Langevin dynamics simulations of polymer translocation through nanopores

We investigate the dynamics of polymer translocation through a nanopore using two-dimensional Langevin dynamics simulations. In the absence of an external driving force, we consider a polymer which is initially placed in the middle of the pore and study the escape time tau(e) required for the polymer to completely exit the pore on either side. The distribution of the escape times is wide and has a long tail. We find that tau(e) scales with the chain length N as tau(e) approximately N(1+2nu), where nu is the Flory exponent. For driven translocation, we concentrate on the influence of the friction coefficient xi, the driving force E, and the length of the chain N on the translocation time tau, which is defined as the time duration between the first monomer entering the pore and the last monomer leaving the pore. For strong driving forces, the distribution of translocation times is symmetric and narrow without a long tail and tau approximately E(-1). The influence of xi depends on the ratio between the driving and frictional forces. For intermediate xi, we find a crossover scaling for tau with N from tau approximately N(2nu) for relatively short chains to tau approximately N(1+nu) for longer chains. However, for higher xi, only tau approximately N(1+nu) is observed even for short chains, and there is no crossover behavior. This result can be explained by the fact that increasing xi increases the Rouse relaxation time of the chain, in which case even relatively short chains have no time to relax during translocation. Our results are in good agreement with previous simulations based on the fluctuating bond lattice model of polymers at intermediate friction values, but reveal additional features of dependency on friction.     

Effect of charge distribution on the translocation of an inhomogeneously charged polymer through a nanopore

We investigate the voltage-driven translocation of an inhomogeneously charged polymer through a nanopore by utilizing discrete and continuous stochastic models. As a simplified illustration of the effect of charge distribution on translocation, we consider the translocation of a polymer with a single charged site in the presence and absence of interactions between the charge and the pore. We find that the position of the charge that minimizes the translocation time in the absence of pore-polymer interactions is determined by the entropic cost of translocation, with the optimum charge position being at the midpoint of the chain for a rodlike polymer and close to the leading chain end for an ideal chain. The presence of attractive and repulsive pore-charge interactions yields a shift in the optimum charge position toward the trailing end and the leading end of the chain, respectively. Moreover, our results show that strong attractive or repulsive interactions between the charge and the pore lengthen the translocation time relative to translocation through an inert pore. We generalize our results to accommodate the presence of multiple charged sites on the polymer. Our results provide insight into the effect of charge inhomogeneity on protein translocation through biological membranes.     

Polymer translocation through a nanopore: a two-dimensional Monte Carlo study

We investigate the problem of polymer translocation through a nanopore in the absence of an external driving force. To this end, we use the two-dimensional fluctuating bond model with single-segment Monte Carlo moves. To overcome the entropic barrier without artificial restrictions, we consider a polymer which is initially placed in the middle of the pore and study the escape time tau required for the polymer to completely exit the pore on either end. We find numerically that tau scales with the chain length N as tau approximately N(1+2nu), where nu is the Flory exponent. This is the same scaling as predicted for the translocation time of a polymer which passes through the nanopore in one direction only. We examine the interplay between the pore length L and the radius of gyration R(g). For LR(g), we find tau approximately N. In addition, we numerically find the scaling function describing crossover between short and long pores. We also show that tau has a minimum as a function of L for longer chains when the radius of gyration along the pore direction R( parallel) approximately L. Finally, we demonstrate that the stiffness of the polymer does not change the scaling behavior of translocation dynamics for single-segment dynamics.     

Nondriven Polymer Translocation Through a Nanopore:Scaling for Translocation Time with Chain Length

We investigated the dynamics of the passage for a polymer chain through a nanopore in the absence of any external driving force with Weeks-Chandler-Andersen potential in two-dimensional simulations, in particular, focused our attention on the scaling law of the mean translocation time. We found that the effect of hydrodynamic interactions is the major factor in determining the scaling exponents with increasing pore size. The scaling close to N1+2v was observed when the hydrodynamic interactions were screened in the cases of small pore sizes, while the scaling close to N3v was obtained when the hydrodynamic interactions were present in the cases of large pore sizes.     

