Some Failures and Successes of Long-Timestep Approaches to Biomolecular Simulations

Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 4)


A personal account of work on long-timestep integration of biomolecular dynamics is presented, emphasizing the limitations, as well as success, of various approaches. These approaches include implicit discretization, separation into harmonic and anharmonic motion, and force splitting; some of these techniques are combined with stochastic dynamics. A Langevin/force-splitting approach for biomolecular simulations termed LN (for its origin in a Langevin/normal-modes scheme) is also described, suitable for general thermodynamic and sampling questions. LN combines force linearization, stochastic dynamics, and force splitting via extrapolation so that the timestep for updating the slow forces can be increased beyond half the period of the fast motions (i.e., 5 fs). This combination of strategies alleviates the severe stability restriction due to resonance artifacts that apply to symplectic force-splitting methods and can yield significant speedup (with respect to small-timestep reference Langevin trajectories). Extensions to sampling problems are natural by this approach.


Molecular Dynamic Molecular Dynamic Simulation Reference Trajectory Resonance Artifact Time Step Method 
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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  1. 1.
    H. S. Chan and K. A. Dill. The protein folding problem. Physics Today, 46: 24–32, 1993.CrossRefGoogle Scholar
  2. 2.
    J. A. McCammon, B. M. Pettitt, and L. R. Scott. Ordinary differential equations of molecular dynamics. Computers Math. Applic., 28: 319–326, 1994.MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    B. J. Leimkuhler, S. Reich, and R. D. Skeel. Integration methods for molecular dynamics. In J. P. Mesirov, K. Schulten, and D. W. Sumners, editors, Mathematical Approaches to Biomolecular Structure and Dynamics, volume 82 of IMA Volumes in Mathematics and Its Applications, pages 161–186, New York, New York, 1996. Springer-Verlag.Google Scholar
  4. 4.
    R. Elber. Novel methods for molecular dynamics simulations. Curr. Opin. Struc. Biol., 6: 232–235, 1996.CrossRefGoogle Scholar
  5. 5.
    T. Schlick, E. Barth, and M. Mandziuk. Biomolecular dynamics at long timesteps: Bridging the timescale gap between simulation and experimentation. Ann. Rev. Biophys. Biomol Struc., 26: 179–220, 1997.CrossRefGoogle Scholar
  6. 6.
    J. M. Sanz-Serna and M. P. Calvo. Numerical Hamiltonian Problems. Chapman & Hall, London, 1994.zbMATHGoogle Scholar
  7. 7.
    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
  8. 8.
    H.C. Andersen. Rattle: a ‘velocity’ version of the SHAKE algorithm for molecular dynamics calculations. J. Comp. Phys., 52: 24–34, 1983.zbMATHCrossRefGoogle Scholar
  9. 9.
    B. Leimkuhler and R. D. Skeel. Symplectic numerical integrators in constrained Hamiltonian systems. J. Comp. Phys., 112: 117–125, 1994.MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    E. Barth, K. Kuczera, B. Leimkuhler, and R. D. Skeel. Algorithms for constrained molecular dynamics. J. Comp. Chem., 16: 1192–1209, 1995.CrossRefGoogle Scholar
  11. 11.
    W. B. Streett, D. J. Tildesley, and G. Saville. Multiple time step methods in molecular dynamics. Mol Phys., 35: 639–648, 1978.CrossRefGoogle Scholar
  12. 12.
    M. E. Tuckerman, B.J. Berne, and G. J. Martyna. Molecular dynamics algorithm for multiple time scales: Systems with long range forces. J. Chem. Phys., 94: 6811–6815, 1991.CrossRefGoogle Scholar
  13. 13.
    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
  14. 14.
    H. Grubmüller, 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
  15. 15.
    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
  16. 16.
    D. E. Humphreys, R. A. Friesner, and B. J. Berne. A multiple-time-step molecular dynamics algorithm for macromolecules. J. Phys. Chem., 98(27): 6885–6892, 1994.CrossRefGoogle Scholar
  17. 17.
