  A dynamical mean-field theory is developed to analyze stochastic single-cell
dynamics of gene expression. By explicitly taking account of nonequilibrium and
nonadiabatic features of the DNA state fluctuation, two-time correlation
functions and response functions of single-cell dynamics are derived. The
method is applied to a self-regulating gene to predict a rich variety of
dynamical phenomena such as anomalous increase of relaxation time and
oscillatory decay of correlations. Effective "temperature" defined as the ratio
of the correlation to the response in the protein number is small when the DNA
state change is frequent, while it grows large when the DNA state change is
infrequent, indicating the strong enhancement of noise in the latter case.
