^{1}and Gary W. Slater

^{1}

### Abstract

Noting the limitations of the standard characterization of translocation dynamics, an incremental mean first passage process methodology is used to more completely map the unbiased translocation of a polymer through a nanopore. In this approach, the average time *t* _{0} required to reach successively increasing displacements for the first time is recorded – a measure shown to be more commensurate with the mean first passage nature of translocation. Applying this methodology to the results of Langevin dynamics simulations performed in three dimensions across a range of viscosities, a rich set of dynamics spanning regular diffusion at low viscosities to sub-diffusion at higher viscosities is revealed. Further, while the scaling of the net translocation time τ with polymer length *N* is shown to be viscosity-dependent, common regimes are found across all viscosities: super-diffusive behaviour at short times, an *N*-independent backbone consistent with τ ∼ *N* ^{2.0} at low viscosities and τ ∼ *N* ^{2.2} at higher viscosities for intermediate times, and *N*-dependent deviations from the backbone near the completion of translocation.

Simulations were performed using the ESPResso package^{33} on the SHARCNET computer system (www.sharcnet.ca) using VMD^{34} for visualization.

I. INTRODUCTION

II. SYSTEM SETUP

III. RESULTS

A. The exponents α and β

B. Limitations of α and β

C. The IMFPP approach

D. IMFPP results

1. = 0.1

2. = 5.0

3. = 1.0

IV. CONCLUSION

##### B82B1/00

## Figures

Schematic of translocation. *s* represents the translocation coordinate by the number of monomers translocated to the right hand side. The translocation time τ is defined by the time required for the polymer to move from half-way *s* _{0} = *N*/2 to either *s* = 0 or *s* = *N*. Although this image is in 2D for clarity, all simulations are done in 3D.

Schematic of translocation. *s* represents the translocation coordinate by the number of monomers translocated to the right hand side. The translocation time τ is defined by the time required for the polymer to move from half-way *s* _{0} = *N*/2 to either *s* = 0 or *s* = *N*. Although this image is in 2D for clarity, all simulations are done in 3D.

α exponent derived from power law fits τ ∼ *N* ^{α} at different viscosities .

α exponent derived from power law fits τ ∼ *N* ^{α} at different viscosities .

The β exponent value derived from ⟨Δ*s*(*t*)^{2}⟩ ∼ *t* ^{β} for four polymer lengths at the selected viscosities: = 0.1, = 1.0, = 5.0.

The β exponent value derived from ⟨Δ*s*(*t*)^{2}⟩ ∼ *t* ^{β} for four polymer lengths at the selected viscosities: = 0.1, = 1.0, = 5.0.

Sample translocation trajectories for *N* = 25 at given by *s*(*t*) plotted with *t* along the vertical axis. The shortest and longest trajectories (labelled) are shown along with eight other trajectories selected from an ensemble of 1000 events. The mean time along with the standard deviation of the mean (shaded region) are also indicated. The spread of this select set is indicative of the full ensemble distribution with the difference between the longest trajectory and the shortest trajectory being much larger than the mean.

Sample translocation trajectories for *N* = 25 at given by *s*(*t*) plotted with *t* along the vertical axis. The shortest and longest trajectories (labelled) are shown along with eight other trajectories selected from an ensemble of 1000 events. The mean time along with the standard deviation of the mean (shaded region) are also indicated. The spread of this select set is indicative of the full ensemble distribution with the difference between the longest trajectory and the shortest trajectory being much larger than the mean.

Normalized probability density function as a function of *s* for times t = 50, 100, 500, 1000, and 2000 at *N* = 49 and = 0.1. Approximate Gaussian fits are shown as dashed black lines. For early times (t = 50 and t = 100), the curves closely match a Gaussian curve with a width increasing as time progresses. Once translocation events begin occurring, data are only available for trajectories in which translocation has not yet occurred and the curves appear to collapse to a single curve with a constant width.

Normalized probability density function as a function of *s* for times t = 50, 100, 500, 1000, and 2000 at *N* = 49 and = 0.1. Approximate Gaussian fits are shown as dashed black lines. For early times (t = 50 and t = 100), the curves closely match a Gaussian curve with a width increasing as time progresses. Once translocation events begin occurring, data are only available for trajectories in which translocation has not yet occurred and the curves appear to collapse to a single curve with a constant width.

Mean square displacement of the translocation coordinate versus time for = 0.1 and *N* = 25.

Mean square displacement of the translocation coordinate versus time for = 0.1 and *N* = 25.

Schematic representing the process of converting *s*(*t*) into *t* _{0}(Δ*s*).

Schematic representing the process of converting *s*(*t*) into *t* _{0}(Δ*s*).

Average initial time *t* _{0} to displacement Δ*s* at a viscosity of for polymer lengths ranging from *N* = 25 to *N* = 299. The top set of curves display data from LD simulations. The bottom set of curves correspond to the equivalent data for the same polymer lengths as generated from using the exact methodology to solve the 1D model of translocation.

Average initial time *t* _{0} to displacement Δ*s* at a viscosity of for polymer lengths ranging from *N* = 25 to *N* = 299. The top set of curves display data from LD simulations. The bottom set of curves correspond to the equivalent data for the same polymer lengths as generated from using the exact methodology to solve the 1D model of translocation.

Average time ⟨*t* _{0}⟩ to displacement Δ*s* at a viscosity of for polymer lengths ranging from *N* = 25 to *N* = 149.

Average time ⟨*t* _{0}⟩ to displacement Δ*s* at a viscosity of for polymer lengths ranging from *N* = 25 to *N* = 149.

Average initial time ⟨*t* _{0}⟩ to displacement |Δ*s*| at a viscosity of for polymers lengths ranging from *N* = 25 to *N* = 299.

Average initial time ⟨*t* _{0}⟩ to displacement |Δ*s*| at a viscosity of for polymers lengths ranging from *N* = 25 to *N* = 299.

Brownian dynamics results for the average time ⟨*t* _{0}⟩ to displacement Δ*s* at a viscosity of for the polymer lengths *N* = 25,49,75,99.

Brownian dynamics results for the average time ⟨*t* _{0}⟩ to displacement Δ*s* at a viscosity of for the polymer lengths *N* = 25,49,75,99.

Normalized probability density function as a function of *s* for times t = 50, 100, 500, 1000, and 2000 at *N* = 49 and = 0.1. The forms predicted from Eq. (A2) are shown as black dashed lines.

Normalized probability density function as a function of *s* for times t = 50, 100, 500, 1000, and 2000 at *N* = 49 and = 0.1. The forms predicted from Eq. (A2) are shown as black dashed lines.

Analytic solution for the normalized peak height and probability of being absorbed plotted as a function of time for *a* = 24 and *D* = 0.25.

Analytic solution for the normalized peak height and probability of being absorbed plotted as a function of time for *a* = 24 and *D* = 0.25.

Analytic solution at *a* = 24 and *D* = 0.25 and LD simulation results at *N* = 49 and for the normalized peak height and probability of being absorbed plotted as a function of time.

Analytic solution at *a* = 24 and *D* = 0.25 and LD simulation results at *N* = 49 and for the normalized peak height and probability of being absorbed plotted as a function of time.

Analytic solution at *a* = 24 and *D* = 0.05 and LD simulation results at *N* = 49 and for the normalized peak height and probability of being absorbed plotted as a function of time.

Analytic solution at *a* = 24 and *D* = 0.05 and LD simulation results at *N* = 49 and for the normalized peak height and probability of being absorbed plotted as a function of time.

## Tables

α and β exponents at the three selected viscosity values.

α and β exponents at the three selected viscosity values.

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