Dynamics of interior loop formation in polymer chains: effect of tail length and confinement

The dynamic process of two distant segments of a polymer chain making a loop is an elementary process for unraveling the complexities of biological functions such as protein folding and gene regulation. This fundamental process incorporates the formation of loops, whether from end-to-end, end-to-interior, or interior-to-interior monomers. While most previous studies have considered end-loop formation, the interior loop formation is more common in vivo. We investigate the loop formations between two inter-segments of length l with dangling tails of length lt for both ends to give the total chain length L=l+2lt. For a given inter-loop length l, in free space and large cavities (R → 16), the loop formation probability Pl decreases with tail length lt and saturates as l_t grows, attributed to steric clashes. Under strong confinement (R →8), however, Pl exhibits a non-monotonic trend, it initially decreases but then increases for higher tail lengths as confinement limits tail movement. Moreover, the looping time increases and eventually saturates at higher tail lengths in the free space and the external confinement. To rationalize this behavior, we analyze the diffusive behavior of the chain monomer in the presence of confinement.