# Polymer Translocation through Nanometer Pores

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theoretical Model

#### 2.1. Charged Polymer Translocation Process

- computation of time-evolution of translocation τ,
- obtaining of the polymer flux through the pore J,

- (I)
- the free energy barrier estimation,
- (II)
- the polymer evolution equations,
- (III)
- the flux probability of translocation across the barrier.

#### 2.1.1. Free Energy

_{1}− μ

_{2}is the chemical potential difference per segment between inside and outside of the cell.

_{m}, in the case of macromolecule suite with m pieces (monomers) located in the cis area, conform to [10,11], will be

#### 2.1.2. Polymer Translocation through Pores

_{0}= ct and

#### 2.1.3. Flux of Polymers

#### 2.1.4. Results and Discussion I

_{1}is the frictional coefficient per pieces (monomer) in the nanopore (more precisely physical reciprocation between monomer and the membrane nanopore).

#### 2.2. Linear Polymers Transport under Constant Force

^{−1}) is dimensional scalar and M (Kg) is the diagonal matrix of masses, respectively. From a dimensional point of view, the product among $\gamma M$ represents the friction general tensor.

#### Results and Discussion II

^{ν}/k

_{B}T ≲ 1, where ν ≈ 0.588 is the Flory exponent for the polymer, we find that τN, the mean time the polymer takes to leave the pore, scales as N

^{2+ν}independent of F, in agreement with our earlier result for F = 0. At strong forces, i.e., for FN

^{ν}/k

_{B}T ≫ 1, the behavior of the passage time crosses over to τN∼N

^{2}/F. We show here that these behaviors stem from the polymer dynamics at the immediate vicinity of the pore, in particular the memory effects in the polymer chain tension imbalance across the pore [26,27].

^{6}, 7 ∗ 10

^{6}], in V/m units) and monomers number N of polymer (in the range [5, 50], [1, 5] and [5, 100]), on the translocation time in a three-dimensional graphic representation, are evaluated.

_{50}, because an undesirable DNA constituent inhabit in the nanopore blocks the current channel. The nanopore escape incidents are arranged in a pair of ably-localized categories with distinct blockage currents (flux intensity, undoubtedly), whose origin is still unclear, inexplicable. Another immediate parallel with the above study is detailed by the certitude that poly(dA) molecules show a top propensity to constitute a single-stranded base-stacked helices in comparison with poly(dC) [29]. The translocation process for two different nucleic acids of the hetero-DNAs poly(dAndCn) type has also been studied. The translocation time was experimentally measured (escape time, analogy) via nanomembrane in a complex investigation, and its histograms were drawn [30].