The effect of hydrodynamic interactions on the dynamics of DNA translocation through pores

In this work, we investigate the effect of hydrodynamic interactions on the dynamics of DNA translocation through micropores. We simulate DNA as a bead-spring chain and use a lattice Boltzmann method to simulate the flow field that arises from the motion of the molecule. We investigate the free-draining entrance of DNA to the pore by diffusion and find that, consistent with experiments, molecules have a higher probability of entering the pore from one end. We then consider the electric-field driven translocation of 21-210 microm DNA with and without hydrodynamic interactions. Consistent with experiments, we study translocation events that are much shorter than the relaxation time of DNA. We find that the effect of hydrodynamic interactions on this process is to cause different regions of a molecule, other than the ones pulled by voltage or chain connectivity into the pore, to move toward the pore. We quantify this effect and show that it is smaller than the difference in the translocation dynamics of chains that arises from different initial configurations of the molecules. A power-law scaling of translocation time with chain length is observed, with exponents of 1.28+/-0.03 and 1.31+/-0.03 in simulations with and without hydrodynamic interactions, respectively. Our results are in good agreement with recent translocation experiments conducted in small pores and show that, for the regime considered in this work, hydrodynamic interactions play a minor role in the relation of the translocation time to chain length. For fast translocation processes, the effect of hydrodynamic interactions is local and the main factor determining the dynamics of DNA is the initial configuration of the molecules.     

Heteropolymer translocation through nanopores

The authors investigate the translocation dynamics of heteropolymers driven through a nanopore using a constant temperature Langevin thermostat. Specifically, they consider heteropolymers consisting of two types of monomers labeled A and B, which are distinguished by the magnitude of the driving force that they experience inside the pore. From a series of studies on polymers with sequences AmBn the authors identify both universal as well as specific sequence properties of the translocating chains. They find that the scaling of the average translocation time as a function of the chain length N remains unaffected by the heterogeneity, while the residence time of each bead is a strong function of the sequence for short repeat units. They further discover that for a symmetric heteropolymer AnBn of fixed length, the pattern exhibited by the residence times of the individual monomers has striking similarity with a double slit interference pattern where the total number of repeat units N/2n controls the number of interference fringes. These results are relevant for designing nanopore based sequencing techniques.     

Dynamics of polymer translocation into a circular nanocontainer through a nanopore

Using Langevin dynamics simulations, we investigate the dynamics of polymer translocation into a circular nanocontainer through a nanopore under a driving force F. We observe that the translocation probability initially increases and then saturates with increasing F, independent of φ, which is the average density of the whole chain in the nanocontainer. The translocation time distribution undergoes a transition from a Gaussian distribution to an asymmetric distribution with increasing φ. Moreover, we find a nonuniversal scaling exponent of the translocation time as chain length, depending on φ and F. These results are interpreted by the conformation of the translocated chain in the nanocontainer and the time of an individual segment passing through the pore during translocation.     

Simulation study on the translocation of a partially charged polymer through a nanopore

The translocation of a partially charged polymer through a neutral nanopore under external electrical field is studied by using dynamic Monte Carlo method on a simple cubic lattice. One monomer in the polymer is charged and it suffers a driving force when it locates inside the pore. Two time scales, mean first passage time τ(FP) with the first monomer restricted to never draw back into cis side and translocation time τ for polymer continuously threading through nanopore, are calculated. The first passage time τ(FP) decreases with the increase in the driving force f, and the dependence of τ(FP) on the position of charged monomer M is in agreement with the theoretical results using Fokker-Planck equation [A. Mohan, A. B. Kolomeisky, and M. Pasquali, J. Chem. Phys. 128, 125104 (2008)]. But the dependence of τ on M shows a different behavior: It increases with f for M < N/2 with N the polymer length. The novel behavior of τ is explained qualitatively from dynamics of polymer during the translocation process and from the free energy landscape.     

Tunneling time in driven double-well system using entangled molecular dynamics method

We calculate quantum tunneling time in the double-well system without perturbation and with symmetry-breaking driven force using the entangled trajectory molecular dynamics method in the present article. Without perturbation, quantum tunneling time decreased with the increase of the energy, and the different contributions of the barrier traversal time and the intrinsic decay time have been shown. The tunneling time dependence on the amplitude and frequency of the symmetry-breaking driven force are present. In the case of weak driven force, tunneling time has a minimum value in the resonant frequency. For strong driven force, chaos brings a huge change to quantum dynamics, tunneling time significantly becomes short. Finally, we directly show the enhancement of quantum tunneling process by chaotic behavior of entangled trajectories and indicate that it was caused by quantum effect.     

