Repton model of gel electrophoresis and diffusion

We analyse the repton model of Rubinstein as adapted by Duke as a model for the gel electrophoresis of DNA. Parameters in the model are the number $N$ of reptons in the chain, a length $a$, a microscopic transition frequency $w$, and the product $eE$ of the electric field $E$ (assumed constant) and the charge $e$ per repton. Formally exact formulas are derived for the dimensionless diffusion coefficient $D/a^{2}w$ and drift velocity $V/aw$, the latter as a function of the field. Calculation of $V/aw$ requires the eigenvector associated with the leading eigenvalue of a $3^{N-1}\times 3^{N-1}$ matrix. For short chains exact results are obtained analytically: $V/aw$ for all $eE$ for $1\leqslant N \leqslant 4$, and $D/a^{2}w$ for $1\leqslant N \leqslant 5$. For large $N$ we deduce that $D/a^{2}w$ vanishes proportionally to $1/N^{2}$, the standard de Gennes reptation result, but we have not evaluated the coefficient analytically. We have determined $D/a^{2}w$ for $N$ up to 150 by simulation and verified the $1/N^{2}$ law