# Translocation, Rejection and Trapping of Polyampholytes

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## Abstract

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## 1. Introduction

## 2. Theory

## 3. Model: Monte Carlo Simulation

## 4. Results and Discussion

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

## Appendix B. The Dependency on the Waiting Time

## References

- Uversky, V.N. Intrinsically disordered proteins from A to Z. Intl. J. Biochem. Cell Biol.
**2011**, 43, 1090–1103. [Google Scholar] [CrossRef] [PubMed][Green Version] - Müller-Späth, S.; Soranno, A.; Hirschfeld, V.; Hofmann, H.; Rüegger, S.; Reymond, L.; Nettels, D.; Schuler, B. Charge interactions can dominate the dimensions of intrinsically disordered proteins. Proc. Natl. Acad. Sci. USA
**2010**, 107, 14609–14614. [Google Scholar] [CrossRef] [PubMed][Green Version] - Bianchi, G.; Longhi, S.; Grandori, R.; Brocca, S. Relevance of Electrostatic Charges in Compactness, Aggregation, and Phase Separation of Intrinsically Disordered Proteins. Int. J. Mol. Sci.
**2020**, 21, 6208. [Google Scholar] [CrossRef] [PubMed] - Sorensen, C.S.; Kjaergaard, M. Effective concentrations enforced by intrinsically disordered linkers are governed by polymer physics. Proc. Natl. Acad. Sci. USA
**2019**, 116, 23124–23131. [Google Scholar] [CrossRef] - Meller, A.; Nivon, L.; Branton, D. Voltage-driven DNA translocations through a nanopore. Phys. Rev. Lett.
**2001**, 86, 3435. [Google Scholar] [CrossRef][Green Version] - Palyulin, V.V.; Ala-Nissila, T.; Metzler, R. Polymer translocation: The first two decades and the recent diversification. Soft Matter
**2014**, 10, 9016–9037. [Google Scholar] [CrossRef][Green Version] - Muthukumar, M. Polymer Translocation; CRC Press: Boca Raton, FL, USA, 2016. [Google Scholar]
- Payet, L.; Martinho, M.; Merstorf, C.; Pastoriza-Gallego, M.; Pelta, J.; Viasnoff, V.; Auvray, L.; Muthukumar, M.; Mathé, J. Temperature effect on ionic current and ssDNA transport through nanopores. Biophys. J.
**2015**, 109, 1600–1607. [Google Scholar] [CrossRef][Green Version] - Kubota, T.; Lloyd, K.; Sakashita, N.; Minato, S.; Ishida, K.; Mitsui, T. Clog and Release, and Reverse Motions of DNA in a Nanopore. Polymers
**2019**, 11, 84. [Google Scholar] [CrossRef][Green Version] - Hsiao, P.Y. Polyelectrolyte Threading through a Nanopore. Polymers
**2016**, 8, 73. [Google Scholar] [CrossRef][Green Version] - Hsiao, P.Y. Conformation Change, Tension Propagation and Drift-Diffusion Properties of Polyelectrolyte in Nanopore Translocation. Polymers
**2016**, 8, 378. [Google Scholar] [CrossRef][Green Version] - Song, L.; Hobaugh, M.R.; Shustak, C.; Cheley, S.; Bayley, H.; Gouaux, J.E. Structure of staphylococcal α-hemolysin, a heptameric transmembrane pore. Science
**1996**, 274, 1859–1865. [Google Scholar] [CrossRef] [PubMed] - Abdolvahab, R.H.; Metzler, R.; Ejtehadi, M.R. First passage time distribution of chaperone driven polymer translocation through a nanopore: Homopolymer and heteropolymer cases. J. Chem. Phys.
**2011**, 135, 245102. [Google Scholar] [CrossRef] [PubMed][Green Version] - Ansalone, P.; Chinappi, M.; Rondoni, L.; Cecconi, F. Driven diffusion against electrostatic or effective energy barrier across α-hemolysin. J. Chem. Phys.
**2015**, 143, 154109. [Google Scholar] [CrossRef] [PubMed] - Wong, C.T.A.; Muthukumar, M. Polymer translocation through α-hemolysin pore with tunable polymer-pore electrostatic interaction. J. Chem. Phys.
**2010**, 133, 07B607. [Google Scholar] [CrossRef][Green Version] - Jeon, B.J.; Muthukumar, M. Electrostatic control of polymer translocation speed through α-hemolysin protein pore. Macromolecules
**2016**, 49, 9132–9138. [Google Scholar] [CrossRef][Green Version] - Jou, I.; Muthukumar, M. Effects of nanopore charge decorations on the translocation dynamics of DNA. Biophys. J.
**2017**, 113, 1664–1672. [Google Scholar] [CrossRef][Green Version] - Katkar, H.H.; Muthukumar, M. Effect of charge patterns along a solid-state nanopore on polyelectrolyte translocation. J. Chem. Phys.
