Calculation of Classical Trajectories with Boundary Value Formulation

  • R. Elber
Part of the Lecture Notes in Physics book series (LNP, volume 703)


An algorithm to compute classical trajectories using boundary value formulation is presented and discussed. It is based on an optimization of a functional of the complete trajectory. This functional can be the usual classical action, and is approximated by discrete and sequential sets of coordinates. In contrast to initial value formulation, the pre-specified end points of the trajectories are useful for computing rare trajectories. Each of the boundary-value trajectories ends at desired products. A difficulty in applying boundary value formulation is the high computational cost of optimizing the whole trajectory in contrast to the calculation of one temporal frame at a time in initial value formulation.


Target Function Slow Mode Classical Trajectory Large Time Step Minimum Energy Path 
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  1. 1.
    L. Verlet (1967) Computer experiments on classical fluids I. Thermodynamics properties of Lennard Jones molecules. Phys. Rev. 98, p. 159Google Scholar
  2. 2.
    D. Chandler (1978) Statistical mechanics of isomerization dynamics in liquids and transition-state approximation. J. Chem. Phys. 68, pp. 2959–2970CrossRefADSGoogle Scholar
  3. 3.
    C. Dellago, P. G. Bolhuis, and D. Chandler (1999) On the calculation of reaction rates in the transition path ensemble. J. Chem. Phys. 110, pp. 6617–6625CrossRefADSGoogle Scholar
  4. 4.
    L. D. Landau and E. M. Lifshitz (2000) Mechanics, third edition, Butterworth- Heinenann, Oxford, Chap. 1Google Scholar
  5. 5.
    L. D. Landau and E. M. Lifshitz (2000) Mechanics, third edition, Butterworth- Heinenann, Oxford, pp. 140–142Google Scholar
  6. 6.
    R. Olender and R. Elber (1996) Calculation of classical trajectories with a very large time step: Formalism and numerical examples. J. Chem. Phys. 105, pp. 9299–9315CrossRefADSGoogle Scholar
  7. 7.
    R. Elber, J. Meller and R. Olender (1999) A stochastic path approach to compute atomically detailed trajectories: Application to the folding of C peptide. J. Phys. Chem. B 103, pp. 899–911CrossRefGoogle Scholar
  8. 8.
    K. Siva and R. Elber (2003) Ion permeation through the gramicidin channel: Atomically detailed modeling by the stochastic difference equation. Proteins, Structure, Function and Genetics 50, pp. 63–80CrossRefGoogle Scholar
  9. 9.
    A. Ulitksy and R. Elber (1990) A new technique to calculate the steepest descent paths in flexible polyatomic systems. J. Chem. Phys. 96, p. 1510ADSGoogle Scholar
  10. 10.
    E. Weinan, R. Weiqing, and E. Vanden-Eijnden (2002) String method for the study of rare events. Physical Review B 66, p. 52301CrossRefGoogle Scholar
  11. 11.
    H. Jonsson, G. Mills, and K. W. Jacobsen (1998) Nudge Elastic Band Method for Finding Minimum Energy Paths of Transitions, in Classical and Quantum Dynamics in Condensed Phase Simulations. Edited by B.J. Berne, G. Ciccotti, D.F. Coker, World Scientific, p. 385.Google Scholar
  12. 12.
    C. Lanczos (1970) The variational principles of mechanics. University of Toronto PressGoogle Scholar
  13. 13.
    L. Onsager and S. Machlup (1953) Phys. Rev. 91, p. 1505; ibid. (1953); 91, p. 1512Google Scholar
  14. 14.
    J. C. M. Uitdehaag, B. A. van der Veen, L. Dijkhuizen, R. Elber, and B. W. Dijkstra (2001) Enzymatic circularization of a malto-octaose linear chain studied by stochastic reaction path calculations on cyclodextrin glycosyltransferase. Proteins Structure Function and Genetics 43, pp. 327–335CrossRefGoogle Scholar
  15. 15.
    K. Siva and R. Elber (2003) Ion permeation through the gramicidin channel: Atomically detailed modeling by the Stochastic Difference Equation. Proteins Structure Function and Genetics 50, pp. 63–80CrossRefGoogle Scholar
  16. 16.
    D. Bai and R. Elber, Calculation of point-to-point short time and rare trajectories with boundary value formulation. J. Chemical Theory and Computation, 2, 484–494(2006)Google Scholar
  17. 17.
    A. Ghosh, R. Elber, and H. Scheraga (2002) An atomically detailed study of the folding pathways of Protein A with the Stochastic Difference Equation. Proc. Natl. Acad. Sci. 99, pp. 10394–10398CrossRefADSGoogle Scholar
  18. 18.
    A. Cárdenas and R. Elber (2003) Kinetics of Cytochrome C Folding: Atomically Detailed Simulations. Proteins, Structure Function and Genetics 51, pp. 245–257CrossRefGoogle Scholar
  19. 19.
    A. Cárdenas and R. Elber (2003) Atomically detailed simulations of helix formation with the stochastic difference equation. Biophysical Journal, 85, pp. 2919–2939CrossRefADSGoogle Scholar

Copyright information

© Springer 2006

Authors and Affiliations

  • R. Elber
    • 1
  1. 1.Department of Computer ScienceCornell UniversityIthaca

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