Repton Model Research Articles

Structural modes of a polymer in the repton model

Using extensive computer simulations, the behavior of the structural modes-more precisely, the eigenmodes of a phantom Rouse polymer-are characterized for a polymer in the three-dimensional repton model and are used to study the polymer dynamics at time scales well before the tube renewal. Although these modes are not the eigenmodes for a polymer in the repton model, we show that numerically the modes maintain a high degree of statistical independence. The correlations in the mode amplitudes decay exponentially with (p∕N)(2)A(t), in which p is the mode number, N is the polymer length, and A(t) is a single function shared by all modes. In time, the quantity A(t) causes an exponential decay for the mode amplitude correlation functions for times <1; a stretched exponential with an exponent 1∕2 between times 1 and τ(R) ∼ N(2), the time-scale for diffusion of tagged reptons along the contour of the polymer; and again an exponential decay for times t > τ(R). Having assumed statistical independence and the validity of a single function A(t) for all modes, we compute the temporal behavior of three structural quantities: the vectorial distance between the positions of the middle monomer and the center-of-mass, the end-to-end vector, and the vector connecting two nearby reptons around the middle of the polymer. Furthermore, we study the mean-squared displacement of the center-of-mass and the middle repton, and their relation with the temporal behavior of the modes.

Polymer diffusion in a lattice polymer model with an intrinsic reptation mechanism

Rubinstein's repton model, extended by van Heukelum and Barkema for the simulation of polymer melts, is studied. This model has two dynamical mechanisms, explicit reptation and sideways motion, each with a prescribed rate. We study how the polymer diffusion coefficient depends on the ratio γ of these rates, while the sum of these rates is kept constant. We find that the diffusion coefficient of the polymers as a function of γ is not simply a linear combination of the contributions of the individual mechanisms; there is a large amount of cooperativity between the two kinds of mechanisms. A better understanding of this cooperativity might benefit research in a wide range of topics.

Exact curvilinear diffusion coefficients in the repton model

The Rubinstein-Duke or repton model is one of the simplest lattice model of reptation for the diffusion of a polymer in a gel or a melt. Recently, a slightly modified model with hardcore interactions between the reptons has been introduced. The curvilinear diffusion coefficients of both models are exactly determined for all chain lengths. The case of periodic boundary conditions is also considered.

Polymer dynamics in repton model at large fields.

Polymer dynamics at large fields in Rubinstein-Duke repton model is investigated theoretically. Simple diagrammatic approach and analogy with asymmetric simple exclusion models are used to analyze the reptation dynamics of polymers. It is found that for polyelectrolytes the drift velocity decreases exponentially as a function of the external field with an exponent depending on polymer size and parity, while for polyampholytes the drift velocity is independent of polymer chain size. However, for polymers, consisting of charged and neutral blocks, the drift velocity approaches the constant limit which is determined by the size of the neutral block. The theoretical arguments are supported by extensive numerical calculations by means of density-matrix renormalization group techniques.

Kinetics of intrachain reactions of supercoiled DNA: Theory and numerical modeling

We considered an irreversible biochemical intrachain reaction of supercoiled DNA as a random event that occurs, with some probability, at the instant of collision between two reactive groups attached to distant sites of the DNA molecule. For sufficiently small intrinsic rate constant kI, the dominant process contributing to the productive collisions is the quasione-dimensional reptation of the strands forming the superhelix. The mean reaction time is then given by τF+1/kIcL, where τF is the mean time of the first collision caused by reptation, and cL is the local concentration of one reactive group around the other. The internal reptation of DNA strands was simulated by the repton model, in which a superhelix branch is approximated by a string of beads placed in a row of cells. This simple model allows semiquantitative estimation of τF and cL (in some arbitrary units) for a large range of the DNA lengths L. The repton chain was calibrated with the help of the data available for small supercoiled plasmids from Monte Carlo and Brownian dynamics simulations. The repton model and the Brownian dynamics give the same form of the distribution of the first collision time. Our estimations show that, for opposite sites of the chain, the mean first collision time τF varies from 5 ms (L=2.5 kb) to 1 s (L=20 kb). The corresponding cL values (for the reaction radius 10 nm) are 3×10−6 and 2×10−7 M.

