Searching circular DNA strands

We introduce and explore a model of an ensemble of enzymes searching, in parallel, a circular DNA strand for a target site. The enzymes performing the search combine local scanning—conducted by a 1D motion along the strand—and random relocations on the strand—conducted via a confined motion in the medium containing the strand. Both the local scan mechanism and the relocation mechanism are considered general. The search durations are analysed, and their limiting probability distributions—for long DNA strands—are obtained in closed form. The results obtained (i) encompass the cases of single, parallel and massively parallel searches, taking place in the presence of either finite-mean or heavy-tailed relocation times, (ii) are applicable to a wide spectrum of local scan mechanisms including linear, Brownian, selfsimilar, and sub-diffusive motions, (iii) provide a quantitative theoretical justification for the necessity of the relocation mechanism, and (iv) facilitate the derivation of optimal relocation strategies.