Event distributions of polymer translocation

We present event distributions for the polymer translocation obtained by extensive Langevin dynamics simulations. Such distributions have not been reported previously and they provide new understanding of the stochastic characteristics of the process. We extract at a high length scale resolution distributions of polymer segments that continuously traverse through a nanoscale pore. The obtained log-normal distributions together with the characteristics of polymer translocation suggest that it is describable as a multiplicative stochastic process. In spite of its clear out-of-equilibrium nature the forced translocation is surprisingly similar to the unforced case. We find forms for the distributions almost unaltered with a common cut-off length. We show that the individual short-segment and short-time movements inside the pore give the scaling relations $\tau \sim N^{\alpha }$ and $\tau \sim f^{-\beta }$ for the polymer translocation.

Figure 1

Log-binned distributions of events assorted by the lengths of polymer segments that traverse in one direction, trans or cis, without pausing and sampled at time intervals, or without reversing for different pore force magnitudes $f$. The length of the polymer for all curves is $N=200$. (a) The number of events $n_E$ vs segment length $\Delta s$. $n_E$ is an average of numbers of events over several polymer translocations. (b) The numbers of events normalized with the maximum ${\widehat{n}}_E=n_E/n_{\text{max}}$ to compare the distributions. All distributions follow closely log-normal forms. The log-normal functions plotted to guide the eye are $\text{func1}=(1/\Delta s)\exp [-\log {(\Delta s/2)}^2/1.2]$ and $\text{func2}=(1/\Delta s)\exp [-\log {(\Delta s/2)}^2/4]$. The apparent power-law decay at $\Delta s>300$ is an artifact due to the log-binning done on very few points there.

Figure 2

The event-time distributions $n_E$ normalized with the maximum number of events. Semilog scale shows the exponential decays. See the text for details.

Figure 3

(a) The distribution (distr) of event times $\Delta t$ with segment lengths $\Delta s$ for $f=0$ and $N=100$. The rest are cumulative distributions for $f=0,0.5$ and 1, $N=50$, 100, and 200. The cut-off segment length $\Delta s_c=95$ and the characteristic scale $\Delta s_0=6$ for all fit functions. The dispersion $\sigma =2$, 7, and $9.5$ for $f=0$, $0.5$, and 1, respectively. Also shown is a fit with the function $\text{pow-exp}\sim$$\Delta s^{-0.3}\exp (-\Delta s/60)$ for the cumulative distribution for $f=1$ and $N=50$. (b) The cumulative distributions of (a) normalized by the maximum value. Lines $\sim$${\left(\Delta s\right)}^{-0.5}$ and $\sim$${\left(\Delta s\right)}^{-5}$ are drawn to guide the eye.

Figure 4

(a) Distributions of total time $t_E$ for constant $N=200$ for transferred segment lengths $\Delta s\in \{1,10,50\}$ and pore force values $f=0$ and $0.25$. (b) The distributions for the constant pore force $f=1$ and different polymer lengths $N$. The bottom three curves with maxima 13, 35, and 100 are for $\Delta s=10$ and $N\in \{50,100,200\}$. The topmost three curves with maxima 130, 350, and 1000 are for $\Delta s=1$ and the same polymer lengths. The marked maxima give the correct scaling $\tau \sim N^{\alpha }$, see the text.

Figure 5

The distributions of the numbers $n_E(\Delta s,\Delta t)$ for $\Delta s=1$, and 10 for pore force $f=1$. The indicated maxima give $\alpha =1.5$.