Twist-Bend Coupling and the Torsional Response of Double-Stranded DNA

Recent magnetic tweezers experiments have reported systematic deviations of the twist response of double-stranded DNA from the predictions of the twistable wormlike chain model. Here we show, by means of analytical results and computer simulations, that these discrepancies can be resolved if a coupling between twist and bend is introduced. We obtain an estimate of $40\pm 10\mathrm{nm}$ for the twist-bend coupling constant. Our simulations are in good agreement with high-resolution, magnetic-tweezers torque data. Although the existence of twist-bend coupling was predicted long ago [J. Marko and E. Siggia, Macromolecules 27, 981 (1994)], its effects on the mechanical properties of DNA have been so far largely unexplored. We expect that this coupling plays an important role in several aspects of DNA statics and dynamics.

Figure 1

Schematic representation of a typical MT experiment. Magnetic fields are used to apply forces and torques (inducing a rotation angle $\theta$) to a paramagnetic bead. A dsDNA molecule is attached at one side to the bead and at the other to a flow-cell surface, separated by a distance $z$ measuring the extension of the molecule. Continuum elastic models describe the double-helix conformation using an orthonormal frame $\{{\widehat{\mathbf{e}}}_1,{\widehat{\mathbf{e}}}_2,{\widehat{\mathbf{e}}}_3\}$ at each point along the molecule, labeled by a coordinate $s$. ${\widehat{\mathbf{e}}}_3$ is tangent to the helical axis, while ${\widehat{\mathbf{e}}}_1$ points from the center of the helix towards the middle of the minor groove, and ${\widehat{\mathbf{e}}}_2={\widehat{\mathbf{e}}}_3\times {\widehat{\mathbf{e}}}_1$.

Figure 2

Force dependence of the effective torsional stiffness from MTT (present work) and FOMT [8] measurements, from simulations of the TWLC and the MS models (where error bars are smaller than the symbols) and from the analytical TWLC approximation [Eq. (5)]. Parameters are $A=43\mathrm{nm}$ and $C=110\mathrm{nm}$ for the TWLC and $A=56\mathrm{nm}$, $C=110\mathrm{nm}$, $ϵ=10\mathrm{nm}$, and $G=40\mathrm{nm}$ for the MS model. The inset shows $C_{\text{eff}}$ as a function of the rescaled variable $\sqrt{k_{B}T/A_{0}f}$ (with $A_0=50\mathrm{nm}$); in these units Eq. (5) becomes a straight line. The experimental data are not well described by the TWLC, but agree quantitatively with the MS model (reduced ${\chi }_{\mathrm{TWLC}}^2=6.1$ and ${\chi }_{\mathrm{MS}}^2=0.74$, respectively). The blue line is an interpolation of the MS simulations points.

Figure 3

(a) Relative extension $z/L$ and (b) torque $\tau$ vs supercoiling density $\sigma$ at $f=0.4\mathrm{pN}$ from MTT experiments (solid squares) and simulations of the TWLC (open circles, $A=43\mathrm{nm}$, $C=110\mathrm{nm}$) and MS (solid triangles, $A=56\mathrm{nm}$, $C=110\mathrm{nm}$, $G=40\mathrm{nm}$, and $ϵ=10\mathrm{nm}$) models. (c),(d) Closeups of $z/L$ and $\tau$ in the prebuckling regime, shaded area in (a) and (b). (e) Snapshots of simulations of the TWLC and MS models, respectively, at $\sigma =0.023$. The arrows point to a plectonemic supercoil in the TWLC and to a solenoidal supercoil in the MS model.