**2014**, 140, 04B602_1. [Google Scholar] [CrossRef][Green Version] - Mirigian, S.; Wang, Y.; Muthukumar, M. Translocation of a heterogeneous polymer. J. Chem. Phys.
**2012**, 137, 064904. [Google Scholar] [CrossRef][Green Version] - Kumar, R.; Chaudhuri, A.; Kapri, R. Sequencing of semiflexible polymers of varying bending rigidity using patterned pores. J. Chem. Phys.
**2018**, 148, 164901. [Google Scholar] [CrossRef][Green Version] - Kapahnke, F.; Schmidt, U.; Heermann, D.W.; Weiss, M. Polymer translocation through a nanopore: The effect of solvent conditions. J. Chem. Phys.
**2010**, 132, 164904. [Google Scholar] [CrossRef] - Noskov, S.Y.; Im, W.; Roux, B. Ion Permeation through the a-Hemolysin Channel: Theoretical Studies Based on Brownian Dynamics and Poisson-Nernst-Plank Electrodiffusion Theory. Biophys. J.
**2004**, 87, 2299–2309. [Google Scholar] [CrossRef] [PubMed][Green Version] - Roux, B.; Allen, T.; Bernèche, S.; Im, W. Theoretical and computational models of biological ion channels. Q. Rev. Biophys.
**2004**, 37, 15–103. [Google Scholar] [CrossRef] [PubMed][Green Version] - Ghosh, B.; Sarabadani, J.; Chaudhury, S.; Ala-Nissila, T. Pulling a folded polymer through a nanopore. J. Phys. Condens. Matter
**2021**, 33, 015101. [Google Scholar] [CrossRef] [PubMed] - Katkar, H.H.; Muthukumar, M. Single molecule electrophoresis of star polymers through nanopores: Simulations. J. Chem. Phys.
**2018**, 149, 163306. [Google Scholar] [CrossRef] - Lenart, W.R.; Kong, W.; Oltjen, W.C.; Hore, M.J. Translocation of soft phytoglycogen nanoparticles through solid–state nanochannels. J. Mater. Chem. B
**2019**, 7, 6428–6437. [Google Scholar] [CrossRef] - Khunpetch, P.; Man, X.; Kawakatsu, T.; Doi, M. Translocation of a vesicle through a narrow hole across a membrane. J. Chem. Phys.
**2018**, 148, 134901. [Google Scholar] [CrossRef] - Saltzman, E.J.; Muthukumar, M. Conformation and dynamics of model polymer in connected chamber-pore system. J. Chem. Phys.
**2009**, 131, 214903. [Google Scholar] [CrossRef][Green Version] - Johner, A.; Joanny, J.F. Translocation of polyampholytes and intrinsically disordered proteins. Eur. Phys. J. E
**2018**, 41, 78. [Google Scholar] [CrossRef] - Lee, N.K.; Jung, Y.; Johner, A.; Joanny, J.F. Globular Polyampholytes: Structure and Translocation. Macromolecules
**2021**, 54, 2394–2411. [Google Scholar] [CrossRef] - Buyukdagli, S.; Sarabadani, J.; Ala-Nissila, T. Dielectric Trapping of Biopolymers Translocating through Insulating Membranes. Polymers
**2018**, 33, 015101. [Google Scholar] [CrossRef][Green Version] - Bouchaud, J.P.; Comtet, A.; Georges, A.; Le Doussal, P. Classical diffusion of a particle in a one-dimensional random force field. Ann. Phys.
**1990**, 201, 285–341. [Google Scholar] [CrossRef] - Sinai, Y.G. The limiting behavior of a one-dimensional random walk in a random medium. Theory Probab. Appl.
**1983**, 27, 256–268. [Google Scholar] [CrossRef] - Kesten, H. The limit distribution of Sinai’s random walk in random environment. Phys. A
**1986**, 138, 299–309. [Google Scholar] [CrossRef] - Comtet, A.; Dean, D.S. Exact results on Sinai’s diffusion. J. Phys. A
**1998**, 31, 8595. [Google Scholar] [CrossRef] - Delyon, F.; Luciani, J.F. Behavior of general one-dimensional diffusion processes. J. Stat. Phys.
**1989**, 54, 1065–1080. [Google Scholar] [CrossRef] - Sun, L.Z.; Cao, W.P.; Wang, C.H.; Xu, X. The translocation dynamics of the polymer through a conical pore: Non-stuck, weak-stuck, and strong-stuck modes. J. Chem. Phys.
**2021**, 154, 054903. [Google Scholar] [CrossRef] - Rowghanian, P.; Grosberg, A.Y. Electrophoretic capture of a DNA chain into a nanopore. Phys. Rev. E
**2013**, 87, 042722. [Google Scholar] [CrossRef][Green Version] - Muthukumar, M. Theory of capture rate in polymer translocation. J. Chem. Phys.
**2010**, 132, 05B605. [Google Scholar] [CrossRef][Green Version] - Buyukdagli, S.; Ala-Nissila, T. Controlling polymer capture and translocation by electrostatic polymer-pore interactions. J. Chem. Phys.
**2017**, 147, 114904. [Google Scholar] [CrossRef]

