Saturation of Enzyme Kinetics in Circadian Clock Models

From the mathematical study of simple models for circadian rhythm, the authors identified a clear effect of saturation in the enzyme kinetics on the promotion or suppression of a sustained oscillation. In the models, a clock gene (pergene) is transcribed to produce mRNAs, which are translated to produce proteins in the cytosol, which are then transported to the nucleus and suppress the transcription of the gene. The negative feedback loop with a long time delay creates sustained oscillation. All the enzymatic reactions (e.g., degradation, translation, and modification) are assumed to be of Michaelis-Menten type. The reaction rate increases with the amount of substrate but saturates when it is very large. The authors prove mathematically that the saturation in any of the reactions included in the feedback loop (in-loop reaction steps) suppresses the oscillation, whereas the saturation of both degradation steps and the back transport of the protein to cytosol (branch reaction steps) makes the oscillation more likely to occur. In the experimental measurements of enzyme kinetics and in published circadian clock simulators, in-loop reaction steps have a small saturation index whereas branch reaction steps have a large saturation index.

[1]  Axel Hunding,et al.  Limit-cycles in enzyme-systems with nonlinear negative feedback , 1974, Biophysics of structure and mechanism.

[2]  M Morales,et al.  Biochemical oscillations in "controlled" systems. , 1967, Biophysical journal.

[3]  R. Allada,et al.  Biological clocks , 2000 .

[4]  A. Goldbeter,et al.  Biochemical Oscillations And Cellular Rhythms: Contents , 1996 .

[5]  Martin A. Lema,et al.  (Too Many) Mathematical Models of Circadian Clocks (?) , 2001 .

[6]  U. Schibler Circadian rhythms: New cogwheels in the clockworks , 1998, Nature.

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

[8]  P. Hardin,et al.  Interlocked feedback loops within the Drosophila circadian oscillator. , 1999, Science.

[9]  A. Goldbeter A model for circadian oscillations in the Drosophila period protein (PER) , 1995, Proceedings of the Royal Society of London. Series B: Biological Sciences.

[10]  J. Tyson,et al.  A proposal for temperature compensation of the circadian rhythm in Drosophila based on dimerization of the per protein. , 1997, Chronobiology international.

[11]  H. Gutfreund Enzymes: Physical Principles , 1972 .

[12]  J. Tyson On the existence of oscillatory solutions in negative feedback cellular control processes , 1975 .

[13]  A. Goldbeter,et al.  A Model for Circadian Rhythms in Drosophila Incorporating the Formation of a Complex between the PER and TIM Proteins , 1998, Journal of biological rhythms.

[14]  R J Konopka,et al.  Clock mutants of Drosophila melanogaster. , 1971, Proceedings of the National Academy of Sciences of the United States of America.

[15]  Jeffrey C. Hall,et al.  Feedback of the Drosophila period gene product on circadian cycling of its messenger RNA levels , 1990, Nature.

[16]  Yoh Iwasa,et al.  Comparative study of circadian clock models, in search of processes promoting oscillation. , 2002, Journal of theoretical biology.

[17]  A. Goldbeter,et al.  Limit Cycle Models for Circadian Rhythms Based on Transcriptional Regulation in Drosophila and Neurospora , 1999, Journal of biological rhythms.

[18]  A Goldbeter,et al.  A minimal cascade model for the mitotic oscillator involving cyclin and cdc2 kinase. , 1991, Proceedings of the National Academy of Sciences of the United States of America.

[19]  M. W. Young The molecular control of circadian behavioral rhythms and their entrainment in Drosophila. , 1998, Annual review of biochemistry.

[20]  C. Walter The occurrence and the significance of limit cycle behavior in controlled biochemical systems. , 1970, Journal of theoretical biology.

[21]  C. Johnson,et al.  Expression of a gene cluster kaiABC as a circadian feedback process in cyanobacteria. , 1998, Science.

[22]  H. Othmer The qualitative dynamics of a class of biochemical control circuits , 1976, Journal of mathematical biology.

[23]  E B Klerman,et al.  Commentary: Model Building, Quantitative Testing, and Model Comparison , 1999, Journal of biological rhythms.

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

[25]  M. Merrow,et al.  The circadian cycle: is the whole greater than the sum of its parts? , 2001, Trends in genetics : TIG.

[26]  J. Dunlap Molecular Bases for Circadian Clocks , 1999, Cell.

[27]  H. Kitano,et al.  Robust oscillations within the interlocked feedback model of Drosophila circadian rhythm. , 2001, Journal of theoretical biology.

[28]  Y. Tounitou Biological Clocks. Mechanisms and Applications. , 1998 .

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

[30]  J. Tyson,et al.  A simple model of circadian rhythms based on dimerization and proteolysis of PER and TIM. , 1999, Biophysical journal.

[31]  J. Dunlap,et al.  Interconnected feedback loops in the Neurospora circadian system. , 2000, Science.

[32]  Kenji Tomioka,et al.  timrit Lengthens Circadian Period in a Temperature-Dependent Manner through Suppression of PERIOD Protein Cycling and Nuclear Localization , 1999, Molecular and Cellular Biology.

[33]  D. Allen,et al.  A circadian enhancer mediates PER-dependent mRNA cycling in Drosophila melanogaster , 1997, Molecular and cellular biology.

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

[35]  P. Lakin-Thomas A beginnner's guide to limit cycles, their uses and abuses , 1995 .

[36]  C. Walter,et al.  The absolute stability of certain types of controlled biological systems. , 1969, Journal of theoretical biology.

[37]  A. Goldbeter,et al.  Modeling the molecular regulatory mechanism of circadian rhythms in Drosophila. , 2000, BioEssays : news and reviews in molecular, cellular and developmental biology.