Determination of Kinetics and Thermodynamics of Biomolecular Processes with Trajectory Fragments

  • Alfredo E. CardenasEmail author
Part of the Springer Series on Bio- and Neurosystems book series (SSBN, volume 8)


Trajectory fragments algorithms are a set of methods that partition the relevant trajectory space between reactants and products into smaller regions of phase space. Many short trajectories are launched to evaluate transition probabilities between these regions. Each of the methods processes this short-trajectory data with different kinetic models and as a result long-time kinetic and thermodynamic information for the overall molecular event can be extracted. This chapter focuses on Milestoning, providing detailed analysis of the approximations involved in the algorithm and its computational implementation. Two other trajectory fragments methods (Partial Path Transition Interface Sampling and Markov State Models) are briefly discussed as well. Finally, two recent applications of trajectory fragments methods are described.


  1. 1.
    Truhlar, D.G., Garrett, B.C., Klippenstein, S.J.: Current status of transition-state theory. J. Phys. Chem. 100(31), 12771–12800 (1996)CrossRefGoogle Scholar
  2. 2.
    Moroni, D., Bolhuis, P.G., van Erp, T.S.: Rate constants for diffusive processes by partial path sampling. J. Chem. Phys. 120(9), 4055–4065 (2004). Scholar
  3. 3.
    van Erp, T.S., Moroni, D., Bolhuis, P.G.: A novel path sampling method for the calculation of rate constants. J. Chem. Phys. 118(17), 7762–7774 (2003)CrossRefGoogle Scholar
  4. 4.
    Bolhuis, P.G., Chandler, D., Dellago, C., Geissler, P.L.: Transition path sampling: throwing ropes over rough mountain passes, in the dark. Ann. Rev. Phys. Chem. 53, 291–318 (2002). Scholar
  5. 5.
    Allen, R.J., Warren, P.B., ten Wolde, P.R.: Sampling rare switching events in biochemical networks. Phys. Rev. Lett. 94(1), 018104 (2005). Scholar
  6. 6.
    Faradjian, A.K., Elber, R.: Computing time scales from reaction coordinates by milestoning. J. Chem. Phys. 120(23), 10880–10889 (2004)CrossRefGoogle Scholar
  7. 7.
    Chodera, J.D., Swope, W.C., Pitera, J.W., Dill, K.A.: Long-time protein folding dynamics from short-time molecular dynamics simulations. Multiscale Model. Simul. 5(4), 1214–1226 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Landau, L.D., Lifshitz, E.M.: Mechanics, vol. 1. Course of Theoretical Physics. Pergamon, Oxford (1976)Google Scholar
  9. 9.
    Machlup, S., Onsager, L.: Fluctuations and irreversible processes. II system with kinetic energy. Phys. Rev. 91, 1512–1515 (1953)CrossRefzbMATHGoogle Scholar
  10. 10.
    Onsager, L., Machlup, S.: Fluctuations and irreversible processes. Phys. Rev. 91, 1505–1512 (1953)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Olender, R., Elber, R.: Calculation of classical trajectories with a very large time step: formalism and numerical examples. J. Chem. Phys. 105(20), 9299–9315 (1996)CrossRefGoogle Scholar
  12. 12.
    Elber, R., Ghosh, A., Cardenas, A.: Long time dynamics of complex systems. Acc. Chem. Res. 35(6), 396–403 (2002)CrossRefGoogle Scholar
  13. 13.
    Elber, R., Cardenas, A., Ghosh, A., Stern, H.A.: Bridging the gap between long time trajectories and reaction pathways. In: Prigogine, I., Rice, S.A. (eds.) Advances in Chemical Physics, vol. 126, pp. 93–129. Wiley & Sons Inc, NJ (2003)CrossRefGoogle Scholar
  14. 14.
    Faccioli, P., Sega, M., Pederiva, F., Orland, H.: Dominant pathways in protein folding. Phys. Rev. Lett. 97(10), 108101 (2006). Scholar
  15. 15.
    Cardenas, A.E., Elber, R.: Kinetics of cytochrome C folding: atomically detailed simulations. Proteins Struct. Funct. Bioinf. 51(2), 245–257 (2003)CrossRefGoogle Scholar
  16. 16.
