Quantification of tension to explain bias dependence of driven polymer translocation dynamics

Motivated by identifying the origin of the bias dependence of tension propagation, we investigate methods for measuring tension propagation quantitatively in computer simulations of driven polymer translocation. Here, the motion of flexible polymer chains through a narrow pore is simulated using Langevin dynamics. We measure tension forces, bead velocities, bead distances, and bond angles along the polymer at all stages of translocation with unprecedented precision. Measurements are done at a standard temperature used in simulations and at zero temperature to pin down the effect of fluctuations. The measured quantities were found to give qualitatively similar characteristics, but the bias dependence could be determined only using tension force. We find that in the scaling relation $\tau \sim N^{\beta }{f}_d^{\alpha }$ for translocation time $\tau$, the polymer length $N$, and the bias force $f_d$, the increase of the exponent $\beta$ with bias is caused by center-of-mass diffusion of the polymer toward the pore on the cis side. We find that this diffusion also causes the exponent $\alpha$ to deviate from the ideal value $-1$. The bias dependence of $\beta$ was found to result from combination of diffusion and pore friction and so be relevant for polymers that are too short to be considered asymptotically long. The effect is relevant in experiments all of which are made using polymers whose lengths are far below the asymptotic limit. Thereby, our results also corroborate the theoretical prediction by Sakaue's theory [Polymers 8, 424 (2016)] that there should not be bias dependence of $\beta$ for asymptotically long polymers. By excluding fluctuations we also show that monomer crowding at the pore exit cannot have a measurable effect on translocation dynamics under realistic conditions.

Figure 1

A two-dimensional cross section of the three-dimensional simulation setup. A bead-spring chain of length $N=25$ is driven through a small pore in a membrane from the cis (bottom) to the trans (top) side. Driving force $f_d$ is applied to the polymer segment in the pore ($f_d$ is divided among the beads currently occupying the pore). The membrane is $3\sigma$ thick and the pore is $1\sigma$ wide at the narrowest point. Polymer beads are indexed as $i=1$ for the first bead to translocate and $i=N$ for the last bead to translocate. In the figure, 14 beads have crossed to the trans side, so the translocation coordinate is $s=14$. The bead radius used in the figure was chosen for clarity. The interaction distance between a monomer and a wall is measured from the center of the bead.

Figure 2

Snapshots of a translocating polymer of $N=400$ beads. The driving force $f_d=32$ and the temperature $kT=1$. The three snapshots from left to right are taken at the start of the simulation, at the time when half of the polymer has translocated, and at the end of the simulation. (Snapshots were created using VMD [23] and POV-Ray [24].)

Figure 3

Illustrations of the five quantities for measuring tension propagation. From left to right: $f_t, v_p, v_c, d_{2\mathrm{bd}}$, and $\gamma$. The examples are given for beads $i=24$ and 23 at the state depicted in Fig. 1, where $s=14$.

Figure 4

Illustration of a color chart on tension propagation. The $x$-axis value $s$ is the translocation coordinate. The $y$-axis value $i$ is the index of the bead. Index 1 corresponds to the end of the polymer that translocates first and index $i=N=200$ corresponds to the end that translocates last. The color gives the ensemble average value of the measured quantity $o(s,i)$ for the given values of $i$ and $s$. The points on the diagonal from the bottom left corner to the top right corner correspond to beads currently in the pore. Points above the diagonal correspond to beads on the cis side and points below the diagonal to beads on the trans side.

Figure 5

Tension propagation charts for the five ensemble averaged quantities. From left to right: tension force $f_t$, velocity towards pore $v_p$, velocity along polymer contour $v_c$, two-bond distance $d_{2\mathrm{bd}}$, and bond angle $\gamma$. From top to bottom: $f_d=2$, 8, and 64.

Figure 6

Derived quantities for the tension propagation data. From left to right: ${\widehat{f}}_t, {\widehat{v}}_p, {\widehat{v}}_c, {\widehat{d}}_{2\mathrm{bd}}$, and $\widehat{\gamma }$. From top to bottom: $f_d=2$, 8, and 64.

Figure 7

The length of the tensed segment $n_d$ as a function of translocation coordinate $s$. (a) $n_d$ curves derived from $\widehat{f_t}$ charts of Fig. 6. (b) $n_d$ curves calculated from waiting times $t_w$ as $n_d=t_w{f}_d/\left(\xi m\sigma \right)$. The curves from top to bottom correspond to forces from $f_d=64$ to 2.

Figure 8

$\widehat{o}$ measured for polymers of length $N=200$ when bead number 100 is in the pore, i.e., $s=100$. ${\widehat{f}}_t, {\widehat{v}}_p, {\widehat{v}}_c, {\widehat{d}}_{2\mathrm{bd}}$, and $\widehat{\gamma }$ are plotted as a function of distance from the pore in number beads $i-s$ along the polymer chain. This way $(i-s)=0$ is the bead in the pore and $(i-s)=100$ is the last bead of the polymer on the cis side.

Figure 9

$\widehat{o}$ for ZFM measured for polymers of length $N=200$ when bead number 100 is in the pore, i.e., $s=100$. ${\widehat{f}}_t, {\widehat{v}}_p, {\widehat{v}}_c, {\widehat{d}}_{2\mathrm{bd}}$, and $\widehat{\gamma }$ are plotted as explained in the caption of Fig. 8.

Figure 10

The $z$ component of the cis side center-of-mass position, i.e., the distance from the pore, as a function of translocation coordinate $s$. The pore midpoint is at $r_{\mathrm{z}}=0$. (a) The curves from top to bottom are for FM and correspond to driving force from $f_d=2$ to 64. (b) Differences between SRD simulations with hydrodynamics (HD) and without hydrodynamics (noHD). For comparison, a curve for Langevin dynamics (LD) using the same friction coefficient as SRD is also given. The three uppermost curves are for $f_d=8$ and the three curves at the bottom are for $f_d=64$. See text for details.

Figure 11

Waiting times $t_w$ for ZFM (a) for $f_d=2$ and (b) for $f_d=64$. Solid lines are for the unmodified ZFM and the dashed lines for ZFM (no trans). $N=50$, 100, 200, and 400.

Figure 12

Ratios of translocation times for the full (FM) and zero-fluctuation models (ZFM) $\tau /{\tau }_{\mathrm{ZFM}}$ for $N=50$, 100, 200, and 400.

Figure 13

Schematic waiting time curves for ZFM (black, upper) and FM [red (gray), lower]. Area $\mathbf{A}$ corresponds to contribution to total translocation time $\tau$ from tension propagation and post propagation in FM. Area $\mathbf{B}$ is the same as $\mathbf{A}$ but for ZFM. Area $\mathbf{C}$ corresponds to contribution to $\tau$ from pore friction.

Figure 14

$f_d·t_w$ vs $s$ shown to compare QSM to simulations of FM. The constant 40 was added to the data for QSM to align the curves for QSM and FM for $f_d=64$. $N=200$.