  The inherent probabilistic nature of the biochemical reactions, and low copy
number of species can lead to stochasticity in gene expression across identical
cells. As a result, after induction of gene expression, the time at which a
specific protein count is reached is stochastic as well. Therefore events
taking place at a critical protein level will see stochasticity in their
timing. First-passage time (FPT), the time at which a stochastic process hits a
critical threshold, provides a framework to model such events. Here, we
investigate stochasticity in FPT. Particularly, we consider events for which
controlling stochasticity is advantageous. As a possible regulatory mechanism,
we also investigate effect of auto-regulation, where the transcription rate of
gene depends on protein count, on stochasticity of FPT. Specifically, we
investigate for an optimal auto-regulation which minimizes stochasticity in
FPT, given fixed mean FPT and threshold.
  For this purpose, we model the gene expression at a single cell level. We
find analytic formulas for statistical moments of the FPT in terms of model
parameters. Moreover, we examine the gene expression model with
auto-regulation. Interestingly, our results show that the stochasticity in FPT,
for a fixed mean, is minimized when the transcription rate is independent of
protein count. Further, we discuss the results in context of lysis time of an
\textit{E. coli} cell infected by a $\lambda$ phage virus. An optimal lysis
time provides evolutionary advantage to the $\lambda$ phage, suggesting a
possible regulation to minimize its stochasticity. Our results indicate that
there is no auto-regulation of the protein responsible for lysis. Moreover,
congruent to experimental evidences, our analysis predicts that the expression
of the lysis protein should have a small burst size.
