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Integration Methods for Molecular Dynamics

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Mathematical Approaches to Biomolecular Structure and Dynamics

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

Abstract

Classical molecular dynamics simulation of a macromolecule requires the use of an efficient time-stepping scheme that can faithfully approximate the dynamics over many thousands of timesteps. Because these problems are highly nonlinear, accurate approximation of a particular solution trajectory on meaningful time intervals is neither obtainable nor desired, but some restrictions, such as symplecticness, can be imposed on the discretization which tend to imply good long term behavior. The presence of a variety of types and strengths of interatom potentials in standard molecular models places severe restrictions on the timestep for numerical integration used in explicit integration schemes, so much recent research has concentrated on the search for alternatives that possess (1) proper dynamical properties, and (2) a relative insensitivity to the fastest components of the dynamics. We survey several recent approaches.

The work of the first author was performed under grant number NSF Grant DMS-9303223 and relied on resources of the Kansas Institute for Theoretical and Computational Science. The work of the second and third authors was performed at the Beck-man Institute of the University of Illinois and supported in part by DOE/NSF grant DE-FG02–91-ER25099/DMS-9304268, by NIH Grant P41RR05969, and by NSF/ARPA Grant ASC-9318159

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References

  1. Allen, M.P. and Tildesley, D.J., Computer Simulation of Liquids, Oxford Science, New York, 1987.

    Google Scholar 

  2. Andersen, H.C., Rattle: a ‘velocity’ version of the shake algorithm for molecular dynamics calculations, J. Comput. Phys. 52, 24–34, 1983.

    Article  CAS  Google Scholar 

  3. Barth, E., A user’s guide to MBLab, 1993 Matlab Conference Proceedings, electronic publication available via anonymous ftp from ftp.mathworks.com.

  4. Barth, E., Kuczera, K., Leimkuhler, B., and Skeel, R.D., Algorithms for constrained molecular dynamics, J. Comput. Chem., 16, 1192–1209, 1995.

    Article  CAS  Google Scholar 

  5. Beeman, D., Some multistep methods for use in molecular dynamics calculations, J. Comput. Phys. 20, 130–139, 1976.

    Article  Google Scholar 

  6. Biesiadecki, J.J. and Skeel, R.D., Dangers of multiple-time-step methods, J. Comput. Phys. 109, 318–328, 1993.

    Article  Google Scholar 

  7. Boothby, W.M., An Introduction to Differentiable Manifolds and Riemannian Geometry, 2nd edition, Academic Press, New York, 1986.

    Google Scholar 

  8. Brooks, B.R., Zhou, J., and Reich, S., Elastic molecular dynamics with flexible constraints, in preparation.

    Google Scholar 

  9. Buneman, O., Time-Reversible Difference Procedures, J. Comput. Phys. 1, 517–535, 1967.

    Article  Google Scholar 

  10. Calvo, M.P. and Sanz-Serna, J.M., The development of variable step symplectic integrators, with application to the two-body problem, SIAM J. Sci. Comp. 14, 936–952, 1993.

    Article  Google Scholar 

  11. Chawla, M.M., On the order and attainable intervals of periodicity of explicit Nystrom methods for y″ = f(t,y), SIAM J. Numer. Anal. 22, 127–131, 1985.

    Article  Google Scholar 

  12. Daubechies, I., Ten Lectures on Wavelets, SIAM, Philadelphia, 1992.

    Google Scholar 

  13. Dragt, A., Neri, F., Rangarajan, G., Douglas, D., Healy, L., and Ryne, R., “Lie algebraic treatment of nonlinear beam dynamics”, Ann. Rev. Nucl. Part. Sci. 38, 455–496, 1988.

    Article  CAS  Google Scholar 

  14. Fixman, M., Classical statistical mechanics and constraints: A theorem and applications to polymers., Proc. Nat. Acad. Sci. 71, 3050–53, 1974.

    Article  CAS  Google Scholar 

  15. Grubmüller, H., Heller, H., Windemuth, A., and Schulten, K., Generalized Verlet algorithm for efficient molecular dynamics simulations with long-range interactions, Molecular Simulation 6, 121–142, 1991.

