^{1}, Gregory C. Rutledge

^{1,a)}and George Stephanopoulos

^{1}

### Abstract

We introduce a new, topologically-based method for coarse-graining polymer chains. Based on the wavelet transform, a multiresolution data analysis technique, this method assigns a cluster of particles to a coarse-grained bead located at the center of mass of the cluster, thereby reducing the complexity of the problem by dividing the simulation into several stages, each with a fraction of the number of beads as the overall chain. At each stage, we compute the distributions of coarse-grained internal coordinates as well as potential functions required for subsequent simulation stages. In this paper, we present the basic algorithm, and apply it to freely jointed chains; the companion paper describes its applications to self-avoiding chains.

This work was supported by the Department of Energy Computational Science Graduate Fellowship Program of the Office of Scientific Computing and Office of Defense Programs in the Department of Energy under Contract No. DE-FG02-97ER25308. This work was also supported by the Engineering Research Centers Programs of the National Science Foundation under NSF Award No. EEC-9731680. Any opinions, findings, conclusions, or recommendations expressed in this material are those of the authors and do not necessarily reflect those of the National Science Foundation.

I. INTRODUCTION

A. Existing coarse-graining techniques

B. The freely jointed chain model

II. COARSE GRAINING OF POLYMER CHAINS

A. The wavelet transform

B. Using wavelets to construct a coarse-grained model

C. “Convergence” of on-lattice and off-lattice coarse-grained computations

III. WAVELET-ACCELERATED MONTE CARLO (WAMC) SIMULATION OF POLYMER CHAINS

A. Hierarchical simulation approach

B. Coarse-grained simulation algorithm

C. Probability distributions for coarse-grained internal coordinates

1. Bond-length distribution

2. Bond-angle distribution

3. Torsion-angle distribution

IV. ILLUSTRATIVE EXAMPLES

V. CONCLUSIONS

### Key Topics

- Polymers
- 23.0
- Wavelet transform
- 15.0
- Wavelets
- 11.0
- Probability theory
- 10.0
- Cumulative distribution functions
- 6.0

## Figures

Coarse graining of a 16-site random walk in two dimensions using “center-of-mass aggregation” of adjacent points along the chain, as suggested by the wavelet transform method. The four sites which would be created after another iteration of this method are indicated on the graph; note that each is at a quarter-integer lattice point.

Coarse graining of a 16-site random walk in two dimensions using “center-of-mass aggregation” of adjacent points along the chain, as suggested by the wavelet transform method. The four sites which would be created after another iteration of this method are indicated on the graph; note that each is at a quarter-integer lattice point.

Coarse graining of a three-dimensional walk on a cubic lattice using the wavelet transform. (a) A 512-step self-avoiding random walk. (b)–(d) The same walk, after one, two, and three iterations of the Haar transform defined by (6) and (7).

Coarse graining of a three-dimensional walk on a cubic lattice using the wavelet transform. (a) A 512-step self-avoiding random walk. (b)–(d) The same walk, after one, two, and three iterations of the Haar transform defined by (6) and (7).

Comparison of the probability distributions for the coarse-grained bond length (distance between adjacent center-of-mass beads) for freely jointed chains for different chain lengths, with each coarse-grained bead representing 32 beads on the original chain.

Comparison of the probability distributions for the coarse-grained bond length (distance between adjacent center-of-mass beads) for freely jointed chains for different chain lengths, with each coarse-grained bead representing 32 beads on the original chain.

Ideal and observed coarse-grained cumulative distribution functions for a coarse-grained Gaussian random walk divided into segments of 32 beads each (*c*=4).

Ideal and observed coarse-grained cumulative distribution functions for a coarse-grained Gaussian random walk divided into segments of 32 beads each (*c*=4).

Cumulative distribution function for the coarse-grained bond angle, in a freely jointed chain with and , as a function of , the “first” bond length forming the angle. The top, solid curve shows bond lengths less than the ideal Gaussian value of ; the middle, dashed curve shows ; and the bottom, dash-dotted curve shows .

Cumulative distribution function for the coarse-grained bond angle, in a freely jointed chain with and , as a function of , the “first” bond length forming the angle. The top, solid curve shows bond lengths less than the ideal Gaussian value of ; the middle, dashed curve shows ; and the bottom, dash-dotted curve shows .

Probability distribution function for the torsion angle of a freely jointed chain of length considered as four beads of effective size .

Probability distribution function for the torsion angle of a freely jointed chain of length considered as four beads of effective size .

The mean-square radius of gyration as a function of bead size for both the detailed and WAMC representations of the freely jointed chain.

The mean-square radius of gyration as a function of bead size for both the detailed and WAMC representations of the freely jointed chain.

## Tables

Mean coarse-grained bond lengths between beads representing 32-mers.

Mean coarse-grained bond lengths between beads representing 32-mers.

Mean-square radius of gyration using bond-angle and torsion-angle distributions

Mean-square radius of gyration using bond-angle and torsion-angle distributions

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