= Solution
Use a short inducer pulse to release <DNA>-bound fluorescent <LacI>, then remove extracellular <IPTG> rapidly and start the search clock after washout. The supplied entry, binding and dissociation times imply a synchronization uncertainty of order a few tenths of a second, which must be measured and separated from the search latency. If rebinding is faster than that dead time, the protocol alone cannot resolve the true mean without a kinetic correction or faster triggering.
Image individual cells rapidly enough to distinguish freely diffusing <proteins> from a persistent immobilized fluorescent spot at the single operator. Prelocate that locus from an independent position marker or a previous binding event. Score the first specific capture after each reset; controls lacking a functional target distinguish stable operator binding from nonspecific <DNA> residence. Correct for localization error, <photobleaching>[bleaching], blinking, missed brief events and target motion, and verify that the fluorescent tag does not substantially change the search kinetics.
With four independent freely searching <molecules>, each having an approximately exponential single-<molecule> target-arrival time of mean $\tau_1$, the first-arrival survival is
$$
S(t)=e^{-4t/\tau_1},\qquad\boxed{\tau_1=4\langle t_{\rm first}\rangle.}
$$
Thus report both the observed first capture among four and the inferred mean for one searcher. If the number of visible available <molecules> varies, use $S(t)=\exp[-\int_0^t n(s)ds/\tau_1]$ and account for censoring rather than blindly multiplying by four. Repeated resets and many cells estimate the distribution; deviations from an exponential test the independent well-mixed approximation. The rapid inducer manipulation defines a synchronization strategy, not an assumption that all subsequent intracellular search paths are identical.
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