Narayan Ganesan 1 , Sandeep Patel 2 , and Michela Taufer 1 Computer and Info. Sciences Dept. 1

1. CHARMM-G, a GPU based MD Simulation code with PME and Reaction Force field for Studying Large Membrane Regions NarayanGanesan1, SandeepPatel2, and Michela Taufer1 Computer and Info. Sciences Dept.1 Chemistry and Biochemistry Dept.2 University of Delaware

2. Outline • Overview of forces in molecular dynamics • Data structures and methodology • PME for long distance electrostatic interactions • Steps involved in PME calculations • Performance and profiling of large membranes • Related work and conclusions

3. Classical Forces Classical Forces Bonded Non-Bonded Bonds Dihedrals Van der Waals Electrostatic Angles Reaction Field (RF) PME

4. Bond Interactions • Bond forces:acts only within pairs of molecules • Angle forces: acts only within a triad of atoms • Torsion or dihedral forces: acts only within quartet of atoms

5. Non-bond Interactions:Van der Waals Potential • Van derWaals or Lennard-Jones potential: Decays rapidly with distance • A cutoff of ~10A, accurately captures the effect of the Van der Waals potential

6. Non-Bond Interactions:Electrostatic Potential • Coulomb Potential: inverse square law • Decays as 1/r with distance • Since 1/r decays rather slowly, the potential can act over long distances • Choosing a cutoff for electrostatic force/potential causes computational errors and inaccuracy • Our solutions to sum long distance electrostatic forces: • Reaction Force Field (RF) • Ewald summation / Particle Mesh Ewald (PME)

7. GPU Implementation: Data Structures • A single thread is assigned to each atom • For each atom a set of lists is maintained: • Bond list stores list of bonds the atom belongs to • Angle list stores list of angles the atom belongs to • Dihedral list stores list of dihedrals the atom belongs to • Nonbond list stores non-bond interactions with atoms within cutoff q1 q2 q3 q4 q6 q5 q7 q8 q9 q2, r2 q6, r6 q8, r8 q9, r9 Nonbond list for q5:

8. MD Simulation • MD simulations are iterative executions of MD steps • Each iteration computes forces on each particle due to: • Bonds – Bond List • Angles – Angle List • Dihedrals – Dihedral List • Electrostatic • Van der Waals • If Ewald summation is used an additional component is added: • Longdistance interaction using PME method Bond, angle, anddihedral lists are unchanged for each atom throughout the simulation Nonbondlist is updated based on acutoff buffer - NonbondList

9. Ways to Update Nonbond List • Global neighbor list • Each thread can iterate through the global list of atoms to build the nonbondlist • Cell-based neighbor list • Divide the domain into equal cells of size = cutoff • Search only in current cell and adjacent cells for neighboring atoms • There are 26 adjacent cells and 1 current cell in 3-dimensions • Cell-based list is computationally very efficient but also needs regular cell updates

10. Cell Updates • Single thread manages a single or a set of cells • Each cell is managed by a list of atoms in the cell called ‘CellList’ • When an atom ‘i’ moves from Cell A to Cell B, the thread responsible for Cell A updates the list of Cell B via thread safe integer atomic intrinsics • Invalid atoms are removed from the cell lists by the ‘CellClean’kernel

11. Periodic Boundary Condition Cell of interest of edge vectors ax, ay q1 q1 q1 q1 q1 q1 q1 q1 q1 q2 q2 q2 q2 q2 q2 q2 q2 q2 q3 q3 q3 q3 q3 q3 q3 q3 q3 q6 q6 q6 q6 q6 q6 q6 q6 q6 q4 q4 q4 q4 q4 q4 q4 q4 q5 q5 q5 q5 q5 q5 q5 q5 q5 q4 q7 q7 q7 q7 q7 q7 q7 q7 q7 q8 q8 q8 q8 q8 q8 q8 q8 q8 q9 q9 q9 q9 q9 q9 q9 q9 q9 Region of influence

12. Reaction Force Field • Any molecule is surrounded by spherical cavity of finite radius • Within the radius, electrostatic interactions are calculated explicitly • Outside the cavity, the system is treaded as a dielectric continuum • This model allows the replacement of the infinite Coulomb sum by a finite sum plus the reaction filed where the second terms is the reaction filed correction and Rc is the radius of the cavity Coulomb potential

13. Ewald Summation Method (I) • Proposed by Paul Peter Ewald in 1921 for crystallographic systems • Has found applications in molecular, astrophysical and crystallographic systems • Used to sum inverse distance potential over long distance efficiently – e.g., Gravity and Coulomb Potential. • Was started to be used in the late 70s for numerical simulations • O(NlogN) instead of O(NxN)

14. Ewald Summation Method • Three contributions to the total energy, depending on the distance of the interaction: • Direct space (Edir) • Reciprocal space (Erec) • Self energy (Eself)

15. Ewald Summation Method (II) • Divide interactions into short range (Direct Space) and long range (Reciprocal Space) Short Range Direct space using NonbondList Long Range Fourier Space V - Volume of the simulation region S(m) – Structure parameters