    R. Zhou and B. J. Berne. A new molecular dynamics method combining the reference system propagator algorithm with a fast multipole method for simulating proteins and other complex systems. J. Chem. Phys., 103: 9444–9459, 1995.CrossRefGoogle Scholar
  18. 18.
    L. Verlet. Computer ‘experiments’ on classical fluids: I. Thermodynamical properties of Lennard-Jones molecules. Phys. Rev., 159(1): 98–103, July 1967.CrossRefGoogle Scholar
  19. 19.
    M. P. Allen and D. J. Tildesley. Computer Simulation of Liquids. Oxford University Press, New York, New York, 1990.Google Scholar
  20. 20.
    P. Deuflhard, M. Dellnitz, O. Junge, and Ch. Schütte. Computation of essential molecular dynamics by subdivision techniques: I. Basic concepts. Technical Report SC 96-45, Konrad-Zuse-Zentrum für Informationstechnik Berlin, Takustraβe 7, D-14195, Berlin-Dahlem, December 1996.Google Scholar
  21. 21.
    D. A. McQuarrie. Statistical Mechanics. Harper & Row, New York, New York, 1976. Chapters 20-21.Google Scholar
  22. 22.
    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, Dordrecht, The Netherlands, 1994.CrossRefGoogle Scholar
  23. 23.
    A. Brünger, C. L. Brooks, III, and M. Karplus. Stochastic boundary conditions for molecular dynamics simulations of ST2 water. Chem. Phys. Lett., 105: 495–500, 1982.CrossRefGoogle Scholar
  24. 24.
    T. Simonson. Accurate calculation of the dielectric constant of water from simulations of a microscopic droplet in vacuum. Chem. Phys. Lett., 250: 450–454, 1996.CrossRefGoogle Scholar
  25. 25.
    D. Beglov and B. Roux. Finite representation of an infinite bulk system: Solvent boundary potential for computer simulations. J. Chem. Phys., 100: 9050–9063, 1994.CrossRefGoogle Scholar
  26. 26.
    D. Beglov and B. Roux. Dominant solvations effects from the primary shell of hydration: Approximation for molecular dynamics simulations. Biopolymers, 35: 171–178, 1994.CrossRefGoogle Scholar
  27. 27.
    D. Beglov and B. Roux. Numerical solutions of the hypernetted chain equation for a solute of arbitrary geometry in three dimensions. J. Chem. Phys., 103: 360–364, 1995.CrossRefGoogle Scholar
  28. 28.
    R. J. Loncharich, B. R. Brooks, and R. W. Pastor. Langevin dynamics of peptides: The frictional dependence of isomerization rates of N-acetylalanyl-N’-methylamide. Biopolymers, 32: 523–535, 1992.CrossRefGoogle Scholar
  29. 29.
    P. Derreumaux and T. Schlick. Long-time integration for peptides by the dynamics driver approach. Proteins: Struc. Func. Gen., 21: 282–302, 1995.CrossRefGoogle Scholar
  30. 30.
    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
  31. 31.
    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
  32. 32.
    D. K. Klimov and D. Thirumalai. Viscosity dependence of the folding rates of proteins. Phys. Rev. Lett., 79: 317–320, 1997.CrossRefGoogle Scholar
  33. 33.
    R. C. Wade, M. E. Davis, B. A. Luty, J. D. Madura, and J. A. McCammon. Gating of the active site of triose phosphate isomerase: Brownian dynamics simulations of flexible peptide loops in the enzyme. Biophys. J., 64: 9–15, 1993.CrossRefGoogle Scholar
  34. 34.
    Hongmei Jian. A Combined Wormlike-Chain and Bead Model for Dynamic Simulations of Long DNA. PhD thesis, New York University, Department of Physics, New York, New York, October 1997.Google Scholar
  35. 35.
    D. A. Case. Normal mode analysis of protein dynamics. Curr. Opin. Struc. Biol., 4: 385–290, 1994.CrossRefGoogle Scholar
  36. 36.