The repton model of gel electrophoresis

We discuss the repton model of agarose gel electrophoresis of DNA. We review previous results, both analytic and numerical, as well as presenting a new numerical algorithm for the efficient simulation of the model, and suggesting a new approach to the model's analytic solution.

Repton model of gel electrophoresis in the long chain limit

Reptation governs motion of long polymers through a confining environment. Slack enters at the ends and diffuses along the polymer as stored length. The rate at which stored length diffuses limits the speed at which the chain can drift. This paper relates the rate of stored length diffusion to the conformation of the tube whthin which the polymer is confined. In the scaling limit of long polymer chains and weak applied electric fields, holding the product of polymer length times field finite, the tube length and stored length density take on their zero-field values. The drift velocity then depends only on the polymer's end-to-end separation in the direction of the field.

Diffusion constant for the repton model of gel electrophoresis

The repton model is a simple model of the "reptation" motion by which DNA diffuses through a gel during electrophoresis. In this paper we show that the model can be mapped onto a system consisting of two types of particles with hard-sphere interactions diffusing on a one-dimensional lattice. Using this mapping we formulate an efficient Monte Carlo algorithm for the model which allows us to simulate systems more than twice the size of those studied before. Our results confirm scaling hypotheses which have previously been put forward for the model. We also show how the particle version of the model can be used to construct a transfer matrix which allows us to solve exactly for the diffusion constant of small repton systems. We give results for systems of up to 20 reptons.

Diffusion and drift of charged polymers

Earlier results of analysis of the Rubinstein-Duke repton model are presented along with a scaling hypothesis and its relation to simulation and experiment.

An invariance property of the repton model

It is required on physical grounds that the eigenvalues of the transition-rate matrix of the Rubinstein-Duke repton model for all values of the spatial wave number q be invariant to the choice of marker repton. This is not immediately obvious from the structure of the transition-rate matrix itself, but we show that these matrices for different choices of marker are related by similarity transformation and therefore have the same eigenvalues.

Electrophoresis of charged polymers: Simulation and scaling in a lattice model of reptation

We report numerical results on the repton model of Rubinstein [Phys. Rev. Lett. 59, 1946 (1987)] as adapted by Duke [Phys. Rev. Lett. 62, 2877 (1989)] as a model for the gel electrophoresis of DNA. We describe an efficient algorithm with which we have simulated chains of N reptons with N several hundred in some instances. The diffusion coefficient D in the absence of an external electric field is obtained for N\ensuremath{\le}100; we find ${\mathit{N}}^{2}$D=1/3(1+5${\mathit{N}}^{\mathrm{\ensuremath{-}}2/3}$) for large N. The coefficient 1/3 is in accord with the analytical results of Rubinstein and of van Leeuwen and Kooiman [Physica A 184, 79 (1992)]. The drift velocity v in a static field of variable strength E is obtained for various N and NE up to N=30 when NE is as small as 0.01 and up to N=400 when NE is as large as 20. We find that v has a finite, nonzero limit as N\ensuremath{\rightarrow}\ensuremath{\infty} at fixed E and that this limit is proportional to \ensuremath{\Vert}E\ensuremath{\Vert}E, in accord with the conclusions of Duke, Semenov, and Viovy [Phys. Rev. Lett. 69, 3260 (1992)] for a different but related model. In a scaling limit in which N\ensuremath{\rightarrow}\ensuremath{\infty} and E\ensuremath{\rightarrow}0 the drift velocity in the repton model is fitted well by the formula ${\mathit{N}}^{2}$v=NE[(1/3${)}^{2}$+(2NE/5${)}^{2}$${]}^{1/2}$ for all values of the scaling variable NE. We present a scaling analysis complementary to that of Duke, Semenov, and Viovy with which we rationalize the \ensuremath{\Vert}E\ensuremath{\Vert}E behavior of the limiting drift velocity.