16. Steps in SMPE Put charges on grids 1 3 2 4 Multiply with structure constants FFT of charge grid FFT back Convolution yields potential at grid points which have to be summed 5 Compute force on atom i by calculating

17. Charge Spreading • Each charge is spread on a 4x4x4 = 64 grid points in 3-D • Grid spacing 1 A by a cardinal B-Spline of order 4 • Create a 3 dimensional Charge Matrix “Q”. • Mesh-based charge density • Approximation by sum of charges at each grid point • Multiple charges can influence a single lattice point Essmanet al., J. Chem. Phys. 1995 Charges xiyizi: position of the ith charge; k1k2k3: index of the lattice point

18. Cardinal B-Spline of Order 4 • B-Splinehas a region of influence of 4 units • Each unit = 1A • During charge spreadingB-Splinehas an impact on the neighboring 4x4x4 cells in 3 dimensions

19. CPU vs. GPU Charge Spreading • Charge Spreading by a cardinal B-Spline of order 4: • CPU implementation is straightforward • Time computation: Natomsx 4 x 4 x 4 time steps • GPU implementation is hard to parallelize • Can lead to racing conditions - need floating point atomic writes • Current version of CUDA supports atomic writes for integers only • Charges need to be converted to fixed point in order to utilize the functionality Unitcell charges

20. CPU vs. GPU Charge Spreading CPU spreading of charges: GPU gathering of charges by a cardinal B-Spline of order 4: Each thread is assigned to a lattice point • Charge spreading on GPU can be parallelized easily by the grid points instead of the atoms • Each thread works on a single or a set of grid points • Need O(ax*ay*az) threads, with each thread parsing through all the atoms within 4x4x4 neighborhood –> O(N)

21. GPU Charge Spreading (I) • Each lattice point maintains a list of atoms within 4x4x4 neighborhood for charge gathering 1 2 Effect of charges 1, 2, 3 are gathered at the lattice point 3 q1, r1 q2, r2 q3, r3 Neighbor list of point:

22. GPU Charge Spreading (II) • When a charge moves, several lattice points need to be updated • The charge is added to the neighbor list of lattice points in dark gray • The charge is removed from the neighbor list of lattice points in light gray • Lattice points in white are not affected • Since there are equal number of light gray and dark gray lattice points, a 1-to-1 mapping was devised 2’ 2 The threads for lattice points in light gray update the list of lattice points in dark gray in a 1-to-1 fashion

23. GPU Charge Spreading (III) 1 2’ 1’ 2 • When a single lattice point is updated by multiple threads, thread safe integer atomic intrinsics are used to update the cell lists

24. Fast Fourier Transform • CUFFT provides library functions to compute FFT and inverse FFT • 3D FFT implemented with series of 1D FFTs and transpositions • CUFFTExeccan be optimized by choosing proper FFT dimensions • Power of 2

25. Scientific Challenge • One-third of the human genome is composed of membrane-bound proteins • Pharmaceuticals target membrane-bound protein receptors e.g., G-protein coupled receptors • Importance of systems to human health and understanding of dysfunction • State-of-the-art simulations only consider small regions (or patches) of physiological membranes • Heterogeneity of the membrane spans length scales much larger than included in these smaller model systems. • Our goal: apply large-scale GPU-enabled computations for the study of large membrane regions

26. DMPC • DiMyristoylPhosphatidylCholine(DMPC) lipid bilayers • Small system: 17 004 atoms,  46. 8A x 46.8 A x 76.0 A • Large system: 68 484 atoms, 93.6 A x 93.6 A x 152.0 A 92A 92A Explicit solvent i.e., water 152A Membrane

27. Performance Small membrane (17 004 atoms) • Largemembrane (68 484atoms) Case studies: Global neighbor list and RF (I), with cell-based list and RF (II), with neighbor list and PME (III), and with cell-based neighbor list and PME (IV)

28. Kernel Profiling (I) • Large membrane – RF method Global neighbor list Cell-based neighbor list

29. Kernel Profiling (II) • Large membrane – PME method Global neighbor list Cell-based neighbor list

30. Related Work • Other MD code including PME method: M. J. Harvey and G. De. Fabritiis, J. Chem. Theory and Comp, 2009” • Our implementation is different in terms of: • Charge spreading algorithm • Force field methods, including RF

31. Conclusions and Future Work • CHARMM-G is a flexible MD codebased on the CHARMM force field integrating • Ewaldsummation • Reaction force field • The code supports explicit solvent representations and enables fast simulations of large membrane regions • Improvements of the CUDA FFT will further improve the performance presented in the paper • Future work include: • Code optimizations and parallelization across multiple GPUs • Scientific characterization of large membranes

32. Acknowledgements GCL Members: Trilce Estrada Boyu Zhang Abel LiconNarayanGanesan LifanXu Philip Saponaro Maria Ruiz Michela Taufer Collaborators: Sandeep Patel, Brad A. Bauer, Joseph E. Davis (Dept. of Chemistry, UD) Related work: Bauer et al, JCC 2010 (In Press) Davis et al., BICoB 2009 More questions: taufer@udel.edu GCL members in Spring 2010 Sponsors: