  Single-cell experiments show that gene expression is stochastic and bursty, a
feature that can emerge from slow switching between promoter states with
different activities. One source of long-lived promoter states is the slow
binding and unbinding kinetics of transcription factors to promoters, i.e. the
non-adiabatic binding regime. Here, we introduce a simple analytical framework,
known as a piecewise deterministic Markov process (PDMP), that accurately
describes the stochastic dynamics of gene expression in the non-adiabatic
regime. We illustrate the utility of the PDMP on a non-trivial dynamical system
by analyzing the properties of a titration-based oscillator in the
non-adiabatic limit. We first show how to transform the underlying Chemical
Master Equation into a PDMP where the slow transitions between promoter states
are stochastic, but whose rates depend upon the faster deterministic dynamics
of the transcription factors regulated by these promoters. We show that the
PDMP accurately describes the observed periods of stochastic cycles in
activator and repressor-based titration oscillators. We then generalize our
PDMP analysis to more complicated versions of titration-based oscillators to
explain how multiple binding sites lengthen the period and improve coherence.
Last, we show how noise-induced oscillation previously observed in a
titration-based oscillator arises from non-adiabatic and discrete binding
events at the promoter site.
