The Goodwin model revisited: Hopf bifurcation, limit-cycle, and periodic entrainment

The three-variable Goodwin oscillator is a minimal model demonstrating the emergence of oscillations in simple biochemical feedback systems. As a prototypical oscillator, this model was extensively studied from a theoretical point of view and applied to various biological systems, including circadian clocks. Here, we reexamine this model, derive analytically the amplitude equation near the Hopf bifurcation and investigate the effect of a periodic modulation of the oscillator. In particular, we compare the entrainment performance when the free oscillator displays either self-sustained or damped oscillations. We discuss the results in the context of circadian oscillators.

[1]  K. Tomita,et al.  Chaotic response of a limit cycle , 1979 .

[2]  Y Sakaki,et al.  Resetting central and peripheral circadian oscillators in transgenic rats. , 2000, Science.

[3]  B. Goodwin.  Oscillatory behavior in enzymatic control processes. , 1965, Advances in enzyme regulation.

[4]  R. D. Bliss,et al.  Role of feedback inhibition in stabilizing the classical operon. , 1982, Journal of theoretical biology.

[5]  K. Tomita,et al.  Entrainment of a Limit Cycle by a Periodic External Excitation , 1977 .

[6]  U. Alon.  Network motifs: theory and experimental approaches , 2007, Nature Reviews Genetics.

[7]  J. Griffith.  Mathematics of cellular control processes. II. Positive feedback to one gene. , 1968, Journal of theoretical biology.

[8]  P. Ruoff,et al.  The Goodwin model: simulating the effect of light pulses on the circadian sporulation rhythm of Neurospora crassa. , 2001, Journal of theoretical biology.

[9]  J. Ross,et al.  A detailed study of a forced chemical oscillator: Arnol'd tongues and bifurcation sets , 1989 .

[10]  Hanspeter Herzel,et al.  How to Achieve Fast Entrainment? The Timescale to Synchronization , 2009, PloS one.

[11]  Carl Hirschie Johnson,et al.  Circadian gene expression in mammalian fibroblasts revealed by real-time luminescence reporting: Temperature compensation and damping , 2003, Proceedings of the National Academy of Sciences of the United States of America.

[12]  U. Alon,et al.  Entrainment of noise-induced and limit cycle oscillators under weak noise. , 2013, Chaos.

[13]  N. Macdonald,et al.  Birfurcation theory applied to a simple model of a biochemical oscillator. , 1977, Journal of theoretical biology.

[14]  John J. Tyson,et al.  The Dynamics of Feedback Control Circuits in Biochemical Pathways , 1978 .

[15]  J. Takahashi,et al.  Circadian Rhythm Generation and Entrainment in Astrocytes , 2005, The Journal of Neuroscience.

[16]  Denis Thieffry,et al.  Dynamical roles of biological regulatory circuits , 2007, Briefings Bioinform..

[17]  R. D. Bliss,et al.  Reconsideration of the theory of oscillatory repression. , 1981, Journal of theoretical biology.

[18]  D. J. Allwright,et al.  A global stability criterion for simple control loops , 1977 .

[19]  J. Griffith,et al.  Mathematics of cellular control processes. I. Negative feedback to one gene. , 1968, Journal of theoretical biology.

[20]  S. Strogatz.  Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry and Engineering , 1995 .

[21]  Yanyan Zhang,et al.  Periodically Forced Hopf Bifurcation , 2011, SIAM J. Appl. Dyn. Syst..

[22]  D. Lathrop.  Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering , 2015 .

[23]  M R Guevara,et al.  Bifurcation analysis of a periodically forced relaxation oscillator: differential model versus phase-resetting map. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[24]  J. Timmer,et al.  Signatures of nonlinearity in single cell noise-induced oscillations. , 2013, Journal of theoretical biology.

[25]  I. H. Segel.  Enzyme Kinetics: Behavior and Analysis of Rapid Equilibrium and Steady-State Enzyme Systems , 1975 .

[26]  Hanspeter Herzel,et al.  Quantification of Circadian Rhythms in Single Cells , 2009, PLoS Comput. Biol..