    Cardenas, A.E., Elber, R.: Atomically detailed Simulations of helix formation with the stochastic difference equation. Biophys. J. 85(5), 2919–2939 (2003)CrossRefGoogle Scholar
  17. 17.
    Bai, D., Elber, R.: Calculation of point-to-point short-time and rare trajectories with boundary value formulation. J. Chem. Theory Comput. 2(3), 484–494 (2006)CrossRefGoogle Scholar
  18. 18.
    Elber, R., Meller, J., Olender, R.: Stochastic path approach to compute atomically detailed trajectories: application to the folding of C peptide. J. Phys. Chem. B 103(6), 899–911 (1999)CrossRefGoogle Scholar
  19. 19.
    Siva, K., Elber, R.: Ion permeation through the gramicidin channel: atomically detailed modeling by the Stochastic Difference Equation. Proteins Struct. Funct. Bioinf. 50(1), 63–80 (2003)CrossRefGoogle Scholar
  20. 20.
    Ghosh, A., Elber, R., Scheraga, H.A.: An atomically detailed study of the folding pathways of protein A with the stochastic difference equation. Proc. Natl. Acad. Sci. U. S. A. 99(16), 10394–10398 (2002)CrossRefGoogle Scholar
  21. 21.
    Tuckerman, M., Berne, B.J., Martyna, G.J.: Reversible multiple time scale molecular-dynamics. J. Chem. Phys. 97(3), 1990–2001 (1992)CrossRefGoogle Scholar
  22. 22.
    Morrone, J.A., Zhou, R.H., Berne, B.J.: Molecular dynamics with multiple time scales: how to avoid pitfalls. J. Chem. Theory Comput. 6(6), 1798–1804 (2010). Scholar
  23. 23.
    Shaw, D.E., Maragakis, P., Lindorff-Larsen, K., Piana, S., Dror, R.O., Eastwood, M.P., Bank, J.A., Jumper, J.M., Salmon, J.K., Shan, Y.B., Wriggers, W.: Atomic-level characterization of the structural dynamics of proteins. Science 330(6002), 341–346 (2010). Scholar
  24. 24.
    Shaw, D.E., Deneroff, M.M., Dror, R.O., Kuskin, J.S., Larson, R.H., Salmon, J.K., Young, C., Batson, B., Bowers, K.J., Chao, J.C., Eastwood, M.P., Gagliardo, J., Grossman, J.P., Ho, C.R., Ierardi, D.J., Kolossvary, I., Klepeis, J.L., Layman, T., McLeavey, C., Moraes, M.A., Mueller, R., Priest, E.C., Shan, Y.B., Spengler, J., Theobald, M., Towles, B., Wang, S.C.: Anton, a special-purpose machine for molecular dynamics simulation. Commun. ACM 51(7), 91–97 (2008). Scholar
  25. 25.
    Valleau, J.: Monte Carlo: changing the rules for fun and profit. In: Berne, B.J., Cicootti, G., Coker, D.F. (eds.) Classical and quantum dynamics in condensed phase simulations. World Scientific, Singapore (1998)Google Scholar
  26. 26.
    Majek, P., Elber, R.: Milestoning without a reaction coordinate. J. Chem. Theory Comput. 6(6), 1805–1817 (2010). Scholar
  27. 27.
    Vanden-Eijnden, E., Venturoli, M.: Markovian milestoning with Voronoi tessellations. J. Chem. Phys. 130(19), 194101 (2009). Scholar
  28. 28.
    West, A.M.A., Elber, R., Shalloway, D.: Extending molecular dynamics time scales with milestoning: Example of complex kinetics in a solvated peptide. J. Chem. Phys. 126(14), 145104 (2007)CrossRefGoogle Scholar
  29. 29.
    Kirmizialtin, S., Elber, R.: Revisiting and computing reaction coordinates with directional milestoning. J. Phys. Chem. A 115(23), 6137–6148 (2011)CrossRefGoogle Scholar
  30. 30.
    Elber, R., West, A.: Atomically detailed simulation of the recovery stroke in myosin by Milestoning. Proc. Natl. Acad. Sci. U. S. A. 107, 5001–5005 (2010)CrossRefGoogle Scholar
  31. 31.
