Dependence of DNA supercoiling on force and salt concentration. (a) Experimental setup and superhelix parameters. (b) DNA supercoiling curves recorded in buffer containing 170 mM Na⁺ at stretching forces of 0.25, 0.5, 1.0, 2.0, 3.0 and 4.0 pN (gray, light blue, dark blue, red, green and dark gray lines). Continuous twisting was carried out at 0.5 Hz. Data was taken at 300 Hz and smoothed to 20 Hz. (c) Slopes after buckling as a function of force obtained from the supercoiling curves for Na⁺ concentrations of 30, 60, 170 and 320 mM (gray, red, black and blue circles) together with the prediction from the theoretical model (solid lines) calculated for a charge adaptation factor χ_CR of 0.42. Inset: Slopes for 60 mM Na⁺ (red circles) together with theoretical predictions for different values of χ_CR (lines). Best agreement is found for χ_CR = 0.42 for all ionic strengths considered. Only slight variations of χ_CR to 0.32 and 0.55 lead to a significant under- or overestimation, respectively, of the slopes.

Coarse-grained Monte-Carlo simulations of DNA supercoiling. (a) Snapshots of the formed plectoneme at 1 pN and 8 turns in the presence of 30 and 320 mM monovalent ions. The linear, stretched parts of the DNA as well as the surface and the bead are not shown. (b) Simulated DNA supercoiling curves for 170 mM Na⁺ at stretching forces of 0.25, 0.5, 1.0, 2.0, 3.0 and 4.0 pN for χ_CR = 0.42 (colors are as in Figure 1b). Simulated curves excellently reproduce the experimentally observed behavior. This includes the abrupt force- and salt-dependent buckling at the onset of the plectonemic phase, which is accompanied by a torque overshoot (see d). Buckling is followed by a linear DNA length decrease with added turns at constant torque. (c) Slopes after buckling obtained from the simulated curves for Na⁺ concentrations of 30, 60, 170 and 320 mM (circles, colors as in Figure 1b) and corresponding predictions from the theoretical model for χ_CR of 0.42 (solid lines). Inset: Slopes from simulated curves at 60 mM Na⁺ for χ_CR = 0.42 (red dots) as well as for χ_CR = 0.32 and χ_CR = 0.55 (gray dots with dashed lines). The prediction for χ_CR = 0.42 is shown as a solid red line. Small variations of χ_CR cause significant differences for the obtained slopes. (d) Torque during DNA supercoiling for the curves shown in b. (e) Torque after buckling as obtained from the simulations (filled circles) and after subtraction of 1.5 pN nm (open squares) together with the corresponding predictions from the theoretical model for χ_CR of 0.42 (solid lines). Torque values from simulations appear globally shifted by 1.5 pN nm compared to the theoretical model since subtraction of this value provides a much better agreement. Na⁺ concentrations and colors are as in c.

Comparison of the predictions from the theoretical model with data from ref. 13. (a) Slopes after buckling versus force for different Na⁺ concentrations. Circles represent experimental data, solid lines the theoretical prediction for χ_CR = 0.42. (b) Torque after buckling. Colors and symbols are as in a.

All-atom molecular dynamics simulations of the effective force between double-stranded DNA. (a) All-atom model used to determine the force. The DNA atoms are depicted as red spheres; the counter and co ions are depicted as blue and purple spheres, respectively; the water is shown as a semi-transparent molecular surface. The distance between the DNA molecules was restrained by a harmonic potential, schematically depicted as a spring. (b) The simulated mean force between the DNA. Data from MD simulations (circles, solid lines) are shown alongside the theoretical predictions using a charge adaptation factor χ_CR of 0.42 (dashed lines) and 1.0 (dotted lines) at 60 (red) and 170 mM (black) bulk ion concentrations. Data for 300 mM (blue) bulk ion concentrations is reproduced from previous work.³² The relative orientation of the DNA did not appear to affect the obtained force within error. The atomic structure of the interacting DNA and nearby solvent cause the force to drop at 300 mM as the separation between the DNA surfaces approaches the atomic scale.³²