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Petrov, Valko; Peifer, Martin; Timmer, Jens

Bistability and self-oscillations in cell cycle control. (English) Zbl 1093.92035

Int. J. Bifurcation Chaos Appl. Sci. Eng. 16, No. 4, 1057-1066 (2006).
Summary: A qualitative model of cell cycle control is presented and its transition from bistability to limit cycle oscillations and vice versa is discussed. The origin of this model is the two-dimensional system of kinetic equations introduced by B. Novak et al. [Chaos 11, No. 1, 277–286 (2001; Zbl 0992.92022)] which is illustrated computationally and analytically. For this purpose a qualitative model is numerically reconstructed from the steady state behavior of the dynamical variables including the bifurcation parameter. Then, the reconstructed cubic polynomial model is generalized to an appropriate canonical form and is analyzed in terms of Lyapunov values. On this basis, the relationship between bistability and self-oscillatory behavior of mitotic cell cycle is approached qualitatively.

MSC:

92C37 Cell biology
34C60 Qualitative investigation and simulation of ordinary differential equation models
34D20 Stability of solutions to ordinary differential equations
93C95 Application models in control theory

Keywords:

stability analysis; Lyapunov values; cell cycle

Citations:

Zbl 0992.92022
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\textit{V. Petrov} et al., Int. J. Bifurcation Chaos Appl. Sci. Eng. 16, No. 4, 1057--1066 (2006; Zbl 1093.92035)
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References:

[1] Andronov A., Theory of Oscillators (1966)
[2] Bautin N., Dynamical System Behaviour near Boundaries of Stability Regions (1984)
[3] Fall C., Computational Cell Biology (2002) · Zbl 1010.92019
[4] DOI: 10.1085/jgp.43.5.867
[5] DOI: 10.1016/S0006-3495(61)86902-6
[6] Georgiev N., l’Academie Bulgare des Sciences 56 pp 25–
[7] Georgiev N., J. Appl. Math. 8 pp 397–
[8] DOI: 10.1098/rspb.1984.0024
[9] Keener J., Mathematical Physiology (1998) · Zbl 0913.92009
[10] DOI: 10.1073/pnas.0305937101
[11] Murray J., Mathematical Biology (1990) · Zbl 0704.92001
[12] Murray A., Sci. Amer. pp 56–
[13] Nagumo J., IRE pp 2061–
[14] DOI: 10.1007/978-94-011-2596-3
[15] DOI: 10.1142/S0218127404009065 · Zbl 1099.37505
[16] DOI: 10.1063/1.1681974
[17] DOI: 10.1063/1.1345725 · Zbl 0992.92022
[18] Petrov V., J. Theor. Appl. Mech. 32 pp 13–
[19] Petrov V., J. Theor. Appl. Mech. 32 pp 101–
[20] DOI: 10.1142/S0218127403008715 · Zbl 1057.37072
[21] Petrov V., J. Theor. Appl. Mech. 2
[22] DOI: 10.1016/S0006-3495(03)74778-X
[23] DOI: 10.1152/ajpcell.00066.2002
[24] DOI: 10.1038/35103078
[25] DOI: 10.1002/bies.10191
[26] DOI: 10.1016/S0955-0674(03)00017-6
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.