Dynamics of the constrained polymer collapse

– The dynamics of polymer collapse with a fixed distance between endpoints is studied analytically and numerically by the Nosé-Hoover algorithm. We find that at the pearling stage of the collapse the number of pearls decays as t−1/2 leading to anomalously long collapse time. To understand the effect of Stokes drag we reduced the problem of longpolymer-chain collapse to the one-dimensional diffusion-limited coalescence of particles with the mass-dependent mobility. In this case the number of pearls decays slower, as t−3/7. The conformational behaviour of long polymer chains under the action of hydrophobic forces is a topic of significant interest [1, 2]. At equilibrium the average size of an isolated polymer molecule depends strongly on the quality of the solvent, and varies from extended conformations in good solvents to collapsed states in poor solvents. The collapse is an integral part of the complex protein folding process. Recent experiments and numerical simulations revealed rich kinetics of the polymer collapse [3–11]. In particular, numerical studies indicate that the collapse of a polymer chain is mediated by the formation of numerous pearls (clusters of monomers) and subsequent coarsening of the pearls leading to the formation of a single large globule. When ends of a polymer chain are free, the coarsening of the pearled state occurs relatively fast [5]. The mechanism of coarsening is a mutual attraction of pearls due to the force exerted by pearls on the polymer chain connecting them. The scaling properties and the kinetics of collapse are not completely understood. The problem becomes especially complex when the polymer is subject to various constraints, e.g. when it is submersed in a shear flow, its points are attached to walls or pulled externally. An interesting behaviour was observed when a biopolymer was stretched by the tip of an atomic-force microscope, see [12, 13]. Various non-trivial regimes in the polymers dynamics due to stretching by external force were studied in recent works [11,14–18]. The kinetics of polymer chains with free ends studied in refs. [5,7] yield a relatively short collapse time tc ∼ N1−1.5, where N is the total number of monomers in the chain. Here we study the dynamics of the homopolymer collapse in case when the endpoints of the chain are fixed. Our numerical and analytical results show that the pearling stage of the homopolymer collapse of a chain with fixed endpoints proceeds much slower than for a free polymer. From