Figure 1

A plectoneme with plectoneme radius $R$, plectoneme angle $\alpha$, and pitch $p$. The standard deviation of the fluctuation channel in the “pitch direction” is $\pi R\sin \left(\alpha \right)$.Reuse & Permissions

Figure 2

The homoclinic solutions. (a) Curves of the homoclinic solutions for the homoclinic parameter $t=0,1/3,2/3$, and 1. (b) The energy of the homoclinic solutions relative to the straight rod energy, where all the linking number is in the twist of the chain. Here $L_c=600$ nm and $f=2$ pN $\gg f_0$.Reuse & Permissions

Figure 3

Force dependence of (a) the plectoneme radius, (b) the plectoneme angle, (c) the energy per writhe difference between loop and plectoneme with logarithmic correction related to the choice of cutoff, and (d) the writhe density ratio between loop and plectoneme for salt concentrations of $30,60,120,210$, and 320 mM. The arrows point in the direction of increasing salt. The range for $\alpha$ was on purpose chosen to be the full allowed range for a stable plectoneme, showing that its value is hardly dependent on the environment.Reuse & Permissions

Figure 4

The decrease of the extension $\overline{z}$ and the contributions into which it decomposes ($\mu$ and $lp$) as functions of scaled linking number density, omitting the straight solution that disappears early on. The plots were generated using the parametrization outlined in Appendix . The green dashed line is the approximation from Eq. (D7). The dotted line corresponds to the fictive one-plectoneme behavior. (a) Typical single-plectoneme behavior at high cost per writhe difference between loop and plectoneme. (b) Lower $\Delta$ increases the number of plectonemes, changing the slope of the turn-extension plot. (c) When the ratio of the writhe densities is 1 the slope does not change even with a large number of plectonemes. (d) When $r_{\omega }$ rises above 1 we end up with a high density of zero-length plectonemes. The force-extension curve also here resembles a one-plectoneme curve but one with modified plectoneme parameters.Reuse & Permissions

Figure 5

The two faces of multiple plectonemes. (a) Contour plot of the multiplectoneme factor as it depends on salt concentration and tension. The thick red line can be interpreted as the border between a single-plectoneme phase on the right and a multiplectoneme phase on the left. The inset shows $\zeta$ as a function of salt concentration for three different forces. Note the crossing of the lines at low salt concentrations. (b) The maximal loop density $\mu$. Note the sharp transition from its maximum ($\mu =1$) to a vanishing number of plectonemes. Since the maximal $\mu$ is reached at the end of the plectoneme slope for $r_{\omega }>1$, it does not reflect the multiplectoneme transition along the beginning of the slope.Reuse & Permissions

Figure 6

Maximal number of plectonemes as a function of tension and salt concentration for a chain length of 1400 nm. The three-dimensional plot (a) shows how the number of plectonemes goes down with decreasing force at low salt but increases at high salt. The anomalous behavior at low salt is a consequence of the growing loop size limiting the maximal number of loops that fit on the chain. The wiggles are an artifact of the interpolation used. The contour plot of the same data (b) shows a clean border between low and high number. The white line is the $\zeta =1$ line that marks the border of single- and multiplectoneme behavior. The difference between the two is caused by the growing number of plectonemes at the end of the plectoneme slope, while $\zeta$ is a measure for the main part of the slope.Reuse & Permissions

Figure 7

Slopes with and without thermal contributions. The dotted curves were calculated from the model as described up to Sec. 3, Eq. (17). The solid thermal curves include multiplectonemes and were calculated using the method outlined in the text.Reuse & Permissions

Figure 8

Influence of multiplectonemes on the turn-extension plot at low salt. For a chain with a contour length of 600 nm and a tension of 3 pN at a 20 mM salt concentration there is a noticeable effect on the slope (a). The growing number of plectonemes and not a growing plectoneme length is responsible for the slope in (b). The experimental data are from Seidel and co-workers [22, 35].Reuse & Permissions

Figure 9

Turns versus extension plots for 20 mM (a) and 320 mM (b) monovalent salt concentrations under varying tension. The torsional persistence length for most salt gave the best fit for 110 nm. For 20 mM a lower value of 90 nm had to taken to get an almost perfect agreement, but as explained in the text it might be a calculational artifact due to the proximity of the bifurcation point.Reuse & Permissions

Figure 10

Long chains. (a) Turns versus extension plots for a chain of 3540 nm with tension from 1 to 4 pN in 320 mM salt. A torsional persistence length of 120 nm was used. (b) Some of the magnetic tweezer measurements from Mosconi et al. [37] with a 5.4–μm-long chain in 100 mM salt for three different forces. The curves are from our model with a torsional persistence length of 115 nm.Reuse & Permissions

Figure 11

(a) Torque as a function of supercoiling density for three different tensions calculated from the model compared to the indirect determination of the torque using Maxwell relations under the assumption of constant plectoneme torque[37]. The conditions are the same as in Fig. 12. (b) Comparison between experimental results from an optical tweezer experiment using a quartz cylinder to directly measure the torque [41] and our theoretical model. The DNA molecule has a contour length of 725 nm. The monovalent salt concentration is 150 mM.Reuse & Permissions

Figure 12

Illustration of the magnetic tweezer measurements from Mosconi et al. [37] as a basis for indirect torque measurements. The curves are calculated from our model for a range of tensions where fluctuations are small enough, using the criterion $K^2\ge 3$. The contour length is 5.6 $\mu \mathrm{m}$ and the monovalent salt concentration is 100 mM. On the left are the resulting turn-extension plots. On the right are torques from our model. Notice how the torque is increasing in the plectoneme region, caused by the growing number of plectonemes. The circuit ABCD results in too high a torque at $C$ when one assume the plectoneme torque to be constant.Reuse & Permissions