Evaluation of Fast Electrostatics Algorithms Alice N. Ko and Jesús A. Izaguirre with Thierry Matthey Department of Computer Science and Engineering University of Notre Dame, USA Department of Informatics University of Bergen, NORWAY

Executive 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 Energy

Motivation u Fast evaluation of full electrostatics in molecular dynamics (MD) of biological molecules important for accuracy in many applications n Structural stability of DNA and proteins n Ionic environments u Many methods exist to do explicit evaluation of fast electrostatics n Fast Multipole Method O(N) Greengard, 1987 n Particle Mesh Ewald O(N log N) Darden, 1993 n Multi-grid summation O(N) Brandt, 1990 Skeel, 2002 u Which one to use for a given system and accuracy?

Objectives 1. Provide practical guidelines for choosing parameters for each algorithm 2. Evaluate competitive algorithms 3. Evaluate suitability of MG to MD simulations

Particle Mesh Ewald u Following Ewald, separates the electrostatic interactions into two parts: n Direct-space short range evaluation n Fourier-space evaluation u The Fourier term is approximated by using fast Fourier transforms on a grid u Method parameters are grid size and cutoff of direct-space

Multigrid I

Multigrid II

Related Work I n Darden et al., J. Chem. Phys l effect of varying parameters of Particle Mesh Ewald n Petersen et al., J. Chem. Phys l accuracy and efficiency of Particle Mesh Ewald (PME) n Krasny et al., J. Chem. Phys l used FMM to compute direct part of Ewald sum n Skeel et al., J. Comp. Chem l study of parameters for multigrid (MG) method. Compared MG to Fast Multipole Method (FMM). MG faster than FMM for low accuracy

Related Work II u Most published results n fail to suggest how to determine the specific values n provide general trends only n contain unknown constants in equations that model performance

Summary u General contributions of this study n Practical guidelines for choosing parameters for each algorithm, and to choose among different algorithms l Implemented important algorithms with reasonable efficiency in ProtoMol l Tested algorithms for various system sizes and accuracy l Tested quality of these methods for MD of solvated proteins n Encapsulated results of this study on a tool called MDSimAid

Experimental protocol u These methods were tested and implemented: 1. Smooth Particle Mesh Ewald 2. Multigrid summation 3. Ewald summation u Testing protocol: n Methods (1) and (2) above were compared against (3) to determine accuracy and relative speedup n Tested on water boxes and protein systems ranging from 1,000 to 100,000 atoms, and low and high accuracies n CHARMM used to prepare systems, NAMD and ProtoMol used for simulations n Determined optimal parameters for each method for a given accuracy and system size n For selected protein systems, structural and transport properties were computed (e.g., Melittin, pdb id 2mlt, in water, atoms)

Solvated Melittin

Results u Big picture n Multi-grid summation is an effective method for low accuracy computation of full electrostatics l For low accuracy, Multi-Grid is faster than PME and Ewald for all system sizes tested (from 1000 to 100,000) l For medium accuracy, Multi-Grid is faster than PME for systems of 8,000 atoms or more n Multi-grid with low accuracy produces correct structural and dynamic properties

Results (10 -4 rPE)

Results (10 -5 rPE)

RDF

Multigrid III u Complex relationship among method parameters: n Cutoff and softening distances for potential evaluation at the particle and grid levels n Grid size and interpolation order n Number of levels u Rules extracted from extensive evaluation encapsulated in MDSimAid n Fine tuned at run-time by running selected tests n Makes method easier to use

Simulation Results for Melittin u PME requires about 3% of the CPU time (17 days 20 hours) when measured against Ewald u MG in pbc requires only about 1% u MG is about 66% faster than PME

Discussion u MG is a competitive method for low accuracy MD simulations u Accuracy not a great concern for long time simulations u MG would be natural choice for multiple time stepping integrators u To choose among methods, and good parameters for each method, MDSimAid is a useful tool u For further reference: n n n n

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