Polymer translocation through a nanopore: a showcase of anomalous diffusion

Polymer Translocation and Nanopore Sequencing: A Review of Advances and Challenges

Various biological processes involve the translocation of macromolecules across nanopores; these pores are basically protein channels embedded in membranes. Understanding the mechanism of translocation is crucial to a range of technological applications, including DNA sequencing, single molecule detection, and controlled drug delivery. In this spirit, numerous efforts have been made to develop polymer translocation-based sequencing devices, these efforts include findings and insights from theoretical modeling, simulations, and experimental studies. As much as the past and ongoing studies have added to the knowledge, the practical realization of low-cost, high-throughput sequencing devices, however, has still not been realized. There are challenges, the foremost of which is controlling the speed of translocation at the single monomer level, which remain to be addressed in order to use polymer translocation-based methods for sensing applications. In this article, we review the recent studies aimed at developing control over the dynamics of polymer translocation through nanopores.

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.

Anomalous fluctuations of renewal-reward processes with heavy-tailed distributions

For renewal-reward processes with a power-law decaying waiting time distribution, anomalously large probabilities are assigned to atypical values of the asymptotic processes. Previous works have revealed that this anomalous scaling causes a singularity in the corresponding large deviation function. In order to further understand this problem, we study in this article the scaling of variance in several renewal-reward processes: counting processes with two different power-law decaying waiting time distributions and a Knudsen gas (a heat conduction model). Through analytical and numerical analyses of these models, we find that the variances show an anomalous scaling when the exponent of the power law is -3. For a counting process with the power-law exponent smaller than -3, this anomalous scaling does not take place: this indicates that if we only consider the standard deviation from the expectation, any anomalous behavior will not be detected. In this case, we argue that anomalous scaling appears in higher order cumulants. Finally, many-body particles interacting through soft-core interactions with the boundary conditions employed in the Knudsen gas are studied using numerical simulations. We observe that the variance scaling becomes normal even though the power-law exponent in the boundary conditions is -3.

Polymer escape through a three dimensional double-nanopore system

We study the escape dynamics of a double-stranded DNA (dsDNA) through an idealized double nanopore geometry subject to two equal and opposite forces (tug-of-war) using Brownian dynamics (BD) simulation. In addition to the geometrical restrictions imposed on the cocaptured dsDNA segment in between the pores, the presence of tug-of-war forces at each pore results in a variation of the local chain stiffness for the segment of the chain in between the pores, which increases the overall stiffness of the chain. We use the BD simulation results to understand how the intrinsic chain stiffness and the tug-of-war forces affect the escape dynamics by monitoring the local chain persistence length ℓ_p, the residence time of the individual monomers W(m) in the nanopores, and the chain length dependence of the escape time ⟨τ⟩ and its distribution. Finally, we generalize the scaling theory for the unbiased single nanopore translocation for a fully flexible chain for the escape of a semi-flexible chain through a double nanopore in the presence of tug-of-war forces. We establish that the stiffness dependent part of the escape time is approximately independent of the translocation mechanism so that ⟨τ⟩∼ℓ_p ^2/D+2, and therefore, the generalized escape time for a semi-flexible chain can be written as ⟨τ⟩=AN^αℓ_p ^2/D+2. We use the BD simulation results to compare the predictions of the scaling theory. Our numerical studies supplemented by scaling analysis provide fundamental insights to design new experiments where a dsDNA moves slowly through a series of graphene nanopores.

Polymer Translocation Across a Corrugated Channel: Fick⁻Jacobs Approximation Extended Beyond the Mean First-Passage Time

Polymer translocation across a corrugated channel is a paradigmatic stochastic process encountered in diverse systems. The instance of time when a polymer first arrives to some prescribed location defines an important characteristic time-scale for various phenomena, which are triggered or controlled by such an event. Here we discuss the translocation dynamics of a Gaussian polymer in a periodically-corrugated channel using an appropriately generalized Fick⁻Jacobs approach. Our main aim is to probe an effective broadness of the first-passage time distribution (FPTD), by determining the so-called coefficient of variation γ of the FPTD, defined as the ratio of the standard deviation versus the mean first-passage time (MFPT). We present a systematic analysis of γ as a function of a variety of system's parameters. We show that γ never significantly drops below 1 and, in fact, can attain very large values, implying that the MFPT alone cannot characterize the first-passage statistics of the translocation process exhaustively well.

