Finite-size scaling arguments naturally lead us to introduce a coordinate-dependent diffusion coefficient in a Fokker-Planck description of the late stage dynamics of unbiased polymer translocation through a membrane pore. The solution for the probability density function of the chemical coordinate matches the initial-stage subdiffusive regime and takes into account the equilibrium entropic drive. Precise scaling relations connect the subdiffusion exponent to the divergence with the polymer length of the translocation time, and also to the singularity of the probability density function at the absorbing boundaries. Quantitative comparisons with numerical simulation data in $d=2$ strongly support the validity of the model and of the predicted scalings.
We perform, with the help of cloud computing resources, extensive Langevin simulations which provide compelling evidence in favor of a general markovian framework for unbiased polymer translocation. Our statistical analysis consists of careful evaluations of (i) two-point correlation functions of the translocation coordinate and (ii) the empirical probabilities of complete polymer translocation (taken as a function of the initial number of monomers on a given side of the membrane). We find good agreement with predictions derived from the Markov chain approach recently addressed in the literatu…
Theoretical derivation and numerical confirmation are provided for the scaling of pore-blockade times as $N^{2+ν}$ for unbiased translocation in two and three dimensions, with $N$ as the polymer length and $ν$ as the Flory exponent of the polymer. The polymer performs Rouse dynamics.
One of the most fundamental quantities associated with polymer translocation through a nanopore is the translocation time $τ$ and its dependence on the chain length $N$. Our simulation results based on both the bond fluctuation Monte Carlo and Molecular Dynamics methods confirm the original prediction $τ\sim N^{2ν+1}$, which scales in the same manner as the Rouse relaxation time of the chain except for a larger prefactor, and invalidates other scaling claims.
We use the Bond Fluctuation Model (BFM) to study the pore-blockade times of a translocating polymer of length $N$ in two dimensions, in the absence of external forces on the polymer (i.e., unbiased translocation) and hydrodynamic interactions (i.e., the polymer is a Rouse polymer), through a narrow pore. Earlier studies using the BFM concluded that the pore-blockade time scales with polymer length as $τ_d \sim N^β$, with $β=1+2ν$, whereas some recent studies with different polymer models produce results consistent with $β=2+ν$, originally predicted by us. Here $ν$ is the Flory exponent of the …
We study the translocation process of a polymer in the absence of external fields for various pore diameters $b$ and membrane thickness $L$. The polymer performs Rouse and reptation dynamics. The mean translocation time $<τ_t>$ that the polymer needs to escape from a cell, and the mean dwell time $<τ_d>$ that the polymer spends in the pore during the translocation process, obey scaling relations in terms of the polymer length $N$, $L$ and $b/R_g$, where $R_g$ is the radius of gyration for the polymer. We explain these relations using simple arguments based on polymer dynamics and the equilibri…
In this paper, we investigate the microscopic dynamics of a polymer of length $N$ translocating through a narrow pore. Characterization of its purportedly anomalous dynamics has so far remained incomplete. We show that the polymer dynamics is anomalous until the Rouse time $τ_{R}\sim N^{1+2ν}$, with a mean square displacement through the pore consistent with $t^{(1+ν)/(1+2ν)}$, with $ν\approx0.588$ the Flory exponent. This is shown to be directly related to a decay in time of the excess monomer density near the pore as $t^{-(1+ν)/(1+2ν)}\exp(-t/τ_{R})$. Beyond the Rouse time translocation beco…
We consider a polymer of length $N$ translocating through a narrow pore in the absence of external fields. Characterization of its purportedly anomalous dynamics has so far remained incomplete. We show that the polymer dynamics is anomalous until the Rouse time $τ_{R}\sim N^{1+2ν}$, with a mean square displacement through the pore consistent with $t^{(1+ν)/(1+2ν)}$, with $ν\approx0.588$ the Flory exponent. This is shown to be directly related to a decay in time of the excess monomer density near the pore as $t^{-(1+ν)/(1+2ν)}\exp(-t/τ_{R})$. Beyond the Rouse time translocation becomes diffusiv…
