We study simple models for circadian rhythm, and examine the condition in which the equilibrium is unstable, generating a sustained oscillation. In the models, a clock gene(s) is transcribed to produce mRNAs, which are translated to produce proteins that suppress the transcription of the clock gene(s). First, using a Lyapunov function, we prove under very general conditions that two-variable models cannot generate a stable oscillation, implying that additional structures are needed for the model to generate a sustainable rhythm. By comparing several models of different complexities using the Routh-Hurwitz criteria of stability, we show that a sustained oscillation is more likely to occur if the cell is compartmentalized and the proteins need to be transported from the cytosol to the nucleus, if the proteins have to be modified before entering the nucleus, if the kinetics of transcription inhibition or the transport to the nucleus have cooperativity with a nonlinear dependence on the substrate concentration, or if the products of two clock genes form a heterodimer that suppresses both of their own genes. We discuss the implications of these results. (C) 2002 Elsevier Science Ltd. All rights reserved.