A fractional optimal control problem with final observation governed by wave equation

In this paper, we deal with the problem of controlling the source function for an optimal control problem involving the fractional wave equation. We show that an optimal solution exists and it is unique for the considered fractional optimal control problem. We calculate the Frechet derivative of the cost functional by means of an adjoint problem and derive necessary optimality conditions. Also, we introduce an efficient numerical approximation for the fractional wave equation with the Atangana-Baleanu derivative.

Trapped and non-trapped polymer translocations through a spherical pore

The polymer translocation through a spherical pore is studied using the Langevin dynamics simulation. The translocation events are classified into two types: one is the trapped translocation in which the entire polymer is trapped in the pore and the other is the non-trapped translocation where the pore cannot hold the whole polymer. We find that the trapped translocation is favored at large spheres and small external voltages. However, the monomer-pore attraction would lead to the non-monotonic behavior of the trapped translocation possibility out of all translocation events. Moreover, both the trapped and non-trapped translocation times are dependent on the polymer length, pore size, external voltage, and the monomer-pore attraction. There exist two pathways for the polymer in the trapped translocation: an actively trapped pathway for the polymer trapped in the pore before the head monomer arrives at the pore exit, and a passively trapped pathway for the polymer trapped in the pore while the head monomer is struggling to move out of the pore. The studies of trapped pathways can provide a deep understanding of the polymer translocation behavior.

Translocation of Charged Polymers through a Nanopore in Monovalent and Divalent Salt Solutions: A Scaling Study Exploring over the Entire Driving Force Regimes

Langevin dynamics simulations are performed to study polyelectrolytes driven through a nanopore in monovalent and divalent salt solutions. The driving electric field E is applied inside the pore, and the strength is varied to cover the four characteristic force regimes depicted by a rederived scaling theory, namely the unbiased (UB) regime, the weakly-driven (WD) regime, the strongly-driven trumpet (SD(T)) regime and the strongly-driven isoflux (SD(I)) regime. By changing the chain length N, the mean translocation time is studied under the scaling form 〈 τ 〉 ∼ N α E - δ . The exponents α and δ are calculated in each force regime for the two studied salt cases. Both of them are found to vary with E and N and, hence, are not universal in the parameter's space. We further investigate the diffusion behavior of translocation. The subdiffusion exponent γ p is extracted. The three essential exponents ν s , q, z p are then obtained from the simulations. Together with γ p , the validness of the scaling theory is verified. Through a comparison with experiments, the location of a usual experimental condition on the scaling plot is pinpointed.

