Using MatCont in a Two-Parameter Bifurcation Study of Models for Cell Cycle Controls

Abstract

A recent application field of bifurcation theory is in modelling the cell cycle. We refer in particular to the work of J.J. Tyson and B. Novak where the fundamental idea is that the cell cycle is an alternation between two stable steady states of a system of kinetic equations. We study and extend the basic model of Tyson and Novak using the Matlab numerical bifurcation software MatCont, in a two-parameter setting and highlight several new features. We show that the limit point curves in the two-variable model behave in an ungeneric way under variation of the natural parameters and that the hysteresis loop in the model is not the usual loop caused by the existence of a codimension-2 cusp point. We continue orbits homoclinic-to-saddle-node (HSN) in the three-variable model and find that these orbits die in a non-central orbit homoclinic-to-saddle-node under a natural parameter variation. As an extension we introduce a model in which cell division appears as a continuous-in-time limit cycle. We perform a continuation of this limit cycle under a natural parameter variation and show that it loses stability in a limit point of cycles bifurcation. Alternatively, we study the cell cycle as a boundary value problem as proposed by Tyson and Novak. This leads to an interpretation of the cell as a slow-fast system and we derive several conclusions on the relation between the growth rate of the cells and the cell size at division, and on the controllability of the process.