Jae Kyoung Kim¹ & Daniel B Forger^1,2

^1.Department of Mathematics, University of Michigan, Ann Arbor, MI, USA
^2.Center for Computational Medicine and Bioinformatics, University of Michigan, Ann Arbor, MI, USA

Correspondence to: Daniel B Forger1,2 Department of Mathematics, University of Michigan, 2074 East Hall, 525 East University, Ann Arbor, MI 48109, USA.

Abstract
Circadian (~24h) timekeeping is essential for the lives of many organisms. To understand the biochemical mechanisms of this timekeeping, we have developed a detailed mathematical model of the mammalian circadian clock. Our model can accurately predict diverse experimental data including the phenotypes of mutations or knockdown of clock genes as well as the time courses and relative expression of clock transcripts and proteins. Using this model, we show how a universal motif of circadian timekeeping, where repressors tightly bind activators rather than directly binding to DNA, can generate oscillations when activators and repressors are in stoichiometric balance. Furthermore, we find that an additional slow negative feedback loop preserves this stoichiometric balance and maintains timekeeping with a fixed period. The role of this mechanism in generating robust rhythms is validated by analysis of a simple and general model and a previous model of the Drosophila circadian clock. We propose a double-negative feedback loop design for biological clocks whose period needs to be tightly regulated even with large changes in gene dosage.

Keywords: biological clocks; circadian rhythms; gene regulatory networks; mathematical model; robustness