^{1}and Dmitrii E. Makarov

^{1,a)}

### Abstract

Translocation of biopolymers through pores is implicated in many biological phenomena. Confinement within a pore often breaks ergodicity on experimental and/or biological time scales by creating large entropic barriers to conformational rearrangements of the chain. Here, we study one example of such hindered rearrangement, in which the chain reverses its direction inside a long pore. Our goal is twofold. First, we study the dependence of the time scale of polymer reversal on the pore size and on the polymer length. Second, we examine the ability of simple one-dimensional theories to quantitatively describe a transition in a system with a complex energy landscape by comparing them with the exact rate constant obtained using brute-force simulations and the forward flux sampling method. We find that one-dimensional transition state theory(TST) using the polymer extension along the pore axis as the reaction coordinate adequately accounts for the exponentially strong dependence of the reversal rate constant on the pore radius and the polymer length , while the transmission factor, i.e., the ratio of the exact rate and the TST approximation, has a much weaker power law and dependence. We have further attempted to estimate the transmission factor from Kramer’s theory, which assumes the reaction coordinate dynamics to be governed by a Langevin equation. However, such an approximation was found to be inadequate. Finally, we examine the scaling behavior of the reversal rate constant with and and show that finite size effects are important even for chains with up to several hundreds.

We thank Venkat Ganesan, Graeme Henkelman and Art Voter for helpful discussions. This work was supported by the Robert A. Welch Foundation (Grant No. F-1514) and by the National Science Foundation (Grant No. CHE 0347862). The CPU time was provided by the Texas Advanced Computer Center.

I. INTRODUCTION

II. THE MODEL

III. TST AND KRAMERS’ THEORY ESTIMATES OF THE REVERSAL RATE

IV. EXACT RATE VERSUS TST AND KRAMERS’ THEORY

V. THE DEPENDENCE OF THE REVERSAL RATE ON THE PORE RADIUS AND THE CHAIN LENGTH

VI. CONCLUDING REMARKS

### Key Topics

- Polymers
- 77.0
- Transition state theory
- 60.0
- Friction
- 23.0
- Reaction rate constants
- 22.0
- Free energy
- 21.0

## Figures

[(a)–(f)] A series of snapshots showing an example of a polymer reversal event observed in simulations. Also shown in (a) is the definition of reversal reaction coordinate, equal to the difference of the positions of the first and last monomers measured along the pore axis. (g) An example of a trajectory showing several chain reversal events.

[(a)–(f)] A series of snapshots showing an example of a polymer reversal event observed in simulations. Also shown in (a) is the definition of reversal reaction coordinate, equal to the difference of the positions of the first and last monomers measured along the pore axis. (g) An example of a trajectory showing several chain reversal events.

The free energy plotted as a function of the reaction coordinate for a polymer confined inside pores of different radius. is a double well symmetric with respect to . Here we show only the left side of the double well. The kink observed at small values of is due to the discrete nature of the end beads, which are in close proximity at .

The free energy plotted as a function of the reaction coordinate for a polymer confined inside pores of different radius. is a double well symmetric with respect to . Here we show only the left side of the double well. The kink observed at small values of is due to the discrete nature of the end beads, which are in close proximity at .

(a) The reversal rate constant plotted as a function of the pore radius for a chain consisting of monomers. Comparison of transition state theory, Kramers’ theory, the forward flux sampling method and brute-force simulations. See text for the details of the calculations. (b) The reversal rate constant plotted as a function of the pore radius for a chain consisting of monomers. Comparison of transition state theory, Kramers’ theory, and the forward flux sampling.

(a) The reversal rate constant plotted as a function of the pore radius for a chain consisting of monomers. Comparison of transition state theory, Kramers’ theory, the forward flux sampling method and brute-force simulations. See text for the details of the calculations. (b) The reversal rate constant plotted as a function of the pore radius for a chain consisting of monomers. Comparison of transition state theory, Kramers’ theory, and the forward flux sampling.

Comparison of the actual dynamics along the reaction coordinate with those approximated by a one-dimensional Langevin equation in the potential of mean force . For one-dimensional Langevin dynamics, the effective friction coefficient was estimated from Eq. (17). The data are for and . (a) The time-dependent transmission factor defined by Eqs. (24) and (25). (b). The position autocorrelation function used to evaluate the friction coefficient according to Eq. (17).

Comparison of the actual dynamics along the reaction coordinate with those approximated by a one-dimensional Langevin equation in the potential of mean force . For one-dimensional Langevin dynamics, the effective friction coefficient was estimated from Eq. (17). The data are for and . (a) The time-dependent transmission factor defined by Eqs. (24) and (25). (b). The position autocorrelation function used to evaluate the friction coefficient according to Eq. (17).

The reversal rate constant plotted as a function of the polymer length for : Comparison of different methods.

The reversal rate constant plotted as a function of the polymer length for : Comparison of different methods.

The reversal free energy barrier plotted as a function of the chain length for different values of the pore radius. See Fig. 2 and Sec. III for the description of .

The reversal free energy barrier plotted as a function of the chain length for different values of the pore radius. See Fig. 2 and Sec. III for the description of .

The mean radial compression force per monomer [Eq. (31)] for a ring polymer and linear polymer confined inside a pore of radius . For the linear polymer, the straight line shown is the best fit for and is described by the equation , which is consistent with the theoretical scaling law. For the ring polymer, the straight line shown fits the data for and is described by .

The mean radial compression force per monomer [Eq. (31)] for a ring polymer and linear polymer confined inside a pore of radius . For the linear polymer, the straight line shown is the best fit for and is described by the equation , which is consistent with the theoretical scaling law. For the ring polymer, the straight line shown fits the data for and is described by .

The TST prefactor as a function of the pore radius for different chain length. The lines are described by the formula .

The TST prefactor as a function of the pore radius for different chain length. The lines are described by the formula .

The transmission factor plotted as a function of the polymer length for . The solid line is described by .

The transmission factor plotted as a function of the polymer length for . The solid line is described by .

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