The infinite dimensional partial delay differential equation is set forth and delay difference state feedback control is considered to describe the cell cycle growth in eukaryotic cell cycles. Hopf bifurcation occurs as varying free parameters and time delay continuously and the multi-layer oscillation phenomena of the homogeneous steady state of a simple gene-protein network module is investigated. Normal form is derived based on normal formal analysis technique combined with center manifold theory, which is further to compute the bifurcating direction and the stability of bifurcation periodical solutions underlying Hopf bifurcation. Finally, the numerical simulation oscillation phenomena is in coincidence with the theoretical analysis results.
References
[1]
Murray, A. and Hunt, T. (1993) The Cell Cycle. Oxford University Press, Oxford.
[2]
Tyson, J.J., Novak, B., Odell, G.M., et al. (1996) Chemical Kinetic Theory: Understanding Cell-Cycle Regulation. Trends in Biochemical Sciences, 21, 89-96. https://doi.org/10.1016/S0968-0004(96)10011-6
[3]
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
[4]
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
[5]
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
[6]
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
[7]
Dubitzky, W. and Kurth, M.J. (2005) Mathematical Models of Cell Cycle Regulation. Briefings in Bioinformatics, 6, 163-177. https://doi.org/10.1093/bib/6.2.163
[8]
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
[9]
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
[10]
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
[11]
Smith, H. (2011) An Introduction to Delay Differential Equations with Applications to the Life Sciences. Springer, New York.
[12]
Kuang, K. (1992) Delay Differential Equations with Application in Population Dynamics. Springer, New York.
[13]
Hale, J. (2003) Theory of Functional Differential Equations. World Publishing Corporation, Beijing.
[14]
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
[15]
Ma, S.Q., Feng, Z.S. and Lu, Q.S. (2009) Dynamics and Double Hopf Bifurtions of the Rose-Hindmarsh Model with Time Delay. International Journal of Bifurcation and Chaos, 19, 3733-3751. https://doi.org/10.1142/S0218127409025080
[16]
Ma, S.Q., Feng, Z.S. and Lu, Q.S. (2008) A Two Parameter Criteria for Delay Differential Equations. Discrete & Continuous Dynamical Systems-B, 9, 397-413. https://doi.org/10.3934/dcdsb.2008.9.397
[17]
Xu, J. and Chung, K.W. (2003) Effects of Time Delayed Position Feedback on a Van der Pol-Duffing Oscillator. Physica D: Nonlinear Phenomena, 180, 17-39. https://doi.org/10.1016/S0167-2789(03)00049-6
[18]
Wang, Z., Hu, H.Y., Xu, Q. and Stepan, G. (2016) Effect of Delay Combinations on Stability and Hopf Bifurcation of an Oscillator with Acceleration-Derivative Feedback. International Journal of Nonlinear Mechanics, 94, 392-399. https://doi.org/10.1016/j.ijnonlinmec.2016.10.008
[19]
Teresa, F. (2000) Normal Forms and Hopf Bifurcation for Partial Diffeential Equations with Delays. Transactions of the American Mathematical Society, 352, 2217-2238. https://doi.org/10.1090/S0002-9947-00-02280-7