    G. Ramachandran and T. Schlick. Beyond optimization: Simulating the dynamics of supercoiled DNA by a macroscopic model. In P. M. Pardalos, D. Shalloway, and G. Xue, editors, Global Minimization of Nonconvex Energy Functions: Molecular Conformation and Protein Folding, volume 23 of DIMACS Series in Discrete Mathematics and Theoretical Computer Science, pages 215–231, Providence, Rhode Island, 1996. American Mathematical Society.Google Scholar
  37. 37.
    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
  38. 38.
    C. W. Gear. Numerical Initial Value Problems in Ordinary Differential Equations. Prentice Hall, Englewood Cliffs, New Jersey, 1971.zbMATHGoogle Scholar
  39. 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, second edition, 1996.Google Scholar
  40. 40.
    D. Janezic and B. Orel. Implicit Runge-Kutta method for molecular dynamics integration. J. Chem. Info. Comp. Sci., 33: 252–257, 1993.CrossRefGoogle Scholar
  41. 41.
    G. Zhang and T. Schlick. Implicit discretization schemes for Langevin dynamics. Mol. Phys., 84: 1077–1098, 1995.CrossRefGoogle Scholar
  42. 42.
    C. S. Peskin and T. Schlick. Molecular dynamics by the backward Euler’s method. Comm. Pure App. Math., 42: 1001–1031, 1989.MathSciNetzbMATHCrossRefGoogle Scholar
  43. 43.
    T. Schlick. Pursuing Laplace’s vision on modern computers. In J. P. Mesirov, K. Schulten, and D. W. Sumners, editors, Mathematical Applications to Biomolecular Structure and Dynamics, volume 82 of IMA Volumes in Mathematics and Its Applications, pages 219–247, New York, New York, 1996. Springer-Verlag.Google Scholar
  44. 44.
    T. Schlick and M. L. Overton. A powerful truncated Newton method for potential energy functions. J. Comp. Chem., 8: 1025–1039, 1987.MathSciNetCrossRefGoogle Scholar
  45. 45.
    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
  46. 46.
    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
  47. 47.
    D. Xie and T. Schlick. Efficient implementation of the truncated Newton method for large-scale chemistry applications. SIAM J. Opt., 1997. In Press.Google Scholar
  48. 48.
    T. Schlick, S. Figueroa, and M. Mezei. A molecular dynamics simulation of a water droplet by the implicit-Euler/Langevin scheme. J. Chem. Phys., 94: 2118–2129, 1991.CrossRefGoogle Scholar
  49. 49.
    A. Nyberg and T. Schlick. Increasing the time step in molecular dynamics. Chem. Phys. Lett., 198: 538–546, 1992.CrossRefGoogle Scholar
  50. 50.
    T. Schlick and C. S. Peskin. Comment on: The evaluation of LI and LIN for dynamics simulations. J. Chem. Phys., 103: 9888–9889, 1995.CrossRefGoogle Scholar
  51. 51.
    T. Schlick and W. K. Olson. Supercoiled DNA energetics and dynamics by computer simulation. J. Mol Biol, 223: 1089–1119, 1992.CrossRefGoogle Scholar
  52. 52.
    T. Schlick and W. K. Olson. Trefoil knotting revealed by molecular dynamics simulations of supercoiled DNA. Science, 257: 1110–1115, 1992.CrossRefGoogle Scholar
  53. 53.
    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
  54. 54.
    G. Liu, W. K. Olson, and T. Schlick. Application of Fourier methods to computer simulation of supercoiled DNA. Comp. Polymer Sci., 5: 7–27, 1995.Google Scholar
  55. 55.
    G. Liu, T. Schlick, A. J. Olson, and W. K. Olson. Configurational transitions in Fourier-series-represented DNA supercoils. Biophys. J., 73: 1742–1762, 1997.CrossRefGoogle Scholar
  56. 56.
    T. Schlick. Modeling superhelical DNA: Recent analytical and dynamic approaches. Curr. Opin. Struc. Biol., 5: 245–262, 1995.CrossRefGoogle Scholar
  57. 57.
    P. Derreumaux and T. Schlick. The loop opening/closing motion of the enzyme triosephosphate isomerase. Biophys. J., 74: 72–81, 1998.CrossRefGoogle Scholar
  58. 58.
