Abstract
A minimal model describing the embryonic cell division cycle at the molecular level in eukaryotes is analyzed mathematically. It is known from numerical simulations that the corresponding three-dimensional system of ODEs has periodic solutions in certain parameter regimes. We prove the existence of a stable limit cycle and provide a detailed description on how the limit cycle is generated. The limit cycle corresponds to a relaxation oscillation of an auxiliary system, which is singularly perturbed and has the same orbits as the original model. The singular perturbation character of the auxiliary problem is caused by the occurrence of small Michaelis constants in the model. Essential pieces of the limit cycle of the auxiliary problem consist of segments of slow motion close to several branches of a two dimensional critical manifold which are connected by fast jumps. In addition, a new phenomenon of exchange of stability occurs at lines, where the branches of the two-dimensional critical manifold intersect. This novel type of relaxation oscillations is studied by combining standard results from geometric singular perturbation with several suitable blow-up transformations.
Similar content being viewed by others
References
Aguda BD, Friedman A (2008) Models of cellular regulation, Oxford Graduate Texts. Oxford University Press, Oxford
Alberts B, Johnson A, Lewis J, Morgan D, Raff M, Roberts K, Walter P (2014) Molecular Biology of the cell. Garland Science, New York
Battogtokh D, Tyson JJ (2004) Bifurcation analysis of a model of the budding yeast cell cycle. Chaos 14:653–661
Broer HW, Kaper TJ, Krupa M (2013) Geometric desingularization of a cusp singularity in slow-fast systems with applications to Zeeman’s examples. J Dyn Differ Equ 25:925–958
Chen KC, Csikasz-Nagy A, Gyorffy B, Val J, Novak B, Tyson JJ (2000) Kinetic analysis of a molecular model of the budding yeast cell cycle. Mol Biol Cell 11:369–391
Chicone C (2006) Ordinary differential equations with applications. Springer Science+Business Media Inc, New York
Dumortier F, Roussarie R (1996) Canard cycles and center manifolds. Mem Am Math Soc 577
Erneux T, Goldbeter A (2006) Rescue of the quasi-steady state approximation in a model for oscillations in an enzymatic cascade. SIAM J Appl Math 67:305–320
Fenichel N (1979) Geometric singular perturbation theory. J Differ Equ 31:53–98
Gérard C, Goldbeter A (2011) A skeleton model for the network of cyclin-dependent kinases driving the mammalian cell cycle. R Soc J Interface Focus 1:24–35
Gérard C, Goldbeter A (2012) The cell cycle is a limit cycle. Math Models Nat Phenom 7:126–166
Goldbeter A, Koshland DE Jr (1981) Ultrasensitivity in biochemical systems controlled by covalent modification. Interplay between zero-order and multistep effects. J Biol Chem 259:14441–14447
Goldbeter A (1991) A minimal cascade model for the mitotic oscillator involving cyclin and cdc2 kinase. Proc Natl Acad Sci USA 88:9107–9111
Goldbeter A (1996) Biochemical oscillations and cellular rhythms: the molecular bases of periodic and chaotic behaviour. Cambridge University Press, Cambridge
Grasman J (1987) Asymptotic methods for relaxation oscillations and applications. Springer, New York
Gucwa I, Szmolyan P (2009) Geometric singular perturbation analysis of an autocatalotor model. Discrete Contin Dyn Syst Ser S 2:783–806
Hek G (2010) Geometric singular perturbation theory in biological practice. J Math Biol 60:347–386
Hunt T (2001) Protein synthesis, proteolysis, and cell cycle transitions, Nobel lecture. http://www.nobelprize.org/nobel_prizes/medicine/laureates/2001/hunt-lecture
Izhikevich EM (2007) Dynamical systems in neuroscience: the geometry of excitability and bursting. The MIT Press, Massachusetts
Jones CKRT (1995) Geometric singular perturbation theory. Springer Lect Notes Math Berlin 1609:44–120
Keener JP, Sneyd J (1998) Mathematical physiology. Springer, New York
Kosiuk I (2012) Relaxation oscillations in slow-fast systems beyond the standard form, PhD thesis, University of Leipzig
Kosiuk I, Szmolyan P (2011) Scaling in singular perturbation problems: blowing up a relaxation oscillator. SIAM J Appl Dyn Syst 10:1307–1343
Krupa M, Szmolyan P (2001) Extending geometric singular perturbation theory to non-hyperbolic points-fold and canard points in two dimensions. SIAM J Math Anal 33:286–314
Krupa M, Szmolyan P (2001) Extending slow manifolds near transcritical and pitchfork singularities. Nonlinearity 14:1473–1491
Kuehn Ch (2015) Multiple time scale dynamical systems. Springer, Berlin. doi:10.1007/978-3-319-12316-5
Lebovitz NR, Schaar JR (1975) Exchange of stabilities in autonomous systems. Studies Appl Math 54:229–260
Mishchenko EF, Kh Rozov N (1980) Differential equations with small parameters and relaxation oscillations. Plenum Press, New York
Morgan DO (2007) The cell cycle: principles of control. New Science Press, Oxford University Press, Sinauer Associates/London, Corby, Sunderland
Novak B, Tyson JJ (1995) Quantitative analysis of a molecular model of mitotic control in fission yeast. J Theor Biol 173:283–305
Novak B, Tyson JJ (1997) Modeling the control of DNA replication in fission yeast. Proc Natl Acad Sci USA 94:9147–9152
Novak B, Csikasz-Nagy A, Gyorffy B, Chen K, Tyson JJ (1998) Mathematical model of the fission yeast cell cycle with checkpoint controls at the G1/S, G2/M and metaphase/anaphase transitions. Biophys Chem 72:185–200
Nurse P (2000) A long twentieth century of the cell cycle and beyond. Cell 100:71–78
Sveiczer A, Tyson JJ, Novak B (2004) Modelling the fission yeast cell cycle. Brief Funct Genomic Proteomic 2:298–307
Szmolyan P (1991) Transversal heteroclinic and homoclinic orbits in singular perturbation problems. J Differ Equ 92:252–281
Szmolyan P, Wechselberger M (2004) Relaxation oscillations in \(\mathbb{R}^3\). J Differ Equ 200:69–104
Tyson JJ, Chen K, Novak B (2003) Sniffers, buzzers, toggles and blinkers: dynamics of regulatory and signaling pathways in the cell
Tyson JJ (1991) Modeling the cell division cycle: cdc2 and cyclin interactions. Proc Natl Acad Sci USA 88:7328–7332
Tyson JJ, Chen K, Novak B (2001) Network dynamics and cell physiology. Nat Rev Mol Cell Biol 2:908–916
Acknowledgments
I. Kosiuk thanks the Max Planck Institute for Mathematics in the Sciences in Leipzig for providing a post-doctoral scholarship funding this research. I. Kosiuk and P. Szmolyan thank the Max Planck Institute for Mathematics in the Sciences in Leipzig and Technische Universität Wien for support and hospitality during several mutual visits. We are also grateful for financial support from the Vienna Science and Technology Fund (WWTF) through Project MA14-049.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Kosiuk, I., Szmolyan, P. Geometric analysis of the Goldbeter minimal model for the embryonic cell cycle. J. Math. Biol. 72, 1337–1368 (2016). https://doi.org/10.1007/s00285-015-0905-0
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00285-015-0905-0
Keywords
- Cell cycle
- Mitotic oscillator
- Enzyme kinetics
- Geometric singular perturbation theory
- Relaxation oscillations
- Blow-up method