### Abstract

Complex networks of interacting proteins control the physiological properties of a cell (metabolism, reproduction, motility, signaling, etc.). Intuitive reasoning about these networks is often sufficient to guide the next experiment, and a cartoon drawing of a network can be useful in codifying the results of hundreds of observations. But what tools are available for understanding the rich dynamical repertoire of such control systems? Why does a control system behave the way it does? What other behaviors are possible? How do these behaviors depend on the genetic and biochemical parameters of the system (gene dosage, enzymatic rate constants, equilibrium binding constants, etc)? Using basic principles of biochemical kinetics, we convert network diagrams into sets of ordinary differential equations and then explore their solutions by analytical and computational methods. We illustrate this approach with a mathematical model of cell cycle transitions in eukaryotes, based on a molecular network controlling the activity of cyclin-dependent kinase (Cdk). In this model, arrest points in the cell cycle correspond to stable steady states of the control system. As biochemical parameters of the control system change, these arrest points are imposed or lifted by transitions called “bifurcations.” During normal growth and division, cell size is the critical parameter that drives progression from G1 to S/G2 to M phase and back to G1. Simple diagrams, which correlate Cdk activity with cell growth, give a new way of thinking about cell cycle control, particularly the role of checkpoint pathways in arresting the cycle. The method is generally applicable to any complex gene-protein network that regulates some behavior of a living cell.

### Related Links

- http://mpf.biol.vt.edu/research/budding_yeast_model/pp/ - Model of the cell cycle in budding yeast
- http://jigcell.biol.vt.edu/generic_model/GenericUC.html - Generic model of cell cycle in eukaryotes
- http://cellcycle.mkt.bme.hu/ - Molecular network dynamics