= Solution
<Magnetic tweezers> apply <force> and controlled turns to a single torsionally constrained <DNA> tether joining a surface and a magnetic bead. A magnetic-field gradient pulls the bead; rotation of the field rotates the bead and changes imposed turns. Imaging measures bead height, hence <DNA> extension. At sufficient excess twisting and suitable <force>, <DNA supercoiling> creates writhed loops that shorten the tether. The extension-versus-turn curve supplies a calibration relating the removal or addition of supercoiling to vertical motion. Multiple attachments at both ends are needed conceptually to constrain rotation; freely swivelling <DNA> would relax the signal.
During <transcription initiation>, <RNA polymerase> binds a <promoter> in a closed complex and then opens a local <DNA> bubble, exposing template for <RNA> synthesis. In bacterial initiation a <sigma factor> helps recognize the <promoter>. Opening changes local helical twist. Because the total <linking number> is constrained,
$$
Lk=Tw+Wr,\qquad \Delta Tw_{\rm rest}+\Delta Wr=-\Delta Tw_{\rm bubble}.
$$
Unwinding the bubble means $\Delta Tw_{\rm bubble}<0$, so the remainder acquires a positive turn increment. A negatively supercoiled tether therefore loses some negative writhe and becomes longer; a positively supercoiled tether gains positive writhe and becomes shorter. Ordinary <DNA> compaction instead shortens both tethers. Comparing both signs separates local unwinding from compaction rather than assigning every height change to opening.
If $q$ is the number of turns unwound, $s$ the magnitude of the extension change per imposed turn, and $c$ the compaction, the symmetric-slope approximation predicts $\Delta z_-=sq-c$ and $\Delta z_+=-sq-c$. Hence
$$
\boxed{q=\frac{\Delta z_--\Delta z_+}{2s},\qquad
c=-\frac{\Delta z_-+\Delta z_+}{2}.}
$$
Measured unequal slopes can be retained in the two separate calibration equations. Multiplying $q$ by the helical repeat converts turns to an estimated number of unwound <base pairs>, subject to the mechanical model and the distinction between twist and writhe.
Repeated extension transitions reveal open-complex lifetimes, closed-complex lifetimes and their dependence on mechanical <torque>. Initial short <RNA> synthesis can occur while <RNA polymerase> remains associated with the <promoter>, including abortive initiation and <DNA> scrunching. Addition of nucleotides allows progression to <promoter escape>; an unwound bubble can then persist with elongation, so loss of the bubble is not the definition of escape. Successive cumulative unwinding events can identify renewed access to the same <promoter> after clearance. These are the measurement principles demonstrated by the paired-sign experiments.<Promoter unwinding and promoter clearance by RNA polymerase: Detection by single-molecule DNA nanomanipulation by Andrey Revyakin and others>[Figures 1–4: extension calibration, opposite-sign opening responses and promoter-clearance readout]{ref}
The experiment observes mechanics rather than directly imaging individual opened <base pairs>. Drift, nonspecific binding, tether integrity, <force>-dependent buckling and limited time resolution can mimic or obscure steps. A control lacking productive <promoter> recognition and agreement of the opposite-sign responses help identify the origin of a signal.
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