Conformation Change, Tension Propagation and Drift-Diffusion Properties of Polyelectrolyte in Nanopore Translocation

Using Langevin dynamics simulations, conformational, mechanical and dynamical properties of charged polymers threading through a nanopore are investigated. The shape descriptors display different variation behaviors for the cis- and trans-side sub-chains, which reflects a strong cis-trans dynamical asymmetry, especially when the driving field is strong. The calculation of bond stretching shows how the bond tension propagates on the chain backbone, and the chain section straightened by the tension force is determined by the ratio of the direct to the contour distances of the monomer to the pore. With the study of the waiting time function, the threading process is divided into the tension-propagation stage and the tail-retraction stage. At the end, the drift velocity, diffusive property and probability density distribution are explored. Owing to the non-equilibrium nature, translocation is not a simple drift-diffusion process, but exhibits several intermediate behaviors, such as ballistic motion, normal diffusion and super diffusion, before ending with the last, negative-diffusion behavior.


Mapping Translocation Time to Real Time
In this study, we choose: σ = 2.38 × 10 −10 m m = 200 g/mol = 3.32 × 10 −25 kg τ u = 2.13 × 10 −12 s e = 1.602 × 10 −19 C as the length, mass, time and charge units of our simulation system, respectively. The mean translocation time is investigated under different conditions [44]. For example, at the weak driving field E = 0.2 k B T/(eσ), the mean translocation time is τ = 88757.9 τ u for N = 384. It yields an average threading time 231.1 τ u per monomer, which corresponds to 0.492 ns in the real time unit. This average threading time is about one to two orders of magnitude shorter than a typical threading time, 5 to 30 ns per base pair, observed in DNA translocation experiments [58]. The reason for this discrepancy can be attributed to the setting of the monomer friction coefficient ζ to a small value of 1.0 mτ −1 u in the Langevin dynamics simulations (refer to Equation (4) in the paper [44]). In an aqueous solution, the friction coefficient for a base pair can be estimated by Stokes' law ζ w = 3πµd, which gives a value of 8.48 × 10 −12 kg · s −1 if we take the water viscosity µ = 9 × 10 −4 Pa · s and set the monomer diameter to d 1 nm. Therefore, an appropriate value for ζ should be ζ w = 54.4 mτ −1 u , which is about 50-times larger than the current value. The small value of ζ was used voluntarily, for the purpose of increasing the particle moving speed, which reduces the needed simulation steps and, thus, accelerates the threading process. Therefore, the results related to time in the simulations should be primarily corrected by multiplying the factor 54.4. The obtained average threading time per monomer is hence corrected to be 26.8 ns, which agrees well with the experiments.

η for an Ideal Chain Forming a Sphere
Let R be the radius of the sphere. Assume that the chain ends are randomly distributed inside the sphere and that the probability density to find a monomer is a constant, P( r) = 1/( 4 3 πR 3 ). We can calculate the mean square of the radius of gyration from the formula: where the triple integral has been performed in the spherical coordinates (r, θ, φ). The mean square of the end-to-end distance can be calculated by: Apply the law of cosines | r 1 − r N | 2 = r 2 1 + r 2 N − 2r 1 r N cos θ 1N , where θ 1N is the angle between vectors r 1 and r N . Change the variable θ 1 to the variable θ 1N and perform the integration for the variables θ N , φ N and φ 1 . We obtain: The shape factor η ≡ R 2 e / R 2 g can be then computed and is equal to two.

η for an Ideal Chain Forming a Disk
Let R be the disk radius. Assume the chain ends are randomly distributed inside the disk and that the probability density to find a monomer is P( r) = 1/(πR 2 ). Similar to the derivation in Section 2.1, the integral can be done in the polar coordinates (r, φ). We obtain: which yields η = 2, as well.

Azimuthal Angle of the Principal Axis of the Sub-chains
The averaged azimuthal angle φ of the principal axis of the sub-chains in the cis region (I) and trans region (III) are plotted in Figure S1. Owing to the symmetry in the transverse direction, the azimuthal curve fluctuates around 0 • .

Variation of the Tension Force at Weak Driving Fields
The tension force f n for each bond n at E = 0.2, 1.0 and 2.0 k B T/(eσ) were calculated and presented in Figure S2. For analysis, the averaged direct distance D n and the averaged contour distance Λ n for each monomer n to the pore were also plotted in the same figure. We can see that the thermal fluctuation blurs out the surge of tension, and therefore, no tension front is observed. Λ n is significantly larger than D n so that the sub-chain is not straightened near the pore entrance. The direction of theñ-axis is reversed so that the monomers entering the trans-region stay on the right-hand side of the plot, while the cis monomers rest on the left-hand side. The sky-blue region indicates the monomers in the pore region. The direct distance D n and the contour distance Λ n to the pore are plotted in red and green colors, respectively. The values of D n and Λ n are read from the right y-axis in the figure.

Notes on the Log-Normal Distribution
Given two parameters µ and σ, the log-normal distribution is defined by: It is the probability density function of a random variable X, whose logarithm satisfies the Gaussian (normal) distribution with the mean µ and the standard deviation σ. The maximum (or called "the mode") of the log-normal distribution occurs at x = x max ≡ µ exp(−σ 2 ), and the value is: The median of the distribution occurs at x = µ because µ 0 p(x) dx = 0.5, and the mean is located at x = x = µ exp(σ 2 /2). In general, the n-th moment of the log-normal variable X is given by: x n ≡ ∞ 0 x n p(x) dx = µ n exp n 2 σ 2 2 (8)