Past exam of the biological physics course of the University of Oxford / 2021 / 6 / a / 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,Unwinding the bubble means $\Delta T{w}_{\mathrm{bubble}}<0$, 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 $\Delta z_{+}=-sq-c$. HenceMeasured 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.^[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.