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

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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

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

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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

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

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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)

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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 q2, r2 q6, r6 q8, r8 q9, r9 q1q2 q3 q9q7q8 q5 q6q4 Nonbond list for q5:

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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: Long distance interaction using PME method - Nonbond List Bond, angle, and dihedral lists are unchanged for each atom throughout the simulation Nonbond list is updated based on a cutoff buffer

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Ways to Update Nonbond List Global neighbor list Each thread can iterate through the global list of atoms to build the nonbond list 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 Cell-based list is computationally very efficient but also needs regular cell updates

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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

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q1q2 q3 q9q7q8 q5 q6 q4 Periodic Boundary Condition q1q2 q3 q9q7q8 q5 q6q4 q1q2 q3 q9q7q8 q5 q6q4 q1q2 q3 q9q7q8 q5 q6q4 q1q2 q3 q9q7q8 q5 q6q4 q1q2 q3 q9q7q8 q5 q6q4 q1q2 q3 q9q7q8 q5 q6q4 q1q2 q3 q9q7q8 q5 q6q4 q1q2 q3 q9q7q8 q5 q6q4 Cell of interest of edge vectors ax, ay Region of influence

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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 R c is the radius of the cavity Coulomb potential

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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)

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Ewald Summation Method Three contributions to the total energy, depending on the distance of the interaction: Direct space (E dir ) Reciprocal space (E rec ) Self energy (E self )

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

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

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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 Charges Essman et al., J. Chem. Phys x i y i z i : position of the i th charge; k 1 k 2 k 3 : index of the lattice point

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

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CPU vs. GPU Charge Spreading Charge Spreading by a cardinal B-Spline of order 4: CPU implementation is straightforward Time computation: N atoms x 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 Unit cell charges

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CPU vs. GPU Charge Spreading CPU spreading of charges:GPU gathering of charges by a cardinal B-Spline of order 4: 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(a x *a y *a z ) threads, with each thread parsing through all the atoms within 4x4x4 neighborhood –> O(N) Each thread is assigned to a lattice point

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

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GPU Charge Spreading (II) 2 2 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 The threads for lattice points in light gray update the list of lattice points in dark gray in a 1-to-1 fashion

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

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

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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

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

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Performance Small membrane ( atoms) 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) Large membrane ( atoms)

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Kernel Profiling (I) Large membrane – RF method Global neighbor listCell-based neighbor list

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Kernel Profiling (II) Large membrane – PME method Global neighbor listCell-based neighbor list

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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

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Conclusions and Future Work CHARMM-G is a flexible MD code based on the CHARMM force field integrating Ewald summation 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

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Acknowledgements 32 Sponsors: 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: GCL Members: Trilce EstradaBoyu Zhang Abel LiconNarayan Ganesan Lifan Xu Philip Saponaro Maria Ruiz Michela Taufer GCL members in Spring 2010

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