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.

We consider the Kramers problem for a long chain polymer trapped in a biased double-well potential. Initially the polymer is in the less stable well and it can escape from this well to the other well by the motion of its N beads across the barrier to attain the configuration having lower free energy. In one dimension we simulate the crossing and show that the results are in agreement with the kink mechanism suggested earlier. In three dimensions, it has not been possible to get an analytical 'kink solution' for an arbitrary potential; however, one can assume the form of the solution of the nonlinear equation as a kink solution and then find a double-well potential in three dimensions. To verify the kink mechanism, simulations of the dynamics of a discrete Rouse polymer model in a double well in three dimensions are carried out. We find that the time of crossing is proportional to the chain length, which is in agreement with the results for the kink mechanism. The shape of the kink solution is also in agreement with the analytical solution in both one and three dimensions.

For highly entangled linear chains, the mean-square displacement of the primitive chain segments [d(n,t)] 2 (n ) segment index) not including a contribution from a nondiffusive part of the contour length fluctuation (CLF) was analyzed on the basis of the tube concept. The analysis was made by just considering the Gaussian nature of the chains (without detailed assumptions about the chain motion). A simple analytic relationship between this [d(n,t)] 2 and the normalized viscoelastic relaxation function µ(t) was derived in a range of µ(t) > 0.5 and at t longer than the Rouse time τRouse for CLF. This relationship was significantly affected by the dynamic tube dilation (DTD) mechanism. For example, the end segment exhibited d 2 ) 〈R 2 〉(1µ) in the absence of DTD and d 2 = 〈R 2 〉(1µ 1/2 ) in the presence of DTD, with 〈R 2 〉 being the mean-square end-to-end distance of the chain. This difference should enable a test of the DTD picture at short t (but still longer than τ Rouse) through comparison of the viscoelastic and segmental displacement data.

We study the thermally activated barrier crossing by long chain molecules, initially confined to one side of an entropic trap. The entropic barrier is assumed to be of Kramers type. The barrier width is considered to be larger than the chain. The latter is in turn assumed to be long enough, so that a continuum description of the chain is applicable throughout the space. The barrier crossing rate is calculated using multidimensional Kramers theory and the functional integral method. For chains having the same total number of segments, the activation energy itself remains constant. However, the preexponential factor depends on the structure of the polymer. Polymers with the same molecular weight but having longer arms can effect larger fluctuations, thereby increasing its chance to cross the barrier. This leads to an almost exponential increase of the rate prefactor with the radius of gyration. The difference in the barrier crossing rates could be effectively exploited for the separation of molecules having architectural differences, for example, DNA of same length but different degrees of supercoiling. This is illustrated by considering star polymers. The Rouse-Ham model is used to analyze the mechanism of the barrier crossing. We show how the rate expression of the Arrhenius type is affected by the long arms of the star chain.