Transition Path Times for Nucleic Acid Folding Determined from Energy-Landscape Analysis of Single-Molecule Trajectories

The duration of structural transitions in biopolymers is only a fraction of the time spent searching diffusively over the configurational energy landscape. We found the transition time, ${\tau }_{\mathrm{TP}}$, and the diffusion constant, $D$, for DNA and RNA folding using energy landscapes obtained from single-molecule trajectories under tension in optical traps. DNA hairpins, RNA pseudoknots, and a riboswitch all had ${\tau }_{\mathrm{TP}}\sim 10\mu \mathrm{s}$ and $D\sim {10}^{-13–14}{\mathrm{m}}^2/\mathrm{s}$, despite widely differing unfolding rates. These results show how energy-landscape analysis can be harnessed to characterize brief but critical events during folding reactions.

Figure 1

Extension trajectories and energy-landscape analysis of DNA hairpins. (a) Schematic free-energy profile for a folding reaction. The folding rate given by Kramers theory ($k$) is set primarily by the length of time spent diffusing within the potential well. The transition path time required to cross the barrier (${\tau }_{\mathrm{TP}}$) is much shorter. ${\kappa }_b$: curvature of the barrier; ${\kappa }_w$: curvature of the well. (b) Single nucleic acid molecules attached to duplex handles are held under tension between beads in two traps. (c) The extension of hairpin 20TS06/T4 as a function of time at a constant force shows sudden changes as the hairpin folds and unfolds (red lines: data sampled at 50 kHz; black lines: filtered at 10 kHz). (d) One-ms records straddling the transitions (red lines) were aligned and averaged to reduce Brownian noise. The averages of 2529 unfolding (black line) and refolding (yellow line, time-reversed) transitions overlap with each other and with the instrument response signal from fast ($<1\mu \mathrm{s}$) motions of the traps (cyan line), indicating that the apparent transition time of $\sim 50\mu \mathrm{s}$ between the inflection points of the extension probability distributions (dashed blue lines) is instrument-limited. (e) The deconvolved landscape profile of hairpin 30R50/T4 (black line) allows the barrier and well curvatures to be measured (red curves: quadratic fits), as well as the barrier heights, thereby determining $D$ and ${\tau }_{\mathrm{TP}}$. (f) Examples of deconvolved landscape profiles for hairpins 20TS06/T4 (blue line), 20TS10/T4 (brown line), and 20TS18/T4 (red line). (g) Fitting the force-dependent unfolding rates for 30R50/T4 (black circles) and 20TS06/T4 (cyan circles) to Equation (S2) (red and brown lines, respectively) reveals that the hairpins have very different unfolding rates at zero force.

Figure 2

Unfolding force distributions for pseudoknots and riboswitches. $D$ and ${\tau }_{\mathrm{TP}}$ were determined from the key landscape parameters obtained by fitting the unfolding force distributions from FECs to Equation (S1). (a) MMTV (blue histograms, at lower force) and PEMV1 (grey histograms, at higher force) pseudoknots. (b) ScYLV (grey histograms, at higher force) and C27A mutant (blue histograms, at lower force) pseudoknots. (c) PT2G32 pseudoknot. (d) The add adenine riboswitch with (blue histograms, at higher force) and without (grey histograms, at lower force) a ligand bound.

Figure 3

Stem-length dependence of ${\tau }_{\mathrm{TP}}$. The transition time for unfolding the DNA hairpins (30- and 20-bp stems) and the riboswitch aptamer (the critical helix unfolded to reach the transition state has 9 bp) varies linearly with the length of the helix stem.