    F. Kang. The Hamiltonian way for computing Hamiltonian dynamics. In R. Spigler, editor, Applied and Industrial Mathematics, pages 17–35. Kluwer Academic, Dordrecht, The Netherlands, 1990.Google Scholar
  59. 59.
    J. C. Simo, N. Tarnow, and K. K. Wang. Exact energy-momentum conserving algorithms and symplectic schemes for nonlinear dynamics. Computer Methods in Applied Mechanics and Engineering, 100: 63–116, 1994.CrossRefGoogle Scholar
  60. 60.
    J. C. Simo and N. Tarnow. The discrete energy-momentum method. Conserving algorithms for nonlinear elastodynamics. ZAMP, 43: 757–793, 1992.MathSciNetzbMATHCrossRefGoogle Scholar
  61. 61.
    O. Gonzales and J. C. Simo. On the stability of symplectic and energymomentum conserving algorithms for nonlinear Hamiltonian systems with symmetry. Comp. Meth. App. Mech. Engin., 134: 197, 1994.CrossRefGoogle Scholar
  62. 62.
    M. Mandziuk and T. Schlick. Resonance in the dynamics of chemical systems simulated by the implicit-midpoint scheme. Chem. Phys. Lett., 237: 525–535, 1995.CrossRefGoogle Scholar
  63. 63.
    J. J. Biesiadecki and R. D. Skeel. Dangers of multiple-time-step methods. J. Comp. Phys., 109: 318–328, 1993.MathSciNetzbMATHCrossRefGoogle Scholar
  64. 64.
    D. I. Okunbor and R. D. Skeel. Canonical numerical methods for molecular dynamics simulations. J. Comp. Chem., 15: 72–79, 1994.CrossRefGoogle Scholar
  65. 65.
    V. I. Arnold. Mathematical Methods of Classical Mechanics. Springer-Verlag, New York, Heidelberg, Berlin, 1989. second edition.Google Scholar
  66. 66.
    R. D. Skeel, G. Zhang, and T. Schlick. A family of symplectic integrators: Stability, accuracy, and molecular dynamics applications. SIAM J. Sci. Comp., 18(1): 202–222, January 1997.MathSciNetCrossRefGoogle Scholar
  67. 67.
    T. Schlick, M. Mandziuk, R.D. Skeel, and K. Srinivas. Nonlinear resonance artifacts in molecular dynamics simulations. J. Comp. Phys., 139: 1–29, 1998.MathSciNetCrossRefGoogle Scholar
  68. 68.
    D. Janezic and B. Orel. Parallelization of an implicit Runge-Kutta method for molecular dynamics integration. J. Chem. Info. Comp. Sci., 34: 641–646, 1994.CrossRefGoogle Scholar
  69. 69.
    M. Zhang and R. D. Skeel. Cheap implicit symplectic integrators. Applied Numerical Mathematics, 25: 297–302, 1997.MathSciNetzbMATHCrossRefGoogle Scholar
  70. 70.
    M. E. Tuckerman, G. J. Martyna, and B. J. Berne. Molecular dynamics algorithm for condensed systems with multiple time scales. J. Chem. Phys., 93: 1287–1291, 1990.CrossRefGoogle Scholar
  71. 71.
    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
  72. 72.
    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
  73. 73.
    E. Barth, M. Mandziuk, and T. Schlick. A separating framework for increasing the timestep in molecular dynamics. In W. F. van Gunsteren, P. K. Weiner, and A. J. Wilkinson, editors, Computer Simulation of Biomolecular Systems: Theoretical and Experimental Applications, volume III, chapter 4, pages 97–121. ESCOM, Leiden, The Netherlands, 1997.Google Scholar
  74. 74.
    R. M. Levy, A. R. Srinivasan, W. K. Olson, and J. A. McCammon. Quasiharmonic method for studying very low frequency modes in proteins. Biopolymers, 23: 1099–1112, 1984.CrossRefGoogle Scholar
  75. 75.
    B. R. Brooks, D. Janežič, and M. Karplus. Harmonic analysis of large systems. I. Methodology. J. Comp. Chem., 16: 1522–1542, 1995.CrossRefGoogle Scholar
  76. 76.