[27]  Johannes Müller,et al.  Modeling the Hes1 Oscillator , 2007, J. Comput. Biol..

[28]  Albert Goldbeter,et al.  Circadian rhythms and molecular noise. , 2006, Chaos.

[29]  J. Tyson,et al.  Design principles of biochemical oscillators , 2008, Nature Reviews Molecular Cell Biology.

[30]  D. Gonze,et al.  Strong feedback limit of the Goodwin circadian oscillator , 2013 .

[31]  P Ruoff,et al.  The Goodwin Oscillator: On the Importance of Degradation Reactions in the Circadian Clock , 1999, Journal of biological rhythms.

[32]  Bard Ermentrout,et al.  Simulating, analyzing, and animating dynamical systems - a guide to XPPAUT for researchers and students , 2002, Software, environments, tools.

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

[34]  K. Aihara,et al.  Synchronization of coupled nonidentical genetic oscillators , 2006, Physical biology.

[35]  Sandeep Krishna,et al.  Inducing phase‐locking and chaos in cellular oscillators by modulating the driving stimuli , 2011, FEBS letters.

[36]  Grégoire Nicolis,et al.  Introduction to Nonlinear Science: References , 1995 .

[37]  S. Invernizzi,et al.  Quantitative analysis of the Hopf bifurcation in the Goodwin n-dimensional metabolic control system , 1991, Journal of mathematical biology.

[38]  Parlitz,et al.  Period-doubling cascades and devil's staircases of the driven van der Pol oscillator. , 1987, Physical review. A, General physics.

[39]  K. Sneppen,et al.  Minimal model of spiky oscillations in NF-κB signaling , 2006 .

[40]  Aronson,et al.  Entrainment regions for periodically forced oscillators. , 1986, Physical review. A, General physics.

[41]  Peter Ruoff,et al.  The relationship between FRQ-protein stability and temperature compensation in the Neurospora circadian clock. , 2005, Proceedings of the National Academy of Sciences of the United States of America.

[42]  Didier Gonze,et al.  The Goodwin Model: Behind the Hill Function , 2013, PloS one.

[43]  B. Goodwin.  Temporal Organization in Cells; a Dynamic Theory of Cellular Control Processes , 2015 .

[44]  B O Palsson,et al.  Mathematical modelling of dynamics and control in metabolic networks: VI. Dynamic bifurcations in single biochemical control loops. , 1985, Journal of theoretical biology.

[45]  J. Tyson,et al.  Periodic enzyme synthesis: reconsideration of the theory of oscillatory repression. , 1979, Journal of theoretical biology.

[46]  M. Lefranc,et al.  Robust entrainment of circadian oscillators requires specific phase response curves. , 2010, Biophysical journal.

[47]  Alexis B. Webb,et al.  Intrinsic, nondeterministic circadian rhythm generation in identified mammalian neurons , 2009, Proceedings of the National Academy of Sciences.

[48]  Achim Kramer,et al.  Synchronization-Induced Rhythmicity of Circadian Oscillators in the Suprachiasmatic Nucleus , 2007, PLoS Comput. Biol..

[49]  Peter Ruoff,et al.  Modeling temperature compensation in chemical and biological oscillators. , 1997, Chronobiology international.

[50]  A. Keller,et al.  Model genetic circuits encoding autoregulatory transcription factors. , 1995, Journal of theoretical biology.

[51]  Albert Goldbeter,et al.  Dependence of the period on the rate of protein degradation in minimal models for circadian oscillations , 2009, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[52]  K. Sneppen,et al.  Oscillations and temporal signalling in cells , 2007, Physical biology.

[53]  R. Toral,et al.  Synchronization and entrainment of coupled circadian oscillators , 2010, Interface Focus.

[54]  L. Glass.  Synchronization and rhythmic processes in physiology , 2001, Nature.

[55]  Stefan Dipl.-Ing. Schuster,et al.  Temperature compensation and entrainment in circadian rhythms , 2012, Physical biology.

[56]  Glass,et al.  Periodic forcing of a limit-cycle oscillator: Fixed points, Arnold tongues, and the global organization of bifurcations. , 1994, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.