Polymer dynamics in rate processes

Abstract

Barrier Crossing and Reaction Dynamics of Polymers: A Kramers Problem Approach

Abstract and Synopsis Introduction The Kramers problem for a particle trapped in a metastable well has been studied extensively (P. Hänggi, P. Talkner, M. Borkovec, Rev. Mod. Phys., 62, 251 (1990)). We generalize this to a long chain polymer confined in a metastable potential well, from which it can escape. This problem is biologically important, as long chain molecules readily translocate across pores in biological membranes (B. Alberts et al., Essential Cell Biology, Garland Publishing, 1998). Experimental studies include DNA/RNA translocation through ion channels (Kasianowicz et al., PNAS, 93, 13770 (1996)) and DNA migration in microfabricated channels (Han, Turner, Craighead, Phys. Rev. Lett., 83, 1688 (1999)).

Barrier Crossing Mechanisms

Using the Rouse model, we analyze polymer escape across a biased double well potential.

Two dominant mechanisms: end crossing and hairpin crossing.

Free energy of activation for hairpin crossing is twice that of end crossing.

Pre-exponential factor scales with polymer length

for hairpin crossing, but is independent of

for end crossing.

Activation energy shows a square root dependence on temperature, leading to a non-Arrhenius rate.

A special time-dependent solution corresponds to a kink confined to the barrier region. Polymer translocation is equivalent to kink motion.

If free energy difference is zero: kink diffuses,

cross

2 .

If free energy difference exists: kink moves with velocity,

cross

.

Predictions agree with experiments of Kasianowicz et al. and Craighead et al.

Simulation Studies

1D Rouse chain in quartic potential: kink forms in barrier region, moves along chain, confirming theory.

3D biased double well: kink mechanism persists,

cross

.

Forced escape under external field: kink mechanism obeyed.

For short chains:

trans

ln

(

) .

For long chains:

trans

.

Simulations confirm both regimes.

Discrete Chain and Traveling Waves

Investigated whether kink solutions exist for discrete chains.

Using inverse problem approach: given a kink solution, derived the potential leading to it.

Found that a double well potential supports kink solutions in discrete Rouse chains.

Demonstrates exact solvability of barrier crossing for certain potentials.

Reaction-Diffusion Problems with Reactive Ends

Considered polymer reactions involving one reactive end.

Modeled as diffusion in N dimensions with reactive surfaces.

Developed approximate method for survival probability and reaction rate.

Example: polymer end reacting at surface

= 0 .

Short-time limit: derived analytical form for survival probability.

Long-time limit: obtained rate constant.

Extended to multiple reactive surfaces (e.g.,

= 0 and

=

).

Conclusions Polymer barrier crossing is governed by kink dynamics, with scaling laws depending on chain length and external bias.

Hairpin vs. end crossing differ in activation energies and prefactors.

Simulation studies validate theoretical predictions.

Discrete chain models can support kink solutions under specific potentials.

Developed approximate methods for multidimensional reaction-diffusion problems involving polymers with reactive ends.