Role of non-equilibrium conformations on driven polymer translocation

One of the major theoretical methods in understanding polymer translocation through a nanopore is the Fokker-Planck formalism based on the assumption of quasi-equilibrium of polymer conformations. The criterion for applicability of the quasi-equilibrium approximation for polymer translocation is that the average translocation time per Kuhn segment, ⟨τ⟩/NK, is longer than the relaxation time τ0 of the polymer. Toward an understanding of conditions that would satisfy this criterion, we have performed coarse-grained three dimensional Langevin dynamics and multi-particle collision dynamics simulations. We have studied the role of initial conformations of a polyelectrolyte chain (which were artificially generated with a flow field) on the kinetics of its translocation across a nanopore under the action of an externally applied transmembrane voltage V (in the absence of the initial flow field). Stretched (out-of-equilibrium) polyelectrolyte chain conformations are deliberately and systematically generated and used as initial conformations in translocation simulations. Independent simulations are performed to study the relaxation behavior of these stretched chains, and a comparison is made between the relaxation time scale and the mean translocation time (⟨τ⟩). For such artificially stretched initial states, ⟨τ⟩/NK < τ0, demonstrating the inapplicability of the quasi-equilibrium approximation. Nevertheless, we observe a scaling of ⟨τ⟩ ∼ 1/V over the entire range of chain stretching studied, in agreement with the predictions of the Fokker-Planck model. On the other hand, for realistic situations where the initial artificially imposed flow field is absent, a comparison of experimental data reported in the literature with the theory of polyelectrolyte dynamics reveals that the Zimm relaxation time (τZimm) is shorter than the mean translocation time for several polymers including single stranded DNA (ssDNA), double stranded DNA (dsDNA), and synthetic polymers. Even when these data are rescaled assuming a constant effective velocity of translocation, it is found that for flexible (ssDNA and synthetic) polymers with NK Kuhn segments, the condition ⟨τ⟩/NK < τZimm is satisfied. We predict that for flexible polymers such as ssDNA, a crossover from quasi-equilibrium to non-equilibrium behavior would occur at NK ∼ O(1000).

Translocation of a Polymer through a Crowded Channel under Electrical Force

The translocation of a polymer chain through a crowded cylindrical channel is studied using the Langevin dynamics simulations. The influences of the field strength F, the chain length N, and the crowding extent ρ on the translocation time are evaluated, respectively. Scaling relation τ ~ F^−α is observed. With the crowding extent ρ increasing, the scaling exponent α becomes large. It is found that, for noncrowded channel, translocation probability drops when the field strength becomes large. However, for high-crowded channel, it is the opposite. Moreover, the translocation time and the average translocation time for all segments both have exponential growth with the crowding extent. The investigation of shape factor 〈δ〉 shows maximum value with increasing of the number of segments outside s. At last, the number of segments inside channel N_in in the process of translocation is calculated and a peak is observed. All the information from the study may benefit protein translocation.

Conformation Change, Tension Propagation and Drift-Diffusion Properties of Polyelectrolyte in Nanopore Translocation

Using Langevin dynamics simulations, conformational, mechanical and dynamical properties of charged polymers threading through a nanopore are investigated. The shape descriptors display different variation behaviors for the cis- and trans-side sub-chains, which reflects a strong cis-trans dynamical asymmetry, especially when the driving field is strong. The calculation of bond stretching shows how the bond tension propagates on the chain backbone, and the chain section straightened by the tension force is determined by the ratio of the direct to the contour distances of the monomer to the pore. With the study of the waiting time function, the threading process is divided into the tension-propagation stage and the tail-retraction stage. At the end, the drift velocity, diffusive property and probability density distribution are explored. Owing to the non-equilibrium nature, translocation is not a simple drift-diffusion process, but exhibits several intermediate behaviors, such as ballistic motion, normal diffusion and super diffusion, before ending with the last, negative-diffusion behavior.

Polyelectrolyte Threading through a Nanopore

Threading charged polymers through a nanopore, driven by electric fields E, is investigated by means of Langevin dynamics simulations. The mean translocation time 〈 τ 〉 is shown to follow a scaling law Nα, and the exponent α increases monotonically from 1.16 (4) to 1.40 (3) with E. The result is double-checked by the calculation of mean square displacement of translocation coordinate, which asserts a scaling behavior tβ (for t near τ) with β complying with the relation αβ = 2. At a fixed chain length N, 〈τ〉 displayed a reciprocal scaling behavior E-1 in the weak and also in the strong fields, connected by a transition E-1.64(5) in the intermediate fields. The variations of the radius of gyration of chain and the positions of chain end are monitored during a translocation process; far-from-equilibrium behaviors are observed when the driving field is strong. A strong field can strip off the condensed ions on the chain when it passes the pore. The total charges of condensed ions are hence decreased. The studies for the probability and density distributions reveal that the monomers in the trans-region are gathered near the wall and form a pancake-like density profile with a hump cloud over it in the strong fields, due to fast translocation.

