# Dynamics of Polymer Translocation: A Short Review with an Introduction of Weakly-Driven Regime

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## Abstract

**:**

## 1. Introduction

## 2. Unbiased Translocation

#### 2.1. Memory Function for Stress Relaxation

#### 2.2. Unbiased Translocation Dynamics

#### 2.3. Traveled Fraction at $t={\tau}_{p}$

#### 2.4. Post-Propagation Stage

## 3. Weakly-Driven Dynamics

#### 3.1. Weakly-Driven Translocation Dynamics

#### 3.2. Post-Propagation Stage

## 4. Strongly-Driven Dynamics

## 5. Summary and Discussions

#### 5.1. Summary of the Scaling Formulae

#### 5.1.1. Unbiased and Weakly-Driven Regimes

#### 5.1.2. Strongly-Driven Regime

#### 5.2. Discussion

## Acknowledgments

## Conflicts of Interest

## Appendix A.

#### Appendix A.1.

#### Appendix A.2.

#### Appendix A.3.

#### Appendix A.4.

## References

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**Figure 1.**Illustration of a translocating polymer and the transport operation ($\Delta n=1$) to measure the stress relaxation associated with the tension propagation. The translocation coordinate is defined as the monomer label $n(t)$ at the pore, which counts the number of monomers already in the trans side at time t in analogy to the reaction coordinate in chemical reaction process [7,8,23,24].

**Figure 2.**Sketch of a translocating polymer driven by a strong force f. (

**left**) Propagation stage: a growing moving domain (with velocity $V(t)$ and the size $R(t)$) on the cis side is shaded, while the chain portion already on the trans side is represented by a dashed curve. (

**right**) Post-propagation stage: the tension has already reached to far end of the polymer; thus, $m(t)=N$ is constant, and the moving domain is shrinking with time. In addition, most of monomers are already on the trans side, so this post-propagation stage adds a finite-size correction to the scaling formula of the translocation time.

**Figure 3.**Dependence of the translocation time on f (

**left**) and N (

**right**) shown in double logarithmic scale. The locations of various regimes, unbiased (UB), weakly-driven (WD) and strongly-driven (SD) are specified; the SD regime is further divided into the trumpet (T) and the stem–flower (S-F) regimes. Note that in the right graph, depicted is the case with $f<{k}_{B}T/a$; otherwise, we have only S-F regime with the slope $1+\nu $. Note also that in these plots, we set $z=2+{\nu}^{-1}$ (free draining dynamics), in which case the plots become particularly simple. The triangles and their nearby numbers designate slopes (exponents), where the numbers after the arrows specify the values for free draining dynamics. For other choices of the dynamical exponent (i.e., $z=3$ for nondraining (Zimm) dynamics), the slope in the SD (T) regime in the left graph is changed. The same applies to UB and WD regimes in the right graph. Comparing these plots with Figure 4 in Reference [16], one finds differences in weak force and short chain length regions.

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Sakaue, T.
Dynamics of Polymer Translocation: A Short Review with an Introduction of Weakly-Driven Regime. *Polymers* **2016**, *8*, 424.
https://doi.org/10.3390/polym8120424

**AMA Style**

Sakaue T.
Dynamics of Polymer Translocation: A Short Review with an Introduction of Weakly-Driven Regime. *Polymers*. 2016; 8(12):424.
https://doi.org/10.3390/polym8120424

**Chicago/Turabian Style**

Sakaue, Takahiro.
2016. "Dynamics of Polymer Translocation: A Short Review with an Introduction of Weakly-Driven Regime" *Polymers* 8, no. 12: 424.
https://doi.org/10.3390/polym8120424