Global Entrainment of Transcriptional Systems to Periodic Inputs

This paper addresses the problem of providing mathematical conditions that allow one to ensure that biological networks, such as transcriptional systems, can be globally entrained to external periodic inputs. Despite appearing obvious at first, this is by no means a generic property of nonlinear dynamical systems. Through the use of contraction theory, a powerful tool from dynamical systems theory, it is shown that certain systems driven by external periodic signals have the property that all their solutions converge to a fixed limit cycle. General results are proved, and the properties are verified in the specific cases of models of transcriptional systems as well as constructs of interest in synthetic biology. A self-contained exposition of all needed results is given in the paper.

[1]  C. A. Desoer,et al.  Nonlinear Systems Analysis , 1978 .

[2]  Eduardo D. Sontag.  An observation regarding systems which converge to steady states for all constant inputs, yet become chaotic with periodic inputs , 2009 .

[3]  Kazuyuki Aihara,et al.  Stochastic synchronization of genetic oscillator networks , 2007, BMC Systems Biology.

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

[5]  T. Ström.  On Logarithmic Norms , 1975 .

[6]  S. Bernard,et al.  Spontaneous synchronization of coupled circadian oscillators. , 2005, Biophysical journal.

[7]  Mario di Bernardo,et al.  An algorithm for the construction of synthetic self synchronizing biological circuits , 2009, 2009 IEEE International Symposium on Circuits and Systems.

[8]  David Angeli,et al.  A Lyapunov approach to incremental stability properties , 2002, IEEE Trans. Autom. Control..

[9]  D. C. Lewis.  Metric Properties of Differential Equations , 1949 .

[10]  Tianshou Zhou,et al.  External Stimuli Mediate Collective Rhythms: Artificial Control Strategies , 2007, PloS one.

[11]  M. Vidyasagar,et al.  Nonlinear systems analysis (2nd ed.) , 1993 .

[12]  G. Dahlquist.  Stability and error bounds in the numerical integration of ordinary differential equations , 1961 .

[13]  Mario di Bernardo,et al.  An Algorithm to Prove Contraction, Consensus, and Network Synchronization , 2009 .

[14]  Eduardo Sontag,et al.  Modular cell biology: retroactivity and insulation , 2008, Molecular systems biology.

[15]  M. Elowitz,et al.  A synthetic oscillatory network of transcriptional regulators , 2000, Nature.

[16]  Nicolas Tabareau,et al.  A Contraction Theory Approach to Stochastic Incremental Stability , 2007, IEEE Transactions on Automatic Control.

[17]  Eduardo D. Sontag,et al.  Deterministic Finite Dimensional Systems , 1988 .

[18]  Jerome T. Mettetal,et al.  The Frequency Dependence of Osmo-Adaptation in Saccharomyces cerevisiae , 2008, Science.

[19]  P. Hartman.  On Stability in the Large for Systems of Ordinary Differential Equations , 1961, Canadian Journal of Mathematics.

[20]  A. Michel,et al.  Stability of Dynamical Systems — Continuous , Discontinuous , and Discrete Systems , 2008 .

[21]  W. Lohmiller,et al.  Contraction analysis of non-linear distributed systems , 2005 .

[22]  J. W. Schmidt.  G. Dahlquist, Stability and Error Bounds in the Numerical Integration of Ordinary Differential Equations. 85 S. Stockholm 1959. K. Tekniska Högskolans Handlingar , 1961 .

[23]  Jean-Jacques E. Slotine,et al.  Methodological remarks on contraction theory , 2004, 2004 43rd IEEE Conference on Decision and Control (CDC) (IEEE Cat. No.04CH37601).

[24]  Jean-Jacques E. Slotine,et al.  On Contraction Analysis for Non-linear Systems , 1998, Autom..

[25]  Jean-Jacques E. Slotine,et al.  On partial contraction analysis for coupled nonlinear oscillators , 2004, Biological Cybernetics.

[26]  J. A. Kuznecov.  Elements of applied bifurcation theory , 1998 .

[27]  Mario di Bernardo,et al.  How to Synchronize Biological Clocks , 2009, J. Comput. Biol..

[28]  Eduardo Sontag,et al.  Forward Completeness, Unboundedness Observability, and their Lyapunov Characterizations , 1999 .

[29]  J. Tyson,et al.  The dynamics of cell cycle regulation. , 2002, BioEssays : news and reviews in molecular, cellular and developmental biology.

[30]  M. Elowitz,et al.  Modeling a synthetic multicellular clock: repressilators coupled by quorum sensing. , 2004, Proceedings of the National Academy of Sciences of the United States of America.

[31]  K. Deimling.  Fixed Point Theory , 2008 .

[32]  Jean-Jacques E. Slotine,et al.  Nonlinear process control using contraction theory , 2000 .

[33]  Alessandro Astolfi,et al.  Stability of Dynamical Systems - Continuous, Discontinuous, and Discrete Systems (by Michel, A.N. et al.; 2008) [Bookshelf] , 2007, IEEE Control Systems.

[34]  Jean-Jacques E. Slotine,et al.  Modular stability tools for distributed computation and control , 2003 .

[35]  Kazuyuki Aihara,et al.  Synchronizing a multicellular system by external input: an artificial control strategy , 2006, Bioinform..

[36]  V. Arnold.  Mathematical Methods of Classical Mechanics , 1974 .

[37]  Nathan van de Wouw,et al.  Convergent dynamics, a tribute to Boris Pavlovich Demidovich , 2004, Syst. Control. Lett..