^{1,a)}, Lanyuan Lu

^{1,2,a)}and Gregory A. Voth

^{1,2,b)}

### Abstract

Many methodologies have been proposed to build reliable and computationally fast coarse-grained potentials. Typically, these force fields rely on the assumption that the relevant properties of the system under examination can be reproduced using a pairwise decomposition of the effective coarse-grained forces. In this work it is shown that an extension of the multiscale coarse-graining technique can be employed to parameterize a certain class of two-body and three-body force fields from atomistic configurations. The use of explicit three-body potentials greatly improves the results over the more commonly used two-body approximation. The method proposed here is applied to develop accurate one-site coarse-grained water models.

This research was supported by the National Science Foundation Collaborative Research in Chemistry (Project No. CHE-0628257). Computer resources were provided by the National Science Foundation through TeraGrid computing resources administered by the Pittsburgh Supercomputing Center, the San Diego Supercomputer Center, the National Center for Supercomputing Applications, the Texas Advanced Computing Center, and Argonne National Laboratories. We thank Hans Andersen and Avisek Das for their insightful comments and suggestions regarding this work. The authors also thank Gary Ayton, Valeria Molinero, Vinod Krishna, Ron Hills, Chris Knight, and Mark Maupin for many valuable discussions. L.L. and L.L. contributed equally to this work.

I. INTRODUCTION

II. METHODS

A. Coarse-graining procedure

B. Limitations of the two-body approximation in CG modeling

C. Generalization to three-body CG potentials

D. Ensemble averages

E. Atomistic simulations

F. CG representation

III. RESULTS AND DISCUSSIONS

A. RDF

B. Angular distribution function

C. Ensemble averages

D. Computational efficiency

IV. CONCLUSIONS

### Key Topics

- Electrostatics
- 5.0
- Molecular dynamics
- 5.0
- Angular distribution
- 4.0
- Particle distribution functions
- 4.0
- Proteins
- 4.0

## Figures

Interaction forces between CG water sites with (red) and without (black) three-body potential. Only the results for the COM representation are shown.

Interaction forces between CG water sites with (red) and without (black) three-body potential. Only the results for the COM representation are shown.

RDF for the one-site CG model for SPC/E water centered in the center of mass of the interacting particle. The use of an explicit three-body CG potential is seen to improve the first shell of solvation greatly.

RDF for the one-site CG model for SPC/E water centered in the center of mass of the interacting particle. The use of an explicit three-body CG potential is seen to improve the first shell of solvation greatly.

RDF for the one-site CG model for SPC/E water centered in the center of geometry of the interacting particle.

RDF for the one-site CG model for SPC/E water centered in the center of geometry of the interacting particle.

The ADF for SPC/E water computed inside the cutoff of the three-body CG potential for the case with the center of mass CG representation. The three-body potential is seen to closely reproduce the atomistic distribution.

The ADF for SPC/E water computed inside the cutoff of the three-body CG potential for the case with the center of mass CG representation. The three-body potential is seen to closely reproduce the atomistic distribution.

The ADF for SPC/E water computed inside the cutoff of the three-body CG potential when the center of geometry CG representation is used.

The ADF for SPC/E water computed inside the cutoff of the three-body CG potential when the center of geometry CG representation is used.

Probability distributions as a function of two-body distance and three-body angle for each triplet, calculated from: (a) All-atom configurations; (b) CG configurations from CG simulation with both two and three-body CG potentials. The angle is and the distance is one of the . The probability is calculated for the area , with and . Only the results for the COM representation are shown.

Probability distributions as a function of two-body distance and three-body angle for each triplet, calculated from: (a) All-atom configurations; (b) CG configurations from CG simulation with both two and three-body CG potentials. The angle is and the distance is one of the . The probability is calculated for the area , with and . Only the results for the COM representation are shown.

A comparison of the internal energy between the atomistic and CG systems for SPC/E water. Details of how the comparison was performed can be found in Sec. II. The center of mass case is shown. An analogous plot can be obtained from the case where the geometric center is employed instead. (a) the internal energy is computed using only a two-body CG force field. The red line is an actual CG simulation. The blue line reports the value of the potential energy obtained when the same two-body CG potential is applied to a previous all-atom trajectory in CG resolution. (b) Same as (a), but using a three-body CG potential. (c) The distribution of the data presented in (a). (d) The distribution of the data shown in (b).

A comparison of the internal energy between the atomistic and CG systems for SPC/E water. Details of how the comparison was performed can be found in Sec. II. The center of mass case is shown. An analogous plot can be obtained from the case where the geometric center is employed instead. (a) the internal energy is computed using only a two-body CG force field. The red line is an actual CG simulation. The blue line reports the value of the potential energy obtained when the same two-body CG potential is applied to a previous all-atom trajectory in CG resolution. (b) Same as (a), but using a three-body CG potential. (c) The distribution of the data presented in (a). (d) The distribution of the data shown in (b).

Comparison of the virial component of the pressure between atomistic and CG simulations for SPC/E water. The case of COM is shown. The case with the center of geometric is analogous. (a) The virial is computed using only a two-body CG force field. The red line is an actual CG simulation. The blue line reports the value of the virial obtained when the same two-body CG potential is applied to a previous all-atom trajectory in CG resolution. (b) Same as (a), but using a three-body CG potential. (c) The distribution of the data presented in (a). (d) The distribution of the data shown in (b).

Comparison of the virial component of the pressure between atomistic and CG simulations for SPC/E water. The case of COM is shown. The case with the center of geometric is analogous. (a) The virial is computed using only a two-body CG force field. The red line is an actual CG simulation. The blue line reports the value of the virial obtained when the same two-body CG potential is applied to a previous all-atom trajectory in CG resolution. (b) Same as (a), but using a three-body CG potential. (c) The distribution of the data presented in (a). (d) The distribution of the data shown in (b).

The virial part of the pressure computed for different densities. The comparison is performed employing the two-body CG potential. For a detailed discussion on how the comparison is performed, see Sec. II D. The curves have the same shape as can be evidenced from shifting the CG simulation data upward (dashed line). The difference is only due to an almost constant shift. Only the center of mass case is shown as the geometric center produces a similar plot.

The virial part of the pressure computed for different densities. The comparison is performed employing the two-body CG potential. For a detailed discussion on how the comparison is performed, see Sec. II D. The curves have the same shape as can be evidenced from shifting the CG simulation data upward (dashed line). The difference is only due to an almost constant shift. Only the center of mass case is shown as the geometric center produces a similar plot.

Same plot as in Fig. 9 but for systems employing an explicit three-body potential. Now the agreement is closer, even though a constant shift can be still observed.

Same plot as in Fig. 9 but for systems employing an explicit three-body potential. Now the agreement is closer, even though a constant shift can be still observed.

## Tables

Computational efficiency. is the CPU time required to finish the simulation. In the last two columns, this value is normalized with respect to GROMACS’ or LAMMPS’ . For the three-body case the cutoff employed is given in parentheses.

Computational efficiency. is the CPU time required to finish the simulation. In the last two columns, this value is normalized with respect to GROMACS’ or LAMMPS’ . For the three-body case the cutoff employed is given in parentheses.

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