    Malnasi-Csizmadia, A., Toth, J., Pearson, D.S., Hetenyi, C., Nyitray, L., Geeves, M.A., Bagshaw, C.R., Kovacs, M.: Selective perturbation of the myosin recovery stroke by point mutations at the base of the lever arm affects ATP hydrolysis and phosphate release. J. Biol. Chem. 282(24), 17658–17664 (2007)CrossRefGoogle Scholar
  32. 32.
    Monticelli, L., Sorin, E.J., Tieleman, D.P., Pande, V.S., Colombo, G.: Molecular simulation of multistate peptide dynamics: a comparison between microsecond timescale sampling and multiple shorter trajectories. J. Comput. Chem. 29, 1740–1752 (2008)CrossRefGoogle Scholar
  33. 33.
    Allen, R.J., Frenkel, D., ten Wolde, P.R.: Forward flux sampling-type schemes for simulating rare events: Efficiency analysis. J. Chem. Phys. 124(19), 194111 (2006). Scholar
  34. 34.
    Allen, R.J., Valeriani, C., ten Wolde, P.R.: Forward flux sampling for rare event simulations. J. Phys.: Condens. Matter. 21(46), 463102 (2009). Scholar
  35. 35.
    Zhang, B.W., Jasnow, D., Zuckerman, D.M.: The “weighted ensemble” path sampling method is statistically exact for a broad class of stochastic processes and binning procedures. J. Chem. Phys. 132(5), 054107 (2010). Scholar
  36. 36.
    Glowacki, D.R., Paci, E., Shalashilin, D.V.: Boxed molecular dynamics: a simple and general technique for accelerating rare event kinetics and mapping free energy in large molecular systems. J. Phys. Chem. B 113(52), 16603–16611 (2009)CrossRefGoogle Scholar
  37. 37.
    Van Erp, T.S.: Dynamical rare event simulation techniques for equilibrium and nonequilibrium systems. In: Nicolis, G., Maes, D. (eds.) Kinetics and Thermodynamics of Multistep Nucleation and Self-Assembly in Nanoscale Materials: Advances in Chemical Physics, vol. 151. Wiley & Sons Inc, Hoboken (2012)Google Scholar
  38. 38.
    Prinz, J.-H., Keller, B., Noe, F.: Probing molecular kinetics with Markov models: metastable states, transition pathways and spectroscopic observables. Phys. Chem. Chem. Phys. 13, 16912–16927 (2011)CrossRefGoogle Scholar
  39. 39.
    Pande, V.S., Beauchamp, K., Bowman, G.R.: Everything you wanted to know about Markov State Models but were afraid to ask. Methods 52, 99–105 (2010)CrossRefGoogle Scholar
  40. 40.
    Bolhuis, P.G., Dellago, C.: Trajectory-based rare event simulations. In: Lipkowitz, K.B. (ed.) Reviews in Computational Chemistry, vol. 27. John Wiley & Sons Inc, Hoboken (2010)Google Scholar
  41. 41.
    Cardenas, A.E., Elber, R.: Enhancing the capacity of molecular dynamics simulations with trajectory fragments. In: Schlick, T. (ed.) Innovations in Biomolecular Modeling and Simulations, vol. 1. RSC Biomolecular Sciences. The Royal Society of Chemistry, Cambridge (2012)CrossRefGoogle Scholar
  42. 42.
    Elber, R.: A milestoning study of the kinetics of an allosteric transition: atomically detailed simulations of deoxy Scapharca hemoglobin. Biophys. J. 92(9), L85–L87 (2007)CrossRefGoogle Scholar
  43. 43.
    Kuczera, K., Jas, G.S., Elber, R.: Kinetics of helix unfolding: molecular dynamics simulations with milestoning. J. Phys. Chem. A 113(26), 7461–7473 (2009). Scholar
  44. 44.
    Shalloway, D., Faradjian, A.K.: Efficient computation of the first passage time distribution of the generalized master equation by steady-state relaxation. J. Chem. Phys. 124(5), 054112 (2006)CrossRefGoogle Scholar
  45. 45.
