Abstract
Poisson and Poisson-Boltzmann equations (PE and PBE) are widely used in molecular modeling to estimate the electrostatic contribution to the free energy of a system. In such applications, PE often needs to be solved multiple times for a large number of system configurations. This can rapidly become a highly demanding computational task. To accelerate such calculations we implemented a graphical processing unit (GPU) PE solver described in this work. The GPU solver performance is compared to that of our central processing unit (CPU) implementation of the solver. During the performance analysis the following three characteristics were studied: (1) precision associated with the modeled system discretization on the grid, (2) numeric precision associated with the floating point representation of real numbers (this is done via comparison of calculations with single precision (SP) and double precision (DP)), and (3) execution time. Two types of example calculations were carried out to evaluate the solver performance: (1) solvation energy of a single ion and a small protein (lysozyme), and (2) a single ion potential in a large ion-channel (α-hemolysin). In addition, influence of various boundary condition (BC) choices was analyzed, to determine the most appropriate BC for the systems that include a membrane, typically represented by a slab with the dielectric constant of low value. The implemented GPU PE solver is overall about 7 times faster than the CPU-based version (including all four cores). Therefore, a single computer equipped with multiple GPUs can offer a computational power comparable to that of a small cluster. Our calculations showed that DP versions of CPU and GPU solvers provide nearly identical results. SP versions of the solvers have very similar behavior: in the grid scale range of 1-4 grids/Å the difference between SP and DP versions is less than the difference stemming from the system discretization. We found that for the membrane protein, the use of a focusing technique with periodic boundary conditions in rough grid provides significantly better results than using a focusing technique with the electric potential set to zero at the boundaries.
[1] J. Antosiewicz, J. A. McCammon, and M. K. Gilson, “Prediction of pH-dependent properties of proteins.,” Journal of Molecular Biology, vol. 238, no. 3, pp. 415–436, 1994. 10.1006/jmbi.1994.1301Search in Google Scholar PubMed
[2] D. Bashford and M. Karplus, “pKa’s of ionizable groups in proteins: atomic detail from a continuum electrostatic model.,” Biochemistry, vol. 29, no. 44, pp. 10219–10225, 1990. 10.1021/bi00496a010Search in Google Scholar PubMed
[3] S. Bhakdi and J. Tranum-Jensen, “Alpha-toxin of Staphylococcus aureus.,” Microbiological reviews, vol. 55, no. 4, pp. 733–751, 1991. 10.1128/mr.55.4.733-751.1991Search in Google Scholar PubMed PubMed Central
[4] P. M. De Biase, C. J. F. Solano, S. Markosyan, L. Czapla, and S. Y. Noskov, “BROMOC-D: Brownian Dynamics/Monte-Carlo Program Suite to Study Ion and DNA Permeation in Nanopores.,” Journal of Chemical Theory and Computation, vol. 8, no. 7, pp. 2540–2551, 2012. 10.1021/ct3004244Search in Google Scholar PubMed PubMed Central
[5] N. Carrascal and D. F. Green, “Energetic decomposition with the generalized-born and Poisson-Boltzmann solvent models: lessons from association of G-protein components.,” The Journal of Physical Chemistry B, vol. 114, no. 15, pp. 5096–5116, 2010. Search in Google Scholar
[6] I.-L. Chern, J.-G. Liu, and W.-C. Wang, “Accurate evaluation of electrostatics for macromolecules in solution,” Methods and Applications of Analysis, vol. 10, no. 2, pp. 309–328, 2003. 10.4310/MAA.2003.v10.n2.a9Search in Google Scholar
[7] R. D. Coalson and M. G. Kurnikova, “Poisson-Nernst-Planck theory approach to the calculation of current through biological ion channels.,” IEEE Transactions on NanoBioscience, vol. 4, no. 1, pp. 81–93, 2005. 10.1109/TNB.2004.842495Search in Google Scholar PubMed
[8] M. L. Connolly, “Molecular Surfaces 5. Solvent Accessible Surfaces,” Network Science Corporation. [Online]. Available: http://www.netsci.org/Science/Compchem/feature14e.html. Search in Google Scholar
[9] B. Eisenberg, “From Structure to Function in Open Ionic Channels,” The Journal of membrane biology, vol. 171, no. 1, pp. 1–24, 2010. 10.1007/s002329900554Search in Google Scholar PubMed
[10] M. K. Gilson and B. H. Honig, “Energetics of charge-charge interactions in proteins.,” Proteins, vol. 3, no. 1, pp. 32–52, Jan. 1988. 10.1002/prot.340030104Search in Google Scholar PubMed
[11] D. F. Green and B. Tidor, “Evaluation of electrostatic interactions.,” Current protocols in bioinformatics editoral board Andreas D Baxevanis et al, vol. Chapter 8, p. Unit 8.3, 2003. 10.1002/0471250953.bi0803s02Search in Google Scholar PubMed
