Unifying model of driven polymer translocation

Fluctuating bottleneck model studies on kinetics of DNA escape from α-hemolysin nanopores

We have proposed a fluctuation bottleneck (FB) model to investigate the non-exponential kinetics of DNA escape from nanometer-scale pores. The basic idea is that the escape rate is proportional to the fluctuating cross-sectional area of DNA escape channel, the radius r of which undergoes a subdiffusion dynamics subjected to fractional Gaussian noise with power-law memory kernel. Such a FB model facilitates us to obtain the analytical result of the averaged survival probability as a function of time, which can be directly compared to experimental results. Particularly, we have applied our theory to address the escape kinetics of DNA through α-hemolysin nanopores. We find that our theoretical framework can reproduce the experimental results very well in the whole time range with quite reasonable estimation for the intrinsic parameters of the kinetics processes. We believe that FB model has caught some key features regarding the long time kinetics of DNA escape through a nanopore and it might provide a sound starting point to study much wider problems involving anomalous dynamics in confined fluctuating channels.

Surface-adsorption-induced polymer translocation through a nanopore: Effects of the adsorption strength and the surface corrugation

The surface corrugation plays an important role in single polymer diffusion on attractive surfaces. However, its effect on dynamics of surface adsorption-induced polymer translocation through a nanopore is not clear. Using three-dimensional Langevin dynamics simulations, we investigate the dynamics of a flexible polymer chain translocation through a nanopore induced by the selective adsorption of translocated segments onto the trans side of the membrane. The translocation probability Ptrans increases monotonically, while the mean translocation time τ has a minimum as a function of the adsorption strength ɛ, which are explained from the perspective of the effective driving force for the translocation. With the surface being smoother, τ as well as the scaling exponent α of τ with the chain length N decreases. Finally, we show that the distributions of the translocation time are non-Gaussian even for strong adsorption at a moderate surface corrugation. A nearly Gaussian distribution of the translocation time is observed only for the smoothest surface we studied.

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.

Dynamics of polymer translocation through kinked nanopores

Polymer translocation through nanopore has potential technological applications for DNA sequencing, where one challenge problem is to slow down translocation speed. Inspired by experimental findings that kinked nanopores exhibit a large reduction in translocation velocity compared with their straight counterparts, we investigate the dynamics of polymer translocation through kinked nanopores in two dimensions under an applied external field. With increasing the tortuosity of an array of nanopores, our analytical results show that the translocation probability decreases. Langevin dynamics simulation results support this prediction and further indicate that with increasing the tortuosity, translocation time shows a slow increase followed by a rapid increase after a critical tortuosity. This behavior demonstrates that kinked nanopores can effectively reduce translocation speed. These results are interpreted by the roles of the tortuosity for decreasing the effective nanopore diameter, increasing effective nanopore length, and greatly increasing the DNA-pore friction.

Ionic current inversion in pressure-driven polymer translocation through nanopores

We predict streaming current inversion with multivalent counterions in hydrodynamically driven polymer translocation events from a correlation-corrected charge transport theory including charge fluctuations around mean-field electrostatics. In the presence of multivalent counterions, electrostatic many-body effects result in the reversal of the DNA charge. The attraction of anions to the charge-inverted DNA molecule reverses the sign of the ionic current through the pore. Our theory allows for a comprehensive understanding of the complex features of the resulting streaming currents. The underlying mechanism is an efficient way to detect DNA charge reversal in pressure-driven translocation experiments with multivalent cations.

Active polymer translocation in the three-dimensional domain

In this work we study the translocation process of a polymer through a nanochannel where a time dependent force is acting. Two conceptually different types of driving are used: a deterministic sinusoidal one and a random telegraph noise force. The mean translocation time presents interesting resonant minima as a function of the frequency of the external driving. For the computed sizes, the translocation time scales with the polymer length according to a power law with the same exponent for almost all the frequencies of the two driving forces. The dependence of the translocation time with the polymer rigidity, which accounts for the persistence length of the molecule, shows a different low frequency dependence for the two drivings.

