Bifurcation Analysis of the Regulatory Modules of the Mammalian <i>M/G<sub>1</sub></i> Phase with Time Delay
- 1 Department of Mathematics, China Agricultural University, Beijing, China
Abstract
G 0 /G 1 “gaps” joint the S phase and M phase to form the cell cycle. The dynamics of enzyme reaction to drive the target protein production in M phase is analyzed mathematically. Time delay is introduced since the signal transmission need time in G 0 /G 1 “gaps” phase. Hopf bifurcation of DDEs model is analyzed by applying geometrical analytical method. The instability oscillating periodic solutions arise as subcritical Hopf bifurcation occurs. The Hysteresis phenomena of the limit cycle are also observed underlying the saddle-node bifurcation of the limit cycle. Due to stability switching, interestingly, the bifurcating periodical solution dies out near the vicinity of Hopf lines. By Lyapunov-Schmidt reduction scheme, the normal form is computed on the center manifold. Finally, it is verified that the theory analytical results are in coincidence with the numerical simulation.
- Murray, A. and Hunt, T. (1993) The Cell Cycle. Oxford University Press, Oxford.
- Novak, B. and Tyson, J. (1993) Numerical Analysis of a Comprehensive Model of M-Phase Control in Xenopus Oocyte Extracts and Intact Enbroys. Journal of Cell Science, 106, 1153-1168. https://doi.org/10.1242/jcs.106.4.1153
- Edelstein-Keshet, L. (1988) Mathematical Models in Biology. McGraW-Hill, Inc., New York.
- Ukai, H. and Ueda, H.R. (2010) Systems Biology of Mammalian Circadian Clocks. Annual Review of Physiology, 72, 579-603. https://doi.org/10.1146/annurev-physiol-073109-130051
- Qu,Z., MacLellan, W.R. and Weiss, J.N. (2003) Dynamics of the Cell Cycle: Checkpoints, Sizers, and Timers. Biophysical Journal, 85, 3600-3611. https://doi.org/10.1016/S0006-3495(03)74778-X
- Angeli, D., Ferrell, J.E. and Sontag, E.D. (2004) Detection of Multistability, Bifurcations, and Hysterisis in a Large Class of Biological Positive-Feedback Systems. Proceedings of the National Academy of Sciences of the United States of America, 101, 1822-1827. https://doi.org/10.1073/pnas.0308265100
- Dubitzky, W. and Kurth, M.J. (2005) Mathematical Models of Cell Cycle Regulation. Bioinformatics, 6, 163-177. https://doi.org/10.1093/bib/6.2.163
- Norel, R. and Agur, Z. (1991) A Model for the Adjustment of the Mitotic Clock by Cyclin and MPF Levels. Science, 251, 1076-1078. https://doi.org/10.1126/science.1825521
- Tyson, J. (1991) Modelling the Cell Division Cycle: Cdc2 and Cyclin Interactions. Proceedings of the National Academy of Sciences of the United States of America, 88, 7328-7332. https://doi.org/10.1073/pnas.88.16.7328
- Goldbeter, A. (1991)A Minimal Cascade Model for the Mitotic Oscillator Invovling Cyclin and Cdc2 Kinase. Proceedings of the National Academy of Sciences of the United States of America, 88, 9107-9111. https://doi.org/10.1073/pnas.88.20.9107
- Smith, H. (2011) An Introduction to Delay Differential Equations with Applications to the Life Sciences. Springer, New York.
- Kuang, K. (1992) Delay Differential Equations with Application in Population Dynamics. Springer, New York.
- Hale, J. (2003) Theory of Functional Differential Equations. World Publishing Corporation, Beijing.
- Ma, S.Q., Wang, X.H., Lei, J.H. and Feng, Z.S. (2010) Dynamics of the Delay Hematological Cell Model. International Journal of Biomathematics, 3, 105-125. https://doi.org/10.1142/S1793524510000829