A theory of the dynamics of DNA loop initiation in condensin/cohesin complexes

The structural maintenance of chromosome complexes exhibit the remarkable ability to actively extrude DNA, which has led to the appealing and popular "loop extrusion" model to explain one of the most important processes in biology: the compaction of chromatin during the cell cycle. A potential mechanism for the action of extrusion is the classic Brownian ratchet, which requires short DNA loops to overcome an initial enthalpic barrier to bending, before favoured entropic growth of longer loops. We present a simple model of the constrained dynamics of DNA loop formation based on a frictional worm like chain, where for circular loops of order, or smaller than the persistence length, internal friction to bending dominates solvent dynamics. Using Rayleighs dissipation function, we show how bending friction can be translated to simple one dimensional diffusion of the angle of the loop resulting in a Smoluchowski equation with a coordinate dependent diffusion constant. This interplay between Brownian motion, bending dissipation and geometry of loops leads to a qualitatively new phenomenon, where the friction vanishes for bends with an angle of exactly 180 degrees, due to a decoupling between changes in loop curvature and angle. Using this theory and given current parameter uncertainties, we tentatively predict mean first passage times of between 1 and 10 seconds, which is of order the cycle time of ATP, suggesting spontaneous looping could be sufficient to achieve efficient initiation of looping.