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Pursuing Laplace’s Vision on Modern Computers

  • Tamar Schlick
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 82)

Abstract

This contribution is an informal essay based on a talk delivered at the Institute for Mathematics and its Applications (IMA) in Minneapolis, under the summer program in molecular biology, July 18–22, 1994. I exclude many technical details, which can be found elsewhere, and instead focus on the basic ideas of molecular dynamics simulations, with the goal of conveying to students and non-specialists the key concepts of the theory and practice of large-scale simulations. Following a description of the basic idea in molecular dynamics, I discuss some of the practical details involved in simulations of large biological molecules, the numerical timestep problem, and approaches to this problem based on implicit-integration techniques. I end with a perspective of open challenges in the field and directions for future research.

Keywords

Molecular Dynamic Simulation Langevin Equation Modern Computer Potential Energy Function Bovine Pancreatic Trypsin Inhibitor 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    P. S. de Laplace. Oeuvres Complètes de Laplace. Théorie Analytique des Probabilités, volume VII Gauthier-Villars, Paris, France, 3 edition, 1820.Google Scholar
  2. [2]
    J. A. McCammon and S. C. Harvey. Dynamics of Proteins and Nucleic Acids. Cambridge University Press, Cambridge, MA, 1987.Google Scholar
  3. [3]
    M. P. Allen and D. J. Tildesley. Computer Simulation of Liquids. Oxford University Press, New York, New York, 1990.Google Scholar
  4. [4]
    C. L. Brooks III, M. Karplus, and B. M. Pettitt. Proteins: A Theoretical Perspective of Dynamics, Structure, and Thermodynamics, volume LXXI of Advances in Chemical Physics. John Wiley & Sons, New York, New York, 1988.Google Scholar
  5. [5]
    F. M. Richards. The protein folding problem. Sci. Amer., 264:54–63, 1991.CrossRefGoogle Scholar
  6. [6]
    H. S. Chan and K. A. Dill. The protein folding problem. Physics Today, 46:24–32, 1993.CrossRefGoogle Scholar
  7. [7]
    R. W. Pastor. Techniques and applications of Langevin dynamics simulations. In G. R. Luckhurst and C. A. Veracini, editors, The Molecular Dynamics of Liquid Crystals, pages 85–138. Kluwer Academic Publishers, Dordrecht, The Netherlands, 1994.Google Scholar
  8. [8]
    M. Vásquez, G. Némethy, and H. A. Scheraga. Conformational energy calculations on polypeptides and proteins. Chemical Reviews, 94:2183–2239, 1994.CrossRefGoogle Scholar
  9. [9]
    J. Phillip Bowen and N. L. Allinger. Molecular mechanics: The art and science of parameterization. In K. B. Lipkowitz and D. B. Boyd, editors, Reviews in Computational Chemistry, volume II, pages 81–97. VCH Publishers, New York, New York, 1991.CrossRefGoogle Scholar
  10. [10]
    U. Burkert & N. L. Allinger. Molecular Mechanics, volume 177 of American Chemical Society Monograph. ACS, Washington, D. C, 1982.Google Scholar
  11. [11]
    S. Lifson. Potential energy functions for structural molecular biology. In D. B. Davies, W. Saenger, and S. S. Danyluk, editors, Methods in Structural Molecular Biology, pages 359–385. Plenum Press, London, 1981.Google Scholar
  12. [12]
    I. K. Roterman, M. H. Lambert, K. D. Gibson, and H. A. Scheraga. A comparison of the CHARMM, AMBER and ECEPP potentials for peptides. J. Biomol. Struct. Dyn., 7:391–452, 1989.Google Scholar
