## Abstract

We measured the mean and variance of end-to-end length in short DNA fragments in solution and reported evidence of DNA stretching that is cooperative over more than two turns of the double helix. Becker and Everaers suggest that the structural fluctuations we observed arise from bending motions of the DNA, rather than stretching. We present three experimental tests of this bending-based explanation.

We recently reported on distance distributions between gold nanocrystals attached to the 3′-hydroxyl groups of short DNA duplexes (*1*, *2*). Our data exhibit two unexpected features: The variance in end-to-end distance grows rapidly with duplex length, and the increase occurs in a nonlinear fashion. We interpret the results to be a manifestation of a soft, correlated stretching motion in DNA. Based on coarse-grained simulations of DNA structural fluctuations, Becker and Everaers (*3*) offer an alternative explanation: that the observed variances arise from bending motions of the DNA that are amplified through the linker attachments to the nanocrystal particles.

As a qualitative interpretation of their simulation results, Becker and Everaers (*3*) suggest that the axial displacement of the nanocrystal centers from the ends of the DNA duplex acts like a lever arm to amplify bending fluctuations. In (*1*), we reported values for the bending-induced variance in end-to-end length computed according to the wormlike chain (WLC) model. When these values are increased by the axial amplification factor of Becker and Everaers [a factor of (*l *+ *axial _{0}*)

^{2}/

*l*

^{2}, where

*axial*is the axial offset of the nanocrystals from the duplex ends, fit to be ~24 Å], they still only account for a fraction of the observed variance: 1% and 19% respectively for the 10– and 35–base-pair (bp) duplexes.

_{0}The simulation results of Becker and Everaers (*3*) appear to be dominated by a different phenomenon: The radial offset of the nanocrystal centers from the helix axis acts as a lever arm to amplify the effect of duplex bending on the internanocrystal distance (Fig. 1A). In our experiments, the nanocrystals were attached to 3′-hydroxyl groups near the edge of the duplex cylinder (fit to be ~9 Å off of the axis; *D*, radial offset). Depending on its direction, a bend in the duplex could either bring the two radially offset nanocrystals closer together or move them farther apart. Thus, the radial-offset phenomenon broadens distributions symmetrically, overcoming one of the objections to a bending-based explanation of our data. This radial-offset amplification is maximal when the two nanocrystals are positioned on the same side of the DNA cylinder and close to zero when the nanocrystals are positioned on opposite sides of the cylinder. Thus, the radially amplified bending model predicts a sinusoidal oscillation in variance with duplex length [proportional to *D*^{2}(1 + cosφ)<α^{2}>; see Fig. 1A]. This oscillation matches the helix period of 10 bp, regardless of assumptions about the linkage between the nanocrystals and the DNA.

To fit their model to our published variance data, Becker and Everaers (*3*) invoke an alternate location for the nanocrystals: a larger radial offset (12.7 Å instead of 9 Å) and a different value for the azimuthal dihedral angle offset φ_{0} (0.91π instead of 1.34π). The larger radial offset increases the magnitude of the modeled radially amplified bending motion, and the alternate dihedral angle offset places all of the experimental measurements near points of inflection in the sinusoidal curves. All of the published data can be situated at points of inflection because the measurements were made on duplexes with lengths incremented in 5-bp intervals, corresponding to a half-turn of the DNA helix.

As a first test of the amplified bending model, we measured additional end-to-end distance distributions for DNA duplexes consisting of 13, 18, 22, and 24 bp (we also remeasured the 15-bp duplex as a reproducibility control) (fig. S1 and table S1). The new duplex lengths do not correspond to half-integral turns of the helix. Figure 1B shows the observed mean lengths for all of the end-labeled duplexes, the lengths predicted by Becker and Everaers, and the lengths predicted by the original nanocrystal position on a DNA cylinder. The chi-squared residual is four times larger for the predictions of Becker and Everaers (*3*) than it is for the original nanocrystal position. Figure 1C shows all of the observed variances, the variances predicted by the amplified bending model of Becker and Everaers, and the variances predicted by the cooperative stretching model [a quadratic curve fit to the data published in (*1*)]. The chi-squared residual to the new variance data points is larger by a factor of 1.9 for the amplified bending model than it is for the stretching model. The cylinder/stretching model better predicts both the length and variance of the new duplexes than does the amplified-bending model. However, neither model accounts for the measurements to within the experimental error, which suggests a dependence of DNA conformational fluctuations on sequence (*4*, *5*).