We perform, with the help of cloud computing resources, extensive Langevin simulations which provide free energy estimates for unbiased three dimensional polymer translocation. We employ the Jarzynski equality in its rigorous setting, to compute the variation of the free energy in single monomer translocation events. In our three-dimensional Langevin simulations, the excluded-volume and van der Waals interactions between beads (monomers and membrane atoms) are modeled through a repulsive Lennard-Jones (LJ) potential and consecutive monomers are subject to the Finite-Extension Nonlinear Elastic…
We consider the dynamics of polymer translocation out of confined environments. Analytic scaling arguments lead to the prediction that the translocation time scales like $τ\sim N^{β+ν_{2D}}R^{1+(1-ν_{2D})/ν}$ for translocation out of a planar confinement between two walls with separation $R$ into a 3D environment, and $τ\sim N^{β+1}R$ for translocation out of two strips with separation $R$ into a 2D environment. Here, $N$ is the chain length, $ν$ and $ν_{2D}$ are the Flory exponents in 3D and 2D, and $β$ is the scaling exponent of translocation velocity with $N$, whose value for the present ch…
We investigate the dynamics of polymer translocation through nanopores under external driving by 3D Langevin Dynamics simulations, focusing on the scaling of the average translocation time $τ$ versus the length of the polymer, $τ\sim N^α$. For slow translocation, i.e., under low driving force and/or high friction, we find $α\approx 1+ν\approx 1.588$ where $ν$ denotes the Flory exponent. In contrast, $α\approx 1.37$ is observed for fast translocation due to the highly deformed chain conformation on the trans side, reflecting a pronounced non-equilibrium situation. The dependence of the transloc…
We determine the scaling exponents of polymer translocation (PT) through a nanopore by extensive computer simulations of various microscopic models for chain lengths extending up to N=800 in some cases. We focus on the scaling of the average PT time $τ\sim N^α$ and the mean-square change of the PT coordinate $<s^2(t)> \sim t^β$. We find $α=1+2ν$ and $β=2/α$ for unbiased PT in 2D and 3D. The relation $αβ=2$ holds for driven PT in 2D, with crossover from $α\approx 2ν$ for short chains to $α\approx 1+ν$ for long chains. This crossover is, however, absent in 3D where $α= 1.42 \pm 0.01$ and $αβ\app…
We study the effect of the crowded nature of the cellular cytoplasm on the translocation of a polymer through a pore in a membrane. By systematically treating the entropic penalty due to crowding, we show that the translocation dynamics are significantly altered, leading to novel scaling behaviors of the translocation time in terms of chain length. We also observe new and qualitatively different translocation regimes depending upon the extent of crowding, transmembrane chemical potential asymmetry, and polymer length.
The translocation dynamics of a polymer chain through a nanopore in the absence of an external driving force is analyzed by means of scaling arguments, fractional calculus, and computer simulations. The problem at hand is mapped on a one dimensional {\em anomalous} diffusion process in terms of reaction coordinate $s$ (i.e. the translocated number of segments at time $t$) and shown to be governed by an universal exponent $α= 2/(2ν+2-γ_1)$ whose value is nearly the same in two- and three-dimensions. The process is described by a {\em fractional} diffusion equation which is solved exactly in the…
We develop a theory for polymer translocation driven by a time-dependent force through an oscillating nanopore. To this end, we extend the iso-flux tension propagation theory (IFTP) [Sarabadani \textit{et al., J. Chem. Phys.}, 2014, \textbf{141}, 214907] for such a setup. We assume that the external driving force in the pore has a component oscillating in time, and the flickering pore is similarly described by an oscillating term in the pore friction. In addition to numerically solving the model, we derive analytical approximations that are in good agreement with the numerical simulations. Our…
We consider the escape of a flexible, self-avoiding polymer chain out of a confined geometry. By means of simulations, we demonstrate that the translocation time can be described by a simple scaling law that exhibits a nonlinear dependence on the degree of polymerization and that is sensitive to the nature of the confining geometry. These results contradict earlier predictions but are in agreement with recently confirmed geometry-dependent expressions for the free energy of confinement.
