The European Physical Journal Special Topics

, Volume 224, Issue 12, pp 2369–2387 | Cite as

Reactive flows and unproductive cycles for random walks on complex networks

  • R. Banisch
  • N. Djurdjevac Conrad
  • Ch. Schütte
Regular Article B. Bridging of Time Scales and Methods for Rare Events
Part of the following topical collections:
  1. Discussion and Debate: Recurrent Problems in Scale Bridging Techniques in Molecular Simulation – What are the Current Options?


We present a comprehensive theory for analysis and understanding of transition events between an initial set A and a target set B for general ergodic finite-state space Markov chains or jump processes, including random walks on networks as they occur, e.g., in Markov State Modelling in molecular dynamics. The theory allows us to decompose the probability flow generated by transition events between the sets A and B into the productive part that directly flows from A to B through reaction pathways and the unproductive part that runs in loops and is supported on cycles of the underlying network. It applies to random walks on directed networks and nonreversible Markov processes and can be seen as an extension of Transition Path Theory. Information on reaction pathways and unproductive cycles results from the stochastic cycle decomposition of the underlying network which also allows to compute their corresponding weight, thus characterizing completely which structure is used how often in transition events. The new theory is illustrated by an application to a Markov State Model resulting from weakly damped Langevin dynamics where the unproductive cycles are associated with periodic orbits of the underlying Hamiltonian dynamics.


Periodic Orbit Transition Rate European Physical Journal Special Topic Directed Network Simple Cycle 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    B. Altaner, S. Grosskinsky, S. Herminghaus, L. Katthän, M. Timme, J. Vollmer, Phys. Rev. E 85, 041133 (2012)CrossRefADSGoogle Scholar
  2. 2.
    G.R. Bowman, V.S. Pande, F. Noé, editors, An Introduction to Markov State Models and Their Application to Long Timescale Molecular Simulation, Vol. 797 of Advances in Experimental Medicine and Biology (Springer, 2014)Google Scholar
  3. 3.
    M. Cameron, E. Vanden-Eijnden, J. Stat. Phys. 156(3), 427 (2014)MathSciNetCrossRefADSGoogle Scholar
  4. 4.
    J. Chodera, N. Singhal, V.S. Pande, K. Dill, W. Swope, J. Chem. Phys., 126 (2007)Google Scholar
  5. 5.
    N. Djurdjevac-Conrad, R. Banisch, Ch. Schütte, J. Comp. Dynamics (2014) [arXiv:1407.8039v2] [math-ph]
  6. 6.
    E. Weinan, E. Vanden-Eijnden, Metastability, conformation dynamics, and transition pathways in complex systems, Multiscale modelling and simulation, Lect. Notes Comput. Sci. Eng., Vol. 39 (Springer, Berlin, 2004), p. 35Google Scholar
  7. 7.
    E. Weinan, E. Vanden-Eijnden, J. Stat. Phys. 123, 503 (2006)MathSciNetCrossRefADSGoogle Scholar
  8. 8.
    E. Weinan, E. Vanden-Eijnden, Ann. Rev. Phys. Chem. 61, 391 (2010)CrossRefGoogle Scholar
  9. 9.
    D. Jiang, M. Qian, M.-P. Quian, Mathematical theory of nonequilibrium steady states: on the frontier of probability and dynamical systems (Springer, 2004)Google Scholar
  10. 10.
    S.L. Kalpazidou, Cycle Representations of Markov Processes (Springer, 2006)Google Scholar
  11. 11.
    K.J. Kohlhoff, D. Shukla, M. Lawrenz, G.R. Bowman, D.E. Konerding, D. Belov, R.B. Altman, V.S. Pande, Nat. Chem. 6(1), 15 (2014)CrossRefGoogle Scholar
  12. 12.
    T. Li, E. Weinan, E. Vanden Eijnden, Proc. Nat. Acad. Sci., 105 (2008)Google Scholar
  13. 13.
    J. Mattingly, A.M. Stuart, D.J. Higham, Stochastic Process Appl. 101(2), 185 (2002)MathSciNetCrossRefGoogle Scholar
  14. 14.
    P. Metzner, Transition Path Theory for Markov Processes: Application to molecular dynamics, Ph.D. thesis, Free University Berlin , 2007Google Scholar
  15. 15.
    P. Metzner, Ch. Schütte, E. Vanden-Eijnden, J. Chem. Phys. 125(8), 084110 (2006)CrossRefADSGoogle Scholar
  16. 16.
    P. Metzner, Ch. Schütte, E. Vanden-Eijnden, Multiscale Modeling Simul. 7(3), 1192 (2009)CrossRefGoogle Scholar
  17. 17.
    F. Noe, S. Fischer, Curr. Opin. Struct. Biol. 18, 154 (2008)CrossRefGoogle Scholar
  18. 18.
    F. Noe, I. Horenko, Ch. Schütte, J.C. Smith, J. Chem. Phys. 126, 155102 (2007)CrossRefADSGoogle Scholar
  19. 19.
    F. Noé, Ch. Schütte, E. Vanden-Eijnden, L. Reich, T. Weikl, PNAS 106(45), 19011 (2009)CrossRefADSGoogle Scholar
  20. 20.
    A.C. Pan, B. Roux, J. Chem. Phys. 129(6), 064107 (2008)CrossRefADSGoogle Scholar
  21. 21.
    J.-H. Prinz, H. Wu, M. Sarich, B. Keller, M. Fischbach, M. Held, J.D. Chodera, Ch. Schütte, F. Noé, J. Chem. Phys. 134, 174105 (2011)CrossRefADSGoogle Scholar
  22. 22.
    H. Risken, The Fokker-Planck Equation, 2nd edition (Springer, New York, 1996)Google Scholar
  23. 23.
    M. Sarich, N. Djurdjevac, S. Bruckner, T.O.F. Conrad, Ch. Schütte, J. Comp. Dyn. 1(1), 191 (2014)CrossRefGoogle Scholar
  24. 24.
    J. Schnakenberg, Rev. Mod. Phys. 48, 571 (1976)MathSciNetCrossRefADSGoogle Scholar
  25. 25.
    Ch. Schütte, M. Sarich, Metastability and Markov State Models in Molecular Dynamics: Modeling, Analysis, Algorithmic Approaches, Vol. 24 of Courant Lecture Notes (American Mathematical Society, 2013)Google Scholar
  26. 26.
    C.R. Schwantes, R.T. McGibbon, V.S. Pande, J. Chem. Phys. 141(9) (2014)Google Scholar
  27. 27.
    E. Vanden-Eijnden, Transition path theory, In Computer Simulations in Condensed Matter Systems: From Materials to Chemical Biology, Vol. 1, edited by M. Ferrario, G. Ciccotti, K. Binder, Vol. 703 of Lecture Notes in Physics (Springer Berlin Heidelberg, 2006), p. 453Google Scholar
  28. 28.
    H. Wang, Ch. Schütte, J. Chem. Theory Computation (2015)Google Scholar
  29. 29.
    R.K.P. Zia, B. Schmittmann, J. Stat. Mechan. Theo. Exper. 7 (2007)Google Scholar

Copyright information

© EDP Sciences and Springer 2015

Authors and Affiliations

  • R. Banisch
    • 1
  • N. Djurdjevac Conrad
    • 2
  • Ch. Schütte
    • 1
    • 2
  1. 1.Institut für MathematikFreie Universität BerlinBerlinGermany
  2. 2.Zuse InstituteBerlinGermany

Personalised recommendations