= Solution
Place a <FRET donor> and <FRET acceptor> on <DNA> positions whose separation reports start-site opening, using an independently calibrated closed/open geometry and a pair whose <Förster distance> is near the geometric mean from part (c). Opening may increase or decrease the measured separation depending on where the dyes are placed; anticorrelated donor/acceptor changes need structural and photophysical controls before being assigned to bubble formation.
Single molecules can be observed by low-background <fluorescence microscopy> while recording <FRET donor> and sensitized <FRET acceptor> <photons>. Correct background, spectral leakage, direct <FRET acceptor> excitation and unequal detection/quantum-yield factors before calculating efficiency, for example $E=I_A/(I_A+\gamma I_D)$ with the corrected intensities. Alternating excitation or independent <FRET acceptor> observation checks that a low-<FRET> interval is not simply a dark or bleached <FRET acceptor>. Intensity anticorrelation, <FRET donor> lifetime and suitable inactive/closed-state controls strengthen the assignment. The original figure below separates the mechanistic schematic from a purely illustrative model trace.
For the <binding-plus-opening first-passage time>, a minimal kinetic scheme is $U\rightleftarrows C\rightleftarrows O$, for unbound, closed bound and open bound states. A binding rate $a=k_{\rm on}[\mathrm{RNAP}]$, dissociation rate $b=k_{\rm off}$ and opening rate $c=k_{\rm open}$ already produce multiple waiting-time contributions. With O treated as absorbing during the measurement and starting from U, the mean first-opening time is
$$
\boxed{\langle t_{U\to O}\rangle=\frac{a+b+c}{ac},}
$$
not simply $1/k_{\rm open}$. This follows from $T_U=1/a+T_C$ and $T_C=1/(b+c)+bT_U/(b+c)$. A bound-state marker, or measurements over <concentration> and a justified kinetic model, helps distinguish binding from opening when unbound and closed <DNA> share a <FRET> level.
Fit transitions and dwell distributions with finite time resolution, missed short states, bleaching and censoring included. An open-state exponential dwell can estimate its exit rate under a simple Markov model, while competing closing and dissociation channels require separate observables or additional assumptions. Starting from a defined bound closed state can isolate opening more directly, but dissociation still competes if appreciable. Resolving repeated opening and closing gives forward and reverse kinetics; irreversibility should only be assumed over a demonstrated observation window. The experiment measures a conformational reporter, so a two-level trace alone does not uniquely prove every microscopic step of open-complex formation.
\Image[/past-exam-of-the-biological-physics-course-of-the-university-of-oxford/2018-transcription-and-fret.png]
{title=Transcription-complex organisation and a calibrated FRET reporter of DNA opening}
{description=Original schematic and FRET-law calculations. Distances of 4 and 7 nm and the state trace are illustrative, not source measurements; labels distinguish the template, non-template and nascent RNA.}
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