    Article  Google Scholar 

  16. Ge, Z. and Marsden, J., Lie-Poisson Hamilton-Jacobi theory and Lie-Poisson integrators, Phys. Lett. A 133, 134–139, 1988.

    Article  Google Scholar 

  17. Hairer, E., Norsett, S., and Wanner, G., Solving Ordinary Differential Equations, I, Springer-Verlag, New York/Berlin, 1987.

    Google Scholar 

  18. Hayli, A., Le problème des N corps dans un champ extérieur application a l’évolution dynamique des amas ouverts-I, Bulletin Astronomique 2, 67–89, 1967.

    Google Scholar 

  19. Janezic, D. and Orel, B., Implicit Runge-Kutta methods for molecular dynamics integration, J. Chem. Inf. Comput. Sci. 33, 252–257, 1993.

    CAS  Google Scholar 

  20. Jay, L., Symplectic partitioned Runge-Kutta methods for constrained Hamiltonian systems, PhD Thesis, Department of Mathematics, University of Geneva, 1993.

    Google Scholar 

  21. Lambert, J. and Watson, D., Symmetricmultistep methods, J. Inst. Maths. Applic. 18, 189–202, 1976.

    Article  Google Scholar 

  22. Leimkuhler, B. and Skeel, R.D., Symplectic Numerical Integrators in Constrained Hamiltonian Systems, J. Comput. Phys. 112, 117–125, 1994.

    Article  Google Scholar 

  23. Marsden, J.R. and Ratiu, T., An Introduction to Mechanics and Symmetry, Springer-Verlag, New York, 1994.

    Google Scholar 

  24. McCammon, J.A. and Harvey, S.C., Dynamics of Proteins and Nucleic Acids, Cambridge University Press, Cambridge, 1987.

    Google Scholar 

  25. Okunbor, D. and Skeel, R.D., Canonical numerical methods for molecular dynamics simulations, J. Comput. Chem. 15, 72–79, 1994.

    Article  CAS  Google Scholar 

  26. Ortega, J.M. and Rheinboldt, W.C., Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, 1970.

    Google Scholar 

  27. Pathria, P.K., Statistical Mechanics, Pergamon Press, Oxford, 1972.

    Google Scholar 

  28. Pear, M.R. and Weiner, J.H., Brownian dynamics study of a polymer chain of linked rigid bodies, J. Chem. Phys. 71, 212–224, 1979.

    Article  CAS  Google Scholar 

  29. Peskin, C.S. and Schlick, T., Molecular dynamics by the backward Euler method, Comm. Pure and Appl. Math. 42, 1001–1031, 1989.

    Article  Google Scholar 

  30. Quinlan, G.D. and Tremaine, S., Symmetric multistep methods for the numerical integration of planetary orbits, Astron. J. 100, 1694–1700, 1990.

    Article  Google Scholar 

  31. Reich, S., Smoothed dynamics of highly oscillatory Hamiltonian systems, Physica D 89, 28–42, 1995.

    Article  CAS  Google Scholar 

  32. Reich, S., Symplectic integration of constrained Hamiltonian systems by composition methods, SIAM J. Numer. Anal., 33, 475–491, 1996.

    Article  Google Scholar 

  33. Reich, S., Molecular dynamics with soft constraints, Technical Report UIUC-BI-TB-94-12, Beckman Institute, Urbana, 1994.

    Google Scholar 

  34. Ryckaert, J.P., Ciccotti, G., and Berendsen, H.J.C., Numerical integration of Cartesian equations of motion of a system with constraints: Molecular dynamics of n-alkanes, J. Comput. Phys. 23, 327–342, 1977.

    Article  CAS  Google Scholar 

  35. Sanders, J.A. and Verhulst, F., Averaging Methods in Nonlinear Dynamical Systems, Springer-Verlag, New York, 1985.

    Google Scholar 

  36. Sanz-Serna, J.M. and Calvo, M.P., Numerical Hamiltonian Problems, Chapman and Hall, London, 1994.

    Google Scholar 

  37. Scully, J.L. and Hermans, J., Multiple time steps: Limits on the speedup of molecular dynamics simulations in aqueous systems, Mol. Simulation 11, 67–77, 1993.

