Pore-blockade Times for Field-Driven Polymer Translocation

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 $\sim N^{(1+2\nu)/(1+\nu)}/E$, where $\nu\simeq0.588$ 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 effects in the polymer chain tension imbalance across the pore. This result, along with the numerical results by several other groups, violates the lower bound $\sim N^{1+\nu}/E$ suggested earlier in the literature. We discuss why this lower bound is incorrect and show, based on conservation of energy, that the correct lower bound for the pore-blockade time for field-driven translocation is given by $\eta N^{2\nu}/E$, where $\eta$ is the viscosity of the medium surrounding the polymer.Comment: 14 pages, 6 figures, slightly shorter than the previous version; to appear in J. Phys.: Cond. Ma

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 such as phantom Rouse, self-avoiding Rouse, Zimm, reptation, and translocation through a narrow pore in a membrane; as well as for many-polymeric system such as polymer melts in the entangled regime --- is robustly described by the Generalized Langevin Equation (GLE). Here I show that the probability distribution of phase space trajectories for all 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.Comment: 8 pages, two figures, 3 eps figure files, minor changes, supplementary material included moved to the appendix, references expanded, to appear in J. Phys.: Condens. Matte

Dynamical Eigenmodes of a Polymerized Membrane

We study the bead-spring model for a polymerized phantom membrane in the overdamped limit, which is the two-dimensional generalization of the well-known Rouse model for polymers. We derive the {\it exact} eigenmodes of the membrane dynamics (the "Rouse modes"). This allows us to obtain exact analytical expressions for virtually any equilibrium or dynamical quantity for the membrane. As examples we determine the radius of gyration, the mean square displacement of a tagged bead, and the autocorrelation function of the difference vector between two tagged beads. Interestingly, even in the presence of tensile forces of any magnitude the Rouse modes remain the exact eigenmodes for the membrane. With stronger forces the membrane becomes essentially flat, and does not get the opportunity to intersect itself; in such a situation our analysis provides a useful and exactly soluble approach to the dynamics for a realistic model flat membrane under tension.Comment: 17 pages, 4 figures, minor changes, references updated, to appear in JSTA

Dynamical Eigenmodes of Star and Tadpole Polymers

The dynamics of phantom bead-spring chains with the topology of a symmetric star with $f$ arms and tadpoles ($f=3$, a special case) is studied, in the overdamped limit. In the simplified case where the hydrodynamic radius of the central monomer is $f$ times as heavy as the other beads, we determine their dynamical eigenmodes exactly, along the lines of the Rouse modes for linear bead-spring chains. These eigenmodes allow full analytical calculations of virtually any dynamical quantity. As examples we determine the radius of gyration, the mean square displacement of a tagged monomer, and, for star polymers, the autocorrelation function of the vector that spans from the center of the star to a bead on one of the arms.Comment: 21 pages in double spacing preprint format, 5 figures, minor changes in the "Discussion" section, to appear in JSTA

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.Comment: 29 pages, 6 figures, 3 tables, to appear in J. Phys.: Condens. Matter as a Topical Revie

Simulations of Two-Dimensional Unbiased Polymer Translocation Using the Bond Fluctuation Model

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 $\tau_d \sim N^\beta$, with $\beta=1+2\nu$, whereas some recent studies with different polymer models produce results consistent with $\beta=2+\nu$, originally predicted by us. Here $\nu$ is the Flory exponent of the polymer; $\nu=0.75$ in 2D. In this paper we show that for the BFM if the simulations are extended to longer polymers, the purported scaling $\tau_d \sim N^{1+2\nu}$ ceases to hold. We characterize the finite-size effects, and study the mobility of individual monomers in the BFM. In particular, we find that in the BFM, in the vicinity of the pore the individual monomeric mobilities are heavily suppressed in the direction perpendicular to the membrane. After a modification of the BFM which counters this suppression (but possibly introduces other artifacts in the dynamics), the apparent exponent $\beta$ increases significantly. Our conclusion is that BFM simulations do not rule out our theoretical prediction for unbiased translocation, namely $\beta=2+\nu$.Comment: minor proofreading corrections, 23 pages (double spacing), 7 figures, published versio

Rouse Modes of Self-avoiding Flexible Polymers

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 several scaling properties for self-avoiding flexible polymers, such as (i) the real-space end-to-end distance, (ii) the end-to-end vector correlation function, (iii) the correlation function of the small spatial vector connecting two nearby monomers at the middle of a polymer, and (iv) the anomalous dynamics of the middle monomer. Importantly, expanding on our recent work on the theory of polymer translocation, we also demonstrate that the anomalous dynamics of the middle monomer can be obtained from the forces it experiences, by the use of the fluctuation-dissipation theorem.Comment: 16 pages (double spaced), 5 figures, small changes and corrections, to appear in J. Chem. Phy

Critical Dynamical Exponent of the Two-Dimensional Scalar ϕ4\phi^4 Model with Local Moves

We study the scalar one-component two-dimensional (2D) $\phi^4$ model by computer simulations, with local Metropolis moves. The equilibrium exponents of this model are well-established, e.g. for the 2D $\phi^4$ model $\gamma= 1.75$ and $\nu= 1$. The model has also been conjectured to belong to the Ising universality class. However, the value of the critical dynamical exponent $z_c$ is not settled. In this paper, we obtain $z_c$ for the 2D $\phi^4$ model using two independent methods: (a) by calculating the relative terminal exponential decay time $\tau$ for the correlation function $\langle \phi(t)\phi(0)\rangle$, and thereafter fitting the data as $\tau \sim L^{z_c}$, where $L$ is the system size, and (b) by measuring the anomalous diffusion exponent for the order parameter, viz., the mean-square displacement (MSD) $\langle \Delta \phi^2(t)\rangle\sim t^c$ as $c=\gamma/(\nu z_c)$, and from the numerically obtained value $c\approx 0.80$, we calculate $z_c$. For different values of the coupling constant $\lambda$, we report that $z_c=2.17\pm0.03$ and $z_c=2.19\pm0.03$ for the two methods respectively. Our results indicate that $z_c$ is independent of $\lambda$, and is likely identical to that for the 2D Ising model. Additionally, we demonstrate that the Generalised Langevin Equation (GLE) formulation with a memory kernel, identical to those applicable for the Ising model and polymeric systems, consistently capture the observed anomalous diffusion behavior.Comment: 14 pages, 4 figures, 6 figure files, to appear in Phys. Rev.