  13. [13]
    P. A. Kollman and K. A. Dill. Decisions in force field development: An alternative to those described by Roterman et al. J. Biomol. Struct. Dyn., 8:1103–1107, 1991.Google Scholar
  14. [14]
    K. B. Gibson and H.A. Scheraga. Decisions in force field development: Reply to Kollman and Dill. J. Biomol. Struct. Dyn., 8:1109–1111, 1991.Google Scholar
  15. [15]
    T. Schlick. Modeling and Minimization Techniques for Predicting Three-Dimensional Structures of Large Biological Molecules. PhD thesis, New York University, Courant Institute of Mathematical Sciences, New York, New York, October 1987.Google Scholar
  16. [16]
    B. R. Brooks, R. E. Bruccoleri, B. D. Olafson, D. J. States, S. Swaminathan, and M. Karplus. CHARMM: A program for macromolecular energy, minimization, and dynamics calculations. J. Comp. Chem., 4:187–217, 1983.CrossRefGoogle Scholar
  17. [17]
    S. J. Weiner, P. A. Kollman, D. T. Nguyen, and D. A. Case. An all atom force field for simulations of proteins and nucleic acids. J. Comp. Chem., 7:230–252, 1986.CrossRefGoogle Scholar
  18. [18]
    Molecular mechanics and modeling, November 1993. Special issue of Chemical Reviews (Volume 93, Number 7).Google Scholar
  19. [19]
    C. A. Schiffer, J. W. Caldwell, P. A. Kollman, and R. M. Stroud. Protein structure prediction with a combined solvation free energy-molecular mechanics force field. Mol. Sim., 10:121–149, 1993.CrossRefGoogle Scholar
  20. [20]
    J. R. Maple, M.-J. Hwang, T. P. Stockfisch, U. Dinur, M. Waldman, C. S. Ewing, and A. T. Hagler. Derivation of class II force fields. I. Methodology and quantum force field for the alkyl functional group and alkane molecules. J. Comp. Chem., 15:162–182, 1994.CrossRefGoogle Scholar
  21. [21]
    P. Derreumaux and G. Vergüten. Influence of the spectroscopic potential energy function SPASIBA on molecular dynamics of proteins: Comparison with the AMBER potential. J. Mol. Struct., 286:55–64, 1993.Google Scholar
  22. [22]
    L. Verlet. Computer ‘experiments’ on classical fluids: I. Thermo dynamical properties of Lennard-Jones molecules. Physical Review, 159(1):98–103, July 1967.CrossRefGoogle Scholar
  23. [23]
    M. P. Calvo and J. M. Sanz-Serna. The development of variable-step symplectic integrators, with application to the two-body problem. SIAM J. Sci. Comput., 14:936, 1993.CrossRefGoogle Scholar
  24. [24]
    P. J. Steinbach and B. R. Brooks. New spherical-cutoff methods for long-range forces in macromolecular simulation. J. Comp. Chem., 15:667–683, 1994.CrossRefGoogle Scholar
  25. [25]
    H. Grubmuller, H. Heller, A. Windemuth, and K. Schulten. Generalized Verlet algorithm for efficient molecular dynamics simulations with long-range interactions. Mol. Sim., 6:121–142, 1991.CrossRefGoogle Scholar
  26. [26]
    J. A. Board Jr., J. W. Causey, T. F. Leathrum Jr., A. Windemuth, and K. Schulten. Accelerated molecular dynamics simulations with the parallel fast multiple algorithm. Chem. Phys. Lett., 198:89–94, 1992.CrossRefGoogle Scholar
  27. [27]
    J. A. Board Jr., L. V. Kale, K. Schulten, R. D. Skeel, and T. Schlick. Modeling biomolecules: Larger scales, longer durations. IEEE Computational Science & Engineering, 1:19–30, Winter 1994.CrossRefGoogle Scholar
  28. [28]
    H. Frauenfelder and P. G. Wolynes. Biomolecules: Where the physics of complexity and simplicity meet. Physics Today, 47:58–64, 1994.CrossRefGoogle Scholar
