Torque correlation length and stochastic twist dynamics of DNA

We introduce a short correlation length for torque in twisting-stiff biomolecules, which is necessary for the physical property that torque fluctuations be finite in amplitude. We develop a nonequilibrium theory of dynamics of DNA twisting which predicts two crossover time scales for temporal torque correlations in single-molecule experiments. Bending fluctuations can be included, and at linear order we find that they do not affect the twist dynamics. However, twist fluctuations affect bending, and we predict the spatial inhomogeneity of twist, torque, and buckling arising in nonequilibrium “rotor-bead” experiments.

Figure 1

(a) Schematic of the equilibrium experiment. A magnetic bead attached to the DNA molecule at the left end is rotated with a mean torque $\langle \tau \rangle$, and the molecule is attached to a substrate at the right end to fix the twist. (b) Schematic of the nonequilibrium experiment. A magnetic bead attached to the DNA at the left end is rotated at a constant angular velocity, ${\omega }_0$, and the molecule is nicked so that it releases twist. Twist release is opposed by the viscous drag, ${\zeta }_B$, of the rotor bead, which is attached to the DNA to the left of the nick.

Figure 2

Solution to Eq. (2) subject to the boundary conditions in Eq. (3). We estimate $\zeta \approx 4\pi \eta d^2$ with viscosity $\eta \approx {10}^{-3}\mathrm{Pa}\mathrm{s}$ and DNA diameter $d\approx 1\mathrm{nm}$, $L\approx 1500\mathrm{nm}$, $\beta {\zeta }_B\approx 0.1\mathrm{s}$, ${\omega }_0\approx 10\pi \mathrm{rad}/\mathrm{s}$ as in experiments [10], and $c_1\approx 100\mathrm{nm}$ [8, 9]. We expect the correlation length to be small, $\xi <c_1$, so that the wormlike chain behavior is observed at large length scales [2, 3, 4, 5]. (a) Steady-state torque in the molecule versus distance from the driving end. The solid line shows $c_2=1000{\mathrm{nm}}^3$ ($\xi =3.16\mathrm{nm}$). Torque curves for other values of $c_2$ are nearly identical, with deviations near the boundaries only noticeable for large $c_2$ (dashed line shows $c_2={10}^5{\mathrm{nm}}^3$). For large $c_2$, $\xi$ is large and the torque clamp boundary condition is felt over a longer length scale. Inset: Time-independent part of the twist angle versus distance from the driving end for $c_2=1000{\mathrm{nm}}^3$. (b) First five spatial eigenmodes. The lowest mode is approximately linear in $s$ and arises from the nonzero mechanical torque at $s=L$. The remaining sinusoidlike modes are nearly identical to those of the equilibrium system. As the mode number $k$ increases, the number of times $S_{\lambda }$ crosses zero increases.

Figure 3

(a) The temporal correlation function, $C_{\tau }(L,L,t)$, has three regimes: approximately constant correlations, $t^{-\frac{1}{2}}$ decay, and exponential decay [Eq. (12)]. $C_{\tau }(L,L,t)$ is plotted for $\xi =1\mathrm{nm}$ (solid line), $\xi =3.16\mathrm{nm}$ (dashed line), $\xi =10\mathrm{nm}$ (dotted line), and $\xi =31.6\mathrm{nm}$ (dash-dotted line). (b) The spatial correlation function, $C_{\tau }(L,L-s,0)$, decays exponentially. Lines show $C_{\tau }(L,L-s,0)$ for the same values of $\xi$ as in panel (a).