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On the Relation Between Projections of the Reweighted Path Ensemble

Abstract

We derive several distribution functions for the recently introduced reweighted path ensemble [Rogal et al. in J. Chem. Phys. 133:174109, 2010]: the configurational and path densities, the reactive current, and the generalized committors for the different path types. We relate these distributions to the free energy and to the expressions for the rate constant in the transition state theory, the reactive flux method, the transition path (interface) sampling framework, and the Bayesian path statistics. In addition, we compute the transmission coefficient (distribution) from the reweighted path ensemble. Finally, we derive the path sampling shooting point distributions. For a simple two dimensional Langevin model we illustrate how these novel distributions can be used as analysis tools in rare event simulations.

References

  1. Bennett, C.H.: Molecular dynamics and transition state theory: the simulation of infrequent events. In: Christofferson, R. (ed.) Algorithms for Chemical Computations. ACS Symposium Series, vol. 46. Am. Chem. Soc., Washington (1977)

    Chapter  Google Scholar 

  2. Best, R., Hummer, G.: Reaction coordinates and rates from transition paths. Proc. Natl. Acad. Sci. USA 102, 6732–6737 (2005)

    ADS  Article  Google Scholar 

  3. Bolhuis, P.G.: Rare events via multiple reaction channels sampled by path replica exchange. J. Chem. Phys. 129, 144108 (2008)

    Article  Google Scholar 

  4. Bolhuis, P.G., Dellago, C., Chandler, D.: Reaction coordinates of biomolecular isomerization. Proc. Natl. Acad. Sci. USA 97(11), 5877–5882 (2000)

    ADS  Article  Google Scholar 

  5. Bolhuis, P.G., Chandler, D., Dellago, C., Geissler, P.L.: Transition path sampling: throwing ropes over rough mountain passes in the dark. Annu. Rev. Phys. Chem. 53, 291–318 (2002)

    ADS  Article  Google Scholar 

  6. Chandler, D.: Statistical mechanics of isomerization dynamics in liquids and the transition state. J. Chem. Phys. 68, 2959 (1978)

    ADS  Article  Google Scholar 

  7. Darve, E., Pohorille, A.: Calculating free energies using average force. J. Chem. Phys. 115(20), 9169–9183 (2001)

    ADS  Article  Google Scholar 

  8. Dellago, C., Bolhuis, P.G.: Transition path sampling and other advanced simulation techniques for rare events. Adv. Polym. Sci. 221, 167–233 (2009)

    Google Scholar 

  9. Dellago, C., Bolhuis, P.G., Csajka, F.S., Chandler, D.: Transition path sampling and the calculation of rate constants. J. Chem. Phys. 108, 1964 (1998)

    ADS  Article  Google Scholar 

  10. Dellago, C., Bolhuis, P.G., Geissler, P.L.: Transition path sampling. Adv. Chem. Phys. 123, 1–78 (2002)

    Article  Google Scholar 

  11. E, W., Ren, W.i, Vanden-Eijnden, E.: String method for the study of rare events. Phys. Rev. B. 66, 052301 (2002)

    ADS  Article  Google Scholar 

  12. E, W., Vanden-Eijnden, E.: Towards a theory of transition paths. J. Stat. Phys. 123, 503–523 (2006)

    MathSciNet  ADS  MATH  Article  Google Scholar 

  13. E, W., Vanden-Eijnden, E.: Transition-path theory and path-finding algorithms for the study of rare events. Annu. Rev. Phys. Chem. 61, 391–420 (2010)

    Article  Google Scholar 

  14. van Erp, T.S.: Reaction rate calculation by parallel path swapping. Phys. Rev. Lett. 98, 268301 (2007)

    ADS  Article  Google Scholar 

  15. van Erp, T.S., Bolhuis, P.G.: Elaborating transition interface sampling methods. J. Comput. Phys. 205, 157 (2005)

    MathSciNet  ADS  MATH  Article  Google Scholar 

  16. van Erp, T.S., Moroni, D., Bolhuis, P.G.: A novel path sampling method for the sampling of rate constants. J. Chem. Phys. 118, 7762–7774 (2003)

    ADS  Article  Google Scholar 

  17. Ferrenberg, A.M., Swendsen, R.H.: Optimized Monte Carlo data-analysis. Phys. Rev. Lett. 63(12), 1195–1198 (1989)

    ADS  Article  Google Scholar 

  18. Grubmüller, H.: Predicting slow structural transitions in macromolecular systems: conformational flooding. Phys. Rev. E 52, 2893–2906 (1995)

