Probability distributions for polymer translocation

We study the passage (translocation) of a self-avoiding polymer through a membrane pore in two dimensions. In particular, we numerically measure the probability distribution $Q\left(T\right)$ of the translocation time $T$, and the distribution $P(s,t)$ of the translocation coordinate $s$ at various times $t$. When scaled with the mean translocation time $⟨T⟩$, $Q\left(T\right)$ becomes independent of polymer length, and decays exponentially for large $T$. The probability $P(s,t)$ is well described by a Gaussian at short times, with a variance of $s$ that grows subdiffusively as $t^{\alpha }$ with $\alpha \approx 0.8$. For times exceeding $⟨T⟩$, $P(s,t)$ of the polymers that have not yet finished their translocation has a nontrivial stable shape.

Figure 1

(Color online) Probability distribution of the translocation coordinate $s$, of a polymer with $N=128$ monomers for MC times $t={10}^4$, $2\times {10}^4,\dots$, and $9\times {10}^4$ (from narrowest to the widest distribution), which are significantly shorter than the mean translocation time. The results are obtained from 10 000 independent runs. Continuous lines represent Gaussian fits to these distributions.

Figure 2

(Color online) The mean squared displacement of the translocation coordinate $s$ as a function of MC time $t$, obtained from ${10}^4$ runs of a 128-monomer polymer. The straight line represents the power-law fit $t^{\alpha }$ in the interval ${10}^4<t<1.2\times {10}^5$ with exponent $\alpha =0.86$.

Figure 3

(Color online) Probability distribution of the translocation variable $s$ of the subset of polymers with $N=128$ monomers that did not complete translocation at MC times $t=40\times {10}^5$, $42\times {10}^5,\dots$, and $48\times {10}^5$ (thin lines), which exceed the mean translocation time. These graphs were obtained by performing ${10}^4$ independent runs out of which only 15–20 % survive to the times when the data is collected. The thick solid line is the average of ten graphs in the range $4\times {10}^6⩽t⩽4.9\times {10}^6$. The dotted line depicts the fit function $A{\sin }^k[s\pi ∕(N+1)]$ with $k=1.44$.

Figure 4

(Color online) Probability distribution of translocation time $T$ for $N=32$, 64, and 128 (left to right), obtained from 100 000, 78 600, and 8600 runs, respectively. The inset demonstrates the collapse of the probabilities when $T$ is scaled with its average value $⟨T⟩$.