Hydrodynamic correlations in the translocation of a biopolymer through a nanopore: Theory and multiscale simulations

We investigate the process of biopolymer translocation through a narrow pore using a multiscale approach which explicitly accounts for the hydrodynamic interactions of the molecule with the surrounding solvent. The simulations confirm that the coupling of the correlated molecular motion to hydrodynamics results in significant acceleration of the translocation process. Based on these results, we construct a phenomenological model which incorporates the statistical and dynamical features of the translocation process and predicts a power-law dependence of the translocation time on the polymer length with an exponent $\alpha \approx 1.2$. The actual value of the exponent from the simulations is $\alpha =1.28\pm 0.01$, which is in excellent agreement with experimental measurements of DNA translocation through a nanopore, and is not sensitive to the choice of parameters in the simulation. The mechanism behind the emergence of such a robust exponent is related to the interplay between the longitudinal and transversal dynamics of both translocated and untranslocated segments. The connection to the macroscopic picture involves separating the contributions from the blob shrinking and shifting processes, which are both essential to the translocation dynamics.

Figure 1

(Color online) Illustration of the interactions in the DNA translocation model: The beads representing the DNA are shown as small black dots connected by straight line segments, the wall is represented by a plane of on-lattice points (small white dots separated by the lattice spacing $\Delta x$) which repel the DNA beads within a range of interaction. The bead-bead interactions are indicated by the blue spheres surrounding the small black ones, defined in Eq. (2), except for the distance between consecutive beads which is fixed at $b$. Interactions between the beads and the wall are indicated by the red spheres surrounding the small white ones, defined in Eq. (2). Interaction between the beads and the constant external field are confined over a shaded (in green) region around the pore (see text for details).

Figure 2

(Color online) Scaling of the translocation time $\tau$ with the number of beads $N_0$, in the presence (circles) or absence (squares) of coupling between the molecule and the solvent. The exponents are $1.28\pm 0.01$ and $1.36\pm 0.03$, respectively.

Figure 3

3D view of an ensemble of 100 polymers with $N_0=300$ at different stages of the process: $A$, $B$, and $C$ correspond to the initial, midpoint, and final translocation times.

Figure 4

(Color online) Longitudinal (filled circles) and transverse (open squares) components of the gyration tensor for (a) the $U$ and (b) the $T$ segment with the number of untranslocated $\left(N_U\right)$ and translocated $\left(N_T\right)$ beads, respectively. The dashed lines show the scaling.

Figure 5

(Color online) (a) Synergy factors with time for the hydrodynamic ($S_H^{\left(T\right)}$, $S_H^{\left(U\right)}$, total $S_H$) and electric field $\left(S_E\right)$. (b) Probability distributions of $S_H,S_E$ during translocation events. Curves are averages over 100 events for $N_0=300$.