Twist-bend coupling and the statistical mechanics of the twistable wormlike-chain model of DNA: Perturbation theory and beyond

The simplest model of DNA mechanics describes the double helix as a continuous rod with twist and bend elasticity. Recent work has discussed the relevance of a little-studied coupling $G$ between twisting and bending, known to arise from the groove asymmetry of the DNA double helix. Here the effect of $G$ on the statistical mechanics of long DNA molecules subject to applied forces and torques is investigated. We present a perturbative calculation of the effective torsional stiffness $C_{\text{eff}}$ for small twist-bend coupling. We find that the “bare” $G$ is “screened” by thermal fluctuations, in the sense that the low-force, long-molecule effective free energy is that of a model with $G=0$ but with long-wavelength bending and twisting rigidities that are shifted by $G$-dependent amounts. Using results for torsional and bending rigidities for freely fluctuating DNA, we show how our perturbative results can be extended to a nonperturbative regime. These results are in excellent agreement with numerical calculations for Monte Carlo “triad” and molecular dynamics “oxDNA” models, characterized by different degrees of coarse graining, validating the perturbative and nonperturbative analyses. While our theory is in generally good quantitative agreement with experiment, the predicted torsional stiffness does systematically deviate from experimental data, suggesting that there are as-yet-uncharacterized aspects of DNA twisting-stretching mechanics relevant to low-force, long-molecule mechanical response, which are not captured by widely used coarse-grained models.

Figure 1

Bottom: The configuration of a twistable elastic rod can be mathematically described with an orthonormal set of vectors $\{{\widehat{\mathbf{e}}}_1,{\widehat{\mathbf{e}}}_2,{\widehat{\mathbf{e}}}_3\}$ assigned to every point $s$ along the rod. The vector ${\widehat{\mathbf{e}}}_3$ is the tangent to the curve and describes the bending fluctuations along the rod. Top: Cross section of the rod, indicating how the remaining vectors ${\widehat{\mathbf{e}}}_1$ and ${\widehat{\mathbf{e}}}_2$, which describe the torsional state, may be chosen in the particular case of DNA.

Figure 2

Typical setup of a magnetic tweezers experiment. A DNA molecule is covalently bound to a substrate at one end and to a paramagnetic bead at the other end. An applied force $f$ stretches the molecule, while an applied rotation $\theta$ twists it. The effective torsional stiffness $C_{\text{eff}}$ is the proportionality constant connecting the applied rotation with the exerted torque [Eq. (4)]. $C_{\text{eff}}$ increases with the force, hence it is higher in (a) than in (b).

Figure 3

Comparison of Monte Carlo simulations of the triad model (points) with various analytical expressions of $C_{\text{eff}}$ (lines) for $A=50$ nm, $C=100$ nm and (a) $G=0$, (b) $G=20$ nm, and (c) $G=40$ nm. The data are plotted as a function of $\sqrt{k_{B}T/fA}$ and correspond to $f\ge 0.25\mathrm{pN}$. The numerical results are in excellent agreement with the analytical expressions, both in the perturbative and nonperturbative regimes (see text). Error bars of Monte Carlo data are smaller than symbol sizes.

Figure 4

Effect of bending anisotropy $\left(A_1\ne A_2\right)$ on the effective torsional stiffness at a fixed force $f=1\mathrm{pN}$ with (a) $G=0$ and (b) $G=30$ nm. Numerical data from Monte Carlo simulations of the triad model (points) are in good agreement with the analytical, nonperturbative predictions of Eq. (28) (solid lines). The vertical dashed lines indicate the isotropic case $(\varepsilon =0)$. A nonvanishing twist-bend coupling (lower panel) induces a $\varepsilon \to -\varepsilon$ symmetry breaking. Error bars of Monte Carlo data are smaller than symbol sizes.

Figure 5

Comparison of oxDNA simulations to Eq. (28). Solid and dashed lines show the nonperturbative result for $C_{\mathrm{eff}}$ for oxDNA1 and oxDNA2 values of Table 1, respectively. Error bars for the oxDNA data are smaller than symbol sizes.

Figure 6

Comparison of the theory from Eq. (28) (lines) with magnetic tweezer experiments (symbols) for $C_{\text{eff}}$ vs. force. The lines have the same parametrization of the solid and dashed lines of Fig. 5, which fit oxDNA and oxDNA2 data, respectively. Two sets of experiments are shown: the freely orbiting magnetic tweezers [22] (circles) and magnetic torque tweezers [14] (triangles).