Evaluating the applicability of the Fokker-Planck equation in polymer translocation: a Brownian dynamics study

Brownian dynamics (BD) simulations are used to study the translocation dynamics of a coarse-grained polymer through a cylindrical nanopore. We consider the case of short polymers, with a polymer length, N, in the range N = 21-61. The rate of translocation is controlled by a tunable friction coefficient, γ0p, for monomers inside the nanopore. In the case of unforced translocation, the mean translocation time scales with polymer length as <τ1> ∼ (N - Np)(α), where Np is the average number of monomers in the nanopore. The exponent approaches the value α = 2 when the pore friction is sufficiently high, in accord with the prediction for the case of the quasi-static regime where pore friction dominates. In the case of forced translocation, the polymer chain is stretched and compressed on the cis and trans sides, respectively, for low γ0p. However, the chain approaches conformational quasi-equilibrium for sufficiently large γ0p. In this limit the observed scaling of <τ1> with driving force and chain length supports the Fokker-Planck (FP) prediction that <τ> ∝ N/fd for sufficiently strong driving force. Monte Carlo simulations are used to calculate translocation free energy functions for the system. The free energies are used with the FP equation to calculate translocation time distributions. At sufficiently high γ0p, the predicted distributions are in excellent agreement with those calculated from the BD simulations. Thus, the FP equation provides a valid description of translocation dynamics for sufficiently high pore friction for the range of polymer lengths considered here. Increasing N will require a corresponding increase in pore friction to maintain the validity of the FP approach. Outside the regime of low N and high pore friction, the polymer is out of equilibrium, and the FP approach is not valid.

Polymer translocation: the first two decades and the recent diversification

Probably no other field of statistical physics at the borderline of soft matter and biological physics has caused such a flurry of papers as polymer translocation since the 1994 landmark paper by Bezrukov, Vodyanoy, and Parsegian and the study of Kasianowicz in 1996. Experiments, simulations, and theoretical approaches are still contributing novel insights to date, while no universal consensus on the statistical understanding of polymer translocation has been reached. We here collect the published results, in particular, the famous-infamous debate on the scaling exponents governing the translocation process. We put these results into perspective and discuss where the field is going. In particular, we argue that the phenomenon of polymer translocation is non-universal and highly sensitive to the exact specifications of the models and experiments used towards its analysis.

Smooth DNA transport through a narrowed pore geometry

Voltage-driven transport of double-stranded DNA through nanoscale pores holds much potential for applications in quantitative molecular biology and biotechnology, yet the microscopic details of translocation have proven to be challenging to decipher. Earlier experiments showed strong dependence of transport kinetics on pore size: fast regular transport in large pores (> 5 nm diameter), and slower yet heterogeneous transport time distributions in sub-5 nm pores, which imply a large positional uncertainty of the DNA in the pore as a function of the translocation time. In this work, we show that this anomalous transport is a result of DNA self-interaction, a phenomenon that is strictly pore-diameter dependent. We identify a regime in which DNA transport is regular, producing narrow and well-behaved dwell-time distributions that fit a simple drift-diffusion theory. Furthermore, a systematic study of the dependence of dwell time on DNA length reveals a single power-law scaling of 1.37 in the range of 35-20,000 bp. We highlight the resolution of our nanopore device by discriminating via single pulses 100 and 500 bp fragments in a mixture with >98% accuracy. When coupled to an appropriate sequence labeling method, our observation of smooth DNA translocation can pave the way for high-resolution DNA mapping and sizing applications in genomics.

