Evaluation of Fast Evaluation of Fast Electrostatics AlgorithmsElectrostatics Algorithms

Alice N. Ko and Jesús A. Izaguirre

with Thierry Matthey

Department of Computer Science and EngineeringUniversity of Notre Dame, USADepartment of InformaticsUniversity of Bergen, NORWAY

Executive summaryExecutive summary

How to choose the best among Particle Mesh Ewald (PME), Multi-Grid (MG) summation, Ewald

sum, for molecular dynamics of biological molecules. Why should your next simulation

consider using MG?

Problem: Full Electrostatic Problem: Full Electrostatic EnergyEnergy

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Motivation Motivation Fast evaluation of full electrostatics in molecular

dynamics (MD) of biological molecules important for accuracy in many applications

Structural stability of DNA and proteins Ionic environments

Many methods exist to do explicit evaluation of fast electrostatics

Fast Multipole Method O(N) Greengard, 1987 Particle Mesh Ewald O(N log N) Darden, 1993 Multi-grid summation O(N) Brandt, 1990

Skeel, 2002 Which one to use for a given system and

accuracy?

ObjectivesObjectives

1. Provide practical guidelines for choosing parameters for each algorithm

2. Evaluate competitive algorithms

3. Evaluate suitability of MG to MD simulations

Particle Mesh EwaldParticle Mesh Ewald Following Ewald, separates the

electrostatic interactions into two parts: Direct-space short range evaluation Fourier-space evaluation

The Fourier term is approximated by using fast Fourier transforms on a grid

Method parameters are grid size and cutoff of direct-space

Multigrid IMultigrid I

Multigrid IIMultigrid II

Related Work IRelated Work I

Darden et al., J. Chem. Phys. 1993 effect of varying parameters of Particle Mesh Ewald

Petersen et al., J. Chem. Phys. 1995 accuracy and efficiency of Particle Mesh Ewald (PME)

Krasny et al., J. Chem. Phys. 2000 used FMM to compute direct part of Ewald sum

Skeel et al., J. Comp. Chem. 2002 study of parameters for multigrid (MG) method.

Compared MG to Fast Multipole Method (FMM). MG faster than FMM for low accuracy

Related Work IIRelated Work II

Most published results fail to suggest how to determine the

specific values provide general trends only contain unknown constants in

equations that model performance

SummarySummary General contributions of this study

Practical guidelines for choosing parameters for each algorithm, and to choose among different algorithms

Implemented important algorithms with reasonable efficiency in ProtoMol

Tested algorithms for various system sizes and accuracy

Tested quality of these methods for MD of solvated proteins

Encapsulated results of this study on a tool called MDSimAid

Experimental protocolExperimental protocol These methods were tested and implemented:

1. Smooth Particle Mesh Ewald2. Multigrid summation3. Ewald summation

Testing protocol: Methods (1) and (2) above were compared against (3)

to determine accuracy and relative speedup Tested on water boxes and protein systems ranging

from 1,000 to 100,000 atoms, and low and high accuracies

CHARMM used to prepare systems, NAMD and ProtoMol used for simulations

Determined optimal parameters for each method for a given accuracy and system size

For selected protein systems, structural and transport properties were computed (e.g., Melittin, pdb id 2mlt, in water, 11845 atoms)

Solvated MelittinSolvated Melittin

ResultsResults Big picture

Multi-grid summation is an effective method for low accuracy computation of full electrostatics

For low accuracy, Multi-Grid is faster than PME and Ewald for all system sizes tested (from 1000 to 100,000)

For medium accuracy, Multi-Grid is faster than PME for systems of 8,000 atoms or more

Multi-grid with low accuracy produces correct structural and dynamic properties

Results (10Results (10-4-4 rPE) rPE)

Results (10Results (10-5-5 rPE) rPE)

RDFRDF

Multigrid IIIMultigrid III Complex relationship among method

parameters: Cutoff and softening distances for potential

evaluation at the particle and grid levels Grid size and interpolation order Number of levels

Rules extracted from extensive evaluation encapsulated in MDSimAid

Fine tuned at run-time by running selected tests

Makes method easier to use

Simulation Results for Simulation Results for MelittinMelittin

PME requires about 3% of the CPU time (17 days 20 hours) when measured against Ewald

MG in pbc requires only about 1% MG is about 66% faster than PME

DiscussionDiscussion MG is a competitive method for low accuracy MD

simulations Accuracy not a great concern for long time simulations MG would be natural choice for multiple time stepping

integrators To choose among methods, and good parameters for

each method, MDSimAid is a useful tool For further reference:

http://www.nd.edu/~lcls/mdsimaid http://www.nd.edu/~lcls/protomol http://www.ks.uiuc.edu/development/namd http://www.nd.edu/~izaguirr

AcknowledgementsAcknowledgements

This research was supported by an NSF Biocomplexity grant and an NSF CAREER award