  We introduce and analyze an extension of the mixed feedback loop model of
Fran\c{c}ois and Hakim. Our extension includes an additional promoter state and
allows for reversible protein sequestration, which was absent from the original
studies of the mixed feedback loop model. Motivated by experimental
observations that link DNA methylation with circadian gene expression, we use
our extended model to investigate the role of DNA methylation(s) in the
mammalian circadian clock. We extend the perturbation analysis of Fran\c{c}ois
and Hakim to determine how methylation affects the presence and the periodicity
of oscillations. We derive a modified Goodwin oscillator model as an
approximation to show that although methylation contributes to period control,
excessive methylation can abolish rhythmicity.