A second test that discriminates between a radially amplified bending model and a stretching model is to increase the radial offset (*D*) of the gold nanocrystals away from the axis of the DNA helix. The bending model predicts that the magnitude of the sinusoidal variance oscillations should increase as the square of the radial offset, whereas the stretching model predicts no dependence of variance on the radial offset. The radial offset of the gold nanocrystals reported in (*1*, *2*) is ~21 Å for the internally labeled duplexes and ~9 Å for the end-labeled duplexes. Thus, the amplified-bending model predicts an ~400% increase in the magnitude of the variance oscillations for the internally labeled duplexes relative to the end-labeled duplexes, and the stretching model predicts no difference. As illustrated in Fig. 1C, the observed variances for the internally labeled and end-labeled duplexes are almost identical.

The third and strongest test of bending-based models for the end-to-end distance variances involves systematically modulating the range of the bending motion. For all bending models, distance variance increases with the mean-squared bend angle (<α^{2}>). The mean-squared bend angle is inversely proportional to the DNA persistence length and can be changed by adjusting solution conditions. In Fig. 2A, we show end-to-end distance distributions for a 24-bp duplex measured at ionic strengths between 45 mM and 1045 mM. In this range of solution conditions, the persistence length of the DNA is expected to decrease from ~552Å to ~480 Å as the salt concentration increases (*6*). The decrease in persistence length corresponds to a 15% increase in mean-squared bend angle. If bending dominates the variation in measured distance, the observed variance should increase by ~15% over the salt series. Instead, we observe a general decrease in variance (Table 1). We also compared the end-to-end distance variance of a 15-bp duplex under our standard buffer conditions (persistence length of ~511 Å) and in a low-salt buffer containing 2.5 mM putrescine (Fig. 2B). Submillimolar putrescine concentrations reduce the persistence length of DNA to ~309 Å as measured by single-molecule stretching experiments (*7*). If bending dominates the variation in measured distance, the observed variance should increase by 65% in the putrescine-containing buffer relative to the standard buffer. Instead of a 65% increase, the putrescine-containing buffer produces a small decrease in observed variance. Similarly, the putrescine-containing buffer decreases rather than increases the variance in end-to-end distance of a 22-bp duplex (Fig. 2C). Assuming that previous measurements of bending persistence length are accurate, the data are unambiguous: The variations in the end-to-end distance of short DNA duplexes reported in (*1*) are not caused by DNA bending.

A soft, correlated stretching motion in DNA remains the simplest consistent explanation of the experimental results and is compatible with independent observations of allosteric behavior in DNA [reviewed in (*8*)]. However, we agree with Becker and Everaers that radially amplified bending fluctuations should have been large enough to detect in the measured variance data. The absence of the predicted oscillations with a period of 10 bp, and of the expected modulation of variances by persistence length, is paradoxical. We can offer two possible explanations. First, the radial offset of the nanocrystals might be smaller than 9 Å, decreasing the magnitude of the variance oscillations below the threshold of detection. The mean end-to-end distance data are not compatible with large radial offsets, but they less clearly eliminate the possibility of a zero radial offset. Alternatively, it has been suggested that short DNA duplexes bend through discrete kinks (*9*). If the kinking idea is correct, then the dominant Gaussian feature in our distance distributions might correspond to an exclusively unbent DNA population. Fits to the Gaussian feature would not be affected by bending phenomena. The kinked DNA molecules would presumably contribute a separate broad probability feature, possibly hidden in the baseline of the distribution. Additional experimental tests should shed light on this exciting possibility.

## Supporting Online Material

↵* Present address: Department of Cell Biology, Harvard Medical School, Boston, MA 02115, USA.