Numerical and theoretical study on the mechanism of biopolymer translocation process through a nano-pore

We conducted a numerical study on the translocation of a biopolymer from the cis side to the trans side of a membrane through a synthetic nano-pore driven by an external electric field in the presence of hydrodynamic interactions (HIs). The motion of the polymer is simulated by 3D Langevin dynamics technique using a worm-like chain model of N identical beads, while HI between the polymer and fluid are incorporated by the lattice Boltzmann equation. The translocation process is induced by electrophoretic force, which sequentially straightens out the folds of the initial random configuration of the polymer chain on the cis side. Our simulation results on translocation time and velocity are in good quantitative agreement with the corresponding experimental ones when the surface charge on the nano-pore and the HI effect are considered explicitly. We found that the translocation velocity of each bead inside the nano-pore mainly depends upon the length of the straightened portion of the polymer in forced motion near the pore. We confirmed this by a theoretical formula. After performing simulations with different pore lengths, we observed that translocation velocity mainly depends upon the applied potential difference rather than upon the electric field inside the nano-pore.     

Polymer translocation in solid-state nanopores: dependence of scaling behavior on pore dimensions and applied voltage

We investigate unforced and forced translocation of a Rouse polymer (in the absence of hydrodynamic interactions) through a silicon nitride nanopore by three-dimensional Langevin dynamics simulations, as a function of pore dimensions and applied voltage. Our nanopore model consists of an atomistically detailed nanopore constructed using the crystal structure of β-Si(3)N(4). We also use realistic parameters in our simulation models rather than traditional dimensionless quantities. When the polymer length is much larger than the pore length, we find the translocation time versus chain length scales as τ ～ N(2+ν) for the unforced case and as τ ～ N((1+2ν)/(1+ν)) for the forced case. Our results agree with theoretical predictions which indicate that memory effects and tension on the polymer chain play an important role during the translocation process. We also find that the scaling exponents are highly dependent on the applied voltage (force). When the length of the polymer is on the order of the length of the pore, we do not find a continuous scaling law, but rather scaling exponents that increase as the length of the polymer increases. Finally, we investigate the scaling behavior of translocation time versus applied voltage for different polymer and pore lengths. For long pores, we obtain the theoretical scaling law of τ ～ 1/V(α), where α ? 1 for all voltages and polymer lengths. For short pores, we find that α decreases for very large voltages and/or small polymer lengths, indicating that the value of α = 1 is not universal. The results of our simulations are discussed in the context of experimental measurements made under different conditions and with differing pore geometries.     

Coupling Effects of Electrostatic Interactions and Salt Concentration Gradient in Polymer Translocation through a Nanopore: A Coarse-Grained Molecular Dynamics Simulations Study

We study the influence of polymer pore interactions and focus on the role played by the concentration gradient of salt in the translocation of polyelectrolytes (PE) through nanopores explicitly using coarse-grained Langevin dynamics simulations. The mean translocation time is calculated by varying the applied voltage, the pH, and the salt concentration gradient. Changing the pH can alter the electrostatic interaction between the protein pore and the polyelectrolyte chain. The polymer pore interaction is weakened by the increase in the strength of the externally applied electric field that drives translocation. Additionally, the screening effect of the salt can reduce the strong charge-charge repulsion between the PE beads which can make translocation faster. The simulation results show there can be antagonistic or synergistic coupling between the salt concentration-induced screening effect and the drift force originating from the salt concentration gradient thereby affecting the translocation time. Our simulation results are explained qualitatively with free energy calculations.     

A Monte Carlo algorithm to study polymer translocation through nanopores. I. Theory and numerical approach

The process during which a polymer translocates through a nanopore depends on many physical parameters and fundamental mechanisms. We propose a new one-dimensional lattice Monte Carlo algorithm that integrates various effects such as the entropic forces acting on the subchains that are outside the channel, the external forces that are pulling the polymer through the pore, and the frictional effects that involve the chain and its environment. Our novel approach allows us to study the polymer as a single Brownian particle diffusing while subjected to a position-dependent force that includes both the external driving forces and the internal entropic bias. Frictional effects outside and inside the pore are also considered. This Monte Carlo method is much more efficient than other simulation methods, and it can be used to obtain scaling laws for various polymer translocation regimes. In this first part, we derive the model and describe a subtle numerical approach that gives exact results for both the escape probability and the mean translocation time (and higher moments of its distribution). The scaling laws obtained from this model will be presented and discussed in the second part of this series.     

