Fractional Brownian motion and the critical dynamics of zipping polymers

We consider two complementary polymer strands of length $L$ attached by a common-end monomer. The two strands bind through complementary monomers and at low temperatures form a double-stranded conformation (zipping), while at high temperature they dissociate (unzipping). This is a simple model of DNA (or RNA) hairpin formation. Here we investigate the dynamics of the strands at the equilibrium critical temperature $T=T_c$ using Monte Carlo Rouse dynamics. We find that the dynamics is anomalous, with a characteristic time scaling as $\tau \sim L^{2.26\left(2\right)}$, exceeding the Rouse time $\sim L^{2.18}$. We investigate the probability distribution function, velocity autocorrelation function, survival probability, and boundary behavior of the underlying stochastic process. These quantities scale as expected from a fractional Brownian motion with a Hurst exponent $H=0.44\left(1\right)$. We discuss similarities to and differences from unbiased polymer translocation.

Figure 1

Sketch of the zipping and unzipping dynamics. The two polymer strands are joined by a common end $O$. Single-stranded polymers are shown as dashed lines, while double stranded are shown as thick solid lines: (a) fully unzipped state, (b) partially zipped state, and (c) fully zipped state. The position of the end of the double-stranded segment is denoted by $Y$. The total length of each strand is $L$. A partially zipped configuration is characterized by two single strands of length $L-s$ and a double strand of length $s$.

Figure 2

A log-log plot of the zipping ${\tau }_z$ and unzipping ${\tau }_u$ times as functions of the strand lengths $L$. The two quantities scale with the same anomalous exponent ${\tau }_z\sim {\tau }_u\sim L^{2.26\left(2\right)}$. Circles and squares are data from simulations with the initial condition $s_0=L/2$, i.e., half-zipped configurations. Diamonds correspond to the zipping time with the initial condition $s_0=3L/4$. Triangles correspond to the unzipping time with the initial condition $s_0=L/4$.

Figure 3

Plot of $P(s,t)/P(L/2,t)$ for strands of length $L=192$ for different simulation times $t=2^9$, $2^{11}$, $2^{13}$, $2^{14}$, $2^{15}$, $2^{16}$, $2^{17}$ and $2^{18}$. The inset shows $P(s=L/2,t)$ versus time on a log-log plot.

Figure 4

Plot of $R=\langle {(s-\langle s\rangle )}^4\rangle /{\langle {(s-\langle s\rangle )}^2\rangle }^2$ as a function of time for various $L$ values. The dashed line indicates the Gaussian value $R=3$. The sampling is over $5\times {10}^5$ configurations for $L=24$, 48, and $2\times {10}^5$ configurations for $L=96$ and ${10}^5$ configurations for $L=192$, 384, and 768.

Figure 5

Plot of ${\sigma }^2$ as a function of time $t$ for different values of $L$. The dashed line is a power-law fit to the data, yielding $H=0.44$.

Figure 6

A log-log plot of minus the velocity autocorrelation as a function of time. The dashed line is a straight line (not a fit) with a slope of $-1.12$.

Figure 7

A linear-log plot of the scaling function of the probability distribution of the unzipping time for sizes $L=24$, 48, 96, and 192. The calculations are done over ${10}^6$ configurations for $L=24$, 48, and 96 and over $2\times {10}^5$ configurations for $L=192$.

Figure 8

Plot of $\ln P(s,t)$ vs $\ln s$ emphasizing the vanishing of the position PDF in the vicinity of the absorbing boundaries (same data as in Fig. 3: $L=192$ and $t=2^9,2^{10},...,2^{17},2^{18}$). For the two longest simulation times the distribution is almost stationary and the probability distribution vanishes, in agreement with a power-law behavior $\sim s^{1.27}$ (dashed line). This is the expected behavior for fractional Brownian motion with $H=0.44$ in the vicinity of a boundary, as derived in Ref. [26].