[12] J. G. Kirkwood, “Theory of solutions of molecules containing widely separated charges with special application to zwitterions,” Journal of Chemical Physics, vol. 2, no. 7, pp. 351–361, 1934. 10.1063/1.1749489Search in Google Scholar
[13] I. Klapper, R. Hagstrom, R. Fine, K. Sharp, and B. Honig, “Focusing of electric fields in the active site of Cu-Zn superoxide dismutase: effects of ionic strength and amino-acid modification.,” Proteins, vol. 1, no. 1, pp. 47–59, 1986. 10.1002/prot.340010109Search in Google Scholar
[14] V. Krishnamurthy and S.-H. C. S.-H. Chung, “Brownian dynamics simulation for modeling ion permeation across bionanotubes.,” IEEE Transactions on NanoBioscience, vol. 4, no. 1, pp. 102–111, 2005. 10.1109/TNB.2004.842494Search in Google Scholar
[15] B. Kuhn and P. A. Kollman, “Binding of a Diverse Set of Ligands to Avidin and Streptavidin: An Accurate Quantitative Prediction of Their Relative Affinities by a Combination of Molecular Mechanics and Continuum Solvent Models,” Journal of Medicinal Chemistry, vol. 43, no. 20, pp. 3786–3791, Oct. 2000. 10.1021/jm000241hSearch in Google Scholar
[16] B. Lu, M. J. Holst, J. A. McCammon, and Y. C. Zhou, “Poisson-Nernst-Planck equations for simulating biomolecular diffusion-reaction processes I: Finite element solutions,” Journal of Computational Physics, vol. 229, no. 19, pp. 6979–6994, 2010. Search in Google Scholar
[17] A. Nicholls and B. Honig, “A rapid finite difference algorithm, utilizing successive over-relaxation to solve the Poisson-Boltzmann equation,” Journal of Computational Chemistry, vol. 12, no. 4, pp. 435–445, 1991. 10.1002/jcc.540120405Search in Google Scholar
[18] NVIDIA, NVIDIA CUDA C Programming Guide, PG-02829-001_v5.0, no. 350. NVIDIA Corporation, 2012. Search in Google Scholar
[19] G. Prévost, L. Mourey, D. A. Colin, and G. Menestrina, “Staphylococcal pore-forming toxins.,” Current Topics in Microbiology and Immunology, vol. 257, pp. 53–83, 2001. 10.1007/978-3-642-56508-3_4Search in Google Scholar
[20] M. Ramanadham, L. C. Sieker, and L. H. Jensen, “Refinement of triclinic lysozyme: II. The method of stereochemically restrained least squares.,” Acta Crystallographica Section B Structural Science, vol. 46, no. 1, pp. 63–69, 1990. 10.1107/S0108768189009195Search in Google Scholar
[21] W. Rocchia, S. Sridharan, A. Nicholls, E. Alexov, A. Chiabrera, and B. Honig, “Rapid grid-based construction of the molecular surface and the use of induced surface charge to calculate reaction field energies: applications to the molecular systems and geometric objects.,” Journal of Computational Chemistry, vol. 23, no. 1, pp. 128–137, 2002. 10.1002/jcc.1161Search in Google Scholar
[22] B. Roux, “Theoretical and computational models of ion channels.,” Current Opinion in Structural Biology, vol. 12, no. 2, pp. 182–189, 2002. 10.1016/S0959-440X(02)00307-XSearch in Google Scholar
[23] N. A. Simakov and M. G. Kurnikova, “Soft wall ion channel in continuum representation with application to modeling ion currents in α-hemolysin.,” The journal of physical chemistry. B, vol. 114, no. 46, pp. 15180–90, Dec. 2010. Search in Google Scholar
[24] D. Sitkoff, K. A. Sharp, and B. Honig, “Accurate Calculation of Hydration Free Energies Using Macroscopic Solvent Models,” Journal of Physical Chemistry, vol. 98, no. 7, pp. 1978–1988, 1994. 10.1021/j100058a043Search in Google Scholar
[25] L. Song, M. R. Hobaugh, C. Shustak, S. Cheley, H. Bayley, and J. E. Gouaux, “Structure of staphylococcal alphahemolysin, a heptameric transmembrane pore.,” Science, vol. 274, no. 5294, pp. 1859–1866, 1996. Search in Google Scholar
[26] Y. Song, J. Mao, and M. R. Gunner, “MCCE2: Improving protein pKa calculations with extensive side chain rotamer sampling,” Journal of Computational Chemistry, vol. 30, no. 14, pp. 2231–2247, 2009. Search in Google Scholar
[27] J. Srinivasan, T. E. Cheatham, P. Cieplak, P. A. Kollman, and D. A. Case, “Continuum Solvent Studies of the Stability of Continuum Solvent Studies of the Stability of DNA , RNA , and Phosphoramidate-DNA Helices,” Structure, vol. 120, no. 37, pp. 9401–9409, 1998. Search in Google Scholar
[28] D. J. Vocadlo, G. J. Davies, R. Laine, and S. G. Withers, “Catalysis by hen egg-white lysozyme proceeds via a covalent intermediate,” Nature, vol. 412, no. 6849, pp. 835–838, 2001. Search in Google Scholar
[29] Y. C. Zhou, M. Feig, and G. W. Wei, “Highly accurate biomolecular electrostatics in continuum dielectric environments.,” Journal of Computational Chemistry, vol. 29, no. 1, pp. 87–97, 2008. 10.1002/jcc.20769Search in Google Scholar PubMed
©2013 Versita Sp. z o.o.
This content is open access.