**Figure 1.**(

**a**) Schematic representation of an $\alpha $-hemolysin pore. The mushroom-shaped complex is approximately 10 nm long. The colored patches represent charged regions inside the pore, a positively charged cis protrusion and a negatively charged trans edge. (

**b**) Translocation free energy (top) and corresponding translocation force (bottom) for small ions, K${}^{+}$ (red) and Cl${}^{-}$ (blue), in $\alpha $-hemolysin, with a transmembrane potential of $+150$ mV (solid lines) and $-150$ mV (dashed lines). The potential is in favor of translocation of negative (positive) ions from cis-to-trans (reverse) direction with $+150$ mV. The free energy values are taken from Figure 7 of Ref. [22] and those values used for the MC simulations are indicated by circles. (

**c**) Schematics showing the initial position of the PA chain in the MC simulation for translocation from the cis to the trans side.

**Figure 2.**(

**a**) The translocation time distributions for 2${}^{20}$ sequences of N = 20. The number of translocated sequences are measured in the time interval [$t-\delta t/2$, $t+\delta t/2$] with $\delta t$ = $1.6\times {10}^{3}$ MCT. The symbols ○ and Δ represent the cis-to-trans and reverse directions, respectively. (

**b**) Distributions of rejection times, measured within the time interval [$t-\delta t/2$, $t+\delta t/2$] with $\delta t$ = 20 MCT. Dashed lines are guides for the power-law relation, $P\left(t\right)\sim {t}^{-1}$. (

**c**) Logarithmic decay of trapped populations, ${\Pi}_{\mathrm{trap}}\left(t\right)$, normalized by the total number of translocation trials.

**Figure 3.**Distributions of translocation times (in the reverse direction) for various Q-ensembles of N = 20. The PDF measures the fraction of translocated sequences at given time interval [$t-\delta t/2$, $t+\delta t/2$] with $\delta t$ = 1600 MCT for each Q-ensemble. Each distribution is normalized by the total number of successfully translocated sequences with ${t}_{w}$ = $1.6\times {10}^{5}$ MCT. Colors from yellow to blue represent charge values of Q = 0, 2, 4, 6, 8, 10, 12 and 14, respectively. The dashed lines are a guide for the eyes indicating power-law relations, $P\left(t\right)\sim {t}^{-(1+\mu )}$ with $\mu =0$ and 1. For each Q, we indicate the average translocation times $\langle {t}_{\mathrm{tr}}\rangle $ by ○. The $\langle {t}_{\mathrm{tr}}\left(Q\right)\rangle $ decreases with increasing net charges. The square symbols for Q = 6, 8, 10, 12 and 14 represent the average translocation times $\langle {t}_{\mathrm{tr}}\left(Q\right)\rangle $ with ${t}_{w}$ = ${10}^{6}$ MCT.

**Figure 4.**The average translocation times $\langle {t}_{\mathrm{tr}}\rangle $ in the cis-to-trans (□) and reverse (○) directions as a function of the net charge Q for N = 20. The + symbols represent the standard deviations of the corresponding data. The averages are obtained from the successful translocation trials with waiting times ${t}_{w}=1.6\times {10}^{5}$ (blue) and ${10}^{6}$ (green) MCTs, respectively. The filled symbols ($Q>15$) indicate the convergence of data independent of the waiting time.

**Figure 5.**The percentages of translocated/rejected/trapped sequences (in the reverse direction) with ${t}_{w}$ = $1.6\times {10}^{5}$ MCT (

**a**) for exactly enumerated N = 20 sequences and (

**b**) for ${10}^{7}$ randomly created sequences of N = 40. The top panels show the dependencies on ${Q}_{h}$ and Q and the bottom panels show the dependencies on ${Q}_{min}$ and Q. Color codes are presented in neighboring color bars.

