  The quasi-equilibrium approximation is acceptable when molecular interactions
are fast enough compared to circuit dynamics, but is no longer allowed when
cellular activities are governed by rare events. A typical example is the
lactose operon (lac), one of the most famous paradigms of transcription
regulation, for which several theories still coexist to describe its behaviors.
The lac system is generally analyzed by using equilibrium constants,
contradicting single-event hypotheses long suggested by Novick and Weiner
(1957). Enzyme induction as an all-or-none phenomenon. Proc. Natl. Acad. Sci.
USA 43, 553-566) and recently refined in the study of (Choi et al., 2008. A
stochastic single-molecule event triggers phenotype switching of a bacterial
cell. Science 322, 442-446). In the present report, a lac repressor
(LacI)-mediated DNA immunoprecipitation experiment reveals that the natural
LacI-lac DNA complex built in vivo is extremely tight and long-lived compared
to the time scale of lac expression dynamics, which could functionally
disconnect the abortive expression bursts and forbid using the standard modes
of lac bistability. As alternatives, purely kinetic mechanisms are examined for
their capacity to restrict induction through: (i) widely scattered derepression
related to the arrival time variance of a predominantly backward asymmetric
random walk and (ii) an induction threshold arising in a single window of
derepression without recourse to nonlinear multimeric binding and Hill
functions. Considering the complete disengagement of the lac repressor from the
lac promoter as the probabilistic consequence of a transient stepwise
mechanism, is sufficient to explain the sigmoidal lac responses as functions of
time and of inducer concentration. This sigmoidal shape can be misleadingly
interpreted as a phenomenon of equilibrium cooperativity classically used to
explain bistability, but which has been reported to be weak in this system.
