Reading Nucleic-Acid Backbones with a Loop-Holonomy Instrument How much does a molecule move when nothing is wrong — and what that number rules out

ABSTRACT
A protein or nucleic acid in solution is not a shape. It is a cloud of shapes — every model in an NMR ensemble is a legitimate fit to the same data, and the molecule really does occupy all of them at room temperature. We measured how wide that cloud is, on three substrates, and the number turns out to decide what any geometric measurement on those substrates is allowed to claim.

Across 1,172 protein molecules and 24,864 conformers, the frame defect angle moves 35.81° when nothing is wrong. The difference between a knotted protein and an unknotted one, on that same channel, is 3.3°.  The feature is one eleventh of the cloud it sits in. Four separate experiments had already been run on that channel; all four were predictable failures, without labels, from a single unattended measurement that takes about a minute.

We report that floor and two more — 0.0640 for RNA |writhe|, 0.1886 for protein, ~0.18 for DNA duplex axes — and note that our registered expectation about which substrate would be noisiest was backwards. Alongside them: a matched positive control (the RydC RNA trefoil, reproduced on 4/4 seeds and confirmed by an independent tool, plus the new result that the knot survives all 24 plausible completions of its unmodelled gap, for a mechanistic reason); a DNA census of 250 curves returning zero knots; a citation trap that stops anyone following the literature; a gap artefact that injects 83 units of fabricated writhe at 460× the floor, large enough to invert a sign; two sign-inverting bugs in our own axis construction, both caught by a synthetic of known answer; a symmetry blind spot that is one line of algebra and the difference between AUC 0.98 and 0.51; and a factor-of-two error in our own linking-number code whose residual resolved to the Bishop frame’s own closure holonomy, agreeing with the
topological route to 1.8×10⁻¹³.

The instrument works. It works at a narrower thing than we hoped, and we can now say exactly how narrow, with a number.

Keywords: holonomy · parallel transport · geometric phase · Berry phase · Wilson loop · winding number · linking
number · Burgers vector · topological invariant · noise floor · loop walk · competence surface · nucleic acid topology ·
writhe · NMR ensemble