  Epigenetic modifications of histones crucially affect the eukaryotic gene
activity, while the epigenetic histone state is largely determined by the
binding of specific factors such as the transcription factors (TFs) to DNA.
Here, the way how the TFs and the histone state are dynamically correlated is
not obvious when the TF synthesis is regulated by the histone state. This type
of feedback regulatory relations are ubiquitous in gene networks to determine
cell fate in differentiation and other cell transformations. To gain insights
into such dynamical feedback regulations, we theoretically analyze a model of
epigenetic gene switching by extending the Doi-Peliti operator formalism of
reaction kinetics to the problem of coupled molecular processes. The spin-1 and
spin-1/2 coherent state representations are introduced to describe stochastic
reactions of histones and binding/unbinding of TF in a unified way, which
provides a concise view of the effects of timescale difference among these
molecular processes; even in the case that binding/unbinding of TF to/from DNA
are adiabatically fast, the slow nonadiabatic histone dynamics give rise to a
distinct circular flow of the probability flux around basins in the landscape
of the gene state distribution, which leads to hysteresis in gene switching. In
contrast to the general belief that the change in the amount of TF precedes the
histone state change, the flux drives histones to be modified prior to the
change in the amount of TF in the self-regulating circuits. The flux-landscape
analyses shed light on the nonlinear nonadiabatic mechanism of epigenetic cell
fate decision making.
