Supercoiling DNA with a free end

TL;DR

This work analyzes torque-driven supercoiling in open, untethered DNA using a coarse-grained Brownian dynamics framework coupled to a mean-field description of twist–writhe interconversion. The authors reveal a non-equilibrium transition from a swollen to a plectonemic state as the applied torque exceeds a threshold, accompanied by a non-linear steady-state twist profile and suppressed writhe diffusion due to localization near the driven end. A linear-chain limit provides a diffusion-dominated baseline with linear twist gradients and a simple relation for the steady-state rotation rate, while a 3D treatment shows how bending and plectoneme formation reshape stress transport and lead to localized structures near the torque source. A reaction–diffusion mean-field theory corroborates the qualitative trends, highlighting a torque-driven transport phenomenon with potential experimental realization in single-molecule assays.

Abstract

In this work, we combine coarse-grained Brownian dynamics simulations and mean-field theory to study supercoiling dynamics, as well as the steady-state profiles of twist and writhe, in an open DNA polymer where one of the free ends is subjected to a constant torque. Even though the other end is free, and hence can spin and release torsional stress, we observe that the entire chain transitions between a swollen and a plectonemic phase as the torque increases beyond a critical threshold. In the plectonemic phase, we observe a non-linear twist profile in the steady state, resulting from the mutual interconversion between the injected twist and geometrical writhe, which distributes inhomogeneously along the chain. We also show that the non-equilibrium dynamics of twist accumulation is diffusive, and that writhe diffusion is negligible in this geometry, as plectonemes remain localised near the end that is being rotated. We discuss the feasibility of testing our results with single-molecule experiments.

Figures (8)

• Figure 1: (a) Initial conformation at $t=0$ of a $2 {\rm kbp}$ polymer. The inset shows an enlargement of the first bead index, with the orthonormal set $\{\mathbf{f}_i, \mathbf{v}_i, \mathbf{u}_i\}$ sketched for the first two beads. Highlighted is the active torque vector $\tau_a$, which is applied only on the bead $i=0$ in the direction of $\mathbf{u}_0$ and approximately parallel to the tangential vector $\mathbf{t}_0$. The circles highlight the direction of rotation caused by $\tau_a$. (b) Snapshots of $2 {\rm kbp}$ long polymers subject to increasing torques $\tau_a=0,1,2,3,6$ (from bottom to top), and for increasing simulation times (from left to right). Chains are colored by chain index, see color bar in (a).
• Figure 2: (a) Radius of gyration as a function of time, for different torques $\tau_a=0,1,2,3,6$, and fixed chain length $N=2\rm kbp$.(b) Radius of gyration, $R_{gyr}$, for various chain lengths and as a function of the active torque $\tau_a$. $R_{gyr}$ is scaled by $N^\nu$, with $\nu=0.588$ the Flory exponent.
• Figure 3: (a) Spin angular velocity profiles in the direction tangential to the chain, $\boldsymbol{\omega}_i \cdot \mathbf{u}_i$ as a function of the chain fraction $i/N$, for different torques $\tau_a=1,3,6$ and at fixed chain length $2 \mathrm{kbp}$. (b) $\boldsymbol{\omega}_i \cdot \mathbf{u}_i$ as a function of $i/N$ for different chain length at fixed torque $\tau_a=6$. (c) $\boldsymbol{\omega}_0 \cdot \mathbf{u}_0$ (triangles), and $\boldsymbol{\omega}_N \cdot \mathbf{u}_N$, diamonds, as a function of the applied torque for various chain lengths. The solid lines are the theoretical values $\omega_L=\frac{\tau_a}{N\gamma_{rot}}$, Eq. \ref{['omega']}, computed for the different chain lengths. The dashed black line is $2.3 \cdot 10^{-2} \tau_a$. (d) Orbital angular velocity $\boldsymbol{\Omega}$ of the chain projected on the direction of the applied torque $\mathbf{u}_0$, as a function of the applied torque for various chain lengths. $\boldsymbol{\Omega}$ has been scaled by the chain length squared $N^2$.
• Figure 4: (a) Total twist ${\rm Tw}$ as a function of time for various active torques $\tau_a$. The dashed line indicates the initial diffusive scaling ${\rm Tw} \propto t^{0.5}$. ${\rm Tw}$ is negative for positive applied torques. (b) Total twist accumulated along the chain, rescaled over the chain length $N$, as a function of $\tau_a$ and for various $N$. The dashed line is a linear trend in $\tau_a$. (c) Total bending angle $\beta$ as a function of time for the same $\tau_a$ values as in (a). (d) $\beta$, rescaled over $N$, as function of $\tau_a$ for various $N$.
• Figure 5: (a) Local twist angle $\alpha_i+\gamma_i$ and (b) local bending angle $\beta_i$ as a function of chain fraction $i/N$, at different active torques and fixed chain length $2{\rm kbp}$. The dashed lines in (a) represent, for each active torque value, the theoretical prediction for a linear chain, Eq. \ref{['eq:twist_distribution_cyl']}.