Using a Péclet number for the translocation of a polymer through a nanopore to tune coarse-grained simulations to experimental conditions

Coarse-grained simulations are often employed to study the translocation of DNA through a nanopore. The majority of these studies investigate the translocation process in a relatively generic sense and do not endeavor to match any particular set of experimental conditions. In this manuscript, we use the concept of a Péclet number for translocation, P(t), to compare the drift-diffusion balance in a typical experiment vs a typical simulation. We find that the standard coarse-grained approach overestimates diffusion effects by anywhere from a factor of 5 to 50 compared to experimental conditions using double stranded DNA (dsDNA). By defining a Péclet control parameter, λ, we are able to correct this and tune the simulations to replicate the experimental P(t) (for dsDNA and other scenarios). To show the effect that a particular P(t) can have on the dynamics of translocation, we perform simulations across a wide range of P(t) values for two different types of driving forces: a force applied in the pore and a pulling force applied to the end of the polymer. As P(t) brings the system from a diffusion dominated to a drift dominated regime, a variety of effects are observed including a non-monotonic dependence of the translocation time τ on P(t) and a steep rise in the probability of translocating. Comparing the two force cases illustrates the impact of the crowding effects that occur on the trans side: a non-monotonic dependence of the width of the τ distributions is obtained for the in-pore force but not for the pulling force.

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.

Controlling polymer translocation and ion transport via charge correlations

We develop a correlation-corrected transport theory in order to predict ionic and polymer transport properties of membrane nanopores under physical conditions where mean-field electrostatics breaks down. The experimentally observed low KCl conductivity of open α-hemolysin pores is quantitatively explained by the presence of surface polarization effects. Upon the penetration of a DNA molecule into the pore, these polarization forces combined with the electroneutrality of DNA sets a lower boundary for the ionic current, explaining the weak salt dependence of blocked pore conductivities at dilute ion concentrations. The addition of multivalent counterions to the solution results in the reversal of the polymer charge and the direction of the electroosmotic flow. With trivalent spermidine or quadrivalent spermine molecules, the charge inversion is strong enough to stop the translocation of the polymer and to reverse its motion. This mechanism can be used efficiently in translocation experiments in order to improve the accuracy of DNA sequencing by minimizing the translocation velocity of the polymer.

Polymer translocation through a nanopore driven by binding particles: influence of chain rigidity

We investigate the influence of chain rigidity on the dynamics of polymer translocation in the presence of binding particles (BPs) through a nanopore using two-dimensional Langevin dynamics simulations. With increasing chain rigidity κ, the mean translocation time 〈τ〉 increases monotonically due to an increase in the radius of gyration and a decrease in the center of mass velocity. Particularly for weak binding, we further find that 〈τ〉 shows a power-law behavior with the persistence length lp. Analysis indicates a scaling relation between the average velocity of the center of mass of a chain 〈vc.m.〉 and lp. As the chain becomes stiffer, the distribution of the translocation time τ approximates the Gaussian distribution and gets broader with the peak position being shifted towards longer translocation time. The corresponding translocation coordinate smax of the maximum waiting time gets smaller with increasing chain rigidity. Finally, under an extremely low BP concentration, 〈τ〉 shows a minimum for small κ, while it decreases monotonically for large κ with increasing binding energy. Our results suggest a nontrivial effect of the intrinsic property of chains on the dynamics of polymer translocation driven by BPs.

Criteria for minimal model of driven polymer translocation

While the characteristics of the driven translocation for asymptotically long polymers are well understood, this is not the case for finite-sized polymers, which are relevant for real-world experiments and simulation studies. Most notably, the behavior of the exponent α, which describes the scaling of the translocation time with polymer length, when the driving force fp in the pore is changed, is under debate. By Langevin dynamics simulations of regular and modified translocation models using the freely jointed-chain polymer model we find that a previously reported incomplete model, where the trans side and fluctuations were excluded, gives rise to characteristics that are in stark contradiction with those of the complete model, for which α increases with fp. Our results suggest that contribution due to fluctuations is important. We construct a minimal model where dynamics is completely excluded to show that close alignment with a full translocation model can be achieved. Our findings set very stringent requirements for a minimal model that is supposed to describe the driven polymer translocation correctly.

Directed translocation of a flexible polymer through a cone-shaped nano-channel

Translocation of a flexible polymer through a cone-shaped channel is studied, theoretically and using computer simulations. Our simulations show that the shape of the channel causes the polymer translocation to be a driven process. The effective driving force of entropic origin acting on the polymer is calculated as a function of the length and the apex-angle of the channel, theoretically. It is found that the translocation time is a non-monotonic function of the apex-angle of the channel. By increasing the apex-angle from zero, the translocation time shows a minimum and then a maximum. Also, it is found that regardless of the value of the apex-angle, the translocation time is a uniformly decreasing function of the channel length. The results of the theory and the simulation are in good qualitative agreement.

