= Solution
In bacterial <transcription initiation>, the <RNA polymerase> holoenzyme uses a <promoter>-specific <sigma factor> to recognize <promoter> elements and form a closed <DNA>-bound complex. Subsequent local <DNA melting> opens a <transcription> bubble, positioning the template strand in the active site: this is an <open promoter complex>. The enzyme can then initiate <RNA> synthesis and escape into elongation. <LacI>, the lac repressor, binds the <lac operator> overlapping or close to the <promoter> and can prevent productive <promoter> binding or its progression to an open complex; operator arrangement and <DNA> looping can strengthen repression. One should not assume every repressor acts at exactly the same microscopic step.
A high-resolution <optical tweezers> assay mechanically connects one trapped bead to <RNA polymerase> and another bead to an upstream or downstream <DNA> handle. Polymerase motion changes the tether length; measuring bead displacement and the calibrated restoring <force> reports elongation under a known load. A feedback <force> clamp can keep that load approximately constant. Reversing the tether geometry permits assisting and opposing loads. The left panel below shows the mechanical arrangement rather than a copied apparatus image.
A base-pair advance is only about $0.34\,\mathrm{nm}$ along B-form <DNA>. Resolving that increment requires exceptional mechanical stability, suppression of drift and laser-pointing noise, calibration of trap and tether compliance, and sufficient temporal sampling to separate steps from thermal bead fluctuations. Position histograms, dwell detection and their spatial correlations can expose a repeated increment; one should also test <nucleotide>-dependent motion, pauses and backtracking. Stabilized dual-trap measurements combined with <transcription>-product analysis established approximately one base-pair advance per incorporated <nucleotide>.<Direct observation of base-pair stepping by RNA polymerase by Elio A. Abbondanzieri and others>[Single-molecule displacement measurement and independent gel analysis supporting one-base-pair stepping]{ref}
\Image[/past-exam-of-the-biological-physics-course-of-the-university-of-oxford/2019-transcription-and-search.png]
{title=Optical trapping of a transcription complex and the competing terms of a facilitated target search}
{description=Original mechanical schematic and dimensionless search-time model. The stepping scale is a physical scale, while the illustrated normalized search curve is not experimental data.}
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