    Article  CAS  Google Scholar 

  38. Simo, J., Rarnow, N., and Wong, K.K., Exact energy-momentum conserving algorithms and symplectic schemes for nonlineaer dynamics, Comp. Meth. Appl. Mech. Eng. 1, 1528–1544, 1992.

    Google Scholar 

  39. Skeel, R.D., Variable step size destabilizes the Stormer/leapfrog/Verlet method, BIT 33, 172–175, 1993.

    Article  Google Scholar 

  40. Skeel, R.D. and Biesiadecki, J.J., Symplectic integration with variable stepsize, Annals of Numer. Math. 1, 191–198, 1994.

    Google Scholar 

  41. Strang, G., On the construction and comparison of difference schemes, SIAM J. Numer. Anal. 5, 506–517, 1968.

    Article  Google Scholar 

  42. Streett, W.B., Tildesley, D.J., and Saville, G., Multiple time step methods in molecular dynamics, Mol. Phys. 35, 639–648, 1978.

    Article  CAS  Google Scholar 

  43. Tuckerman, M., Berne, B.J., and Martyna, G.J., Reversible multiple time scale molecular dynamics, J. Chem. Phys. 97, 1990–2001, 1992.

    Article  CAS  Google Scholar 

  44. van Gunsteren, W.F. and Karplus, M., Effects of constraints on the dynamics of macromolecules, Macromolecules 15, 1528–1544, 1982.

    Article  Google Scholar 

  45. van Gunsteren, W.F., Constrained dynamics of flexible molecules, Molecular Physics 40, 1015–1019, 1980.

    Article  Google Scholar 

  46. Wisdom, J. and Holman, M., Symplectic Maps for the N-body problem, Astron. J. 102, 1528–1538, 1991.

    Article  Google Scholar 

  47. Zhang, G. and Schlick, T., LIN: A new algorithm to simulate the dynamics of biomolecules by combining implicit-integration and normal mode techniques, J. Comput. Chem. 14, 1212–1233, 1993.

    Article  CAS  Google Scholar 

  48. Zhang, M.Q. and Skeel, R.D., Symplectic integrators and the conservation of angular momentum, J. Comput. Chem. 16, 365–369, 1995.

    Article  CAS  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics, University of Kansas, Lawrence, KS, 66045, USA

    Benedict J. Leimkuhler

  2. Konrad-Zuse Zentrum, Heilbronner Str. 10, D-10711, Berlin, Germany

    Sebastian Reich

  3. Department of Computer Science, University of Illinois, 1304 West Springfield Avenue, Urbana, IL, 61801-2987, USA

    Robert D. Skeel

Authors
  1. Benedict J. Leimkuhler
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  2. Sebastian Reich
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  3. Robert D. Skeel
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Editor information

Editors and Affiliations

  1. Computer Science Department, Boston University, 111 Cummington Street, Boston, MA, 02215, USA

    Jill P. Mesirov

  2. Beckman Institute, Theoretical Biophysics Group, University of Illinois, 405 N. Mathews Avenue, Urbana, IL, 61801, USA

    Klaus Schulten

  3. Department of Mathematics, Florida State University, Tallahassee, FL, 32306-3027, USA

    De Witt Sumners

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© 1996 Springer-Verlag New York, Inc.

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Leimkuhler, B.J., Reich, S., Skeel, R.D. (1996). Integration Methods for Molecular Dynamics. In: Mesirov, J.P., Schulten, K., Sumners, D.W. (eds) Mathematical Approaches to Biomolecular Structure and Dynamics. The IMA Volumes in Mathematics and its Applications, vol 82. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-4066-2_10

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  • DOI: https://doi.org/10.1007/978-1-4612-4066-2_10

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-0-387-94838-6

  • Online ISBN: 978-1-4612-4066-2

  • eBook Packages: Springer Book Archive

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