  29. [29]
    H. Frauenfelder, S. G. Sligar, and P. G. Wolynes. The energy landscapes and motions of proteins. Science, 254:1598–1603, 1991CrossRefGoogle Scholar
  30. [29a]
    P.G. Wolynes, J.N. Onuchic, and D. Thirumalai, Navigating the Folding Routes, Science, 267:1619–1620, 1995.CrossRefGoogle Scholar
  31. [30]
    T. Schlick, B. Li, and W. K. Olson. The influence of salt on DNA energetics and dynamics. Biophys. J., 67:2146–2166, 1994.CrossRefGoogle Scholar
  32. [31]
    W. F. van Gunsteren and P. K. Weiner, editors. Computer Simulation of Biomolecular Systems. ESCOM, Leiden, The Netherlands, 1989.Google Scholar
  33. [32]
    P. Derreumaux and T. Schlick. Long-time integration for peptides by the dynamics driver approach. Proteins, Structure, Function and Genetics, 21:282–302, 1995.CrossRefGoogle Scholar
  34. [33]
    R. W. Pastor, B. R. Brooks, and A. Szabo. An analysis of the accuracy of Langevin and molecular dynamics algorithms. Mol. Phys., 65:1409–1419, 1988.CrossRefGoogle Scholar
  35. [34]
    A. M. Stuart and A. R. Humphries. Model problems in numerical stability theory for initial value problems. SIAM Review, 36:226–257, 1994.CrossRefGoogle Scholar
  36. [35]
    H. A. Scheraga. Predicting three-dimensional structures of oligopeptides. In K. B. Lipkowitz and D. B. Boyd, editors, Reviews in Computational Chemistry, volume III, pages 73–142. VCH Publishers, New York, New York, 1992.CrossRefGoogle Scholar
  37. [36]
    Mathematical challenges from theoretical/computational chemistry, National Research Council Report, National Academy Press, Washington, D.C., 1995.Google Scholar
  38. [37]
    Z. Wu. The effective energy transformation scheme as a special continuation approach to global optimization with application to molecular conformation. SIAM J. Opt., 6, 1996.Google Scholar
  39. [38]
    E. Hairer, S. P. Nørsett, and G. Wanner. Solving Ordinary Differential Equations I. Nonstiff Problems, volume 8 of Springer Series in Computational Mathematics. Springer-Verlag, New York, New York, 2 edition, 1993.Google Scholar
  40. [39]
    E. Hairer and G. Wanner. Solving Ordinary Differential Equations II. Stiff and Differential-Algebraic Problems, volume 14 of Springer Series in Computational Mathematics. Springer-Verlag, New York, New York, 1991.Google Scholar
  41. [40]
    A. Brünger, C. B. Brooks, and M. Karplus. Stochastic boundary conditions for molecular dynamics simulations of ST2 water. Chem. Phys. Lett., 105:495–500, 1982.CrossRefGoogle Scholar
  42. [41]
    W. F. van Gunsteren. Constrained dynamics of flexible molecules. Mol. Phys., 40:1015–1019, 1980.CrossRefGoogle Scholar
  43. [42]
    W. F. van Gunsteren and H.J.C. Berendsen. Algorithms for macromolecular dynamics and constraint dynamics. Mol. Phys., 34:1311–1327, 1977.CrossRefGoogle Scholar
  44. [43]
    J. P. Ryckaert, G. Ciccotti, and H. J. C. Berendsen. Numerical integration of the Cartesian equations of motion of a system with constraints: Molecular dynamics of n-alkanes. J. Comp. Phys., 23:327–341,1977.CrossRefGoogle Scholar
  45. [44]
    S. Miyamoto and P. A. Kollman. SETTLE: An analytical version of the SHAKE and RATTLE algorithm for rigid water models. J. Comp. Chem., 13:952–962, 1992.CrossRefGoogle Scholar
  46. [45]
    W. F. van Gunsteren and M. Karplus. Effect of constraints on the dynamics of macromolecules. Macromolecules, 15:1528–1543, 1982.CrossRefGoogle Scholar
