Cell Cycle Vignettes Analysis of Cell Cycle Dynamics by Bifurcation Theory

Bifurcation theory provides a classification of the expected ways in which the number and/or stability of invariant solutions (‘attractors’ and ‘repellors’) of nonlinear ordinary differential equations may change as parameter values are changed. The most common qualitative changes are ‘saddle-node’ bifurcations, ‘Hopf’ bifurcations, and ‘SNIPER’ bifurcations. At a saddle-node bifurcation, a pair of steady states, usually a stable node and an unstable saddle point, coalesce and disappear. At a Hopf bifurcation, a stable focus changes to an unstable focus and makes way for a small amplitude periodic solution (‘limit cycle’). At a SNIPER bifurcation, the coalescence of a saddle point and a stable node creates an infinite-period limit cycle solution. These bifurcations have clear physiological correlates in the regulation of DNA replication, mitosis and cell division. Saddle-node bifurcations are related to checkpoints in the cell cycle: the establishment and removal of checkpoints correspond to the creation and annihilation of stable steady states at saddle-node bifurcations. The repetitive nature of the cell cycle (G1-S-G2-MG1-etc.) is related to limit cycle solutions of the underlying kinetic equations: the ability to oscillate spontaneously arises at either a Hopf or a SNIPER bifurcation.