    R. M. Levy, O. de la Luz Rojas, and R. A. Friesner. Quasi-harmonic method for calculating vibrational spectra from classical simulations on multidimensional anharmonic potential surfaces. J. Phys. Chem., 88: 4233–4238, 1984.CrossRefGoogle Scholar
  77. 77.
    B. R. Brooks and M. Karplus. Normal modes for specific motions of macromolecules: Application to the hinge-bending mode of lysozyme. Proc. Natl. Acad. Sci. USA, 82: 4995–4999, 1985.CrossRefGoogle Scholar
  78. 78.
    M. Hao and S. C. Harvey. Analyzing the normal mode dynamics of macromolecules by the component synthesis method. Biopolymers, 32: 1393–1405, 1992.CrossRefGoogle Scholar
  79. 79.
    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
  80. 80.
    D. Perahia and L. Mouawad. Computation of low-frequency normal modes in macromolecules: Improvements to the method of diagonalization in a mixed basis and application to hemoglobin. Comput. Chem., 19: 241–246, 1995.CrossRefGoogle Scholar
  81. 81.
    A. Amadei, A. B. M. Linssen, and H. J. C. Berendsen. Essential dynamics of proteins. Proteins: Struc. Func. Gen., 17: 412–425, 1993.CrossRefGoogle Scholar
  82. 82.
    A. Amadei, A. B. M. Linssen, B. L. deGroot, D. M. F. van Aalten, and H. J. C. Berendsen. An efficient method for sampling the essential subspace of proteins. J. Biomol. Struct Dynam., 13: 615–625, 1996.CrossRefGoogle Scholar
  83. 83.
    M. A. Balsera, W. Wriggers, Y. Oono, and K. Schulten. Principal component analysis and long time protein dynamics. J. Phys. Chem., 100: 2567–2572, 1996.CrossRefGoogle Scholar
  84. 84.
    D. Janezic and F. Merzel. An efficient symplectic integration algorithm for molecular dynamics simulations. J. Chem. Info. Comp. Sci., 35: 321–326, 1995.CrossRefGoogle Scholar
  85. 85.
    J. D. Turner, P. K. Weiner, H. M. Chun, V. Lupi, S. Gallion, and U. C. Singh. Variable reduction techniques applied to molecular dynamics simulations. In W. F. van Gunsteren, P. K. Weiner, and A. J. Wilkinson, editors, Computer Simulation of Biomolecular Systems: Theoretical and Experimental Applications, volume 2, chapter 24. ESCOM, Leiden, The Netherlands, 1993.Google Scholar
  86. 86.
    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
  87. 87.
    B. Garcia-Archilla, J.M. Sanz-Serna, and R.D. Skeel. Long-time-step methods for oscillatory differential equations. SIAM J. Sci. Comp., 1996. To appear, [Also Tech. Rept. 1996/7, Dep. Math. Applic. Comput., Univ. Valladolid, Valladolid, Spain].Google Scholar
  88. 88.
    E. Barth and T. Schlick. Overcoming stability limitations in biomolecular dynamics: I. combining force splitting via extrapolation with Langevin dynamics in LN. J. Chem. Phys., 109: 1617–1632, 1998.CrossRefGoogle Scholar
  89. 89.
    E. Barth and T. Schlick. Extrapolation versus impulse in multiple-timestepping schemes: II. linear analysis and applications to Newtonian and Langevin dynamics. J. Chem. Phys., 109: 1632–1642, 1998.Google Scholar
  90. 90.
    M. E. Tuckerman, B. J. Berne, and G. J. Martyna. Reversible multiple time scale molecular dynamics. J. Chem. Phys., 97: 1990–2001, 1992.CrossRefGoogle Scholar
  91. 91.
    A. Sandu and T. Schlick. Further analysis of impulse and extrapolation force splitting in molecular dynamics. J. Comp. Phys., 1998. SubmittedGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  1. 1.Department of Chemistry and Courant Institute of Mathematical SciencesNew York University and The Howard Hughes Medical InstituteNew YorkUSA

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