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

Abstract

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.