Encounter dynamics of a small target by a polymer diffusing in a confined domain

We study the first passage time for a polymer, that we call the narrow encounter time (NETP), to reach a small target located on the surface of a microdomain. The polymer is modeled as a Freely Joint Chain (beads connected by springs with a resting non zero length) and we use Brownian simulations to study two cases: when (i) any of the monomer or (ii) only one can be absorbed at the target window. Interestingly, we find that {in the first case} the NETP is an increasing function of the polymer length until a critical length, after which it decreases. Moreover, in the long polymer regime, we identified an exponential scaling law for the NETP as a function of the polymer length. {In the second case, the position of the absorbed monomer along the polymer chain strongly influences the NETP}. Our analysis can be applied to estimate the mean first time of a DNA fragment to a small target in the chromatin structure or for mRNA to find a small target.