Interplay of tail length and confinement in the formation of interior loops in flexible chains

Loop formation between distant interior segments of a polymer is a fundamental process to biological functions such as gene regulation and protein folding. While prior studies predominantly focus on end-to-end looping, interior loop formation is more relevant in-vivo. Using Langevin dynamics simulations, we investigate the kinetics of interior loop formation in confined flexible polymers with specific internal segments having attractive interaction, focusing on the effect of tail length (lt) and spatial confinement. The probability distribution function of the distance between attractive beads forming the interior loop, P(ra), and the corresponding free energy profile, F(ra), exhibit a bimodal structure, due to the coexistence of two distinct conformational states: a compact folded state and an unlooped relaxed configuration. We observe a non-monotonic dependence of the looping probability (Pl ) and looping time (Tl) on lt under strong confinement. We identify the optimal combination of the cavity size and the tail length that leads to maximizing the looping time Tl and minimizing the looping probability Pl . Additionally, the interior loop dynamics for distinct loop lengths (ll) of a fixed polymer chain (L) showcase that Tl changes rapidly with the addition of the first few monomers and then plateaus, as the tail grows, which is exactly verified with analytical results. The observed coexistence of looped and extended states is a hallmark of intermediate ε, disappearing for weak or strong attractions, highlighting the tunability of looping dynamics via interaction strength.