  47. [46]
    C. S. Peskin and T. Schlick. Molecular dynamics by the backward Euler’s method. Comm. Pure App. Math., 42:1001–1031, 1989.CrossRefGoogle Scholar
  48. [47]
    T. Schlick and C. S. Peskin. Can classical equations simulate quantum-mechanical behavior? A molecular dynamics investigation of a diatomic molecule with a Morse potential. Comm. Pure App. Math., 42:1141–1163, 1989.CrossRefGoogle Scholar
  49. [48]
    T. Schlick and A. Fogelson. TNPACK — A truncated Newton minimization package for large-scale problems: I. algorithm and usage. ACM Trans. Math. Softw., 14:46–70, 1992.CrossRefGoogle Scholar
  50. [49]
    P. Derreumaux, G. Zhang, B. Brooks, and T. Schlick. A truncated-Newton method adapted for CHARMM and biomolecular applications. J. Comp. Chem., 15:532–552, 1994.CrossRefGoogle Scholar
  51. [50]
    T. Schlick, S. Figueroa, and M. Mezei. A molecular dynamics simulation of a water droplet by the implicit-Euler/Langevinscheme. J. Chem. Phys., 94:2118–2129, 1991.CrossRefGoogle Scholar
  52. [51]
    A. Nyberg and T. Schlick. Increasing the time step in molecular dynamics. Chem. Phys. Lett., 198:538–546, 1992.CrossRefGoogle Scholar
  53. [52]
    T. Schlick and W. K. Olson. Supercoiled DNA energetics and dynamics by computer simulation. J. Mol. Biol., 223:1089–1119, 1992.CrossRefGoogle Scholar
  54. [53]
    T. Schlick and W. K. Olson. Trefoil knotting revealed by molecular dynamics simulations of supercoiled DNA. Science, 257:1110–1115, 1992.CrossRefGoogle Scholar
  55. [54]
    G. Ramachandran and T. Schlick. Solvent effects on supercoiled DNA dynamics explored by Langevin dynamics simulations. Phys. Rev. E, 51:6188–6203, 1995.CrossRefGoogle Scholar
  56. [55]
    G. Zhang and T. Schlick. LIN: A new algorithm combining implicit integration and normal mode techniques for molecular dynamics. J. Comp. Chem., 14:1212–1233, 1993.CrossRefGoogle Scholar
  57. [56]
    W. B. Streett, D. J. Tildesley, and G. Saville. Multiple time step methods in molecular dynamics. Mol. Phys., 35:639–648, 1978.CrossRefGoogle Scholar
  58. [57]
    M. E. Tuckerman and B. J. Berne. Molecular dynamics in systems with multiple time scales: Systems with stiff and soft degrees of freedom and with short and long range forces. J. Comp. Chem., 95:8362–8364, 1992.Google Scholar
  59. [58]
    M. Watanabe and M. Karplus. Dynamics of molecules with internal degrees of freedom by multiple time-step methods. J. Chem. Phys., 99:8063–8074, 1993.CrossRefGoogle Scholar
  60. [59]
    J. J. Biesiadecki and R. D. Skeel. Dangers of multiple-time-step methods. J. Comp. Phys., 109:318–328, 1993.CrossRefGoogle Scholar
  61. [60]
    G. Zhang and T. Schlick. The Langevin/implicit-Euler/Normal-Mode scheme (LIN) for molecular dynamics at large time steps. J. Chem. Phys., 101:4995–5012, 1994.CrossRefGoogle Scholar
  62. [61]
    J. O’Neil and D. B. Szyld. A block ordering method for sparse matrices. SIAM J. Sci. Stat. Comp., 11:811–823, 1990.CrossRefGoogle Scholar
  63. [62]
    I. S. Duff, A. M. Erisman, and J. K. Reid. Direct Methods for Sparse Matrices. Oxford University Press, New York, New York, 1986.Google Scholar
  64. [63]
    G. H. Golub and C. F. van Loan. Matrix Computations. John Hopkins University Press, Baltimore, MD, 2 edition, 1986.Google Scholar
  65. [64]
    Z. Zlatev. Computational Methods for General Sparse Matrices. Kluwer Academic Publishers, Dordrecht, The Netherlands, 1991.Google Scholar