We investigate the dynamics of pore-driven polymer translocation by theoretical analysis and molecular dynamics (MD) simulations. Using the tension propagation theory within the constant flux approximation we derive an explicit equation of motion for the tension front. From this we derive a scaling relation for the average translocation time $τ$, which captures the asymptotic result $τ\propto N_0^{1+ν}$, where $N_0$ is the chain length and $ν$ is the Flory exponent. In addition, we derive the leading correction-to-scaling term to $τ$ and show that all terms of order $N_0^{2ν}$ exactly cancel o…
Polymer translocation in three dimensions out of planar confinements is studied in this paper. Three membranes are located at $z=-h$, $z=0$ and $z=h_1$. These membranes are impenetrable, except for the middle one at $z=0$, which has a narrow pore. A polymer with length $N$ is initially sandwiched between the membranes placed at $z=-h$ and $z=0$ and translocates through this pore. We consider strong confinement (small $h$), where the polymer is essentially reduced to a two-dimensional polymer, with a radius of gyration scaling as $R^{\tinytext{(2D)}}_g \sim N^{ν_{\tinytext{2D}}}$; here, $ν_{\ti…
The transport of polymers with folded configurations across membrane pores is investigated theoretically by analyzing simple discrete stochastic models. The translocation dynamics is viewed as a sequence of two events: motion of the folded segment through the channel followed by the linear part of the polymer. The transition rates vary for the folded and linear segments because of different interactions between the polymer molecule and the pore. It is shown that the translocation time depends non-monotonously on the length of the folded segment for short polymers and weak external fields, whil…
Iso-flux tension propagation (IFTP) theory and Langevin dynamics (LD) simulations are employed to study the dynamics of channel-driven polymer translocation in which a polymer translocates into a narrow channel and the monomers in the channel experience a driving force $f_{\rm c}$. In the high driving force limit, regardless of the channel width, IFTP theory predicts $τ\propto f_{\textrm{c}}^β$ for the translocation time, where $β=-1$ is the force scaling exponent. Moreover, LD data show that for a very narrow channel fitting only a single file of monomers, the entropic force due to the subcha…
Using Langevin dynamics simulations, we investigate the influence of polymer-pore interactions on the dynamics of biopolymer translocation through nanopores. We find that an attractive interaction can significantly change the translocation dynamics. This can be understood by examining the three components of the total translocation time $τ\approx τ_1+τ_2+τ_3$ corresponding to the initial filling of the pore, transfer of polymer from the \textit{cis} side to the \textit{trans} side, and emptying of the pore, respectively. We find that the dynamics for the last process of emptying of the pore ch…
We investigate the dynamics of polymer translocation through a nanopore under an externally applied field using the 2D 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 $τ$ with the chain length from $τ\sim N^{2ν}$ for relatively short polymers to $τ\sim N^{1 + ν}$ for longer chains, where $ν$ is the Flory exponent. We demonstrate that this crossover is due to the change in …
We study the dynamics of the passage of a polymer through a membrane pore (translocation), focusing on the scaling properties with the number of monomers $N$. The natural coordinate for translocation is the number of monomers on one side of the hole at a given time. Commonly used models which assume Brownian dynamics for this variable predict a mean (unforced) passage time $τ$ that scales as $N^2$, even in the presence of an entropic barrier. However, the time it takes for a free polymer to diffuse a distance of the order of its radius by Rouse dynamics scales with an exponent larger than 2, a…
We present a theoretical description of the dynamics of a semi-flexible polymer being pulled through a nanopore by an external force acting at the pore. Our theory is based on the tensile blob picture of Pincus in which the front of the tensile force propagates through the backbone of the polymer, as suggested by Sakaue and recently applied to study a completely flexible polymer with self-avoidance, by Dubbledam et al. For a semi-flexible polymer with a persistence length P , its statistics is self-avoiding for a very long chain. As the local force increases, the blob size starts to decrease. …
We consider the passage of long polymers of length N through a hole in a membrane. If the process is slow, it is in principle possible to focus on the dynamics of the number of monomers s on one side of the membrane, assuming that the two segments are in equilibrium. The dynamics of s(t) in such a limit would be diffusive, with a mean translocation time scaling as N^2 in the absence of a force, and proportional to N when a force is applied. We demonstrate that the assumption of equilibrium must break down for sufficiently long polymers (more easily when forced), and provide lower bounds for th…
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…
We study the translocation dynamics of a polymer chain threaded through a nanopore by an external force. By means of diverse methods (scaling arguments, fractional calculus and Monte Carlo simulation) we show that the relevant dynamic variable, the translocated number of segments $s(t)$, displays an {\em anomalous} diffusive behavior even in the {\em presence} of an external force. The anomalous dynamics of the translocation process is governed by the same universal exponent $α= 2/(2ν+2 - γ_1)$, where $ν$ is the Flory exponent and $γ_1$ - the surface exponent, which was established recently fo…
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/micro-fluidics. During the the translocation process if the monomers in the channel can sufficiently quickly assume steady state motion, we observe the scaling $τ\sim N/F$ of the translocation time $τ$ with the driving force $F$ per bead and the number $N$ of monomers per chain. With sma…