Driven translocation of a semi-flexible chain through a nanopore: a Brownian dynamics simulation study in two dimensions

We study translocation dynamics of a semi-flexible polymer chain through a nanoscopic pore in two dimensions using Langevin dynamics simulation in presence of an external bias F inside the pore. For chain length N and stiffness parameter κb considered in this paper, we observe that the mean first passage time <τ> increases as <τ(κb)>~<τ(κb=0)>lp(aN) , where κb and lp are the stiffness parameter and persistence length, respectively, and aN is a constant that has a weak N dependence. We monitor the time dependence of the last monomer xN(t) at the cis compartment and calculate the tension propagation time (TP) ttp directly from simulation data for <xN(t)> ~ t as alluded in recent nonequlibrium TP theory [T. Sakaue, Phys. Rev. E 76, 021803 (2007)] and its modifications to Brownian dynamics tension propagation theory [T. Ikonen, A. Bhattacharya, T. Ala-Nissila, and W. Sung, Phys. Rev. E 85, 051803 (2012); and J. Chem. Phys. 137, 085101 (2012)] originally developed to study translocation of a fully flexible chain. We also measure ttp from peak position of the waiting time distribution W(s) of the translocation coordinate s (i.e., the monomer inside the pore), and explicitly demonstrate the underlying TP picture along the chain backbone of a translocating chain to be valid for semi-flexible chains as well. From the simulation data, we determine the dependence of ttp on chain persistence length lp and show that the ratio ttp∕<τ> is independent of the bias F.

Polymer translocation dynamics in the quasi-static limit

Monte Carlo (MC) simulations are used to study the dynamics of polymer translocation through a nanopore in the limit where the translocation rate is sufficiently slow that the polymer maintains a state of conformational quasi-equilibrium. The system is modeled as a flexible hard-sphere chain that translocates through a cylindrical hole in a hard flat wall. In some calculations, the nanopore is connected at one end to a spherical cavity. Translocation times are measured directly using MC dynamics simulations. For sufficiently narrow pores, translocation is sufficiently slow that the mean translocation time scales with polymer length N according to <τ> ∝ (N - N(p))(2), where N(p) is the average number of monomers in the nanopore; this scaling is an indication of a quasi-static regime in which polymer-nanopore friction dominates. We use a multiple-histogram method to calculate the variation of the free energy with Q, a coordinate used to quantify the degree of translocation. The free energy functions are used with the Fokker-Planck formalism to calculate translocation time distributions in the quasi-static regime. These calculations also require a friction coefficient, characterized by a quantity N(eff), the effective number of monomers whose dynamics are affected by the confinement of the nanopore. This was determined by fixing the mean of the theoretical distribution to that of the distribution obtained from MC dynamics simulations. The theoretical distributions are in excellent quantitative agreement with the distributions obtained directly by the MC dynamics simulations for physically meaningful values of N(eff). The free energy functions for narrow-pore systems exhibit oscillations with an amplitude that is sensitive to the nanopore length. Generally, larger oscillation amplitudes correspond to longer translocation times.

Through the eye of the needle: recent advances in understanding biopolymer translocation

In recent years polymer translocation, i.e., transport of polymeric molecules through nanometer-sized pores and channels embedded in membranes, has witnessed strong advances. It is now possible to observe single-molecule polymer dynamics during the motion through channels with unprecedented spatial and temporal resolution. These striking experimental studies have stimulated many theoretical developments. In this short theory-experiment review, we discuss recent progress in this field with a strong focus on non-equilibrium aspects of polymer dynamics during the translocation process.

First-passage characteristics of biased diffusion in a planar wedge

We obtain compact, exact, analytical expressions for the first-passage-time distribution for a particle undergoing biased diffusion in a planar wedge for wedge angles π/p, where p is a positive integer. We then provide the long-time limit of the first-passage time and found it to be dependent on the drift direction and wedge angle. We finally provide exact expressions for the mean first-passage time for specific cases.

Translocation dynamics of freely jointed Lennard-Jones chains into adsorbing pores

Polymer translocation into adsorbing nanopores is studied by using the Fokker-Planck equation of chain diffusion along the energy landscape calculated with Monte Carlo simulations using the incremental gauge cell method. The free energy profile of a translocating chain was found by combining two independent sub-chains, one free but tethered to a hard wall, and the other tethered inside an adsorbing pore. Translocation dynamics were revealed by application of the Fokker-Planck equation for normal diffusion. Adsorption of polymer chains into nanopores involves a competition of attractive adsorption and repulsive steric hindrance contributions to the free energy. Translocation times fell into two regimes depending on the strength of the adsorbing pore. In addition, we found a non-monotonic dependence of translocation times with increasing adsorption strength, with sharp peak associated with local free energy minima along the translocation coordinate.

