISTATIONARITY IN DYNAMICAL SYSTEMS , 1995 .

[10]  Bruce Hannon,et al.  Positive and Negative Feedback , 1994 .

[11]  F R Adler,et al.  How to make a biological switch. , 2000, Journal of theoretical biology.

[12]  J. Gouzé Positive and Negative Circuits in Dynamical Systems , 1998 .

[13]  Hal L. Smith Periodic tridiagonal competitive and cooperative systems of differential equations , 1991 .

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

[15]  Eduardo D. Sontag,et al.  Mathematical Control Theory: Deterministic Finite Dimensional Systems , 1990 .

[16]  James E. Ferrell,et al.  Bistability in cell signaling: How to make continuous processes discontinuous, and reversible processes irreversible. , 2001, Chaos.

[17]  John Ian Ferrell,et al.  Detection of multi-stability , 2004 .

[18]  R. Engelhardt,et al.  EARLY BIOLOGICAL MORPHOGENESIS AND NONLINEAR DYNAMICS , 1995 .

[19]  M Laurent,et al.  Multistability: a major means of differentiation and evolution in biological systems. , 1999, Trends in biochemical sciences.

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

[21]  El Houssine Snoussi Necessary Conditions for Multistationarity and Stable Periodicity , 1998 .

[22]  S. Smale On the differential equations of species in competition , 1976, Journal of mathematical biology.

[23]  Rafael de la Llave,et al.  On Irwin’s proof of the pseudostable manifold theorem , 1995 .

[24]  Carlo Piccardi,et al.  Remarks on excitability, stability and sign of equilibria in cooperative systems , 2002, Syst. Control. Lett..

[25]  J E Ferrell,et al.  The biochemical basis of an all-or-none cell fate switch in Xenopus oocytes. , 1998, Science.

[26]  H. Kunze,et al.  A graph theoretical approach to monotonicity with respect to initial conditions II , 1999 .

[27]  David Angeli,et al.  Monotone control systems , 2003, IEEE Trans. Autom. Control..

[28]  Eduardo Sontag,et al.  Building a cell cycle oscillator: hysteresis and bistability in the activation of Cdc2 , 2003, Nature Cell Biology.

[29]  M. Hirsch Systems of Differential Equations that are Competitive or Cooperative II: Convergence Almost Everywhere , 1985 .

[30]  J. Smillie Competitive and Cooperative Tridiagonal Systems of Differential Equations , 1984 .

[31]  Eduardo Sontag,et al.  Untangling the wires: A strategy to trace functional interactions in signaling and gene networks , 2002, Proceedings of the National Academy of Sciences of the United States of America.

[32]  Chris Cosner,et al.  Book Review: Monotone dynamical systems: An introduction to the theory of competitive and cooperative systems , 1996 .

[33]  W. Burridge,et al.  “Excitability” , 1933 .