Measuring the Local Velocity along Transition Paths during the Folding of Single Biological Molecules

Transition paths are the most interesting part of folding reactions but remain little studied. We measured the local velocity along transition paths in DNA hairpin folding using optical tweezers. The velocity distribution agreed well with diffusive theories, yielding the diffusion coefficient. We used the average velocity to calculate the transmission factor in transition-state theory (TST), finding observed rates that were $\sim {10}^5$-fold slower than predicted by TST. This work quantifies the importance of barrier recrossing events and highlights the effectiveness of the diffusive model of folding.

Figure 1

Transition paths in folding reactions. A transition path (dashed line) represents the part of the trajectory (gray) that crosses the energy barrier (shaded region) separating the folded ($F$) and unfolded ($U$) states.

Figure 2

Measuring transition paths in DNA hairpins with optical tweezers. (a) The end-to-end extension of the hairpin held under tension fluctuates between the folded and unfolded states (dashed lines). Upper inset: Cartoon of measurement. A hairpin attached to DNA handles is held between two beads trapped by laser beams. Right inset: Probability distribution of hairpin extension. (b) Zooming in on a single unfolding transition, the transition path is seen as that part of the trajectory crossing between the boundaries defining the barrier region (dotted lines). Inset: The velocity along the transition path is found by differentiating the smoothed transition path. (c) Selected unfolding transitions and corresponding velocity profiles showing a wide range of velocities. The trajectory sometimes reverses course along the transition path, reflecting barrier recrossing.

Figure 3

Transition-path velocity distributions. (a) The probability density for the velocity as a function of position within the barrier region for the hairpin 30R50/T4. Inset: sequence of the hairpin 30R50/T4. (b) The distribution of velocities measured at all positions in the barrier region for the hairpin 30R50/T4 is close to Gaussian. Inset: Fitting the distribution of velocities observed at the barrier peak to a Gaussian yields $D$. (c) The average velocity profile within the barrier region is the same for folding (black) as for unfolding (red) but generally nonuniform. Cyan: barrier region boundaries ($x_1$, $x_2$). Blue: position of barrier peak (double dagger symbol). Insets: hairpin sequences. Error bars represent standard error of the mean.