Experimental determination of the boundary between the topologically frustrated dynamical state and reptation regime

The movement of a polymer chain trapped in spatially restricted media is usually described by the classical Ogston model, entropic barrier model, and reptation model sequentially as the degree of confinement is increased. The newly discovered topologically frustrated dynamical state emerges at intermediate confinements before the onset of reptation regime, where the relaxation time of the whole polymer chain far exceeds the practical time scales of experimentation. The dynamically localized topologically frustrated state at intermediate confinements has recently been shown to arise from the collective behavior of multiple entropic traps acting on a single chain. Particle tracking in the presence of an external electric field has revealed that the net entropic barrier is on the orders of tens of k_BT. Under what conditions does this entropic barrier become small so as to enable entanglement effects to dominate? To address this crossover between the topologically frustrated dynamical state and the entangled (reptation) regime, we have experimentally measured the entropic barrier of λ-DNA trapped in the polyacrylamide-co-sodium acrylate (PAM-co-NaAc) gel matrix with a wide range of cross-link densities based on single-molecule electrophoresis. We report that the dependence of the entropic barrier on cross-link density is nonmonotonic with a maximum at an intermediate cross-link density. The entropic barrier becomes weak (comparable to k_BT) at a higher cross-link density which marks the boundary between the topologically frustrated dynamical state and the reptation regime.