Monte Carlo study of the translocation of a polymer chain through a hole

The translocation of a polymer chain through a narrow hole in a rigid obstacle has been studied by the static Monte Carlo simulations. A modified self-avoiding walk on a cubic lattice has been used to model the polymer in an athermal solution. The entropy of the chain before, in the course, and after the translocation process has been estimated by the statistical counting method. The thermodynamic generalized forces governing the translocation have been calculated. The influence of the system geometry on the entropic barrier landscape is discussed.     

Modeling of polynucleotide translocation through protein pores and nanotubes

In an effort to understand recent experiments, we have performed Brownian dynamics simulations of polymer translocation through nanometer-scale protein pores under the influence of an external applied electric field. Multiple peaks in the translocation time distribution are observed in agreement with experiments. Under the same conditions, but replacing the protein pore with a rigid cylindrical tube of comparable size, only a single peak is observed in the translocation time distribution. These results directly show that the geometry of the protein pores is mainly responsible for multiple peaks observed in experiments. In the case of alpha-hemolysin channel, we find the vestibule, by confining many conformations of the translocating polymer, to be responsible for the second peak with longer translocation time.     

Unforced translocation of a polymer chain through a nanopore: the solvent effect

The authors have performed the Langevin dynamics simulation to investigate the unforced polymer translocation through a narrow nanopore in an impermeable membrane. The effects of solvent quality controlled by the attraction strength lambda of the Lennard-Jones cosine potential between polymer beads and beads on two sides of the membrane on the translocation processes are extensively examined. For polymer translocation under the same solvent quality on both sides of the membrane, the two-dimensional and three-dimensional simulations confirm the scaling law of tautrans approximately N1+2upsilon for the translocation in the good solvent, where tautrans is the translocation time, N is the chain length, and upsilon is the Flory exponent. For the three-dimensional polymer translocation under different solvent qualities on two sides of the membrane, the translocation efficiency may be notably improved. The scaling law between tautrans and N varies from tautrans approximately N1+2upsilon to tautrans approximately N with the increase of the difference of solvent qualities, and the crossover occurs at the theta temperature point, where a scaling law of tautrans approximately N1.27 is found. The simulation results here also show that the translocation time changes from a wide and asymmetric distribution with a long tail to a narrow and symmetric distribution with the increase of the difference of the solvent qualities.     

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.     

Kinetics of polymer translocation through a pore

We theoretically study kinetics of a polymer threading through a pore embedded in a flat membrane. We numerically solve three coupled kinetic equations for the number n(1) of polymer segments in one side of the membrane and expansion factors of the polymer chain in each side of the membrane. We find the time evolution n(1) proportional to t(1/(1+nu)) at late stages and the translocation time tau(t) is scaled as tau(t) proportional to 1+nu) for large number n of the polymer segments, where nu is the effective size exponent of the radius of gyration of the polymer. When the polymer is translocated into a region with a good solvent condition (nu=3/5), we obtain n(1) proportional to t(5/8) and tau(t) proportional to n(8/5).     

Chaperone-assisted translocation of a polymer through a nanopore

Using Langevin dynamics simulations, we investigate the dynamics of chaperone-assisted translocation of a flexible polymer through a nanopore. We find that increasing the binding energy ε between the chaperone and the chain and the chaperone concentration N(c) can greatly improve the translocation probability. Particularly, with increasing the chaperone concentration a maximum translocation probability is observed for weak binding. For a fixed chaperone concentration, the histogram of translocation time τ has a transition from a long-tailed distribution to a gaussian distribution with increasing ε. τ rapidly decreases and then almost saturates with increasing binding energy for a short chain; however, it has a minimum for longer chains at a lower chaperone concentration. We also show that τ has a minimum as a function of the chaperone concentration. For different ε, a nonuniversal dependence of τ on the chain length N is also observed. These results can be interpreted by characteristic entropic effects for flexible polymers induced by either the crowding effect from a high chaperone concentration or the intersegmental binding for the high binding energy.     

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.     

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.     

Simulation study on the translocation of polymer chains through nanopores

The translocation of polymer chains through nanopores is simulated by dynamical Monte Carlo method. The free energy landscape for the translocation of polymer is calculated by scanning method. The dependence of the free energy barrier Fb and the chemical difference Deltamu on the concentration of chains can explain the behavior of polymer translocation at low and high concentration limits. The relationship between Deltamu and the escaping time tau(2) is in good agreement with the theoretical conclusions obtained by Muthukumar [J. Chem. Phys. 111, 10371 (1999)]. Our simulation results show that the relaxation time is mainly dominated by Fb, while the escaping time is mainly dominated by Deltamu.     

