* *

*Figure
1.** Dumbbell experimental geometry
with DNA hairpin. From Woodside,
et al., PNAS 2006.*

We used DNA "handles" to attach a hairpin to two small beads, obtaining a “dumbbell” which we could then manipulate with our dual-beam optical tweezers. As illustrated in Figure 1, pulling on the bead with the tweezers applies a force to the hairpin, eventually causing it to unzip. In addition to measuring the hairpin in a non-equilibrium force ramp, we can also observe the hairpin as a bi-stable system at equilibrium by holding it at constant force in our optical force-clamp, as described in Greenleaf, Woodside, Abbondanzieri and Block, PRL 2005. Figure 2 is a record of the molecular extension we measured between the trapped beads of a dumbbell, showing a hairpin “flickering” between the folded and unfolded states at constant force. This set-up allowed us to directly measure quantities such as the hairpin’s opening distance (to sub-nanometer precision) and the lifetimes of the two states. We could then calculate the energy difference between the folded and unfolded states, the height and position of energy barrier separating the states, and the unfolding rate at zero force.

*Figure
2.** DNA extension vs. time at constant force reveals
two-state behavior. From Woodside,
et al., PNAS 2006.*

To better understand our data, we developed a model for the folding energy landscape of a DNA hairpin that took into account the sequence-dependent energies of breaking individual DNA base-pairs (obtained from the literature) as well as the physics of our experimental set-up. The initial landscape is simply a running sum of the base-pair energies (Figure 3A). The horizontal axis is then transformed from number of base-pairs (sequentially) unzipped to end-to-end extension (a more natural reaction coordinate for this experiment), and the landscape is tilted to favor the unzipped state because the traps do work on the hairpin. Potentials are then added at the limits of high and low extension, and the landscape is smoothed due to the elasticity of the unzipped single-stranded DNA (Figure 3B).

*Figure
3.** Hairpin energy landscape model.
A) Computed free energy for sequential unzipping of base pairs in the
hairpin stem at zero force based on literature data.
B) The landscape from A) is transformed to a different coordinate system,
tilted in proportion to the work done by the traps, and augmented with potentials
for high and low extension to obtain the landscape in green.
This is smoothed due to the elasticity of unzipped single-stranded
DNA to obtain the model energy landscape (black).
The landscape is exponetiated to obtain the
probability distribution in blue, which can be further smoothed (red) for
comparison with experimental extension histograms. From
Woodside,
et al., PNAS 2006.*

Upon exploring 20 hairpin sequences having “pseudo-random”
stem sequences, and differing by stem length, loop length, and GC content, we
found remarkable agreement between our data and model predictions for all
quantities reflecting hairpin stability. Free energies of folding varied by two orders
of magnitude, and lifetimes at F_{1/2} (the force at which a hairpin
spends equal time in the folded and unfolded states) spanned six orders of
magnitude. Our most interesting finding
was that the energy barrier between the two states corresponded to a distance ~1-2
bp away from the loop (Figure 4).

*Figure
4.** Distances to the transition state (energy barrier)
from the folded *(Dx^{‡}_{f})* and unfolded *(Dx^{‡}_{u})* states. The data are consistent
with each hairpin having a transition state located ~1-2 bp from the loop. From **Woodside,
et al., PNAS 2006**.*

We then turned to hairpins in which we had engineered mismatches or blocks of AT and GC base-pairs, in order to ask how well our model worked for arbitrary stem sequences. The mismatched hairpins appeared to have an intermediate folding state in addition to the fully folded and unfolded states, and the location of this state corresponded to the hairpin fraying up to the mismatched bases. This was indeed quantitatively predicted by the model (Figure 5). Similarly, by moving a block of GC bases down an otherwise-AT-rich stem, we could move the energy barrier away from the loop, and by gradually enriching the lower half of the hairpin stem in GC base-pairs we could raise the height of the barrier, in both cases in agreement with the model.

*Figure
5.** A) Extension record of hairpin with a T:T mismatch 7 bp from the base of the stem. Fit of extension
histogram to two Gaussian peaks reveals a third Gaussian residual (fits in
red, residual in blue), indicating three states: unfolded (U), intermediate
(I), and folded (F). ( B) Blow-up of the extension record shows rapid,
5-nm fluctuations between F and I states. (C) Distance from F to U
(black), F to I (blue), and I to U (red) plotted versus the mismatch location.
The intermediate state location moves in concert with the mismatch, in good
agreement with model predictions (light-colored solid circles in gray, purple,
orange), which indicates that the intermediate state consists of a hairpin
folded up to the mismatch. From Woodside,
et al., Science 2006.*

As the model was now fully validated for predicting key
features of the energy landscape of any DNA hairpin, such as the relative
positions (in the energy vs. extension plane) of stable folding states and
transition states, we asked whether we could predict the *entire* energy landscape of the hairpin. To measure the full landscape, we recognized first
that energy and probability are related by an exponential Boltzmann factor, and
second that an extension histogram (Figure 2) is effectively a probability
distribution. A model extension
probability distribution can be obtained from a model energy landscape just by exponentiating it, but the resulting function is not
comparable with a measured histogram until it’s smoothed to account for
elasticity of the DNA handles, uncertainty in the bead positions, and noise in
the instrument. To work backward from a
measured histogram toward a landscape, it was necessary to remove these sources
of smoothing by deconvolution.
Furthermore, not just any histogram would do: we measured hairpins at
constant force over periods of up to 30 minutes, at the maximum possible data
rate, in order to obtain histograms well-populated at extensions between the
peaks (corresponding to the energy barrier), and worked to minimize
instrumental drift in order to prevent the histograms from becoming skewed over
time. The resulting experimentally-derived
energy landscapes agreed quite well with the model landscapes (Figure 6),
suggesting that the model is a powerful tool for characterizing the folding of
nucleic acid secondary structures.

*Figure
6.** A
potential landscape determined by deconvolution of the extension histogram
based on thousands of folding transitions. Experimental probability distributions
and associated free-energy landscapes (black) were deconvolved to remove the effects of blurring caused by thermal
motions of beads attached to the hairpin via elastic handles. Probability
distributions (left column) and landscapes (right column) recovered by Gaussian
deconvolution (red) have small residual errors (green) and agree well with
energy landscapes predicted by the model (blue). From Woodside,
et al., Science 2006.*