We propose a simple scaling theory describing the variation of the mean first passage time (MFPT) $τ(N,M)$ of a regular block copolymer of chain length $N$ and block size $M$ which is dragged through a selective liquid-liquid interface by an external field $B$. The theory predicts a non-Arrhenian $τ$ vs. $B$ relationship which depends strongly on the size of the blocks, $M$, and rather weakly on the total polymer length, $N$. The overall behavior is strongly influenced by the degree of selectivity between the two solvents $χ$. The variation of $τ(N,M)$ with $N$ and $M$ in the regimes of weak a…
We study pore blockade times for a translocating polymer of length $N$, driven by a field $E$ across the pore in three dimensions. The polymer performs Rouse dynamics, i.e., we consider polymer dynamics in the absence of hydrodynamical interactions. We find that the typical time the pore remains blocked during a translocation event scales as $\sim N^{(1+2ν)/(1+ν)}/E$, where $ν\simeq0.588$ is the Flory exponent for the polymer. In line with our previous work, we show that this scaling behaviour stems from the polymer dynamics at the immediate vicinity of the pore -- in particular, the memory ef…
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 a viral ejection is completed. By detailed high-resolution simulations we show that such capsi…
Motivated by identifying the origin of the bias dependence of tension propagation we investigate methods for measuring tension propagation quantitatively in computer simulations of driven polymer translocation. Here the motion of flexible polymer chains through a narrow pore is simulated using Langevin dynamics. We measure tension forces, bead velocities, bead distances, and bond angles along the polymer at all stages of translocation with unprecedented precision. Measurements are done at a standard temperature used in simulations and at zero temperature to pin down the effect of fluctuations.…
We propose to model the stochastic dynamics of a polymer passing through a pore (translocation) by means of a fractional Brownian motion, and study its behavior in presence of an absorbing boundary. Based on scaling arguments and numerical simulations, we present a conjecture that provides a link between the persistence exponent $θ$ and the Hurst exponent $H$ of the process, thus sheding light on the spatial and temporal features of translocation. Furthermore, we show that this conjecture applies more generally to a broad class of self affine processes undergoing anomalous diffusion in bounded…
We consider two complementary polymer strands of length $L$ attached by a common end monomer. The two strands bind through complementary monomers and at low temperatures form a double stranded conformation (zipping), while at high temperature they dissociate (unzipping). This is a simple model of DNA (or RNA) hairpin formation. Here we investigate the dynamics of the strands at the equilibrium critical temperature $T=T_c$ using Monte Carlo Rouse dynamics. We find that the dynamics is anomalous, with a characteristic time scaling as $τ\sim L^{2.26(2)}$, exceeding the Rouse time $\sim L^{2.18}$.…
From scaling arguments and numerical simulations, we investigate the properties of the generalized elastic model (GEM), that is used to describe various physical systems such as polymers, membranes, single-file systems, or rough interfaces. We compare analytical and numerical results for the subdiffusion exponent beta characterizing the growth of the mean squared displacement <(delta h)^2> of the field h described by the GEM dynamic equation. We study the scaling properties of the qth order moments <|delta h|^q> with time, finding that the interface fluctuations show no intermittent behavior. …
Using a lattice-based Monte Carlo code for simulating self-avoiding flexible polymers in three dimensions in the absence of explicit hydrodynamics, we study their Rouse modes. For self-avoiding polymers, the Rouse modes are not expected to be statistically independent; nevertheless, we demonstrate that numerically these modes maintain a high degree of statistical independence. Based on high-precision simulation data we put forward an approximate analytical expression for the mode amplitude correlation functions for long polymers. From this, we derive analytically and confirm numerically severa…
Repetitive measurements can cause freezing of dynamics of a quantum state, which is known as quantum Zeno effect. We consider an interacting one-dimensional fermionic system and study the fate of the many-body quantum Zeno transition if the system is allowed to evolve repetitively under the unitary dynamics, followed by a measurement process. Measurement induced phase transitions can be accessed by tuning a suitably defined parameter representing measurement strength (frequency). We use different diagnostics, such as long-time evolved entanglement entropy, purity and their fluctuations in orde…
We consider the probability of knotting in equilateral random polygons in Euclidean 3-dimensional space, which model, for instance, random polymers. Results from an extensive Monte Carlo dataset of random polygons indicate a universal scaling formula for the knotting probability with the number of edges. This scaling formula involves an exponential function, independent of knot type, with a power law factor that depends on the number of prime components of the knot. The unknot, appearing as a composite knot with zero components, scales with a small negative power law, contrasting with previous…
We consider a lattice version of the Bisognano-Wichmann (BW) modular Hamiltonian as an ansatz for the bipartite entanglement Hamiltonian of the quantum critical chains. Using numerically unbiased methods, we check the accuracy of the BW-ansatz by both comparing the BW Rényi entropy to the exact results, and by investigating the size scaling of the norm distance between the exact reduced density matrix and the BW one. Our study encompasses a variety of models, scanning different universality classes, including transverse field Ising, Potts and XXZ chains. We show that the Rényi entropies obtain…