    ADS  Article  Google Scholar 

  19. Hummer, G.: From transition paths to transition states and rate coefficients. J. Chem. Phys. 120, 516–523 (2004)

    ADS  Article  Google Scholar 

  20. Juraszek, J., Bolhuis, P.G.: Sampling the multiple folding mechanisms of Trp-cage in explicit solvent. Proc. Natl. Acad. Sci. USA 1030(43), 15859 (2006)

    ADS  Article  Google Scholar 

  21. Juraszek, J., Bolhuis, P.G.: Rate constant and reaction coordinate of Trp-cage folding in explicit water. Biophys. J. 95, 4246–4257 (2008)

    ADS  Article  Google Scholar 

  22. Juraszek, J., Bolhuis, P.G.: Effects of a mutation on the folding mechanism of beta-hairpin. J. Phys. Chem. B 113(50), 16184–16196 (2009)

    Article  Google Scholar 

  23. Keck, J.C.: Statistical investigation of dissociation cross-sections for diatoms. Discuss. Faraday Soc. 33, 173 (1962)

    Article  Google Scholar 

  24. Laio, A., Parrinello, M.: Escaping free-energy minima Proc. Natl. Acad. Sci. USA 99, 12562 (2002)

    ADS  Article  Google Scholar 

  25. Lechner, W., Rogal, J., Juraszek, J., Ensing, B., Bolhuis, P.G.: Nonlinear reaction coordinate analysis in the reweighted path ensemble. J. Chem. Phys. 133(17), 174110 (2010)

    ADS  Article  Google Scholar 

  26. Moroni, D.: Efficient sampling of rare event pathways. Ph.D. thesis, Universiteit van Amsterdam (2005)

  27. Peters, B., Trout, B.L.: Obtaining reaction coordinates by likelihood maximization. J. Chem. Phys. 125, 054108 (2006)

    ADS  Article  Google Scholar 

  28. Peters, B., Beckham, G.T., Trout, B.L.: Extensions to the likelihood maximization approach for finding reaction coordinates J. Chem. Phys. 127, 034109 (2007)

    ADS  Article  Google Scholar 

  29. Rogal, J., Lechner, W., Juraszek, J., Ensing, B., Bolhuis, P.G.: The reweighted path ensemble. J. Chem. Phys. 133(17), 174109 (2010)

    ADS  Article  Google Scholar 

  30. Sørensen, M.R., Voter, A.F.: Temperature-accelerated dynamics for simulation of infrequent events. J. Chem. Phys. 112, 9599 (2000)

    ADS  Article  Google Scholar 

  31. Torrie, G.M., Valleau, J.P.: Monte Carlo free energy estimates using non-Boltzmann sampling. Chem. Phys. Lett. 28, 578 (1974)

    ADS  Article  Google Scholar 

  32. Voter, A.F.: Hyperdynamics: accelerated molecular dynamics of infrequent events. Phys. Rev. Lett. 78, 3908 (1997)

    ADS  Article  Google Scholar 

  33. Voter, A.F.: A method for accelerating the molecular dynamics simulation of infrequent events. J. Chem. Phys. 106, 4665 (1997)

    ADS  Article  Google Scholar 

  34. Voter, A.F., Sørensen, M.R.: Accelerating atomistic simulations of defect dynamics: hyperdynamics, parallel replica dynamics, and temperature-accelerated dynamics. Mater. Res. Soc. Symp. Proc. 538, 427 (1999)

    Article  Google Scholar 

  35. Vreede, J., Juraszek, J., Bolhuis, P.G.: Predicting the reaction coordinates of millisecond light-induced conformational changes in photoactive yellow protein. Proc. Natl. Acad. Sci. USA 107, 2397–2402 (2010)

    ADS  Article  Google Scholar 

  36. Wang, F., Landau, D.P.: Efficient multiple-range random walk algorithm to calculate the density of states. Phys. Rev. Lett. 86(10), 2050 (2001)

    ADS  Article  Google Scholar 

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Correspondence to Peter G. Bolhuis.

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Bolhuis, P.G., Lechner, W. On the Relation Between Projections of the Reweighted Path Ensemble. J Stat Phys 145, 841–859 (2011). https://doi.org/10.1007/s10955-011-0324-6

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Keywords

  • Rare event simulations
  • Transition path sampling
  • Rate constants