The Mean Looping Time of DNA

Abstract

In many gene expression systems, a protein located on the DNA can affect the expression of a gene far along the chain. It has been recognized that the DNA can form transient loops, bringing a specific region of the gene close to another. Thus, transcription can be activated when a transcription factor is positioned far away from its site. The frequency of bending is a characteristic time scale of the activation process.The mean time for a DNA molecule to loop, bringing together two sites, is a fundamental factor that we studied. Various approximations have been used to model polymers. Interestingly, dsDNA has been found to be well described by the standard Rouse model, in which the polymer is described as a collection of bead monomers connected by harmonic springs. The Rouse model is relevant when the sites are at a distance considerably bigger than the DNA persistence length. When the distance between the sites is of several persistence lengths, the semi-flexible chain model is better suited to model the DNA dynamics. The polymer chain is subjected to random independent motion (Brownian motion). When the two monomers come closer than a certain distance, interaction takes place and the monomers connect. We assumed that the interaction rate is much faster than the encounter time, thus the process ends with the first encounter of the monomers. This allowed us to compute the asymptotic formula for the mean encounter time in the two models. We obtained precise estimates for this mean first encounter time in two and three dimensions. Brownian simulations confirm our formulas and we discuss consequences of our results for random gene activation in the nucleus.