Single-molecule electrophoresis of beta-hairpin peptides by electrical recordings and Langevin dynamics simulations

We used single-channel electrical recordings and Langevin molecular dynamics simulations to explore the electrophoretic translocation of various beta-hairpin peptides across the staphylococcal alpha-hemolysin (alphaHL) protein pore at single-molecule resolution. The beta-hairpin peptides, which varied in their folding properties, corresponded to the C terminal residues of the B1 domain of protein G. The translocation time was strongly dependent on the electric force and was correlated with the folding features of the beta-hairpin peptides. Highly unfolded peptides entered the pore in an extended conformation, resulting in fast single-file translocation events. In contrast, the translocation of the folded beta-hairpin peptides occurred more slowly. In this case, the beta-hairpin peptides traversed the alphaHL pore in a misfolded or fully folded conformation. This study demonstrates that the interaction between a polypeptide and a beta-barrel protein pore is dependent on the folding features of the polypeptide.     

Adsorption-driven translocation of polymer chain into nanopores

The polymer translocation into nanopores is generally facilitated by external driving forces, such as electric or hydrodynamic fields, to compensate for entropic restrictions imposed by the confinement. We investigate the dynamics of translocation driven by polymer adsorption to the confining walls that is relevant to chromatographic separation of macromolecules. By using the self-consistent field theory, we study the passage of a chain trough a small opening from cis to trans compartments of spherical shape with adsorption potential applied in the trans compartment. The chain transfer is modeled as the Fokker-Plank diffusion along the free energy landscape of the translocation pass represented as a sum of the free energies of cis and trans parts of the chain tethered to the pore opening. We investigate how the chain length, the size of trans compartment, the magnitude of adsorption potential, and the extent of excluded volume interactions affect the translocation time and its distribution. Interplay of these factors brings about a variety of different translocation regimes. We show that excluded volume interactions within a certain range of adsorption potentials can cause a local minimum on the free energy landscape, which is absent for ideal chains. The adsorption potential always leads to the decrease of the free energy barrier, increasing the probability of successful translocation. However, the translocation time depends non-monotonically of the magnitude of adsorption potential. Our calculations predict the existence of the critical magnitude of adsorption potential, which separates favorable and unfavorable regimes of translocation.     

Modeling the selective partitioning of cations into negatively charged nanopores in water

Partitioning and transport of water and small solutes into and through nanopores are important to a variety of chemical and biological processes and applications. Here we study water structure in negatively charged model cylindrical [carbon nanotube (CNT)-like] nanopores, as well as the partitioning of positive ions of increasing size (Na+, K+, and Cs+) into the pore interior using extensive molecular dynamics simulations. Despite the simplicity of the simulation system-containing a short CNT-like nanopore in water carrying a uniformly distributed charge of qpore=-ne surrounded by n (=0,...,8) cations, making the overall system charge neutral-the results provide new and useful insights on both the pore hydration and ion partitioning. For n=0, that is, for a neutral nanopore, water molecules partition into the pore and form single-file hydrogen-bonded wire spanning the pore length. With increasing n, water molecules enter the pore from both ends with preferred orientations, resulting in a mutual repulsion between oriented water molecules at the pore center and creating a cavity-like low density region at the center. For low negative charge densities on the pore, the driving force for partitioning of positive ions into the pore is weak, and no partitioning is observed. Increasing the pore charge gradually leads to partitioning of positive ions into the pore. Interestingly, over a range of intermediate negative charge densities, nanopores display both thermodynamic as well as kinetic selectivity toward partitioning of the larger K+ and Cs+ ions into their interior over the smaller Na+ ions. Specifically, the driving force is in the order K+>Cs+>Na+, and K+ and Cs+ ions enter the pore much more rapidly than Na+ ions. At higher charge densities, the driving force for partitioning increases for all cations-it is highest for K+ ions-and becomes similar for Na+ and Cs+ ions. The variation of thermodynamic driving force and the average partitioning time with the pore charge density together suggest the presence of free energy barriers in the partitioning process. We discuss the role of ion hydration in the bulk and in the pore interior as well as of the pore hydration in determining the barrier heights for ion partitioning and the observed thermodynamic and kinetic selectivities.     

