Promoter methylation in a mixed feedback loop circadian clock model

We investigate how epigenetic modifications to clock gene promoters affect transcriptomic activity in the circadian clock. Motivated by experimental observations that link DNA methylation with the behavior of the clock, we introduce and analyze an extension of the mixed feedback loop (MFL) model of François and Hakim. We extend the original model to include an additional methylated promoter state and allow for reversible protein sequestration, an important feature for circadian applications. First, working with the general form of the MFL model, we find that the qualitative behavior of the model is dictated by the promoter state with the highest transcription rate. We then build on the work of Kim and Forger, who analyzed the stability of the mammalian circadian clock by using a reduced form of the MFL model. We present a rigorous procedure for translating between the MFL model and the reduction of Kim and Forger. We then propose a model reduction more appropriate for the study of oscillatory promoter states, making use of a fully coupled quasi-steady-state approximation rather than the standard partially uncoupled quasi-steady-state approach. Working with the novel reduced form of the model, we find substantial differences in the transcription function and show that, although methylation contributes to period control, excessive methylation can abolish rhythmicity. Altogether our results show that even in a minimal clock model, DNA methylation has a nontrivial influence on the system's ability to oscillate.