## 3. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Ambjörnsson, T.; Apell, S.P.; Konkoli, Z.; Di Marzio, E.A.; Kasianowicz, J.J. Charged polymer membrane translocation. J. Chem. Phys.
**2002**, 117, 4063–4073. [Google Scholar] [CrossRef] - Sung, W.; Park, P.J. Polymer Translocation through a Pore in a Membrane. Phys. Rev. Lett.
**1996**, 77, 783. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Paun, V.P. An estimation of the polymer translocation time through membrane articol. Mater. Plast.
**2006**, 43, 57–58. [Google Scholar] - Muthukumar, M. Polymer translocation through a hole. J. Chem. Phys.
**1999**, 111, 10371–10374. [Google Scholar] [CrossRef] - Kong, C.Y.; Muthukumar, M. Monte Carlo study of adsorption of a polyelectrolyte onto charged surfaces. J. Chem. Phys.
**1998**, 109, 1522. [Google Scholar] [CrossRef] - Muthukumar, M. Translocation of a Confined Polymer through a Hole. Phys. Rev. Lett.
**2001**, 86, 3188–3191. [Google Scholar] [CrossRef] [PubMed] - Sofos, F.; Karakasidis, T.E.; Liakopoulos, A. How wall properties control diffusion in grooved nanochannels: A molecular dynamics study. Heat Mass Transf.
**2013**, 49, 1081–1088. [Google Scholar] [CrossRef] - Sofos, F.; Karakasidis, T.E.; Spetsiotis, D. Molecular Dynamics simulations of ion separation in nano-channel water flows using an electric field. Mol. Simul.
**2019**, 45, 1395–1402. [Google Scholar] [CrossRef] - Sofos, F.; Karakasidis, T.E.; Liakopoulos, A. Fluid flow at the nanoscale: How fluid properties deviate from the bulk. Nanosci. Nanotechnol. Lett.
**2013**, 5, 457–460. [Google Scholar] [CrossRef] - Doi, M.; Edwards, S.F. Theory of Polymer Dynamics; Clarendon Press: Oxford, UK, 1986. [Google Scholar]
- Ma, S.K. Statistical Mechanic; World Scientific: Singapore, 1985. [Google Scholar]
- Kasianowicz, J.J.; Brandin, E.; Branton, D.; Deamer, D.W. Characterization of individual polynucleotide molecules using a membrane channel. Proc. Natl. Acad. Sci. USA
**1996**, 93, 13770–13773. [Google Scholar] [CrossRef] [Green Version] - Meller, A.; Nivon, L.; Branton, D. Voltage-driven DNA translocations through a nanopore. Phys. Rev. Lett.
**2001**, 86, 3435–3438. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Paun, V.P. Theoretical Study of the Polymer Transport through Nanopores. Rev. Chim.
**2006**, 57, 221–223. [Google Scholar] - Buyukdagli, S.; Sarabadani, J.; Ala-Nissila, T. Theoretical Modeling of Polymer Translocation: From the Electrohydrodynamics of Short Polymers to the Fluctuating Long. Polymers
**2019**, 11, 118. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Ghosh, B.; Chaudhury, S. Translocation Dynamics of an Asymmetrically Charged Polymer through a Pore under the Influence of Different pH Conditions. J. Phys. Chem. B
**2019**, 123, 4318–4323. [Google Scholar] [CrossRef] [PubMed] - Buyukdagli, S.; Ala-Nissila, T. Controlling polymer capture and translocation by electrostatic polymer-pore interactions. J. Chem. Phys.
**2017**, 147, 114904. [Google Scholar] [CrossRef] - Paun, V.P. Polymer dynamics simulation at nanometer scale in a 2D diffusion model. Mater. Plast.
**2007**, 44, 393–395. [Google Scholar] - Paun, V.-P.; Chiroiu, V.; Munteanu, L. Polymer Transport Process through Biological Membranes with Nanometric Pores. In “Progress in Nanoscience and Nanotechnologies” Monograph; Kleps, I., Ion, A.C., Dascalu, D., Eds.; Romanian Academy Press: Bucharest, Romanian, 2007; pp. 267–275. [Google Scholar]
- Hamidabad, M.N.; Abdolvahab, R.H. Translocation through a narrow pore under a pulling force. Sci. Rep.
**2019**, 9, 17885. [Google Scholar] [CrossRef] [Green Version] - Menais, T. Polymer translocation under a pulling force: Scaling arguments and threshold forces. Phys. Rev. E
**2018**, 97, 022501. [Google Scholar] [CrossRef] [Green Version] - Paun, V.P. Two-dimensional diffusion model for the biopolymers dynamics at nanometer scale. Open Phys.
**2009**, 7, 607–613. [Google Scholar] [CrossRef] - Ermak, D.L.; Buckholz, H. Numerical integration of the Langevin equation: Monte Carlo simulation. J. Comput. Phys.
**1980**, 35, 169–182. [Google Scholar] [CrossRef] - Van Gunsteren, W.F.; Berendsen, H.J.C. Algorithms for Brownian dynamics. Mol. Phys.
**1982**, 45, 637–647. [Google Scholar] [CrossRef] - Brünger, A.; Brooks III, C.L.; Karplus, M. Stochastic boundary conditions for molecular dynamics simulations of ST2 water. Chem. Phys. Lett.
**1984**, 105, 495–500. [Google Scholar] [CrossRef] - Bou-Rabee, N. Time Integrators for Molecular Dynamics. Entropy
**2014**, 16, 138–162. [Google Scholar] [CrossRef] [Green Version] - Hamidabad, M.N.; Asgari, S.; Abdolvahab, R.H. Nanoparticle-assisted polymer translocation through a nanopore. Polymer
**2020**, 204, 122847. [Google Scholar] [CrossRef] - Meller, A.; Nivon, L.; Brandin, E.; Golovchenko, J.A.; Branton, D. Rapid nanopore discrimination between single polynucleotide molecules. Proc. Natl. Acad. Sci. USA
**2000**, 97, 1079. [Google Scholar] [CrossRef] [Green Version] - Meller, A.; Branton, D. Single Molecule Measurements of DNA Transport through a Nanopore. Electrophoresis
**2002**, 23, 2583. [Google Scholar] [CrossRef] - Luo, K.F.; Ala-Nissila, T.; Ying, S.C.; Bhattacharya, A. Sequence dependence of DNA translocation through a nanopore. Phys. Rev. Lett.
**2008**, 100, 058101. [Google Scholar] [CrossRef] [Green Version]

**Figure 1.**Polymer in transition through nanometer pore: (

**a**) The charged polymer is found in the cis zone (region I), (

**b**) The polymer has m monomers in the trans area and N-m monomers in the cis area, (

**c**) The charged polymer is entirely found in the trans zone (region II).

**Figure 2.**(

**a**) Polymer escape in transition, (

**b**) Associated free energy barrier (m segments in region II). m* is the value for which F(m*) = F*, respectively Fmaxim, meaning the maximum value of the function F, associated free energy barrier.

**Figure 4.**The polymer configurations of numerical computation, for N = 50. Initial position: polymer in cis zone (

**left side**); final position: polymer in trans zone (

**right side**).

**Figure 5.**Simultaneous influence of both external electric field E (in the range [10

^{6}, 8 ∗ 10

^{6}]) and monomers number N of polymer (in the range [1, 5]), on the translocation time.

**Figure 6.**Simultaneous influence of both external electric field E (in the range [10

^{6}, 7 ∗ 10

^{6}]) and monomers number N of polymer (in the range [5, 50]), on the translocation time. Two different angles of presentation.

**Figure 7.**Simultaneous influence of both external electric field E (in the range [10

^{6}, 7 ∗ 10

^{6}]) and monomers number N of polymer (in the range [5, 100]), on the translocation time. Two different angles of presentation.

**Figure 8.**Translocation times histogram: (

**a**) poly(dAdC)

_{64}and (

**b**) poly(dA

_{64}dC

_{64}) under F = 0.5.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Paun, M.-A.; Paun, V.-A.; Paun, V.-P.
Polymer Translocation through Nanometer Pores. *Polymers* **2022**, *14*, 1166.
https://doi.org/10.3390/polym14061166

**AMA Style**

Paun M-A, Paun V-A, Paun V-P.
Polymer Translocation through Nanometer Pores. *Polymers*. 2022; 14(6):1166.
https://doi.org/10.3390/polym14061166

**Chicago/Turabian Style**

Paun, Maria-Alexandra, Vladimir-Alexandru Paun, and Viorel-Puiu Paun.
2022. "Polymer Translocation through Nanometer Pores" *Polymers* 14, no. 6: 1166.
https://doi.org/10.3390/polym14061166