Nonsmooth dynamical systems: Application to a cell cycle model

Abstract

The cycle consisting of repeating growth(Intermediate stages G1, S and G2) and division (Mitosis) is an extremely complex process. Among the key regulators of the cell cycle are Cyclin-Dependent Kinase 1 (CDK1) and the Anaphase-Promoting Complex (APC). We consider a existing mathematical model of the cell cycle, which is based on the dynamics CDK1 and APC. This model considers the cell cycle as a cyclin driven process and is convenient to investigate anti-cancer treatments which block this compound. In this model the Hill function is smoothly switching on and off the terms in which it is involved. One may remark that any switching function can be used and that we can also consider the Heaviside function.

We show that by using this discontinuous switch function, it is possible to get a better insight into the possible values of the involved rates and to have a better understanding of the impact of the rate of cyclin synthesis on the cycle period.

The righthand side of the original model represents a continuous vector field. There is a well developed theory, which among other things includes the Poincare-Bendinxon theorem applied to mathematically prove the existence of a unique limit cycle of the model. When the Hill function is replaced by the Heaviside function, we have a discontinuous righthand side and the model is within the realm of Nonsmooth Dynamical Systems. Overall, the talk promotes the idea that modelling should be guided by biological insight and not be restricted by known or popular mathematical theory.