^{1}, Alessandro Fiasconaro

^{2}, Dominique Persano Adorno

^{1,3}and Bernardo Spagnolo

^{1,3,a)}

### Abstract

We study the translocation dynamics of a short polymer moving in a noisy environment and driven by an oscillating force. The dynamics is numerically investigated by solving a Langevin equation in a two-dimensional domain. We consider a phenomenological cubic potential with a metastable state to model the polymer-pore interaction and the entropic free energy barrier characterizing the translocation process. The mean first translocation time of the center of inertia of polymers shows a nonmonotonic behavior, with a minimum, as a function of the number of the monomers. The dependence of the mean translocation time on the polymer chain length shows a monotonically increasing behavior for high values of the number of monomers. Moreover, the translocation time shows a minimum as a function of the frequency of the oscillating forcing field for all the polymer lengths investigated. This finding represents the evidence of the resonant activation phenomenon in the dynamics of polymer translocation, whose occurrence is maintained for different values of the noise intensity.

This work was partially supported by MIUR and partially by the Spanish DGICYT Project No. FIS2011-25167, co-financed by FEDER funds.

I. INTRODUCTION

II. POLYMER DYNAMICS MODEL

III. RESULTS AND DISCUSSION

IV. CONCLUSIONS

### Key Topics

- Polymers
- 90.0
- Free energy
- 12.0
- DNA
- 10.0
- Probability density functions
- 7.0
- Thermal noise
- 7.0

##### C12

## Figures

3D-view of the phenomenological potential energy *U* _{Ext}, which is included in our system to simulate the presence of a barrier to be surmounted by the polymer during the translocation dynamics. The inset shows the projection of *U* _{Ext} on the *z*-*x* plane (solid line); dashed lines indicate the maximum and the minimum shape of the potential barrier caused by the presence of the oscillating forcing field.

3D-view of the phenomenological potential energy *U* _{Ext}, which is included in our system to simulate the presence of a barrier to be surmounted by the polymer during the translocation dynamics. The inset shows the projection of *U* _{Ext} on the *z*-*x* plane (solid line); dashed lines indicate the maximum and the minimum shape of the potential barrier caused by the presence of the oscillating forcing field.

MFTT vs. frequency of the forcing field for seven different values of the number of monomers, namely, *N* = 15, 18, 20, 32, 40, 50, 60. The noise intensity is *D* = 1.0. The values of the potential energy parameters are: *K* _{r} = *K* _{θ} = 20, ε_{LJ} = 0.1, σ = 3, and *d* = 5, in arbitrary units (AU). The amplitude of the forcing field is *A* = 2 × 10^{−2} (AU).

MFTT vs. frequency of the forcing field for seven different values of the number of monomers, namely, *N* = 15, 18, 20, 32, 40, 50, 60. The noise intensity is *D* = 1.0. The values of the potential energy parameters are: *K* _{r} = *K* _{θ} = 20, ε_{LJ} = 0.1, σ = 3, and *d* = 5, in arbitrary units (AU). The amplitude of the forcing field is *A* = 2 × 10^{−2} (AU).

Mean first translocation time (MFTT) as a function of the number of monomers for three different values of the angular frequency of the forcing field, namely, ω = 0.0001, 0.01, 0.1. For polymer chain length *N* > 40, a monotonic increasing behavior of MFTT vs *N* is observed. All other parameter values are the same as those of Fig. 2 .

Mean first translocation time (MFTT) as a function of the number of monomers for three different values of the angular frequency of the forcing field, namely, ω = 0.0001, 0.01, 0.1. For polymer chain length *N* > 40, a monotonic increasing behavior of MFTT vs *N* is observed. All other parameter values are the same as those of Fig. 2 .

Standard deviation (SD) of the first translocation time (FTT) as a function of the frequency of the oscillating field, for the same values of polymer length plotted in Fig. 2 , except for *N* = 60, whose curve essentially overlaps that with *N* = 50 and, for this reason, it was not plotted, and *N* = 24 which instead was added. (Inset) The SD of FTT versus the number of monomers for three different values of the frequency of the forcing field, namely, ω = 0.0001, 0.1, 1. All other parameter values are the same as those of Fig. 2 .

Standard deviation (SD) of the first translocation time (FTT) as a function of the frequency of the oscillating field, for the same values of polymer length plotted in Fig. 2 , except for *N* = 60, whose curve essentially overlaps that with *N* = 50 and, for this reason, it was not plotted, and *N* = 24 which instead was added. (Inset) The SD of FTT versus the number of monomers for three different values of the frequency of the forcing field, namely, ω = 0.0001, 0.1, 1. All other parameter values are the same as those of Fig. 2 .

Probability density function (PDF) of the first translocation time (FTT). Each panel shows three PDFs, each one characterized by a specific value of the number of monomers, namely, *N* = 15, 30, 60. The three panels differs for the frequency of the forcing field: (a) low frequency domain (ω = 0.0001); (b) resonant activation region (ω = 0.01); (c) high frequency domain (ω = 1.0). All other parameter values are the same as those of Fig. 2 .

Probability density function (PDF) of the first translocation time (FTT). Each panel shows three PDFs, each one characterized by a specific value of the number of monomers, namely, *N* = 15, 30, 60. The three panels differs for the frequency of the forcing field: (a) low frequency domain (ω = 0.0001); (b) resonant activation region (ω = 0.01); (c) high frequency domain (ω = 1.0). All other parameter values are the same as those of Fig. 2 .

Mean first translocation time of polymers having constant length (*N* = 30) as a function of the frequency of the oscillating field, for three different values of the noise intensity, namely *D* = 0.25, 0.5, 1.0. The inset shows the standard deviation (SD) of the FTTs. All other parameter values are the same as those of Fig. 2 .

Mean first translocation time of polymers having constant length (*N* = 30) as a function of the frequency of the oscillating field, for three different values of the noise intensity, namely *D* = 0.25, 0.5, 1.0. The inset shows the standard deviation (SD) of the FTTs. All other parameter values are the same as those of Fig. 2 .

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