Escape of a chain molecule over a barrier - the kink mechanism

Abstract

We consider the generalization of the Kramers escape over a barrier problem to the case of a long chain molecule. The problem involves the motion of a chain molecule of $N$ segments across a region where the free energy per segment is higher, so that it has to cross a barrier. We consider the limit where the length of the molecule is much larger than the width of the barrier. The width is taken to be sufficiently wide that a coninuum description is applicable to even to the portion over the barrier. We use the Rouse model and analyze the mechanism of crossing a barrier. There can be two dominant mechanisms. They are: end crossing and hairpin crossing. We find the free energy of activation for the hairpin crossing is two times that 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. 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.