Translocation time of periodically forced polymer chains

In this work we study the presence of both a minimum and clear oscillations in the frequency dependence of the translocation time of a polymer described as a unidimensional Rouse chain driven by a spatially localized oscillating linear potential. The observed oscillations of the mean translocation time arise from the synchronization between the very mean translocation time and the period of the external force. We have checked the robustness of the frequency value for the minimum translocation time by changing the damping parameter, finding a very simple relationship between this frequency and the correspondent translocation time. The translocation time as a function of the polymer length has been also evaluated, finding a precise $L^2$ scaling. Furthermore, the role played by the thermal fluctuations described as a Gaussian uncorrelated noise has been also investigated, and the analogies with the resonant activation phenomenon are commented.

Figure 1

(Color online) Scheme of the oscillating pushing force acting on a polymer chain formed by N monomers. The force drives only a small fraction of the whole chain.

Figure 2

(Color online) Log plot of the mean first passage time $\tau$ as a function of frequency of the oscillating force for the damping parameter $\gamma =1$. The minimum and the oscillating behavior is showed in the inset. The thermal noise intensity is $D=0.01$. The first minimum, which is also the global one, satisfies the very robust condition: ${\nu }_m^{-1}=T_m=\alpha {\tau }_m$, were $\alpha$ is a constant.

Figure 3

(Color online) Standard deviation $\sigma$ (left), mean velocity (center), and mean difference between the rest distance $d_0$ and the distance of the monomers during the dynamics (right) related to Fig. 2. The negative values of $d_{med}$ means that the springs are compressed. The oscillation appears in both the mean velocity and the elongation difference.

Figure 4

(Color online) Mean first passage time $\tau$ as a function of frequency of the oscillating force for the damping parameter $\gamma =1$ with various values of the initial phase $\varphi$, namely, $\varphi =0.01,\pi /4,\pi /2,3\pi /4,5\pi /4,3\pi /2,7\pi /4$. A minimum is always present. Moreover a very strong oscillating behavior appears in the region before the high frequency saturation value. The thermal noise intensity is $D=0.01$.

Figure 5

(Color online) Resonant MFPT $\left({\tau }_m\right)$ vs resonant period $\left(T_m\right)$ for all the case investigated. The data concerns the parameters $\gamma =0.01,0.1,1,10,15,20$ and also the overdamped case with $N=12$. Furthermore, for $\gamma =1$, the different polymer length $N=12,18,24,36$. All the points lie on the linear relation given by Eq. (9).

Figure 6

(Color online) The minimum and the oscillating behavior is present independently on the damping parameter $\gamma$. The figure shows the calculations for different $\gamma$ values: 0.5, 1, 10, 20, plotted following the scaling indicated in the legend of the axis. $\gamma =10$ approaches very well the overdamped limit (also shown). In the inset we show the curves without scaling.

Figure 7

(Color online) MFPT as a function of $\nu$ for different values of the polymer length $L$, namely $N=12$, 18, 24, 36, and 120. When axis is properly scaled all the original curves (shown in the inset) superimpose upon each other.

Figure 8

(Color online) Range of the oscillating effect for different thermal noise intensity, namely: $D=0,0.1,0.5,0.1$. The oscillating behavior tends to disappear by increasing the temperature. The maxima decrease and the minima increase.

Figure 9

(Color online) A force pull is applied at the first monomer to measure the stall force of the potential.

Figure 10

(Color online) Stall force as a function of the frequency of the oscillating driving for $\gamma =0.1$. The other parameters are the same of Fig. 7. The inset shows the mean velocity versus frequency curves for two values of the pull force, $F_p=-0.446$ and $-0.452$.

Figure 11

(Color online) Stall force as a function of the frequency of the oscillating driving for $\gamma =1$. The two minima displayed in Fig. 10 have disappeared, and now only a minimum appears. The other parameters are the same of Fig. 7. The inset shows the mean velocity versus frequency curves for three values of the pull force, $F_p=-0.446$, $-0.450$, and $-0.454$.