American Institute of Mathematical Sciences

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2005, 2(3): 625-642. doi: 10.3934/mbe.2005.2.625

The Role Of Time Delays, Slow Processes And Chaos In Modulating The Cell-Cycle Clock

E.V. Presnov 1, and  Z. Agur 2,

1. 

Volcani Center, Gilat Experiment Station, Negev 4, 85280, Israel
2. 

Institute for Medical Biomathematics, PO Box 282, 10, Hate'ena St., Bene Ataroth, 60991, Israel

Received  January 2005 Revised  July 2005 Published  August 2005

The regulation of the cell cycle clock is examined using a theoretical model for the embryonic cell cycle, where the clock is described as a single-limit cycle [1]. By taking the coefficient of the autocatalytic reaction as proportional to the deviation of the system from its equilibrium state, we show how such clocks can be adjusted to function on several time scales. This feedback control, causing a periodic change in the sign of the autocatalytic reaction, may be interpreted as a periodic change in the ratio of cdc25/wee1 activity. Its introduction results in the appearance of a double limit cycle, signifying the acquisition of the G1 phase and the G2 phase, during embryonic development. Following the loss of stability of the double cycle, through a period-doubling bifurcation, another limit set—a strange attractor—is born. The complicated geometry of this strange attractor can be viewed as an unlimited reservoir of periods in the phase space.

We hypothesize that the existence of such a reservoir is advantageous in morphogenetic tissues, such as the bone marrow, as it enables time- and site-specific selection of the optimal cell-cycle period for any specific micro- environment. This can be obtained by the addition of a time delay in the autocatalytic reaction, reflecting, for example, the influence of external molecular signals on cell-cycle progression.
Keywords: autocatalysis., limit cycle, CDK, Hopf bifurcation, morphogenesis, cyclin.
Mathematics Subject Classification: 92D3.
Citation: E.V. Presnov, Z. Agur. The Role Of Time Delays, Slow Processes And Chaos In Modulating The Cell-Cycle Clock. Mathematical Biosciences & Engineering, 2005, 2 (3) : 625-642. doi: 10.3934/mbe.2005.2.625
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