  66. [65]
    P. Dauber-Osguthorpe and D. J. Osguthorpe. Partitioning the motion in molecular dynamics simulations into characteristic modes of motion. J. Comp. Chem., 14:1259–1271, 1993.CrossRefGoogle Scholar
  67. [66]
    A. Amadei, A. B. M. Linssen, and H. J. C. Berendsen. Essential dynamics of proteins. Proteins, Structure, Function and Genetics, 17:412–425, 1993.CrossRefGoogle Scholar
  68. [67]
    J. C. Simo, N. Tarnow, and K. K. Wong. Exact energy-momentum conserving algorithms and symplectic schemes for nonlinear dynamics. Computer Methods in Applied Mechanics and Engineering, 100:63–116, 1991.CrossRefGoogle Scholar
  69. [68]
    G. Zhang and T. Schlick. Implicit discretization schemes for Langevin dynamics. Mol. Phys., 84:1077–1098, 1995.CrossRefGoogle Scholar
  70. [69]
    M. Mandziuk and T. Schlick. Resonance in chemical-system dynamics simulated by the implicit-midpoint scheme. Chem. Phys. Lett., 237:525–535, 1995.CrossRefGoogle Scholar
  71. [70]
    N. Grønbech-Jensen and S. Doniach. Long-time overdamped Langevin dynamics of molecular chains. J. Comp. Chem., 15:997–1012, 1994.CrossRefGoogle Scholar
  72. [71]
    J. A. McCammon, B. R. Gelin, and M. Karplus. Dynamics of folded proteins. Nature, 267:585–590, 1977.CrossRefGoogle Scholar
  73. [72]
    R. S. Struthers, J. Rivier, and A. T. Hagler. Theoretical simulation of conformation, energetics, and dynamics in the design of GnRH analogs. Transactions of the American Crystallographic Association, 20:83–96, 1984. Proceedings of the Symposium on Molecules in Motion, University of Kentucky, Lexington, Kentucky, May 20–21, 1984.Google Scholar
  74. [73]
    M. Levitt. Computer simulation of DNA double-helix dynamics. Cold Spring Harbor Symp. Quant. Biol., 47:251–275, 1983.Google Scholar
  75. [74]
    G. L. Seibel, U. C. Singh, and P. A Kollman. A molecular dynamics simulation of double-helical B-DNA including counterions and water. Proc. Natl. Acad. Sci. USA, 82:6537–6540, 1985.CrossRefGoogle Scholar
  76. [75]
    J. J. Wendoloski, S. J. Kimatian, C. E. Schutt, and F. R. Salemme. Molecular dynamics simulation of a phospholipid micelle. Science, 243:636–638, 1989.CrossRefGoogle Scholar
  77. [76]
    H. Heller, M. Schaefer, and K. Schulten. Molecular dynamics simulation of a bilayer of 200 lipids in the gel and in the liquid-crystal phases. J. Phys. Chem., 97:8343–8360, 1993.CrossRefGoogle Scholar
  78. [77]
    P. J. Kraulis. Molscript: A program to produce both detailed and schematic plots of protein structures. J. App. Crystallogr., 24:946–950, 1991.CrossRefGoogle Scholar
  79. [78]
    M.-H. Hao, M.R. Pincus, S. Rackovsky, and H.A. Scheraga. Unfolding and refolding of the native structure of bovine pancreatic trypsin inhibitor studied by computer simulations. Biochemistry, 32:9614–9631, 1993.CrossRefGoogle Scholar
  80. [79]
    B. Mishra and T. Schlick. Error analysis in numerical integration of the Langevin equation: 1. Linear analysis for five explicit and implicit schemes. Preprint, 1995.Google Scholar

Copyright information

© Springer-Verlag New York, Inc. 1996

Authors and Affiliations

  • Tamar Schlick
    • 1
    • 2
  1. 1.Howard Hughes Medical InstituteNew York UniversityNew YorkUSA
  2. 2.Chemistry DepartmentCourant Institute of Mathematical SciencesNew YorkUSA

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