    Noe, F., Schutte, C., Vanden-Eijnden, E., Reich, L., Weikl, T.R.: Constructing the equilibrium ensemble of folding pathways from short off-equilibrium simulations. Proc. Natl. Acad. Sci. U. S. A. 106(45), 19011–19016 (2009). Scholar
  46. 46.
    Swope, W.C., Pitera, J.W.: Describing protein folding kinetics by molecular dynamics simulations. 1. Theory. J. Phys. Chem. B 108(21), 6571–6581 (2004)CrossRefGoogle Scholar
  47. 47.
    Chodera, J.D., Singhal, N., Pande, V.S., Dill, K.A., Swope, W.C.: Automatic discovery of metastable states for the construction of Markov models of macromolecular conformational dynamics. J. Chem. Phys. 126(15), 155101 (2007)CrossRefGoogle Scholar
  48. 48.
    Prinz, J.-H., Wu, H., Sarich, M., Keller, B., Senne, M., Held, M., Chodera, J.D., Schutte, C., Noe, F.: Markov models of molecular kinetics: generation and validation. J. Chem. Phys. 134(17), 174105 (2011)CrossRefGoogle Scholar
  49. 49.
    Noe, F., Horenko, I., Schutte, C., Smith, J.C.: Hierarchical analysis of conformational dynamics in biomolecules: transition networks of metastable states. J. Chem. Phys. 126(15), 155102 (2007)CrossRefGoogle Scholar
  50. 50.
    Buch, I., Giorgino, T., De Fabritiis, G.: Complete reconstruction of an enzyme-inhibitor binding process by molecular dynamics simulations. Proc. Natl. Acad. Sci. U. S. A. 108(25), 10184–10189 (2011)CrossRefGoogle Scholar
  51. 51.
    Voelz, V.A., Bowman, G.R., Beauchamp, K., Pande, V.S.: Molecular simulation of ab initio protein folding for a millisecond folder NTL9(1-39). J. Am. Chem. Soc. 132(5), 1526–1528 (2010)CrossRefGoogle Scholar
  52. 52.
    Scalco, R., Caflisch, A.: Equilibrium distribution from distributed computing (Simulations of protein Folding). J. Phys. Chem. B 115(19), 6358–6365 (2011)CrossRefGoogle Scholar
  53. 53.
    Singhal, N., Pande, V.S.: Error analysis and efficient sampling in Markovian state models for molecular dynamics. J. Chem. Phys. 123(20), 204909 (2005)CrossRefGoogle Scholar
  54. 54.
    Schutte, C., Noe, F., Lu, J.F., Sarich, M., Vanden-Eijnden, E.: Markov state models based on milestoning. J. Chem. Phys. 134(20), 204105 (2011). Scholar
  55. 55.
    Cardenas, A.E., Jas, G.S., DeLeon, K.Y., Hegefeld, W.A., Kuczera, K., Elber, R.: Unassisted transport of N-Acetyl-L-tryptophanamide through membrane: experiment and simulation of kinetics. J. Phys. Chem. B 116, 2739–2750 (2012)CrossRefGoogle Scholar
  56. 56.
    Lane, T.J., Bowman, G.R., Beauchamp, K., Voelz, V.A., Pande, V.S.: Markov State Model reveals folding and functional dynamics in ultra-long MD trajectories. J. Am. Chem. Soc. 133, 18413–18419 (2011)CrossRefGoogle Scholar
  57. 57.
    Berezhkovskii, A., Hummer, G., Szabo, A.: Reactive flux and folding pathways in network models of coarse-grained protein dynamics. J. Chem. Phys. 130(20), 205102 (2009). Scholar
  58. 58.
    Metzner, P., Schutte, C., Vanden Eijnden, E.: Transition path theory for Markov jump processes. Multiscale Model. Simul. 7, 1192–1219 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  59. 59.
    Bowman, G.R., Beauchamp, K., Boxer, G., Pande, V.S.: Progress and challenges in the automated construction of Markov state models for full protein systems. J. Chem. Phys. 131(12), 124101 (2009)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Institute for Computational Engineering and SciencesUniversity of TexasAustinUSA

Personalised recommendations