**Figure 6.**The PDFs of translocation times of PA (sequence (d) in Table 3) and IDP IN sequences (sequence (h) and (i) in Table 3) engaging in the cis-to-trans direction with ${t}_{w}$ = $1.6\times {10}^{5}$ MCT. The number of translocated sequences are measured in the time interval [$t-\delta t/2$, $t+\delta t/2$] with $\delta t$ = $1.0\times {10}^{3}$ MCT. The PDFs of sequence (d) and (h) clearly show exponential decay as a function of time.

**Table 1.**Free energy values of ions relative to free solution. The positions in the pore are indicated in Figure 1.

Position | $-\mathit{a}$ | 0 | a | $2\mathit{a}$ | $3\mathit{a}$ | $4\mathit{a}$ | $5\mathit{a}$ |
---|---|---|---|---|---|---|---|

Free energy of cations (${k}_{\mathrm{B}}T$) | 5.53 | 6.98 | 5.00 | 4.65 | 1.59 | 2.32 | 0.07 |

Free energy of anions (${k}_{\mathrm{B}}T$) | 0.07 | 3.35 | 8.82 | 9.08 | 7.47 | 7.00 | 5.87 |

**Table 2.**Comparison of translocation times for the given populations of N = 20 engaging in the cis-to-trans and reverse directions. The time resolution is given by the binning size $\delta t$ = 400 MCT.

Translocated Population | 1% | 2% | 4% | 6% | 8% | 10% |
---|---|---|---|---|---|---|

${t}_{\mathrm{tr}}$ (MCT), cis-to-trans | 400 | 800 | 2400 | 7200 | 35,600 | 300,000 |

${t}_{\mathrm{tr}}$ (MCT), reverse | 400 | 800 | 3600 | 9600 | 39,200 | 436,000 |

**Table 3.**Comparison of translocation times of specific sequences with Q = 8 and N = 20 (a–g) and sequences of IDP IN (h,i) with Q = 4 and N = 16, engaging in the cis-to-trans direction under $+150$ mV of electric potential. (Favorable and antagonistic charges are labeled as 1 and $-1$, respectively.) Antagonistic charges are highlighted as red. The statistics were obtained with ${t}_{w}=1.6\times {10}^{5}$ from 10,000 different translocation trials for each sequence. (a) Regular sequence, (b) 1 block of (-1-1), (c) reverse sequence of (b), (d) 2 blocks of (-1-1), (e) 3 blocks of (-1-1), (f) 1 block of (-1-1-1), (g) reverse sequence of (f), (h) sequence of IN, (i) sequence of IN, reverse of (h).

Sequences | ${\mathit{Q}}_{\mathbf{min}}$ | ${\mathit{Q}}_{\mathit{h}}$ | $\overline{{\mathit{t}}_{\mathbf{tr}}}$ (MCT) | ${\mathit{\sigma}}_{\mathbf{th}}$ (MCT) | Success Rate (%) | Trapped Rate (%) | |
---|---|---|---|---|---|---|---|

a | 11-111-111-111-111-111-111 | 1 | 3 | 66 | 18 | 86 | 0 |

b | 11-1-11111-1111-111-11-111 | 1 | 1 | 1050 | 860 | 55 | 0 |

c | 11-11-111-1111-11111-1-111 | 1 | 1 | 340 | 170 | 66 | 0 |

d | 11-1-11111-1-111-111-11111 | $-1$ | 1 | 2250 | 1590 | 54 | 0 |

e | 11-1-1111-1-11111-1-111111 | 1 | 1 | 4270 | 2490 | 31 | 0 |

f | 111-1-1-111-11111-11111-11 | $-1$ | 1 | 69,740 | 45,450 | 34 | 47 |

g | 111-11111-1111-1111-1-1-11 | $-1$ | 3 | 1440 | 1280 | 99 | 0 |

h | 11-1111-1-11 -111-1-111 | $-1$ | 3 | 3670 | 3460 | 84 | 0 |

i | 11-1-111-11-1-1111-111 | $-1$ | 1 | 7070 | 6360 | 37 | 0 |

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**MDPI and ACS Style**

Kim, Y.-B.; Chae, M.-K.; Park, J.-M.; Johner, A.; Lee, N.-K. Translocation, Rejection and Trapping of Polyampholytes. *Polymers* **2022**, *14*, 797.
https://doi.org/10.3390/polym14040797

**AMA Style**

Kim Y-B, Chae M-K, Park J-M, Johner A, Lee N-K. Translocation, Rejection and Trapping of Polyampholytes. *Polymers*. 2022; 14(4):797.
https://doi.org/10.3390/polym14040797

**Chicago/Turabian Style**

Kim, Yeong-Beom, Min-Kyung Chae, Jeong-Man Park, Albert Johner, and Nam-Kyung Lee. 2022. "Translocation, Rejection and Trapping of Polyampholytes" *Polymers* 14, no. 4: 797.
https://doi.org/10.3390/polym14040797