Kramers problem for a polymer in a double well
We consider a long chain molecule, initially confined to the metastable side of a biased double well potential. It can escape from this side to the other by the motion of its N segments across the barrier. We assume that the length of the molecule is much larger than the width w of the barrier. The width w is taken to be sufficiently large that a continuum description is applicable to even the portion over the barrier. We use the Rouse model and analyze the mechanism of crossing the barrier. There can be two dominant mechanisms: end crossing and hairpin crossing. We find the free energy of activation for the hairpin crossing to be two times that for end crossing. The pre-exponential factor for hairpin crossing is proportional to N, while it is independent of N for end crossing. In both cases, the activation energy has a square root dependence on the temperature T, leading to a non-Arrhenius form for the rate. We also show that there is a special time dependent solution of the model, which corresponds to a kink in the chain, confined to the region of the barrier. The movement of the polymer from one side to the other is equivalent to the motion of the kink on the chain in the reverse direction. If there is no free energy difference between the two sides of the barrier, then the kink moves by diffusion and the time of crossing t(cross) approximately N(2)/T(3/2). If there is a free energy difference, then the kink moves with a nonzero velocity from the lower free energy side to the other, leading to t(cross) approximately N/sqrt[T]. We also discuss the applicability of the mechanism to the recent experiments of Kasianowicz [Proc. Natl. Acad. Sci. USA 93, 13 770 (1996)], where DNA molecules were driven through a nanopore by the application of an electric field. The prediction that t(cross) approximately N is in agreement with these experiments. Our results are in agreement with the recent experimental observations of Han, Turner, and Craighead [Phys. Rev. Lett. 83, 1688 (1999)]. We also consider the translocation of hydrophilic polypeptides across hydrophobic pores, a process that is quite common in biological systems. Biological systems accomplish this by having a hydrophobic signal sequence at the end that goes in first. We find that for such a molecule, the transition state resembles a hook, and this is in agreement with presently accepted view in cell biology.