Periodically driven DNA: Theory and simulation

We propose a generic model of driven DNA under the influence of an oscillatory force of amplitude $F$ and frequency $\nu$ and show the existence of a dynamical transition for a chain of finite length. We find that the area of the hysteresis loop, $A_{\mathrm{loop}}$, scales with the same exponents as observed in a recent study based on a much more detailed model. However, towards the true thermodynamic limit, the high-frequency scaling regime extends to lower frequencies for larger chain length $L$ and the system has only one scaling $\left(A_{\mathrm{loop}}\approx {\nu }^{-1}{F}^2\right)$. Expansion of an analytical expression for $A_{\mathrm{loop}}$ obtained for the model system in the low-force regime revealed that there is a new scaling exponent associated with force $\left(A_{\mathrm{loop}}\approx {\nu }^{-1}{F}^{2.5}\right)$, which has been validated by high-precision numerical calculation. By a combination of analytical and numerical arguments, we also deduce that for large but finite $L$, the exponents are robust and independent of temperature and friction coefficient.

Figure 1

Schematic representations of DNA: (a) zipped, (b) partially zipped, and (c) unzipped state. One end is kept fixed (indicated by the solid circle), while the other end may move in positive (shown by the solid line) or negative direction (shown by the dashed line) depending on the force direction.

Figure 2

Comparative plots of the area of hysteresis loop for (a) staircase and (b), (c) sinusoidal force for $F=0.33$ and chain length $L=24$. In (c) the overdamped limit is taken. For the sinusoidal force, $\nu$ was taken as $\nu /2$ to compare the positive branch of the hysteresis loop with the staircase force. Here $\nu$ is given in units of ${10}^{-3}$.

Figure 3

(a) Variation of $A_{\mathrm{loop}}$ with $\nu$ for a staircase (LD1) and sinusoidal force (LD2) using LD simulations, and sinusoidal force in the overdamped limit using BD simulations. (b) After rescaling $\left(A_{\mathrm{loop}}^{*},{\nu }^{*}\right)$, all three curves collapse onto a single master curve, which justifies the use of Eq. (3) for the further understanding.

Figure 4

(a) and (b) show the $F-x$ curves at different frequencies obtained from Eq. (3) for $F=0.25$ (low) and $0.65$ (high) amplitudes, respectively. (c) and (d) show the collapse of the data in (a) and (b) onto a single hysteresis curve. (e) Comparison of numerical (RK4) values of $t_2$ as a function of $t_1$ with the approximate analytical form for $\omega =1$. (f) shows the comparison of $\nu A_{\mathrm{loop}}-F$ curves obtained from different approaches [Eqs. (3), (4), and (6)]. The inset of Fig. 4 is a log-log plot showing that the numerical data (RK4) agree with the expansion of Eq. (6) in the limit $y\to 1(F\to F_c)$, i.e., Eq. (4) scales with $\alpha =2.5$ instead of 2.

Figure 5

Scaling of $A_{\mathrm{loop}}$ with respect to (a) ${\nu }^{0.5}{(F-F_c)}^{0.33}$ in the low-frequency regime and (b) ${\nu }^{-1}{(F-F_c)}^2$ in the high-frequency regime for a fixed length $L=24$. (c) Scaling of $\frac{A_{\mathrm{loop}}}{2L}$ with respect to ${\nu }^{0.5}{(F-F_c)}^{0.33}{L}^{0.5}$ in the low-frequency regime. Each color represents here three different values of $L$. (d) Scaling of $A_{\mathrm{loop}}$ with respect to ${\nu }^{-1}{(F-F_c)}^2$ for a particular value of $F(=0.3)$, demonstrating the length independence in the high-frequency limit.