^{1}, James F. Dama

^{1}, Marissa G. Saunders

^{1}, Gregory A. Voth

^{1}, Jonathan Weare

^{1,a)}and Aaron R. Dinner

^{1,b)}

### Abstract

Coarse-graining a molecular model is the process of integrating over degrees of freedom to obtain a reduced representation. This process typically involves two separate but related steps, selection of the coordinates comprising the reduced system and modeling their interactions. Both the coordinate selection and the modeling procedure present challenges. Here, we focus on the former. Typically, one seeks to integrate over the fast degrees of freedom and retain the slow degrees of freedom. Failure to separate timescales results in memory. With this motivation, we introduce a heuristic measure of memory and show that it can be used to compare competing coordinate selections for a given modeling procedure. We numerically explore the utility of this heuristic for three systems of increasing complexity. The first example is a four-particle linear model, which is exactly solvable. The second example is a sixteen-particle nonlinear model; this system has interactions that are characteristic of molecular force fields but is still sufficiently simple to permit exhaustive numerical treatment. The third example is an atomic-resolution representation of a protein, the class of models most often treated by relevant coarse-graining approaches; we specifically study an actin monomer. In all three cases, we find that the heuristic suggests coordinate selections that are physically intuitive and reflect molecular structure. The memory heuristic can thus serve as an objective codification of expert knowledge and a guide to sites within a model that requires further attention.

This material is based upon work supported by the National Science Foundation (NSF) through the Center for Multiscale Theory and Simulation (CHE-1136709).

I. INTRODUCTION

II. METHODS

A. Generalized Langevin equation

B. Estimating the memory timescale

C. Mapping schemes

III. LINEAR MASS-SPRING MODEL

IV. NONLINEAR MOLECULAR MODEL

V. ACTIN MODEL

VI. CONCLUSIONS

### Key Topics

- Chemical bonds
- 10.0
- Eigenvalues
- 9.0
- Trajectory models
- 8.0
- Integrable systems
- 7.0
- Normal modes
- 7.0

## Figures

Plot of λ = 2*I* _{1}/*I* _{0} for β(*t*) = *Ae* ^{−t/τ}cos (ω*t*). λ is a good approximation for τ in the region ξ ≫ 1.

Plot of λ = 2*I* _{1}/*I* _{0} for β(*t*) = *Ae* ^{−t/τ}cos (ω*t*). λ is a good approximation for τ in the region ξ ≫ 1.

Estimated memory timescale λ = 2*I* _{1}/*I* _{0} as a function of the mapping parameters for a four-particle linear mass-spring model coarse-grained to a single effective particle using an internal coordinate style mapping. Coarse-grained variables corresponding to eigenmodes of the fine-grained system are marked by red upward triangles and optima of the estimated memory timescale are marked by green downward triangles. The function is plotted relative to its minimum attained value, 1.29. The memory-optimal coarse-grainings are quite close to the eigenmodes of the system, as conjectured, despite statistical errors in the estimation of the memory timescale and the sensitivity of the minima to noise.

Estimated memory timescale λ = 2*I* _{1}/*I* _{0} as a function of the mapping parameters for a four-particle linear mass-spring model coarse-grained to a single effective particle using an internal coordinate style mapping. Coarse-grained variables corresponding to eigenmodes of the fine-grained system are marked by red upward triangles and optima of the estimated memory timescale are marked by green downward triangles. The function is plotted relative to its minimum attained value, 1.29. The memory-optimal coarse-grainings are quite close to the eigenmodes of the system, as conjectured, despite statistical errors in the estimation of the memory timescale and the sensitivity of the minima to noise.

Coarse-grainings for the nonlinear molecular system. The angle terms involving the bond between atoms 7 and 8 (red) are varied as described in the text. Particles are shaded according to their groupings into effective particles. (a) Memory-optimal coarse-graining when the maps are required to divide the fine-grained particles evenly between the two effective particles and one of the particles must contain a contiguous segment of bonded atoms; (b) a coarse-graining with identical memory timescale to (a) when angles involving bond 7-8 are stiffer than others in the system; (c) an arbitrarily chosen coarse-graining used as a comparison to the memory-optimal ones.

