^{1}and Dmitrii E. Makarov

^{1,a)}

### Abstract

Motivated by recent experimental efforts to measure the duration of individual folding/unfolding transitions in proteins and RNA, here we use simulations to study the duration of a simple transition mimicking an elementary step in biopolymer folding: the closure of a loop in a long polymer chain. While the rate of such a transition is well approximated by a one-dimensional Smoluchowski model that views the end-to-end distance dynamics of a polymer chain as diffusion governed by the one-dimensional potential of mean force, the same model fails rather dramatically to describe the duration of such transitions. Instead, the latter timescale is well described by a model where the chain ends diffuse freely, uninfluenced by the average entropic force imposed by the polymer chain. The effective diffusion coefficient then depends on the length scale of the loop closure transition. Our findings suggest that simple one-dimensional models, when applied to estimate the duration of reactive events in complex molecular systems, should be used with caution.

R.R.C. was supported by the Robert A. Welch Foundation (through grant F-1514 to D.E.M). D.E.M. acknowledges partial support from the National Science Foundation (NSF) grant CHE 0848571. CPU time was provided by the Texas Advanced Computing Center.

I. INTRODUCTION

II. RESULTS

A. Polymer cyclization transit times are independent of chain length

B. 1D Smoluchowski picture fails to describe the mean transit time

C. Free-diffusion model for the transition paths in polymer cyclization

III. DISCUSSION AND CONCLUSION

### Key Topics

- Polymers
- 52.0
- Diffusion
- 22.0
- Conformational dynamics
- 13.0
- Friction
- 11.0
- Protein folding
- 8.0

## Figures

End-to-end distance trajectory of a polymer chain. Transition paths from A to B are the trajectory segments, where the end-to-end distance, *R*, enters the transition region at *R* _{ A } and reaches *R* _{ B } without first returning to *R* _{ A }. The transit time *t* _{ AB } is the time spent by a transition path within the transition region *R* _{ B } < *R* < *R* _{ A }. In contrast, trajectory segments that start at *R* _{ A } and later exit the transition region into A prior to reaching *R* _{ B } (duration *t* _{ AA }) will contribute to the mean first passage time from A to B but not to the mean transit time.

End-to-end distance trajectory of a polymer chain. Transition paths from A to B are the trajectory segments, where the end-to-end distance, *R*, enters the transition region at *R* _{ A } and reaches *R* _{ B } without first returning to *R* _{ A }. The transit time *t* _{ AB } is the time spent by a transition path within the transition region *R* _{ B } < *R* < *R* _{ A }. In contrast, trajectory segments that start at *R* _{ A } and later exit the transition region into A prior to reaching *R* _{ B } (duration *t* _{ AA }) will contribute to the mean first passage time from A to B but not to the mean transit time.

Transit times vs first passage times. The mean transit time from A to B (filled symbols) is independent of the chain length *N*, in contrast to the strongly chain-length-dependent mean first passage from A to B (empty symbols). Here both times are plotted as a function of the spatial extent of the transition region *R* _{ A } − *R* _{ B }, with *R* _{ B } fixed at 2.5 equilibrium bond lengths.

Transit times vs first passage times. The mean transit time from A to B (filled symbols) is independent of the chain length *N*, in contrast to the strongly chain-length-dependent mean first passage from A to B (empty symbols). Here both times are plotted as a function of the spatial extent of the transition region *R* _{ A } − *R* _{ B }, with *R* _{ B } fixed at 2.5 equilibrium bond lengths.

Polymer model vs Free diffusion model vs 1D Smoluchowski model for polymer cyclization transit times. The mean transit times for cyclization of a polymer chain (filled circles) are independent of the chain length. For small values of *R* _{ A } − *R* _{ B } they agree with the free diffusion model (open circles), which ignores the polymer chain and assumes that the end monomers move freely. In contrast, the model of Eq. (2) (solid, dashed, and dotted lines) that assumes 1D diffusion in a one-dimensional potential of mean force is inconsistent with the polymer data and shows significant dependence on the chain length. As in Fig. 2, the distance *R* _{ B } is fixed here at 2.5 equilibrium bond lengths.

Polymer model vs Free diffusion model vs 1D Smoluchowski model for polymer cyclization transit times. The mean transit times for cyclization of a polymer chain (filled circles) are independent of the chain length. For small values of *R* _{ A } − *R* _{ B } they agree with the free diffusion model (open circles), which ignores the polymer chain and assumes that the end monomers move freely. In contrast, the model of Eq. (2) (solid, dashed, and dotted lines) that assumes 1D diffusion in a one-dimensional potential of mean force is inconsistent with the polymer data and shows significant dependence on the chain length. As in Fig. 2, the distance *R* _{ B } is fixed here at 2.5 equilibrium bond lengths.

Cyclization of a polymer chain can be achieved via straightening of a chain segment containing *n* ≪ *N* monomers.

Cyclization of a polymer chain can be achieved via straightening of a chain segment containing *n* ≪ *N* monomers.

The effective number of monomers rearranging in a cyclization transition [estimated from Eqs. (5) and (6)]. The effective number of monomers *n*(*R* _{ A }, *R* _{ B }) involved in the cyclization of a chain increases with the change in the end-to-end distance *R* _{ A } − *R* _{ B } required to close the loop. For small values of *R* _{ A } − *R* _{ B } loop closure is accomplished by moving only *n*(*R* _{ A }, *R* _{ B }) = 2 end monomers. Here *R* _{ B } is fixed at 2.5 bond lengths. Dashed line shows Eq. (4), which predicts that *n*(*R* _{ A }, *R* _{ B }) is proportional to the distance *R* _{ A } − *R* _{ B } traveled during a transition.

The effective number of monomers rearranging in a cyclization transition [estimated from Eqs. (5) and (6)]. The effective number of monomers *n*(*R* _{ A }, *R* _{ B }) involved in the cyclization of a chain increases with the change in the end-to-end distance *R* _{ A } − *R* _{ B } required to close the loop. For small values of *R* _{ A } − *R* _{ B } loop closure is accomplished by moving only *n*(*R* _{ A }, *R* _{ B }) = 2 end monomers. Here *R* _{ B } is fixed at 2.5 bond lengths. Dashed line shows Eq. (4), which predicts that *n*(*R* _{ A }, *R* _{ B }) is proportional to the distance *R* _{ A } − *R* _{ B } traveled during a transition.

The distribution of transit times obtained from the free diffusion model, when rescaled according to Eq. (7) to account for the effective number of monomers involved, is nearly identical to the actual distribution obtained from simulations of polymer chains. (a): *N* = 160, *R* _{ A } = 10.39, and *R* _{ B } = 2.5. (b): N = 160, *R* _{ A } = 25.97, and *R* _{ B } = 2.5.

The distribution of transit times obtained from the free diffusion model, when rescaled according to Eq. (7) to account for the effective number of monomers involved, is nearly identical to the actual distribution obtained from simulations of polymer chains. (a): *N* = 160, *R* _{ A } = 10.39, and *R* _{ B } = 2.5. (b): N = 160, *R* _{ A } = 25.97, and *R* _{ B } = 2.5.

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