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

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Abstract
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.