Coarse-grainings for the nonlinear molecular system. The angle terms involving the bond between atoms 7 and 8 (red) are varied as described in the text. Particles are shaded according to their groupings into effective particles. (a) Memory-optimal coarse-graining when the maps are required to divide the fine-grained particles evenly between the two effective particles and one of the particles must contain a contiguous segment of bonded atoms; (b) a coarse-graining with identical memory timescale to (a) when angles involving bond 7-8 are stiffer than others in the system; (c) an arbitrarily chosen coarse-graining used as a comparison to the memory-optimal ones.

Comparison for the nonlinear molecular model of the memory heuristic λ = 2*I* _{1}/*I* _{0} to the standard deviation of the fine-grained forces in each bin of the coarse-grained interaction, weighted by how often the system is observed in each bin. Each point corresponds to one possible mapping; data shown are for coarse-grainings of models with central bond angles that are more flexible (*B* _{78k } = 0.1) (red circles), as flexible (*B* _{78k } = 1) (green upward triangles), and less flexible (*B* _{78k } = 2) (blue downward triangles) than all other angles. (Inset) Same comparison for the case that the coarse-grained sites are not constrained to be the same size.

Comparison for the nonlinear molecular model of the memory heuristic λ = 2*I* _{1}/*I* _{0} to the standard deviation of the fine-grained forces in each bin of the coarse-grained interaction, weighted by how often the system is observed in each bin. Each point corresponds to one possible mapping; data shown are for coarse-grainings of models with central bond angles that are more flexible (*B* _{78k } = 0.1) (red circles), as flexible (*B* _{78k } = 1) (green upward triangles), and less flexible (*B* _{78k } = 2) (blue downward triangles) than all other angles. (Inset) Same comparison for the case that the coarse-grained sites are not constrained to be the same size.

Comparison between coarse-grained trajectories projected from the fine-grained nonlinear molecular model and generated directly from GLE models for mappings 1111111100000000 (a), 1110110000000000 (b), and 0001111111001100 (c), illustrated in Figs. 3(a)–3(c) , respectively.

Comparison between coarse-grained trajectories projected from the fine-grained nonlinear molecular model and generated directly from GLE models for mappings 1111111100000000 (a), 1110110000000000 (b), and 0001111111001100 (c), illustrated in Figs. 3(a)–3(c) , respectively.

Comparison between the memory function and the autocorrelation of the force residual for the coarse-graining 1111111100000000 of the nonlinear molecular model. Note that λ = 2*I* _{1}/*I* _{0} is independent of the vertical scale.

Comparison between the memory function and the autocorrelation of the force residual for the coarse-graining 1111111100000000 of the nonlinear molecular model. Note that λ = 2*I* _{1}/*I* _{0} is independent of the vertical scale.

Convergence of the Monte Carlo scheme used to search for memory-optimal mappings for heteroENM models of the actin monomer. In the contiguous case, effective particles are composed of atoms of residues that are contiguous in sequence, while in the discontiguous case, no such restriction is placed on the mappings.

Convergence of the Monte Carlo scheme used to search for memory-optimal mappings for heteroENM models of the actin monomer. In the contiguous case, effective particles are composed of atoms of residues that are contiguous in sequence, while in the discontiguous case, no such restriction is placed on the mappings.

Structural comparison of coarse-grainings of an actin monomer: handpicked (a), memory-optimal sequence contiguous (b), and essential dynamics (c). Sequence ranges for colors are given in Fig. 9 .

Structural comparison of coarse-grainings of an actin monomer: handpicked (a), memory-optimal sequence contiguous (b), and essential dynamics (c). Sequence ranges for colors are given in Fig. 9 .

Sequential comparison of coarse-grainings of an actin monomer. Colors correspond to Fig. 8 . Effective particles in the memory-optimal coarse-graining that correspond to subdivisions of effective particles in the hand-picked coarse-graining are indicated in bold font and discussed further in the text.

Sequential comparison of coarse-grainings of an actin monomer. Colors correspond to Fig. 8 . Effective particles in the memory-optimal coarse-graining that correspond to subdivisions of effective particles in the hand-picked coarse-graining are indicated in bold font and discussed further in the text.

## Tables

Relative error in positions between projected fine-grained dynamics and coarse-grained model dynamics (χ) at two time intervals (δ*t*) for the indicated coarse-grainings of the nonlinear molecular model.

Relative error in positions between projected fine-grained dynamics and coarse-grained model dynamics (χ) at two time intervals (δ*t*) for the indicated coarse-grainings of the nonlinear molecular model.

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