Biomolecule transport across biomembranes in the presence of crowding: polymer translocation driven by concentration and disorder gradients

The transport of biomolecules across biomembranes occurs in a complex environment where the fluid on both sides of the membrane contains many inclusions. We investigate the translocation of a polymer through a nanopore in such crowded environments via computer simulations. Modeling intracellular and extracellular inclusions as spherical obstacles, the emergence of an entropic driving force is demonstrated for (i) a gradient in the number of obstacles on either side of the pore and (ii) having the obstacles in an ordered arrangement on one side and disordered on the other. With a directional preference of ≈90% of events occurring towards the disordered side and a reduction in the translocation time of ≈30%, the latter case represents a manifestation of entropic trapping. Simple estimates for the magnitude of the driving force are developed for both cases. The effect of allowing the obstacles to move as well as systems containing opposing concentration and disorder gradients are also explored. Demonstrating a robust conclusion that translocation to lower obstacle densities and higher degrees of disorder will be accelerated, these results suggest that accounting for such effects will be critical in developing a realistic model of in vivo translocation.

Driven polymer transport through a periodically patterned channel

We study the driven transport of polymers in a periodically patterned channel using Langevin dynamics simulations in two dimensions. The channel walls are patterned with periodically alternating patches of attractive and non-attractive particles that act as trapping sites for the polymer. We find that the system shows rich dynamical behavior, observing giant diffusion, negative differential mobility, and several different transition mechanisms between the attractive patches. We also show that the channel can act as an efficient high-pass filter for polymers longer than a threshold length Nthr, which can be tuned by adjusting the length of the attractive patches and the driving force. Our findings suggest the possibility of fabricating polymer filtration devices based on patterned nanochannels.

Dynamics of polymer ejection from capsid

Polymer ejection from a capsid through a nanoscale pore is an important biological process with relevance to modern biotechnology. Here, we study generic capsid ejection using Langevin dynamics. We show that even when the ejection takes place within the drift-dominated region there is a very high probability for the ejection process not to be completed. Introducing a small aligning force at the pore entrance enhances ejection dramatically. Such a pore asymmetry is a candidate for a mechanism by which viral ejection is completed. By detailed high-resolution simulations we show that such capsid ejection is an out-of-equilibrium process that shares many common features with the much studied driven polymer translocation through a pore in a wall or a membrane. We find that the ejection times scale with polymer length, τ ∼ N(α). We show that for the pore without the asymmetry the previous predictions corroborated by Monte Carlo simulations do not hold. For the pore with the asymmetry the scaling exponent varies with the initial monomer density (monomers per capsid volume) ρ inside the capsid. For very low densities ρ ≤ 0.002 the polymer is only weakly confined by the capsid, and we measure α = 1.33, which is close to α=1.4 obtained for polymer translocation. At intermediate densities the scaling exponents α = 1.25 and 1.21 for ρ = 0.01 and 0.02, respectively. These scalings are in accord with a crude derivation for the lower limit α = 1.2. For the asymmetrical pore precise scaling breaks down, when the density exceeds the value for complete confinement by the capsid, ρ ⪆ 0.25. The high-resolution data show that the capsid ejection for both pores, analogously to polymer translocation, can be characterized as a multiplicative stochastic process that is dominated by small-scale transitions.

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.

Cis-trans dynamical asymmetry in driven polymer translocation

During polymer translocation driven by, e.g., voltage drop across a nanopore, the segments in the cis side are incessantly pulled into the pore, which are then pushed out of it into the trans side. This pulling and pushing of polymer segments are described in the continuum level by nonlinear transport processes known, respectively, as fast and slow diffusions. By matching solutions of both sides through the mass conservation across the pore, we provide a physical basis for the cis and trans dynamical asymmetry, a feature repeatedly reported in recent numerical simulations. We then predict how the total driving force is dynamically allocated between cis (pulling) and trans (pushing) sides, demonstrating that the trans-side event adds a weak finite-chain length effect to the dynamical scaling.

Memory effect and fluctuating anomalous dynamics of a tagged monomer

We analyze the anomalous dynamics of a tagged monomer under external navigation. The memory effect causing the anomaly is elucidated, which depends on the magnitude of the force. In particular, the nonlinear and nonequilibrium memory effect under strong force is characterized by the force-dependent self-affine process for the tension transmission along the connectivity. Utilizing such knowledge, a generalized Langevin equation approach is proposed to quantify the fluctuating dynamics of driven anomalous walkers.