Polymer translocation under time-dependent driving forces: resonant activation induced by attractive polymer-pore interactions

We study the driven translocation of polymers under time-dependent driving forces using N-particle Langevin dynamics simulations. We consider the force to be either sinusoidally oscillating in time or dichotomic noise with exponential correlation time, to mimic both plausible experimental setups and naturally occurring biological conditions. In addition, we consider both the case of purely repulsive polymer-pore interactions and the case with additional attractive polymer-pore interactions, typically occurring inside biological pores. We find that the nature of the interaction fundamentally affects the translocation dynamics. For the non-attractive pore, the translocation time crosses over to a fast translocation regime as the frequency of the driving force decreases. In the attractive pore case, because of a free energy well induced inside the pore, the translocation time can be a minimum at the optimal frequency of the force, the so-called resonant activation. In the latter case, we examine the effect of various physical parameters on the resonant activation, and explain our observations using simple theoretical arguments.

Influence of polymer-pore interaction on the translocation of a polymer through a nanopore

The translocation of a bond fluctuation polymer through an interacting nanopore is studied using dynamic Monte Carlo simulation. A driving force F is applied only for monomers inside the pore. The influence of polymer-pore interaction on the scaling relation τ~N(α) is studied for both unbiased and biased translocations, with τ the translocation time and N the polymer length. Results show that the exponent α is dependent on the polymer-pore interaction. For a noninteracting pore, we find α=2.48 for unbiased translocation and α=1.35 for strong biased translocation; for strong attraction, we find α=2.35 for unbiased translocation and α=1.22 for strong biased translocation. The unbiased translocation corresponds to the low-NF regime whereas the strong biased translocation corresponds to the high-NF regime.

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.

Markov chain modeling of polymer translocation through pores

We solve the Chapman-Kolmogorov equation and study the exact splitting probabilities of the general stochastic process which describes polymer translocation through membrane pores within the broad class of Markov chains. Transition probabilities, which satisfy a specific balance constraint, provide a refinement of the Chuang-Kantor-Kardar relaxation picture of translocation, allowing us to investigate finite size effects in the evaluation of dynamical scaling exponents. We find that (i) previous Langevin simulation results can be recovered only if corrections to the polymer mobility exponent are taken into account and (ii) the dynamical scaling exponents have a slow approach to their predicted asymptotic values as the polymer's length increases. We also address, along with strong support from additional numerical simulations, a critical discussion which points in a clear way the viability of the Markov chain approach put forward in this work.

Force-driven polymer translocation through a nanopore: an old problem revisited

We consider DNA translocation through a pore in a planar membrane. The pore is so narrow that only one DNA segment can fit in. Assuming that the biasing force f acts inside the pore only, and that the DNA monomer number N is asymptotically large, we modify the previously developed treatment of the stretched part of the pre-translocated polymer by introducing the concept of "iso-flux trumpet". We show that friction of a moving chain in the trumpet, although it determines the speed of the process, provides only a marginal fraction of overall dissipation in the process. The dominant dissipation turns out to be due to irreversible entropic squeezing of the chain into the small pore. We also discover that because of the role of the membrane a much larger amount of heat of order k(B)T per monomer gets transferred from the heat bath on the post-translocation side to that on the pre-translocation side.