Dynamics of a droplet imbibing on a rough surface

We consider the imbibition of a liquid droplet of finite size on a rough surface and theoretically show that the imbibition dynamics is significantly slower than the familiar Washburn diffusive dynamics, ～t(0.5). The imbibition does not follow a simple power law. The droplet starts to imbibe with ～t(0.5) dynamics but progressively becomes slower with time. The slower imbibition is mainly attributed to the finite size of the droplet, resulting in a limited capillary driving force as compared to a steady capillary driving force in the case of imbibition from a steady source.     

Capillarity-induced resonance of blobs in porous media: analytical solutions, Lattice-Boltzmann modeling, and blob mobilization

Theoretical considerations and experiments in capillary tubes suggest that blobs exhibit resonance in porous media when they are trapped because of interfacial tension. Here, we investigate the hypothesis that such blobs can be mobilized by exploiting a phenomenon entitled capillarity-induced resonance, that is, by exciting the blobs at their resonant frequency. We used Lattice-Boltzmann (LB) modeling to perform numerical experiments, and we validated the LB model using analytical solutions that approximate the linear response of blobs with pinned menisci in straight and polygonal pore channels to an oscillatory body force. The LB simulations agree well with the quasistatic response, which the analytical solutions describe correctly. Furthermore, the frequency response, particularly the resonant frequency, agrees well, even though the analytical solutions do not accurately estimate viscous pressure drops. Numerical experiments in polygonal and sinusoidal pore channels, as well as disc packings, show that blobs, which are trapped even though a constant body force is applied, can indeed be mobilized by exploiting capillarity-induced resonance. Moreover, the resonant frequency can be estimated in numerical experiments by determining the dominant frequency in the blob amplitude in response to a force pulse. This is of great practical relevance for complex geometries, for which the resonant frequency cannot be easily predicted theoretically.     

The chain sucker: translocation dynamics of a polymer chain into a long narrow channel driven by longitudinal flow

Using analytical techniques and Langevin dynamics simulations, we investigate the dynamics of polymer translocation into a narrow channel of width R embedded in two dimensions, driven by a force proportional to the number of monomers in the channel. Such a setup mimics typical experimental situations in nano/microfluidics. During the translocation process if the monomers in the channel can sufficiently quickly assume steady state motion, we observe the scaling τ ～ N∕F of the translocation time τ with the driving force F per bead and the number N of monomers per chain. With smaller channel width R, steady state motion cannot be achieved, effecting a nonuniversal dependence of τ on N and F. From the simulations we also deduce the waiting time distributions under various conditions for the single segment passage through the channel entrance. For different chain lengths but the same driving force, the curves of the waiting time as a function of the translocation coordinate s feature a maximum located at identical s(max), while with increasing the driving force or the channel width the value of s(max) decreases.     

Polymer translocation through a nanopore under an applied external field

We investigate the dynamics of polymer translocation through a nanopore under an externally applied field using the two-dimensional fluctuating bond model with single-segment Monte Carlo moves. We concentrate on the influence of the field strength E, length of the chain N, and length of the pore L on forced translocation. As our main result, we find a crossover scaling for the translocation time tau with the chain length from tau approximately N2nu for relatively short polymers to tau approximately N1+nu for longer chains, where nu is the Flory exponent. We demonstrate that this crossover is due to the change in the dependence of the translocation velocity v on the chain length. For relatively short chains v approximately N-nu, which crosses over to v approximately N(-1) for long polymers. The reason for this is that with increasing N there is a high density of segments near the exit of the pore, which slows down the translocation process due to slow relaxation of the chain. For the case of a long nanopore for which R parallel, the radius of gyration Rg along the pore, is smaller than the pore length, we find no clear scaling of the translocation time with the chain length. For large N, however, the asymptotic scaling tau approximately N1+nu is recovered. In this regime, tau is almost independent of L. We have previously found that for a polymer, which is initially placed in the middle of the pore, there is a minimum in the escape time for R parallel approximately L. We show here that this minimum persists for weak fields E such that EL is less than some critical value, but vanishes for large values of EL.