Acceleration of Markov chain Monte Carlo simulations through sequential updating

Strict detailed balance is not necessary for Markov chain Monte Carlo simulations to converge to the correct equilibrium distribution. In this work, we propose a new algorithm which only satisfies the weaker balance condition, and it is shown analytically to have better mobility over the phase space than the Metropolis algorithm satisfying strict detailed balance. The new algorithm employs sequential updating and yields better sampling statistics than the Metropolis algorithm with random updating. We illustrate the efficiency of the new algorithm on the two-dimensional Ising model. The algorithm is shown to identify the correct equilibrium distribution and to converge faster than the Metropolis algorithm with strict detailed balance. The main advantages of the new algorithm are its simplicity and the feasibility of parallel implementation through domain decomposition.     

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.     

Excluded volume effect on confined polymer translocation through a short nanochannel

We simulated the translocation process of a polymer chain from a source container to a drain container through a short nanochannel. We utilized the bond fluctuation model coupled with Monte Carlo dynamics in our simulations. The calculation results show that the excluded volume effect significantly affects the polymer's translocation time tau. This time depends nonmonotonically on the polymer length N. For a fixed nanochannel length, tau decreases when the polymer length increases. tau, however, increases when the polymer length exceeds a certain threshold. This observation differs from those predicated for a Gaussian chain. In this paper, we will further present our findings to explain this phenomenon. The knowledge we gain from this research can enhance the understanding of complex transport processes in many biological systems.     

Memory effects during the unbiased translocation of a polymer through a nanopore

Through a detailed Langevin dynamics simulation study, the role of memory effects during unbiased translocation is explored. Tests are devised to uncover the presence of memory effects by directly measuring forward/backward-correlated motion as well as the associated change in the dynamics. Conducting these tests at low and high viscosities, a range of behaviours across different time scales is revealed: short-time forward correlations at all viscosities, quasi-static behaviour at low viscosity, and long-time backward correlations at high viscosity. By applying these tests at different portions of the translocation process, these memory effects are also shown to vary as translocation proceeds. Combining this information with standard measurements, a physical picture of unbiased translocation as the diffusion of a local minimum is proposed.     

A second-order Markov process for modeling diffusive motion through spatial discretization

A new "mesoscopic" stochastic model has been developed to describe the diffusive behavior of a system of particles at equilibrium. The model is based on discretizing space into slabs by drawing equispaced parallel planes along a coordinate direction. A central role is played by the probability that a particle exits a slab via the face opposite to the one through which it entered (transmission probability), as opposed to exiting via the same face through which it entered (reflection probability). A simple second-order Markov process invoking this probability is developed, leading to an expression for the self-diffusivity, applicable for large slab widths, consistent with a continuous formulation of diffusional motion. This model is validated via molecular dynamics simulations in a bulk system of soft spheres across a wide range of densities.     

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.     

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.     

Threading synthetic polyelectrolytes through protein pores

We have measured the ionic current signatures of sodium poly(styrene sulfonate) as its single molecules translocate through an alpha-hemolysin pore embedded into a bilayer in a salty aqueous medium under an externally applied electric field. As in the previous experiments involving DNA and RNA, the pore current, which is a measure of the ionic conductivity of the low molar mass electrolyte ions, is significantly reduced when the polymer molecule translocates through the pore. The magnitude and the duration of the reduction in the pore current are measured for each of the translocation events. By studying thousands of events of reduction in the ionic current, we have constructed distribution functions for the extent of the reduced current and for the translocation time. The details of these distribution functions are significantly different from those for DNA and RNA. By investigating over two orders of magnitude in the molecular weight of the polymer, the average translocation time is found to be proportional to the molecular weight and inversely proportional to the applied voltage. This demonstration of threading a synthetic polyelectrolyte through a protein pore opens up many opportunities to systematically explore the fundamental physical principles behind translocation of single macromolecules, by resorting to the wide variety of synthetically available polymers without the complexities arising from the sequences of biological polymers. In addition, the present experiments suggest yet another experimental protocol for separation of polymer molecules directly in aqueous media.     

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.     

A Monte Carlo algorithm to study polymer translocation through nanopores. II. Scaling laws

In the first paper of this series, we developed a new one-dimensional Monte Carlo approach for the study of flexible chains that are translocating through a small channel. We also presented a numerical scheme that can be used to obtain exact values for both the escape times and the escape probabilities given an initial pore-polymer configuration. We now present and discuss the fundamental scaling behaviors predicted by this Monte Carlo method. Our most important result is the fact that, in the presence of an external bias E, we observe a change in the scaling law for the translocation time tau as function of the polymer length N: In the general expression tau approximately N(beta)E, the exponent changes from beta=1 for moderately long chains to beta=1+nu or beta=2nu for very large values of N (for Rouse and Zimm dynamics, respectively). We also observe an increase in the effective diffusion coefficient due to the presence of entropic pulling on unbiased polymer chains.     

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.