Analytical Description of Extension, Torque, and Supercoiling Radius of a Stretched Twisted DNA

We study the mixture of extended and supercoiled DNA that occurs in a twisted DNA molecule under tension. Closed-form asymptotic solutions for the supercoiling radius, extension, and torque of the molecule are obtained in the high-force limit where electrostatic and elastic effects dominate. We demonstrate that experimental data obey the extension and torque scaling laws apparent in our formulas, in the regime where thermal fluctuation effects are quenched by applied force.

Figure 1

Supercoiled DNA under force and torque. Molecule length is partitioned between two phases: an extended phase and a plectonemic phase where strong self-interaction occurs.

Figure 2

Comparison of experimental and theoretical slopes as a function of the applied force, for 50, 100, 200, and 500 mM salt (top to bottom): (a) experimental data from Ref. 1 (circles); (b) theoretical solution of the full equations $\mathbf{\nabla }\mathcal{G}=0$ (continuous lines); (c) ratio of the experimental slopes to the formula in Eq. (16) (filled circles). Experimentally given values of $A/kT=46,47,44,45\mathrm{nm}$ and $C/kT=94\mathrm{nm}$ were used.

Figure 3

Comparison of experimental and theoretical torques as a function of the applied force for 50, 100, 200, and 500 mM salt: (a) experimental data from Ref. 1 (circles); (b) theoretical solution of the full equations $\mathbf{\nabla }\mathcal{G}=0$ (continuous lines); (c) ratio of the experimental torque to the formula in Eq. (17) (filled circles). Experimentally given values of $A/kT=46,47,44,45\mathrm{nm}$ and $C/kT=94\mathrm{nm}$ were used.

Figure 4

Supercoiling radius as a function of the force function $g(f)$, computed with (a) the full nonlinear equations $\mathbf{\nabla }\mathcal{G}=\mathbf{0}$ (plain lines, red); (b) the formula in Eq. (11) (dashed lines, black) for the four salt concentrations 50, 100, 200, and 500 mM (top to bottom). The separation of the curves at high force is due to the nonlinearities of $\mathbf{\nabla }\mathcal{G}=\mathbf{0}$, which are omitted in Eq. (11).