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

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

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

Asymptotic Scaling of the Diffusion Coefficient of Fluctuating "Pulled" Fronts

We present a (heuristic) theoretical derivation for the scaling of the diffusion coefficient $D_f$ for fluctuating ``pulled'' fronts. In agreement with earlier numerical simulations, we find that as $N\to\infty$, $D_f$ approaches zero as $1/\ln^3N$, where $N$ is the average number of particles per correlation volume in the stable phase of the front. This behaviour of $D_f$ stems from the shape fluctuations at the very tip of the front, and is independent of the microscopic model.Comment: Some minor algebra corrected, to appear in Rapid Comm., Phys. Rev.

Fluctuating "Pulled" Fronts: the Origin and the Effects of a Finite Particle Cutoff

Recently it has been shown that when an equation that allows so-called pulled fronts in the mean-field limit is modelled with a stochastic model with a finite number $N$ of particles per correlation volume, the convergence to the speed $v^*$ for $N \to \infty$ is extremely slow -- going only as $\ln^{-2}N$. In this paper, we study the front propagation in a simple stochastic lattice model. A detailed analysis of the microscopic picture of the front dynamics shows that for the description of the far tip of the front, one has to abandon the idea of a uniformly translating front solution. The lattice and finite particle effects lead to a ``stop-and-go'' type dynamics at the far tip of the front, while the average front behind it ``crosses over'' to a uniformly translating solution. In this formulation, the effect of stochasticity on the asymptotic front speed is coded in the probability distribution of the times required for the advancement of the ``foremost bin''. We derive expressions of these probability distributions by matching the solution of the far tip with the uniformly translating solution behind. This matching includes various correlation effects in a mean-field type approximation. Our results for the probability distributions compare well to the results of stochastic numerical simulations. This approach also allows us to deal with much smaller values of $N$ than it is required to have the $\ln^{-2}N$ asymptotics to be valid.Comment: 26 pages, 11 figures, to appear in Phys. rev.

Polymers grafted to porous membranes

We study a single flexible chain molecule grafted to a membrane which has pores of size slightly larger than the monomer size. On both sides of the membrane there is the same solvent. When this solvent is good, i.e. when the polymer is described by a self avoiding walk, it can fairly easily penetrate the membrane, so that the average number of membrane crossings tends, for chain length $N\to\infty$, to a positive constant. The average numbers of monomers on either side of the membrane diverges in this limit, although their ratio becomes infinite. For a poor solvent, in contrast, the entire polymer is located, for large $N$, on one side of the membrane. For good and for theta solvents (ideal polymers) we find scaling laws, whose exponents can in the latter case be easily understood from the behaviour of random walks.Comment: 4 pages, 6 figure

Amplitude and Frequency Spectrum of Thermal Fluctuations of A Translocating RNA Molecule

Using a combination of theory and computer simulations, we study the translocation of an RNA molecule, pulled through a solid-state nanopore by an optical tweezer, as a method to determine its secondary structure. The resolution with which the elements of the secondary structure can be determined is limited by thermal fluctuations. We present a detailed study of these thermal fluctuations, including the frequency spectrum, and show that these rule out single-nucleotide resolution under the experimental conditions which we simulated. Two possible ways to improve this resolution are strong stretching of the RNA with a back-pulling voltage across the membrane, and stiffening of the translocated part of the RNA by biochemical means.Comment: Significantly expanded compared to previous version, 13 pages, 4 figures, to appear in J. Phys.: Condens. Matte

Approximate Solution of the effective mass Klein-Gordon Equation for the Hulthen Potential with any Angular Momentum

The radial part of the effective mass Klein-Gordon equation for the Hulthen potential is solved by making an approximation to the centrifugal potential. The Nikiforov-Uvarov method is used in the calculations. Energy spectra and the corresponding eigenfunctions are computed. Results are also given for the case of constant mass.Comment: 12 page