Nucleation of Multiple Buckled Structures in Intertwined DNA Double Helices

We study the statistical-mechanical properties of intertwined double-helical DNAs (DNA braids). In magnetic tweezers experiments, we find that torsionally stressed stretched braids supercoil via an abrupt buckling transition, which is associated with the nucleation of a braid end loop, and that the buckled braid is characterized by a proliferation of multiple domains. Differences between the mechanics of DNA braids and supercoiled single DNAs can be understood as an effect of the increased bulkiness in the structure of the former. The experimental results are in accord with the predictions of a statistical-mechanical model.

Figure 1

(a) Schematic of the magnetic tweezers setup. Two DNA molecules are attached (using only one of the DNA strands as attachments) to the glass surface and a paramagnetic bead via digoxigenin-antidigoxigenin and biotin-streptavidin interactions, respectively. The single-strand attachments ensure no twisting of individual DNA molecules upon rotation of the bead [8, 24]. (b) Experimentally measured end-to-end extension of a braid as a function of the catenation number for 0.8 (black points) and 1.2 pN (gray cross marks) applied forces in 100 mM NaCl, where the error bars represent standard deviation. The solid lines represent theoretical predictions for $1.6\mu \mathrm{m}$ DNA molecules, with intertether distance $0.19\mu \mathrm{m}$ and under 0.8 (black line) and 1.2 pN (gray line) force at 100 mM monovalent salt [25]. The small peak in extension at zero catenation is due to the small intertether distance, i.e., the close proximity of the two DNA molecules for this particular DNA pair [28].

Figure 2

(a) Time series of the end-to-end extension of a braid constituted of two torsionally unconstrained 6 kb DNAs, under 0.8 pN force at 100 mM NaCl [Fig. 1], for three catenation numbers ($\mathrm{Ca}=-23$, $-24$, and $-25$) near the buckling transition point [the point of the slope change in the extension curve, Fig. 1]. Data were collected at 200 Hz (light gray dots) and then median filtered (dark points) using a 0.1 sec time window to show dynamic switching between discrete extension states. The panels on the right of each time series plot show histograms of the raw data using 10 nm bins (gray shaded area); the $y$ axis is the same for left and right panels. The histograms were fit to a sum of multiple Gaussian distributions, where the dark line is the best-fit distribution and corresponds to the sum of the individual Gaussians shown in gray lines. The sum of two Gaussian distributions fit the data better at low catenation ($\mathrm{Ca}=-23$), indicating nucleation of the first plectoneme domain, whereas the sum of at least three Gaussians is required to fit the histograms at higher catenation ($\mathrm{Ca}=-24$, $-25$), due to the appearance of multiple plectoneme domains. (b) Schematic diagram of three braid-extension states accessible via thermal fluctuations near the buckling transition. (c) Theoretically predicted extension histograms near the buckling transition, where the black dotted line shows the total extension distribution ($P_{\text{tot}}$); also plotted are the individual contributions from the straight braid ($P_0$, cyan solid line) and the buckled braid with one ($P_1$, red dot-dashed line) and two ($P_2$, blue dashed line) plectoneme domains [see Eq. (3)]. Contributions from the three- ($P_3$) and four-domain ($P_4$) plectonemes are also plotted; however, the negligible statistical weight of those states for the plotted range of catenation renders them almost invisible in the predicted histograms.

Figure 3

Comparison of the theoretically predicted change in the extension upon nucleation of the first (red open triangles) and the second (blue open squares) plectoneme domain with experimental observations. Red points and blue cross marks represent the difference in the extension between successive peaks (inset) in the experimental best-fit histograms, where the second jump (blue cross markers) is measured one unit catenation after the first jump (red points). The error bars represent the binning error. The red and blue dashed lines show the expected $f^{-1/2}$ power law [Eq. (1)], best fit to red open triangles and blue open squares, respectively.