Discontinuities at the DNA supercoiling transition

While slowly turning the ends of a single molecule of DNA at constant applied force, a discontinuity was recently observed at the supercoiling transition when a small plectoneme is suddenly formed. This can be understood as an abrupt transition into a state in which stretched and plectonemic DNA coexist. We argue that there should be discontinuities in both the extension and the torque at the transition and provide experimental evidence for both. To predict the sizes of these discontinuities and how they change with the overall length of DNA, we organize a phenomenological theory for the coexisting plectonemic state in terms of four parameters. We also test supercoiling theories, including our own elastic rod simulation, finding discrepancies with experiment that can be understood in terms of the four coexisting state parameters.

Figure 1

(Color online) Extension and torque as a function of linking number $K$, for $L=2.2\text{kbp}$ at $F=2\text{pN}$. Black lines show data from Ref. [4], smoothed using a “boxcar” average of nearby points. The green (gray) lines show WLC predictions below the transition (in the unsupercoiled SS) and fits to the data after the transition (in the CS), linear for the extension and constant for the torque. The size of the torque jump, not visible in the smoothed data, is implied by the coexisting torque $\tau$, the CS fit, and the transition linking number $K^{\ast }$ in the extension data. Inset: simulated DNA showing the CS of a plectoneme and straight DNA, ignoring thermal fluctuations. The ends are held with fixed orientation and pulled with a constant force $F$, here 2 pN.

Figure 2

(Color online) Directly measuring the torque jump by observing thermal hopping, for the same conditions as Fig. 1. As linking number $K$ is slowly increased near $K^{\ast }$, thermal fluctuations induce hopping between states with (CS) and without (SS) a plectoneme. Averaging over these two states gives a direct way of measuring the torque jump: analogously to a lock-in amplifier, we set a threshold in the extension signal to separately average the SS (black) and CS [red (gray)] data near the transition. Using multiple traces, we find an average torque jump of $\Delta \tau =2.9\pm 0.7\text{pN}\text{nm}$ for $L=2.2\text{kbp}$ at $F=2\text{pN}$. Additionally, this value of $\Delta \tau$ implies (see text) that the transition should happen over a range of linking number $K$ (a) of about 0.9 turns, agreeing with the observed range.

Figure 3

(Color online) The four parameters describing the CS (coexisting torque $\tau$, extension versus linking number slope $q$, and the extra free energy ${\mathcal{F}}_0$ and extension $z_0$ necessary to form the end loop and tails of the plectoneme) as a function of applied force. The circles show values calculated from experimental data taken at two different overall DNA lengths $L$. Model predictions for our simulation [21] and Marko’s model [6] are shown as solid and dashed lines, respectively (using $S=0$ for ${\mathcal{F}}_0$ predictions). The circular end-loop model uses average $\tau$ and $q$ values from the experiment to predict ${\mathcal{F}}_0$ and $z_0$, shown as dotted lines.

Figure 4

(Color online) [(a) and (b)] Predicted length dependence of the extension and torque jumps at $F=2\text{pN}$. Using the CS parameters extracted from the experiment at two different lengths, Eqs. (4, 5) predict the $L$ dependence of $\Delta z$ and $\Delta \tau$. The circles show experimentally measured values [with the torque jump here calculated from $K^{\ast }$ using Eq. (4)]. Without entropic corrections to ${\mathcal{F}}_0$ ($S=0$; dot-dashed lines) $\Delta z$ depends noticeably on $L$ but including an initial estimate of $S$ (solid lines) shows that entropic effects can significantly reduce this length dependence. [(c) and (d)] Force dependence of the extension and torque jumps and predictions from two models. Disagreements with experimental data can be understood in terms of the four CS parameters in Fig. 3. Also plotted as a diamond is $\Delta \tau$ measured using the direct method depicted in Fig. 2.