Single-polymer dynamics under constraints: scaling theory and computer experiment

The relaxation, diffusion and translocation dynamics of single linear polymer chains in confinement is briefly reviewed with emphasis on the comparison between theoretical scaling predictions and observations from experiment or, most frequently, from computer simulations. Besides cylindrical, spherical and slit-like constraints, related problems such as the chain dynamics in a random medium and the translocation dynamics through a nanopore are also considered. Another particular kind of confinement is imposed by polymer adsorption on attractive surfaces or selective interfaces--a short overview of single-chain dynamics is also contained in this survey. While both theory and numerical experiments consider predominantly coarse-grained models of self-avoiding linear chain molecules with typically Rouse dynamics, we also note some recent studies which examine the impact of hydrodynamic interactions on polymer dynamics in confinement. In all of the aforementioned cases we focus mainly on the consequences of imposed geometric restrictions on single-chain dynamics and try to check our degree of understanding by assessing the agreement between theoretical predictions and observations.

Probabilistic phase space trajectory description for anomalous polymer dynamics

It has been recently shown that the phase space trajectories for the anomalous dynamics of a tagged monomer of a polymer--for single polymeric systems and phenomena such as phantom Rouse, self-avoiding Rouse, and Zimm ones, reptation, and translocation through a narrow pore in a membrane, as well as for many polymeric systems such as polymer melts in the entangled regime--are robustly described by the generalized Langevin equation. Here I show that the probability distribution of phase space trajectories for all of these classical anomalous dynamics for single polymers is that of a fractional Brownian motion (fBm), while the dynamics for polymer melts between the entangled regime and the eventual diffusive regime exhibits small but systematic deviations from that of a fBm.

Polymer translocation induced by a bad solvent

We study polymer translocation through a nanopore subject to conformational differences created by putting two different solvents at the cis and trans compartments using Langevin dynamics in three dimensions (3D). Initially a fraction of the chain is placed in a good solvent at the cis side and the rest of the chain at the trans side is immersed in a bad solvent where it forms a globule. We study several aspects of the translocating chain as a function of the strength of the interaction ɛ/k(B)T for the bad solvent, where the temperature T is kept below the Θ temperature for the specific bead-spring model that we have used to describe the chain. For ɛ/k(B)T≥1 we find the mean first passage time (τ)~(ɛ/k(B)T)(-1) and (τ)~N(1.1±0.05). In that regard, translocation under solvent asymmetry is similar to the case of driven translocation under a bias inside the pore. However, the globule formed at the trans in the immediate vicinity of the pore readily absorbs the incoming particles making the translocation process faster than the driven translocation. Our simulation results for long chains and ɛ/k(B)T≥1 agree well with a theoretical prediction by Muthukumar [M. Muthukumar, J. Chem. Phys. 111, 10371 (1999).].

Fractional Brownian motion approach to polymer translocation: the governing equation of motion

We suggest a governing equation that describes the process of polymer-chain translocation through a narrow pore and reconciles the seemingly contradictory features of such dynamics: (i) a Gaussian probability distribution of the translocated number of polymer segments at time t after the process has begun, and (ii) a subdiffusive increase of the distribution variance Δ(t) with elapsed time Δ(t)∝t(α). The latter quantity measures the mean-squared number s of polymer segments that have passed through the pore Δ(t)=([s(t)-s(t=0)](2)), and is known to grow with an anomalous diffusion exponent α<1. Our main assumption [i.e., a Gaussian distribution of the translocation velocity v(t)] and some important theoretical results, derived recently, are shown to be supported by extensive Brownian dynamics simulation, which we performed in 3D. We also numerically confirm the predictions made recently that the exponent α changes from 0.91 to 0.55 to 0.91 for short-, intermediate-, and long-time regimes, respectively.

Cooperative translocation dynamics of biopolymer chains through nanopores in a membrane: Slow dynamics limit

The cooperative translocation dynamics of two complementary single-stranded DNA chains through two nanopores located in a membrane is investigated theoretically. The translocation process is considered to be quasi-equilibrium, and then under the limit of slow dynamics the average translocation times are numerically presented under different conditions. It is shown that the effects of the chemical potential gradient, the recombination energy and the distance between the two nanopores on the cooperative translocation are significant. The present model predicts that the cooperative translocation of such two chains can shorten the translocation time, reduce the backward motion and thus improve the translocation efficiency.