Influence of non-universal effects on dynamical scaling in driven polymer translocation

We study the dynamics of driven polymer translocation using both molecular dynamics (MD) simulations and a theoretical model based on the non-equilibrium tension propagation on the cis side subchain. We present theoretical and numerical evidence that the non-universal behavior observed in experiments and simulations are due to finite chain length effects that persist well beyond the relevant experimental and simulation regimes. In particular, we consider the influence of the pore-polymer interactions and show that they give a major contribution to the non-universal effects. In addition, we present comparisons between the theory and MD simulations for several quantities, showing extremely good agreement in the relevant parameter regimes. Finally, we discuss the potential limitations of the present theories.

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.

Topological jamming of spontaneously knotted polyelectrolyte chains driven through a nanopore

The advent of solid state nanodevices allows for interrogating the physicochemical properties of a polyelectrolyte chain by electrophoretically driving it through a nanopore. Salient dynamical aspects of the translocation process have been recently characterized by theoretical and computational studies of model polymer chains free from self-entanglement. However, sufficiently long equilibrated chains are necessarily knotted. The impact of such topological "defects" on the translocation process is largely unexplored, and is addressed in this Letter. By using Brownian dynamics simulations on a coarse-grained polyelectrolyte model we show that knots, despite being trapped at the pore entrance, do not per se cause the translocation process to jam. Rather, knots introduce an effective friction that increases with the applied force, and practically halts the translocation above a threshold force. The predicted dynamical crossover, which is experimentally verifiable, ought to be relevant in applicative contexts, such as DNA nanopore sequencing.

Translocation of a heterogeneous polymer

We present results on the sequence dependence of translocation kinetics for a partially charged heteropolymer moving through a very thin pore using theoretical tools and Langevin dynamics simulational techniques. The chain is composed of two types of monomers of differing frictional interaction with the pore and charge. We present exact analytical expressions for passage probability, mean first passage time, and mean successful passage times for both reflecting/absorbing and absorbing/absorbing boundary conditions, showing rich and unexpected dependence of translocation behavior on charge fraction, distribution along the chain, and electric field configuration. We find excellent qualitative and good quantitative agreement between theoretical and simulation results. Surprisingly, there emerges a threshold charge fraction of a diblock copolymer beyond which the success rate of translocation is independent of charge fraction. Also, the mean successful translocation time of a diblock copolymer displays non-monotonic behavior with increasing length of the charged block; there is an optimum length of the charged block where the mean translocation rate is the slowest; and there can be a substantial range of higher charge fractions which make the translocation slower than even a minimally charged chain. Additionally, we find for a fixed total charge on the chain, finer distribution along the backbone significantly decreases mean translocation time.

Dragging a polymer in a viscous fluid: steady state and transient

We study the conformation and dynamics of a single polymer chain that is pulled by a constant force applied at its one end with the other end free. Such a situation is relevant to the growing technology of manipulating individual macromolecules, which offers a paradigm research for probing far-from-equilibrium responses of long flexible biological polymers. We first analyze the Rouse model for the Gaussian chains for which the exact analytical results can be obtained. More realistic features such as the finite extensibility, the excluded volume, and the hydrodynamic interactions are taken into account with the help of the scaling argument, which leads to various nontrivial predictions such as the force-dependent friction constants. We elucidate (i) generalized dynamical equations of state describing extension and friction laws in steady-state and (ii) the tension propagation laws in the transient process. We point out that the time evolutions of the dynamic friction in the transient process crucially depend on the experimental protocol, i.e., either constant force or constant velocity ensemble. These predictions could be verified in experiments using giant DNAs and chromosomes.

Process time distribution of driven polymer transport

We discuss the temporal distribution of dynamic processes in driven polymer transport inherent to flexible chains due to stochastic tension propagation. The stochasticity originates from the disordered initial configuration of an equilibrium polymer coil, which results in random paths for tension propagation. We consider the process time for when translocation occurs across a fixed pore and when stretching occurs by pulling the chain end. A scaling argument for the mean and standard deviation of the process time is provided using the two-phase picture for stochastic propagation. The two cases are found to differ remarkably. The process time distribution of the translocation exhibits substantial spreading even in the long-chain limit, unlike that found for the dynamics of polymer stretching. In addition, the process time distribution in the driven translocation is shown to have a characteristic asymmetric shape.