Driven polymer translocation in good and bad solvent: Effects of hydrodynamics and tension propagation

We investigate the driven polymer translocation through a nanometer-scale pore in the presence and absence of hydrodynamics both in good and bad solvent. We present our results on tension propagating along the polymer segment on the cis side that is measured for the first time using our method that works also in the presence of hydrodynamics. For simulations we use stochastic rotation dynamics, also called multiparticle collision dynamics. We find that in the good solvent the tension propagates very similarly whether hydrodynamics is included or not. Only the tensed segment is by a constant factor shorter in the presence of hydrodynamics. The shorter tensed segment and the hydrodynamic interactions contribute to a smaller friction for the translocating polymer when hydrodynamics is included, which shows as smaller waiting times and a smaller exponent in the scaling of the translocation time with the polymer length. In the bad solvent hydrodynamics has a minimal effect on polymer translocation, in contrast to the good solvent, where it speeds up translocation. We find that under bad-solvent conditions tension does not spread appreciably along the polymer. Consequently, translocation time does not scale with the polymer length. By measuring the effective friction in a setup where a polymer in free solvent is pulled by a constant force at the end, we find that hydrodynamics does speed up collective polymer motion in the bad solvent even more effectively than in the good solvent. However, hydrodynamics has a negligible effect on the motion of individual monomers within the highly correlated globular conformation on the cis side and hence on the entire driven translocation under bad-solvent conditions.

Figure 1

Schematic depiction of the geometry used in the translocation simulations.

Figure 2

Snapshots of translocating polymers of length $N=1600$. Initial conformations on the left. Pore force $F=1$. Hydrodynamics is included. Top row, good solvent; bottom row, bad solvent. The polymer translocates from the bottom cis side to the top trans side. On the bottom row the pore region is shown by a blue rectangle. The wall whose thickness is equal to the pore length extends horizontally to the left and right from the pore (not shown). Due to a different length scale the pore does not show on the first row. In the leftmost snapshot only three monomers are inside the pore on the top. In the following snapshots the pore and the wall lie just below the upper region of crowded monomers.

Figure 3

Translocation times $\tau$ as a function of polymer length $N$ for good solvent (GS) and bad solvent (BS) with and without hydrodynamics (HD). (a) $F=1$; (b) $F=3$.

Figure 4

Measured radii of gyration of the translocating polymers on the cis side. The equilibrium $R_g$ measured for seven discrete polymer lengths and the solid line giving the obtained scaling $R_g\sim N^{\nu \left(\mathrm{measured}\right)}$ are shown for reference. Hydrodynamics is included. $F=10$. (a) Good solvent, $\nu \left(\mathrm{measured}\right)\approx 0.63$. (b) Bad solvent, $\nu \left(\mathrm{measured}\right)\approx 0.32$. The chain lengths are [from bottom to top in (a)] $N=25,50,100,200,400,800,1600$.

Figure 5

Good solvent. Measured $R_g$ as a function of the translocation coordinate on the trans (plots increasing with $s$) and cis (plots decreasing as $s$ increases) sides. The equilibrium values for the radius of gyration $R_g^{\text{eq}}$ are are plotted with solid black lines. Hydrodynamics is included, $N=400$. (a) $F=1$; (b) $F=3$.

Figure 6

A schematic five-bead polymer segment showing the measured drag distance over the three middle beads.

Figure 7

(Top row and bottom left) The drag distances $d_i$ of each bead $i$ as a function of the translocation coordinate $s$. $F=10$ and $N=400$. (Top left) Good solvent with hydrodynamics. (Top right) Good solvent without hydrodynamics. (Bottom left) Bad solvent with hydrodynamics. (Bottom right) A logarithmic plot of the number of beads in the tensed segment $n_t$ on the cis side as a function of $s$. $F=3$ and $N=1600$. Curves from top to bottom: Good solvent without hydrodynamics, good solvent with hydrodynamics, bad solvent without hydrodynamics, and bad solvent with hydrodynamics. The dotted line shows the scaling $\sim s^{0.5}$.

Figure 8

Comparison of the number of beads in the tensed segment $n_t$ and the waiting times $t_w$ as a function of the translocation coordinate $s$ in good solvent. $F=3$. (a) Hydrodynamics not included. $N=400$. (b) Hydrodynamics included. $N=400$. (c) Logarithmic plots of $n_t\left(s\right)$ and $t_w\left(s\right)$ with (H) and without hydrodynamics (NH). $N=800$. (d) $n_t$ as a function of the normalized reaction coordinate $s/N$ with no hydrodynamics. $F=3$. From bottom to top $N=50$, 100, 200, 400, 800, and 1600.

Figure 9

Good solvent, $N=800$. (a) The length of the tensed segment on the cis side during translocation $n_t\left(s\right)$. Curves from bottom to top: (i) $F=1$, with hydrodynamics (HD); (ii) $F=1$, no HD; (iii) $F=3$, with HD; (iv) $F=3$, no HD; (v),(vi) $F=10$, with and without HD. (b) Tension profiles $n_t\left(s\right)$ with and without HD. $n_t\left(s\right)$ with HD are scaled, $A{n}_t\left(s\right)$, such that perfect alignment with no-HD profiles are obtained. $A=1.5$ for $F=1$ (lower curves) and $A=1.1$ and $F=3$ (upper curves).

Figure 10

Terminal velocity times polymer length $Nv$ as a function of dragging force $f_{\mathrm{drag}}$ for polymers of different lengths. Upper plots, hydrodynamics included; lower plots, hydrodynamics not included. The dashed lines are linear fits to the data for $N=25$. (a) Good solvent; (b) bad solvent. In the case of bad solvent with hydrodynamics the least-squared fitted lines are plotted to show the deviation from the relation $Nv\sim f_{\mathrm{drag}}$ [Eq. (6)].

Figure 11

Translocation in bad solvent, no hydrodynamics. Logarithmic plot of cumulative waiting times $T_w\left(s\right)$ for $F=1$ (upper curves) and 3 (lower curves). Polymer lengths $N=100$, 200, 400, 800, and 1600.

Figure 12

A snapshot of a simulation with bad solvent at the reaction coordinate value $s=1400$. A trefoil knot (emphasized in red) is seen at the pore entrance. $N=1600$ and $F=3$. For improved visibility the polymer is depicted with a tube of diameter 0.2 that is smaller than the diameter of 1 implied by the Lennard-Jones interaction range.